OPTICAL FIBERSRESEARCH ADVANCES OPTICAL FIBERS RESEARCH ADVANCES JÜRGEN C. SCHLESINGER EDITOR Nova Science Publishers, Inc. New York Copyright © 2007 by Nova Science Publishers, Inc. All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. For permission to use material from this book please contact us: Telephone 631-231-7269; Fax 631-231-8175 Web Site: http://www.novapublishers.com NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers’ use of, or reliance upon, this material. Any parts of this book based on government reports are so indicated and copyright is claimed for those parts to the extent applicable to compilations of such works. Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS. LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA Schlesinger, Jürgen C. Optical fibers research advances / Jürgen C. Schlesinger, Editor. p. cm. Includes index. ISBN-13: 978-1-60692-607-9 1. Optical communications. 2. Fiber optics. 3. Optical fibers. I. Title. TK5103.59.S35 2008 621.36'92--dc22 2007031168 Published by Nova Science Publishers, Inc. New York CONTENTS Preface vii Short Communication 1 Ignition with Optical Fiber Coupled Laser Diode 3 Shi-biao Xiang, Xu Xiang , Wei-huan Ji and Chang-gen Feng Research and Review Studies 13 Chapter 1 Evanescent Field Tapered Fiber Optic Biosensors (TFOBS): Fabrication, Antibody Immobilization and Detection 15 Angela Leung, P. Mohana Shankar and Raj Mutharasan Chapter 2 New Challenges in Raman Amplification for Fiber Communication Systems 51 P.S. André, A.N. Pinto, A.L.J. Teixeira, B. Neto, S. Stevan Jr., Donato Sperti, F. da Rocha, Micaela Bernardo, J.L. Pinto, Meire Fugihara, Ana Rocha and M. Facão Chapter 3 Fiber Bragg Gratings in High Birefringence Optical Fibers 83 Rogério N. Nogueira, Ilda Abe and Hypolito J. Kalinowski Chapter 4 Applications of Hollow Optical Fibers in Atom Optics 119 Heung-Ryoul Noh and Wonho Jhe Chapter 5 Advances in Physical Modeling of Ring Lasers 161 Vittorio M.N. Passaro and Francesco De Leonardis Chapter 6 Investigation of Optical Power Budget of Erbium-Doped Fiber 187 Hideaki Hayashi, Setsuhisa Tanabe and Naoki Sugimoto Chapter 7 Recent Developments in All-Fibre Devices for Optical Networks 205 Nawfel Azami and Suzanne Lacroix Contents vi Chapter 8 Advances in Optical Differential Phase Shift Keying and Proposal for an Alternative Receiving Scheme for Optical Differential Octal Phase Shift Keying 231 M. Sathish Kumar, Hosung Yoon and Namkyoo Park Chapter 9 A New Generation of Polymer Optical Fibers 257 Rong-Jin Yu and Xiang-Jun Chen Chapter 10 Dissipative Solitons in Optical Fiber Systems 279 Mário F.S. Ferreira and Sofia C.V. Latas Chapter 11 Bright - Dark and Double - Humped Pulses in Averaged, Dispersion Managed Optical Fiber Systems 301 K.W. Chow and K. Nakkeeran Chapter 12 Dynamics and Interactions of Gap Solitons in Hollow Core Photonic Crystal Fibers 315 Javid Atai and D. Royston Neill Chapter 13 Multiwavelength Optical Fiber Lasers and Semiconductor Optical Amplifier Ring Lasers 335 Byoungho Lee and Ilyong Yoon Chapter 14 Aging and Reliability of Single-Mode Silica Optical Fibers 355 M. Poulain, R. El Abdi and I. Severin Index 369 PREFACE An optical fiber is a glass or plastic fiber designed to guide light along its length by confining as much light as possible in a propagating form. In fibers with large core diameter, the confinement is based on total internal reflection. In smaller diameter core fibers, (widely used for most communication links longer than 200 meters) the confinement relies on establishing a waveguide. Fiber optics is the overlap of applied science and engineering concerned with such optical fibers. Optical fibers are widely used in fiber-optic communication, which permits transmission over longer distances and at higher data rates than other forms of wired and wireless communications. They are also used to form sensors, and in a variety of other applications. The term optical fiber covers a range of different designs including graded-index optical fibers, step-index optical fibers, birefringent polarization-maintaining fibers and more recently photonic crystal fibers, with the design and the wavelength of the light propagating in the fiber dictating whether or not it will be multi-mode optical fiber or single-mode optical fiber. Because of the mechanical properties of the more common glass optical fibers, special methods of splicing fibers and of connecting them to other equipment are needed. Manufacture of optical fibers is based on partially melting a chemically doped preform and pulling the flowing material on a draw tower. Fibers are built into different kinds of cables depending on how they will be used. This new book presents the latest research in the field. Optical fibers, an important and promising material, have attracted more and more attention and extended their applications to various scientific and practical aspects. In the short communication, the key role of fibers, as the carriers of information and energy in our times, was briefly summarized. Afterwards, the configuration of fiber coupled laser diode ignition system was elucidated as well as the advantages, developments and applications of this technology. Furthermore, the energy-transmitting characteristics of single-mode fibers and multi-mode ones and the key points of fiber-coupled technology were analyzed. In a practical case, the effect of the diameters of core on laser ignition, from both theory and experiments, was studied. The findings suggest that the smaller the diameters of core, the lower the ignition threshold under the same laser power. That is to say, the ignition becomes easier while using fibers with smaller core. Finally, the issue on selection of core was clarified based on the consideration of both laser power density and the endurance of fibers. Tapered Fiber Optic Biosensors (TFOBS) are sensors that operate based on fluctuations in the evanescent field in the tapered region. In the laboratory, TFOBS are made by heat Jürgen C. Schlesinger viii pulling commercially-available single mode optical fibers. They have been investigated for various applications, including measurement of physical characteristics (refractive index, temperature, pressure, etc.), chemical concentrations, and biomolecule detection. In this chapter, an up-to-date review of TFOBS research is provided, with emphasis on applications in biosensing such as pathogen, proteins, and DNA detection. The physics of sensing and optical behavior based on taper geometry is discussed. Methods of fabrication, antibody immobilization, sample preparation, and detection from our laboratory are described. This chapter presents results on the non-specific response, simulation, and detection of E.coli O157:H7 and BSA. Chapter 1 will conclude with an analysis of the future direction of the Tapered Fiber Optic Biosensors. Raman fiber amplifiers (RFA) are among the most promising technologies in lightwave systems. In recent years, Raman optical fiber amplifiers have been widely investigated for their advantageous features, namely the transmission fiber can be itself used as the gain media reducing the overall noise figure and creating a lossless transmission media. The introduction of RFA based on low cost technology will allow the consolidation of this amplification technique and its use in future optical networks. Chapter 2 reviews the challenges, achievements, and perspectives of Raman amplification in optical communication systems. In Raman amplified systems, the signal amplification is based on stimulated Raman scattering, thus the peak of the gain is shifted by approximately 13.2 THz with respect to the pump signal frequency. The possibility of combining many pumps centered on different wavelengths brings a flat gain in an ultra wide bandwidth. An initial physical description of the phenomenon is presented as well as the mathematical formalism used to simulate the effect on optical fibers. The review follows with one section describing the challenging developments in this topic, such as using low cost pump lasers, in-fiber lasing, recurring to fiber Bragg grating cavities or broadband incoherent pump sources and Raman amplification applied to coarse wavelength multiplexed networks. Also, one of the major issues on Raman amplifier design, which is the determination of pump powers in order to realize a specific gain will be discussed. In terms of optimization, several solutions have been published recently, however, some of them request extremely large computation time for every interaction, what precludes it from finding an optimum solution or solve the semi-analytical rate equation under strong simplifying assumptions, which results in substantial errors. An exhaustive study of the optimization techniques will be presented. This paper allows the reader to travel from the description of the phenomenon to the results (experimental and numerical) that emphasize the potential applications of this technology. Fiber Bragg gratings (FBG) are a key element in optical communication devices and in fiber sensors. This is mainly due to its intrinsic characteristics, which include low insertion loss, passive operation and immunity to electromagnetic interferences. Basically a FBG is a periodic modulation of the core refractive index formed by exposure of a photosensitive fiber to a spatial pattern of ultraviolet light in the region of 244–248 nm. The lengths of FBGs are normally within the region of 1–20 mm. Usually a FBG operates as a narrow reflection filter, where the central wavelength is directly proportional to the periodicity of the spatial modulation and to the effective refractive index of the fiber. The production technology of these devices is now in a mature state, which enables the design of gratings with custom- Preface ix made transfer functions, crucial for all-optical processing. Recently, some work has been done in the application of FBG written in highly birefringent fibers (HiBi). Due to the birefringence, the effective refractive index of the fiber will be different for the two transversal modes of propagation. Therefore, the reflection spectrum of a FBG will be different for each polarization. This unique property can be used for advanced optical processing or advanced fiber sensing. Chapter 3 will describe in detail this unique device. The chapter will also analyze the device and demonstrate different applications that take advantage of its properties, like multiparameter sensors, devices for optical communications or in the optimization of certain architectures in optics communications systems. A hollow optical fiber (HOF) has a lot of interesting applications in atom optics experiments such as atom guiding and the generation of hollow laser beam (HLB). In this article the authors present theoretical and experimental works on the use of hollow optical fibers in atom optics. Chapter 4 is divided into two parts: One is devoted to the atom guide using HOFs and the other describes the atom optics researches that utilizes laser lights emanated from the HOF. First, the authors describe the electromagnetic fields inside the HOF and characterize the electromagnetic modes diffracted from the HOF. Then they describe two guiding schemes using red and blue detuned laser lights. Finally, they describe the various relevant experiments using LP 01 or LP 11 modes such as the generation of HLB from the HOF, funneling atoms using the diffracted fields, diffraction-limited dark laser spot, and a dipole trap using LP 01 mode of the diffracted field from the HOF. In Chapter 5, an overview on fiber ring lasers and III/V semiconductor integrated ring lasers is presented. In particular, some aspects of mathematical modelling of these devices are reviewed. In the first part of the chapter, the authors have focused our attention on the more recent theoretical and experimental studies concerning fiber ring laser architectures. Then, a complete quantum-mechanical model for integrated ring lasers is presented, including the evaluation of all the involved physical parameters, such as self and cross saturation and backscattering. Finally, the influence of sidewall roughness on either unidirectional or bidirectional regime in multi-quantum-well III/V semiconductor ring lasers is demonstrated. In Chapter 6, the authors investigated optical power budget of an erbium-doped fiber (EDF). In addition to the output signal and amplified spontaneous emission (ASE) powers from the fiber end, lateral spontaneous emissions and scattering laser powers in the EDF were measured quantitatively by using an integrating sphere. Compared with the signal and ASE powers, it was found that considerable powers were consumed by the laterally emitting lights. As an optically undetected loss which limits power conversion efficiency (PCE) of the fiber amplifier, the effect of nonradiative decay from the termination level of pump excited state absorption (pump ESA) was estimated from decay rate analyses of the relevant levels. The nonradiative loss was comparable to amplified signal power in the EDF when pumped with a 980 nm LD. Nonradiative decay following cooperative upconversion (CUP) process is also discussed using rate equations analysis. All-fibre components are essential components of optical networks systems. Development of such devices is of great importance to allow network functions to be performed in the glass of the optical fibre itself. Among of all fabrication techniques, the Fused Fibre Biconical Taper (FBT) technique allows optical devices with high performances. Although fibre devices are mainly based on the passive directional coupler basic structure, research is made to design components that perform complex functionalities in today optical Jürgen C. Schlesinger x networks systems. Recent developments on all-fibre devices in network systems are presented. Research is mainly focused on enhanced fabrication and stability of FBT fabrication technique, passive thermal compensation for stable interferometer optical structure, broadband spectral operation for multi-wavelength operations and new interferometer designs. An overview of recent fused fibre devices for optical telecommunications is presented to understand the main functionalities of these fibre devices. The limiting factors are explained in Chapter 7, to understand challenges on fibre devices development. Optical Differential Phase Shift Keying (oDPSK) with delay interferometer based direct detection receiver was proposed as an alternative for the conventional On-Off Keying (OOK) modulation schemes. Compared to OOK, oDPSK was predicted to have a 3dB improvement in performance due to its balanced detection receiver structure. It was also predicted that due to the optical signal occupying all the symbol slots, unlike in OOK, symbol pattern dependent fiber nonlinear effects will make less of an impact on long haul optical transmission schemes based on oDPSK. Subsequent successful demonstrations of these positive attributes of oDPSK resulted in active investigations into multilevel formats of oDPSK namely, optical Differential Quadrature Phase Shift Keying (oDQPSK) and optical Differential Octal Phase Shift Keying (oDOPSK). Significant developments in theoretical models of optically amplified lightwave communication systems based on the Karhunen-Loeve Series Expansion (KLSE) method assisted such investigations. In Chapter 8, the authors discuss some of the recent advances in oDPSK and its multilevel formats that have been achieved such as proposals for receiver schematics, theoretical analysis of receiver schematics, electronic techniques to counter polarization mode dispersion induced penalties, and application of coded modulation techniques. The chapter also proposes an alternative receiver schematic for oDOPSK which can separately detect the three constituent bits from an oDOPSK symbol. Chapter 9 describes the background to the development of Polymer Optical fibers (POFs), discusses the optical and temperature resistant properties of polymers while emphasizing the intrinsic high attenuation of them. The first generation of POFs which consists of a solid-core surrounded by cladding and transmits light by total internal reflection, is puzzled by the difficulty of high attenuation. Then, the method of using a specific structure (i.e. hollow-core Bragg fiber) to solve the problem is presented. A new generation of POFs based on the hollow-core Bragg fibers with cobweb-structured cladding can guide light with low transmission loss and high bandwidth in the wavelength range of visible to terahertz (THz ) radiation. Efficient hollow-core guiding for delivery of power laser radiation and solar radiation can be achieved by replacing the traditional polymethylmethacrylate (PMMA) with heat-resistant polymers. Lastly, this chapter concludes with a discussion of applications in diverse areas. Chapter 10 introduces the concept of dissipative solitons, which emerge as a result of a double balance: between nonlinearity and dispersion and also between gain and loss. Such dissipative solitons have many unique properties which differ from those of their conservative counterparts and which make them similar to living things. The authors focus our discussion on dissipative solitons in optical fiber systems, which can be described by the cubic-quintic complex Ginzburg-Landau equation (CGLE). The conditions to have stable solutions of the CGLE are discussed using the perturbation theory. Several exact analytical solutions, namely in the form of fixed-amplitude and arbitrary-amplitude solitons, are presented. The numerical solutions of the quintic CGLE include plain pulses, flat-top pulses, and composite pulses, Preface xi among others. The interaction between plain and composite pulses is analyzed using a two- dimensional phase space. Stable bound states of both plain and composite pulses are found when the phase difference between them is 2 / π ± . The possibility of constructing multisoliton solutions is also demonstrated. As explained in Chapter 11, the envelope of the axial electric field in a dispersion managed (DM) fiber system is governed by a nonlinear Schrödinger model. The group velocity dispersion (GVD) varies periodically and thus realizes both the anomalous and normal dispersion regimes. Kerr nonlinearity is assumed and a loss / gain mechanism will be incorporated. Due to the big changes in the GVD parameter, the correspondingly large variation in the quadratic phase chirp of the DM soliton will be identified. An averaging procedure is applied. In many DM systems, an amplifier at the end of the dispersion map will compensate for the energy dissipated in that map. Here the case of gain not exactly compensating the loss is considered, in other words, a small residual amplification / attenuation is permitted. The present model differs from other similar ones on variable coefficient NLS, as the inhomogeneous features involve both time and the spatial coordinate. The goal here is to extend the model further, by permitting coupled modes or additional degree of freedom. More precisely, the coupling of fiber loss and initial chirping is considered for a birefringent fiber. The corresponding dynamics is governed by variable coefficient, coupled NLS equations for the components of the orthogonal polarization of the pulse envelopes. When the self phase and cross phase modulation coefficients are identical for special angles, several new classes of wave patterns can be found. New stationary wave patterns which possess multiple peaks within each period are found, similar to those found for the classical Manakov model. For situations where the self- and cross-phase modulation coefficients are different, symbiotic solitary pulses are studied. A pair of bright-dark pulses exists, where either or both pulse(s) cannot propagate in that waveguide if coupling is absent. The existence and stability of gap solitons in a model of hollow core fiber in the zero dispersion regime are analyzed in Chapter 12. The model is based on a recently introduced model where the coupling between the dispersionless core mode and nonlinear surface mode (in the presence of the third order dispersion) results in a bandgap. It is found that similar to the anomalous and normal dispersion regimes, the family of solitons fills up the entire bandgap. The family of gap solitons is found to be formally unstable but in a part of family the instability is very weak. Consequently, gap solitons belonging to that part of the family are virtually stable objects. The interactions and collisions of in-phase and the π -out-of- phase quiescent solitons and moving solitons in different dispersion regimes are investigated and compared. Chapter 13 reviews various schemes for multiwavelength fiber lasers and semiconductor optical amplifier (SOA) ring lasers. Multiwavelength fiber lasers have applications in wavelength division multiplexing (WDM) optical communication systems, optical fiber sensors and optical spectroscopy. Erbium-doped fiber amplifiers (EDFAs), Raman amplifiers and SOAs are mainly used as gain media for multiwavelength fiber lasers. Because EDFAs are homogeneously broadened gain media, various methods have been researched to enable the multiwavelength generation. Due to the introduction of liquid nitrogen cooling, four-wave mixing, frequency shifted feedback, and so on, multiwavelength erbium-doped fiber lasers could become realized. Jürgen C. Schlesinger xii On the other hand, because SOA and Raman amplifiers are gain media with inhomogeneous broadening, multiwavelength generation is relatively easy. The useful features of the multiwavelength lasers are mainly dependent on a comb filter. One of the most important features of multiwavelength lasers is tunability. The tunability of wavelengths and channel spacing is required for WDM optical communication systems. Much research has been conducted to enable implementation of tunable multiwavelength fiber lasers. Various comb filters such as Fabry-Perot filters, fiber Bragg gratings, and polarization-maintaining fiber loop mirrors can be used for multiwavelength fiber lasers. The authors review several schemes for multiwavelength SOA-fiber and Raman fiber lasers in this chapter. The optical fiber reliability in telecommunication networks has been still an issue, that’s why the question of how long an optical fibers might been used without a significant probability of failure isn’t out of interest. Much work was developed around this issue, but the optical fiber fatigue and aging process has not been yet fully understood. The reliability of the optical fibers depends on various parameters that have been identified: time, temperature, applied stress, initial fiber strength and environmental corrosion. The major and usually unique corrosion reagent is water, either in the liquid state or as atmospheric moisture. Glass surface contains numerous defects, either intrinsic, the so- called “Griffith’s flaws and extrinsic, in relation to fabrication process. Under permanent or transient stress, microcracks grow from these defects, and growth kinetics depend on temperature and humidity. Although polymeric coating efficiently protects glass surface from scratches, it does not prevent water to reach glass fiber. The work carried out during the last years made possible to apprehend in a more coherent way the problems of failure and rupture of fibers subjected to severe aging conditions. In Chapter 14, some informations on the used characterization methodology for the silica optical fibers are given. In addition, Optical fibers analysis advantages, expected percussions and theoretical background are given to enlighten the potential concerned persons. The principal optical fiber test benches are described and some results are commented. Finally, final remarks are noted. SHORT COMMUNICATION In: Optical Fibers Research Advances ISBN: 1-60021-866-0 Editor: Jurgen C. Schlesinger, pp. 3-11 © 2007 Nova Science Publishers, Inc. IGNITION WITH OPTICAL FIBER COUPLED LASER DIODE Shi-biao Xiang 1,2* , Xu Xiang 3 , Wei-huan Ji 2 and Chang-gen Feng 4 1 Department of Technical Physics, Zhengzhou Institute of Light Industry, No.5 Dongfeng Road, Zhengzhou 450002, P.R. China 2 Key Laboratory of Informationalized Electric Apparatus of Henan Province, Zhengzhou 450002, P.R. China 3 State Key Laboratory of Chemical Resource Engineering, Beijing University of Chemical Technology, P.O. BOX 98, Beijing 100029, P.R. China 4 School of Mechanics and Engineering, Beijing Institute of Technology, Beijing 100081, P.R. China Abstract Optical fibers, an important and promising material, have attracted more and more attention and extended their applications to various scientific and practical aspects. In this article, the key role of fibers, as the carriers of information and energy in our times, was briefly summarized. Afterwards, the configuration of fiber coupled laser diode ignition system was elucidated as well as the advantages, developments and applications of this technology. Furthermore, the energy-transmitting characteristics of single-mode fibers and multi-mode ones and the key points of fiber-coupled technology were analyzed. In a practical case, the effect of the diameters of core on laser ignition, from both theory and experiments, was studied. The findings suggest that the smaller the diameters of core, the lower the ignition threshold under the same laser power. That is to say, the ignition becomes easier while using fibers with smaller core. Finally, the issue on selection of core was clarified based on the consideration of both laser power density and the endurance of fibers. * E-mail address:
[email protected]. Tel: 86-371-63557226 (Corresponding author: S. B. Xiang) Shi-biao Xiang, Xu Xiang, Wei-huan Ji et al. 4 1. Introduction Optical fibers as carrier of information and energy have intrigued intensive interest worldwide due to its scientific and technological significance in various practical fields. For instance, in optical communications, fibers have received tremendous attention from both experimental and theoretical aspects not only on the type of fiber materials but also on various communicating techniques [1-4], in which the most primary function of fibers is to transmit information like voice, images and videos from one place to another. A wide variety of optical fiber devices have been designed and exploited in the field of fiber-based communications, such as fiber optical amplifiers, frequency or phase modulators, planar waveguides and fiber polarizers. Furthermore, the developments of microstructured optical fibers (MOFs) and photonic crystal fibers [5-9] enable a number of potential functionalities including tunability and enhanced nonlinearity, and extend novel fiber device applications to fiber Bragg gratings, tunable resonant filters, variable optical attenuators and nonlinear optics devices owing to their unique characteristics [10-15]. More interestingly, chemical sensors based on optical fibers have been widely explored in the past few years [16-18]. For example, sensors for gases or vapors [19-20], humidity [21- 22], metallic ions, specific chemical compounds [23], viscosity [24], intensity [25] and miniature pressure [26] have been delicately designed and rapidly developed. Also, biosensors [27, 28] for enzymes, antibodies or antigens, DNA [29] and bacteria are becoming a prevailing research topic on the basis of fiber materials. They have been exhibiting promising applications in a variety of fields such as chemical analysis, biological monitoring and environmental detection. In this article, the emphasis has been highlighted on the fundamental principles and the important practice of fiber-coupled laser diode ignition. 2. Fiber Coupled Laser Diode Ignition 2.1. Brief Review on Laser Diode (LD) Ignition Laser ignition is a kind of ignition technique, which refers to detonation or ignition of energetic materials such as solids or fluids [30-33] by laser beam. At early stage of laser ignition technique, the types of laser used for the experimental and application research are mostly Nd:YAG, Nd: GSGG, Nd: glass laser and CO 2 laser [34-40]. These lasers possess the characteristics of high output power or energy, small radiation angle of light, long life-span and low price. However, the obvious disadvantages of this kind of laser are low energy conversion efficiency, in which the ratio of output light energy and input electric energy is usually lower than 3%, as well as large volume and heavy weight. With the born of LD and the naissance of LD ignition, the research and evolution of laser ignition technique come into a new era. The experimental studies for laser diode ignition began in the middle of 1980s. Ewick, Kunz, Kramer, Jungst, Merson, Glass and Roman et al have made great devotion to the field of LD ignition, of which Ewick [41] and Kunz [42] published their literatures firstly. LD belongs to a kind of semiconductor laser stimulated by current. In LD ignition, LD is utilized as energy source, and the energy is transmitted to powders by using optical fiber, which detonates or ignites the energetic materials. This ignition configuration has the Ignition with Optical Fiber Coupled Laser Diode 5 characteristics of safety, reliability, and strong capability of anti-interference of electromagnetism. In addition, the following advantages are also realized. (1) It is easy for LD ignition system to realize miniaturization of apparatus due to its small volume and light weight. (2) LD ignition system has excellent adaptability to the ambient environment because of the input of low voltage and electric energy. (3) LD ignition system can output multi-channel laser signals by using LD arrays and consequently control multi-point ignition through the selection to time and order of signals. As a result, LD ignition has received extensive attention, and exhibits promising application especially in the field of aviation and aerospace. Fig. 1 illustrates the schematic diagram of ignition system induced by laser diode. Laser diode is employed as light source, and energy is transmitted to powders by optical fibers. The powders are ignited and subsequently exploded while enough energy is provided. powder fiber fiber coupler connecting laser aperture lock device Figure 1. Schematic illustration of ignition system induced by laser diode. 2.2. Optical Fiber and Fiber Coupled Technology As a carrier to transmit laser, optical fiber plays a crucial role in LD ignition. The materials of optical fiber should possess the favorable characteristics of optical and mechanical properties as well as the characteristic of temperature. The widely used fibers are made of silica glass or plastic. The fibers can be classified into two types, one is step-index fiber and the other is grade-index one according to the distribution of refractive-index of fiber core. The refractive index of core is a constant for step-index fiber, schematically shown in Fig. 2. However, for the grade-index fiber, the refractive index of core gradually decreases outwards along the radial direction. Due to the self-focusing characteristic of the grade-index fiber, the output beam has higher energy density close to the axis of fiber. As a consequence, the laser power density can be enhanced by using the grade-index fiber. Shi-biao Xiang, Xu Xiang, Wei-huan Ji et al. 6 n y n 2 n 1 Cladding Core z y r φ Fiber axis Figure 2. The schematic diagram of step-index fibers. Both theoretical analysis and experimental results indicate that the increase of power density is considerably favorable to LD ignition. That is to say, the combination of thin diameter, low attenuation, small numerical aperture and grade-index fiber is advantageous to LD ignition. Ewick and coworkers found that the threshold of ignition using grade-index fiber was decreased by around 30% than that using step-index fiber in the ignition experiments of Ti/KClO 4 and CP/carbon black. Generally, optical fibers can be classified into single-mode fibers (SMFs) and multi-mode fibers (MMFs) according to the transmission modes. SMFs exhibit excellent capability in optical communications. And the light energy transmitted by SMFs presents to be Gauss distributions, which means the more centralized energy can be obtained, and is thus favorable to LD ignition. Nevertheless, the diameter of core in SMFs is confined to a large extent. The fiber waveguide parameters can be expressed as 2 / 1 2 2 2 1 ) ( n n kr V − = , where 1 n and 2 n are the refractive indices of the core and the cladding, respectively, and r is the core radius. And 0 / 2 λ π = k is the wave number, where λ 0 represents the wavelength in vacuum. Single mode operation is obtained for V <2.405, and it can be observed for the wavelength longer than λ c ( 405 . 2 0 = = c V λ λ ). For example, in order to obtain V =2.405 at λ =10.6 μm, a silver halide fiber can have core diameter 2r = 8 μm while the normalized difference between the refractive indices of the core and the cladding [ 1 2 1 1 / ) ( / n n n n n − = = Δ Δ ] should be 0.1. Alternatively, V =2.405 will be obtained for 2r = 80 μm and Δ = 0.001. In both cases one can observe a SMF operation. In typical silica SMFs, the value is of the order of 0.002. Shalem et al [43] selected to design silver halide SMFs which have 2r = 60 μm and Δ = 0.002, for which the estimated waveguide parameter was at least 10% lower than 2.405 (at λ = 10.6 μm). To make the laser power transmitted by fiber enough high in the transmitting process, it is necessary to ensure that the fiber has enough thick core, otherwise the fiber will be inevitably damaged by laser. Based on the findings of Schmidt-Uhlig and his colleagues [44], the feasibility of transmitting 20 mW, 5 ns laser pulse from a frequency doubled Nd:YAG laser through a standard 1500 μm multi-mode optical fiber has been demonstrated. Furthermore, the experiments on the delivery of more than 20000 pulses with mean energy of 110 mJ with no damage to fiber have been performed. Consequently, multi-mode optical fibers are prevailing in LD ignition. Ignition with Optical Fiber Coupled Laser Diode 7 Commonly, the specification of optical fiber used in LD ignition is as follows, the diameter of core (2r) of 100-200 μm, numerical aperture (NA) <0.3, attenuation per kilometer dB km <3dB and output power of fiber P ≥ 1W. Additionally, the end-face quality of fiber is an important aspect, which includes the perpendicularity of end-face to axes, degree of levelness and cleanliness of end-face, etc. These factors have a significant effect on the transmission of laser. Surface defects and contamination not only debase the transmission efficiency but also result in the strong electric field and large thermal stress in local area under higher power density, and eventually damage fiber itself. Accordingly, the cleansing and polishing treatment to the end-face of fiber is a serious issue. Ewick adopted tailor-made polishing apparatus for fiber to polish the end-face and directly observed it by a microscope with 400× magnifications in order to determine the quality of polishing. Besides transmitting energy, optical fiber can also be applied in coupling and splitting. The fiber-coupled technology was firstly used in the coupling between fiber and LD. LD emits the elliptic radiation, and thus the proper convergent lens is required to effectively couple laser into output fiber. The difficulty of coupling is greatly increased because of the small fiber core used in LD ignition, and the loss of coupling can even be as high as 5dB. Apparently, it is crucial to enhance the coupling efficiency at coupling sites. The second problem of this technology is the coupling between fibers, which includes two types of coupling: one is the coupling between single and single fiber (one in and one out), and the other is single and multi fiber (one in and several out). The latter is indispensable for multi- point ignition of LD [45]. The coupling between fibers is realized by linkers and commutators. To reduce the loss of coupling, it is necessary to improve the techniques of collimation, tight- fittings, and fixed-airproof. Roman and coworkers [46] used STC linkers in LD ignition to link two fibers for light in and light out, one of which has a core diameter of 100 μm and the other has an outer diameter of 140 μm. The linkers, having an attenuation of as low as 0.56dB and jack diameter of 144 μm, can operate in the temperature range of –40 to +80 o C. This is a kind of linker with low attenuation, easy to manipulate, and good performance. Further, it is also a noticeable issue on the coupling technique between fibers and ignited powders. Kramer et al [47] designed and developed two kinds of components i.e., the fiber foot and the optical window, to resolve the coupling between fibers and ignited powders. These components make LD ignition system more convenient and practical in operation. They are required not only to meet the demand of mechanical strength but also to reduce the energy loss as could as possible. The fiber foot with high mechanical strength is prepared through the following steps, firstly envelop a short fiber into glass preform within a metal shell under high temperatures, and secondly polish two ends of the shell. The advantages of the component are small cross-sections of fiber, high mechanical strength, and meanwhile fiber itself plays a role as waveguide with the characteristic of high transmission quality. However, the disadvantage is the energy attenuation caused by surface reflection and non- collimation. The optical window is a kind of transparent solid made of glass material, which is fixed between the fiber and ignited powders. There is a little probability resulting from non- collimation. However, the material for windows can absorb laser and disperse radiation of beam, thus leading to the decrease of power density. By selecting a proper material and appropriate thickness of windows combined with the convergence method, the above- Shi-biao Xiang, Xu Xiang, Wei-huan Ji et al. 8 mentioned loss can evidently be decreased. Also, they compared sapphire glass with phosphor glass as window material. When the thickness of window is the same 0.4 mm, the ignition energy for carbon black-doped powders is 3.4 mJ by using sapphire glass as window material, while the ignition energy is 2.3 mJ using phosphor glass as window, and correspondingly the ignition energy is 1.6 mJ with no window. Compared with sapphire glass window, phosphor glass window has a better performance due to its lower thermal conductivity and lower refractive index. 2.3. Experimental Study The dependence of ignition threshold on diameter of core (2r) can be demonstrated by experimental studies and numerical calculations. The experimental setup, schematically illustrated in Fig. 3, is divided into two main parts: A for ignition part and B for testing one. T-type fiber junction separates the emitted laser into two ways, one of which directly delivers laser into photoelectric detector and then to oscillograph. The other transmits laser to ignitor, where energetic powders are ignited and shone. The time difference of two ways of light reaching the oscillograph is defined as ignition delay time t i , measured by a photoelectric detector. The threshold energy of ignition E i can be calculated in the following formula: E i = P i ·t i (1) where P i refers to laser power imported ignitor. Figure 3. Schematic diagram of experimental setup of laser ignition. The maximal output power of LD used in the experiments is 1W and the wavelength is 808 nm. The laser is continuously output and is power-adjustable. The diameter of the coupled fibers is 100 μm, 200 μm and 400 μm, respectively. And the powder is Zr/KClO 4 with a mass ratio of 1:1. The ignition experiments were carried out at room temperature. Fig. 4 shows the relationship of the ignition threshold and laser power with regard to three types of fibers with different diameters of core. Ignition with Optical Fiber Coupled Laser Diode 9 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0 10.5 11.0 2r =400μm 2r =200μm 2r =100μm E i / m J P/W Figure 4. The relationship of ignition threshold and laser power. The three plots in Fig. 4 from bottom to top correspond to the diameter of core of 100 μm, 200 μm and 400 μm, respectively. According to the three plots, one can conclude that the ignition threshold decreases with the increasing laser power when the diameter of core has a fixed value. And the threshold increases as the diameter of core becomes thicker under the same laser power. This suggests that the ignition threshold decreases as the increasing power density under certain conditions. 3. Conclusion Both theoretical and experimental results indicate that two issues need to be considered for the selection of diameters of core in fiber coupled laser ignition system. One is to enhance power density as could as possible i.e., to select the core with thinner diameters, which can decrease the ignition threshold. And the other is to take into account the endurance of fiber itself i.e., the fiber with too thin diameter of core is not suitable. As a result, the commonly used fibers are multi-mode fibers with diameters of core of 100-200 μm. References [1] Litchinitser, N. M.; Sumetsky, M.; Westbrook, P. S. J. Opt. Fiber. Commun. Rep. 2007, 4, 41-85. [2] Li, X. Y.; Voss, P. L.; Chen. J.; Sharping, J. E.; Kumar, P. Opt. Lett. 2005, 30, 1201- 1202. [3] Hutchinson, D. P.; Richards, R. K. J. Opt. Fiber. Commun. Rep. 2007, 4, 1-11. [4] Wong, K. K.Y.; Lu, G. W.; Chen, L. K. Opt. Commun. 2007, 270, 429-432. Shi-biao Xiang, Xu Xiang, Wei-huan Ji et al. 10 [5] Monro, T. M.; Ebendorff-Heidepriem, H. Annu. Rev. Mater. Res. 2006. 36, 467-495. [6] Cregan, R. F.; Mangan, B. J.; Knight, J. C.; Birks, T. A.; Russell, P. St. J; Roberts, P. J.; Allan, D. C. Science 1999, 285, 1537-1539. [7] Knight, J. C.; Broeng, J.; Birks, T. A.; Russell, P. St. J. Science 1998, 282, 1476 [8] Benabid, F.; Couny, F.; Knight, J. C.; Birks, T. A.; Russell, P. St J. Nature 2005, 434, 488-491. [9] Talukder, A. I.; Totsuka, K.; Tomita. M. Appl. Phys. Lett. 2006, 89, 054103. [10] Eggleton, B. J.; Kerbage. C.; Westbrook, P. S.; Windeler, R. S.; Hale, A. Opt. Express 2001, 9, 698-713. [11] Villatoro, J.; Minkovich, V. P.; Pruneri, V.; Badenes, G. Opt. Express 2007, 15, 41491- 1496. [12] Hu, J.; Marks, B. S.; Menyuk, C. R.; Kim, J.; Carruthers, T. F.; Wright, B. M.; Taunay, T. F.; Friebele, E. J. Opt. Express 2006, 14, 4026-4036. [13] Bian, S. P.; Zhang, W. J.; Kuzyk, M. G. Opt. Lett. 2003, 28, 929-931. [14] Akimov, D. A.; Schmitt, M.; Maksimenka, R.; Dukel’skii, K. V.; Kondrat’ev, Y. N.; Khokhlov, A. V.; Shevandin, V. S.; Kiefer, W.; Zheltikov, A. M. Appl. Phys. B 2003, 77, 299-305. [15] Skryabin, D. V.; Luan, F.; Knight, J. C.; Russell, P. St. J. Science 2003, 301, 1705-1708. [16] Wolfbeis, O. S. Anal. Chem. 2004, 76, 3269-3284. [17] Lee S. T.; George, N. A.; Sureshkumar, P.; Radhakrishnan, P.; Vallabhan, C. P. G.; Nampoori, V. P. N. Opt. Lett. 2001, 26, 1541-1543. [18] Hassani, A.; Skorobogatiy, M. Opt. Express 2006, 14, 11616-11621. [19] Minkovich, V. P.; Monzón-Hernández, D.; Villatoro, J.; Badenes, G. Opt. Express 2006, 14, 8413- 8418. [20] Pickrell, G.; Peng, W.; Wang, A. Opt. Express 2004, 29, 1476-1478. [21] Xu, L.; Fanguy, J. C.; Soni, K.; Tao, S. Q. Opt. Express 2004, 29, 1191-1193. [22] Kronenberg, P.; Rastogi, P. K.; Giaccari, P.; Limberger, H. G. Opt. Lett. 2002, 27, 1385- 1387. [23] Santoyo, A. T.; Shlyagin, M. G.; Jimenez, F. J. M.; Oyarzabal, L. N. R. Opt. Commun. 2007, 271, 386-390. [24] Haidekker, M.A.; Akers, W.J.; Fischer, D.; Theodorakis, E.A. Opt. Lett. 2006, 31, 2529- 2531. [25] Baptista, J. M.; Marques, J. M.; Frazao, O.; Santos, S.; Santos, J. L.; Marques, M. B. Opt. Commun. 2007, 271, 224-227. [26] Wang, X. W.; Xu, J. C.; Zhu, Y. Z.; Cooper, K. L.; Wang, A. B. Opt. Lett. 2006, 31, 885-887. [27] Emiliyanov, G.; Jensen, J. B.; Bang, O.; Hoiby, P. E.; Pedersen, L. H.; Kjær, E. M. Opt. Lett. 2007, 32, 460-462. [28] Ye, J. Y.; Myaing, M. T.; Norris, T. B.; Thomas, T.; Baker, Jr. J. Opt. Lett. 2002, 27, 1412-1414. [29] Wang, X. W.; Cooper, K. L.; Wang, A. B.; Xu, J. C.; Wang, Z.; Zhang, Y.; Tu, T. Z. Appl. Phys. Lett. 2006, 89, 163901. [30] Starik, A. M.; Lukhovitskii, B. I. Kinet. Katal. 2006, 47, 333-340. [31] Krauns, Ch.; Barelko, V.; Fabre, G.; Tredicce, J.; Krinsky, V. Catal. Lett. 2001, 72(3-4), 161-165. [32] El-Rabii, H.; Gaborel, G. Appl. Phys. B 2007, DOI: 10.1007/s00340-006-2563-9. Ignition with Optical Fiber Coupled Laser Diode 11 [33] Korepanov, V. I.; Lisitsyn, V. M.; Oleshko, V. I.; Tsipilev1, V. P. Combust. Expl. Shock Waves, 2006, 42, 94-106. [34] Ostrowski, P. P.; Grant, J. F.; Petrow, E. D.; Downs. D. S. Proceedings of the 8th International Pyrotechnics Seminar, 1982, 526. [35] Hill, R. A. Appl. Opt. 1981, 20, 2239-2242. [36] Holy, J. A.; Girmann, T. C. Proceedings of the 13th international pyrotechnic seminar, Grand Junction, CO, 11-15 July 1988, 449-469. [37] Carleton, F. B.; Krallis, K.; Weinberg, F. J. AD-A212 342, 1989. [38] Barrows, A. W.; Forch, B. E.; Beyer, R. A.; Cohen, A.; Newberry, J. E. AD-A261 049. 1993. [39] Phuoc, T. X.; Mathur, M. P.; Ekmann, J. M.; Durbetaki, P. High-Energy Flame, 1993, 94, 349-362. [40] Ostrowski, P. P.; Petrick, J. T.; Petrow, E. D.; Smith, T. C. Proceedings of the 8th International Pyrotechnics Seminar, 1982, 543. [41] Ewick, D. W.; Dosser, L. R.; Mcconb, S. R.; Brodsky, L. P. Proceedings of the thirteenth international pyrotechnic seminar, 263-277, Grand Junktion,CO, 1988 [42] Kunz, S. C.; Salas, F. J. DE88008771/HDM, 1988. [43] Shalem, S.; Tsun, A.; Rave, E.; Millo, A.; Nagli, L.; Katzir, A. Appl. Phys. Lett. 2005, 87, 091103. [44] Schmidt-Uhlig, T.; Karlitschek, P.; Marowsky, G.; Sano, Y. Appl. Phys. B 2001, 72, 183-186. [45] Boucher, C.; Schulze, N. 31AIAA95-2982, 1995. [46] Oestmark, H.; Roman, N. J. Appl. Phys. 1993, 73, 1993-2003. [47] Kramer, D. P.; Beckman, T. M.; Ewick, D. W. Proceedings of the 15th international pyrotechnic seminar, 1990, 569-579. RESEARCH AND REVIEW STUDIES In: Optical Fibers Research Advances ISBN: 1-60021-866-0 Editor: Jurgen C. Schlesinger, pp. 15-49 © 2007 Nova Science Publishers, Inc. Chapter 1 EVANESCENT FIELD TAPERED FIBER OPTIC BIOSENSORS (TFOBS): FABRICATION, ANTIBODY IMMOBILIZATION AND DETECTION Angela Leung 1 , P. Mohana Shankar 2 and Raj Mutharasan 1* 1 Department of Chemical and Biological Engineering 2 Department of Electrical and Computer Engineering, Drexel University, Philadelphia, PA 19104 Abstract Tapered Fiber Optic Biosensors (TFOBS) are sensors that operate based on fluctuations in the evanescent field in the tapered region. In our laboratory, TFOBS are made by heat pulling commercially-available single mode optical fibers. They have been investigated for various applications, including measurement of physical characteristics (refractive index, temperature, pressure, etc.), chemical concentrations, and biomolecule detection. In this chapter, an up-to-date review of TFOBS research is provided, with emphasis on applications in biosensing such as pathogen, proteins, and DNA detection. The physics of sensing and optical behavior based on taper geometry is discussed. Methods of fabrication, antibody immobilization, sample preparation, and detection from our laboratory are described. We present results on the non-specific response, simulation, and detection of E.coli O157:H7 and BSA. This chapter will conclude with an analysis of the future direction of the Tapered Fiber Optic Biosensors. 1.0. Introduction Tapered fiber optic biosensors (TFOBS) are made from optical fibers, and, are capable of detecting specific analytes using optical responses. They have been used for the measurement of physical and chemical properties [4-8], [9-14] of biological molecules [2, 15-19] and have several applications including environmental monitoring, drug screening, clinical diagnostics, and defense. TFOBS offer many advantages including flexibility, ease of use, affordability, * E-mail address:
[email protected]. Tel.: (215) 895-2236. Fax: (215) 895-5837. (Corresponding author) Angela Leung, P. Mohana Shankar and Raj Mutharasan 16 and ability to perform sensing using a small amount of sample. These sensors are based on the evanescent field associated with fiber, and, often are also referred to as Evanescent Field Tapered Fiber Sensors. In this chapter, the basics of TFOBS are discussed, along with an up- to-date literature review of TFOBS. Experimental methods and recent results from our laboratory are also presented. 2.0. Physics of Evanescent Field Sensing in Tapered Fibers Optical fibers are cylindrical waveguides, and, are made of a silica core surrounded by a silica cladding. The core refractive index is higher than the cladding refractive index (RI) because it is doped with Ge. Light propagates through the core by total internal reflection (TIR). Besides the light propagating in the core, there is a small component of light, known as the evanescent field, which decays into the cladding. Evanescent light penetration is described by its penetration depth (d p ), which is the position away from the core/cladding interface at which the light decays to 1/e of its value at the core-cladding interface, and is given by: 2 2 2 2 sin p co cl d n n λ π θ = − (1) In eqn. (1), λ is the operating wavelength of light, n co the index of the core and n cl the index of the cladding. The angle of incidence at the core cladding interface is θ. The evanescent field in a uniform diameter fiber does not interact with the outside environment because it decays to a negligible value as it reaches beyond the cladding. This is due to the fact that in typical fibers the cladding thickness is several times that of the core. However, if the cladding is removed or the fiber is tapered down to a diameter less than the original core diameter, evanescent field can interact with the external medium affecting the transmission through the fiber. The penetration depth in a tapered fiber depends on the local diameter of the tapered fiber, the RI of the core, and RI of the external medium. Since there is a continuous change in the diameter along the fiber in the tapered region (except in the waist), coupling of light among the modes can occur [20]. Coupling in the tapered region causes the transmission properties of the fiber to change. Presence of analytes in the tapered region can lead to RI changes in the taper. This results in changes in the coupling characteristics and causes changes in the optical throughput. Physical characteristics of the fiber such as RI of the core and cladding, core diameter, and operating wavelength determine the number and type of modes that propagate through the fiber. The lowest order mode has the tightest confinement of the field, and hence the weakest evanescent field. As the mode order goes up, the associated evanescent field also increases. In a tapered or de-cladded fiber, the optical characteristics of the surrounding medium such as its index, absorption, etc. can affect the optical throughput. Evanescent Field Tapered Fiber Optic Biosensors (TFOBS)… 17 2.1. Wave Propagation in Absorption Sensors The shape of the optical field in a fiber is determined by the number of modes present. Figure 1 shows a tapered fiber with a short region of constant thickness (waist) and contracting and expanding regions. The number of modes that can be supported in a fiber is determined by the V-number, 2 2 0 2 co cl a V n n π λ = − (2) where a 0 is the radius of the core. When V<2.405, only the lowest order mode is supported, and as V increases, number of modes increase. Although in a single-mode (SM) fiber only the lowest order is supported, in the tapered region higher order modes can potentially be supported because of the larger difference in refractive index between the core and the sample (~0.12) compared to a regular fiber (~0.01). Reduction of the fiber radius increases the evanescent field strength, and enhances the interaction of the evanescent field with the analyte leading to variations in optical throughput (transmitted light). Figure 1. Photograph of a TFOBS. The region of interest in a tapered fiber is identified by the region where V core <1. 2.2. Wave Propagation in Continuous Bi-conical Tapered Fibers In our laboratory, tapered fibers were made by heat-pulling an optical fiber without removing the cladding. Unlike uniform fibers, the V-number changes along the length of a tapered fiber. When V-number becomes less than unity, the core is too small to contain the light and light guidance is determined by the original cladding which acts as the core and the external medium of RI n ext which serves as the cladding. The new V-number is called V clad where the parameter, a 0 in Eq. (2) is replaced by the radius of the overall fiber, b(z), and is given by 2 2 2 ( ) ( ) clad cl ext b z V z n n π λ = − (3) Angela Leung, P. Mohana Shankar and Raj Mutharasan 18 In tapered fibers, the V-value is generally referred to as V clad to distinguish it from V core , given in Eq. (2). Note that the diameter b in eqn. (3) is a function of the location (z), indicating the existence of tapering. 2.3. Numerical Simulation of Light Transmission in a Tapered Fiber Numerical simulation of light transmission through a tapered fiber can provide useful insight into its properties. To simplify the analyses, the simplified mode theory based on linearly polarized modes (LP) can be used to determine the transmission behavior [21]. Assuming that light enters into the fiber parallel to the axis, the only modes that are excited are the LP 0m modes. The transverse components of the electrical field inside the fiber are: ( ) ( ) 0 0 0 0 0 0 0 0 ( ), 1 ( ) ( ), 1 m X m J U R R E r A J U K W R R K W ≤ ⎧ ⎫ ⎪ ⎪ = ⎨ ⎬ > ⎪ ⎪ ⎩ ⎭ (3) where R is the normalized radial coordinate, r/a 0, whereas U and W are constants They depend on the wavelength and RI of the core and cladding, 0 2 2 2 2 2 0 0 2 m co m U a n π β λ ⎡ ⎤ ⎛ ⎞ = − ⎢ ⎥ ⎜ ⎟ ⎝ ⎠ ⎢ ⎥ ⎣ ⎦ (4a) 0 2 2 2 2 2 0 0 2 m m cl W a n π β λ ⎡ ⎤ ⎛ ⎞ = − ⎢ ⎥ ⎜ ⎟ ⎝ ⎠ ⎢ ⎥ ⎣ ⎦ (4b) where c is the speed of light and β is the propagation constant. When m=1, only the fundamental mode exists. In eqn. (3) J 0 (.) and K 0 (.) are the Bessel and modified Bessel functions of zero th order, respectively. A is a constant determined from orthogonality principle [21]. The subscript m represents the various circularly symmetric LP 0m modes that may be present in the fiber. When V core <1 and V clad >2.405, many modes are supported since the index difference between the cladding and the external medium (n ext ) is large. As mentioned previously, tapering leads to coupling among LP 0m modes [20, 22] . A simple means to visualize the taper is to model taper geometry by approximating the slopes of the taper by stepwise linear approximation. At each step ‘i’, the parameters U 0m and W 0m are analogous to the constants U and W in a uniform diameter fiber. They can be expressed using the local radius i ρ as 0 0 2 2 2 2 2 2 m m i i i cl U n π ρ β λ ⎡ ⎤ ⎛ ⎞ = − ⎢ ⎥ ⎜ ⎟ ⎝ ⎠ ⎢ ⎥ ⎣ ⎦ (5a) Evanescent Field Tapered Fiber Optic Biosensors (TFOBS)… 19 0 0 2 2 2 2 2 2 m m i i i ext W n π ρ β λ ⎡ ⎤ ⎛ ⎞ = − ⎢ ⎥ ⎜ ⎟ ⎝ ⎠ ⎢ ⎥ ⎣ ⎦ (5b) The V-number for each step is given by: 1/ 2 2 2 2 i i cl ext V n n π ρ λ ⎡ ⎤ = − ⎣ ⎦ (6) The values of U, W and β are calculated using the LP mode approximation [21]. The relationship between the modal amplitudes of the LP 0m modes of the i th and (i+1) th step is: 1 1 1 1 1 1 ( ) ( ) i i i i q m j z j z i i i i m m q q m q A E r e B E r e α β + + − − + + = = = ∑ ∑ (7) where E(r) is the electric field, β m is the propagation constant on the left and α q is the propagation constant on the right. It has been assumed that E(r) are orthonormal [21]. A m is the amplitude of the modes on the left and B q is the amplitude of the modes on the right. That is, 2 2 0 0 ( ) 1 E r rdrd π φ ∞ = ∫ ∫ (8) The amplitude on the right is obtained by applying the orthogonality principle: 1 1 1 ; + + − − + = ∑∑ i i i i q m j z j z i i q m nm pq n m B e A e C α β (9) 1 ; 0 2 ( ) ( ) ∞ + = ∫ i i nm pq q m C E r E r rdr π (10) In the tapered region, light is coupled among the various LP 0m modes. When V core =1, power in the LP 01 cladding mode is transferred to LP 01 core mode and appears at the output end of the fiber. The light remaining in other modes stays in the cladding and is lost. A MATLAB ® program was used to estimate the amplitude and output power. The taper geometry, wavelength and number of steps were varied to determine the resulting changes in power. Sample simulation results, illustrated in Figure 2, show changes in transmission vs. waist diameters for two taper geometries. In Panel A, the taper geometry resembles a symmetric taper made by the fusion splicer, while in Panel B, the simulation is for a long taper similar to a heat-drawn taper. The transmission is normalized with respect to air, so that a value of 1.2 indicates a transmission increase of 20% in water compared to air. Figure 2 show that as the waist radius increases, the difference in transmission between water and air decreases. However, at intermediate values the ratio may be higher or lower than unity, particularly at smaller waist diameters. For certain values of the radius, the transmission through water is Angela Leung, P. Mohana Shankar and Raj Mutharasan 20 higher than in air. At a longer wavelength (550 nm, for example) and for the same waist diameter values, the difference in transmission between air and water differ by less than 10%. To explore further, simulation was undertaken by varying the diameter of the waist in much smaller steps of 0.001 μm. These results are shown in Figure 3 for two starting diameters, 5 μm and 6.25 μm. The transmission characteristics change significantly for small changes in diameter. For example, a 5.54 μm diameter taper exhibits 30% higher transmission in water 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 5 9 13 17 Waist diameter (μm) N o r m a l i z e d T r a n s m i s s i o n 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 5 9 13 17 Waist diameter (μm) N o r m a l i z e d t r a n s m i s s i o n 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 5 9 13 17 Waist diameter (μm) N o r m a l i z e d T r a n s m i s s i o n 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 5 9 13 17 Waist diameter (μm) N o r m a l i z e d t r a n s m i s s i o n Figure 2. Transmission in water normalized with respect to air as waist diameter is altered. Top: A short symmetric taper: a= 0.425 mm, b=0.325 mm, c=0.500 mm. Operating wavelength = 470 nm. Bottom: A longer asymmetric taper: a=2.25 mm, b=0.245 mm, c=4.5 mm. Operating wavelength = 550 nm. Adapted from [3]. Evanescent Field Tapered Fiber Optic Biosensors (TFOBS)… 21 while a 5.58 μm waist diameter taper transmits 20% less transmission. The differences, however, become smaller for larger diameters. The example of 6.25 μm in Figure 3 shows that the changes in transmission were less than 10 %. 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 0 0.02 0.04 0.06 0.08 0.1 Change in waist diameter (mm) N o r m a l i z e d T r a n s m i s s i o n at 6.25 mm at 5.5 mm Figure 3. Transmission characteristics of a taper (a= 2.25 mm, b=0.25 mm, c=4.5 mm) in water at 470 nm as a function of change in waist diameter. Transmission is normalized with respect to transmission in air at the corresponding geometric values. Smaller starting diameter tapers show large changes in transmission for small (0.01 μm) changes is waist diameter. Adapted from [3]. These simulation results can serve as a guide to the analysis and interpretation of the experimental data on the tapered fibers. It is important to recognize that in the simulation we considered only the effect of refractive index in the waist region. In actual sensing experiments, the cells absorb at the operating wavelength, and the resulting sensor response is a complex interplay of these two phenomena. Furthermore, cells do not have homogeneous RI because the cells constitute particulate matter. Finally, the cell attachment onto the taper surface is often not uniform as we showed in our earlier report [2]. 3.0. Literature Review In this section, the applications of TFOBS for pathogen detection, toxins measurements, clinical measurements, and DNA detection are presented. In tables 1 to 3, we summarize the analytes detected, matrices in which they were detected, detection principle, basis of sensors, and detection limits. Table 1. TFOBS For Pathogen and Toxin Measurement Target Analyte LOD Matrix Taper Geometry Fiber Type Detection Principle References Bacillus anthracis 3.2E5 spores/mL buffer BT Polystyrene MM Fluorescent sandwich assay [32] Bacillus subtilis var. niger 8 x 10(4) spores/mL buffer NA (chip) NA (chip) Leaky wave (SPR) [59] LacZ DNA in Escherichia coli 25 pM buffer Uniform MM Fluorescent intercalating agents [55] Staphylococcus aureus Protein A 1 ng/mL ND ND MM plastic Fluorescent sandwich assay [35] Escherichia coli O157:H7 0.016 dB/h/N o , Initial number (N o ): 10-800 * buffer BT MM Absorption [23] Escherichia coli O157:H7 70 cells/mL Buffer BT SM Intensity [2] Escherichia coli O157:H7 1 CFU/ml ground beef samples Uniform MM polystyrene Fluorescent sandwich assay [25, 26] Salmonella 50 CFU/g irrigation water used in the sprouting of seeds RAPTOR – uniform Waveguide Fluorescent sandwich assay [27] Salmonella 10(4) CFU/ml Hotdog samples RAPTOR – uniform Waveguide Fluorescent sandwich assay [60] Salmonella 10(4) CFU/mL Nutrient broth TT MM Fluorescent sandwich assay [28] Table 1. Continued Target Analyte LOD Matrix Taper Geometry Fiber Type Detection Principle References Staphylococcal Enterotoxin B Min: 0.5 ng/ml (buffer) Buffer, human serum, urine, and aqueous extract of ham CTT ND Fluorescent sandwich assay [33] C. Botulinum toxin A, Pseudexin Toxin Min: 30 pM (C. Botulinum toxin A), 60 pM (Pseudexin) ND CTT MM Fluorescent sandwich assay [61] Clostridium-Botulinum Toxin-A 5 ng/mL buffer TT MM Fluorescent sandwich assay [36] E. coli lipopolysaccharide endotoxin Min: 10 ng/ml Buffer and plasma CTT ND Fluorescent sandwich assay [38] Ricin Concentration Min: 100 pg/ml (buffer) Max: 1 ng/mL (river water) Buffer, river water CTT MM (plastic clad silica) Fluorescent sandwich assay [37] Listeria monocytogenes 5 x 10(5) CFU/ml frankfurter sample RAPTOR – uniform Waveguide Fluorescent sandwich assay [31] Listeria monocytogenes 5.4 x 10(7) CFU/ml Hotdog samples RAPTOR – uniform Waveguide Fluorescent sandwich assay [30] Listeria monocytogenes 4.3x10(3) CFU/ml Buffer Uniform MM polystyrene Fluorescent sandwich assay [29] Abbreviations: BT = Biconical Taper, TT = Tapered Tip, CTT = Combination Taper Tip, SM = Single Mode, MM = Multimode, ND = Not Described, * = the change in dB per hour per number of cells at inoculation, NA = Not Applicable. Table 2. TFOBS For Biochemical Measurements Target Analyte LOD Matrix Taper Geometry Fiber Type Detection Principle References NADH, NADPH Concentration Min: 0.2 μM (NADH), 0.5 μM (NADPH) buffer BT SM Absorption [18] Chinese Hamster Ovary Cell Concentration Min: 10 5 cells/ml buffer BT SM Absorption [18] Paraoxon Sub ppm buffer TT MM Chemiluminescence [62] STAT3 ND Buffer Uniform MM Fluorescent sandwich assay [63] Abbreviations: BT = Biconical Taper, TT = Tapered Tip, CTT = Combination Taper Tip, SM = Single Mode, MM = Multimode, ND = Not Described. Table 3. TFOBS For Clinical Measurements Target Analyte LOD Matrix Taper Geometry Fiber Type Detection Principle References Protein A 1 μg/mL ND NA (chip) NA Leaky wave (SPR) [64] BSA 10 fg/mL Buffer BT SM Intensity [1] BSA 7.4 ng/mL buffer Chip (NA) Chip (NA) SPR [65] BSA 2.5 μg/ml Buffer BT ND (plastic clad silica) Dye-protein complex absorption [43] Ovalbumin 2.5 μg/ml Buffer BT ND (plastic clad silica) Dye-protein complex absorption [43] Table 3. Continued Target Analyte LOD Matrix Taper Geometry Fiber Type Detection Principle References Hemoglobin 2.5 μg/ml Buffer BT ND (plastic clad silica) Dye-protein complex absorption [43] IgG 20 fM Buffer TT MM Fluorescent competitive assay [44] IgG 75 pg/mL Serum and jejunal fluids diluted with buffer BBT SM Fluorescent sandwich assay [45] Protein C 0.1 μg/mL Buffer TT ND Fluorescent sandwich assay [16] Protein C 0.5 μg/mL Plasma TT MM Fluorescent sandwich assay [46] Protein C 0.5 μg/mL Plasma Uniform MM Fluorescent sandwich assay [66] Protein S 0.5 μg/mL Plasma Uniform MM Fluorescent sandwich assay [66] Antithrombin III (ATIII) 30 μg/mL Plasma Uniform MM Fluorescent sandwich assay [66] Plasminogen (PLG) 30 μg/mL Plasma Uniform MM Fluorescent sandwich assay [66] B-type natriuretic peptide (BNP) 0.1 ng/mL Plasma Uniform MM Fluorescent sandwich assay [66] cardiac troponin I (cTnI) 1 ng/mL Plasma Uniform MM Fluorescent sandwich assay [66] C-reactive protein (CRP) 1 μg/mL Plasma Uniform MM Fluorescent sandwich assay [66] Table 3. Continued Target Analyte LOD Matrix Taper Geometry Fiber Type Detection Principle References Myoglobin (MG) 75 ng/mL Plasma Uniform MM Fluorescent sandwich assay [66] L. donovani Antibody Concentration Min: 0.244 ng/ml Serum CTT MM (plastic clad silica) Fluorescent sandwich assay [17] Progesterone ng/mL Buffer ND ND Fluorescent sandwich assay [42] Adriamycin 0.01 μg/mL blood Straight core tip MM Fluorescence quenching [49] Cytochrome c ND Cell TT MM Fluorescent sandwich assay [50] Cytochrome c 2.5 μg/ml buffer BT ND (plastic clad silica) Dye-protein complex absorption [43] Yersinia pestis fraction 1 50 ng/mL Buffer, serum, plasma, and whole blood BT MM Fluorescent sandwich assay [15] cTnI 31 pM plasma TT MM quartz Nano gold particle enhanced fluorescence [67] BNP 26 pM plasma TT MM quartz Nano gold particle enhanced fluorescence [67] Intracellular Benzopyrene Tetrol 6.4 pM cell TT ND Autofluorescence [51] Benzo\c\phenanthridinium alkaoids ND buffer Chip NA SPR [68] Fumonisin B 1 10 ng/ml methanol/water- extracted corn TT MM (plastic clad silica) Fluorescent sandwich assay [69] Myoglobin 2.9 ng/mL buffer tip MM SPR [39] Table 3. Continued Target Analyte LOD Matrix Taper Geometry Fiber Type Detection Principle References Myoglobin 5 nmol/L buffer Uniform probe MM Fluorescent Energy Transfer [40] Thrombin 1 nM Buffer Spheres (NA) NA Fluorescent competitive assay [48] Thrombin 1nM Buffer Uniform MM Coagulation of fluorescently labeled fibrinogen to unlabelled fibrinogen bound to the surface of the fibre optic [47] RNA pM Buffer TT SM Fluorescence [56] DNA 70 fM Buffer Uniform MM Fluorescence [52] interleukin-1 (IL-1), interleukin-6 (IL-6), and tumor necrosis factor-a. (TNF-alpha) 1 ng/mL Buffer and spiked cell culture medium (CCM) ND MM Fiber-optic surface plasmon resonance (SPR) [41] DNA 5 nM buffer ND MM Fluorescence [53] Abbreviations: BT = Biconical Taper, TT = Tapered Tip, CTT = Combination Taper Tip, SM = Single Mode, MM = Multimode, ND = Not Described, NA = Not Applicable. Angela Leung, P. Mohana Shankar and Raj Mutharasan 28 3.1. Pathogen Detection Escherichia coli O157:H7 [2, 23-26], Salmonella typhimurium [27, 28], Listeria monocytogenes [29-31], and Bacillus anthracis [32] are some of the pathogens which have been detected using TFOBS. Most pathogen detection studies done to date used fluorescence TFOBS [25-32], but a few of them used intensity-based TFOBS [2, 23, 24]. Ferreira et al. developed an intensity-based evanescent sensor, to be used with a 840 nm light source, to detect Escherichia coli O157:H7 growth [23]. This evanescent sensor was fabricated by chemically etching. Transmission is reduced due to light absorption by the bacteria, and the power loss is proportional to the intrinsic bulk absorption and scattering, which depends on the concentration of the bacteria. The sensitivity of this sensor was 0.016 dB / hour-N o , where N o is initial cell concentration and ranges from 10 to 800. Similarly, Maraldo et al. used TFOBS to detect Escherichia coli JM 101 growth on poly-L-lysine [24]. E.coli JM 101 expressing green fluorescent protein was immobilized on the poly-L-lysine coated fibers, and growth was monitored by light transmission at 480 nm. The transmission decreased exponentially with cell growth on the tapered surface. In a follow up study by Rijal et al., Escherichia coli O157:H7 (EC) was covalently bonded to the surface of a TFOBS via an antibody, and concentrations as low as 70 cells/mL was detected by changes in intensity at 470 nm [2]. Detection of EC in real samples is of great interest and was investigated by DeMarco et al. [25]. EC in seeded ground beef samples was prepared and detected by a sandwich immunoassay using cyanine 5 dye-labeled polyclonal anti-E. coli O157:H7. Light was launched at 635 nm and the fluorescence was emitted at 670 to 710 nm. Responses were obtained within 20 minutes, and E. coli O157:H7 at 3 to 30 CFU/mL were detected. A similar study was recently conducted by Geng et al., where a sandwich immunoassay was used with FOBS to detect EC in ground beef [26]. Light was launched at 635 nm and the fluorescence was emitted at 670 to 710 nm. The sensor detected 10(3) CFU/ml of pure cultured EC grown in culture broth. Artificially inoculated EC at concentration of 1 CFU/ml in ground beef was detected after 4 hours of enrichment. Kramer et al. [27] studied the detection of Salmonella typhimurium in sprout rinse water using RAPTOR™, an evanescent fluorescence sensor developed by Research International, Monroe, Washington.. Alfalfa seeds contaminated with various concentrations of Salmonella typhimurium were sprouted, and the sprout water was measured by the instrument. Salmonella typhimurium was identified for seeds that were contaminated with 50 CFU/g. Zhou et al. [28] also used a sandwich immunoassay to detect Salmonella. Light was launched at 650 nm and the fluorescence was emitted at 680 nm. Tapered fiber tips with various geometries and treatments were studied and optimized, and Salmonella was detected at 10(4) CFU/mL. An antibody-based sandwich fluorescence FOBS was developed by Geng et al. to detect Listeria monocytogenes [29]. Light was launched at 635 nm and the fluorescence emission was in the range of 670 to 710 nm. The sensor was specific, as shown by the significantly lower signals caused by other Listeria species or microorganisms. The LOD was 4.3x10(3) CFU/ml for a pure culture of L. monocytogenes. In less than 24 h, L. monocytogenes in hot dog or bologna was detected at 10 to 1,000 CFU/g after enrichment. Recently, Kim et al. also detected L. monocytogenes using the RAPTOR™ sensor [30]. This method achieved a LOD of 5.4 x 10(7) CFU/ml. L. monocytogenes was detected in phosphate buffered saline (PBS) by Nanduri et al. using RAPTOR™ to evaluate the effect of flow on antibody immobilization Evanescent Field Tapered Fiber Optic Biosensors (TFOBS)… 29 [31]. Light was launched at 635 nm and the fluorescence was emitted at around 670 nm. It was found that both the static and the flow through mode method had a LOD of 1 x 10(3) CFU/ml. However, the effective disassociation constant and the binding valences for static modes were higher than for flow through method of antibody immobilization. The flow through mode was chosen to test real samples, and the LOD was 5 x 10(5) CFU/ml. Bacillus anthracis, is a serious threat to national security. Tims et al. addressed the need to detect Bacillus anthracis, and achieved detection at a concentration of 3.2 x 10(5) spores/mg in spiked powders in less than 1 hour [32]. The method used was based on fluorescent sandwich assay and a polystyrene tapered fiber. The excitation wavelength was 635 nm. 3.2. Toxin Measurement TFOBS have been used to detect toxins such as enterotoxins [33-36], ricin [37], and endotoxins [38]. Fluorescence was used for all the toxins measurements which are discussed here. Staphylococcal enterotoxins are a major cause of food poisoning. Tempelman et al. quantified Staphyloccoccal enterotoxin B (SEB) in a fluorescent sandwich immunoassay on a fiber optic biosensor [33]. A 635 nm diode laser was used to excite the labeled antibody. The fluorescence level was measured and gave a detection limit of 0.5 ng/mL. Shriver-Lake et al. used an array biosensor to detect SEB at a LOD of 0.5 ng/mL in buffer and six different types of food samples [34]. Staphylococcus aureus is the only species which produces protein A and was detected by Chang et al. using a fluorescent sandwich FOBS at a LOD of 1 ng/mL [35]. Excitation of this sensor was at 488 nm. Similar to SEB, Clostridium botulinum toxin A was detected by a fluorescent sandwich FOBS at 5 ng/mL [36]. A light source at 514 nm was used in this case. Narang et al. reported a sandwich fluorescent TFOBS ricin detection in buffer and in river water [37]. The light source was 635 nm. Antibody to ricin was immobilized onto tapered fiber surface using silanization and avidin-biotin linkage. The avidin-biotin method had a higher sensitivity and wider linear dynamic range. The response of the avidin-biotin sensor was linear in the range of 100 pg/mL to 250 ng/mL. The LOD for ricin in buffer solution was 100 pg/mL, and in river water it is 1 ng/ml. At concentrations greater than 50 ng/ml, there was a strong interaction between ricin and avidin due to the lectin activity of ricin. This interaction was reduced for fibers coated with neutravidin or with the addition of galactose. James et al. developed a method to detect lipopolysaccharide (LPS) endotoxin, which is the most powerful immune stimulant and causes sepsis [38]. LPS from E. coli was detected at a LOD of 10 ng/mL using fluorescent FOBS based on the competitive assay. Polymyxin B was used as a recognition molecule and was covalently immobilized onto the surface of the probe. Fluorescent labeled LPS was introduced to the fiber and attached to the Polymyxin B. Unlabeled LPS was then introduced and competed with the labeled LPS for the binding sites on the Polymyxin B. As LPS concentration increases, fluorescence decreases. Angela Leung, P. Mohana Shankar and Raj Mutharasan 30 3.3. Clinical Measurements Most clinical measurements done with TFOBS used proteins as analytes. Notable examples include cardiac markers [39, 40], cytokines [41], and hormones [42]. Investigators have detected model proteins using TFOBS in order to characterize TFOBS’ potential. Preejith et al. detected model proteins using fiber optic evanescent wave spectroscopy [43]. They immobilized Comassie Blue on a multimode fiber surface using a porous glass coating. Comassie Blue normally absorbs at 467 nm, but it forms a dye-protein complex with the protein when exposed to an acidic environment, and such a complex absorbs at 590 nm. The protein concentration is inversely proportional to the output power at 590 nm, because increase in protein concentrations causes the evanescent absorption to increase. Calibration curves were obtained for BSA, hemoglobin, ovalbumin, and cytochrome c in the range of 0 to 20 μg/mL. In our laboratory, BSA was recently detected at 10 fg/mL in stagnant condition using intensity-based TFOBS [1]. Tromberg et al. detected antibody to IgG at 20 fM on a fluorescent FOBS tip using a competitive assay [44]. Light was launched at 488 nm and the fluorescence was emitted at 520 nm. Rabbit IgG was immobilized on the fiber tip, and exposed to fluorescein isothiocyanate (FITC) labeled and unlabeled anti-IgG. The response was inversely proportional to the amount of unlabeled anti-IgG, because the unlabeled anti- IgG displaced the labeled one. Hale et al. developed a fluorescent optical fiber loop sensor to detect antibody to IgG [45]. The sensor was used with a two-step sandwich assay. IgG was labeled with the fluorescent dyes fluorescein isothiocyanate or tetramethyl rhodamine. Antibody to IgG was detected at 75 pg/mL with this method. Deficiency in Protein C (PC), if left untreated, may result in thrombotic complications, and, thus presents an important clinical challenge. Spiker et al. detected PC at 0.1 μg/mL in buffer using a sandwich fluorescent fiber optic sensor [16]. Real-time detection of PC in plasma is an important challenge in the clinical setting. Convective flow plays a vital role in the transport of PC in a viscous medium such as plasma. Tang et al. who examined PC detection in plasma with fluorescent sandwich FOBS and obtained a detection limit of 0.5 μg/mL [46]. Cardiac markers myoglobin (MG) and cardiac tropinin I (cTnI) can be measured to predict the occurrence of myocardial infarction, because they are released from cardiac muscles when they are damaged. A fiber-optic SPR sensor was developed by Masson et al. to detect MG and cTnI at 3 ng /mL [39]. A direct fluorescence FOBS was also used to detect myoglobin at 5nM [40]. An excitation wavelength of 425 nm was used to excite the Cascade Blue-labeled antibody, which was entrapped in the sensing element and fluoresces at 425 nm. Fluorescence quenching occurred when myoglobin attaches to the labeled antibody. Recently, Tang et al. developed a fiber-optic multi-analyte system which simultaneously quantifies two groups of multi-biomarkers related to cardiovascular diseases (CVD): anticoagulants (protein C, protein S, antithrombin III, and plasminogen) for deficiency diagnosis; and cardiac markers (B-type natriuretic peptide, cardiac troponin I, myoglobin, and C-reactive protein) for coronary heart disease diagnosis. Garden et al. detected thrombin at 1 nM using fluorescent FOBS [47]. Excitation was at 495 nm and emission was at 520 nm. Unlabeled fibrinogen was first attached to the FOBS surface. Then, coagulation of solution phase fluorescently labeled fibrinogen to unlabelled fibrinogen bound to the surface was observed. Lee et al. detected thrombin at 1 nM using a Evanescent Field Tapered Fiber Optic Biosensors (TFOBS)… 31 fluorescent FOBS immobilized with an antithrombin DNA aptamer receptor [48]. The aptamer was immobilized on the surface of silica microspheres, which were distributed in microwells on the distal tip of an imaging fiber that was coupled to a modified epifluorescence microscope system. Another set of microspheres was prepared with a different oligonucleotide to measure the non specific binding. The distal end of the imaging fiber was incubated with fluorescein-labeled thrombin (F-thrombin), and the non-labeled thrombin was detected using the competitive method. Progesterone was found to have evidence of carcinogenicity based on animal studies. Progesterone can be found in various surface waters commonly used for drinking water. In a study by Tschmelak et al., a fluorescence FOBS was immobilized with a labeled-antibody and used successfully to detect progesterone at concentrations lower than ng/L [42]. A fluorescent tip FOBS was used to measure adriamycin (ADM) at 10 ng/mL in vivo in a blood vessel [49]. A polymeric fluorescent D-70 membrane with pore sizes of 1-2 μm was immobilized on the fiber tip. Fluorescence was quenched by ADM present in the blood and the fluorescence signal was measured by a photomultiplier tube (PMT) at a wavelength of 530 nm. The protein cytochrome c is involved in apoptosis and was detected by a sandwich fluorescent nanobiosensor fabricated by Song et al. [50]. δ-Aminolevulinic acid (5-ALA), a photodynamic therapy (PDT) drug, was activated by a He-Ne laser at 632.8 nm to induce apoptosis in MCF-7 human breast carcinoma cells. When mitochondria are damaged by PDT, cytochrome c is released into the cytoplasm; therefore cytochrome c concentration is an indication of apoptosis. Results indicate that 5-ALA PDT-treated cells had a much higher fluorescence signal, pointing to high cytochrome c concentrations in the treated cells. Yersinia pestis is an etiologic agent of plague. A sandwich fluorescent FOBS devised by Cao et al. was used to detect Yersinia pestis Fraction 1 antigen at a limit of 5 ng/mL [15]. The light source was a 514 nm argon ion laser. This system detected 50 – 400 ng/mL of protein in serum, and the results were in excellent agreement with ELISA results. Nath et al. developed a fluorescent FOBS to detect L. donovani specific antibodies [17]. The sensor was made by de-cladding an optical fiber so that the evanescent wave propagated outside the tapered region. The sensor was used with a 488 nm light source. Cell surface protein of L. donovani was immobilized covalently on the sensing region. Then, the sensor was incubated with patient serum for 10 minutes, followed by incubation with goat anti- human IgG tagged with FTIC, which excites at 525 nm. The amount of L. donovani specific antibodies in the patient serum was proportional to the fluorescence. There were no false positive results from leprosy, tuberculosis, typhoid, and malaria serum. Cullum et al. detected benzo[a]pyrene tetrol (BPT) at 6.4 ± 1.7 E pM in mammary carcinoma cells using a sandwich fluorescent fiber-optic nanosensor tip [51]. BPT is a metabolite of benzo[a]pyrene. Using a 325 nm light source, the authors were able to calibrate the sensor and obtain an unknown concentration by observing the level of fluorescence. This technique is useful for cancer screening since carcinogens bind to DNA and form substances such as BPT. Three cytokines related to chronic wound healing are interleukin-1 (IL-1), interleukin-6 (IL-6), and tumor necrosis factor-α (TNF-alpha) [41]. A fiber-optic SPR sensor was modified with antibodies at the surface, and detected these proteins with LOD of 1 ng/mL in buffered saline solution and spiked cell culture medium (CCM). Angela Leung, P. Mohana Shankar and Raj Mutharasan 32 3.4. DNA Hybridization Kleinjung et al. detected DNA hybridization at 3.2 attomoles (70 fM) using a fluorescent multimode FOBS with 13-mer probe attached to the de-cladded core [52]. The complementary strands were labeled and detected when introduced to the sensor. This sensor was able to distinguish between matching sequences, single nucleotide mismatch, and mismatch caused by additional deviations. Zeng et al. examined the interfacial hybridization kinetics of oligonucleotides immobilized onto silica using a fluorescent FOBS that was excited at 632 nm [53]. A dT20 DNA probe was used as recognition molecules, while target fluorescein-labeled non- complementary DNA (ncDNA) dT20 and fluorescein-labeled dA20 were detected. The target DNA concentrations were 5 nM to 0.1 μM. The response of the sensor fit the second order Langmuir model. Molecular beacons (MB) are oligonucelotide probes that fluoresces upon hybridization with target DNA or RNA molecules [54]. Liu et al. immobilized MB on a fluorescent FOBS and determined the effects of ionic strength and target DNA concentration on hybridization kinetics. Using an excitation wavelength of 514 nm, they found the LOD was 1.1 nM of DNA. The sensor showed selectivity by distinguishing between 100 nM of ncDNA, 100 nM of one-base mistmatch, and 100 nM of cDNA [54]. 3.4.1. Pathogen Detection via DNA A fluorescent FOBS was developed by Almadidy et al. to detect short sequences of oligonucleotides that identify E. coli microbial contamination [55]. DNA probes were first immobilized to silica surface via a silane reagent. Then, stepwise synthesis of oligonucleotides by the β-cyanoethyl-phosphoramidite protocol took place on the surface. The sensor was exposed to both complementary (cDNA) and non-complementary (ncDNA) 20-mers, as well as genomic DNA from E.coli. The cDNA and ncDNA were introduced at a concentration of about 1.7 nM, whereas genomic DNA was introduced at 1.7 pM to 170 pM. Fluorescent intercalating dye was used to detect hybridization. Quantities as low as 100 fM were detected using this method. Pilevar et al. detected Helicobacter pylori total RNA using a fluorescent FOBS that had probes immobilized on its surface [56]. IRD-41 is a near-infrared fluorophore which is excited by 785 nm light. Real-time hybridization measurement of IRD 41-labeled oligonucleotide at various concentrations to the surface bound probes was performed. Complementary DNA at lower than nM concentration was detected. Sandwich assays were performed with Helicobacter pylori total RNA, and results showed that this sensor could detect H. pylori RNA in a sandwich assay at 25 pM. 4.0. Methods 4.1. Fabrication Corguide fibers (Corning Glass Works, NY, attenuation at 1300 and 1500 nm of 0.36 and 0.26 dB km−1, respectively) with a core diameter of 8 μm and total diameter of 125 μm were used in all the fabrication methods described here. The fabrication methods commonly used Evanescent Field Tapered Fiber Optic Biosensors (TFOBS)… 33 in our laboratory are chemical etching, heat pulling by flame, and heat pulling by fusion splicer. 4.1.1. Chemical Etching Chemical etching using hydrofluoric acid (HF) is one of the simplest ways to create tapers with a step change in radius. Acrylic (Plexiglas) was used to construct the etching reactor because HF does not attack most plastic materials. In order to monitor the etching, a spectrofluorometer was used to detect the transmission through the fiber as etching took place. This instrument has a compact 75 W Xenon arc lamp (Ushio Inc., Japan) coupled to a monochromator and a PMT (model R1527P in housing 710, PTI Inc.) coupled to a monochromator. The plastic sheathing of the fiber was removed by immersing the fiber in acetone (Fisher Scientific) for 15–20 min followed by mechanical removal with a fiber optic stripper (NO- NIK). A fiber optic cleaver (NO-NIK) was used to make a clean-cut fiber tip so as to enhance the efficiency of light collection into the fiber. HF (Fisher Scientific, Philadelphia) at a concentration of 49.5 wt.% was used. Two hundred microliter of HF was introduced into the reaction chamber. Once HF was injected, the transmission was monitored at 350 nm. When the diameter of the fiber was etched to a certain fraction of the initial diameter, the etching process was stopped by first removing the HF and then washing the chamber twice with 5 N NaOH as rapidly as possible. The fiber was then immersed in a 200 mL of 0.1 N NaOH bath for 60 min to stabilize the fiber. If this step was not carried out, any remaining HF would have continued to etch the fiber until it dissolved completely. It was found that the length of the etching time needed at room temperature was about 40 min. 4.1.2. Heat Pulling Using a Manual Propane Torch Heat pulling using a micro-propane torch is another method of obtaining tapers. The apparatus for this method is illustrated in Figure 4. The polymeric sheathing of a 30-35 cm long fiber was removed similarly as in the chemical etching method. The fiber was then mounted on the apparatus with two paper clips of identical weights (2.8 g) on either ends to provide tension to the fiber. A micro-flame (Model 6000, Microflame, Inc., MN) was positioned such that the fiber was approximately one third distance from the top end of the visible end of the flame, and the flame was removed as soon as the paper clip touched the stop. The ends of the tapered fiber were cleaved using a fiber-optic cleaver (NO-NIK) to give clean cut ends. The fiber was placed in an optical fiber holder to be used in the experiments. The dimensions of a fiber were measured after taking micrographs of the taper using an IMT- 2 optical microscope (Olympus, Japan) equipped with a video camera (Cohu Corp., Japan) linked to a computer. The dimensions were measured in the Scion Image software (Scion Corp., MD) after a calibration was performed according to the microscope objective used. Angela Leung, P. Mohana Shankar and Raj Mutharasan 34 Fiber Heat 4 mm 2 mm Magnification Microflame torch Weight Fiber Tapering stand Fiber Heat 4 mm 2 mm Magnification Microflame torch Weight Fiber Tapering stand Figure 4. Some of the fibers discussed in this paper were fabricated using a simple fiber tapering device. Fiber without sheathing was mounted on a stand and two pieces of weights were attached to the ends to provide tension required for tapering. The fiber was then heated with a flame while carefully monitoring the diameter of the taper. When the desired diameters were reached, the two ends of the fibers were clipped and the tapered fiber were placed on an optical fiber holder to be used in the experiments. Adapted from [2]. 4.1.3. Heat Pulling Using a Fusion Splicer The polymeric sheathing was removed over a distance of 5 cm at the center and both ends of the fiber. The fiber was cleaned with isopropanol and the ends were cut clean using a fiber cleaver (Ericsson EFC 11-4). The fiber was inserted into the programmable fusion splicer (Ericsson FSU975), where electric current was applied via a pair of electrodes for up to 60 seconds while the taper was pulled automatically. Various current levels (3-13 mA) and pull times (2-30 s) were used to produce fibers of varying taper diameters and lengths. A micrograph of the fiber was taken via a camera inside the fusion splicer, and the dimensions were measured in the Scion Image software (Scion Corp., MD). 4.2. Optical Characterization of Tapers 4.2.1. Preliminary Characterization Using Water The preliminary characterization method used in our laboratory for determining the evanescent field strength is the comparison of the transmission in water to that in air. The reason for the choice of these two media was that they provide the most difference in refractive index that is expected to be present in the waist region for biological samples. If a Evanescent Field Tapered Fiber Optic Biosensors (TFOBS)… 35 taper exhibited little or no transmission change going from air to water, its transmission was not expected to change significantly at the presence of dilute analyte solutions. 4.2.2. Characterization in the Visible Range Using E.coli JM101 Characterization in the visible range was performed on fusion spliced and torch heat-drawn tapers using E.coli JM101 (ECJ) as the analyte. Tapers that showed little or no transmission change in water compared to air, also showed no or low response to the presence of ECJ suspensions. Both symmetric and asymmetric tapers of small and large waist diameters had this behavior. Several tapers exhibited a significant transmission difference (~50%) in water compared to air. These tapers also showed little or no change in the presence of the ECJ. Small RI changes due to the presence of ECJ suspension were not sufficient to produce the transmission changes, resulting in poor sensitivity. Any impact on the light through the fiber was already saturated from change due to water itself, such that the presence of ECJ suspension had little further impact. Other tapers that had this characteristic property, but were of smaller waist diameter showed weak sensitivity. Most of such tapers were symmetric tapers. On the other hand, the asymmetric tapers showed lower relative transmission through water, but allowed further modulation in transmission from the presence of ECJ. We compared relative transmission at 470 nm for fusion splicer tapered fibers. In general, there were two types of responses. In the first type, the transmission increased or decreased monotonically, as ECJ concentration increased. In the second type, an initial increase for low cell concentration is followed by a decrease at higher cell concentration. There were tapers which also showed an increase in transmission at low ECJ concentrations, followed by decrease at intermediate concentrations and then an increase at 7 million cells/mL. Heat drawn tapers have typically a much longer convergent, waist and divergent sections, each on the order of millimeters. Similar to the fusion spliced tapers, HD tapers showed two basic characteristics. In one case, tapers showed a decrease in transmission as concentration increased. In the other case, tapers showed a slight increase and then a decrease in transmission for higher concentrations. At low concentration, the RI of cellular suspension influenced transmission response, and caused the increase in transmission. At higher concentrations, the evanescent light absorption by the cells dominated the response. These results suggest that torch-drawn tapers have excellent potential as biosensors. 4.2.3. Characterization in the RI Range Using Glucose Solutions In order to characterize the tapered fibers in the IR region, we measured transmission properties under various RI fluids in the tapered region, using an experimental setup similar to Figure 5 but in flow condition. The transmissions at 1310 and 1550 nm were monitored and recorded simultaneously using a spectrum analyzer and LabView program. Once the transmission stabilized in air, de-ionized (DI) water was flowed in at 0.5 mL/min. Glucose solutions of various concentrations were then flowed past the taper, with de-ionize (DI) water flowed in to rinse out the taper in between glucose solutions. Angela Leung, P. Mohana Shankar and Raj Mutharasan 36 Ando AQ-6310B Spect rum Analyzer Anrit su GB5A016 1550nm Laser sensor Thorlabs TED200 Temperat ure cont roller FC-FC adapt er Thorlabs LDC202 Laser diode cont roller Const ant t emperat ure wat er bat h or incubat or Reservoir Ando AQ-6310B Spect rum Analyzer Anrit su GB5A016 1550nm Laser sensor Thorlabs TED200 Temperat ure cont roller FC-FC adapt er Thorlabs LDC202 Laser diode cont roller Const ant t emperat ure wat er bat h or incubat or Reservoir Figure 5. Experimental setup at 1550 nm in stagnant condition. 4.3. Antibody Immobilization The most commonly used antibody immobilization method in our laboratory was adapted from Hermanson [57] with modification for the fiber surface and geometry. Prior to immobilization, the taper was cleaned with 1 M hydrochloric acid for 30 minutes, sulfuric acid for 10 minutes, and 1 M sodium hydroxide for 10 minutes. The sample holder and taper were rinsed several times with de-ionized water between cleaning steps. The cleaning procedure produced reactive hydroxyl groups on tapered surface. The surface was then silanylated with 3-aminopropyl-triethoxysilane (APTES; Sigma-Aldrich) in de-ionized water for 2-24 hours. The fiber was then dried overnight in a vacuum oven at 40 o C, or in a regular oven at 75 o C. The APTES reaction creates amine groups at the surface, which can further react with carboxylic groups in the antibody to form a peptide bond. The polyclonal antibody to BSA (anti-BSA; Sigma Catalog # B1520) contains carboxyl groups which were activated using 1-ethyl-3-(3-dimethylaminopropyl)-carbodiimide (EDC; Sigma-Aldrich) and stabilized by sulfo-N-hydroxysuccinimide (Sigma-Aldrich). EDC converts carboxylic groups into reactive unstable intermediates which are susceptible to hydrolysis. However, Sulfo-NHS replaces the EDC, resulting in a more stable reactive intermediate which catalyzes reaction with amine groups. To prepare the antibody, 0.4 mg of EDC and 1.1 mg of sulfo-NHS was added to each mL of antibody solution and the reaction was left on for 30 minutes at room temperature. Then, 1.4 μL of 2-mercaptoethanol was added to quench the EDC. This intermediate was added to the silanylated tapered fiber surface and covalent bonding was carried out at room temperature for 2 hours, in stagnant condition. At the end of antibody immobilization, Hydroxylamine was added to regenerate the carboxylic groups of the antibody. Transmission through the fiber was recorded during antibody immobilization and is shown in Figure 6. Evanescent Field Tapered Fiber Optic Biosensors (TFOBS)… 37 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 20 40 60 80 time (min) C h a n g e i n T r a n s m i s s i o n ( d B ) 0 5 10 15 20 25 30 35 40 T e m p e r a t u r e ( C ) Temperature Transmission Figure 6. Transmission change vs. time for antibody immobilization at 1550 nm. Temperature was held constant at 30 o C ± 0.5 o C as indicated. Adapted from [1]. Alternatively, the antibody can be activated via carbohydrate groups. For this protocol, 1 mg/ml of antibody was dissolved in PBS and protected from light. Then, 100 μL of 0.1 M NaIO4 solution was added to antibody and allowed to react for 30 minutes. The silanized fibers were exposed to the solution for 2 hours. Then, 10 μL of NaCNBH3 was added for 30 minutes to reduce the Schiff Base to a second amine. Another possible method of functionalization which is currently under investigation is the use of Protein G with gold. In this method, the taper was first coated with a 1:1000/v:v Polyurethane/Toluene mixture and dried overnight. The taper was then coated with 10 to 100 nm of gold using Denton Vacuum Desk IV® system. After gold coating, the taper was enclosed in the fiber holder by epoxy. During the first step of immobilization, Protein G was flowed into the sample chamber and left there in stagnant condition for 90 minutes. The sample chamber was then rinsed thoroughly with PBS, and antibody was flowed in and left there for 90 minutes. Then, the chamber was rinsed thoroughly prior to using it in a detection experiment. 4.4. Sample Preparation All biological samples were prepared as per the instructions of the manufacturer using solutions of 0.1% Sodium Azide in PBS as the solvent. Usually a bulk solution of antibody is made and then aliquots of 4 mL are dispensed into sterilized scintillation vials. The vials are then stored at -30 C freezer until use. The antibody vials were for single use and were disposed at the end of the experiment. As for the analytes such as BSA and E.coli, they were prepared in bulk in sterile centrifuge tubes each holding a maximum of 50 mL. Each tube contained one concentration of analyte, and they were all stored at 4 o C. The tubes were placed back refrigerated at 4 o C after each use. Angela Leung, P. Mohana Shankar and Raj Mutharasan 38 4.5. Detection 4.5.1. E.coli O157:H7 in Stagnant Condition E.coli O157:H7 (EC) was detected using a wavelength of 470 nm in stagnant conditions. Tapers were fabricated using heat pulling by torch or fusion splicer. The surfaces of the tapers were functionalized with antibody to E.coli O157:H7 using APTES and carboxylic linkage. The taper was exposed to various concentrations of pathogen, and showed transmission changes as the antigen attached. 0.97 0.99 1.01 1.03 1.05 1.07 0 5 10 15 20 25 Time, min N o r m a l i z e d T r a n s m i s s i o n E coli 0157:H7 at 1e06/mL 0.81 0.82 0.83 0.84 0.85 0.86 0.87 0.88 27 29 31 33 35 37 39 Time, min N o r m a l i z e d T r a n s m i s s i o n Release of E coli 0157:H7 Figure 7. Detection and release of 1 million cells/mL of EC on antibody immobilized tapered fiber. EC detection and release experiment were performed on a 8.8 μm diameter TFOBS. After attachment (top panel), release buffer (glycine-HCl/ethylene glycol buffer, pH 1.7) was injected into the chamber to release EC (bottom panel). Evanescent Field Tapered Fiber Optic Biosensors (TFOBS)… 39 An EC stock solution (7x10 9 cells/mL) was prepared as per the vendor’s (KPL) rehydration protocol in 10 mM PBS at pH 7.4. Lower concentrations (7x10 7 cells/mL, 7x10 5 cells/mL, 7x10 3 cells/mL, and 70 cells/mL) were prepared in PBS (pH 7.4) by serial dilution. 150 μL of each sample was injected into the sample chamber. After EC attachment, the sample was removed and the chamber was loaded with either HCl/PBS buffer at pH of 2.3 or Glycine-HCl/ethylene glycol (1:1 v/v) buffer at pH 1.7. The response due to attachment and release of EC cells are shown in Figure 6. Immediately upon addition of the 1E6 cells/mL of EC sample, there was a rapid increase in transmission due to the RI change of the medium. Subsequently, a gradual and exponential decrease in transmission occurred due to EC attachment. Cells change the RI surrounding the fiber and absorb light from the evanescent field. When the attachment reached equilibrium, no further light is absorbed and the transmission remained constant. The antigen attached to the sensor may be released by altering the pH as the antibody- antigen binding is pH-dependent. The response due to release was equal in magnitude and opposite in direction, as shown in Figure 7. This change occurred because cells released into the bulk are too far away from the taper surface to influence light transmission. When cells released reached equilibrium, the transmission reached a constant value. 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 0 10 20 30 40 Time, minutes N o r m a l i z e d T r a n s m i s s i o n a t 4 7 0 n m PBS 7000 #/mL 70 #/mL Figure 8. The response at 470 nm due to different concentrations of EC cells. Adapted from [2]. Intuitively, one would imagine that the transmission change would be directly proportional to the concentration. However, results show that the magnitude of the change is inversely proportional to the pathogen concentration, as shown in Figure 8. In addition, the response for this experiment was an increase in transmission, contrary to the experiment shown in Figure 7. The cause of this is not entirely clear, but we believe that it is due to the combined effects of evanescent absorption and scattering of the evanescent light. As cells Angela Leung, P. Mohana Shankar and Raj Mutharasan 40 cover the taper surface, the evanescent light is absorbed by the cells in proportion to the surface coverage. On the other hand, the RI is increased due to cell attachment. If the sample were homogeneous, increase in refractive index tends to increase transmission through the core due to reduction in the evanescent field. Hence, cell attachment result in transmission increase or decrease. 0.92 0.94 0.96 0.98 1.00 1.02 1.04 1.06 0 1000 2000 3000 4000 Time, S N o r m a l i z e d T r a n s m i s s i o n 50% Wild 70% Pathogen 30% Wild 100% Wild 50% Pathogen 50% Wild Figure 9. Effect of pathogenic and non-pathogenic EC mixture. Experiments were performed on a 9.5 μm diameter TFOBS. When 0% pathogen (100% wild strain JM101) was injected around the taper, there was no significant transmission change through the taper. When a solution containing an EC and the wild strain is added to the solution, EC bind to the antibody thus resulting in a decrease in transmission through the fiber. As the concentration of EC is increased to 50% and 70%, there is a greater binding of pathogen to the antibody on the surface and thus greater change in transmission occurs. Adapted from [2]. In order to evaluate specificity, the response to non-pathogenic E. coli was measured. Stock solution containing EC was mixed with a wild strain of E. coli (JM101) in volumetric proportions of 0%, 50% and 70%. The total bacterial count was 7x10 7 cells/mL. The detection experiments were carried out in the same manner as with pure EC. The sensor showed good selectivity to the pathogenic antigen as shown in Figure 9. It is useful to obtain the kinetics EC attachment on antibody-immobilized surfaces. The immobilization and detection responses show exponential behavior, similar to the adsorption process often referred to as Langmuir kinetics. The Langmuir kinetics model can be expressed as [58]: 1 o b s k t e θ − = − (11) Evanescent Field Tapered Fiber Optic Biosensors (TFOBS)… 41 where θ ( ) 1 0 ≤ ≤ θ is the fractional coverage of the reactive sites at time t. The parameter, obs k , is the observed binding rate constant, which depends on the bulk concentration of the reactant. We hypothesize that the transmission is indicative of attachment, and express the Langmuir model as follows: ( ) ( ) ( ) 1 b kC t I I e − ∞ Δ = Δ − (12) where ( ) I Δ is the transmission change at time, t , ( ) ∞ ΔI is the steady state transmission change, and C b is the bulk concentration. Taking the natural log on both sides of Eq. (12) we obtain: ( ) ( ) ( ) ln b I I kC t I ∞ ∞ ⎛ ⎞ Δ − Δ = − ⎜ ⎟ ⎜ ⎟ Δ ⎝ ⎠ (13) -4 -3 -2 -1 0 0.E+00 1.E+08 2.E+08 3.E+08 4.E+08 Cb t, # bacteria-min/mL l n [ ( Δ I * – Δ I ) / Δ I * ) ] Slope = 9.2 E-7 min -1 (#/mL) -1 Figure 10. Calculation of rate of attachment (slope k) for EC. The above suggests that the characteristic rate constant k during initial time can be determined from a plot of the left hand side versus C b *t in Eq. (13). Figure 10 is an example of a graph displaying Eq. (13). The kinetic constant (k) was found to be in the range of 4x10 -9 min -1 (pathogen/mL) -1 to 7x10 -9 min -1 (pathogen/mL) -1 . 4.5.2. BSA in Stagnant Condition Although we were able to detect E.coli O157:H7 at 470 nm, the sensitivity of the sensors was limited due to the diameter in relation to the wavelength. Because the fibers are very fragile, we are unable to fabricated tapers that are less than 5 μm in diameter. However, at 470 nm the penetration of the evanescent field is limited. Also, cells are relatively large compared to the evanescent field generated at 470 nm. On the other hand, according to Eq. (1), there are reasons to believe that the evanescent field would be larger at a longer wavelength. Therefore, Angela Leung, P. Mohana Shankar and Raj Mutharasan 42 detection of BSA was detected similarly to EC but performed mostly using near-IR wavelengths. We first reported the use of a 1550 nm laser with TFOBS to monitor the real- time attachment of BSA to the antibody-immobilized surface [1]. While cuvette measurements established that BSA was non-absorbing at 1550 nm, antibody-immobilized TFOBS showed transmission changes at bulk concentrations of 10 fg/mL of BSA. The experimental setup for near-IR detection is shown in Figure 5. Solutions of BSA from 10 fg/mL to 1mg/mL were prepared. After antibody was immobilized, it was rinsed with PBS, and 200 μL of BSA was injected into the sample chamber. Only one concentration was used in each experiment of attachment and release. After attachment, the BSA was removed and the sensor was rinsed with PBS. Then, PBS adjusted to a pH of 2 by H 2 SO 4 was added to release the BSA. The acidic PBS weakens the binding of BSA to the antibody because it changes the conformation of the protein. After this the fiber surface was regenerated with the cleaning sequence, followed by modification by APTES. -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0 5 10 15 20 25 30 35 Time (minutes) C h a n g e i n T r a n s m i s s i o n ( d B ) Attachment Release Figure 11. BSA attachment and release of 10 pg/mL sample. The attachment response was obtained when the tapered region was first exposed to 10 pg/mL of BSA. Data was collected for 30 minutes, rinsed with PBS, and then the tapered region was exposed to pH2 PBS for BSA release. The transmission changes due to attachment and release are in opposite direction and have approximately the same magnitude. Adapted from [1]. When BSA was injected into the sample holder, transmission decreased due to change in surface refractive index caused by the presence of BSA. When BSA was replaced by low pH PBS, transmission increased back almost to the starting value. Similar experiments were performed with many tapers using different concentrations, and we conclude that experimental results are reproducible with the different fibers. The results of the attachment and release of 10 pg/mL of BSA are shown in Figure 11. Multiple step attachment experiments was performed on three TFOBS with up to five different solutions of BSA, with concentration ranging from 100 fg/mL to 10 ng/mL. The experiments were initially performed with a starting concentration of 100 fg/mL. In the last Evanescent Field Tapered Fiber Optic Biosensors (TFOBS)… 43 experiment, shown in Figure 12, the initial concentration was set at 10 fg/mL. The BSA solutions were added in order from the lowest to the highest concentration, with removal of each sample after collection of data for up to 40 minutes. The transmission decreased as a function of time as BSA attached to the antibody. The steady state transmission for each concentration also decreased. -34.3 -34.1 -33.9 -33.7 -33.5 -33.3 -33.1 -32.9 -32.7 50 100 150 200 250 300 Time (minutes) T r a n s m i s s i o n ( d B ) 19 21 23 25 27 29 31 T e m p e r a t u r e ( o C ) Temperature 10 fg/mL 100 fg/mL 1 pg/mL Transmission Figure 12. Semi-batch staircase experiment showing attachment of BSA from10 fg/mL to 10 pg/mL. Temperature was maintained at 30 o C ± 0.5 o C using an incubator. The BSA solutions were added sequentially from lowest to highest upon removal of the previous solution. The purple line with peaks represents transmission through the fiber. The peaks correspond to time instants when the samples were introduced. The dotted line at the bottom represents the trend exhibited by the steady state transmission with respect to time. Adapted from [1]. Like the EC results, transmission change is not linearly proportional to concentration. We believe that the reason for this is that at low concentrations, the surface of the fiber is not saturated with the antigen BSA. An estimate of the antibody/antigen surface coverage can be made with a few simplifying assumptions, and it was suggested that the concentration required for saturation is less than 4 ng/mL. Transmission changes are caused by the evanescent field interaction with the surface layer of antigen. Once the concentration approaches ng/mL levels, the surface is saturated with BSA. Additional BSA molecules would attach on top of the surface layer. However, the evanescent field magnitude decays away from the surface. Therefore the effect of BSA on top of the first layer results in much smaller changes. In addition, the condition for immobilization varies from one experiment to another. It is possible that nonlinearity was observed because at the lowest concentration, the bulk refractive index is approximately the same as that of PBS, and the BSA molecules on the fiber surface act as isolated points of high refractive index. When the concentration increases to saturation point, the fiber surface is covered with a layer of BSA which has a higher refractive index than PBS. Angela Leung, P. Mohana Shankar and Raj Mutharasan 44 5.0. Conclusion It is seen that TFOBS have several advantages in in terms of detection, including sensitivity, selectivity, ease of use, affordability, ability for remote sensing, and small sample volumes. They have been used for many applications such as pathogen detection, medical diagnostics based on protein or cell concentration, and detection of DNA hybridization. As far as the sensor physics is concerned, intensity-based sensors have been used to a limited extent in cell detection. On the other hand, fluorescence based TFOBS are widely used for protein and DNA detection because amplification is a convenient tool, and often necessary to achieve low LODs. In addition, SPR is commonly used for protein characterization and has also been used for the detection of DNA hybridization. The ng/mL LOD of SPR makes it suitable for many medical applications. While fluorescence is very selective, its LOD is higher than SPR’s. In addition, fluorescence requires multiple steps for the preparation of the sensor or the sample. In terms of target analytes, one possible area of growth is the use of SPR or intensity- based TFOBS protein and DNA detection. Another application may be drug screening using TFOBS. Because of recent concerns of homeland security, there will likely be a significant push for research in bio-threat detection. Pathogen detection also remains important in maintaining a safe environment and food supply. Clinical applications of TFOBS will likely be important as medical professionals seek convenient methods to diagnose diseases. TFOBS have been used as intensity-based sensors in our laboratory. We have used three methods of fabrication: step-etching using hydrofluoric acid, heat pulling by flame, and heat pulling by fusion splicer. The sensing ability of TFOBS was characterized by measuring the transmission in water, E.coli JM101 solutions, and glucose solutions. TFOBS were functionalized with antibodies using covalent bonding or surface coating with gold and Protein G. TFOBS was used in our laboratory to measure E.coli O157:H7 in stagnant condition. One surprising finding was that concentration had an inverse effect on the transmission. TFOBS was shown to be selective to the pathogens. BSA was detected at 10 fg/mL in stagnant condition at 1550 nm. Transmission data was fitted to the Langmuir absorption model to determine the attachment rate. As TFOBS evolves, new efforts will be focused on enhancing the sensitivity and selectivity. Improved surface chemical modification and stability of the recognition molecule can increase the sensitivity and robustness of TFOBS, especially for intensity-based TFOBS because it is the most sensitive when molecules are bound to its surface. As was shown in this chapter, there is a solid foundation of work to support the use of TFOBS and a wide variety of applications. Given its promising advantages, it is likely that TFOBS will remain a popular choice for detection in the future. Acknowledgements This work was supported through the National Science Foundation Grant # CBET-0329793, “Ultra Sensitive Continuous Tapered Fiber Biosensors for Pathogens and Bioterrorism Agents”. Evanescent Field Tapered Fiber Optic Biosensors (TFOBS)… 45 References [1] Leung, A., Shankar, P.M., Mutharasan, R. (2006) Real-time monitoring of bovine serum albumin at femtogram/mL levels on antibodyimmobilized tapered fibers Sens. Actuators B: Chem. in press, doi:10.1016/j.snb.2007.03.010 . [2] Rijal, K., Leung, A., Shankar, P.M., Mutharasan, R. (2005) Detection of vathoizen Escherichia coli O157 : H7 AT 70 cells/mL using antibody-immobilized biconical tapered fiber sensors. Biosensors & Bioelectronics 21: 871-880. [3] Leung, A., Rijal, K.S., Shankar, P.M., Mutharasan, R. (2006) Effects of geometry on transmission and sensing potential of tapered fiber sensors. Biosensors & Bioelectronics 21: 2202-2209. [4] Arregui, F.J., Matias, I.R., Lopez-Amo, M. (2000) Optical fiber strain gauge based on a tapered single-mode fiber. Sensors and Actuators a-Physical 79: 90-96. [5] Datta, P., Matias, I., Aramburu, C., Bakas, A., LopezAmo, M., Oton, J.M. (1996) Tapered optical-fiber temperature sensor. Microwave and Optical Technology Letters 11: 93-95. [6] Bariain, C., Matias, I.R., Arregui, F.J., Lopez-Amo, M. (2000) Optical fiber humidity sensor based on a tapered fiber coated with agarose gel. Sensors and Actuators B- Chemical 69: 127-131. [7] Diaz-Herrera, N., Navarrete, M.C., Esteban, O., Gonzalez-Cano, A. (2004) A fibre-optic temperature sensor based on the deposition of a thermochromic material on an adiabatic taper. Measurement Science & Technology 15: 353-358. [8] Bariain, C., Matias, I.R., Arregui, F.J., Lopez-Amo, M. (2000) Tapered optical-fiber- based pressure sensor. Optical Engineering 39: 2241-2247. [9] Villatoro, J., Luna-Moreno, D., Monzon-Hernandez, D. (2005) Optical fiber hydrogen sensor for concentrations below the lower explosive limit. Sensors and Actuators B- Chemical 110: 23-27. [10] Villatoro, J., Diez, A., Cruz, J.L., Andres, M.V. (2003) In-line highly sensitive hydrogen sensor based on palladium-coated single-mode tapered fibers. Ieee Sensors Journal 3: 533-537. [11] Bariain, C., Matias, I.R., Romeo, I., Garrido, J., Laguna, M. (2000) Detection of volatile organic compound vapors by using a vapochromic material on a tapered optical fiber. Applied Physics Letters 77: 2274-2276. [12] Kim, Y.C., Peng, W., Banerji, S., Booksh, K.S. (2005) Tapered fiber optic surface plasmon resonance sensor for analyses of vapor and liquid phases. Optics Letters 30: 2218-2220. [13] McCue, R.P., Walsh, J.E., Walsh, F., Regan, F. (2006) Modular fibre optic sensor for the detection of hydrocarbons in water. Sensors and Actuators B-Chemical 114: 438-444. [14] Zalvidea, D., Diez, A., Cruz, J.L., Andres, M.V. (2006) Hydrogen sensor based on a palladium-coated fibre-taper with improved time-response. Sensors and Actuators B- Chemical 114: 268-274. [15] Cao, L.K., Anderson, G.P., Ligler, F.S., Ezzell, J. (1995) Detection of Yersinia-Pestis Fraction-1 Antigen with a Fiber Optic Biosensor. Journal of Clinical Microbiology 33: 336-341. Angela Leung, P. Mohana Shankar and Raj Mutharasan 46 [16] Spiker, J.O., Kang, K.A. (1999) Preliminary study of real-time fiber optic based protein C biosensor. Biotechnology and Bioengineering 66: 158-163. [17] Nath, N., Jain, S.R., Anand, S. (1997) Evanescent wave fibre optic sensor for detection of L-donovani specific antibodies in sera of kala azar patients. Biosensors & Bioelectronics 12: 491-498. [18] Haddock, H.S., Shankar, P.M., Mutharasan, R. (2003) Evanescent sensing of biomolecules and cells. Sensors and Actuators B-Chemical 88: 67-74. [19] Ko, S.H., Grant, S.A. (2006) A novel FRET-based optical fiber biosensor for rapid detection of Salmonella typhimurium. Biosensors & Bioelectronics 21: 1283-1290. [20] Bobb, L.C., Shankar, P.M., Krumboltz, H.D. (1990) Bending Effects in Biconically Tapered Single-Mode Fibers. Journal of Lightwave Technology 8: 1084-1090. [21] Okamoto, K. (2000): In Fundamentals of Optical Waveguides, Academic Press, pp. 323-330. [22] Moar, P.N., Huntington, S.T., Katsifolis, J., Cahill, L.W., Roberts, A., Nugent, K.A. (1999) Fabrication, modeling, and direct evanescent field measurement of tapered optical fiber sensors. Journal of Applied Physics 85: 3395-3398. [23] Ferreira, A.P., Werneck, M.M., Ribeiro, R.M. (2001) Development of an evanescent- field fibre optic sensor for Escherichia coli O157 : H7. Biosensors & Bioelectronics 16: 399-408. [24] Maraldo, D., Shankar, P.M., Mutharasan, R. (2006) Measuring bacterial growth by tapered fiber and changes in evanescent field. Biosensors & Bioelectronics 21: 1339-1344. [25] DeMarco, D.R., Saaski, E.W., McCrae, D.A., Lim, D.V. (1999) Rapid detection of Escherichia coli O157 : H7 in ground beef using a fiber-optic biosensor. Journal of Food Protection 62: 711-716. [26] Geng, T., Uknalis, J., Tu, S.I., Bhunia, A.K. (2006) Fiber-optic biosensor employing Alexa-Fluor conjugated antibody for detection of Escherichia coli O157 : H7 from ground beef in four hours. Sensors 6: 796-807. [27] Kramer, M.F., Lim, D.V. (2004) A rapid and automated fiber optic-based biosensor assay for the detection of salmonella in spent irrigation water used in the sprouting of sprout seeds. Journal of Food Protection 67: 46-52. [28] Zhou, C.H., Pivarnik, P., Auger, S., Rand, A., Letcher, S. (1997) A compact fiber-optic immunosensor for Salmonella based on evanescent wave excitation. Sensors and Actuators B-Chemical 42: 169-175. [29] Geng, T., Morgan, M.T., Bhunia, A.K. (2004) Detection of low levels of Listeria monocytogenes cells by using a fiber-optic immunosensor. Applied and Environmental Microbiology 70: 6138-6146. [30] Kim, G., Morgan, M.T., Ess, D., Hahm, B.K., Kothapalli, A., Valadez, A., Bhunia, A. (2006) Detection of Listeria monocytogenes using an automated fiber-optic biosensor: RAPTOR. Advanced Nondestructuve Evaluation I, Pts 1 and 2, Proceedings 321-323: 1168-1171. [31] Nanduri, V., Kim, G., Morgan, M.T., Ess, D., Hahm, B.K., Kothapalli, A., Valadez, A., Geng, T., Bhunia, A.K. (2006) Antibody immobilization on waveguides using a flow- through system shows improved Listeria monocytogenes detection in an automated fiber optic biosensor: RAPTOR (TM). Sensors 6: 808-822. Evanescent Field Tapered Fiber Optic Biosensors (TFOBS)… 47 [32] Tims, T.B., Lim, D.V. (2004) Rapid detection of Bacillus anthracis spores directly from powders with an evanescent wave fiber-optic biosensor. Journal of Microbiological Methods 59: 127-130. [33] Tempelman, L.A., King, K.D., Anderson, G.P., Ligler, F.S. (1996) Quantitating staphylococcal enterotoxin B in diverse media using a portable fiber-optic biosensor. Analytical Biochemistry 233: 50-57. [34] Shriver-Lake, L.C., Shubin, Y.S., Ligler, F.S. (2003) Detection of staphylococcal enterotoxin B in spiked food samples. Journal of Food Protection 66: 1851-1856. [35] Chang, Y.H., Chang, T.C., Kao, E.F., Chou, C. (1996) Detection of protein a produced by Staphylococcus aureus with a fiber-optic-based biosensor. Bioscience Biotechnology and Biochemistry 60: 1571-1574. [36] Ogert, R.A., Brown, J.E., Singh, B.R., Shriverlake, L.C., Ligler, F.S. (1992) Detection of Clostridium-Botulinum Toxin-a Using a Fiber Optic-Based Biosensor. Analytical Biochemistry 205: 306-312. [37] Narang, U., Anderson, G.P., Ligler, F.S., Burans, J. (1997) Fiber optic-based biosensor for ricin. Biosensors & Bioelectronics 12: 937-945. [38] James, E.A., Schmeltzer, K., Ligler, F.S. (1996) Detection of endotoxin using an evanescent wave fiber-optic biosensor. Applied Biochemistry and Biotechnology 60: 189-202. [39] Masson, J.F., Obando, L., Beaudoin, S., Booksh, K. (2004) Sensitive and real-time fiber- optic-based surface plasmon resonance sensors for myoglobin and cardiac troponin I. Talanta 62: 865-870. [40] Hanbury, C.M., Miller, W.G., Harris, R.B. (1997) Fiber-optic immunosensor for measurement of myoglobin. Clinical Chemistry 43: 2128-2136. [41] Battaglia, T.M., Masson, J.F., Sierks, M.R., Beaudoin, S.P., Rogers, J., Foster, K.N., Holloway, G.A., Booksh, K.S. (2005) Quantification of cytokines involved in wound healing using surface plasmon resonance. Analytical Chemistry 77: 7016-7023. [42] Tschmelak, J., Proll, G., Gauglitz, G. (2004) Sub-nanogram per litre detection of the emerging contaminant progesterone with a fully automated immunosensor based on evanescent field techniques. Analytica Chimica Acta 519: 143-146. [43] Preejith, P.V., Lim, C.S., Kishen, A., John, M.S., Asundi, A. (2003) Total protein measurement using a fiber-optic evanescent wave-based biosensor. Biotechnology Letters 25: 105-110. [44] Tromberg, B.J., Sepaniak, M.J., Vo-Dinh, T., Griffin, G.D. (1997) Fiber-optic chemical sensors for competitive binding fluoroimmunoassay. Anal. Chem. 59: 1226-1230. [45] Hale, Z.M., Payne, F.P., Marks, R.S., Lowe, C.R., Levine, M.M. (1996) The single mode tapered optical fibre loop immunosensor. Biosensors & Bioelectronics 11: 137-148. [46] Tang, L., Kwon, H.J., Kang, K.A. (2004) Theoretical and experimental analysis of analyte transport in a fiber-optic, Protein C immuno-biosensor. Biotechnology and Bioengineering 88: 869-879. [47] Garden, S.R., Doellgast, G.J., Killham, K.S., Strachan, N.J.C. (2004) A fluorescent coagulation assay for thrombin using a fibre optic evanescent wave sensor (vol 19, pg 737, 2004). Biosensors & Bioelectronics 19: 1775-1775. [48] Lee, M., Walt, D.R. (2000) A fiber-optic microarray biosensor using aptamers as receptors. Analytical Biochemistry 282: 142-146. Angela Leung, P. Mohana Shankar and Raj Mutharasan 48 [49] Wen-xu, L., Jian, C. (2003) Continuous monitoring of adriamycin in vivo using fiber optic-based fluorescence chemical sensor. Analytical Chemistry 75: 1458-1462. [50] Song, J.M., Kasili, P.M., Griffin, G.D., Vo-Dinh, T. (2004) Detection of cytochrome c in a single cell using an optical nanobiosensor. Analytical Chemistry 76: 2591-2594. [51] Cullum, B.M., Griffin, G.D., Miller, G.H., Vo-Dinh, T. (2000) Intracellular measurements in mammary carcinoma cells using fiber-optic nanosensors. Analytical Biochemistry 277: 25-32. [52] Kleinjung, F., Bier, F.F., Warsinke, A., Scheller, F.W. (1997) Fibre-optic genosensor for specific determination of femtomolar DNA oligomers. Analytica Chimica Acta 350: 51-58. [53] Zeng, J., Almadidy, A., Watterson, J., Krull, U.K. (2003) Interfacial hybridization kinetics of oligonucleotides immobilized onto fused silica surfaces. Sensors and Actuators B-Chemical 90: 68-75. [54] Liu, X., Tan, W. (1999) A fiber-optic evanescent wave DNA biosensor based on novel molecular beacons. Anal. Chem. 71: 5054-5059. [55] Almadidy, A., Watterson, J., Piunno, P.A.E., Foulds, I.V., Horgen, P.A., Krull, U. (2003) A fibre-optic biosensor for detection of microbial contamination. Canadian Journal of Chemistry-Revue Canadienne De Chimie 81: 339-349. [56] Pilevar, S., Davis, C.C., Portugal, F. (1998) Tapered optical fiber sensor using near- infrared fluorophores to assay hybridization. Analytical Chemistry 70: 2031-2037. [57] Hermanson, G.T.: Bioconjugate Teshniques. Elsevier, San Diego 1996. [58] Yan, D., Saunders, J.A., Jennings, K. (2002) Kinetics of formation for n-alkanethiolates self-assembled monolayers onto gold in aqueous micellar solutions of C12E6 and C12E7. Langmuir 18: 10202. [59] Zourob, M., Mohr, S., Brown, B.J.T., Fielden, P.R., McDonnell, M.B., Goddard, N.J. (2005) An integrated metal clad leaky waveguide sensor for detection of bacteria. Analytical Chemistry 77: 232-242. [60] Morgan, M.T., Kim, G., Ess, D., Kothapalli, A., Hahm, B.K., Bhunia, A. (2006) Binding inhibition assay using fiber-optic based biosensor for the detection of foodborne pathogens. Advanced Nondestructuve Evaluation I, Pts 1 and 2, Proceedings 321-323: 1145-1150. [61] Shriverlake, L.C., Ogert, R.A., Ligler, F.S. (1993) A Fiberoptic Evanescent-Wave Immunosensor for Large Molecules. Sensors and Actuators B-Chemical 11: 239-243. [62] Chen, Z., Kaplan, D.L., Gao, H., Kumar, J., Marx, K.A. (1996) Molecular assembly of multilayer enzyme: toward the development of a chemiluminescent-based fiber optic biosensor. Materials Science and Engineering C4: 155-159. [63] Kapoor, R., Kaur, N., Nishanth, E.T., Halvorsen, S.W., Bergey, E.J., Prasad, P.N. (2004) Detection of trophic factor activated signaling molecules in cells by a compact fiber- optic sensor. Biosensors & Bioelectronics 20: 345-349. [64] Zourob, M., Goddard, N.J. (2005) Metal clad leaky waveguides for chemical and biosensing applications. Biosensors & Bioelectronics 20: 1718-1727. [65] Wu, S.Y., Ho, H.P., Law, W.C., Lin, C.L., Kong, S.K. (2004) Highly sensitive differential phase-sensitive surface plasmon resonance biosensor based on the Mach- Zehnder configuration. Optics Letters 29: 2378-2380. Evanescent Field Tapered Fiber Optic Biosensors (TFOBS)… 49 [66] Tang, L., Ren, Y.J., Hong, B., Kang, K.A. (2006) Fluorophore-mediated, fiber-optic, multi-analyte, immunosensing system for rapid diagnosis and prognosis of cardiovascular diseases. Journal of Biomedical Optics 11: 21011-21011 to 21011-21010. [67] Hong, B., Kang, K.A. (2006) Biocompatible, nanogold-particle fluorescence enhancer for fluorophore mediated, optical immunosensor. Biosensors & Bioelectronics 21: 1333-1338. [68] Minunni, M., Tombelli, S., Mascini, M., Bilia, A., Bergonzi, M.C., Vincieri, F.F. (2005) An optical DNA-based biosensor for the analysis of bioactive constituents with application in drug and herbal drug screening. Talanta 65: 578-585. [69] Thompson, V.S., Maragos, C.M. (1996) Fiber-optic immunosensor for the detection of fumonisin B-1. Journal of Agricultural and Food Chemistry 44: 1041-1046. In: Optical Fibers Research Advances ISBN: 1-60021-866-0 Editor: Jurgen C. Schlesinger, pp. 51-81 © 2007 Nova Science Publishers, Inc. Chapter 2 NEW CHALLENGES IN RAMAN AMPLIFICATION FOR FIBER COMMUNICATION SYSTEMS P.S. André 1,2 , A.N. Pinto 1,3 , A.L.J. Teixeira 1,3 , B. Neto 1,2 , S. Stevan Jr. 1,3 , Donato Sperti 1,3,4 , F. da Rocha 1,3 , Micaela Bernardo 2,5 , J.L. Pinto 1,2 , Meire Fugihara 1,3 , Ana Rocha 1,2 and M. Facão 2 1 Instituto de Telecomunicações, Aveiro Portugal 2 Departamento de Física, Universidade de Aveiro, Aveiro, Portugal 3 Departamento de Electrónica, Telecomunicações e Informática, Universidade de Aveiro, Aveiro, Portugal 4 Università Degli Studi di Parma, Parma, Italy 5 Portugal Telecom Inovação SA, Aveiro, Portugal Abstract Raman fiber amplifiers (RFA) are among the most promising technologies in lightwave systems. In recent years, Raman optical fiber amplifiers have been widely investigated for their advantageous features, namely the transmission fiber can be itself used as the gain media reducing the overall noise figure and creating a lossless transmission media. The introduction of RFA based on low cost technology will allow the consolidation of this amplification technique and its use in future optical networks. This paper reviews the challenges, achievements, and perspectives of Raman amplification in optical communication systems. In Raman amplified systems, the signal amplification is based on stimulated Raman scattering, thus the peak of the gain is shifted by approximately 13.2 THz with respect to the pump signal frequency. The possibility of combining many pumps centered on different wavelengths brings a flat gain in an ultra wide bandwidth. An initial physical description of the phenomenon is presented as well as the mathematical formalism used to simulate the effect on optical fibers. The review follows with one section describing the challenging developments in this topic, such as using low cost pump lasers, in-fiber lasing, recurring to fiber Bragg grating cavities or broadband incoherent pump sources and Raman amplification applied to coarse wavelength multiplexed networks. Also, one of the major issues on Raman amplifier design, P.S. André, A.N. Pinto, A.L.J. Teixeira et al. 52 which is the determination of pump powers in order to realize a specific gain will be discussed. In terms of optimization, several solutions have been published recently, however, some of them request extremely large computation time for every interaction, what precludes it from finding an optimum solution or solve the semi-analytical rate equation under strong simplifying assumptions, which results in substantial errors. An exhaustive study of the optimization techniques will be presented. This paper allows the reader to travel from the description of the phenomenon to the results (experimental and numerical) that emphasize the potential applications of this technology. 1. Introduction The deployment of optical communication systems through long haul networks required the development of transparent optical amplifiers, for replacement of the expensive and limitative optoelectronic regeneration. The increasing distance between amplification sites saves amplification huts reducing by this way the investment and operational cost in the network management. The first choice for transparent optical amplification pointed out to the Erbium Doped Fiber amplifiers (EDFA), which was a mature technology by the beginning of the last decade of the XX century. However, the growing demand in terms of transmission capacity has been increasing dramatically, fulfilling the entire spectral band of the EDFA, and wideband amplifiers are now required. Raman fiber amplifiers (RFA) have emerged as a key technology for the optical networks. In lumped amplified systems (using for example EDFAs) the amplification modules are placed every 40~50 km of span. This module amplifies back to the initial power level, the transmission signal attenuated during propagation. The distance between amplifiers is determined by the span loss, by the limit imposed from the maximum admissible power allowed in the fiber without inducing nonlinear effects and by the minimum acceptable power that avois a degradation of the signal-to-noise-ratio. The use of Raman amplification allows the confinement of the signal inside the limits imposed by the nonlinearities and of the signal-to-noise-ratio degradation resulting from higher span distances. This advantage of the distributed (Raman) over lumped amplification is illustrated in figure 1. Figure 1. Distributed and lumped amplification signal evolution. New Challenges in Raman Amplification for Fiber Communication Systems 53 The distributed amplification scheme can be used to cover very long span links or to increase the distance of ultra-long haul systems. Raman fiber amplifiers are based on the power transfer from pump(s) signal(s) to information carrying signals (usually described as probes) due to stimulated Raman scattering (SRS) which occurs when there is sufficient pump power within the fiber. Since the gain peak of this amplification is obtained for signals downshifted approximately by 13.2 THz (for Silica), relative to the pump frequency, to achieve gain at any wavelength we need to select a pump whose frequency complies with this relation. In this way, it is possible to optimize the number of pumps to obtain a wide and flat gain [1-3]. However, it is necessary to bear in mind, that due to the pump-to-pump interaction, the shorter wavelength pumps demand more power to be effective [4,5]. From a telecommunications point of view, the pump wavelengths must be placed around 1450 nm because the signal wavelengths used on the so called 3 rd transmission window are centered around 1550 nm and the maximum gain occurs for a Stokes frequency shift of 13.2 THz. The RFA had become attractive just after the development and commercialization at a reasonable cost of a key component: the high power pump laser [6]. Typically a high power laser for Raman amplification, provides an optical power of 300 mW, launched over an optical fiber, which for a standard single mode fiber (SMF) is equivalent to a power density of 3.75 GW/m 2 . This high power injected into the fiber, especially when multipump lasers are utilized, imposes new concerns in terms of safety. Therefore, the use of RFAs requires the utilization of automatic power reduction or automatic laser shut down systems to prevent the hazard of high power leakage from the optical cables or service cabinets. Also, as the optical power rises, the nonlinear effects, such as the fiber fuse effect, start to become relevant. This effect has threshold intensity of 10~30 GW/m 2 and it is responsible by a catastrophic destruction of the fiber core. This destruction once started propagates in direction to the optical source, resulting also in the destruction of the pumping laser [7]. For operating wavelengths of 1550 nm, the fuse effect power threshold is ~1.5 W for SMF fibers, while for dispersion shift fibers (DSF) this power is reduced to ~1.2 W [8]. This effect is also responsible by the damage of the optical connectors interface [8]. In terms of implemented systems, several architectures have been proposed, based in all Raman or hybrid Raman/EDFA amplification [10]. The use of bidirectional Raman amplification has also been reported for long reach access networks. Experimental results have shown the feasibility of systems with symmetric up-and-downstream signals with bitrates up to 10 Gb/s, supported by distributed Raman amplification over 80 km of fiber [11]. Field transmission experiments have been reported with 8 × 170 Gb/s over 210 km of single mode standard fiber, achieving spectral efficiency of 0.53 bit/s/Hz [12]. As the traffic increases, wavelength division multiplexing (WDM) arises to enlarge the transmission capacity. This, in turn, requires flexible and broadband architectures which reinforces the interest in Raman amplification. Nowadays, WDM exists in two formats: Dense WDM (DWDM) working at C and L spectral windows, allocating a maximum of 150 channels spaced by 0.8 nm [13], and Coarse WDM (CWDM) working at O, E, S, C, and L spectral windows, allocating a maximum of 18 channels spaced by 20 nm [14]. The DWDM solution is extensively used in long haul systems, sending as much information as possible. CWDM is a good solution whenever less information is transmitted over short distances in a P.S. André, A.N. Pinto, A.L.J. Teixeira et al. 54 less expensive way than DWDM. As CWDM works with far apart signals, it can make use of uncooled distributed feedback (DFB) lasers [15,16] needing multiplexing components with flexible tolerances. However, as the channels in CWDM systems are far apart, optical amplification is still a matter of concern. Traditional EDFA bandwidth (20~40 nm) cannot support the full band of CWDM channels [17]. Other technical solution to amplification of signals is the semiconductor optical amplifier, which presents a low saturation power (around 13 dBm) when compared with other fiber based amplifiers, but with a signal-to-noise ratio degradation quite considerable. A good solution for the amplification of both DWDM and CWDM relies on Raman amplifiers. A wide and flat spectral gain profile is achievable thanks to the combination of several pumping lasers operating at specific powers and wavelengths. The composite amplification is determined from the mutual interactions among the pump and signal wavelengths. Gain spectra as large as 100 nm were obtained using multiple pumps. Emori et al. have presented an experimental Raman amplifier with a 100 nm bandwidth using a WDM laser diode unit with 12 wavelengths ranging from 1405 to 1510 nm, whose maximum total power was equal to 2.2 W [4, 18]. Therefore, a gain equal to 2 dB is obtained over a 25 km SMF link and a 6.5 dB gain using a 25 km DSF link, both with 0.5 dB of maximum ripple. Kidorf et al. provided a mathematical model to implement a 100 nm Raman amplifier using low power pumps with maximum power of each pump equal to 130 mW [14]. They used 8 pumps from 1416 nm to 1502 nm along 45 km of SMF, obtaining a gain around 4 dB with a maximum ripple equal to 1.1 dB. The growing maturity of high pump module technologies is providing competitive solutions based on Raman amplification and currently many alternative techniques are being developed to overcome the ordinary one pump and dual pumping methods [19, 20]. In particular, we report here two major techniques. First, the use of low power pumping lasers provides gain comparable to the ordinary one pump Raman amplification. This technique is especially interesting for combining commercial and low cost lasers [21]. The second particular technique corresponds to an evolution of the cascaded Raman amplification. Actually, a sixth order cascade Raman amplifier was recently proposed [22]. In the cascade Raman amplification, the pump power is downshifted in frequency by using a pair of fiber Bragg gratings (FBG) placed in spectral positions multiples of 13 THz, from the pump frequency. In a particular case, the generation of the fiber pump laser is obtained by using only one passive reflector element and distributed reflectors over the long optical fiber, established by a nonlinear fiber intrinsic effect called Rayleigh backscattering. The enlargement of the bandwidth of Raman amplifiers is also achieved using incoherent pumping instead of multi-pump schemes [23-27]. Vakhshoori et al. proposed a high-power incoherent semiconductor pump prototype that uses a low-power seed optical signal, coupled into a long-cavity semiconductor amplifier. It was achieved 400mW of optical power over a 35nm spectral window [27]. A 50 nm bandwidth amplifier was obtained with an on/off gain equal to 7 dB. It was also demonstrated that the use of six coherent pumps is less efficient, in terms of flatness, than the use of two incoherent pumps [24]. The signal wavelengths were comprised between 1530 nm and 1605 nm and the transmission occurs over 100 km of optical fiber. Another advantage of using incoherent pumping is the reduction of nonlinear effects, such as Brillouin scattering, four wave mixing of pump-pump, pump-signal and pump-noise [28]. New Challenges in Raman Amplification for Fiber Communication Systems 55 RFAs have become a crucial component for the implementation of fiber optic communication systems [9]. An exponential increase on the product distance × capacity of the transmission experiments on optical communication systems was observed in the last decade. The majority of these experiments, especially since the year 2000, have employed RFA as amplification technology [9]. This survey attempts to cover the most recent aspects in the field of Raman amplification for fiber communication systems. 2. Theoretical Description of Raman Scattering In 1928 Raman scattering was discovered independently and almost simultaneously by two research groups, one working in India and lead by Sir C. V. Raman [29], and the other by G. S. Landsberg and L. I. Mandelstam working in Russia [30]. In 1930, the Nobel committee distinguished Sir C. V. Raman for his discovery of the molecular scattering of light and since then this effect has been known as the Raman effect. Raman effect is a scattering effect of light. Light scattering occurs as a consequence of fluctuations in the optical properties of a medium. In optical fibers three types of scattering effects are relevant: Rayleigh, Brillouin and Raman scattering. Rayleigh scattering is an elastic process, i.e., the incident and the scattered photon have the same energy, therefore the same frequency. Rayleigh scattering in fibers couples light from guided modes to unguided ones leading to optical attenuation. Indeed, in modern fibers operating in the near infrared, Rayleigh scattering is the major source of attenuation, as absorption is practically negligible. In fact, Silica lattice and electronic resonances are in the mid infrared and in the ultra-violet, respectively. Therefore in the near infrared, fibers operate, essentially in an off-resonance regime, apart from impurities, which in nowadays fibers are reduced to an extremely low level [31]. However, besides the off-resonant interaction with bound electrons, optical waves also interact with molecules inside Silica fibers, through scattering. Raman and Brillouin scattering are both inelastic processes, i.e., the incident and scattering photons have different energies. The energy lost by the incident field is stored into the medium in the form of vibrational energy, named phonons. Indeed, the origin of both Raman and Brillouin scattering effects resides in the interaction of light with these vibrational states (phonons). In the Brillouin scattering low frequency vibrational states are involved, usually referred as acoustic phonons. In the Raman process high frequency vibrational states are presented, named as optical phonons. Raman scattering can occur in two distinguished forms: Spontaneous Raman Scattering, and Stimulated Raman Scattering (SRS). In the spontaneous form, Raman scattering occurs when the incident field interacts with vibrational modes, mainly excited by thermal effects, of the molecules constituting the medium. From this interaction, it can result another optical phonon, with frequency Ω, and a down shifted optical photon with frequency 0 = −Ω S ν ν , or a up shifted photon of frequency 0 = + Ω A ν ν and in this case an optical phonon is annihilated, υ 0 is the frequency of the incident signal. As the frequency Ω is related to the normal vibrational modes of the molecules constituents of the medium, by analyzing the scattered light, information about the medium can be retrieved. This is the main idea behind Raman spectroscopy, a widely used P.S. André, A.N. Pinto, A.L.J. Teixeira et al. 56 technique for materials characterization. In amorphous materials, like Silica, Ω can assume a value belonging to a broad spectral range, starting from zero and going up to 40 THz. Experimentally both down shifted and up shifted frequencies waves have been observed and have been named as Stokes and anti-Stokes, respectively. Stimulated Raman Scattering was discovered by E. J. Woodbury and W. K. Ng, almost accidentally in 1962, when working with a Ruby laser [32]. They observed a strong spectral line not coincident with any spectral line of the fluorescence spectrum of Ruby. To understand this process let us assume that an incident photon is scattered by an optical phonon in the medium, and in this process a down shifted photon and an optical phonon are created. We can see that we have two ways of creating phonons, the scattering process and the thermal mechanism. If the intensity of the incident light is small, the rate of phonons created by scattering is low and due to thermal equilibrium the density of phonons in the medium is unchanged, and therefore the medium maintains the same optical properties. If the intensity of light is increased above a certain threshold, the optical properties of the medium can be changed in a way that the scattering process is enhanced [33]. In this situation, the incident light stimulates the scattering process and we are in the presence of Stimulated Raman Scattering. Through this positive feedback the scattering process can be enhanced by several orders of magnitude. Due to the Bosonic nature of the photons, this process can indeed provide gain. The photon emission process by a scattering center, it can be stimulated by the presence of another photon, and this stimulated emission is the origin of the gain. The term emission is used in this context in a quite abusive way because there is no absorption to a real state, but this process can be treated considering that the scattering photon is initially absorbed to a virtual state and after re-emitted. If we consider that the decay from the virtual states only occurs spontaneously, the Stokes power grows linearly with the pump power. In the other way, if we consider that the decays from the virtual states must be triggered by another photon, the Stokes power grows exponentially with the pump power. Off course, in reality both spontaneous and stimulated emission occurs. If the photon that triggers the stimulated emission is part of a signal we are in the presence of optical gain, which can be beneficial for optical communication systems [34]. If this photon was initially generated by spontaneous emission we are in the presence of amplified spontaneous emission noise which is usually considered as harmful, at least for telecommunications purposes. The spontaneous emission process always leads to an excess of noise in the system. The optical gain provided by the Raman process can be completely characterized by the Raman-gain coefficient ( ) Ω R g , which is related with the imaginary part of the third-order nonlinear susceptibility. The characterization of the amplified spontaneous emission process requires, besides the Raman-gain coefficient ( ) Ω R g , another coefficient named noise spontaneous emission factor ( ) Ω sp n . However, it turns out that another source of noise must be also considered to characterize the noise in Raman amplifiers. This source of noise arises from Rayleigh scattering. Most of the Rayleigh scattered photons are lost through non-guided modes, but some of them are coupled to the counter-propagating mode. Those photons can be amplified and through another Rayleigh scattering process can appear as extra-noise at the amplifier output. This effect is usually named as double Rayleigh scattering and will be described in more detail in section 4.3. New Challenges in Raman Amplification for Fiber Communication Systems 57 3. Modeling of Raman Amplifiers The implementation of RFA, using an optical fiber as gain medium, requires that the pump and information signals must be injected into the same fiber. A basic scheme for a RFA architecture is displayed in figure 2. The signal and pump waves are launched into the optical fiber (the gain medium) by a coupler, so, that stimulated Raman scattering can occur. Since the SRS effect occurs uniformly for all the orientations between pumps and signals, Raman amplifiers can work both in forward and/or backward pumping configuration. Figure 2. General scheme for a distributed Raman amplifier. For simplicity the optical isolators used to protect the pumps and signals sources, were omitted. The model for power evolution in Raman amplifiers assuming a multipump multisignal configuration is often based on an unified treatment of channels, pumps and spectral components of the amplified spontaneous emission (ASE). The major interactions can be reasonably drawn by considering the pump-to-pump, signal-to-signal and pump-to-signal power transfer, attenuation, Rayleigh back scattering, spontaneous Raman emission and its temperature dependence. Other effects, such as noise generation due to spontaneous anti- Stokes scattering, polarization and nonlinear index are neglected, but they can reach considerable importance in certain regimes of transmission. It must be noted that signal channels and Raman pumps are treated as fields at single frequencies, so ignoring the interactions due to the spectral shape of signals and pumps. In a general approach, the power evolution of pumps, signals and ASE (in forward and backward directions), with time along the fiber distance is given by the following set of coupled differential equation [35]. For N p pumps, N s probe signals and N ASE spectral components for ASE, the system is formed by N p +N s +2N ASE equations. ( ) ( ) ( ) ( ) [ ] ( ) ( ) [ ] ( ) ( ) ( ) ( ) [ ]( ) υ η υ γ υ η υ υ υ α Δ + + Γ + + ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ Δ + Γ − + − + + − = ∂ ∂ ∂ ∂ ± − + − = ± + = + = − + − + − = ± ± ∑ ∑ ∑ ∑ ji j j i j ji i i i i m i j ij ij i m i j j j ij j i j j i j ji i i i i t z P t z P g h t z P t z P g h t z P t z P g t z P t z P g t t z P V z t z P 1 , , ) , ( , 1 2 , , , , , 1 , 1 1 1 1 1 1 m m (1) V i is the frequency dependent group velocity. The ± signs stand for the forward or backward propagating waves, being α i and γ i the coefficients of attenuation and Rayleigh of P.S. André, A.N. Pinto, A.L.J. Teixeira et al. 58 the i th wave at frequency υ i . h and k B are the Planck and Boltzmann constants, respectively, and T is the fiber absolute temperature. The phonon occupancy factor is given by: ( ) 1 1 exp − ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − = T k h B j i ij υ υ η (2) The frequencies υ i are numbered by their decreasing value (the lower order corresponds to the higher frequency). Thus, the terms in the summation in expression 1, from j=1 to j=i-1 cause amplification since the wave i is receiving power from the lower order waves (with higher frequency). For the same reason, the terms in the summation from j=i+1 to j=m originate depletion. For mathematical convenience the gain spectrum was divided into slices of width Δυ, spanning the range over which ASE spectral components are significant. The terms that contain a product of powers describe the coupling via stimulated Raman Scattering, being its strength determined by the Raman gain coefficient of the fiber, g ij obtained by equation 3. ( ) eff j i R ij A g g Γ − = υ υ (3) where A eff is the effective area of the fiber and the factor Γ is a dimensionless quantity comprised between 1 and 2 that takes into account the polarization random effects. The achieved gain, as well as the slope of the gain spectrum, depends on the transmission fiber [36, 37]. In figure 3, two Raman gain coefficient spectra are displayed, showing the different strengths of the Raman coupling of a SMF fiber and a dispersion compensating fiber (DCF). -30 -20 -10 0 10 20 30 -4 -3 -2 -1 0 1 2 3 4 R a m a n g a i n c o e f f i c i e n t ( W - 1 k m - 1 ) pump-signal frequency difference (THz) SMF DCF Figure 3. Raman gain coefficient spectra for two germanosilicate fibers: Single mode fiber (SMF) and dispersion compensating fiber (DCF), for a pump wavelength of 1450 nm. New Challenges in Raman Amplification for Fiber Communication Systems 59 As a matter of fact, the small effective area of the DCF (15 μm 2 ) is determinant for its higher Raman gain coefficient when compared to the SMF (80 μm 2 ) or when compared with DSF fibers (50 μm 2 ). Those spectra also show peaks that are broader than those presented by crystalline materials, since the amorphous nature of Silica allows a continuum of molecular vibrational frequencies. To obtain a steady-state power distribution, the time derivative in equation 1 is settled equal to zero, and the set of equation takes the form of expression 4. [ ] [ ] ( ) [ ]( ) υ η υ γ υ η υ υ υ α Δ + + Γ + + + ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ Δ + Γ − + − + + − = ± − + − = ± + = + = − + − + − = ± ∑ ∑ ∑ ∑ ji j j i j ji i i i i m i j ij ij i m i j j j ij j i j j i j ji i i P P g h P P g h P P g P P g dz dP 1 1 2 1 1 1 1 1 1 m (4) In spite of the simplification, the modeling is still computationally intensive, especially for the situation of backward or bidirectional pumping. In those situations, the mathematical problem that describes the power evolution of pumps and signals along the fiber is a boundary value problem (BVP) which is more difficult to solve than the initial value problem (IVP) in the forward pumping scheme. An immediate approach to the numerical solution of such problem is the shooting method [38]. There are other allowable numerical methods, such as relaxation methods, or collocation methods [39]. Generally, shooting methods are faster than relaxation ones. In shooting methods, we choose values for all the dependent variables at one boundary, solve the system of ordinary differential equation (ODE) as an IVP and verify if the obtained values on the other boundary are consistent with the stipulated values (boundary conditions) [40]. Then, the parameters are repeatedly changed using some correction scheme until this goal is attained. The selection of the correction scheme is crucial for stability and efficiency of the resulting algorithm. An other variant of the shooting method, we can guess boundary values at both ends of the domain, integrate the equation to a common midpoint and repeatedly adjust the guessed boundary values so that the solution tends to the same value at the middle point. This adjustment task is usually performed by the Newton-Raphson method. Recently, some shooting algorithms with different correction schemes for the design of Raman fiber amplifiers have been proposed in order to improve convergence of the solutions even for larger fiber lengths [41]. This scheme is obtained by modifying the numerical method used to perform the IVP integration (fourth order Runge-Kutta, Runge-Kutta- Felhberg, etc). Other approaches to solve the equation 4 propose a shooting method to a fitting point using a correction scheme based on a modified Newton approach. Therefore, by introducing the Broydens rank-one method into the modified Newton method, the algorithm becomes more efficient and stable. This happens because the intensive numerical calculations of the Jacobi matrix are substituted by simpler algebraic calculations [41]. The use of projection methods such as collocation, gives a continuous approximation of the solution as a function of the fiber length. The basic idea is to approximate the BVP solution by a simpler function that represents an approximation. P.S. André, A.N. Pinto, A.L.J. Teixeira et al. 60 Nevertheless, the Raman equations (equation 4) are also solvable through semi-analytical methods, using the average power analysis (APA) presented by Min et al. [42]. The amplifier is split into n small segments, in order to avoid the position dependency of the powers of equation 4. The equations are then solved analytically in each segment, considering as input conditions the outputs provided by the solution on the previous segment. Equations 5 to 8 show how the powers are iteratively computed. The output pump/signal power at each section end is given by: ) , ( υ z G P P in out ± ± = (5) being G(z,υ) the section gain, [ ] z B A z G Δ − + − = )) ( ) ( ) ( ( exp ) , ( υ υ υ α υ (6) The constants, A(υ) and B(υ) are obtained through: j m i j ji j i j ij P g B P g A ∑ ∑ + = − = = = 1 1 1 ) ( ) ( υ υ (7) The optical power term in each section can be substituted by its length averaged values given by: )) ( ln( 1 ) ( υ υ G G P P in − = ± (8) For a RFA, the net gain is usually defined as the ratio between the signal powers at the end and at the beginning of the fiber link, as defined in equation 9: ) 0 ( ) ( = = = z P L z P G signals signals net (9) The so-called on/off gain is another useful quantity that measures the increase in signal powers at the amplifier output when the pumps are turned on, as follows: off pumps with ) ( on pumps with ) ( / L z P L z P G signals signals off on = = = (10) The numerical issues due to the backward pumping can be surpassed by assuming that the pump inputs are located at the same fiber end that the signal inputs. Therefore, the pump equations are integrated reversely as if they were backward by multiplying them by (−1). A guessed initial input is necessary to perform the integration, but the algorithm is able to adjust it using an optimization routine that adjust the initial input until the output at the fiber end New Challenges in Raman Amplification for Fiber Communication Systems 61 reaches the real backward pump input. The use of the APA approach has shown a reduction of two orders of magnitude in the computation time, being the obtained results in agreement with the ones resulting from traditional numerical methods. To demonstrate the numerical resolution of the steady-state Raman propagation equations, we assume the scheme in figure 4, where three bidirectional pumps (two backward and one forward) and four probe signals are considered. The counterpropagated pumps have power levels set equal to 0.1W, working at 1450 nm and 1460 nm, respectively. The copropagated pump is working at 1470 nm with an output power also equal to 0.1W. The forward pumping signal are then injected into 40 km of SMF fiber and combined with 4×1000 GHz spaced C band probe signals with an initial optical power equal to 1μW. The spatial evolution of pumps and probe signals are displayed in figure 4. 0 10 20 30 40 0.6 0.7 0.8 0.9 1.0 1.1 1.2 P r o b e P o w e r ( μ W ) Fiber length (km) 1530 nm 1538 nm 1546 nm 1554 nm 0 10 20 30 40 0.00 0.02 0.04 0.06 0.08 0.10 1450 nm 1460 nm 1470 nm P u m p P o w e r ( W ) Fiber length (km) Figure 4. Spatial evolution of two counterpropagated pumps, one copropagate pump and four probe signal along a 40 km SMF fiber span amplifier. Probe signals evolution (top) and pump signals evolution (bottom). P.S. André, A.N. Pinto, A.L.J. Teixeira et al. 62 The implementation of equation 4 also allows the calculation of the total noise for each signal (forward and backward ASE) within the amplifier, whose spatial evolution for this system can be followed in figure 5. 0 10 20 30 40 1 2 3 4 5 6 7 8 9 T o t a l N o i s e P o w e r ( n W ) Fiber length (km) 1530 nm 1538 nm 1548 nm 1554 nm Figure 5. Spatial evolution of the total noise (forward and backward ASE) power along 40 km SMF fiber span. The noise figure of an optical amplifier amounts the degradation of the signal to noise ratio (SNR) when the signals are amplified. The most important source of noise in optical amplifiers is ASE, which, for Raman amplifiers is due to spontaneous scattering. Assuming that the signals are initially as noiseless as possible, and that their degradation is due to signals spontaneous beat noise produced by ASE, the noise figure, in linear units, is given by equation 11 [36]: net ASE G h L z P 1 1 ) ( 2 NF ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + Δ = ≈ + υ υ (11) where hυ is the photon energy and P ASE + is the forward ASE measured over the reference bandwidth Δυ. The first term corresponds to the noise from the signal spontaneous beating and the second one to shot noise. Another quantity, named effective noise figure, accounts the noise that a discrete amplifier placed at the end of an unpumped fiber link would need to have the same noise performance that a distributed Raman amplifier. In decibel units, the effective noise figure is computed using: ( ) dB dB dB eff L NF NF α − = (12) New Challenges in Raman Amplification for Fiber Communication Systems 63 Typically, WDM systems allocate a large number of channels spaced over wide bandwidths. Considering the previous system but doubling the pump powers and using 64 probe signals (instead of 4), we obtained the spectra of the gain and noise figure which are plotted in figure 6. 1530 1540 1550 1560 1570 1580 0 2 4 6 8 10 Signal Wavelengths (nm) N e t G a i n ( d B ) -2.0 -1.5 -1.0 -0.5 0.0 E f f e c t i v e N o i s e F i g u r e ( d B ) Figure 6. Net gain and effective noise figure spectra for a system with two counterpropagated pumps, one copropagate pump and 64×100 GHz probe signals along 40 km SMF fiber span. As depicted in this section and despite some remaining numerical issues, the modeling of a multipump Raman amplifier anticipates many valuable applications for WDM systems, namely the broadband gain. It is important to notice that gain spectra as wide as 100 nm are achievable and that the gain value can be kept quite constant by an appropriate tailoring of the amplifier architecture. This procedure involves solely the proper dimensioning of the pump power levels and operating wavelengths, as discussed more extensively in section 4. Another interesting feature of RFA is the noise performance. The ASE noise in RFA is intrinsically low (as suggested by the negative effective noise figure presented above). The reason relies in the fast relaxation of the optical phonons, the absorption of signal photon to the upper virtual state is extremely small. The inversion of population is almost complete. 4. Challenges in Raman Amplification 4.1. Gain Profile Optimization One of the most impressive features of Raman fiber amplifiers is assuredly the possibility to achieve gain at any wavelength, by selecting the appropriate pump wavelength. Therefore, it is possible to operate in spectral regions outside the Erbium doped fiber amplifiers bands over a wide bandwidth (encompassing the S, C and the L spectral transmission bands). P.S. André, A.N. Pinto, A.L.J. Teixeira et al. 64 Nevertheless, some studies have been reporting that despite the Raman gain dependence is essentially due to the pump-signal frequency difference; there is also some weaker dependence on the pump absolute frequency [43]. However, since a deeper study of this topic is beyond the scope of this work, we will not consider it in the gain optimization. 1400 1420 1440 1460 1480 1500 1520 1540 1560 1580 1600 0.20 0.22 0.24 0.26 0.28 0.30 0.32 0.34 Net gain (around 5 dB) Gain contribution from each individual pump pumps powers and wavelengths P u m p p o w e r ( W ) Wavelength (nm) 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 N o r m a l i z e d g a i n c o e f f c i e n t Figure 7. Numerical simulation of broadband Raman amplifier gain. Bars show backward input pump powers and wavelengths. Ticker line show 14×400 GHz probe signals optimized net gain and thin lines the gain contribution of each individual pump. The simulation was carried out through 25 km of SMF fiber. A flat spectral gain profile is achievable with the combination of several pumping lasers operating at specific powers and wavelengths. The Raman gain created by pumps at different frequencies is slightly shifted from each other to partly overlap and form a composite gain. When the pump powers and frequencies are properly chosen, this wide gain can also be considerably flat. Another important feature to take into account when designing a flat gain scheme, is the strong Raman interaction between the pumps, since the higher frequency pump is responsible for the amplification of the lower frequency signals, more pumping power is needed, as some will also be transferred to the lower frequency pumps. This interaction between pumps also affects the noise properties of the amplifier. However, some novel pumping schemes have been recently proposed in order to prevent those unwanted effects: copumping, time dependent Raman pumping, higher order pumping and broad-band pumping [44]. Typically, laser diodes with output powers in the 100-200 mW range can be used in a multipump scheme. This scheme is normally composed of a set of laser diodes operating in the 14XX nm region, whose spectral width is narrowed and stabilized by a FBG. Optical couplers are used to combine and depolarized them, in order to suppress the polarization New Challenges in Raman Amplification for Fiber Communication Systems 65 dependent gain. The multipumping allows bandwidth upgradeability by the addition of new laser diodes. Theoretically, the larger the number of pumps the better the gain ripples. Nevertheless, there are economic issues that prohibit the use of an arbitrary number of pumps. For this reason, we have to find a balance between the system performance and the cost of amplification. Optimization of the gain spectrum has been widely performed making use of several global search methods, such as neural networks [45], simulated annealing [46] and genetic algorithm (GA) [47]. During the search process, the pump powers and frequencies are directly substituted into the system of propagation equations to calculate de gain profile. Depending on the speed of the numerical method used to integrate the system of equations, the amount of numerical computations involved can be considerably large and the optimization inevitably time consuming. Those solutions are not suitable for practical applications where the real optimal solution must be provided in a short time. However, some alternatives can be found by replacing the usual intensive numerical integrations with simpler algebraic calculations using the APA method while integrating the Raman propagation equations. Another simple but important issue when using a global optimization method relies in a proper dimensioning of the search domain. Using the APA method, all the inputs are located at the same fiber end, even for the counter pump situations. Therefore, the pump power inputs are chosen by presuming a typical propagation profile. By this way, it is advisable to try lower power values for the higher frequency pumps and higher power values for the lower frequency pumps (the opposite happens at other fiber end). Regarding to the optimization of the pump frequencies, it is advisable to divide our spectral range into the number of pumps and then shift those values by 13 THz. A second approach to speed up the search of the optimal pump configuration uses the genetic algorithm (GA) method only to search the pump frequencies and a quadratic programming method to solve the power integral [48]. The search domain of the GA method is by this way reduced to a half, enabling faster convergence. Another approach combines GA with the Nelder-Mead search. This so called hybrid GA can be useful in certain situations for the purpose of saving some function evaluations and consequently to perform the optimization in the least time possible [49]. The hybrid GA follows the routine depicted in Figure 8. Firstly, the initial population, as well as the other GA operators are dimensioned: selection, crossover and mutation. The selection, together with the crossover, is responsible for the bulk of GA processing power. The mutation is an operator that plays a secondary role in the GA. Since, the genetic operators can be performed by different methodologies, it is important to choose the ones that are more adequate to the problem we are dealing with, in order to improve the GA search procedure [50]. It must be noted that if the search space is not large, it can be searched exhaustively and the best possible solution will be probably found. The maximum number of allowed generations is also an important feature because, when carefully chosen, it can save a large number of function evaluations. The Nelder-Mead method uses a simplex in a n-dimensional space, characterized by the n+1 distinct vectors that are its vertices. At each step of the search, a new point in or near the current simplex is generated. The function value at the new point is compared with the function values at the vertices of the simplex and one of the vertices is replaced by the new point, giving a new simplex. This step is repeated until the diameter of the simplex is less than the specified tolerance P.S. André, A.N. Pinto, A.L.J. Teixeira et al. 66 Figure 8. Scheme of the hybrid GA implementation. 0 5 10 15 20 25 30 35 40 45 0.5 0.6 0.7 0.8 0.9 1.0 B A τ Number of Generations Figure 9. Total simulation time for a hybrid GA against the number of generations for a population size of 50 individuals. The line is a visual guide. By determining properly the right moment to switch from one method to another, it is possible to reduce the simulation time to a half when compared to simple GA. This result can be observed in figure 9. The normalized total simulation time (τ) against the number of generations is plotted. Here, the reference is the slowest simulation, the one that use 40 generations, identified by the letter B. The best situation (tagged as A) tooks a simulation time New Challenges in Raman Amplification for Fiber Communication Systems 67 equal to about a half of the time of the worse situation and it was attained with 17 generations. An heuristic explanation relies on the intrinsic nature of the GA. We verify that for small number of generations (bellow 15) the GA time is small but the system reaches a worse fittest solution. Thus, the Nelder-Mead method needs more time to reach a desirable solution. When the number of generations increases the GA reaches a best solution but the needed computation time increase accordingly. 0 5 10 15 20 0 20 40 60 80 100 120 140 160 1410.0 nm 1424.2 nm 1437.9 nm 1452.1 nm 1465.5 nm 1494.8 nm 1502.4 nm P u m p P o w e r ( m W ) Fiber Length (km) Figure 10. Power evolution of optimized pumps along 20 km of SMF (lines). The geometric shapes stand for the used experimental values. 1520 1530 1540 1550 1560 1570 1580 1590 1600 1610 7.5 7.6 7.7 7.8 7.9 8.0 O n / O f f G a i n ( d B ) Channel Wavelength (nm) Figure 11. Experimental (arrows) and simulation (line) on/off spectral gain for the 20 probe signals and 7 counter propagated pumps, over 20 km of SMF fiber. P.S. André, A.N. Pinto, A.L.J. Teixeira et al. 68 In order to enlighten the conclusions provided by the hybrid GA algorithm, a laboratorial implementation was carried out to test the optimization results. A Raman amplified system with 20 km of SMF fiber, 20 probe signals and 7 backward pumps was implemented. Since the pump wavelengths are already settled, only the optimization of the power levels is needed. The simulation used the stochastic uniform method for selection, the scattered crossover method and the uniform mutation. A population of 50 individuals and a number of generations equal to 35 were considered. The spatial evolution of the pumps signals optimized values are displayed in figure 10 jointly with the pump signal experimental values. In figure 11, the optimized and experimental on/off gain spectra are presented. This is a good agreement between the optimization modeling and the experiment. The maximum ripple attained by the optimization is 0.41 dB being the experimental maximum ripple equal to 0.23 dB. The mean square deviation between simulation and experimental results is equal to 0.0036. Indeed, a flat gain over a wide bandwidth (~80 nm) was attained, using seven pumps with a total input power equal to 453 mW. 4.2. Raman Amplification Using Multiple Low Power Lasers One of the main issues in Raman amplification is related to the stability of the high power lasers, the costs and the need for efficient cooling. To go around these problems, the usual solution is the use of several pump signals, what results in added advantages, like high, flat and wide-gain bandwidth [51-53]. The technology evolution allowed that high power pumps are nowadays commercially available, although some problems still limited [54]. The pressure on optical components prices, lead to the creation of CWDM standards [55]. This is reflected specially on price dropping of uncooled lasers with relatively high powers (>10mW). The price to pay is wavelength wondering, however, neither for CWDM nor for Raman, wavelength stability is not a stringent requirement, allowing simple control. With this technology the possibility of achieving Raman gain by combining multiple of these low power lasers was successfully implemented [21]. Teixeira et el proposed the use of an array of low cost lasers to achieve wideband Raman amplification, providing both experimental and simulation results [21]. In this work a counterpropagating topology was implemented, using 40 C band lasers with 20 mW output power spaced by 0.8 nm (1533 nm – 1557 nm). These lasers were combined using a multiplexer, bringing up a total power of more than 200mW (23 dBm). This power is enough to generate SRS. Several fibers were tested: True Wave and dispersion compensating. To characterize the gain profile, an array of 40 L-band 0.8nm spaced probe lasers (1565 nm - 1605 nm) with a total optical power of 1 mW was used. Figure 12 (a) shows the implemented setup. Figure 12 (b) presents the simulation results for the implemented system to four different pumping configurations. The first curve corresponds to the traditional approach, where one high power pump (23.6 dBm) at a single wavelength (located at 1530 nm) is used. In the second case, three lasers spaced by 0.8 nm starting at 1530 nm having total power of 23.6 dBm were multiplexed. The results for the two above pumping configurations are approximately equal, having only a wavelength shift of 0.8 nm as expected due to the average pumps wavelength difference. Similar simulation was experienced considering 40 lasers, each with 7.6 dBm (after the multiplexer), resulting in a total power of 23.6 dBm. In this case, the New Challenges in Raman Amplification for Fiber Communication Systems 69 gain curve appears even smothered and the peak shifted by ~16 nm; the peak gain is similar, however a small enhancement on the 3 dB gain bandwidth was obtained and the gain profile smothered. In order to explore the advantages of the methodology (gain flatness), the power distribution for the pumps was optimized to reach an equalized gain, while maintaining the same total pump comb. The peak gain was decreased at the expense of an increased flattened profile. a) 1600 1620 1640 1660 1680 -2 -1 0 1 2 3 4 5 G a i n ( d B ) Wavelength (nm) 1 laser 3 laser 40 laser 40 laser eq. b) Figure 12. a) Implement setup for the simulation and experimental systems, b) simulated Raman gain profiles for several sets of pumping configurations, with 23.6 dBm of total power. PM demotes an optical power meter, OSA denotes an optical spectrum analyzer and MUX is an optical multiplexer. Due to limitations on available probe and pump signals, the experimental implemented system only can scope part of the spectral bands used in simulation. The gain is only measurable when the pump powers go above 10 dBm. A maximum of 3 dB net gain was achieved in the L band for full pump power, 23.6 dBm, as displayed in figure 13 b). Also, in the same figure, a minimum of 2dB gain over more than 30nm, with 1dB ripple, was achieved without any power distribution optimization. The results have demonstrated the effectiveness of the technique to achieve Raman amplification. P.S. André, A.N. Pinto, A.L.J. Teixeira et al. 70 1530 1535 1540 1545 1550 1555 1560 0 5 10 15 20 25 1575 1580 1585 1590 1595 1600 1605 1610 0 1 2 3 4 R e l a t i v e P u m p P o w e r ( d B m ) Wavelength (nm) G a i n ( d B ) Wavelength (nm) 0dBm 10dBm 23.6dBm a) b) Figure 13. a) Pump to pump Raman effect; b) experimental Raman gain achieved for several values of the pump power, with probes at 0 dBm. In figure 13 (a) it is illustrated the pump to pump effect which is commonly occurring in dense WDM systems. This effect starts to be noticeable above 10 dBm of total power and is evident for 7.6 dBm per channel. This phenomenon can be harmful due to uneven distribution of power during transmission, however, if correctly considered can be used to obtain beneficial extra gain in the system, if a pre equalization is also implemented . 4.3. Raman Amplification Using Rayleigh Backscattering Raman amplification pumping can also be achieved by recurring to the traditional methods of shifted gain [19, 20]. In these methods, several FBG reflector pairs are used to generate resonant cavities in the maximum of the Raman gain spectrum. Thus, with a Ytterbium laser operating in the vicinity of 1090 nm, where it exhibits its maximum efficiency, it is possible to generate pumps in the E band, as demonstrated by Papernyi et al, where a set of 6 FBG reflector pairs were used to generate pumping in the E-band [22]. The latter amplifies the C band, where the probe signals transmission usually occurs. The main penalties of traditional Raman amplification are associated with intrinsic nonlinear phenomena such as nonlinear refraction and Rayleigh backscattering, since it is required to use high powers and long fiber spans. This last effect occurs when a fraction of scattered light is backreflected towards the launch end of the optical waveguide. This reflection is called single Rayleigh backscattering (SRB). Part of this scattered light is also backreflected in the forward direction and it is called double Rayleigh backscattering (DRB), as shown in figure 14 [56]. SRB and DRB can be controlled by actuating properly on the fiber drawing process or by a correct power design [57]. The Rayleigh backscattering has been studied, modeled and characterized by many authors [56-59]. It is known that the process results from multiple reflections of light inside the fiber and therefore spontaneous and unstable lasing can occur [60]. However, this phenomenon has been observed as an impairment to signal transmission [61, 62]. New Challenges in Raman Amplification for Fiber Communication Systems 71 Figure 14. Simple Rayleigh backscattering (SRB) and double Rayleigh backscattering (DRB) representations over an infinitesimal length of fiber. Recently, a method that, up to some extent, allows the control of this phenomenon was reported [56,63]. With the possibility of controlling the SRB and DBR effect, novel applications can be drafted. One suggestion is the use of this effect to generate distributed resonant cavities, which will degenerate in lasing if enough gain is achieved. These are achieved with the help of only one end FBG set [63]. This is advantageous when compared to the previously described methods to obtain cascaded Raman amplification, since it needs only one FBG set, minimizing the need for identical FBG to be used and tuned at different sites which can be not colocated. In order to demonstrate the application of this technique to control SRB and DRB, the experimental system reported in figure 15 was implemented. A Raman pump in the E-band, at 1428 nm, was coupled to the transmission fiber, with controllable power up to 1.5 W. A circulator was used to protect the laser from back reflections and, simultaneously, to allow the measurements of the back reflected power spectrum. Two different scenarios were observed: the FBGs are absent between the fiber and the coupler; and the setup was complete as described in figure 15. These two scenarios target to show the controlling effect achieved by the FBGs. A set of three FBG with wavelengths centered at: 1520 nm, 1531.6 nm, 1535.6 nm all having 95% reflectivity, were placed after the pump and act as reflective elements. In a first setup, a 14 km DCF fiber with dispersion parameter equal to -1393 ps/nm and Raman coefficient of 3.05 x 10 -3 m -1 W -1 was used as transmission medium. Figure 15. Experimental setup for the double shifted Raman experiments; WDM denotes a band coupler and Att denotes an optical attenuator. Considering the first scenario, where no FBGs were present, the common Raman effect in the C band was observed, figure 16 a) for a pump power of 350 mW. When the power of P.S. André, A.N. Pinto, A.L.J. Teixeira et al. 72 the pump was increased to 600 mW, Rayleigh backscattering spontaneous lasing effect is observed, as displayed in figure 16 b). This effect presents random behavior, both in wavelength and power, being the spectrum time dependent. In a second scenario the FBGs were present, control of the random process generated by the SRB and DRB was achieved and the lasing was stabilized in the FBGs wavelengths. In this situation a virtual cavity was established, formed by the FBG and the Rayleigh backscattered light. To generate more than one laser in the C-band a set of cascaded wavelength mismatched FBGs were used. These gratings are responsible for a multipeak frequency dependent reflection back into the fiber of the amplified spontaneous emission and DRB light from the fiber. This, in conjunction with the FBG, create resonant cavities, which generate stable wavelength constant lasing actions, from now on called as FBG-DRB lasing. Due to the different reflectivities of the FBGs and the Raman gain profile, different lasing powers for each configuration occur in the C-band. Whenever the power of the generated lasers in the C-band is high, cascaded Raman effects will occur that generate gain in the far L and U-band. The FBG-DRB lasing and consequent stabilization process with the simultaneous L-U band spontaneous emission is reported in figure 16 c), where a pump power of 1.2 W was used [63]. 1400 1450 1500 1550 1600 1650 1400 1450 1500 1550 1600 1650 -90 -80 -70 -60 -50 -40 -30 -20 -10 1400 1450 1500 1550 1600 1650 U-band ASE C-band FBG-DRS lasing (c) (a) C-band ASE P o w e r ( d B m ) Pump (b) C-band Spontaneous Lasing Pump Pump Wavelength (nm) Figure 16. Transmission spectra for 14 km DCF fiber: a) Spontaneous ASE for a pump of 300mW; b) spontaneous lasing for a pump of 600mW; c) C-Band FBG-DRB lasing and far L and U-band Raman generated ASE for a 1.2 W pump. The results show a 38 nm flattened ASE bandwidth in the U-band, generated by the FBG- DRB. By introducing a copropagating probe at 1625 nm, a gain of 10 dB was measured for an E-band pump power of 1 W. In a second setup, different optical fibers were tested in order to compare the pump power laser threshold. A 14 km long DCF fiber, a 50 km long DSF fiber and a 50 km long non zero dispersion shift fiber (NZDSF) were used [60]. Figure 17 shows the different lasing thresholds and curve shapes resulting from the intrinsic differences between the optical fibers. From figure 17, it can be observed that this process is more efficient in the DCF fibers, where the threshold power is 350 mW, while for the NZDSF fiber is 650mW and 1W for the DSF fiber. New Challenges in Raman Amplification for Fiber Communication Systems 73 0,2 0,4 0,6 0,8 1,0 1,2 -50 -40 -30 -20 -10 0 10 0,2 0,4 0,6 0,8 1,0 1,2 0,2 0,4 0,6 0,8 1,0 1,2 P e a k P o w e r ( d B m ) 1520.0nm 1531.6nm 1535.4nm 1428.0nm 14 km DCF Fiber Selected Input Power (W) 50 km NZD Fiber 50 km DSF Fiber Figure 17. Depletion of the E-band pump and peak power of the C band lasers as a function of the pump power for several fiber types; from left to right: DSF, NZD and DCF. Usually, the simulation of Raman amplification as convergence and stability problems, especially for high pump powers, has reported in previous sections. The simulation of the lasing effect with high pump power has similar difficulties. To avoid such problems the solving method for the differential equation system is simplified to an analytical method based on the transfer matrix (APA) as proposed in section 4.1. Inside these fiber slices, the parameters are considered to have small variations and the solution of the equation system is obtained by stabilization after multiple passes along the length of the fiber [64]. The initial solution uses an analytical approach that was based in the undepleted case. The approach to the pump depletion is included in the attenuation of the pump. In each fiber slice, the Rayleigh backscattering is calculated at the boundary and this backscattering power is added to the signals in the same direction and wavelength, that also suffer amplification and depletion. 0.2 0.4 0.6 0.8 1.0 -50 -40 -30 -20 -10 0 10 Simulated Experimental O u t p u t P e a k P o w e r ( d B m ) Input Power (W) Figure 18. (a) Optical power density spectra for 14 km DCF fiber from E-band to U-band; (b) Output power evolution of the lasing effect of the FBG-DRB at 1520 nm. P.S. André, A.N. Pinto, A.L.J. Teixeira et al. 74 Figure 18 (a) presents the simulation results of such algorithm for 29.03 dBm of pump power. The evolution of the power densities from E-band to U-band spectra is shown for a long fiber span. The E-band pump signal suffers depletion in the long propagation fiber. This pump works as a seed of the C-Band FBG-DRB lasing, which generate the L-U-Band Raman gain. Since the process of lasing is not stable, the simulation process presents a slow stabilization, but, the boundary powers over the FBG are quickly stabilized. Figure 18 (b) presents a comparison of the threshold laser power obtained by experimental and semi- analytical methods. The output peak power of the FBG-DRB signal at 1520 nm is related with the input pump power. As observed for the DCF fiber, stable multiple laser actions were achieved for moderate pump powers (350 mW) for both simulation and experiment. 4.4. Amplification with Incoherent Pumps A technique to increase the bandwidth and decrease the spectral ripple of RFA is available with incoherent pump lasers. A Raman amplifier with incoherent pumps can be modeled as a multipump Raman amplifier. In such case, the spectrum of the incoherent pump is well approximated by a large number of pumps of infinitesimal spectral width and whose power sum equals the integral power of the incoherent pump. Therefore, the theoretical model used for incoherent pump schemes is based on the model, previously presented, for coherent multipump configurations. 1470 1480 1490 1500 1510 1520 0.0 0.2 0.4 0.6 0.8 1.0 Wavelenght (nm) O p t i c a l p o w e r ( a u ) Figure 19. Pump spectrum for the incoherent pump. An incoherent pump spectrum, as displayed in figure 19, with 10 nm FWHM, can be approximated by 100 pumps of infinitesimal spectral width, having an aggregate power equal to the integral power of the incoherent pump. The incoherent pump here considered was New Challenges in Raman Amplification for Fiber Communication Systems 75 obtained from a high power FBG (Fiber Bragg Grating) laser, from which the stabilization grating was removed [65]. To evaluate the advantages of this technique, the Raman on/off gain and the noise figure were measured for coherent and incoherent pumping over 40 km of SMF fiber. The probe signal combo consists of 13 channels, with 1 mW power, spaced by 100 GHz over the 1546- 1556 nm spectral region. Both co-propagating and counter-propagating architectures were considered. The coherent pumping source was a high power FBG laser with a wavelength of 1490 nm. In both cases the pump power was 290 mW. The results of Raman on/off gain and effective noise figure are shown in figure 20. The relatively low on/off gain is due to the fact that the pump wavelengths have not been optimized for this signal band. 1546 1548 1550 1552 1554 1556 0 2 4 6 8 10 12 14 incoherent co puming incoherent counter pumping coherent co pumping coherente counter punping E f f e c t i v e N o i s e f i g u r e ( d B ) Wavelenght (nm) 1546 1548 1550 1552 1554 1556 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 Incoherent co pumping Coherent co pumping Incoherent counter pumping Coherent counter pumping G a i n ( d B ) Wavelenght (nm) Figure 20. Raman gain and Effective Noise Figure. Lines are simulated results and points represents to experimental data. The incoherent pumping gain slopes are 0.015±0.008 dB/nm and 0.017±0.004 dB/nm for co and counter propagation configurations, respectively. For coherent pumping, the gain slopes are 0.042±0.01 (co-propagation) and 0.052±0.005 dB/nm (counter-propagation). Such results show that the incoherent pumping configuration presents a flatter gain. The noise figure is approximately the same for coherent and incoherent pumping in the counter-propagating configuration. However, in the co-propagating case, the noise figure is considerably lower for coherent pumping. In agreement with previous works [23-26], these results indicate that the incoherent pumping technique can be used to decrease the spectral ripple of the Raman gain. 4.5. Raman in CWDM Systems Another important challenge is the deployment of RFA for access networks, namely for CWDM networks. Since CWDM systems require large bandwidths to guarantee the transmission of a reasonable number of channels, spaced by 20 nm, wide band Raman amplifiers are well suited for this purpose. P.S. André, A.N. Pinto, A.L.J. Teixeira et al. 76 The Raman amplifier bandwidth can be enlarged by using multiple pumps. Optimization of the number of pumps and their wavelengths enables the large needed gain spectra and that could be placed in any range of wavelengths used in optical communications. The design of an amplifier that fits more than two CWDM channels can be achieved, with the following procedure. The number of channels to be transmitted is determined in order to define the required bandwidth. The optical fiber characteristics impose a minimum to the required gain, and finally the number of pumps as well as their characteristics are decided. The scheme of figure 21 illustrates the important issues to be considered to design a multi- pumped Raman amplifier for a CWDM system. Since the number of CWDM channels and the length of the link as well as its losses are defined, the minimum required gain to compensate the transmission losses and the minimum bandwidth to transmit all the required channels may be determined using the rectangle shown in figure 21. The gain has to be high enough to compensate the losses caused by the optical fiber and the bandwidth should be large enough to support all the transmitted channels. The purpose is to obtain a spectrum that encloses this rectangle. Bandwidth Ripple Gain Bandwidth Ripple Gain Figure 21. Design concerns for a multi-pumped Raman amplifier for CWDM systems. Another point is to guarantee that the maximum deviation between the values of the designed and the needed gain as smallest as possible. The curved line in figure 21 represents the obtained spectrum after optimization of pumps characteristics. The ripple represents the maximum deviation cited above. Another concern in designing the spectrum is to make it flat, with all the channels at the same level, in order to avoid reception constrains. The multipumped Raman amplifier can be designed using a set of coupled nonlinear equations as equation 4. Solving the coupled equations for one signal and one pump, may be simplified when pump depletion is ignored. This approximation is valid because the pump power is higher than the signal power, P p » P s [66]. However, whenever multiple pumps are used this simplification cannot be used due to the interaction between pumps which enhances the effect of depletion due to the higher powers involved. New Challenges in Raman Amplification for Fiber Communication Systems 77 1350 1400 1450 1500 1550 1600 1650 1700 1750 0 5 10 15 20 G a i n ( d B ) Wavelength (nm) Figure 22. Example of a multi-pumped Raman amplifier applied to CWDM systems. Figure 22 shows an example of an optimized Raman amplifier spectrum applied to CWDM systems. It was designed to transmit five probe channels at 1490 nm, 1510 nm, 1530 nm, 1550 nm and 1570 nm. The transmission link is based on 80 km SMF fiber, with 0.23 dB/km losses, which implies a 18.4 dB gain with a minimum spectral bandwidth of 80 nm. The graph in figure 22 is the result of a forward pumping configuration. The number of pumps used in this example was six. The continuous line represents the gain spectrum obtained with the six pumps the arrows represent the transmitted probe channels. The bandwidth is 100 nm, 20 nm larger than the minimum required bandwidth, in order to guarantee that all the signals are amplified. The gain is around 18.4 dB with a maximum deviation between the designed and needed gain equal to 1 dB, and a maximum gain deviation for each channel being 0.9 dB. The six pumps used are centered at 1380 nm, 1393 nm, 1405 nm 1428 nm, 1444 nm, and 1468 nm with powers of 450 mW, 200 mW, 330 mW, 160 mW, 45 mW, and 55 mW, respectively. The pump of lower wavelength needs the highest power due to the interactions between pumps: The lower wavelength pump loses energy to the higher wavelengths, causing its depletion. This optimization scheme was verified experimentally, with 3 probe channels CDWM system, pumped with 3 pump signals at 1470 nm, 1490 nm and 1510 nm. For the optimization the hybrid GA algorithm, previously presented, was used. This pump allocation problem is less exigent, in terms of ripple, than for a DWDM system, since the probe signal are far apart. The implemented scenario consists of a 40 km SMF fiber, with a counterpropagating pump scheme and a 7 dB gain target. The optimized pump powers were 128.1 mW, 65.0 mW and 146.9 mW, respectively. The maximum gain excursion was 0.002 dB and 0.12 dB for the simulation and experimental systems, respectively. Experimentally, we can observe that the Raman amplification improves the eye opening penalty of a signal transmitted along a fiber link allowing a good reception at the end of the P.S. André, A.N. Pinto, A.L.J. Teixeira et al. 78 transmission path. To illustrate this behavior, the eye diagram of a signal after a 40 km link is shown in figure 23. Figure 23. Comparison between eye diagrams with and without Raman amplification. Figure 23 illustrates a real case where there is a signal centered at 1567 nm and two pumps centered at 1508.8 nm, one in forward configuration and another in backward configuration. The powers of the pumps are chosen to be 100 mW each. The eye openings are given in Volt and the gain was obtained using the on/off definition (equation 10). Using the relation between power and voltage, P=V 2 /R, the on/off gain becomes G Voltage = 10log 10 (V with pump /V without pump ). It is notorious that an increase of the eye opening obtained when both pumps are turned on. The scale is the same to all the graphs in figure 23 to allow comparisons of the eye opening amplitude. The eye opening to the bidirectional configuration is higher due to the higher pump power, while the forward and backward systems have 100 mW, the bidirectional system uses 200 mW. The respective gains of the eye openings are 1.94 dB, 2.35 dB, and 2.98 dB to the forward, backward and bidirectional systems, respectively. The results show a higher gain for the counter propagating situation. 5. Conclusion Raman fiber amplifiers are a technological key component that fulfill the challenging strict requirements of the beginning of this century, enabling applications not feasible with conventional EDFAs. New Challenges in Raman Amplification for Fiber Communication Systems 79 In this contribution, we have discussed the origin of Raman scattering and the critical properties for system design, such as pumping allocation, cascade pump and broadband amplification for multiple CDWM networks. It was also presented solutions that provide that gain, such as the use of low power pumps or incoherent pumps. These issues are, in the authors point of view, the relevant questions and challenges associated with Raman amplification on communication systems. Acknowledgments This work was supported by the POSC program, financed by the European Union FEDER fund and by the Portuguese scientific program. The authors also greatly acknowledge the ARPA (POSI/EEA-CPS/55781/ 2004) and TECLAR (POCI/A072/2005) projects and to FCT and ALBAN scholarship program. References [1] Hu, J.; Marks, B. S.; Menyuk, C. R. J Lightwave Technol. 2004, 22, 1519-1522. [2] Giltrelli, M.; Santagiustina, M. IEEE Photon Technol Lett. 2004, 16, 2454-2456. [3] Cui, S.; Liu, J.; Ma, X. IEEE Photon Technol Lett. 2004, 16, 2451-2453. [4] Kidorf, H.; Rottwitt, K.; Nissov, M.; Ma, M.; Rabarijaona, R. IEEE Photon Technol Lett. 1999, 11, 530-532. [5] Bromage, J. J Lightwave Technol. 2004, 22, 79-93 [6] Namiki, S.; Seo, K.; Tsukiji, N.; Shikii, S. Proceedings IEEE 2006, 94, 1024-1035. [7] Shuto, Y.; Yanagi, S.; Asakawa, S.; Kobayashi, M.; Nagase, R. IEEE Journal Quantum Electronics 2004, 40, 1113-1121. [8] Seo, K.; Nishimura, N.; Shiino, M.; Yuguchi, R.; Sasaki, H. Furukawa Review 2003, 24, 17-22. [9] Bromage, J. J Lightwave Technol. 2004, 22, 79-93. [10] Chen D. Z., Wellbrock G., Peterson D. L., Park S. Y., Thoen E., Burton C., Zyskind J., Penticost S. J., Mamyshev P., ECOC 2006, Cannes. [11] Monroy I. T., Kjaer R., Palsdottir B., Koonen A: M. J., Jeppesen P., ECOC 2006, Cannes. [12] Scheiders, M.; Vorbeck, S.; Leppla, R.; Lach, E.; Schimidt, M.; Papernyi, S.; Sanapi, K, Optical fiber conference 2005, Anaheim, Post-deadline paper. [13] ITU -T G.694.1, 2002. [14] ITU -T G.694.2, 2002. [15] Eichenbaum, B. R.; Das S. K. National Fiber Optic Engineers Conference 2001, Baltimore, USA, 1444-1448. [16] Wang, D.; He, C.; Li, Y. FibreSystems Europe/Lightwave Europe. 2005, 13-15 [17] Desurvire E. Erbium-Doped Fiber Amplifiers - Principles and Applications; John Wiley & Sons, New York, NY: 1994. [18] Emori, Y.; Tanaka, K.; Namiki, S. Electron Lett. 1999, 35, 1355-1356. [19] D. I. Chang, H. K. Lee and K. H. Kim, Electron. Lett. 1999, 35, 1951-1952. P.S. André, A.N. Pinto, A.L.J. Teixeira et al. 80 [20] S.B.Papernyi,V.I.Karpov,andW.R.L.Clements, Optical Fiber Communication (OFC) 2002, Anaheim, USA, PostdeadlinePaperFB4-1. [21] Teixeira, A.; Andre, P.; Stevan, S.; Silveira, T.; Tzanakaki, A.; Tomkos, I. International Conference on Internet and Web Applications and Services/Advanced International Conference AICT-ICIW 2006, on 19-25 Feb. 2006 Page(s):85 - 85 ,Digital Object Identifier 10.1109/AICT-ICIW.2006.158 [22] Papernyi, S.B., Ivanov, V.B. ,Koyano, Y., Yamamoto, H., ”Sixth order cascaded Raman Amplification .- Ivanov”, Optical Fiber Communication Conference, 2005. Technical Digest. OFC/NFOEC [23] Han, B.; Zhang, X. P.; Zhang, G. D.; Lu, Z. G.; Yang, G. X. Opt Express. 2005, 13, 6023-6032. [24] Zhang, T.; Zhang, X.; Zhang, G. IEEE Photon Technol Lett. 2005, 17, 1175-1177. [25] Wen, S. F. Opt Express. 2006, 14, 3752-3762. [26] Wen, S. F. Opt Express. 2006, 15, 45-55. [27] Vakhshoori, D.; Azimi, M.; Chen, P.; Han, B.; Jiang, M.; Knopp, K. J.; Lu, C. C.; Shen, Y.; Vander Rhodes, G.; Vote, S.; Wang, P. D.; Zhu, X. OFC. 2003, 3, PD47-P1-3. [28] X. Zhou, M. Birk and S. Woodward, IEEE Photon. Technol. Lett 2002, vol 14, 1686- 1688. [29] C. V. Raman, K. S. Krishnan, "A New Type of Secondary Radiation", Nature, Vol. 121, pp. 501-502, March, 1928. [30] G. S. Landsberg, L. I. Mandelstam, "Eine neue Erscheinung bei der Lichtzerstreuung in Krystallen", Naturwissenschaften, Vol. 16, pp. 557-558, July, 1928. [31] P. S. André, A. N. Pinto, Chromatic Dispersion Fluctuations in Optical Fibers Due to Temperature and Its Effects in High-Speed Optical Communication Systems, Optics Communications, Vol. 246, Issues 4-6, 15 February 2005, pp. 303-311, 2005; [32] E. J. Woodbury, W. K. Ng, "Ruby Laser Operation in the Near IR", Procedings of IRE (correspondence), Vol. 50, No. 11, pp. 2367, November, 1962. [33] R. W. Boyd, Nonlinear Optics, second edition, San Diego, Academic Press, 2003; [34] C. Headley, G. P. Agrawal, Raman Amplification in Fiber Optical Communication Systems, San Diego, Academic Press, 2005. [35] Karásek M., Kanka J., Honzátko P., Peterka P., Int. J. Numerical Modelling: Electronic Network, Devices and Fields , 2004, vol. 17, nº2, 165-176. [36] Bromage J., J. Lightwave Technol, 2004, vol. 22, nº1, 79-93. [37] André, P., Teixeira, A., Kalinowsky, H., Pinto, J. L. Optica Aplicatta 2003, 33 559 – 573. [38] Liu X., Lee B., Opt. Express, 2003, vol. 11, nº.12, 1452-1461. [39] Neto B.; Stevan S; Teixeira A. T; André P. S.; ICT 2006, Funchal. [40] Han Q., Ning J., Chen Z., Shang L., Fan G., J. Opt A Pure Appl. Opt., 2005, 7, 386-390. [41] Han Q., Ning J., Zhang H., Chen Z., J. Lightwave Technol, 2006, 24, 1946-1952. [42] Min B., Lee W. J., Park N., IEEE Photon. Technol Lett., 2002, 12, 1486-1488. [43] Newbury N. R., , J. Lightwave Technol, 2003, 21, 3364-3373. [44] Mollenauer L. F., Grant A. R., Mamyshev P. V., Opt. Lett., 2001, 592-594. [45] Xiao P. C., Zeng Q. J., Huang J., Liu J. M, IEEE Photon. Technol Lett. 2003, 15, 206-208. [46] Yan M., Chen J., Jiang W, Li J., Chen J., Li X., IEEE Photon. Technol Lett. 2001, 13, 948-950. New Challenges in Raman Amplification for Fiber Communication Systems 81 [47] Zhou X., Lu C., Shum P., Cheng T. H., IEEE Photon. Technol Lett. 2001, 13, 945-947. [48] Cui S, Liu L, Ma X, IEEE Photon. Technol Lett. 2004, 16, 2451-2453. [49] Neto, B., Junior, S., Teixeira, A., André, P. European Conf. on Networks and Optical Communications - NOC 2006, Berlin, Germany. [50] Goldberg D. E., Genetic Algorithms in Search, Optimisation and Machine Learning, Massachusetts: Addison-Wesley co, 1989, pp 28-56 [51] V. E. Perlin and H. G. Winful J. Lightwave Technol. 2002, 20, 250–254. [52] X. Liu, J. Chen, C. Lu, and X. Zhou, Opt. Express 2004, 12, 6053-6066. [53] X. M. Liu, et al., J. Lightwave Technol. 2003, 21, 3446-3455. [54] M. Islam, IEEE Journal of Selected Topics in Quantum Electronics 2002, 8. [55] T. Miyamoto, R. Lindsay, Lightwave Magazine 2003, 1. [56] Kobyakov, A., Gray S. and Vasilyev M. Electronics Letters 2003, 39, 732 – 733. [57] Tsujkawa K., Tajima K., Ohashi, M. J. Lightwave Technol. 2000, 18, 1528-1532. [58] Essambre R.; Winzer P.; Bromage J.; Kim C. H., Photonics Tech. Letters 2002, 14, 914 – 916. [59] Park, J., Kim N. Y., Choi W.; Lee, H., Park N. Photonics Tech. Letters 2004, 16, 1459 – 1461. [60] Teixeira A., Stevan Jr., S. Silveira T.; Nogueira R.; Tosi Beleffi G. M., Forin D., Curti F. European Conf. on Networks and Optical Communications - NOC 2005,London, UK. [61] P. B. Hansen, L. Eskildsen, A. J. Stentz, T. A. Strasser, J. Judkins, J. J. DeMarco, R. Pedrazzani, and D. J. DiGiovanni, IEEE Photon. Technol. Lett. 1998, 10, 159-161. [62] Faralli, S. Di Pasquale, F., IEEE Photonics Technology Letters 2003, 15, 804- 806. [63] S. Stevan Jr., A. Teixeira, T. Silveira, P. André, G. M. Tosi Beleffi, A. Reale and A. Pohl, Double shifted Raman amplification by means of spontaneous Rayleigh Backsattering lasing control, ITC 2006. [64] S. Stevan Jr. , A. Teixeira, P. André, G. M. Tosi Beleffi, A. Pohl, simulation of Raman amplification and Rayleigh Scattering laser using the transference matrix method, MTPT 2006, Leiria, Portugal. [65] André, P., A. N. Pinto, Teixeira, A.T., Neto, B., Junior, S., Spertti, D., Rocha, F., Bernardo, M., Fujiwara, M., Rocha, A., Facão, M. ICTON 2007, Rome, Italy. [66] Agrawal, G. P. Nonlinear Fiber Optics, 3rd ed. San Diego: Academic Press, 2001. In: Optical Fibers Research Advances ISBN: 1-60021-866-0 Editor: Jurgen C. Schlesinger, pp. 83-117 © 2007 Nova Science Publishers, Inc. Chapter 3 FIBER BRAGG GRATINGS IN HIGH BIREFRINGENCE OPTICAL FIBERS Rogério N. Nogueira, Ilda Abe and Hypolito J. Kalinowski Instituto de Telecomunicacoes, polo de Aveiro, Aveiro, Portugal Abstract Fiber Bragg gratings (FBG) are a key element in optical communication devices and in fiber sensors. This is mainly due to its intrinsic characteristics, which include low insertion loss, passive operation and immunity to electromagnetic interferences. Basically a FBG is a periodic modulation of the core refractive index formed by exposure of a photosensitive fiber to a spatial pattern of ultraviolet light in the region of 244–248 nm. The lengths of FBGs are normally within the region of 1–20 mm. Usually a FBG operates as a narrow reflection filter, where the central wavelength is directly proportional to the periodicity of the spatial modulation and to the effective refractive index of the fiber. The production technology of these devices is now in a mature state, which enables the design of gratings with custom-made transfer functions, crucial for all-optical processing. Recently, some work has been done in the application of FBG written in highly birefringent fibers (HiBi). Due to the birefringence, the effective refractive index of the fiber will be different for the two transversal modes of propagation. Therefore, the reflection spectrum of a FBG will be different for each polarization. This unique property can be used for advanced optical processing or advanced fiber sensing. The chapter will describe in detail this unique device. The chapter will also analyze the device and demonstrate different applications that take advantage of its properties, like multiparameter sensors, devices for optical communications or in the optimization of certain architectures in optics communications systems. 1. Introduction The development of the fiber optical technology was an important step in the revolution of global communications and in information technology. One of these developments happened Rogério N. Nogueira, Ilda Abe and Hypolito J. Kalinowski 84 in the 70’s with the first optical fibers with low attenuation [1], a feature that enabled long- distance communication with high bandwidth. The intrinsic optical bandwidth of the optical fibers has also allowed the propagation of different simultaneous channels, allowing the transmission of data at Tbit/s rates [2]. In these systems, in addition to transmission and amplification, it is often necessary to do all-optical processing to the signal. This is due to the inherent advantages of the optical processing, relative to the optic-electric-optic processing, like the higher flexibility to operate at different bit rates and modulation formats and also at the higher bandwidth. The evolution of the fiber optical technology has also enabled the development of devices for all optical processing. In this way, the insertion loss is reduced and the processing quality improved. One of the factors contributing to all-fiber optical processing devices was the discovery of the photosensitivity in optical fibers. It was documented for the first time in 1978 by Hill et al. [3] and led to the development of fiber Bragg gratings (FBG). A FBG is, generally speaking, a periodic perturbation, along the longitudinal axis, of the refractive index in the fiber core. The production of the refractive index perturbation is done optically in a photosensitive fiber. With the current techniques, it is possible to produce fiber Bragg gratings with different optical properties, which can be designed according to the desired optical processing. In addition to the high flexibility in the production of gratings with custom amplitude and phase responses, the compatibility with common transmission fiber also reduces the insertion loss and decreases the production costs. The application in optical sensors is also a large potential market for FBG. Their intrinsic low immunity to electromagnetic interference, high dynamic range, passive operation, resistance to corrosion and the possibility of multiplexing hundreds of sensors have made FBG a quite interesting sensor for different applications including medicine, civil, aeronautics or biomechanics. Their properties enable the measurement of temperature and also deformation with extremely high resolution. Nevertheless, it can also be used to measure other parameters using indirect measurements [4-7]. The high potential of these devices has also induced the creation of several companies dedicated to the production and installation of fiber sensors. There are already good references for the study of FBGs [8,9]. The purpose of this chapter is not to study in detail these devices, but to describe a special case when a fiber Bragg grating is written in high birefringence fibers (HiBi FBG). These special gratings have unique polarization properties that give them exclusive capabilities for optical communications. This is due to the possibility of applying a different optical processing for different polarization components of the signal being transmitted. HiBi FBGs are also quite interesting for multiparameter sensors, due to their response to temperature variations and deformation. Sensors capable of measuring simultaneously several physical parameters have increased in importance in today’s technological world. In particular, there are various applications of such sensors in civil, mechanical, biomedical or aeronautical engineering, where measurements of different parameters are required [10]. Engineering structures are an example of an application area for the multiparameters sensors, where strain sensing can lead to better understanding about their lifetime and failure. Such knowledge can be critical for some applications like smart skins for airplanes and aeronautical vehicles. Fiber Bragg Gratings in High Birefringence Optical Fibers 85 2. Fiber Bragg Gratings A FBG is an optical device produced within the core of a standard optical fiber (figure 1). Basically, it is a periodic modulation of the core refractive index formed by exposure of a photosensitive fiber to a spatial pattern of ultraviolet light. The length of a FBG is dependent on its application, but it generally varies between a few millimeters to a few centimeters. Fiber Bragg grating Λ Fiber Figure 1. Scheme of a Fiber Bragg grating written in an optical fiber. The periodic modulation of the refraction index generates a resonant condition at the Bragg’s wavelength (λ B ) which is given by the Bragg’s condition: 2 B eff n λ = Λ (1) where n eff is the effective refraction index of the fiber and Λ is the modulation period. Therefore, when a FBG is illuminated by a broadband source, a spectral band centered at λ B will be reflected back. The reflection function can be determined using the coupled mode theory [11-14], since it is difficult to determine analytically. The exception is the uniform FBG, where it is possible to calculate the reflectivity in an analytical way. Considering a uniform periodic modulation of the refractive index, with amplitude Δn, the reflection coefficient of the grating can be given by ( ) ( ) ( ) ( ) sinh sinh cosh L L i L κ ϕ ρ λ δ ϕ ϕ ϕ − = + (2) where L is the length of the FBG, the propagation constant mismatch, δ , is given by 2 eff n π π δ λ = − Λ , (3) 2 2 ϕ κ δ = − , and κ is the coupling constant given by Rogério N. Nogueira, Ilda Abe and Hypolito J. Kalinowski 86 n π κ η λ Δ = (4) where η is the overlap integral and can be approximated as η≈1 for single mode fibers with step index. The reflectivity is given by ( ) ( ) 2 2 2 2 2 sinh cosh L R L ϕ ρ δ ϕ κ = = − (5) and the phase by ( ) ( ) Im arctan Re R ρ φ ρ ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦ (6) Figure 2 shows the calculated reflectivity and the phase of a uniform FBG with L =5 mm and Δn = 2x10 -4 as given by the above equations. -1 -0.5 0 0.5 1 0 0.2 0.4 0.6 0.8 1 -1 -0.5 0 0.5 1 -4 -2 0 2 4 P h a s e ( r a d ) Δλ [nm] Δλ [nm] R e f l e c t i v i t y Figure 2. Reflectivity and phase of a uniform FBG. Parameters: L=mm and Δn= 2x10 -4 . If the period changes linearly with the length of the grating, the FBG is said to have a linear chirp. Figure 3 shows the simulation of the reflectivity and group delay of a linear chirped FBG. The simulation method is based on the coupled mode theory. Fiber Bragg Gratings in High Birefringence Optical Fibers 87 z 1546 1547 1548 1549 1550 1551 1552 1553 1554 -20 -18 -16 -14 -12 -10 -8 -6 -4 -2 0 Wavelength [nm] R e f l e c t i v i t y ( d B ) -50 0 50 100 150 200 250 300 G r o u p d e l a y [ p s ] Figure 3. Reflectivity and group delay of a linear chirped FBG. 3. High Birefringence Fibers In an ideal monomode fiber, with a perfect cylindrical core, and with uniform diameter, the fundamental propagation mode is a degenerated combination of two orthogonal propagation modes. However, in real fibers, that degeneration does not exist. In fact, small variations of diameter in the fiber’s core generate a birefringence in the optical fiber. The birefringence can also be a result of an anisotropic stress in the fiber. The local birefringence, B, in each position of the fiber, is defined as ( ) x y f x y B n n C σ σ = − = − (7) where x n and y n are the mean refractive index of the orthogonal polarization modes, σ x and σ y are the main stress on the polarization axes and C f is the photoelastic constant of the fiber. In monomode silica fibers C f is around 3.08 x10 -6 mm 2 /N for wavelengths near 1500 nm, while B is typically B ≈ 10 -7 . Due to this small birefringence value, the two polarization components of the light propagating in the fiber have a propagation velocity very similar. Therefore, small environmental perturbations will lead to an energy coupling between one polarization to another. As a result, a linearly polarized light will rapidly evolve to a random polarization. This situation can be avoided with high birefringence fibers. In these fibers, the core has an anisotropic stress, which is generated due to the geometric properties of the fiber. Due to the photoelastic effect, the stress induces a birefringence in the core. Typical values are B ≈ 10 -4 [15]. Due to the high birefringence, the propagation constant is different for the two orthogonal propagation modes, which means that the coupling between both transversal propagation modes is far lower as compared to standard fibers. Therefore, the higher the birefringence, the easier will be for a linearly polarized light, propagating in one of the Rogério N. Nogueira, Ilda Abe and Hypolito J. Kalinowski 88 orthogonal modes, to maintain its state of polarization. Due to this feature, HiBi fibers are also known as polarization maintaining fibers. Figure 4 shows the main structure of the most common HiBi fibers. Fast axis (Y) Slow axis (X) air PANDA IEC Bow-Tie Elliptical Core Side-Hole D-Shaped Elliptical Core Figure 4. Schematic of the transversal section of some of the most known HiBi fibers. The PANDA (Polarization-maintaining AND Attenuation-reducing), IEC (Internal Elliptical Cladding) and Bow tie fibers have anisotropic glass structures around the core, with a Poisson coefficient different from the rest of the fiber. These structures create the anisotropic stress in the core, which produces the birefringence. The Side-Hole, the Elliptical Core and the D-Shaped Elliptical Core fibers have an elliptical core to generate the birefringence, aided by two air structures, in the case of the Side-Hole or by the shape of the cladding, in the case of the D-Shaped elliptical core. The main axes of the HiBi fibers are designated as fast axis (Y) for the lower refraction index and slow axis (X) for the higher refraction index. Coherence Length If a linearly polarized light propagates in a monomode fiber, with a polarization angle of 45º, relatively to the main axes of the fiber, both orthogonal polarization modes will be excited with equal power. If the fiber has a constant birefringence, the mismatch, Ф HB (z), between the Fiber Bragg Gratings in High Birefringence Optical Fibers 89 orthogonal polarization components will change as a function of the propagation distance on the fiber, z, and it’s given by ( ) ( ) HB x y z z β β Φ = − (8) where β x and β y are the propagation constants in the X and Y axes respectively. The mismatch will change periodically with the fiber, leading to a change in the state of polarization from linear to elliptical and back again to linear (figure 5). ? =0 ? =π/2 ? =π ? =3π/2 ? =2π Figure 5. Evolution of the state of polarization in a birefringence fiber. The spatial periodicity of the evolution of the state of polarization is designated as coherence length (L B ). It is determined by the birefringence of the fiber and can be expressed as / B L B λ = (9) where λ is the operating wavelength. Typical coherence lengths for HiBi fibers are in the millimeter scale [16]. 4. Fiber Bragg Gratings Written in HiBi Fibers HiBi fibers can have two linear polarization modes with refractive indexes n x and n y for the slow and fast modes respectively. When a FBG is written in one of these fibers, the periodic modulation will be the same for the two orthogonal polarization modes; however since the effective refraction index is different for the two polarizations, the Bragg wavelength will also be different for each mode. Consequently, expression (1) can be rewritten for the two orthogonal modes: 2 , X,Y i i n i λ = Λ = (10) where λ i are the Bragg wavelengths for each polarization mode. Rogério N. Nogueira, Ilda Abe and Hypolito J. Kalinowski 90 The wavelength difference between the two reflection peaks, Δλ HB , can be calculated by 2 2 HB x y x y n n λ λ λ Δ = − = Λ− Λ (11) The reflectivity of a HiBi FBG will be given by the linear sum of the reflectivity of the two polarization components, i.e. R(λ)=R x (λ)+R y (λ). R x and R y are the reflectivity for each polarization given by ( ) ( ) 2 2 2 2 2 sinh cosh i i i i L R L ϕ ρ δ ϕ κ = = − , i=x,y (12) where 2 i i n π π δ λ = − Λ , i=x,y (13) and 2 2 i i ϕ κ δ = − , i=x,y (14) Figure 6 shows a simulation, using the previous model, for the reflectivity of a HiBi FBG with birefringence of B = 3.2 × 10 -4 . 1547.5 1548.0 1548.5 1549.0 0.0 0.2 0.4 0.6 0.8 1.0 Polarization x R e f l e c t i v i t y Wavelength [nm] Polarization y Figure 6. Reflectivity of a simulated HiBi FBG. Simulation parameters: B=3.2 × 10 -4 , Λ=535 nm, L=10 mm. Fiber Bragg Gratings in High Birefringence Optical Fibers 91 If the HiBi FBG is illuminated with light having the two orthogonal components, the reflection spectrum will have those two peaks at orthogonal polarizations. This feature can be very important in some applications, namely in optical communications, as it will be confirmed further in this chapter. The production of HiBi FBGs uses the same techniques as the ones used in regular FBGs. The only difference will be in the utilization of photosensitive HiBi fiber. Generally it is used a hydrogenated HiBi fiber. Table 1 shows the dimensions of the anisotropic glass structures around the core of some HiBi fibers obtained through the photographs of the transverse section. The table also displays the main characteristics of HiBi fibers obtained from the manufacturers data sheet. Table 1. Characteristics of different HiBi fibers. The structures of the HiBi fibers were obtained by microphotography. Fiber type Commercial provider Wavelength (nm) Core diameter (μm) Cladding diameter (μm) Intrinsic stress-applying region IEC (FS- PM-6621) 3M 1300 8 125 Ellipse: Major axis: 75 μm; Minor axis: 30 μm Bow tie (F- SPPC-15) Newport 1550 8 125 From core center to extremity of bow tie lobe: 18.4 μm Bow tie (HB- 1500G) Fibercore 1550 8 80 From core center to extremity of bow tie lobe: 16.5 μm PANDA (SM-13-P-7) Fujikura 1300 8 125 From core center to opposite extremity of side cylinder: 41 μm; Diameter of side cylinder: 32 μm The reflection spectra for gratings written in the above fibers are shown on figure 7, where the plots of the best-fitted bands are also presented [19]. All the gratings were produced with the phase mask technique. The estimated length of the grating is 10 mm. From these spectra it can be seen the effect of the intrinsic birefringence of the HiBi fibers. The IEC fiber has the higher birefringence, corresponding to larger spectral splitting between both polarizations bands, while the bow tie fiber presents the lowest birefringence. 1546.0 1546.5 1547.0 1547.5 1548.0 0.0 0.2 0.4 0.6 0.8 1.0 I N O R M A L I Z E D I N T E N S I T Y WAVELENGTH (nm) Figure 7. Continued on next page. Rogério N. Nogueira, Ilda Abe and Hypolito J. Kalinowski 92 1548.0 1548.5 1549.0 1549.5 0.0 0.2 0.4 0.6 0.8 1.0 I N O R M A L I Z E D I N T E N S I T Y WAVELENGTH (nm) 1548.0 1548.5 1549.0 1549.5 0.0 0.2 0.4 0.6 0.8 1.0 I N O R M A L I Z E D I N T E N S I T Y WAVELENGTH (nm) Figure 7. Reflection spectra of Bragg gratings written in different HiBi fibers: IEC ( ∇), Panda ( Ο) and bow tie ( Δ). The continuous line represents the simulated best fit. Table 2 shows the best-fit parameters obtained in the simulation process. From the fit it is also possible to obtain the values of birefringence of the HiBi fibers. Table 2. Parameters of FBGs written in HiBi fibers obtained for the best fit for the experimental data. HiBi Fiber Bands λ (nm) n eff kL Λ (nm) B IEC (FS-PM-6621) λ Y λ X 1546.57 1547.29 1.44539 1.44606 1.7212 1.7196 535 6.7 x 10 -4 @ 1550 nm PANDA (15P8) λ Y λ X 1548.39 1548.82 1.44709 1.44750 1.7172 1.7162 535 4.1 x 10 -4 @ 1550 nm Bow tie (SPPC-15) λ Y λ X 1548.61 1548.95 1.44730 1.44762 1.7167 1.7159 535 3.2 x 10 -4 @ 1550 nm The Bragg wavelength peaks of the optical spectrum for both polarizations can change with temperature and strain. Therefore, considering a HiBi FBG under a temperature variation of ΔT and under a strain aligned with the main axes of the fiber Δε X , Δε Y and Δε Z , the Fiber Bragg Gratings in High Birefringence Optical Fibers 93 resultant wavelength shift, Δλ x and Δλ y of both wavelength peaks, λ x and λ y , can be expressed as ( ) T n T n )] ( p p [ 2 n X Y Z 12 X 11 2 X Z X X Δ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ∂ ∂ + α + ε Δ + ε Δ + ε Δ − ε Δ = λ λ Δ (15) ( ) T n T n )] ( p p [ 2 n Y X Z 12 Y 11 2 Y Z Y Y Δ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ∂ ∂ + α + ε Δ + ε Δ + ε Δ − ε Δ = λ λ Δ (16) where p 11 and p 12 are the components of the photoelastic tensor and α is the thermal expansion coefficient of the fiber, α= 0.55×10 -6 K -1 [17]. For a fiber based on germanium and silica, p 11 =0.113, p 12 =0.252 and the thermo-optic coefficient is ( ) n T n ∂ ∂ =8.6×10 -6 [18]. Figure 8 shows schematically the effect on the reflection spectrum of a HiBi FBG when it is under temperature variations, under transversal strain or longitudinal strain. The effect of temperature variations or longitudinal strain in the reflection spectrum is equivalent to a translation in the wavelength. On the other hand, when under a transversal strain, the peak separation will change. This difference can be used in multiparameter sensors as it will be discussed further in this chapter. Wavelength P o w e r Wavelength P o w e r Increasing longitudinal stress and/or temperature Increasing transversal strain Figure 8. Evolution of the reflection spectrum of a HiBi FBG when under a longitudinal stress, temperature variation or transversal strain. Rogério N. Nogueira, Ilda Abe and Hypolito J. Kalinowski 94 4.1. Characterization of Bragg Gratins Written in High-Birefringence Fiber Optics 4.1.1. Transverse Strain The sensitivity of HiBi FBG to transversal strain can be characterized using a mechanical set- up, like the one shown in figure 9. The transversal load is applied using a micro scratch mechanical system. The system uses an arm to apply a load with a precision of 0.1 N. A grating written in HiBi fiber was placed between two plates having a length of 13 mm. The apparatus arm applies the load to the upper plate. The transverse loads were made for several orientations of the birefringence axis with respect to the direction of the applied load through two fiber rotators. Figure 9 also shows the optical system used to analyze the FBG reflection spectrum. Optical spectra were recorded using an amplified spontaneous emission (ASE) of an erbium doped fiber amplifier as light source, an optical circulator and conventional optical spectrum analyzer (OSA). Circulator Figure 9. Set-up for the characterization of HiBi FBGs under a transversal load. The detail shows the transverse section of a HiBi fiber oriented along the angle ϕ.when subjected to applied force F. ASE: Broadband optical source (amplified spontaneous emission); OSA: optical spectrum amplifier. Figure 10 (a), (b) and (c) shows an example of the reflection spectra of a FBG written in a IEC HiBi fiber as a function of an applied load of 0°, 45° and 90°, respectively. The results show that, if a load is applied to one of the main axes, fast or slow, it leads to a change in the wavelength of the spectral band associated with the orthogonal axis, while the band associated with the correspondent axis will show a smaller variation. For the applied load angle of 45º both polarization bands present similar evolution. The figure also shows, for different applied load angles (ϕ), the evolution of the peak wavelength of each reflection band with the transverse strain applied to the sample. The strain calibration points in the spectra deformed areas were obtained by identifying and measuring local maximum, minimum and inflexion points. The band split that occurs in some of the spectra is due to a phase shift induced by the applied load. The resulting complex structure is known to be responsible for spectral changes of FBG subject to mechanical stress [9]. Fiber Bragg Gratings in High Birefringence Optical Fibers 95 F X Y 1546.0 1546.5 1547.0 1547.5 1548.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 WAVELENGTH (nm) L O A D ( N /m m ) N O R M A L I Z E D I N T E N S IT Y -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 1546.4 1546.6 1546.8 1547.0 1547.2 1547.4 1547.6 1547.8 Y X W A V E L E N G T H ( n m ) LOAD (N/mm) (a) -0.1 0.0 0. 1 0.2 0. 3 0.4 0.5 0.6 0.7 1546.4 1546.6 1546.8 1547.0 1547.2 1547.4 1547.6 1547.8 Y X W A V E L E N G T H ( n m ) LOAD (N/mm) 1546.0 1546.5 1547.0 1547.5 1548.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 L O A D ( N / m m ) WAVELENGTH (nm) N O R M A L I Z E D I N T E N S I T Y F Y X (b) -0.1 0. 0 0.1 0.2 0.3 0. 4 0.5 0.6 0.7 1546.4 1546.6 1546.8 1547.0 1547.2 1547.4 1547.6 1547.8 Y X W A V E L E N G T H ( n m ) LOAD (N/ mm) 1546.0 1546.5 1547.0 1547.5 1548.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 WAVELENGTH (nm) L O A D ( N / m m ) N O R M A L IZ E D I N T E N S I T Y F Y X (c) Figure 10. Left: Changes in the spectral response of a FBG written in an IEC HiBi fiber when subjected to an applied load oriented along the angle ϕ: (a) 0° (X-axis); (b) 45º and (c) 90° (Y-axis). Right: Peak position of each band as a function of the applied load. The lines represent the linear best fit for the experimental data. -90 -60 -30 0 30 60 90 -0.2 0.0 0.2 0.4 0.6 λ Y λ X W A V E L E N G T H S E N S I T I V I T Y ( n m / N / m m ) ϕ (degrees) Figure 11. Curves of peak sensitivities of the FBG in IEC HiBi fiber as a function of the applied load angle. Rogério N. Nogueira, Ilda Abe and Hypolito J. Kalinowski 96 Figure 11 shows the wavelength sensitivity curves obtained for both polarization bands. The graph also displays the periodic evolution of the bands as a function of the applied load angle. Identifying and measuring the reflection peaks as a function of the applied load can be used to obtain the calibration line for each polarization band. The respective slopes can be evaluated and, from them, the dependence of the Bragg wavelength position with the strain can be obtained. Table 3 shows some measurements obtained with FBGs written in IEC and PANDA fibers. Table 3. Slopes and strain sensitivities of FBGs written in IEC and PANDA HiBi fibers as a function of the direction of applied load (module values). Both fibers have a diameter of 125 μm. X- polarization band Y- polarization band HiBi Fiber Angle of applied load Slope (nm/N/mm) Strain sensitivity (pm/με) Slope (nm/N/mm) Strain sensitivity (pm/με) ϕ = 90° 0.51 7.02 0.07 1.02 IEC ϕ = 0° 0.02 0.29 0.11 1.55 ϕ = 90° 0.46 3.78 0.02 0.24 PANDA ϕ = 0° 0.01 0.11 0.13 2.80 4.1.2. Longitudinal Strain The Bragg wavelength dependence with the longitudinal strain can be measured by gluing one extremity of the fiber in a holder, while the other is glued to a translation stage, which applies a known deformation using a calibrated micrometer. Figure 12 shows the reflection spectra of a FBG and peak position of each band, written in an IEC fiber as a function of longitudinal strain. 1545.0 1545.5 1546.0 1546.5 1547.0 0.0 0.2 0.4 0.6 0.8 1.0 0 83.7 167.5 251.2 334.9 418.7 L O N G I T U D I N A L S T R A I N ( μ ε ) WAVELENGTH (nm) N O R M A L I Z E D I N T E N S I T Y 0 100 200 300 400 1545.2 1545.6 1546.0 1546.4 1546.8 λ Y λ X . . . W A V E L E N G T H ( n m ) LONGITUDINAL STRAIN (με) Figure 12. Left: Changes in the spectral response of a FBG written in IEC HiBi fiber when subjected to a longitudinal strain. Right: Peak position of each band as a function of the longitudinal strain. The lines represent the linear best fit for the experimental data. Fiber Bragg Gratings in High Birefringence Optical Fibers 97 Both bands show the same behavior, which is an increase of peak wavelengths as the strain increases. The slopes and the Bragg wavelength sensitivity to longitudinal strain are given in table 4. The obtained ratios between strain and applied load were 758 με/N (X-axis) and 755 με/N (Y-axis). Table 4. Slopes and longitudinal strain sensitivity of a FBG written in an IEC HiBi fiber. Bands Slope (nm/N) Longitudinal strain sensitivity (pm/με) X – polarization 1.44 1.9 Y - polarization 1.51 2.0 4.1.3. Temperature The temperature dependence of the reflection bands of HiBi FBGs can be characterized using a cooling/heating system. Figure 13 shows the evolution of the reflection bands and peak position of each band of a Bragg grating written in an IEC fiber as a function of temperature. 1546.0 1546.5 1547.0 1547.5 1548.0 0.0 0.2 0.4 0.6 0.8 1.0 15 20 25 30 35 40 45 50 55 T E M P E R A T U R E ( º C ) WAVELENGTH (nm) N O R M A L I Z E D I N T E N S I T Y 10 20 30 40 50 60 1546.5 1546.8 1547.1 1547.4 λ Y λ X W A V E L E N G T H ( n m ) TEMPERATURE (ºC) Figure 13. Left: Changes in the spectral response of a FBG written in an IEC HiBi fiber when subjected to different temperatures. Right: Peak position of each polarization band as a function of the temperature. The lines represent the linear best fit for the experimental data. Table 5 shows the temperature sensitivity values for a FBG written in IEC, PANDA and bow tie HiBi fibers. Table 5. Slopes of temperature for FBGs written in HiBi fibers. Slope (pm/°C) HiBi fiber X – polarization band Y – polarization band IEC 125 μm 6.76 6.71 PANDA 125 μm 3.28 3.40 Bow Tie 125 μm 10.93 11.12 Bow Tie 80 μm 8.02 8.46 The results show that there are quite large variations between the sensitiveness to temperature for different HiBi fibers. The values changed between ~ 3 pm/°C for the PANDA fiber to ~ 11 pm/°C for the IEC fiber. There are also differences in the Rogério N. Nogueira, Ilda Abe and Hypolito J. Kalinowski 98 coefficients between polarization bands of the same fiber. For example, in the PANDA fiber this difference is 0.24 pm/°C. These results can be used for simultaneous measurements of temperature and longitudinal strain with only one FBG in a HiBi fiber. This approach, along with others, will be described in the next section. 5. Application in Multiparameter Sensors FBG sensors are generally based on a unique grating written in a standard fiber optic. The wavelength shift in the reflection spectrum may be used to measure a single component of strain or temperature variation, but not both simultaneously. An adequate measurement of both temperature and strain requires a suitable sensor with a differential sensitivity between parameters. HiBi FBGs can be used as sensors to simultaneously measure one component of transverse strain, temperature and/or longitudinal strain. As it was shown previously in this chapter there are differences in the calibration coefficients of both polarization bands, which can be used to simultaneous measure the temperature and longitudinal strain with only one HiBi FBG. Since the variations of temperature or longitudinal strain causes both bands to shift, and the variation of strain causes asymmetric spectral response in the polarization bands depending of the direction of the applied load, allows the FBG in the HiBi fiber to measure simultaneously transverse strain and temperature or transverse strain and longitudinal strain. Several types of optical sensors using FBG written in HiBi fibers, which simultaneously measure longitudinal strain and temperature have been proposed and demonstrated [20-24]. Some of the methods include the recording by a CCD camera of the LP 01 and LP 11 spatial modes [22], using a HiBi FBG partially exposed to chemical etching [20] or by using a quasi- rectangular HiBi fiber to increase the birefringence [21]. In those works, only the longitudinal strain component was measured in simultaneous with temperature. But, there are many applications where it is desirable to determine the transverse strain components in addition to longitudinal strain. Several techniques based in HiBi FBG have already been reported for transverse strain sensing [19, 25-29]. However, when a transverse strain is applied to a HiBi FBG, depending of the fiber orientation relatively to the applied load, the separation of the two Bragg wavelengths can be quite low, so it becomes impossible to resolve the two peaks. To overcome this problem, it can be used an interrogation system capable of detecting independently and simultaneously the two orthogonally polarized signals reflected from the HiBi FBG [26]. There are many applications where it is necessary an ultra small sensor to measure simultaneously components of transverse strain, longitudinal strain and temperature. The use of two superimposed Bragg gratings in HiBi fiber have been described in the literature like potential sensors for monitoring four parameters, two components of transverse strain, longitudinal strain and temperature. [30-34]. 5.1. Simultaneous Measurement of Transverse Strain and Temperature The change in the Bragg wavelength of a HiBi FBG, for each polarization, due to a temperature change ΔT and a transversal strain Δε, is given by Fiber Bragg Gratings in High Birefringence Optical Fibers 99 X X X T T λ λ λ ε ε ∂ ∂ Δ = Δ + Δ ∂ ∂ (17) Y Y Y T T λ λ λ ε ε ∂ ∂ Δ = Δ + Δ ∂ ∂ (18) were ∂λ X /∂T and ∂λ Y /∂T are the temperature coefficients and ∂λ X /∂ε and ∂λ Y /∂ε are the transverse deformation coefficients. Expressions (17) and (18) can be rearranged and written in matrix form in order to calculate the transverse strain and temperature, given the measured wavelength shifts for each polarization band: 1 X Y T λ ε λ − Δ Δ ⎡ ⎤ ⎡ ⎤ = Κ ⎢ ⎥ ⎢ ⎥ Δ Δ ⎣ ⎦ ⎣ ⎦ (19) where K is a matrix given by ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ = ε λ λ ε λ λ Y Y X X , T , T K (20) A simultaneous measurement of transverse strain and temperature can be obtained by determining the coefficients of K, which are determined with previous characterization. Two examples of simultaneous measurement of these parameters are shown in the table 6 for an IEC fiber and table 7 for a PANDA fiber. The results were obtained using the values of the Bragg wavelength changes for both polarizations bands. Table 6. Simultaneous measurements of temperature and transverse strain using a FBG written in a IEC HiBi fiber. The set values were determined by the experimental system equipment. Direction of applied load: 0º. [19]. Set values 12 °C 23 °C 31 °C 46 °C 11.8 °C 24.6 °C 31.5 °C 45.5 °C 61 µε 65 µε 71 µε 70 µε 75 µε 12.0 °C 25.4 °C 32.6 °C 46.3 °C 76 µε 69 µε 79 µε 73 µε 81 µε 12.7 °C 26.8 °C 34.0 °C 48.0 °C 91 µε 76 µε 83 µε 78 µε 79 µε Rogério N. Nogueira, Ilda Abe and Hypolito J. Kalinowski 100 Table 7. Simultaneous measurements of temperature and transverse strain using a FBG in written in a PANDA HiBi fiber. The set values were determined by the experimental system equipment. Direction of applied load: 90º. Set values 7 °C 21 °C 22 °C 40 °C 53 °C 7 °C 20 °C 23 °C 39 °C 51 °C 11 µε 12 µε 12 µε 9 µε 11 µε 10 µε 7 °C 19 °C 22 °C 38 °C 51 °C 21 µε 21 µε 20 µε 17 µε 16 µε 18 µε 6 °C 18 °C 22 °C 38 °C 53 °C 31 µε 32 µε 31 µε 31 µε 22 µε 24 µε 5 °C 18 °C 21 °C 37 °C 55 °C 41 µε 39 µε 44 µε 44 µε 37 µε 39 µε 5 °C 18 °C 21 °C 37 °C 55 °C 51 µε 43 µε 42 µε 48 µε 41 µε 43 µε 5.2. Simultaneous Measurement of Transverse Strain and Longitudinal Strain For the measurement of the longitudinal (Δε Z ) and transverse (Δε X or Δε Y ) strain, the equations can also be written in matrix form, given the measured wavelength shifts for each polarization band: 1 , Z X X Y Y ε λ ε λ − Δ Δ ⎡ ⎤ ⎡ ⎤ = Κ ⎢ ⎥ ⎢ ⎥ Δ Δ ⎣ ⎦ ⎣ ⎦ (21) where K is now given by: , , , , X X Z X Y Y Y Z X Y K λ λ ε ε λ λ ε ε ∂ ∂ ⎡ ⎤ ⎢ ⎥ ∂ ∂ ⎢ ⎥ = ⎢ ⎥ ∂ ∂ ⎢ ⎥ ∂ ∂ ⎣ ⎦ (22) Table 8 shows an example of simultaneous measurements of longitudinal and transversal strain obtained using the wavelength changes of both polarizations bands. Fiber Bragg Gratings in High Birefringence Optical Fibers 101 Table 8. Simultaneous measurements of longitudinal and transverse strain using an FBG written in a IEC HiBi fiber. The set values were determined by the experimental system equipment. Direction of applied load: 90º. Set values 0 µε 9 µε 14 µε 1 µε 8 µε 13 µε 83 µε 64 µε 66 µε 68 µε 1 µε 8 µε 13 µε 167 µε 160 µε 161 µε 154 µε 0 µε 7 µε 12 µε 251 µε 240 µε 246 µε 244 µε 1 µε 8 µε 13 µε 335 µε 320 µε 316 µε 313 µε 5.3. Simultaneous Measurement of Transverse Strain, Longitudinal Strain and Temperature Two superimposed Bragg gratings can be written in high birefringence fiber optics to measure simultaneously temperature, transverse and longitudinal strain. This section demonstrates the use of a pair of Bragg gratings written in high birefringence fiber optics to measure, simultaneously, three physical parameters [31]. The Bragg gratings are superimposed in the same position of the fiber optic, in order to behave as a single sensor with reduced dimension. 5.3.1. Superimposed Bragg Gratings 1534 1536 1546 1548 0 8 16 24 32 X 2 X 1 Y 2 Y 1 I N T E N S I T Y ( n W ) WAVELENGTH (nm) Figure 14. Optical reflection spectrum of two superimposed Bragg gratings written in HiBi IEC fiber [31]. Rogério N. Nogueira, Ilda Abe and Hypolito J. Kalinowski 102 Figure 14 shows an optical reflection spectrum of two gratings recorded at the same fiber position. The two FBG were written with different periods in an IEC HiBi optical fiber with 125 µm diameter. The figure shows the polarization bands (Y-polarization and X- polarization) of each pair. Their relative intensity is not the same as the optical source was not flat along the full wavelength range. The superimposed HiBi FBGs were characterized by longitudinal, transversal strain and temperature. The measurements of transversal load were made with the fiber oriented with the fast or slow birefringence axis in the direction of the applied load. Figure 15 shows the dependence of the peak position of each reflection band against the transversal strain applied to the sample (load applied along the Y-axis direction). The best- fitted lines are not parallel; their slopes are different depending on the polarization band. This asymmetric behavior can be used to distinguish the effects of longitudinal and transversal strain acting upon the grating pair. 0 20 40 60 80 1534 1535 1536 1546 1547 1548 W A V E L E N G T H ( n m ) TRANSVERSE STRAIN (με) Figure 15. Dependence of the peak wavelength on transverse strain for the reflection bands [X (V) and Y (U)] of the two superimposed FBGs written in an IEC HiBi fiber. Direction of applied load: 90º. The lines represent the linear best fit to the experimental data [31]. The behavior of the reflection bands, when the sensor is under longitudinal strain, is the same for both gratings. The temperature dependence of the reflection bands of the both FBGs has also approximately the same behavior, which is an increase in the wavelength with an increase of temperature. Table 9. Slopes of temperature, longitudinal and transverse strain for the two superimposed FBGs in an IEC HiBi fiber. Direction of applied transverse load: Y-axis [31]. Polarization bands Slopes Y 1 X 1 Y 2 X 2 ∂λ/∂T (pm/°C) 8.4 7.8 7.8 7.5 ∂λ/∂ε Y (pm/με) 0.08 4.02 0.19 4.11 ∂λ/∂ε Z (pm/με) 1.3 1.39 1.39 1.36 Fiber Bragg Gratings in High Birefringence Optical Fibers 103 The corresponding slopes of temperature, longitudinal and transversal strain for both polarization bands, for the best-fitted lines of superposing FBGs in IEC HiBi fiber, are given in table 9. 5.3.2. Simultaneous Measurements The change in the Bragg wavelength of the reflection spectrum of the both FBGs, due to a temperature change ΔT, a transverse strain (Δε X or Δε Y ) and longitudinal strain Δε Z , for each polarization, is given by 1 1 1 1 , , X X X X X Y Z X Y Z T T λ λ λ λ ε ε ε ε ∂ ∂ ∂ Δ = Δ + Δ + Δ ∂ ∂ ∂ (23) 1 1 1 1 , , Y Y Y Y X Y Z X Y Z T T λ λ λ λ ε ε ε ε ∂ ∂ ∂ Δ = Δ + Δ + Δ ∂ ∂ ∂ (24) 2 2 2 2 , , X X X X X Y Z X Y Z T T λ λ λ λ ε ε ε ε ∂ ∂ ∂ Δ = Δ + Δ + Δ ∂ ∂ ∂ (25) 2 2 2 2 , , Y Y Y Y X Y Z X Y Z T T λ λ λ λ ε ε ε ε ∂ ∂ ∂ Δ = Δ + Δ + Δ ∂ ∂ ∂ (26) where ∂λ X1 /∂T, ∂λ X2 /∂T, ∂λ Y1 /∂T and ∂λ Y2 /∂T are the temperature coefficients, ∂λ X1 /∂ε X,Y , ∂λ X2 /∂ε X,Y , ∂λ Y1 /∂ε X,Y and ∂λ Y2 /∂ε X,Y are the transversal deformation coefficients, and ∂λ X1 /∂ε Z , ∂λ X2 /∂ε Z , ∂λ Y1 /∂ε Z and ∂λ Y2 /∂ε Z are the longitudinal deformation coefficients. Equations (23) to (26) can be rearranged and written in matrix form, in order to calculate the transverse, longitudinal strain and temperature, given the measured wavelength shifts for each polarization band. In this way, the calculation of the three parameters being measured can be made using the following (the choice of reflection bands was arbitrary): 1 , 1 2 Y X Y X Z Y T K λ ε λ ε λ − Δ Δ ⎡ ⎤ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ Δ = Δ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ Δ Δ ⎣ ⎦ ⎣ ⎦ 1 (27) where K is assembled from the several sensitivities for temperature and deformation: 1 1 1 , 1 1 1 , 2 2 2 , K Y Y Y X Y Z X X X X Y Z Y Y Y X Y Z T T T λ λ λ ε ε λ λ λ ε ε λ λ λ ε ε ⎡ ⎤ ∂ ∂ ∂ ⎢ ⎥ ∂ ∂ ∂ ⎢ ⎥ ⎢ ⎥ ∂ ∂ ∂ = ⎢ ⎥ ∂ ∂ ∂ ⎢ ⎥ ⎢ ⎥ ∂ ∂ ∂ ⎢ ⎥ ∂ ∂ ∂ ⎢ ⎥ ⎣ ⎦ (28) Rogério N. Nogueira, Ilda Abe and Hypolito J. Kalinowski 104 After a previous characterization, in order to obtain K, the temperature, longitudinal and transversal strain components can be simultaneously measured. Some of the obtained results with the grating pair described above are given in table 10. Table 10. Simultaneous measurements of temperature, transverse and longitudinal strain using two superimposed FBGs in IEC HiBi fiber. The set values were determined by the experimental system equipment [31]. 167 με 251 με Set values 15 °C 45 °C 15 °C 45 °C 12 °C 42 °C 12 °C 37 °C 13 με 10 με 16 με 10 με 12 με 117 με 99 με 228 με 177 με 16 °C 33 °C 16 °C 43 °C 18 με 16 με 16 με 24 με 22 με 141 με 132 με 252 με 187 με 16 °C 36 °C 18 °C 43 °C 32 με 29 με 21 με 32 με 32 με 116 με 139 με 236 με 178 με 5.4. Bragg Gratings in Reduced Diameter High Birefringence Fiber Optics Bragg gratings written in reduced diameter high birefringence fiber optics can also be used for multiparameter sensing. Changes in the stress profile of HiBi fibers due to reduced diameter can modify the response of a FBG sensor system to strain or temperature optimizing the simultaneous measurement of those parameters. Chemical etching can be a good tool to reduce the fiber diameter. The changes in the birefringence properties of HiBi fibers as a function of fiber diameter can be analyzed using fiber samples chemically etched in hydrofluoric acid (HF), while the optical spectra of pre-recorded gratings are measured [34]. 0 10 20 30 40 50 70 80 90 100 110 120 D I A M E T E R ( μ m ) EXPOSURE TIME (min) Figure 16. Diameter of an IEC HiBi fiber as a function of the exposure time. HF concentration: 20% [34]. Fiber Bragg Gratings in High Birefringence Optical Fibers 105 The diameter of the fibers during the etching can be measured by having several samples of the fiber in the acid. The samples are removed successively from the acid, rinsed in distilled water, dried, and then measured under a microscope with a calibrated scale. The evolution of the diameter, as a result of etching, for an IEC fiber is presented in figure 16. HF acid was diluted to 20 % (parts per volume) in order to reduce the velocity of chemical etching and to increase the sampling points along the process. Figure 17 shows the changes in the transversal section of the IEC fiber, with 125 μm of diameter (left) and after etching (right), with 86 μm of diameter. The internal elliptical cladding can be observed in these photographs. The major axis of the ellipse has approximately 75 μm. The etched IEC fiber shows a higher asymmetry on the borders close to the axes along the major axis of the internal elliptical cladding. Figure 17. Microphotographs of the transverse section of an IEC HiBi fiber. Left: standard HiBi fiber with 125 μm of diameter. Right: etched HiBi fiber with 86 μm of diameter [34]. 1545.2 1545.6 1546.0 1546.4 0 50 100 150 200 4 12 20 28 36 44 P O W E R ( p W ) E X P O S U R E T I M E ( m i n ) WAVELENGTH (nm) 70 80 90 100 110 120 130 1545.2 1545.4 1545.6 1545.8 1546.0 1546.2 W A V E L E N G T H ( n m ) DIAMETER (μm) Figure 18. Left: evolution of the reflection bands of a FBG written in an IEC HiBi fiber as a function of the etching time. Right: peak position of the polarized bands (Y-polarized (∇) and X- polarized (Δ))as a function of the fiber diameter. The lines represent the linear best fit for the experimental data. HF concentration: 20 % [34]. Rogério N. Nogueira, Ilda Abe and Hypolito J. Kalinowski 106 Figure 18 (left) illustrates the optical reflection spectra of the FBG in IEC fiber, obtained as a function of HF exposure time. After 36 minutes of exposition time, the optical spectrum had a single band, which means that, the fiber birefringence was almost zero. That is a consequence of the stress release due to the etching. Figure 18 (right) shows the changes in the peak position of the reflected polarized bands as a function of the IEC fiber diameter. The different slopes for the X and Y polarized bands can be related to asymmetric changes of the internal stress applied by the internal elliptical cladding. The evolution of the birefringence, as a function of the diameter, can be seen in figure 19. 70 80 90 100 110 120 130 0.0 1.0x10 -4 2.0x10 -4 3.0x10 -4 4.0x10 -4 5.0x10 -4 6.0x10 -4 B I R R E F R I N G E N C E DIAMETER (μm) Figure 19. Calculated birefringence of the IEC HiBi fiber as a function of diameter. HF concentration: 40 % (∇) and 20 % (Δ) [34]. 1545.6 1545.9 1546.2 1546.5 0.0 2.0x10 -5 4.0x10 -5 6.0x10 -5 8.0x10 -5 90 110 130 150 170 183 187 E X P O S U R E T I M E ( m i n ) WAVELENGTH (nm) P O W E R ( m W ) 40 50 60 70 80 90 100110120130 1545.6 1545.8 1546.0 1546.2 1546.4 W A V E L E N G T H ( n m ) DIAMETER (μm) Figure 20. Left: evolution of the polarized bands of a FBG written in a bow tie HiBi fiber as a function of the etching time. Right: peak position of polarized bands (Y-polarized (∇) and X-polarized (Δ)) as a function of the fiber diameter. The lines represent the linear best fit for the experimental data. HF concentration: 20 % [34]. A similar characterization can be made to other types of HiBi fibers. For example, figure 20 (left) shows the effect of chemical etching in the optical spectrum of a Bragg grating written in a bow tie fiber. The etching rate is lower and it is possible to observe that the two polarization bands collapse. Initially both bands show a trend to longer wavelengths on their Fiber Bragg Gratings in High Birefringence Optical Fibers 107 peak position, as the diameter changes from 100 µm to 65 µm (figure 20 (left)). Further etching now causes the X polarized band to shift sharply to shorter wavelengths, until both bands collapse when the diameter reaches approximately 40 µm. This value agrees with the intrinsic stress-applying region dimensions, where the distance between the boundaries of the two internal side-lobes is approximately 37 μm. Figure 21 shows the birefringence for a bow tie fiber as a function of the diameter. The results show that IEC and bow tie fibers have vanishing birefringence for diameters that are close to the value of the maximum dimension of the stress-applying region. B I R E F R I N G E N C E DIAMETER (μm) Figure 21. Measured birefringence of bow tie HiBi fiber as a function of diameter. HF concentration: 20 % [34]. 5.4.1. Reduced Diameter for the Simultaneous Measure of Transverse Strain and Temperature A FBG in an etched HiBi fiber can be applied as a sensor to simultaneously measure the transverse strain and temperature. Once again, a previous calibration of the different sensitivities must be made. The temperature and transverse strain coefficients for an etched IEC fiber is shown in table 11. It also displays the coefficients for a non-etched bow tie fiber with a similar diameter. Table 11. Slopes of temperature and transverse strain of a FBG written in etched IEC and non-etched bow tie HiBi fibers [34]. Temperature Transversal strain Fiber (diameter) ∂λ x /∂T (pm/°C) ∂λ y /∂T (pm/°C) ∂λ x /∂ε (pm/με) ∂λ y /∂ε (pm/με) Etched IEC (82 μm) 7.00 6.90 0.7 (X -axis) 3.4 (Y -axis) 2.23(X -axis) 0.1 (Y -axis) Bow tie (80 μm) 8.02 8.46 0.02 (X -axis) 1.2 (Y -axis) 1.16 (X -axis) 0.3 (Y -axis) Rogério N. Nogueira, Ilda Abe and Hypolito J. Kalinowski 108 The results of simultaneous transversal strain and temperature measurements obtained with matrix K and the values of X λ Δ and Y λ Δ of the reflection spectra are displayed in table 12. Table 12. Simultaneous measurements of temperature and transverse strain using etched FBGs in IEC HiBi fiber (diameter of 82 μm). The set values are determined by the experimental system equipment. Direction of applied load: 90º [34]. Set values 16 °C 26 °C 36 °C 46 °C 56 °C 15 °C 28 °C 33 °C 44 °C 56 °C 33 με 37 με 37 με 39 με 42 με 42 με 17 °C 29 °C 36 °C 47 °C 56 °C 48 με 54 με 57 με 56 με 54 με 59 με 17 °C 28 °C 36 °C 46 °C 57 °C 64 με 49 με 48 με 48 με 54 με 53 με 17 °C 29 °C 36 °C 48 °C 58 °C 79 με 66 με 80 με 71 με 65 με 73 με 17 °C 29 °C 36 °C 48 °C 58 °C 94 με 80 με 100 µε 94 με 91 με 85 με The errors obtained using a FBG in normal and reduced diameter HiBi fibers as a sensor are of comparable magnitude, but the dynamic range for strain measurements with the later ones is almost doubled as compared to the former sensors. This fact is important for technological applications where FBG can be tailored to attend a specific measurement range. 6. Applications to Optical Communications All optical processing devices are becoming a key element in the next generation of optical communication systems, since they play a critical role in pulse formatting, spectral shaping and optimized all-optical routing and switching. These devices don’t have the typical bottleneck associated to the optical-electrical-optical conversion and the majority is transparent to modulation format and bit-rate. FBGs are quite interesting for these applications, due to their low insertion loss and due to the avoidance of the decoupling of the signal outside the fiber. Moreover, the production technology is now in a mature state, which enables the design of gratings with custom made transfer functions, crucial for all-optical processing. Some advanced processing can be made if the transfer function is different for the two transversal modes of propagation in the fiber. This can be achieved by a HiBi FBG. One of the devices that take full advantage of the optical processing capabilities of the HiBi FBG is the orthogonal pumps source [35-37], which can be used in all optical wavelength converters [38, 39]. A tunable PMD compensator can also be developed based on the polarization processing properties of these special gratings [40, 41]. Also, a tunable microwave-photonic notch filter that makes use of a time delay element based on tunable HiBi chirped FBG has been demonstrated [42, 43] In addition, the interference due to laser Fiber Bragg Gratings in High Birefringence Optical Fibers 109 coherence, typical in those micro-wave photonic filters was also reduced due to the polarization properties of the HiBi FBGs. The following sections describe some example application of HiBi FBG in optics communications. 6.1. Optical Delay Line for PMD Compensation In a linearly chirped grating, written in a HiBi fiber, each position of the grating will reflect two wavelengths at orthogonal polarizations (figure 22). This means that the group delay of these gratings is a combination of two linear functions, one for each polarization, with the same slope (D FBG ) and shifted by Δλ HB : ( ) ( ) ( ) y FBG x FBG HB D b D b τ λ λ τ λ λ λ = + = − Δ + (29) where b in (29) is a constant. Therefore, the relative group delay induced by a linearly chirped FBG written in a HiBi fiber (∆τ=τ x -τ y ) is calculated using the following expression 2 FBG FBG D D B τ λ Δ = − Δ ≈ − Λ (30) Expression (30) shows that the dynamic tuning of the induced PMD can be made by adjusting the birefringence of the fiber, which can be done by applying a transversal stress in the fiber, as shown before in this chapter. 1546 1548 1550 1552 1554 -50 -40 -30 -20 -10 0 y polarization x polarization Wavelength [nm] R e f l e c t i v i t y [ d B ] 0 50 100 150 200 250 300 Δλ HB G r o u p d e l a y [ p s ] Δτ Figure 22. Reflectivity and group delay of a linearly chirped HiBi FBG for both transversal propagation modes [47]. Rogério N. Nogueira, Ilda Abe and Hypolito J. Kalinowski 110 6.1.1. Compensation Using a Linear Chirp As can also be observed in expression (30), it is also possible to tune the PMD by adjusting the dispersion of the grating. That can be done using different methods [44-46]. One of them is by using thermal gradients to induce a linear chirp to a uniform FBG. Let us consider a uniform HiBi FBG put in a thermal contact with metal substrate. By applying different temperatures to the substrate, different linear temperature gradients will be generated. This gradient will induce a linear chirp to the FBG, due to thermo-optic and photoelastic effects. By changing the temperature gradient, the dispersion will also change, inducing a tunable differential delay line [47]. Figure 23 shows the experimental results of the evolution of ∆τ as a function of the applied temperature gradient to a 24 mm uniform HiBi FBG. 10 20 30 40 50 -50 0 50 100 Δ τ [ p s ] ΔT[ºC] Figure 23. Relative group delay as a function of the applied linear gradient to a uniform HiBi FBG with 24 mm length. Therefore, the presented device can be included in a PMD compensator as a tunable optical relative group delay line. 6.1.2. Compensation Using a Nonlinear Chirp Let us now suppose that we have a HiBi FBG with a quadratic chirp. The group delay is now composed by two parabolic functions (one for each polarization) shifted by Δλ HB . If the grating is tuned by temperature or longitudinal stress, the relative induced delay between the orthogonal polarizations, for a specific wavelength will change [40]. Figure 24 shows a simulation of a quadratic chirped FBG, with a length of 25 mm, written in a HiBi fiber with birefringence B = 5x10 -4 . For a tuning of 4.5 nm in the central wavelength, the relative group delay at 1550 nm changed from 41.6 ps to 12.1 ps. In this way, with this method, it is possible to do small corrections in the relative group delay. The advantages of this method are its tuning simplicity and the flexibility in the operation range. However, the technique needs a FBG with a nonlinear chirp, which is quite complex to produce. It is generally produced with a custom made phase mask with a nonlinear chirp. Fiber Bragg Gratings in High Birefringence Optical Fibers 111 1544 1546 1548 1550 1552 1554 1556 0 50 100 150 200 250 Δτ = 12.1 ps G r o u p d e l a y [ p s ] Wavelength [nm] Δτ = 41.6 ps Tuning Figure 24. Simulation of the group delay of a HiBi FBG with quadratic chirp. Line: Y polarization; dots: X polarization [47]. 6.2. Tunable Multiwavelength Linear Polarized Fiber Lasers Fiber lasers have different applications in sensors and telecommunications due to their reduced linewidth, power and spectral profile. Like other lasers, fiber lasers need two components: a gain medium and a resonant cavity. For a fiber laser operating around 1550 nm, it is generally based on an optical pump with 980 or 1480 nm of wavelength, an erbium-doped fiber and an optical filter. The gain is obtained from the amplified spontaneous emission due to the optical pump. Generally, fiber optical lasers based on an optical ring with erbium-doped fiber don’t enable the generation of more than one laser line [48,49]. This is a consequence of the fact that erbium is a medium with homogeneous gain at room temperature, resulting in strong mode competition, which induces laser instability. A method was proposed to reduce the homogeneity of the fiber by cooling the fiber to 77 K [50, 51]. However, by obvious reasons, it is not very practical. Other methods used special fibers like the elliptical core fibers [52] or the twincore fibers [53]. PC Circulator Two Tunable HiBi FBG Output Optical Pump (980 nm) EDF WDMc Figure 25. Diagram of a multiwavelength fiber laser based on HiBi FBGs. EDF: Erbium doped fiber; PC: Polarization controller; WDMc: WDM optical coupler; Rogério N. Nogueira, Ilda Abe and Hypolito J. Kalinowski 112 Another way to reduce the homogeneity of the fiber is to use different laser lines operating at different longitudinal modes. For this kind of implementation, HiBi FBGs can have an important role, since they will reflect two wavelengths at orthogonal polarizations. An implementation method for a tunable laser with up to four laser lines is depicted in figure 25. -60 -50 -40 -30 -20 -10 0 -60 -50 -40 -30 -20 -10 0 1530 1535 1540 1545 1550 1530 1535 1540 1545 1550 -60 -50 -40 -30 -20 -10 0 x x yy x x x x x y y y P o w e r [ d B m ] Wavelength [nm] Figure 26. Optical spectra at the output of the fiber laser with different operation modes. The operating laser lines are at a linear polarization (x or y). The two tunable HiBi FBGs enable the selection of 4 different wavelengths. By tuning the polarization controller (PC) inside the optical cavity, it is possible to select the appropriate laser lines. Figure 26 shows some of the possibilities that can be achieved with just two HiBi FBGs. One of the advantages of this technique is its ability to generate two laser lines at orthogonal polarizations (see last spectrum of figure 26). Therefore, it can be used as two orthogonal pumps in a polarization insensitive wavelength converter [38]. 6.3. Optical Networks Architectures Using HiBi FBG for Performance Improvement 6.3.1. Optical Code Division Multiple Access Metro optical code division multiple access (OCDMA) networks can benefit from the polarization multiplexing, since two users using codes in the same time-wavelength chip can be given orthogonal polarizations to operate, therefore reducing interference. One of the Fiber Bragg Gratings in High Birefringence Optical Fibers 113 implementation techniques is the “polarization assisted OCDMA with HiBi FBG” [54]. The technique uses the polarization properties of the HiBi FBG along with a special code generation scheme to improve the performance of OCDMA based networks. The coders are based on HiBi FBGs. To implement the suggested polarization assisted OCDMA, each HiBi FBG will reflect a pair of wavelengths λ i λ j , which are consecutive and cross polarized. In the case of the proposed method, a set of three of these HiBi FBGs, spaced by the fiber length needed for achieving the corresponding time chip spacing, results in two subsequent codes. Here, λ i corresponds to a X polarized reflection and λ j to a Y polarized one. This allows two consecutive user spreading sequences to share the same encoder. An implementation example is depicted in figure 27 . User A PBC Circ HiBi FBG λ 24, 25 λ 3,4 λ 11, 12 X User B Y Encoder HiBi fiber X Y A B Figure 27. Schematic of the proposed encoder implementation showing the use of the polarization to encode simultaneously two users with different wavelengths at orthogonal polarizations. Legend: PBC: polarization beam combiner; Circ: optical circulator [54] (© 2006 IEEE). X Y λ 11,12 λ 3,4 HiBi fiber Circ PBS Decoder HiBi FBG λ 24,25 Figure 28. Schematic of the proposed implementation for the decoder based on HiBi FBG. Legend: PBS: polarization beam splitter; Circ: optical circulator [54] (© 2006 IEEE). Each bit of information from users A and B is a wavelength comb which includes at least the wavelengths of the correspondent code (or a standard modulated broadband source can be used). Both bit sequence signals are multiplexed using a polarization beam combiner, thereby Rogério N. Nogueira, Ilda Abe and Hypolito J. Kalinowski 114 ensuring that they enter the encoder at the correct orthogonal polarizations. The encoder is based on three HiBi FBG reflecting the wavelengths λ 3 , λ 11 and λ 24 for X polarization and λ 4 , λ 12 and λ 25 for Y polarization. To achieve networking operation many such encoders need to operate simultaneously, and due to the properties of the technique, the number of encoders needed reduces to almost half. The decoder can be imprinted with standard FBG, since each user has its own code. However, for reduced user interference and to reduce the number of decoders needed, it can be based on HiBi FBG like the one exemplified in figure 28. The decoding process is similar to the encoding, where the HiBi FBG correlates two codes simultaneously. Afterwards a polarization beam splitter is used to separate both users. If no polarization maintaining fiber is used in the transmission link between the encoder and the decoder, the former must be preceded by a polarization rotator to ensure correct polarization coupling to the receiver. The polarization rotator can be automatically controlled by the receiver of one of the users using simple electronics. Even if no alignment of the polarization is made between non adjacent users, on average, only half the power will induce interference since the decoder will process only one of the two available polarizations. In the same way, the sensitivity to heterodyne crosstalk is also reduced since the power of the adjacent user, generated by the same encoder, is orthogonally polarized. In opposition to other coding/decoding techniques, like the ones based on arrayed waveguide gratings and optical delay lines, the proposed coder/decoders are quite compact, simple to use and have low insertion losses. On the other hand, since the gratings can have a length down to 1-2 mm and still have a high reflectivity, the time slots can be as low as a few picoseconds which can be considered enough for the majority of applications. 6.3.2. Radio over Fiber In radio over fiber systems (RoF), using the same central station to transmit to different local stations, one can use frequency interleaving to improve the bandwidth efficiency, exploiting the unused band between the carrier and data when high modulation frequencies are used with single side band (SSB) format. However, frequency interleaving also increases the bit error rate (BER), due to the interference of adjacent carriers. This drawback can be minimized if polarization multiplexing is used, i.e., the carriers and data are at orthogonal polarizations (figure 29). Interference: same polarization (40 GHz spacing) Traditional Implementation Implementation with HiBi FBG Data of interest Interference: both polarizations (20 GHz spacing) o o o o o o o o X Y X Y X Y X Y o o o o o o o o x y x y x y x y Figure 29. Diagram of the concept of interleaving using polarization multiplexing between carriers and data. Fiber Bragg Gratings in High Birefringence Optical Fibers 115 The implementation of this concept can be made using a HiBi FBG filter at the transmission which creates the SSB format and, at the same time, selects one polarization for the carrier and the orthogonal one for the data. At the local station another HiBi FBG removes the selected channel with reduced interference, since the interference will only be made by the data of the adjacent channels, which are at higher wavelength spacing and with lower power, relatively to the adjacent carriers. This technique has an impact on the overall performance of the system since the bandwidth efficiency can be improved without increasing the BER [55]. 7. Conclusion This chapter described some of the characteristics and functionalities associated with HiBi FBG. Their anisotropic behavior, relative to stress and/or strain, make them well suited for multiparameter sensors, including temperature, transversal strain and longitudinal strain. Moreover, their polarization processing capabilities also give them an interesting potential for different applications in optics communications. These applications include the development of new devices, like multiwavelength fiber lasers or in the optimization of certain architectures, like OCDMA. Some of the applications for sensing and optical communications were described but many more are yet to come. References [1] Kapron, F. P.; Keck, D. B. Appl. Phys. Lett. 1970, vol.17, 423-425. [2] Yamada, Y.; Nakagawa, S. I.; Kurosawa, Y.; Kawazawa, T.; Taga, H.; Goto, K. Elect. Lett. 2002, vol. 38, 328-330. [3] Hill, K. O.; Fufii Y.; Johnson D. C.; Kawasaki, B. S. Appl. Phys. Lett. 1978, vol. 32, 647-649. [4] Mora, J.; Díez, A.; Cruz, J. L.; Andrés, M. V. IEEE Photon. Technol. Lett. 2000, vol. 12, 1680-1682. [5] Guan, B.; Tam, H.; Liu, S. IEEE Photon. Technol. Lett. 2004, vol. 16, 224-226. [6] Tjin, S. C.; Suresh, R.; Ngo, N. Q. J. Lightwave Technol. 2004, vol. 22, 1728–1733. [7] Liu, Y.; Chiang, K. S.; Chu, P. L. IEEE Photon. Technol. Lett. 2005, vol. 17, 450-452. [8] Kashyap, R. Fiber Bragg gratings, Publisher: Academic Press, San Diego, CA, 1999. [9] Othonos, A.; Kalli, K. Fiber Bragg gratings, Publisher: Artech House, Norwood, NA, 1999. [10] Higuera, J. M. L. Handbook of Optical Fiber Sensing Technology; Publisher: John Wiley & Sons, New York, NY, 2002. [11] Erdogan, T. J. Lightwave. Technol. 1997, vol. 15, 1277-1294. [12] Lam, D. K. W.; Garside, B. K. Appl. Optics. 1981, vol. 20, 440-456. [13] Yariv, A. J. Quantum Electron. 1973, vol. 9, 919-933. [14] Yariv, A.; Nakamura, M. J. Quantum Electron. 1977, vol. 13, 233-253. [15] Dyott, R. B. Elliptical Fiber Waveguides; Publisher: Artech House, Boston, MA, 1995. [16] Rashleigh, S. C. J. Lightwave. Technol. 1983, vol. LT-1, 312-330. Rogério N. Nogueira, Ilda Abe and Hypolito J. Kalinowski 116 [17] Kersey, A. D.; Davis, M. A.; Patrick, H. J.; LeBlanc, M.; Koo, K. P.; Askins, C. G.; Putnam, M. A.; Friebele, E. J. J. Lightwave Technol. 1997, vol. 15, 1442-1463. [18] Othonos, A. Rev. Sci. Instrum. 1997., vol. 68, 4309-4341, [19] Abe, I.; Kalinowski, H. J.; Nogueira, R. N.; Pinto, J. L.; Frazão, O. IEE P-Circ. Dev. Syst. 2003, vol. 12, 495-500. [20] Frazão, O.; Pereira, D. A.; Santos, J. L.; Araújo, F. M.; Ferreira, L. A. Proc. SPIE 2005, vol.5855, 765-758. [21] Chen, G; Liu, L; Jia, H; Yu, J; Xu, L.; Wang, W. IEEE Photon. Technol. Lett. 2004, vol. 16, 221-223. [22] Urbanczyk, W.; Chmielewska, E.; Bock, W. Meas. Sci. Technol. 2001, vol. 12, 800-804. [23] Ferreira, L. A.; Araújo, F. M.; Santos, J. L.; Farahi, F. Opt. Eng. 2000, vol. 39, 2226- 2234. [24] Sudo, M.; Nakai, M.; Himeno, K.; Suzaki, S.; Wada, A.; Yamauchi, R. Proc. 12 th Int. Conf. Optical Fiber Sensors 1997, vol. 16, 170-173. [25] Chehura, E.; Ye, C. C.; Staines, E. S.; James, S. W.; Tatam, R. P. Smart Mater. Struct. 2004, vol.13, 888-895. [26] Ye, C.C.; Staines, S. E.; James, S. W.; Tatam, R. P. Meas. Sci. technol. 2002, vol. 13, 1446-1449. [27] Bosia, F.; Giaccari, P.; Facchini, M.; Botsis, J.; Limberger, H.; Salathé, R. Proc. SPIE 2002, vol. 4694, 175-186. [28] Lawrence, C. M.; Nelson, D. V.; Udd, E.; Bennett, T. Exp. Mech. 1999, vol. 39,202-209. [29] Lawrence, C. M.; Nelson, D. V.; Udd, E. Proc. SPIE 1997, vol. 3042, 218-228. [30] Caucheteur, C.; Ottevaere, H.; Nasilowski, T.; Chah, K.; Statkiewicz, G.; Urbanczyk, W.; Berghmans, F.; Thienpont, H.; Mégret, P. Proc. SPIE 2005, vol. 5952, 1-10. [31] Abe, I.; Kalinowski, H. J.; Frazão, O.; Santos, J. L.; Nogueira, R. N.; Pinto, J. L. Meas. Sci. Technol. 2004, vol. 15, 1453-1457. [32] Udd, E.; Schulz, W. L.; Seim, J. Proc. SPIE 1999, vol. 3538, 206-214. [33] Udd, E.; Nelson, D., Lawrence, C.; Ferguson, B. Proc. SPIE 1996, vol. 2718,104-107. [34] Abe I.; Frazão O.; Schiller M. W.; Nogueira R. N.; Kalinowski H. J.; Pinto, J. L. Meas. Sci. Technol. 2006, vol.17, 1477–1484. [35] Nogueira, R. N.; Teixeira, A. L. J.; André, P. S.; Rocha, J. F.; Pinto, J. L. Proc. Conf. Lasers Electro-Optics 2003, 545. [36] André, P. S.; Nogueira, R. N.; Teixeira, A. L.; Lima, M. J. N.; Rocha, R. F.; Pinto, J. L. Laser Physics Lett. 2004, vol. 1, 1 - 4. [37] Zhao, C. L.; Yang, X.; Ng, J. H.; Dong, X.; Guo, X.; Wang, X.; Zhou, X.; Lu, C. Microw. Opt. Technol. Lett. 2004, vol. 41, 73 – 75. [38] Nogueira, R. N.; Teixeira, A. L. J.; André, P. S; Rocha, J. F.; Pinto, J. L. Opt. Commun. 2006, vol. 262/1, 38-40. [39] Nogueira, R. N.; Teixeira, A. L. J.; Pinto, J. L.; Rocha, R. F. IEE Elect. Lett. 2004, vol. 40, 616–617. [40] Lee, S.; Khosravant, R.; Peng, J.; Grubsky, V.; Starodubov, D. S.; Willner, A. E.; Feinberg, J., IEEE Photon. Technol. Lett. 1999, vol. 11, 1277–1279. [41] Willner, A. E.; Feng, K.-M.; Cai, J.; Lee, S.; Peng, J.; Sun, H. IEEE J. Select. Topics Quantum Electron. 1999, vol. 5, 1298-1311. [42] Yi, X.; Lu, C.; Yang, X.; Zhong, W.-D.; Wei, F.; Ding, L.; Wang, Y. IEEE Photon. Technol. Lett. 2003, vol. 15, 754-756. Fiber Bragg Gratings in High Birefringence Optical Fibers 117 [43] Zhang, W.; Williams, J. A. R.; Bennion, I. IEEE Photon. Technol. Lett. 2001, vol. 13, 523-525. [44] Eggleton, B. J.; Mikkelsen, B.; Raybon, G.; Ahuja, Rogers, A.; J. A.; Westbrook, P. S.; Nielsen, T. N.; Stulz, S.; Dreyer, K. IEEE Photon. Technol. Lett. 2000, vol. 12, 1022- 1024. [45] Mora, J.; Ortega, B.; Andrés, M. V.; Capmany, J.; Cruz, J. L.; Pastor, D.; Sales, S. IEEE Photon. Technol. Lett. 2003, vol. 15, 951-953. [46] Lauzon, J.; Thibault, S.; Martin J.; Ouellettet, F. Opt. Lett. 1994, vol. 19, 2027-2029. [47] Nogueira, R. N.; Pinto, J. L.; Rocha, J. F. Microw. Opt. Tech. Lett. 2006, vol. 48 , 2357- 2359. [48] Gloag, A. J.; Langford, N.; Bennion I.; Zhang, L. Opt. Commun. 1996, vol. 123, 553-557. [49] Inaba, H.; Akimoto, Y.; Tamura, K.; Yoshida, E.; Komukai T.; Nakazawa, M. Opt. Commun. 2000, vol. 180, 121–125. [50] Yamashita S.; Hotate, K. Electron. Lett. 1996, vol. 32, 1298-1299. [51] Wei, D.; Li, T.; Zhao Y.; Jian, S. Opt. Lett. 2000, vol. 25, 1150-1152. [52] Das G.; Lit, J. W. Y. IEEE Photon. Technol. Lett. 2002, vol. 14, 606-608. [53] Graydon, O.; Loh, W. H.; Laming, R. I.; Dong, L. IEEE Photon. Technol. Lett. 1996, vol. 8, 63-65. [54] Nogueira, R. N.; Teixeira, A. L. J.; Pinto, J. L.; Rocha, J. F. IEEE Photon. Technol. Lett. 2006, vol. 18, 841 – 843. [55] Teixeira, A. T.; Nogueira, R. N.; André, P. S.; Lima, M. J. N.; Rocha, J. F., Electron. Lett. 2005, vol. 41, 30 - 32. In: Optical Fibers Research Advances Editor: Jurgen C. Schlesinger, pp. 119-159 ISBN 1-60021-866-0 c 2007 Nova Science Publishers, Inc. Chapter 4 APPLICATIONS OF HOLLOW OPTICAL FIBERS IN ATOM OPTICS Heung-Ryoul Noh 1 and Wonho Jhe 2∗ 1 Department of Physics and Institute of Opto-Electronic Science and Technology, Chonnam National University, Gwangju 500-757, Korea 2 School of Physics and Astronomy, Seoul National University, Seoul 151-742, Korea Abstract A hollow optical fiber (HOF) has a lot of interesting applications in atom optics experiments such as atom guiding and the generation of hollow laser beam (HLB). In this article we present theoretical and experimental works on the use of hollow optical fibers in atom optics. This article is divided into two parts: One is devoted to the atom guide using HOFs and the other describes the atom optics researches that utilizes laser lights emanated from the HOF. Firstly, we describe the electromagnetic fields inside the HOF and characterize the electromagnetic modes diffracted from the HOF. Then we describe two guiding schemes using red and blue detuned laser lights. Finally, we describe the various relevant experiments using LP 01 or LP 11 modes such as the generation of HLB from the HOF, funneling atoms using the diffracted fields, diffraction-limited dark laser spot, and a dipole trap using LP 01 mode of the diffracted field from the HOF. PACS: 32.80.Pj, 42.50.Vk, 39.25.+k, 32.80.-t, 03.75.Be 1. Introduction For the last two decades, there has been much progress on atom optics that manipulates atoms by using laser lights [1, 2, 3, 4, 5, 6, 7, 8, 9]. This field includes the studies such as focusing, reflecting, diffracting, and guiding atoms. In particular, it is an atom guide that provides a high spatial resolution of atom manipulation. So far several types of atom ∗ E-mail address:
[email protected]. (Corresponding author) 120 Heung-Ryoul Noh and Wonho Jhe guides using the optical, magnetic means, and using hollow optical fibers (HOFs) have been demonstrated. Of these, guidance of atoms by using HOFs is widely investigated, since it enables atoms to be controlled very accurately and guided over a long distance at high spatial accuracy without much atomic loss. A hollow optical fiber has many interesting applications in sensor [10] and harmonic generation [11], and optical communications [12, 13, 14, 15]. In this article, we describe the application of the HOF in atom optics. The basic principle of an optical atom guide is that atoms in laser beams are either attracted to or repelled from the regions of high intensity laser light, depending on the sign of the laser frequency detuning with respect to the atomic resonance. For two level atoms, the optical dipole potential due to the guiding laser is described by [16] U(r) = ∆ 2 ln 1 + I(r)/I s 1 + 4∆ 2 /Γ 2 , (1) where ∆ = ω L − ω 0 is the laser detuning from the atomic resonance, I s is the saturation intensity, I(r) is the guiding laser intensity, and Γ is the spontaneous decay rate of the upper level. When the laser frequency is tuned slightly below the atomic resonance, or red-detuned, atoms are attracted to the high-intensity regions. One the other hand, atoms are repelled from the high-intensity regions when the light is tuned above the resonance, or blue-detuned. One way to guide atoms is to launch the laser light inside the hollowregion of the hollow fiber and tune the laser frequency to the red side of the atomic resonance as proposed by Ol’shanni et al. [17]. This guiding scheme was successfully realized by Renn et al. at JILA [18]. In that study, a red-detuned laser was coupled to the lowest-order grazing incidence mode inside the glass capillary. The second method is to introduce the laser light into the glass core of the hollow fiber and tune the laser frequency to the blue side of the atomic resonance. This method was first proposed by Savage et al. [19, 20] and later by Jhe et al. [21] for different waveguide configurations. This guiding scheme was demonstrated for the capillary fiber by Renn et al. [22] and then subsequently demonstrated for the micron- sized hollow optical fiber by Ito et al. [23]. A similar experiment was later performed by Workurka et al. [24] and also by Dall et al. [25, 26]. In addition to the atom guidance, the HOF has another application in atom optics, which is the generation of laser beams diffracted from the facet of the HOF. The first use of the HOF for the generation of a hollow laser beam (HLB) by imaging the field distribution of LP 01 mode was demonstrated by Yin et al. [27]. Such hollow laser beams have been used for atom guidance [28], atom fountains [29], and atom traps [30]. The characterization of the output-field distribution of the hollow optical fiber was described in detail [31, 32]. The hollow laser beams made by a combination of two orthogonal LP 11 modes without an imaging lens can be used for funneling and guiding atoms [32]. They can also be used for generation of the diffraction-limited dark laser spot [33]. Furthermore the diffracted output of the LP 01 mode has a bright focused spot, which can be used for a tight optical dipole trap when a red-detuned laser is used [34]. This article is organized as follows: In the next section, we characterize the electromag- netic fields inside the HOF and the field distributions diffracted from it. In Secs. 3. and 4., the atom guidance by using the red and blue-detuned laser lights are presented, respectively. The experimental works with diffracted LP 11 modes from the HOF are described in Sec. Applications of Hollow Optical Fibers in Atom Optics 121 5., and the discussions on the applications of diffracted LP 10 modes such as atom guiding, atom fountain, crossed HLB trap, and single optical dipole trap follow in Sec. 6. The final section presents the summary of the work. 2. Characteristics of Electromagnetic Field for a Hollow Optical Fiber 2.1. Electromagnetic Field Modes Inside the Hollow Optical Fibers We describe the electromagnetic field modes for both inside and outside the hollow optical fibers. The electromagnetic fields inside HOF are given in the first subsection, and then the discussion on the electromagnetic fields diffracted fromHOF follows in the next subsection. The schematic diagram of the HOF with the hollow diameter of 2a and the core thickness d ≡ b − a is shown in Fig. 1 [20, 35]. Since the difference of the refractive indices of core and hollow region is not small, the weakly-guiding approximation seems to be not applicable. However, this approximation proved to be well applicable for the HOF [36, 37]. Therefore, instead of discussing the cumbersome vectorial approach, we will use the scalar theory for the analysis of electromagnetic fields modes. A capillary fiber composed of a hole of radius a and outer glass part is also used in the guiding experiment. The discussions on the guiding modes for the capillary fiber with red-detuned laser beam will be given in Sec. 4.1.. Figure 1. The schematic diagram of the hollow optical fiber. The diameter of the hollow region and the thickness of the cylindrical core are 2a and d, respectively. The refractive in- dices of the hollow, the core, and the cladding are 1, n 1 , and n 2 , respectively. The thickness of cladding can be taken to be infinite. In the cylindrical coordinate (r, θ, z), the longitudinal component of the electric field 122 Heung-Ryoul Noh and Wonho Jhe E z (r, θ) with E z (r, θ, z, t) = E z (r, θ) exp[i(ωt−βz)] satisfies the Maxwell equation given by ∂ 2 E z ∂r 2 + 1 r ∂E z ∂r + 1 r 2 ∂ 2 E z ∂θ 2 + k 2 n 2 −β 2 E z = 0 , (2) where ω, β, k and n are the angular frequency, the propagation constant, the wave number, and the refractive index, respectively. The solution of Eq. (2) is given by E z (r, θ) = AI ν (vr) sin(νθ +φ) (r < a), (BJ ν (ur) +CN ν ) sin(νθ +φ) (a ≤ r ≤ b), DK ν (wr) sin(νθ +φ) (r > b), (3) where J ν and N ν (I ν and K ν ) are the (modified) Bessel functions of the first and the sec- ond kind of order ν, respectively, n 1 (n 2 ) is the refractive index of the core (cladding), and φ is a phase constant. The constants A, B, C, and D can be determined by the con- tinuity conditions of E z at r = a and b. The quantity u = k 2 n 2 1 −β 2 is the transverse propagation constant, whereas v = β 2 −k 2 and w = β 2 −k 2 n 2 2 are the transverse attenuation constants. The magnetic longitudinal components, H z , can be obtained by re- placing sin(νθ + φ) in Eq. (3) with cos(νθ + φ). The transverse components (E r , E θ , H r , and H θ ) can be derived from these longitudinal components [38, 39]. The solutions of Eq. (3) are called HE νµ (EH µν ) modes, when E z and H z have the same (different) signs [40]. The dispersion equations describing the light propagation modes of the HOF can then be derived from the secular equation obtained by applying the boundary conditions to the tangential components E z , E θ , H z , and H θ [39]. The explicit expressions of the dispersion relation are presented in Eqs. (5)-(7) of Ref. [35]. Let us assume the silica-glass HOF with 2a = 7µm, d = 3.8µm, and n 2 = 1.45, in which the core is germanium doped with the relative refractive index difference ∆n = n 2 1 −n 2 2 /2n 2 1 = 0.0018. We also consider the guiding of rubidium atoms with the wave- length λ = 780 nm of D 2 line. From the dispersion equation, one can find that six propa- gation modes, TE 01 , TM 01 , HE 11 , HE 21 , HE 31 , and EH 11 can be excited at the wavelength of 780 nm (Fig. 2). The lowest mode is the HE 11 mode, whereas the TE 01 , TM 01 , and HE 21 modes exhibit almost the same dispersion curves, so that they form the second group of propagation modes. In the same way, EH 11 and HE 11 modes consist of the third group. When the refractive indices of the core and the cladding are nearly the same in a step- index solid fiber, the weakly guiding approximation is generally used [39, 41]. In this case, since one of the tangential field components is far larger than the other orthogonal transverse or longitudinal components, the guided mode can be approximately described only by the dominant transverse component, for which the ”linearly polarized” LP lm -mode description is usually employed [38, 39, 41] where l is the azimuthal mode number and m is the radial mode number. In case of an HOF, despite the large difference of the refractive index between the core and the hollow region, one can still use the LP lm modes due to the relatively small intensity in the hollow region. The transverse component for an LP lm mode is given in the same form as the longitudinal one [Eq. (3)] and thus we use it as a guided mode in the following calculations due to its simplicity over the traditional mode description method. From the continuity conditions at the boundaries r = a and r = a + d, one can derive Applications of Hollow Optical Fibers in Atom Optics 123 the following simple dispersion equation describing the LP modes as [42] J m (ua) I m (va) − u v J m (ua) I m (va) N m (ub) K m (wb) − u w N m (ub) K m (wb) = N m (ua) I m (va) − u v N m (ua) I m (va) J m (ub) K m (wb) − u w J m (ub) K m (wb) . (4) The dispersion equation in Eq. (4) yields the LP m1 modes (m = 0, 1, 2, · · · ) for the hollow fibers considered here. In fact, from the numerical analysis of Eq. (4), one observes that three LP modes can be excited in the 7-µm hollow core at the wavelength of 780 nm. Figure 2 shows the dispersion curves with respect to several lower modes. The solid circles indicate three LP modes: LP 01 , LP 11 , and LP 21 mode. Comparing with the the exact numerical results described above, one can find that: (i) LP 01 mode is approximately equal to HE 11 mode, (ii) LP 11 mode is made up of TE 01 , TM 01 , and HE 21 modes, and (iii) LP 21 mode consists of EH 11 and HE 31 modes. It should be noted that in Fig. 2(a), the points where each dispersion curve intersects the horizontal axis represent the cutoff frequencies. According to Eq. (4), the 7- and 2-µm hollow-core fibers become multi-moded at 780 nm, whereas the 1.4- and 0.3-µm hollow fibers become single-moded. Figure 2. (a) Dispersion curves of the propagating modes at the wavelength of 780 nm in the 7-µm hollow optical fiber with the core thickness of 3.8 µm and the relative refractive index difference of 0.18%. The cross-sectional intensity profiles for (b) LP 01 mode, (c) LP 11 mode, and (d) LP 21 mode (Figure from Ref. [35]). Figures 2(b-d) showthe cross-sectional mode-patterns of the 7-µmhollowfiber; (b), (c), and (d) present the LP 01 , LP 11 , and LP 21 modes, respectively. Note that one can selectively excite one of these LP modes by careful alignment of the incident angle of the laser beam. 124 Heung-Ryoul Noh and Wonho Jhe These CCD camera images show that the LP 01 mode is suitable for guiding atoms: LP 01 mode has no nodes around the cylindrical inner wall of the hollow core, as shown in Fig. 2(b). 0 2 4 6 8 10 12 14 0.00 0.05 0.10 0.15 0.20 0.25 Cladding Core Hollow I n t e n s i t y ( a r b . u n i t ) r (mm) LP 01 LP 11 Figure 3. The radial intensity distributions of LP 01 (solid curve) and LP 11 (dotted curve) modes inside the HOF. The intensity distributions of LP 01 and LP 11 modes along the radial direction are shown in Fig. 3 as solid and dotted curves, respectively. In Fig. 3, the both the diameter of the hole and the core thickness are typically assumed to be 4.5 µm. While the intensity distribution of LP 01 mode is azimuthally symmetric as shown in Fig. 2(b), that of LP 11 mode has an angular dependence sin 2 θ as shown in Fig. 2(c). Therefore the real two-dimensional intensity distribution of LP 11 mode should be obtained by multiplying the angular factor sin 2 θ to the function in Fig. 3. The guided LP 01 mode produces the blue-detuned optical evanescent fields on the core-vacuum interface, which then generates the optical potential barrier so that atoms can be guided in the dark hollow region of the core. 2.2. Characterization of Diffracted Fields from a Hollow Optical Fiber In the previous subsection, we have described the electromagnetic field modes inside the hollow optical fibers. The various applications using the inside modes will be described in Secs. 3. and 4. In this subsection, we will provide a theory of the diffracted laser lights from an HOF. As was mentioned in the previous section, we will adopt a simple scalar approach for the calculation. Since we know the electric field, Eq. (3), on the facet of the HOF (z = 0), we can calculate the diffraction pattern at (x, y, z) using the Huygens-Fresnel integral [43] E(x, y, z) = z 2π E 0 (x 0 , y 0 ) 1 ρ + ik e ikρ ρ 2 dx 0 dy 0 , (5) where (x 0 , y 0 ) is the coordinate of a source point, ρ = z 2 + (x −x 0 ) 2 + (y −y 0 ) 2 1/2 , and E 0 (x 0 , y 0 ) is the electric field at the fiber facet. In the near-field region where z Applications of Hollow Optical Fibers in Atom Optics 125 (kb 2 /2) 200 µm, using the Rayleigh-Sommerfeld theory [44, 45], one can calculate the diffraction pattern without any approximation on ρ as in the Fresnel diffraction calculation. For a given LP lm mode represented by E 0 lm at z = 0, we then obtain E lm (x, y, z) = E 0 lm (x 0 , y 0 ) ∗ h(x, y, z) (6) where h(x, y, z) = _ exp _ −ikz _ 1 + (x/z) 2 + (y/z) 2 __ × _ −iλz _ 1 + (x/z) 2 + (y/z) 2 __ −1 (7) and ‘∗’ denotes a two-dimensional convolution integral. Eq. (6) becomes a normal product in the transformed space, U lm (ξ, η; z) = U 0 lm (ξ, η) ×H(ξ, η; z) (8) where U lm (U 0 lm ) is the Fourier transformation of E lm (E 0 lm ), and H(ξ, η; z) = exp _ −ikz _ 1 −λ 2 (ξ 2 + η 2 ) _ . (9) The main task is to calculate the diffracted field in the transformed space and then convert the results to real space by an inverse transformaton. In the cylindrical coordinates (r,θ) and (ζ,ψ) in each space, the source field in the transformed space is given as: U 0 lm (ζ, ψ) = 2πi l sin(lψ + φ 0 ) _ +∞ 0 r 0 E 0 lm (r 0 )J l (2πζr 0 )dr 0 ≡ U 0 lm (ζ) sin(lψ + φ 0 ), (10) where E 0 lm (r 0 , θ 0 ) ≡ E 0 lm (r 0 ) sin(lθ 0 + φ 0 ). Substituting U 0 lm into Eq. (8), we inverse- transform U lm to obtain E lm (r, θ, z) = 2π(−i) l sin (lθ + φ 0 ) × _ ∞ 0 U 0 lm (ζ) exp[−ikz _ 1 −λ 2 ζ 2 ]J l (2πζr)ζdζ. (11) Here, U 0 lm is given analytically by U 0 lm (ζ) = 2πi l × [+ aA _ ¯ ζJ 1+l (a ¯ ζ)I l (av) + vI 1+l (av)J l (a ¯ ζ) _ v 2 + ¯ ζ 2 − aB _ ¯ ζJ −1+l (a ¯ ζ)J l (au) −uJ −1+l (au)J l (a ¯ ζ) _ u 2 − ¯ ζ 2 + bB _ ¯ ζJ −1+l (b ¯ ζ)J l (bu) −uJ −1+l (bu)J l (b ¯ ζ) _ u 2 − ¯ ζ 2 − aC _ ¯ ζJ −1+l (a ¯ ζ)N l (au) −uN −1+l (au)J l (a ¯ ζ) _ u 2 − ¯ ζ 2 126 Heung-Ryoul Noh and Wonho Jhe + bC ¯ ζJ −1+l (b ¯ ζ)N l (bu) −uN −1+l (bu)J l (b ¯ ζ) u 2 − ¯ ζ 2 − bD ¯ ζJ 1+l (b ¯ ζ)K l (bw) −wK 1+l (bw)J l (b ¯ ζ) w 2 + ¯ ζ 2 ] (12) where ¯ ζ ≡ 2πζ. Using these results, one can calculate the general profile of the diffracted beam at any position z. The numerical results of the radial intensity distributions for small z’s are presented in steps of 5 µm in Fig. 4. Figure 4(a) explains how the LP 01 mode diffracts in free space near the HOF: the two peaks (which represents a cross-section of ring-shaped mode) at z = 0 diminish away while an additional central peak grows up. In Fig. 4(b), one can see that the peaks of LP 11 also diminish whereas another pair of peaks grow. Nevertheless, there still does exist a dark column along the central axis. -10 -5 0 5 10 LP01 distance(um) 0 10 r(mm) 0 70 z (mm) (a) LP -10 -5 0 5 10 LP01 distance(um) -10 -5 0 5 10 LP01 distance(um) 0 10 r(mm) 0 70 z (mm) (a) LP -10 -5 0 5 10 LP11 distance(um) 0 10 r(mm) 0 70 z (mm) (b)LP -10 -5 0 5 10 LP11 distance(um) -10 -5 0 5 10 LP11 distance(um) 0 10 r(mm) 0 70 z (mm) (b)LP Figure 4. Development of the radial intensity distributions due to the diffraction of (a) LP 01 and (b) LP 11 mode near the facet of HOF (z = 0 at the facet). We now describe the experimental verification of theoretical calculated results for the developments of LP 10 and LP 11 modes using a simple imaging technique. In wave optics, the light propagating through a lens from a source plane to a screen can be described by equation (4.3-18) of reference [38], which shows that the field distribution at the screen is identical to the original distribution within a magnification constant m ≡ −d o /d i when (d i −1 + d o −1 − f −1 ) = 0, where d i is the distance between the lens and the source field, d o , the distance between the lens and the screen, and f, the focal length of the lens. This condition is easily realized in experiments when d i = f d o . Therefore, one can observe the intensity distribution of z = 0 at the screen far enough from the lens by locating the lens at a distance of the focal length from the tip of the HOF. In addition, one can observe the intensity distribution (diffraction pattern) of an arbitrary z by moving the lens by z toward the screen. The experimental results are shown in Fig. 5. We measured the intensity distribution of z = 0 at the screen 100 cm away from the lens with a focal length of 4.3 mm, and repeated the measurement at several values of z by moving the lens. As shown in Fig. 5, the experimental results show good agreement with the theoretical results. The output intensity distribution of LP 01 mode at z = 100 µm has a maximum at the center with the Applications of Hollow Optical Fibers in Atom Optics 127 10 m m (a)LP (z=0 m) 01 m (b)LP (z=150 m) 01 m (c)LP (z=0 m) 11 m (d)LP (z=150 m) 11 m -20 -10 0 10 20 LP 01 (z=150 mm) I n t e n s i t y ( a r b . u n i t ) Radial Distance (mm) (e) Figure 5. Output intensity distributions of the LP 01 mode ((a) at z = 0; (b) at z = 150 µm) and the LP 11 mode ((c) at z = 0; (d) at z = 100 µm), and (e) beam profile of the output of the LP 01 mode at z = 150 µm in a radial direction. The solid curve in the graph represents the fitting curve obtained from the calculation. full width at half maximum(FWHM) of 7.6 µm although it is in the shape of a ring inside the HOF. In case of LP 11 mode, on the other hand, both the dark center and the angular node line are preserved continuously even after propagation into free space. 3. Atom Guidance by Hollow Optical Fiber with Red-Detuned Laser In this section, we describe the atom guidance experiment by a hollow optical fiber with the red-detuned Gaussian laser lights. This scheme was suggested by Ol’Shanii et al. [17], and experimentally realized by Renn et al. for the hollow glass capillary using the red-detuned Gaussian laser beam [18, 46]. We first describe the experiment performed by Renn et al., and then the parametric excitation experiment will be briefly discussed [47]. Let us briefly summarize the theory of atom guidance by the red-detuned laser in the hollow capillary fiber which is composed of a hole of radius a and a glass with the refractive index of n. The thickness of the surrounding glass is assumed to infinite compared to radius of the hole. In the cylindrical coordinate (r, θ, z), the electric field of the lowest guided EH 11 mode is given by E(r, z, t) = ˆ eE 0 (r)e i(ωt−βz) , (13) where ˆ e is a unit transverse vector, and E 0 (r) is given by E 0 (r) = AK 0 (γa)J 0 (χr) (r < a), AJ 0 (χa)K 0 (γr) (r > a), (14) where J ν (K ν ) is the first (second) kind of the (modified) Bessel functions of order ν, A is a normalization constant, χ 2 = k 2 −β 2 , γ 2 = β 2 −n 2 k 2 , and k = 2π/λ . 128 Heung-Ryoul Noh and Wonho Jhe The value χ can be determined by the following characteristic equation, γJ 0 (χa)K 1 (γa) + χJ 1 (χa)K 0 (γa) = 0 , (15) When ka 1, Eq. (15) can be simplified as J 0 (χa) = iχ 2k n 2 + 1 √ n 2 −1 J 1 (χa) . (16) χa is 2.405 + 0.022i for a 40-µm hollow-core diameter capillary at λ = 780 nm. The imaginary part of β is the attenuation coefficient of the mode amplitude and is given by Im(β) = χa 2π 2 λ 2 2a 3 n 2 + 1 √ n 2 −1 , (17) for ka 1. For the 40-µm-diameter capillary fiber, the attenuation length, [Im(β)] −1 , was 6.2 cm. Figure 6 shows the intensity profile of EH 11 mode. As can be seen from Fig. 6 and Eq. (14), the intensity profile of the laser beam undergone by the atoms inside the hollow region is approximately given by I(r) = I 0 J 2 0 (χr), where I 0 is the peak laser intensity. When the guiding laser is red-detuned to the atomic resonance frequency, the atoms can be guided along the capillary hole owing to the attractive dipole force exerted by the laser light. -40 -30 -20 -10 0 10 20 30 40 0.0 0.2 0.4 0.6 0.8 1.0 Glass Glass Hole I n t e n s i t y ( a r b . u n i t ) r (mm) Figure 6. Radial intensity profile of EH 11 mode inside the capillary fiber with the inner diameter of 40 µm. The experiment of Renn et al. [18] consisted of two separate vacuum chambers con- nected by a 3.1-cm-long capillary fiber with an outer diameter of 144 µm and capillary-core diameter of 40 µm (Fig. 7). The guiding light from a Ti-Sapphire laser was coupled into the EH 11 mode. They observed a 50% coupling efficiency and an attenuation loss of 7% per cm of fiber length for the EH 11 mode. The guided atoms leaving the capillary fiber are surface ionized on a heated Pt or Re wire in the second chamber. Then the resulting ions are detected with a channeltron electron multiplier and recorded with a pulse counter. Applications of Hollow Optical Fibers in Atom Optics 129 Figure 7. Experimental apparatus. Figure 8. Guided atom signal versus laser detuning. The intensities of the guiding laser are I 0 = 3.6 × 10 3 W/cm 2 (upper curve) and I 0 = 1.3 × 10 3 W/cm 2 (lower curve) (Figure from Ref. [18]). 130 Heung-Ryoul Noh and Wonho Jhe Figure 8 shows the detuning dependence of the number of guided atoms for two laser intensities. The zero point of the detuning (δ) was taken to be the average transition fre- quency of the 5S 1/2 − 5P 3/2 multiplet of 85 Rb and 87 Rb. The guiding signal rose to half maximum at δ −2 GHz and full maximum at δ −3 GHz. For the laser detunings larger than a few GHz, the guided signal decreased approximately as δ −1 . With increasing intensity the signal increased and the position of maximum flux shifts to larger negative detunings. Figure 9. Guided atomflux vs laser detuning fromresonance at several laser intensities. The fiber lengths are 3.1 and 6.2 cm for (a) and (b)−(d), respectively, and the inner diameter is 40 µm (Figure from Ref. [46]). In a subsequent paper, Renn et al. reported on the improved results for the dependence of the guided atom signals on the laser detuning for several laser intensities [46]. Figure 9 shows the typical dependence of guided atom flux on the laser detunings. At an intensity of ∼ 0.6 MW/m 2 [Fig. 9(a)], the curve follows the similar trend as those shown in Fig. 8: The flux reaches a maximum near a detuning of about -1 GHz and then falls off rapidly for larger detunings. As the intensity increased, as shown in Figs. 9(b)−9(d), they observed a substantial flux at significantly higher detunings and found a dip formed in the profile at intermediate detunings. As a function of increasing intensity the dip grows deeper and broader [Figs. 9(c)−9(d)]. They found that the formation of the hole was attributed to viscous dipole forces which heat the atoms to larger transverse energies than can be guided [46]. Recently Hayashi et al. reported on the parametric excitation of laser guided Cs atoms Applications of Hollow Optical Fibers in Atom Optics 131 in an HOF [47]. Periodic modulation of the optical potential achieved by the intensity mod- ulation of the red-detuned guiding laser resonantly enhances the amplitude of the atomic motion. Eventually the atoms hit the inner wall of the HOF and absorbed on the surface. Thus the parametric resonance causes a reduction in the number of atoms coming out of the HOF. The experimental apparatus consists of two separate vacuum chambers connected by a 3-cm long borosilicate glass capillary having an outer diameter of 2 mm and a hollow-core diameter of 86 µm. One chamber serves as a Cs oven, which contains a Cs ampoule and is heated to 310 K to produce Cs vapor at 1023 Pa. The other chamber is a high vacuum detection chamber having a background pressure of 1026 Pa. Guided atoms leaving the capillary outlet are surface-ionized on a heated Pt wire. The generated ions are detected by a Channeltron electron multiplier and counted by a pulse counter. A Ti:sapphire laser with a wavelength of 852 nm was used to guide Cs atoms. For parametric excitation, the laser intensity is modulated by an acousto-optical modulator (AOM). Figure 10. Dependence of the count rate of Cs atoms on the modulation frequency of the laser intensity (Figure from Ref. [47]). Figure 10 shows the count rate of Cs atoms as a function of the modulation frequency of the laser intensity. The modulation depth is set at 0%, 10%, 40%, and 60%. The guiding laser is -15 GHz red-detuned from D 2 resonance line. As the modulation depth is increased, the guided atom flux lowers appreciably. For 10% modulation, a broad resonant structure appears in the parametric excitation spectrum. As the modulation frequency is increased, the number of guided Cs atoms starts to decrease around 100 Hz, takes its minimum at 5 132 Heung-Ryoul Noh and Wonho Jhe kHz, and then grows for higher frequencies. Above 100 kHz, the flux recovers to its original level. For 40% modulation, parametric excitation occurs over a much wider frequency range. Still, the dip of the spectrum is centered around 5 kHz. When the modulation depth is increased to 60%, severe guiding loss takes place over the whole frequency range, and the resonant structure disappears. 4. Atom Guidance by Hollow Optical Fiber with Blue-Detuned Evanescent Light 4.1. Atomic Guide by Glass Capillary Fiber In this section, the atom guidance experiments by the blue-detuned laser beams through capillary fibers or micron-sized hollow optical fibers are presented. First, in this subsection, we will describe the guiding experiments by means of hollow capillary fibers, performed by two groups Renn et al. [22] and Baldwin et al. [25] for the Rb and metastable He atoms, respectively. Renn et al. [22] at JILA reported the first atom guidance with the blue-detuned evanes- cent waves. They used a hollow glass capillary whose hollow-core diameter (2a) was 10 µm and outer diameter was 77 µm. In the absence of the laser field, atoms are supposed to be attracted to the glass surface due to the long-range van der Walls forces. At low inten- sities, the optical repulsive potential is weaker than the van der Waals potential, which is given by [48, 49] U vdW = − 3 4f n 2 −1 n 2 + 1 Γ (kr) 3 , (18) where f is the oscillator strength, k is the wave number, n is the refractive index, so that the total potential is always attractive. Therefore, one can define the threshold intensity above which atoms feel the net repulsive potential, which is the position of 3/(2k) apart from the wall. The experimental setup consisted of two vacuum chambers that were connected by the hollow capillary fiber like the red-detuned case. A blue-detuned high-power laser with 500 mW was focused on the annular region of the fiber end and a weaker, red-detuned escort laser with 10 mW was focused in the hollow region where it was coupled to the EH 11 grazing incidence mode. The escort laser facilitates atom loading into the capillary guide, which might be hampered by the scattered light near the core facet. Figure 11 shows the enhancement of the atomic flux due to optical guidance by a 6-cm-long glass capillary with the 20-µm core diameter. With the red-detuned escort laser alone, 200 atoms/s were guided in the fiber. When the evanescent light was excited on the glass interface, on the other hand, the flux increased at least by a factor of 3 at the optimum guide-laser detuning of +3 GHz and for the escort laser detuning of -1.6 GHz. When the escort laser detuning was decreased to -9.4 GHz, the number of atoms injected into the evanescent guide was also reduced. On the other hand, tuning the escort laser to the blue side of the resonance inhibited injection of atoms into the guide and thus completely suppressed the guided atom flux. They also measured the intensity dependence of the atomic guidance signal. Unlike the red-detuned guidance discussed in the previous subsection, there seems to exist threshold Applications of Hollow Optical Fibers in Atom Optics 133 Figure 11. The dependence of the guided atomic flux on the guide-laser detuning in the presence of the red-detuned escort laser. The detuning of the escort laser is -1.6 GHz (-9.4 GHz) for the upper (lower) curve (Figure from Ref. [22]). behaviour near the intensity of about 6 MW/m 2 , and the flux increases roughly linearly above this value. However, it was not straightforward to account for this behaviour in terms of the cavity potential because the hollow diameter was rather large, unlike the micron-sized HOFs where the van der Waals interaction near the core wall becomes significant (refer to next section for quantitative discussions of the cavity quantum electrodynamic effects). The JILA group also demonstrated evanescent-light guidance of laser cooled 87 Rb atoms in the hollow-core fibers [50] using the similar apparatus. The flux of an atomic beam generated from the source chamber by the low velocity intense source (LVIS) method [51] was approximately ∼ 10 9 atoms/s and the brightness was ∼ 10 13 atoms/(sr·s). The transverse velocity of LVIS was in the range v t = 8.0±1.5 cm/s and its longitudinal veloc- ity was measured to be v r = 10.0 ±2.0 m/s. The fibers used in their experiment were glass capillary tubes with 100-µm inner and 160-µm outer diameter. They have guided atoms by using several fibers of different lengths varying from 17 to 30.5 cm. In the 30.5-cm- long fiber, in particular, the atomic flux of 7 × 10 4 atoms/s was measured and 5.9 × 10 5 atoms/s was the highest flux obtained in the 23.5-cm-long fiber. The transverse velocity of the guided atoms was found to be 9.4 ±1.7 cm/s. Dall et al. demonstrated guidance of metastable helium atoms by blue-detuned evanes- cent waves in hollow capillary fibers [25]. A bright helium atomic beam in the metastable 2 3 S 1 state is generated by the nitrogen-cooled discharge source. The atomic beam is initially collimated in transverse directions by using a diode laser operating at 1083 nm 500 400 300 200 100 0 -4 -2 0 2 4 6 8 Guide Laser Detuning (GHz) F l u x ( H z ) Escort Laser Detuning -1.6 GHz -9.4 GHz 134 Heung-Ryoul Noh and Wonho Jhe (2 3 S 1 → 2 3 P 2 ). Then the atoms are cooled longitudinally by a second laser using a Zee- man slower, where the longitudinal velocity is decreased from ∼ 900 m/s down to ∼ 100 m/s. Finally, the atoms are compressed in two-dimensional magneto-optical trap by a sepa- rate laser, which results in the beam flux of up to 10 10 atoms/s over 1 cm 2 area. The guided atoms are detected by the channeltron detector. To couple the light into the capillary fiber, one end of the capillary is optically polished at 45 ◦ angle so that the focused guiding laser can be coupled from the side as discussed in Section 4.2. They have used three types of cap- illaries: square-section capillary (350 µm wide, 49 µm hole), round-section capillary (110 µm wide, 40 µm hole), and further round-section capillary (150 µm wide, 10 µm hole). With the input light power of 23 mW, the coupling efficiency exceeds 70% for the square capillary, whereas it is as low as 10% for the 110/40 µm thin-walled capillary. Figure 12 shows the guided atom signals with respect to the copropagating guide-laser detuning for the 350/49 µm capillary with the guide-laser power of 15 mW (the saturated absorption signal is also recorded for frequency reference). As can be seen, the guiding signal increases rapidly and decreases gradually as the detuning is increased. Figure 12(b) shows the effect of reducing the laser intensity to ∼ 1/3 with respect to that shown in Fig. 12(a). The reduced width of the transmitted signal is consistent with the decrease of the evanescent-wave potential. The signal in Fig. 12(b) can be well fitted with the ∆ −2 dependent function rather than ∆ −1 , as expected fromthe expression of the dipole potential. Note that the rapid decline of the transmitted signal with respect to ∆ in Fig. 12(b) was not due to capillary curvature. Rather, it was attributed to the rapid change of the area of the bright regions associated with the speckle pattern (formed by the multimode guide light) in the capillary. They also performed the atomic guidance experiment in the 110/40 µm and 150/10 µm round capillaries and obtained qualitatively similar results with respect to the case of the square 350/49 µm capillary [25]. 4.2. Atomic Guide in Micron-Sized Hollow Optical Fiber In a series of works, the Japan-Korea collaboration has reported on optical guidance of thermal rubidium atoms by the blue-detuned evanescent waves induced in the micron-sized HOFs. Figure 13 shows the schematic diagram of the experimental setup [23]. In the vacuum chamber, a rubidium atomic beam from the hot oven, well collimated by several apertures, is introduced into the HOF that is coaxially placed behind a holed mirror. The fiber has the hollow diameter of 7 µm (2 µm), core thickness of 3.8 µm (4 µm), and length of 3 cm. The typical incident atomic flux was of the order of 10 6 atom/s. The transmitted Rb atoms through the hollow fiber were detected by a channel electron multiplier, via two-step photoionization detection with two overlapping lasers; a diode laser tuned to the Rb D 2 line and a high-power Ar-ion laser at the wavelength of 476.5 nm (the ionization energy is 4.177 eVabove the 5S 1/2 ground state). The condition for efficient two- step photoionization of the ground-state atom is given by P i ∼ P 0 (σ 0 /σ i ) where P i is the light intensity required for ionizing atoms in the excited state and σ i is the ionization cross- section from the excited state to the ionization level [52]. For the Ar-ion laser intensity of 0.5 GW/m 2 , assuming the resonant excitation cross-section σ 0 = (3/2π)λ 2 = 3×10 −9 cm 2 for the 5S 1/2 →5P 3/2 transition [53] and σ i = 2.5 ×10 −17 cm 2 [54], the photoionization efficiency of about 30% was estimated. Applications of Hollow Optical Fibers in Atom Optics 135 Figure 12. (a) The transmitted atom signal (dots) and saturated absorption signal (solid curve) as functions of the laser detuning for the square 350/49 µm capillary in the copropa- gating configuration. The inset shows a detailed view at small detunings. (b) Same as(a) but with laser power reduced by ∼ 1/3. The dashed curve shows the ∆ −2 fit for the far-detuned blue wing (Figure from Ref. [25]). 136 Heung-Ryoul Noh and Wonho Jhe Figure 13. Schematic of the experimental setup. Figure 14 shows the typical photoionization spectrum of the 85 Rb atoms guided by the 7-µm HOF over the distance of 3 cm as a function of the frequency detuning δ of the guiding laser with respect to the 5S 1/2 , F = 3 upper ground state. As expected, the guided atomic flux is greatly enhanced in the blue-detuning region. The foot level of the photoionization signal extends to the detuning over +20 GHz. The broken line in Fig. 14 shows the background transmission level that was obtained without the guiding laser, which came from those atoms ballistically flying through the hollow fiber [23]. By comparison of the maximum guided atomic flux with the background transmission, the atomic-guide enhancement factor was found to be about 20. From a similar experiment with the 2-µm hollow fiber, in which the axis of the optical fiber was slightly tilted against that of the Rb atomic beam, a higher enhancement factor of 80 was obtained [55]. Figure 15 shows the novel characteristics of atomic isotope separation achieved in the 7-µm hollow fiber. The upper curve of Fig. 15 shows the case where the guide laser is blue-detuned for both isotopes. In this case, both isotopes can be guided in the hollow fiber. On the other hand, the lower curve of Fig 15 shows the case where the guide laser is blue-detuned for 87 Rb atoms but nearly red-detuned for 85 Rb atoms. As is clear, the 87 Rb atoms are guided by the hollow fiber, while the transmission of the 85 Rb atoms is greatly suppressed, which represents the interesting feature of an in-line atomic-state filter. In a subsequent experiment Ito et al. investigated a novel atomic guiding scheme in which the guide light beam is coupled to the hollow core sideways at a 45 ◦ angle via total internal reflection near the edge [56]. There are several advantages for this method: First, the scheme can easily remove unwanted light scattering such as the propagating modes inside the hollow region. Second, this scheme prevents the leak of the incident light near the entrance of the hollow fiber, so that heating or optical pumping of atoms near the fiber entrance can be minimized. Moreover, this scheme also enables one to couple the guide light from the rear direction. Recently, Fatemi et al. reported a side-coupling method for the mutitimode HOF using embedded microprism [57]. Microprisms embedded into a multimode, double-clad hollowfiber, allowlaser light to be coupled into the fiber at multiple locations along the length of the fiber. In the atom guidance through HOF, in addition to the repulsive optical dipole interac- tion, the attractive cavity quantum electrodynamic (QED) interaction is also induced be- Applications of Hollow Optical Fibers in Atom Optics 137 Figure 14. Two-step photoionization spectrum of the 85 Rb atoms in the 5S 1/2 , F = 3 state guided by the 7-µm hollow fiber over a distance of 3 cm (solid line). The broken line shows the background transmission level without the guide laser (Figure from Ref. [23]). Figure 15. In-line spatial separation of 85 Rb and 87 Rb in the 7-µm hollow optical fiber. The upper curve shows the case where a guide laser is blue detuned for both isotopes while the lower curve shows the case where a guide laser is blue-detuned for 87 Rb but red-detuned for 85 Rb (Figure from Ref. [23]). 138 Heung-Ryoul Noh and Wonho Jhe tween the dielectric wall and the nearby atoms [48, 58, 59]. Moreover, when the cavity potential exceeds the optical potential near the surface, atoms will be attracted to the inner wall and will be lost. Therefore, one can expect to observe the threshold behaviour of the atomic transmission in the cylindrical dielectric cavity [49, 60, 61, 62, 63]. Figure 16. Two-step photoionization signal of 87 Rb atoms guided by the 1.4-µm hollow- core optical fiber as a function of the guide-laser power (Figure from Ref. [65]. Figure 17. Atomic transmission signal of the guided 87 Rb atoms in the 0.3-µm hollow fiber near the low-intensity threshold region (Figure from Ref. [64]). Ito et al. have measured the threshold guide-laser intensities where the atomic trans- mission starts increased [55, 64]. The hollow-core diameter is only of the order of or less than the resonant wavelength of 780 nm for the 87 Rb D 2 transition, and consequently the induced cavity effects are much significant so that the small threshold intensities are now easily measurable even in the 300-nm hollow fiber. Figure 16 shows experimental data of Applications of Hollow Optical Fibers in Atom Optics 139 photoionization signal due to the atoms guided in the 1.4 µm fiber near the threshold [65]. As can be seen, the threshold behaviour in the atomic transmission is clearly observed at the small power of 125 µW. The laser power can be equivalently expressed in terms of the pure optical potential U op (r = a) normalized to the mean transverse kinetic energy K av = mv 2 tr of the guided 87 Rb atoms (m is the atomic mass and v tr is the transverse root-mean-square velocity of 87 Rb atoms). Then the threshold intensity for the case of 1.4 µm fiber corre- sponded to the ratio of U op (r = a)/K av = 2.5. When the total potential including the cavity potential is considered, on the other hand, the ratio of the total potential to the trans- verse kinetic energy becomes approximately one, as observed in the case of plane surface [66]. The same experiment is also done in the slightly larger HOF with the core diame- ter of 2.0 µm. The results show very similar threshold behaviours and the corresponding threshold laser-power is 40 µW, which is only 30% of the value for the 1.4-µm case. The atomic guidance experiment was also performed in the much smaller 0.3-µm hollow fiber. From the atomic transmission data near the threshold region presented in Fig. 17, one can obtain the threshold guide-laser power of about 2.6 mW. This laser power is much larger than those for the other two cases of larger HOFs. Note that the larger values of the thresh- old intensities for the smaller hollow-core fibers can provide the direct manifestation of the cavity QED effects in the cylindrical cavity. Figure 18. Spatial distribution of Rb atoms guided by the 7-µm hollow fiber (Figure from Ref. [55]). Ito et al. reported on the possibility of fabricating an arbitrary pattern by using the atomic waveguide with the hollow optical fiber, which may lead to a novel lithographic technique of optically controlled atom deposition [55]. The experimental setup is similar to that in Fig. 13. Figure 18 shows the surface-ionization signal of Rb atoms guided by the 7-µm hollow optical fiber over the distance of 3 cm. The spatial distribution of the 140 Heung-Ryoul Noh and Wonho Jhe Figure 19. Two-step photoionization spectrum of the 87 Rb atoms in the 5S 1/2 , F = 2 state guided through the 1.4-µm hollow fiber at a low oven temperature (Figure from Ref. [55]). guided atomic flux is obtained by the cross-sectional scan of the hot-wire at the distance of 12 mm downstream from the exit facet of the hollow fiber. The guide-laser frequency is blue-detuned at the optimal value of +3 GHz with respect to the 85 Rb, 5S 1/2 , F = 3 upper ground state. The FWHM of the spatial distribution shown in Fig. 19 is 20 µm. Considering the quantum efficiency of 0.9 of the channeltron and the cross section of the hot wire, the guided Rb flux Φ is measured to be 10 5 atom/s above the background level. 5. Experiments with Diffracted LP 11 Modes 5.1. Generation of a Hollow Laser Beam Diffracted from a Hollow Optical Fiber In this subsection, we describe the method of generation of hollow laser beams (HLBs) by means of diffracted LP 11 modes from an HOF. Although a doughnut-shaped divergent beam generated directly from an HOF is needed, for the simultaneous realization of an atomic funnel and an HOF atomic guiding, however, the calculations and experimental results in section 2.2. reveal that such an HLB cannot be produced in a simple method. These problems can be solved by using one of the TE 01 , TM 01 , and HE 21 modes which result in LP 11 modes. All of these modes are, like the LP 01 mode, ring-shaped inside the HOF, and, in addition, each output beam forms a doughnut-shaped hollow beam since they are only the superposition of the output-field pair of the LP 11 modes (For example, for TE 01 mode, the pair is shown in the upper row of Fig. 20(a)). Consequently we have superposed two LP 11 modes instead of exciting the desired mode directly. In this combination, one LP 11 mode has a polarization and a node line orthogonal to those of the other mode. Applications of Hollow Optical Fibers in Atom Optics 141 TE 01 +HE 21 TE 01 -HE 21 TM 01 +HE 21 TM 01 -HE 21 APP hal f-wave PBS1 M1 M2 PBS2 4X HOF 40X CCD SCREEN pl ate camera Screen Figure 20. Generation of an HLB by a proper superposition of LP 11 modes: (a) diagram for four degenerate configurations of LP 11 modes and (b) a sketch of the experimental setup. LD, APP, M, and PBS, in (b) stand for laser diode, anamorphic prism pair, mirror, and polarizing beam splitter, respectively. The experimental setup is presented in Fig. 20(b). If one wants a specific mode to be excited dominantly in a multimode fiber, the transverse distribution of the incident light should resemble that of the mode as much as possible and, in particular, its polarization should be also matched [67]. A half-wave plate just after the laser makes it possible to balance a relative intensity ratio of one mode to the other, which allows generation of a more symmetric mode in an azimuthal direction. The resulting combined beam in front of the fiber may look like a single linearly-polarized beam with its plane of polarization rotated 45 ◦ with respect to the horizontal plane, but one should note that each beam can be adjusted separately, which was important in exciting modes different from each other. The first two pictures in Fig. 21, which were obtained by blocking one of the two optical paths, represent the patterns of perpendicular modes at z = 0 before they are merged, and 142 Heung-Ryoul Noh and Wonho Jhe 5 m m (a) (b) (c) Figure 21. Superposition of two orthogonal LP 11 modes. Transverse intensity profiles at z = 0 are shown: (a) and (b) before, and (c) after superposition. Polarization and angular variation of the corresponding electric field can be described by, for example, (a) ˆ xsin(θ + φ) and (b) ˆ y cos(θ + φ). the last one shows their combined pattern, which is similar to that of LP 01 mode. Figure 22 shows its output intensity distribution at z = 250 µm. The peak-to-peak distance is about 17 µm and the dark spot size is about 8.2 µm. We have checked its azimuthal isotropy by measuring the beam profiles along eight different radial axes and they showed good uniformity within the maximum error of about 7%. -20.0 -10.0 0.0 10.0 20.0 10 m m Radialdistance( m) m I n t e n s i t y ( a r b . u n i t ) (a) (b) Figure 22. HLB made of the diffracted output of the superposed mode. (a) CCD images of the intensity distribution (z = 250 µm) and (b) its profile in a radial direction. Let us now discuss briefly on the application of this beam to an atomic funnel. Figure 23 shows a possible configuration of our overall atom guiding system. It consists of three main parts: a pyramidal or axicon mirror trap [68, 69], a HLB atomic funnel, and an HOF evanescent-wave atomic guiding. 85 Rb atoms trapped in a pyramidal or an axicon mirror trap are pushed through a small hole by the power imbalance of the laser light along the mirror axis, and guided inside the HLB. With the HLB converging into the hollow region of the HOF, they are finally guided through the HOF. The guiding laser light is coupled into the core of the HOF from the side by the total reflection at the glass-vacuum interface which is ground and polished at an angle of 45 ◦ as employed in reference [56]. The cold Applications of Hollow Optical Fibers in Atom Optics 143 Pyramid Mirror Trap AtomicFunnel madeby DarkHollowBeam Hollow-core Optical Fiber Evanascent Field Blue-detuned LaserBeam TrappingBeam Channeltron Figure 23. Set-up of a novel atomic guiding system. The guiding beam is launched into the fiber from the side, and channeltron is used to detect the guided atoms. atom funnel by employing similar apparatus with the red-detuned Gaussian funneling beam rather than a blue-detuned HLB was recently realized [70, 71]. 5.2. Diffraction-Limited Dark Laser Spot Produced by a Hollow Optical Fiber Shin et al. reported on the generation of the diffraction-limited HLB having submicron- sized dark spot by using the diffracted LP 11 mode from the HOF [33]. From the practical point of view, leakage of light fromthe cladding region on the fiber facet is the main obstacle to obtaining the good optical quality of diffracted HLB since a short fiber is generally used for atom optical experiments, where contamination due to the leaked light is inevitable. To produce an ideal HLB with HOF, it is required to block the cladding-guided light so that unwanted scattering can be avoided. For this purpose, a microsphere is employed as an evaporation mask for the core of HOF. Figure 24(a) shows the image of the field distribution of the LP 01 mode on the fiber facet observed by the imaging method [27]. The scattered light on the cladding surface is clearly observed. They blocked the cladding-mode light by selectively metal-coating the cladding on the fiber’s facet with a microsphere used as an evaporation mask. They first attach the 144 Heung-Ryoul Noh and Wonho Jhe Figure 24. Intensity distribution of the LP 01 mode on the HOF-end facet imaged on the screen for (a) the normal HOF and (b) the metal-coated HOF. (c) scanning electron micro- scope (SEM) image of the microsphere attached to the HOF center after metal evaporation. (d) Enlarged SEM picture near the hollow-core region with the sphere removed. microsphere on the fiber center masking the hollow core, evaporate thin film of aluminum, and then remove the microsphere afterwards. A 20-µm-diameter microsphere is used to effectively mask the hollow core having the diameter of 2a + 2d 12.3µm. In Fig. 24(c), the scanning electron microscope (SEM) image shows that the microsphere is positioned well on the fiber center. Once it is attached to the fiber, the van der Waals force holds the sphere tightly in its position. The more-detailed SEM picture of the facet after removing the microsphere, i.e. the metal-coated HOF facet, is presented in Fig. 24(d). Fig. 24(b) shows that the cladding-mode light is completely blocked by the described procedures. They measured the beam profiles and the dark hollow size of the diffracted LP 11 mode as shown in Fig. 25. In Fig. 25(a), one can observe the beam-propagation characteristics of the freely-diffracting HLB: whereas the bright ring on the fiber-end facet is diminished, the new peaks are developed from the center at around 30 µm, maintaining the dark region along the axis (the inner peaks diverge with a diffraction angle of 40 mrad). In particular, the dark spot is preserved along the central axis even when HLB is spontaneously focused due to diffraction. The experimental results are in good agreement with the numerical simulations obtained by the Rayleigh-Sommerfeld theory, as also shown in Fig. 25(a). Fig. 25(b) shows the radial intensity distributions at z = 15, 20, 25, 30 µm, respectively. Note that the central dark region near z = 30 µm is slightly contaminated, which may be associated with the intensity imbalance of two peaks, a slight excitation of LP 01 mode, or the resolution limit of imaging system. To estimate the size of the dark hollow region, they fitted the measured profile for a given z with two independent Gaussian curves and define the radius of dark spot R max as a half the distance between the central maxima of each curve. As shown in Fig. 26, the Applications of Hollow Optical Fibers in Atom Optics 145 Figure 25. Characteristic dimensions of the dark hollow region, measured in terms of the dark-spot radius R max and the half-width w, which are in good agreement with the numer- ical simulation. 0 20 40 60 80 100 0 1 2 3 4 5 hole radius of hollowfiber ( m m ) z (mm) R max (simulation) R max (experiment) w Figure 26. (a) Experimental and numerical results of the radial-intensity profiles of the diffracted LP 11 mode measured in steps of 5 µm from the HOF-end facet (z = 0). (b) Intensity profiles measured near the focus at z = 15, 20, 25, 30 µm. 146 Heung-Ryoul Noh and Wonho Jhe minimum radius of dark spot is about 2 µm which is similar to the hollow radius a itself. As an alternative definition characterizing the dark spot, on the other hand, we also have fitted the dark region profile between the two curves with an inverted Gaussian curve, and estimate the half width of dark spot w as the half width at half maximum of the single inverted Gaussian. In this way, as in Fig. 26, we obtain that the smallest value of w is less than 1 µm (about 0.8 µm around z = 35 µm). 6. Experiments with Diffracted LP 10 Modes In this section, we describe the various experiments performed by using the hollow laser beams produced by a hollow optical fiber. First, we describe the micro-imaging method to generate HLBs and discuss the experiments of atom guiding, atom fountain guided by an HLB, and crossed atom trap. Finally, as an application of Diffracted LP 10 modes, the discussions on the optical dipole trap is presented. 6.1. Micro-Imaging Method for Hollow Fiber Modes Yin et al. [27] obtained an HLB by using a micro-collimation technique for the output beam of a micron-sized hollow optical fiber. The principle of this method is very simple: for a fiber waveguide consisting of a hollow cylindrical core, some low-order modes can be guided in the hollow core, such as the LP 01 , LP 11 , LP 21 , and LP 31 mode [35]. Therefore, when one uses a microscope objective with a short focal length to image the output intensity distribution at the facet of a hollow fiber, a simple HLB can be obtained. The inner and outer diameter of the hollow-core of the fiber was 7 µm and 14.6 µm, respectively and the outer diameter of the cladding of the fiber was 123.4 µm. The relative refractive index difference, ∆n = (n 2 1 − n 2 2 )/(2n 2 1 ) = 0.0018 and n 2 = 1.45, where n 1 and n 2 are the refractive index of the core and the cladding, respectively. The numerical aperture is about 0.124. Figure 27 shows the relationship between the dark spot size (DSS) and the propagation distance Z of the dark HLB. It can be observed that (i) the DSS of the dark hollow beam collimated by a M-20× objective is about 50 µm at Z = 100 mm and about 100 µm at Z = 500 mm, and (ii) the relative divergent angle in the near field of HOF is about 6.5 ×10 −5 , whereas the divergent angle in the far field is 4.0 ×10 −4 . If one uses an HOF having a slightly larger hollow-core, an HLB with a smaller DSS and better propagation invariance may be obtained. The HLB generated by the micro-imaging method was used to couple into the HOF to increase the coupling efficiency by Takamizawa et al. [72]. In order to eliminate the undesirable preinteraction before atoms enter the hollow region, they used an annular beam whose shape and diameter are approximately the same as those of the core. The use of annular beam also makes it feasible to excite the LP 01 mode with a relatively low power of ∼10 mW to reflect atoms. The evanescent wave produced with the LP 01 mode has a cylindrical shape around the inner-wall surface without nodes, and consequently it is suitable for the atom guidance. A Gaussian light beam coupled to the HOF excites LP 01 mode. The output light beam from the HOF, which has an annular shape just after exiting the HOF, is collimated and recoupled to another HOF with two convex lenses. Then, annular light beam with the same Applications of Hollow Optical Fibers in Atom Optics 147 Figure 27. The relationship between the DSS and the propagation distance Z of the HLB measured with (a) M-20 × and (b) M-40 × objective lens. intensity profile as that of LP 11 mode is reproduced at the focal point behind the second lens. The typical values of the focal length and the distance of the two lenses are 25.2 mm and 500 mm, respectively. The coupling efficiency was increased to 75% when the annular- shaped beam was used compared to 59%, which was the maximum coupling efficiency when a Gaussian beam was coupled to the HOF [72]. The hole diameter, core thickness, and the length of the used HOF was 7 µm, 3.8 µm, and 3 m, respectively. 6.2. Atom Guiding with Hollow Laser Beams Xu et al. performed optical guiding of trapped cold atoms by a hollow laser beam produced by micro-collimation and micro-imaging technique as discussed in the previous subsection [28]. The atomic guiding direction was downward along the gravity (+z direction), whereas the HLB propagated along the −z direction (counterpropagating scheme) or along the +z direction (copropagating scheme). A Ti:sapphire laser was used as the guiding laser source with a maximum output power of 1.8 W. It was coupled to the core of HOF with a coupling efficiency of about 30%. The typical HLB power used for guiding atoms was 250 mW. They used a micron-sized HOF that has a hollow diameter of 4 µm, core thickness of 2 µm, and length of 25 cm. In both guiding schemes, they obtained the identical radius of the maximum-intensity ring ρ m (z) that varies linearly with the distance z [ρ m (z) = ρ m (0) − 148 Heung-Ryoul Noh and Wonho Jhe αz, where ρ m (0) = 1.4 mm is the value at the trap center (z=0) and α = 1.27(4) ×10 −3 ]. They used a standard vapor-cell magneto-optical trap (MOT) of 85 Rb atoms. The num- ber of trapped atoms was typically 2 × 10 7 and the trap diameter was about 1 mm so that the loading efficiency of the trapped atoms into HLB was 98%. By time-of-flight measure- ment, the temperature of atoms in the MOT was found to be about 140 µK, which was further cooled down to 16 µK by the polarization-gradient cooling. After the sub-Doppler cooling, the cooling and repumping lasers were blocked by mechanical shutters, and the HLB was simultaneously introduced to the atoms to guide their gravitational falling. The number and the temperature of guided atoms were detected by observing the probe-induced fluorescence with a photomultiplier. The probe laser beam was placed horizontally at 105 mm below the trap center. 0.00 0.05 0.10 0.15 0.0 0.5 1.0 1.5 2.0 2.5 Free 16 GHz 10 GHz 6 GHz 2 GHz 1 GHz G u i d e d A t o m F l u x ( a r b . u n i t s ) Time of Flight (s) 0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24 0.0 0.5 1.0 1.5 2.0 2.5 Free 16 GHz 10 GHz 6 GHz G u i d e d A t o m F l u x ( a r b . u n i t s ) Time of Flight (s) Figure 28. Typical TOF signals of atoms guided by a single HLB. (a) In the copropagating scheme, the laser detuning δ 2 is 1, 2, 6, 10, and 16 GHz, respectively. (b) In the counter- propagating scheme, δ 2 is 6, 10, and 16 GHz, respectively. Figure 28 shows time-of-flight signals of guided cold atoms in both guiding schemes at various laser detunings with respect to the 5S 1/2 , F = 2 → 5P 3/2 transition line. For comparison, the detected signal without the HLB for the freely falling atoms is also shown. In particular, it is observed that the number of atoms guided by the copropagating HLB is Applications of Hollow Optical Fibers in Atom Optics 149 about 20-fold enhanced with respect to that without the HLB at 2 GHz detuning. In this case, the guided atoms also become accelerated along the +z direction due to the increased radiation pressure at small detunings [Fig. 28(a)]. In the counterpropagating case, on the other hand, the guided atoms are decelerated as the detuning is decreased [Fig. 28(b)]. -4 -2 0 2 4 6 8 10 12 14 16 0 10 20 30 40 50 Copropagating Counterpropagating (b) (a) G u i d i n g E f f i c i e n c y h ( % ) Detuning d (GHz) Figure 29. Guiding efficiency as a function of the detuning δ 2 in the copropagating (a) as well as the counterpropagating (b) scheme. The solid curves represent numerical simulation results. Figure 29 presents experimental and numerical guiding efficiencies versus detuning in the copropagating (a) as well as in the counterpropagating (b) scheme. Note that in numer- ical simulation, the HLB was assumed, to a good approximation, as the lowest Laguerre- Gaussian (LG 1 0 ) mode given by I(ρ, z) = 4P πw 2 ρ 2 w 2 exp − 2ρ 2 w 2 , (19) where P is the laser power, w = w 0 1 + ξ(z/z R ) 2 is the beam waist at distance z, w 0 is the beam waist at z = 0, and z R = πw 2 0 /λ is the Rayleigh length. It can be observed that at small detuning, atoms are most efficiently guided in the copropagating scheme (for example, the maximum guiding efficiency is about 50% at the detuning of 2 GHz). On the other hand, the counterpropagating guiding is generally less efficient as found in Fig. 29. However, for large detunings, both schemes provide similar guiding efficiencies and the maximum efficiency of 23% is obtained at 10 GHz detuning in the counterpropagating scheme. 6.3. Atom Fountain with Hollow Laser Beams The development of an atomic fountain based on laser-cooled atoms [73, 74] has created prospects for an improved accuracy and stability of frequency standards. In such a clock, one approach to solve the line shift due to cold collisions is to use laser light for guiding the upward launched atoms [75]. This is because the guiding can enhance the number of atoms which come back into the microcavity without increasing the atomic densities. 150 Heung-Ryoul Noh and Wonho Jhe Optical guiding of an atomic fountain by using a cylindrical HLB was demonstrated by Kim et al. [29]. The generated HLB by using micro-imaging method was collimated by the objective lens and propagated downwards toward the center of the Rb MOT. The power of the guiding laser was 250 mW and the beam waist was 3 mm. The HLB was nearly collimated in order to remove the dipole force of the guiding direction, which can cause broadening of the spatial distribution of guided atoms. With an intensity of 3 mW/cm 2 in each beam, the typical diameter of an atomic cloud in the MOT was about 1 mm, and the number of trapped atoms was typically 2 × 10 7 . Cold atoms were then launched upwards in a rather simple way by varying rapidly the vertical magnetic field resulting in the atomic Zeeman shift. After 1-ms acceleration, the detuning of the laser beams was changed from −2.5 Γ to −70 Γ, lowering the atomic temperature to 33.7 µKin the frame moving upwards. A typical launching velocity of ascending atoms was 1.4 m/s and the atoms were launched up to 10 cm. The number of guided atoms was detected by observing the fluorescence with a photo- multiplier tube, which was induced by a horizontally placed probe laser at 10.5 cm below the center of the MOT. They observed that 0.5% of the launched atoms were detected with- out the HLB. On the other hand, a tenfold enhancement of the HLB-guided atomic fountain was clearly obtained without appreciable heating. In Fig. 30, the line (a) is the time-of- flight (TOF) signal of atoms that are launched without the HLB, while the other line (b) is the TOF signal with the HLB at a detuning of 19 GHz. From this TOF signal, one can deduce the guiding efficiency of atoms and the temperature. Without the HLB, the temper- ature was about 33.7 (±2.1) µK and about 34.4 (±1.7) µK with the HLB. 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.0 0.2 0.4 0.6 0.8 1.0 P=250 mW d 2 =19 GHz v launch =1.4 m/s HLBoff (a) HLBon (b) N u m b e r o f A t o m s ( a r b . u n i t s ) Time of Flight (s) Figure 30. TOF signals in the HLB-guided atomic fountain experiment. The line (a) is for the case without the HLB whereas the line (b) is for the case with the HLB. To characterize the enhancement due to the guiding HLB, they introduced the enhance- ment factor, defined as the ratio of the number of atoms guided with the HLB to that without HLB. In Fig. 31, the line with filled squares shows the relationship between the enhance- ment factor and the detuning measured with respect to the 5S 1/2 , F = 2 →5P 3/2 transition line. The inset shows the enhancement of the guiding efficiency for larger detuning. The line with empty circles shows how the number of scatterings, or the heating, is changed with Applications of Hollow Optical Fibers in Atom Optics 151 0 5 10 15 20 0 10 20 30 40 0 500 1000 1500 2000 0 50 100 0 10 20 30 40 N u m b e r o f S c a t t e r i n g s E n h a n c i n g F a c t o r d 2 [GHz] Enhancing Factor Number of Scattering Figure 31. Dependence of the enhancement factor (•) and the number of scatterings (◦) on the detuning δ. The inset shows the enhancement factor for larger detuning. the detuning. They observe that for small detuning, the enhancement factor is more than 35, but there is serious heating. As the detuning increases, the enhancement factor as well as the heating decrease. Note that the number of scatterings decreases more rapidly (∼ δ −2 ) than the enhancement factor. At a detuning of 19 GHz, the enhancement factor is over 10 and an atom experiences spontaneous emissions about 40 times during the launching and falling processes. According to the calculation, however, they found that the heating due to spontaneous emissions was not so serious. In order to reduce a loss of atomic coherence, an HLB with a large detuning may be used. For example, if a 15-W Ar-ion laser is used for a tenfold enhancement of guiding efficiency, the average rate of spontaneously scattered photons is calculated to be 10 −3 Hz. While the number of atoms being guided in the fountain is increased, the HLB introduces inhomogeneous energy shifts of the ground-state hyperfine levels. In a trap based on a sheet of blue-detuned light supporting against gravity, a Stark shift of 270 mHz is obtained for 4-s trapping time, which is larger than the line shift due to cold atom collisions [76]. One possibility for reducing the light shift in the HLB is to use a much higher-order Bessel beam for the HLB or to use the evanescent waves of a hollow optical fiber. Since the evolution of atoms in an HLB depends on the shape of the HLB, it is suggested that the ensemble- averaged heating and the light shift will be changed with the shape of the HLB [77]. In principle, if the potential of the HLB is square, then atoms in the HLB may not feel any scattering or light shift. 6.4. Crossed-HLB Trap of Rb Xu et al. constructed a blue-detuned optical dipole trap by intersecting two horizontal, cylindrical HLBs at a right angle in the center of a Rb MOT [30]. The polarizations of the beams were chosen to be orthogonal in the crossed region in order to suppress standing- wave effects. The detuning of HLBs was 20 GHz from 5S 1/2 , F = 3 → 5P 3/2 , F = 4 transition line of 85 Rb and the trap depths were about 10 µK in the x-direction and 90 µK in the z-direction, respectively. They estimated that 60% of the atoms in the MOT were 152 Heung-Ryoul Noh and Wonho Jhe initially loaded in the HLB trap. 0 20 40 60 80 100 0 20 40 60 80 100 Experimental Data Simulation Results T r a p p i n g E f f i c i e n c y ( % ) Trapping Time (ms) Figure 32. The trapping efficiency of a crossed-HLB trap as a function of the trapping time. The two horizontal HLBs have a power of 200 and 400 mW, the detuning of each HLB is 20 GHz, and the initial temperature of atoms is 16 µK. The number and the temperature of the trapped atoms could be deduced by the TOF measurements. They found that the temperature of the trapped atoms was about 7 µK and this value was close to the minimum height of potential barrier of 10 µK, where about 10 5 atoms stayed inside the trap for 100 ms and the lifetime of trapped atoms was about 20 ms as shown in Fig. 32. Figure 32 also shows the trapping efficiency as a function of the trapping time. For comparison, the simulation results are also shown as the solid line in Fig. 32 [78]. Since the detuning was much larger than the splitting between the hyperfine- structure levels of the excited state, the three-level interaction mode was quite good for the simulation. 6.5. Optical Dipole Trap Shin et al. proposed a three-dimensional, microscopic, and diffraction-limited far-off- resonance optical dipole trap (DFORT) for neutral atoms operating in the Lamb-Dicke regime, which is produced by employing the diffracted LP 01 mode of an HOF [34]. DFORT provides, in particular, a large trap volume so that it can be loaded with a large number of cold atoms. Moreover, the 3D DFORT can be also operated as an elongated 1D optical trap. Such a microscopic and tight optical trap can be realized with moderate laser power and a simple experimental setup. As the LP 01 mode propagates in free space, its initial annular intensity distribution, which is represented by the two peaks on the HOF exit facet diminishes away while the central bright peak starts to appear. The resultant generation of a tightly focused bright spot near the fiber facet can be qualitatively understood as the diffraction of the plane wave (i.e., the uniform LP 01 mode) by a ring-shaped aperture and the subsequent constructive interference near the central axis. Note that the intensity along the axial direction first increases to a maximum and then gradually decreases down to zero [Fig. 33(b)], and this axial asymmetry of the intensity gradient may be useful for efficient Applications of Hollow Optical Fibers in Atom Optics 153 loading of the nearby precooled atoms into DFORT. Note also that the axial intensity gradi- ent is larger than that of a typical focused Gaussian beam and the spot is even more tightly focused in the transverse radial direction [Fig. 33(b)]. r/l z/l r/l -10 0 10 0.0 0.5 1.0 r / l z/l z/l 0 200 400 Figure 33. (a) Typical diffraction profile of the LP 01 mode of HOF when a = 4λ and d = 3.5λ. (b) The contour plots of the produced optical potential. A, the radial distribution of DFORT at z = z 0 = 50.9λ, B, axial distribution from z = 0 to z = 600λ. As a specific application, they consider the 87 Rb atoms and the laser power P = 100 mW at the wavelength λ = 800 nm, which is far detuned from the D 1 and D 2 resonance lines. Then they obtain a cigar-shaped DFORT having the following basic parameters: the maximum trap depth U 0 = 7.9 mK, the axial trap frequency f z = 4.9 kHz, the radial trap frequency f r = 125 kHz, the trap volume V trap 10 −8 cm 3 , and the scattering rate at the trap center Γ sc = 2π × 310 Hz. In this case, the Lamb-Dicke parameters in the axial and the radial direction, defined by the ratio of the corresponding ground-state size a 0i to the laser wavelength, η i = 2πa 0i /λ (i = z, r), are given by η z = 0.86 and η r = 0.18, respectively. For 133 Cs atoms at λ = 900 nm, on the other hand, the trap depth is U 0 = 7.0 mK, the oscillation frequencies are f z = 3.3 kHz and f r = 842 kHz, the Lamb-Dicke parameters are η z = 0.75 and η r = 0.15, and the scattering rate is Γ sc = 2π × 317 Hz. 154 Heung-Ryoul Noh and Wonho Jhe The DFORT has the unique feature of a tightly focused trap with a large trap volume and convenient loading and cooling of the precooled atoms, and may be also operated as an elongated one-dimensional optical trap. 7. Conclusion We have reviewed various atom optical experiments using the hollow optical fibers. The detailed study on characteristics of electromagnetic fields inside and outside the HOF is presented. We review the neutral atom guidance by means of red- and blue-detuned laser beams, and the applications of the laser beams emanating from the HOF. In particular, an atomic guide in small hollow-core fibers has interesting applications. For example, by using submicron-sized hollow fibers, it is possible to carry cold atoms at arbitrary points on a substrate for precise control of atomic deposition. By using bent hollow fibers, an opto- gravitational trap [79] and an atom-laser cavity [80] can be also constructed. Furthermore, the nonlinear dynamics has been theoretically studied for the guided atoms inside HOF with the potential depth periodically modulated [81]. Moreover, cold atoms can be manipulated by a sharpened, nanometric optical fiber tip or trapped by an evanescent field induced near the tip [82], just like an optical tweezer is used for manipulating nanometric particles [83]. In particular, this may lead to the possibility of single atom manipulation so that atomic- scale crystal growth can be realized. Cold atom guidance by hollow fiber can be also applicable to crystal growth of silicon on the atomic scale with a near-field optical device if a laser of 252 nm wavelength is available. The cavity QED effect in the near-field region is also an interesting subject: hollow fiber or sharpened fiber with an induced optical near-field can be a unique tool for experimental study of the cavity QED effects by measuring the atomic deflection and comparing it with the dipole forces [64]. Recently, the scope of the atomic guidance through the HOF is extended: As well as cold atoms, the micro- or nano-sized particles were guided through the HOF [84]. Furthermore, in stead of hollow optical fibers, a photonic band-gap fiber (PBG) is also used for guiding atoms [85] or Bose-Einstein condensed atomic sample [86]. In the second part of this article, we have investigated the characteristics of the output intensity distribution of each LP lm mode in HOF using the diffraction theory, and compared them with experiments. We have observed that the LP 01 or LP 11 mode itself cannot satisfy the basic requirement for an atomic funnel since the former does not support a dark column along the central axis while the latter causes loss of atoms due to the line of nodes. To overcome these limitations, two LP 11 modes were excited separately with their node lines and polarizations orthogonal to each other and then combined at an even fraction. The resultant mode has an annular intensity distribution nearly similar to that of LP 01 and its output forms a divergent doughnut-shaped beam with the minimum dark spot of a few µm, and this HLB can provide a deep optical potential for an atomic funnel which focuses atoms from their source onto a micron-sized hollow region of an HOF with a high efficiency, and the evanescent field associated with the corresponding mode can allow atoms guided through an HOF. Moreover, the microscopic dark spot may be useful for the increase of the atomic density in the optical funnel trap suggested in reference [87, 88]. They can also be used for generation of the diffraction-limited dark laser spot [33]. Small focused dark spot Applications of Hollow Optical Fibers in Atom Optics 155 of HLB may be useful in atom optical experiments such as atomic lens, atom trap, and atom switch. The hollow laser beam generated by imaging the guided laser output of the hollow fiber has been used for atom guidance [28], atom fountains [29], and atom traps [30]. Further- more, the diffracted output of the LP 01 mode has a bright focused spot, which can be used for a tight optical dipole trap when a red-detuned laser is used [34]. References [1] Adams, C. S; Siegel, M.; Mlynek, J. Phys. Rep. 1994, 240, 143−210. [2] Balykin, V. I.; Letokhov, V. S. Atom Optics with Laser Light; Harwood Academic Publishers: Chur, 1995. [3] Dowling, J. P.; Geo-Banacloche, J. Adv. At. Mol. Opt. Phys. 1996, 34, 1−94. [4] Berman, P. R. Atom Interferometry; Academic Press: New York, 1997. [5] Dholakia, K. Contemp. Phys. 1998, 39, 351−369. [6] Balykin, V. I. Adv. At. Mol. Opt. Phys. 1999, 41, 181−260. [7] Hinds, E. A.; Hughes, I. G. J. Phys. D 1999, 32, R119−R146. [8] Friedman, N.; Kaplan, A.; Davidson, N. Adv. At. Mol. Opt. Phys. 2002, 48, 99−151. [9] Folman, R.; Kr¨ uger, P.; Schmiedmayer, J.; Denschlag, J.; Henkel, C. Adv. At. Mol. Opt. Phys. 2002, 48, 263−356. [10] Sudo, S.; Yokoyama, I.; Yaska, H.; Sakai, Y.; Ikegami, T. IEEE Photon. Tech. Lett. 1990, 2, 128−131. [11] Christov, I.; Kapteyn, H.; Murnane, M. Optics Exp. 1998, 3, 360−365. [12] Choi, S.; Eom, T. J.; Yu, J. W.; Lee, B. H.; Oh, K. IEEE Photon. Rechnol. Lett. 2002, 14, 1701−1703. [13] Choi, S.; Oh, K.; Shin, W.; Park, C. S.; Paek, U. C.; Park, K. J.; Chung, Y. C.; Kim, G. Y.; Lee, Y. G. IEEE Photon. Technol. Lett. 2002, 14, 248−250. [14] Choi, S.; Oh, K. Opt. Commun. 2003, 221, 307−312. [15] Oh, K.; Choi, S.; Jung, Y.; Lee, J. W. J. Lightw. Technol. 2005, 23, 524−532. [16] Ashkin, A. Phys. Rev. Lett. 1978, 40, 729−732. [17] Ol’shanii, M. A.; Ovchinnikov, Yu. B.; Letokhov,V. S. Opt. Commun. 1993, 98, 77−79. [18] Renn, M. J.; Montgomery, D.; Vdovin, O.; Anderson, D. Z.; Wieman, C. E.; Cornell, E. A. Phys. Rev. Lett. 1995, 75, 3253−3256. 156 Heung-Ryoul Noh and Wonho Jhe [19] Savage, C. M.; Marksteiner, S.; Zoller, P. In Fundamentals of Quantum Optics III; Ehlotzky F.; Ed.; Springer-Verlag: Berlin, 1993; pp 60−74 . [20] Marksteiner, S.; Savage, C. M.; Zoller, P.; Rolston, S. Phys. Rev. A 1994, 50, 2680−2690. [21] Jhe, W.; Ohtsu, M.; Hori, H.; Friberg, S. R. Jpn. J. Appl. Phys. 1994, 33, L1680−L1682. [22] Renn, M. J.; Donley, E. A.; Cornell, E. A.; Wieman, C. E.; Anderson, D. Z. Phys. Rev. A 1996, 53, R648−R651. [23] Ito, H.; Nakata, T.; Sakaki, K.; Ohtsu, M.; Lee, K. I.; Jhe, W. Phys. Rev. Lett. 1996, 76, 4500−4503. [24] Workurka, G.; Keupp, J.; Sengstock, K.; Ertmer, W. IQEC ’98, Technical Digest 1998, 235. [25] Dall, R. G.; Hoogerland, M. D.; Baldwin, K. G. H.; Buckman, S. J. J. Opt. B 1999, 1, 396−401. [26] Dall, R. G.; Hoogerland, M. D.; Tierney, D.; Baldwin, K. G. H.;Buckman, S.J. Appl. Phys. B 2002, 74, 11−18. [27] Yin, J. P.; Noh, H. R.; Lee, K.; Kim, K.; Wang, Y.; Jhe, W. Optics Comm. 1997, 138, 287−292. [28] Xu, X.; Kim, K.; Jhe, W.; Kwon, N. Phys. Rev. A 2001, 63, 063401. [29] Kim, K.; Noh, H. R.; Yeon, Y. H.; Xu, X.; Jhe, W.; Kwon, N. J. Korean Phys. Soc. 2001, 39, 877−880. [30] Xu, X.; Kim, K.; Jhe, W. J. Korean Phys. Soc. 2000, 37, 661−664. [31] Won, C.; Yoo, S. H.; Oh, K.; Paek, U. C.; Jhe, W. Opt. Commun. 1999, 161, 25−30. [32] Yoo, S. H.; Won, C. Y.; Kim, J. A.; Kim, K.; Oh, K.; Paek, U. C.; Jhe, W. J. Opt. B 1999, 1, 364−370. [33] Shin, Y. I.; Kim, K.; Kim, J. A.; Noh, H. R.; Jhe, W. Opt. Lett. 2001, 26, 119−121. [34] Shin, Y. I.; Heo, M.; Kim, J. W.; Shim, W.; Noh, H. R.; Jhe, W. J. Opt. Soc. Am. B 2003, 20, 937−941. [35] Ito, H.; Sakaki, K.; Nakata, T.; Jhe, W.; Ohtsu, M. Opt. Commun. 1995, 115, 57−64. [36] Chaudhuri, P. R.; Lu, C.; Xiaoyan, W. Opt. Commun. 2003, 228, 285−293. [37] Hautakorpi, M.; Kaivola, M. J. Opt. Soc. Am. A 2005, 22, 1163−1169. [38] Marcuse, D. Light Transmission Optics; Van Nostrand Reinhold: New York, 1982. Applications of Hollow Optical Fibers in Atom Optics 157 [39] Okoshi, T. Optical Fibers; Academic Press: New York, 1989. [40] Snitzer, E. J. Opt. Soc. Am. 1981, 51, 491−498. [41] Gloge, D. Appl. Opt. 1971, 10, 2252−2258. [42] Ito, H.; Sakaki, K.; Nakata, T.; Jhe, W.; Ohtsu, M. Ultramicroscopy 1995, 61, 91−97. [43] Akhmanov, S. A.; Nikitin, S. Y. Physical Optics; Oxford University Press: New York, 1997. [44] Gaskill, J. D. Linear Systems, Fourier Transform, and Optics; John Wiley & Sons: New York, 1978. [45] Sheppard, C. J. R.; Gan, X. S.; Fatemi, H. Electron. Lett. 1996, 32, 1128−1130. [46] Renn, M. J.; Zozulya, A. A.; Donley, E. A.; Cornell, E. A.; Anderson, D. Z. Phys. Rev. A 1997, 55, 3684−3696. [47] Hayashi, S.; Ishimizu, A.; Tohei, T.; Tachikawa, M. Phys. Rev. A 2003, 68, 053408. [48] Chevrollier, M.; Fichet, M.; Oria, M.; Rahmat, G.; Bloch, D.; Ducloy, M. J. Phys. II France 1992, 2, 631−657. [49] Courtois, J. Y.; Courty, J. M.; Mertz, J. C. Phys. Rev. A 1996, 53, 1862−1878. [50] M¨ uller, D.; Cornell, E. A.; Anderson, D. Z.; Abraham, E. R. I. Phys. Rev. A 2000, 61, 033411. [51] Lu, Z. T.; Corwin, K. L.; Renn, M. J.; Anderson, M. H.; Cornell, E. A.; Wieman, C. E. Phys. Rev. Lett. 1996, 77, 3331−3334. [52] Letokhov, V. S. Laser Photoionization Spectroscopy; Academic Press: San Diego, 1987. [53] Drake, G.W. Atomic, Molecular, and Optical Physics Handbook; AIP Press: New York, 1996. [54] Dinneen, T. P.; Wallace, C. D.; Tan, K.-Y. N.; Gould, P. L. Opt. Lett. 1992 17, 1706−1708. [55] Ito, H.; Sakaki, K.; Jhe, W.; Ohtsu, M. Appl. Phys. Lett. 1997, 70, 2496−2498. [56] Ito, H.; Sakaki, K.; Jhe, W.; Ohtsu, M. Opt. Commun. 1997, 141, 43−47. [57] Fatemi, F. K.,; Bashkansky, M.; Moore, S. Opt. Express 2005, 13, 4890−4895. [58] Casimir, H. B. G.; Polder, D. Phys. Rev. 1948, 73, 360−372. [59] Berman, P. R. Cavity Quantum Electrodynamics; Academic Press: San Diego, 1994. [60] Jhe, W.; Kim, J. W. Phys. Rev. A 1995, 51, 1150−1153. 158 Heung-Ryoul Noh and Wonho Jhe [61] Jhe, W.; Kim, J. W. Phys. Lett. A 1995, 197, 192−196. [62] Wylie, J. M.; Sipe, J. E. Phys. Rev. A 1984, 30, 1185−1193. [63] Nha, H.; Jhe, W. Phys. Rev. A 1996, 54, 3505−3513. [64] Ohtsu, M. Near-Field Nano/Atom Optics and Technology; Springer-Verlag: Tokyo, 1998. [65] Jhe, W.; Noh, H. R.; Ito, H.; Ohtsu, M. unpublished. [66] Landragin, A.; Courtois, J. Y.; Labeyrie, G.; Vansteenkiste, N.; Westbrook, C. I.; Aspect, A. Phys. Rev. Lett. 1996, 77, 1464−1467. [67] Saleh, B. E. A.; Teich, M. C. Fundamentals of Photonics; Wiley Interscience Publica- tion: New York, 1991. [68] Lee, K. I.; Kim, J. A.; Noh, H. R.; Jhe, W. Opt. Lett. 1996, 21, 1177−1179. [69] Kim, J. A.; Lee, K. I.; Noh, H. R.; Jhe, W.; Ohtsu, M. Opt. Lett. 1997, 22, 117−119. [70] Takamizawa, A.; Ito, H.; Ohtsu, M. Appl. Phys. Lett. 2004, 85, 1790−1792. [71] Takamizawa, A.; Ito, H.; Yamada, S.; Ohtsu, M. Jpn. J. Appl. Phys. Pt. 1 2002, 41, 6215−6218. [72] Takamizawa, A.; Ito, H.; Ohtsu, M. Jpn. J. Appl. Phys. Pt. 1 2000, 39, 6737−6742. [73] Kasevich, M.; Riis, E.; DeVoe, R. G.; Chu, S. Phys. Rev. Lett. 1989, 63, 612−615. [74] Clairon, A.; Salomon, C.; Guellati, S.; Phillips, W. D. Europhys. Lett. 1991, 16, 165−170. [75] Szymaniec, K.; Davies, H. J.; Adams, C. S. Europhys. Lett. 1999, 45, 450−455. [76] Davidson, N.; Lee, H. J.; Adams, C. S.; Chu, S. Phys. Rev. Lett. 1995, 74, 1311−1314. [77] Song, Y.; D. Milam, D.; Hill III, W. T. Opt. Lett. 1999, 24, 1805−1807. [78] Xu, X.; Minogin, V. G.; Lee, K.; Wang, Y. Z.; Jhe, W. Phys. Rev. A 1999, 60, 4796−4804. [79] Harris, D. J.; Savage, C. M. Phys. Rev. A 1995, 51, 3967−3971. [80] Moy, G. M.; Hope, J. J.; Savage, C. M. Phys. Rev. A 1997, 55, 3631−3638. [81] Liu, X. M.; Milburn, G. J. Phys. Rev. A 2000, 61, 053401. [82] Kobayashi, K., Sangu, S.; Ito, H.; Ohtsu, M. Phys. Rev. A 2001, 63, 013806. [83] Novotny, L.; Bian, R. X.; Xie, X. S. Phys. Rev. Lett. 1997, 79, 645−648. [84] Renn, M. J.; Pastel, R.; Lewandowski, H. J. Phys. Rev. Lett. 1999, 82, 1574−1577. Applications of Hollow Optical Fibers in Atom Optics 159 [85] Bigelow, M.; Takekoshi, T.; Maes, C.; Knize, R. Abstract of DAMOP Meeting 2005, 42. [86] Will, S.; Christensen, C.; Jo, G.-B.; Pasquini, T. A.; Saba, M.; Shin, Y.; Ketterle, W.; Pritchard, D. E. Abstract of ICAP Meeting 2006, 506. [87] Morsch, O.; Meacher, D. R. Opt. Comm. 1998, 148, 49−53. [88] Yin, J. P.; Zhu, Y. Optics Comm. 1998, 152, 421−428. In: Optical Fibers Research Advances ISBN: 1-60021-866-0 Editor: Jurgen C. Schlesinger, pp. 161-185 © 2007 Nova Science Publishers, Inc. Chapter 5 ADVANCES IN PHYSICAL MODELING OF RING LASERS Vittorio M.N. Passaro 1* and Francesco De Leonardis 2** 1 Photonics Research Group, Dipartimento di Elettrotecnica ed Elettronica, Politecnico di Bari, via Edoardo Orabona n. 4, 70125 Bari, Italy 2 Photonics Research Group, Dipartimento di Ingegneria dell’Ambiente e per lo Sviluppo Sostenibile, Politecnico di Bari, viale del Turismo n. 8, 74100 Taranto, Italy Abstract In this chapter, an overview on fiber ring lasers and III/V semiconductor integrated ring lasers is presented. In particular, some aspects of mathematical modelling of these devices are reviewed. In the first part of the chapter, we have focused our attention on the more recent theoretical and experimental studies concerning fiber ring laser architectures. Then, a complete quantum-mechanical model for integrated ring lasers is presented, including the evaluation of all the involved physical parameters, such as self and cross saturation and backscattering. Finally, the influence of sidewall roughness on either unidirectional or bidirectional regime in multi-quantum-well III/V semiconductor ring lasers is demonstrated. Keywords: Semiconductor Ring lasers, Fiber Ring Lasers, Multi quantum well, Modeling Introduction Nowadays, fiber ring lasers are typical fiber devices obtained by placing a fiber amplifier inside a cavity to provide optical feedback, so involving a number of linear and non linear physical effects. Moreover, semiconductor ring lasers are of great interest for monolithic and integrated optoelectronic applications. Since these lasers do not require the use of cleaved facets or difficult Bragg gratings for optical feedback, they can be conveniently integrated so reducing the occupation area. Moreover, the semiconductor ring lasers offer new capabilities * E-mail address:
[email protected] ** E-mail address:
[email protected] Vittorio M.N. Passaro and Francesco De Leonardis 162 in the travelling-wave unidirectional oscillation. Unidirectional operation is desired because it offers the advantages of enhanced mode purity (high side-mode suppression ratio), reduced sensitivity to feedback and higher single beam power. Unidirectional ring lasers are used for telecommunications systems, feedback laser diodes, multi-wavelength and all-optical flip flop operation. In this work we present both fiber and semiconductor ring lasers, with some recent advances on their physical models. The chapter is substantially divided in two parts. The first part is essentially a review of fiber ring laser architectures, their relevant physical effects, technologies and main applications, including erbium doped devices, and continuous wave and mode-locking operations. In the second part, we focus on integrated architectures of semiconductor ring lasers and their physical models. In particular we investigate the influence of some technological parameters (ring sidewall roughness, ring radius) over the backscattering coefficient influencing the operation regime and performance of any semiconductor ring laser. The theory of the physical model based on quantum mechanical approach is briefly summarized and some numerical results and simulations applied to multi- quantum-well III/V semiconductor ring lasers are presented. Here how either unidirectional or bidirectional regime is related to the ratio between the injection current and the backscattering coefficient value is shown. Finally simulation results relevant to an architecture of the ring laser with asymmetrical output coupler to obtain a stable unidirectional regime are presented. In general, our model is based on four differential equations in total, two coupled equations for the counter-propagating modes, one rate equation for the carriers injected in the laser active region and one equation describing the phase difference dynamics between the two modes. Both the self- and cross- saturation effects and the backscattering effect over both microscopic and macroscopic scale have been taken into account, where with microscopic scale we mean the effect of the backscattered wave on the gain medium, while as macroscopic scale the effect of the backscattering on the Maxwell's equations is considered. By our model we can evaluate all the physical coefficients of the rate equations by means of a full quantum mechanical analysis. In fact, starting by the energy band information of the MQW structure and assuming typical values for the time constants of the electron scattering, our model is able to evaluate the linear and non linear gain coefficients by using a compact density matrix formalism. Then, these parameters are put inside the rate equations to calculate the dynamic evolution of the oscillating modes in the ring resonator. By the study of this dynamics, the investigation of any ring laser operating regime, bidirectional or unidirectional, should be possible. Fiber Ring Lasers: An Overview Fiber ring lasers have attracted tremendous interest because of their many important applications in fiber-optic test and measurement [1], fiber communications [2], fiber sensor systems [3], and high-resolution spectroscopy. This is mainly due to their features, such as wide tunable range, narrow linewidth, and tuning at high speed allowing fast component characterization. However, despite of these advantages, the fiber ring lasers present a number of drawbacks essentially related to the possibility of multimode operation, which induces the beat-noise generated as a result of beating effect between the lasing longitudinal modes and could severely limit the lasers applications. Advances in Physical Modeling of Ring Lasers 163 Differents methods to realise a fiber laser have been proposed in recent years. All the experimental architectures can be classified in two main groups: fiber lasers based on nonlinear effects, and fiber lasers based on doped fiber amplifiers. Basically, both ring and Fabry-Perot geometries have been used to realise the resonant cavity in which a piece of nonlinear or doped fiber as active medium must be inserted. Nonlinear effects play an important role in high power fiber lasers. They are induced by the large amount of power density in the small area fiber core. Among the nonlinear effects, the Stimulated Raman Scattering (SRS) and the Stimulated Brillouin Scattering (SBS) have shown significant potentials. The fiber-Raman lasers not only have a lower threshold compared with the single pass SRS, but also they can be tuned over a wide frequency range (10 THz). Fig. 1 schematically shows a typical fiber Raman laser [4]. Fiber Pump Pump M 1 M 2 Prism Lens Lens Figure 1. Schematic architecture of a tunable fiber Raman laser. A piece of single-mode fiber is placed inside a Fabry-Perot cavity formed by the partially reflecting mirrors M1 and M2. The cavity provides a resonant, wavelength-selective feedback for Stokes light generated inside the fiber by means of the SRS effect. The intracavity prism is needed to tune the laser wavelength by spatially dispersing various Stokes wavelengths which can be selected by rotating the mirror M2. It is evident that the laser threshold corresponds to the pump power at which the Stokes amplification during a round trip is large enough to balance the cavity losses. As mentioned in [4], a threshold level of about 1 W can be obtained using a fiber length of about 10 m. By adding separate mirrors for each order of Stokes waves, the fiber-Raman laser can be operated at several wavelengths simultaneously, each of which can be independently tuned by tuning the mirrors [5]. Several experiments have demonstrated that it is relatively easy to achieve synchronization in fiber-Raman lasers. The reason is that the laser can select a particular wavelength satisfying the synchronous- pumping requirement among the wide range of possible wavelengths near the peak of the Raman-gain spectrum. Moreover, the laser wavelength can be tuned by simply changing the cavity length. This technique is referred to as time-dispersion tuning to distinguish it from prism tuning based on spatial dispersion provided by the prism. Thus, synchronously pumped fiber –Raman lasers have attracted attention for generating ultrashort optical pulses. In addition, if the Raman pulse falls in the anomalous group velocity dispersion (GVD) regime of the fiber, the soliton effects can create pulses with widths of about 100 fs or less. Similar to the SRS case, the Brillouin gain in optical fibers can be used to make fiber- Brillouin lasers by placing the fiber inside a cavity. To this aim, both ring cavity and Fabry- Perot geometries have been used. Fiber-Brillouin can lead to obtain a threshold input power of 0.56 mW by using an all-fiber ring resonator, as shown schematically in Fig. 2 [4] Vittorio M.N. Passaro and Francesco De Leonardis 164 Fiber-Brillouin lasers consisting of a Fabry-Perot cavity exhibit features which are qualitatively different from those of a ring cavity configuration. The difference arises from the simultaneous presence of forward and backward propagating components associated with the pump and Stokes waves. The simultaneous presence of many equi-spaced spectral lines in the output of a fiber-Brillouin laser indicates the possibility of obtaining ultrashort optical pulses if the laser can be mode-locked. Thus, an intra-cavity modulator could be useful to realise the mode-locking process [4]. However, Brillouin backscattering can be the limitation of the output power for narrow- band signals; and Raman scattering can generate a frequency shift which decreases the pump power and signal power. For these reasons, a great attention has been reserved to fiber lasers based on doped fiber amplifier. Pump Laser Spectrum Analyzer Lens Directional Coupler Fiber ring Polarization controller Figure 2. Schematic architecture of a fiber-Brillouin ring laser. The erbium-doped fiber amplifier (EDFA) was invented by D.N. Payne and co-workers in 1987 [6]. The EDFA commercialization made long haul optical communications inexpensive and reliable and optical fiber became the standard for long-haul telecommunication systems. However, in situations where high-power lasers are needed, an EDFA does not work very well because the high power density damages the fiber. Nd-doped fiber lasers (NDFL) and Yb-doped fiber lasers (YDFL) were developed for scaling output power of fiber lasers. Neodymium can be pumped at 808 nm to get good absorption, while ytterbium can be pumped at 975 nm. Both of these elements can emit light at around 1060 nm with slightly different energy transition mechanisms. Initially, YDFLs attracted less attention with respect to NDFLs essentially because Nd 3+ has the advantage of a four-level pumping scheme, while Yb 3+ works with a three or quasi four-level scheme. A four level laser system tends to easier lasing because it has a lower threshold. However, ytterbium offers higher power conversion efficiencies and larger output powers. In fact, even if its energy level structure and the re-absorption effect make the threshold pump power relatively high, ytterbium does not have self-quenching effects [7]-[8] as neodymium, and it can have a higher ion concentration. Moreover, ytterbium can be more efficient due to the small quantum defects. Several theoretical works have been presented in literature to model and design fiber ring lasers. Generally speaking, these models are mainly focused on the study of the gain medium (in case of doped fiber lasers) and on nonlinear effects that origin in the ring cavity. In the Advances in Physical Modeling of Ring Lasers 165 former case, the mathematical modelling is constituted from a system of rate equations describing the time evolution of the carrier population and photon densities. These two kinds of rate equations contain complete information about the dynamics of the laser system. Since they are non linear coupled equations, a straightforward analytical solution is not feasible. A numerical solution of these set of equations is presented in [9]. They can be analyzed under certain approximations corresponding to real physical situations and an analytical solution is possible in some cases [9]. A generalization of the mathematical model to the case of multi-frequency erbium-doped fiber ring lasers employing a periodic filter and a frequency shifter has been also presented [10]. In this work, the rate equation systems describing the laser dynamics are coupled with three spatial differential equations related to the pump propagation inside the fiber ring and both forward and backward propagating amplified spontaneous emission (ASE) powers. An iterative solution of the rate equations and propagation equations for two counter-propagating ASE powers and pump has been implemented using a fourth-order Runge–Kutta routine in which appropriate boundary conditions were been imposed at beginning and end of the active fiber, according with the experimental setup. More recently, a steady-state model has been proposed to analyse and design a quasi- continuous wave injection-seeded ring laser [11]. As it is known, frequency stabilized laser diodes can achieve a high degree of wavelength stability and are well-suited as master oscillators for spectroscopic lidar systems. However, diode lasers alone do not have sufficient power to provide for a high signal-to-noise ratio in these systems. One method to efficiently scale the system power is to use the stabilized laser as a seeder in an injection-locked fiber ring laser. Two factors can limit the performance of seeded ring lasers. First, the spectral overlap between the seed laser and a ring laser cavity mode. The use of a low-finesse ring cavity has been shown to relax this spectral requirement making the seeding process relatively easy. The second issue is that the laser self oscillation can dominate the seeding process. Self-seeding can be overcome through the incorporation of intracavity filters or using sufficient seed power. The latter condition has been assumed in a model proposed in literature [11]. However, it is worth noting that the above cited mathematical models do not require heavy computational efforts, essentially because they do not include optical nonlinear effects, that can originate in the ring fiber in high power regime. Generally, the nonlinear effects in fiber ring lasers can be modelled by means of the nonlinear Schroedinger equation (NLS) [4]. In particular, the NLS equation represents a powerful mathematical tool not only to design the fiber-Raman or fiber-Brilluoin lasers [4], but also to analyse the wave behaviour of an optical fiber exhibiting a weak Kerr nonlinearity. The NLS has been extensively investigated in this context, with particular emphasis given to the robust and stable soliton solutions that result from a fundamental balance between linear dispersion and cubic nonlinearity [4]. Thus, soliton pulses are ideal carriers for transmitting optical data. For applications for which polarization effects are important, one must consider a system of coupled NLSs, that is generally not integrable. Despite the increased numerical complexity, in order to manage soliton like solutions it is not only useful to optimise the design of long-haul communications systems [12], but also to develop efficient optical fiber ring lasers [13]-[16]. For the ring laser configuration, Kerr nonlinearity of the birefringent optical fiber generates a nonlinear rotation of the polarization state that depends on the pulse intensity. Then, the insertion of a passive polarizer provides an effective intensity filter that stabilizes, or mode locks, a propagating Vittorio M.N. Passaro and Francesco De Leonardis 166 pulse by periodically attenuating all components of the pulse that are not aligned with the polarizer. Simple devices such as these have been shown experimentally to generate stable and robust soliton-like pulse trains, that can be used for a wide variety of telecommunications purposes [13]-[17]. Recently, an advanced mathematical model has been proposed in literature [18] for fiber ring lasers. This model includes in the standard NLS equations two key modeling elements that describe the mode-locking dynamics: the nonlinear polarization rotation induced by cross-phase modulation, and the polarization control through the passive polarizer. In particular, the theoretical model consists of the coupled NLSs with periodic perturbations that are due to the polarizer. In deriving this model, the contributions from continuum radiation have been neglected. This is due to the filtering function of the passive polarizer in the mode- locking process. Although a residual amount of radiation is expected from the periodicity of the cavity, it remains negligible in comparison with the energy in the localized mode-locked pulse. In addition, a source of amplitude fluctuations arising from the interplay between nonlinearity and polarization control has been considered. In conclusion, the mathematical model proposed in [18] represents a useful theoretical tool to examine the underlying mode- locking mechanism of the fiber ring and to describe the systematic and predictable amplitude fluctuations that result from this interaction. As mentioned above, the main drawback in the fiber ring laser is related to the multimode longitudinal behaviour. This disadvantage has induced many research groups to investigate experimental solutions and search different configurations to optimise the fiber ring laser performance. A number of studies have been presented in literature to reduce the beat noise, and to realise a fiber ring laser with single-longitudinal-mode (SLM) operation. For example, a compound-ring cavity to reduce the beat-noise has been proposed [19] where, due to the short ring-length, the dual-coupler fiber ring acts as a small free spectral range (FSR) etalon filter and, combined with the tunable optical bandpass filter, selects one longitudinal mode. However, this proposed solution needs PZT to accurately control the length of the cavity. Recently, the research on SLM operation using saturable absorbers has been reported to overcome the limited spectral width of the mode selection filters [19]. Additionally, the longitudinal lasing mode become unstable when the fiber ring cavity and the filter are highly sensitive to the temperature drift and other external disturbances. To prevent the unstable operation, the fiber lasers of various schemes have been reported so that length of tunable filter and fiber ring cavity were stabilized [20]-[21], or a saturable absorber was used to suppress multimode operation [22]-[24]. As it is known, a large free spectrum range FSR is also required to facilitate the SLM operation. In this sense, an architecture based on S-band ring lasers has been recently proposed [25]. Fig. 3 shows the proposed configuration and experimental setup of an S-band erbium fiber laser with a triple-ring cavity resonator. The architecture is constituted by the main ring (Ring-1), coupled with two rings having different lengths (Ring-2, and Ring-3) by means of two 50:50 optical couplers (C). The Ring- 1 is composed of an S-band EDFA module, a 90:10 optical coupler (C 1 ) to couple out the optical beam, a fiber Fabry-Perot tunable filter (FFP-TF), and two polarization controllers (PCs). It is worth noting that the presence of two PCs lead to align the state of polarization of the Ring-1 cavity to guarantee a SLM oscillation. In addition, the FFP-TF is necessary not only to determine a lasing wavelength, but also it serves as a mode-restricting component to provide the first restriction on the possible laser modes. Advances in Physical Modeling of Ring Lasers 167 C 980 nm Pump LD W W C C C 1 PC PC FFP-TF S-Band EDFA Module Output Ring 3 Ring 2 Ring 1 EDF Isolator Figure 3. Experimental setup of an S-band erbium fiber laser with a multiple-ring cavity structure. Finally, the S-band amplifier, constituted by three isolators, a pump laser at 980 nm, and two erbium doped fibers (EDF) with a saturated output power of 16.1 dBm at 1498 nm, has a depressed-cladding design and a power-sharing 980 nm pump laser to generate EDF gain extension effect [26]. The behaviour of this architecture is based on Vernier effect. In fact, indicating with FSRm and FSRs the free spectral range of the main and sub-ring cavities, respectively, the value of effective FSR becomes the least common multiple number of both FSRm and FSRs. As a result, the mode suppression can be achieved and governed by the length of the main- ring and sub-ring cavities. Thus, the laser mode oscillates only at a frequency that simultaneously satisfies the resonant conditions of main cavity and all the sub-ring cavities. Due to the combination of a FFP-TF with a triple-ring cavity, a SLM selection in this fiber laser is successfully achieved. The polarization state of proposed laser should be maintained by adjusting the PCs as that of the Ring-1. The experimental results performed by means of an optical spectrum analyzer (OSA) with a 0.05 nm resolution [25], indicate that an output power larger than 0 dBm, a power stability less than 0.05 dB, a wavelength variation less than 0.02 nm and a side-mode suppression ratio (SMSR) larger than 54.3 dB / 0.05 nm can be obtained over an operating range of 1481 to 1513 nm. As mentioned before, one of the main problems considered in the fiber ring laser is the beat-noise. Recently, a novel method to suppress the beat-noise fiber ring laser using a Fabry– Pérot laser diode (FP-LD) has been proposed in [27]. As reported there, the beat-noise of fiber ring lasers is primarily in the low-frequency region of about 10 MHz due to the long ring cavity length, which is typically of the order of several tens of meters. Thus, basically an injected FP-LD can act as a high-pass filter to suppress the low-frequency beat-noise of fiber ring lasers [28]. This is mainly due to the fact that the FP-LD has a fast carrier recovery rate Vittorio M.N. Passaro and Francesco De Leonardis 168 (1 ns) and experiments the gain saturation effect. Fig. 4 shows the experimental set-up used [27]. Pump Fiber Fabry-Perot Filter FP-LD Circulator 1 Circulator 2 Output 3 dB Coupler Angled Splices Fusion Splice Bi-EDF A B 1 2 3 3 2 1 Figure 4. Configurations of the highly polarized, low beat-noise, tunable fiber ring laser. The architecture is constituted by an Bismuth oxide-based EDF (Bi-EDF) [29] pumped by one 1480-nm semiconductor laser diode via Port 1 of an optical circulator (circulator 1), which fairly exhibits a flat pass-band in the wavelength range from 1460 to 1630 nm. The length of the Bi-EDF was of 51.4 cm. In the experiments, the refractive index of the Bi-EDF core and cladding were 2.03 and 2.02, while the diameters of core and cladding were 3.9 and 124.7 m, respectively. The erbium concentration in the Bi-EDF was 6500 wt ppm and Boron and Lanthanum are co-doped in the Bi 2 O 3 -based fiber to increase the pump efficiency. The Bi-EDF peak absorption measured at 1480 and 1530 nm were 141 and 219 dB/m, respectively. Both ends of Bi-EDF were first angle spliced to high numerical aperture fiber (Nufern 980-HP fiber) before splicing to Port 2 (SMF-28 fiber) of Circulator 1 and to Port 3 (SMF-28 fiber) of Circulator 2, providing better mode field diameter matching. The splicing loss attained was less than 0.2 dB for the angled splices. The angled splices reduce the reflection in the laser cavity to less than 60 dB. The set-up presents also a large FSR fiber Fabry–Pérot (FFP) filter employed to tune the laser wavelength and a double-channel planar-buried hetero-structure FP-LD with cavity mode spacing and threshold current of 1.12 nm and 10.9 mA, respectively. The optical bandwidth and FSR of FFP filter were about 50 pm ( 6.25 GHz) and 110 nm, respectively. Thus, since the gain bandwidth of Bi-EDF is less than 110 nm [14], only one wavelength in the fiber ring laser cavity is excited. The FP-LD was biased at 12 mA, slightly above the threshold current, to realize the low beat-noise laser output. Finally, a 3-dB fused fiber taper was included in the laser cavity to provide the laser output. Advances in Physical Modeling of Ring Lasers 169 The experimental results [27] show that the configuration described in Fig. 4 leads to obtain high performance, including a fiber ring laser wavelength tuning from 1536.82 to 1570.72 nm in 1.12 nm steps, with a maximum output power of about +3 dBm. The polarization degree and extinction ratio of the laser output are about 99% and 60 dB, respectively. Finally, the beat-noise, with and without FP-LD, was dramatically reduced by 50 dB. In parallel to the amount of research focused to realise SLM fiber ring laser, a number of studies has been proposed about multi-wavelength erbium-doped fiber lasers (MW-EDFLs). They have attracted considerable interest for potential applications in optical test and measurement, and optical wavelength-division-multiplexing communication and sensing systems [30]. Compared with compact semiconductor-based lasers, EDFLs are competitive because of their all-fiber structure, as well as their capacity to provide high power and ultrashort pulse width [31]. They can be used in applications that require multiple wavelength sources, with small equal-wavelength spacing, a large number of peaks within a broad band, and high output uniformity across the channels. These requirements pose a very challenging task for building a cost-effective multi-wavelength EDFL for continuous-wave or pulsed operation. Previously, due to the homogeneously broadened gain property, many MW-EDFLs were developed with wavelength spacing larger than homogeneous linewidth of about 3.5nm, to overcome gain competition. Many different approaches have also been explored for developing MW-EDFLs, including use of polarization or spatial hole burning, use of independent gain media, frequency shifting, and phase modulation [32]-[36]. However, recently a novel room-temperature-operated MW-EDFL with wavelength spacing less than the homogeneous broadening linewidth, based on interchannel four-wave mixing (FWM), has been proposed [37]. The EDF gain-clamping effect is compensated by the parametric four wave mixing (FWM) between multi-wavelength channels in a highly nonlinear fiber section that is inserted into the fiber ring cavity and it is based on highly nonlinear photonic crystal fiber (HNL-PCF). Finally, sampled-fiber Bragg grating (SFBG) is used in input to one port of the circulator (see Fig. 5). EDFA Sampled FBG Circulator 10% Output Polarization controller HNL-PCF Output coupler Figure 5. Schematic diagram of the multi-wavelength erbium-doped fiber laser. Vittorio M.N. Passaro and Francesco De Leonardis 170 The multiple wavelength operation has been initiated, by adjusting the polarization controller and a specific sampled fiber Bragg grating (SFBG), with 0.8nm wavelength spacing. The architecture uses the inter-channel four-wave mixing-induced dynamic gain- flattening mechanism to stabilize the output. Thus, the FWM processes created by means of the HNL-PCF suppress the EDFL homogeneous line broadening and stabilize the multiple wavelength oscillation. By tuning the intracavity polarization controller and then the FWM efficiency, the number of concurrent lasing wavelengths can be changed from two to five, and the peak power differences for the main oscillation wavelengths are less than 2.0 dB. Channel spacing of 0.5nm operation of 10GHz dual-wavelength mode locking with the help of fiber birefringence, has been obtained with cavity structure with 60 m HNL-PCF [37]. In addition, a supermode suppression ratio higher than 60dB, and a time-bandwidth products ranging from 0.39 to 0.41, have been measured. Semiconductor Ring Lasers Nowadays, multi-quantum-well (MQW) semiconductor ring lasers are of great interest for monolithic and integrated optoelectronic applications [38]. Since these lasers do not require the use of cleaved facets or difficult Bragg gratings for optical feedback, they can be conveniently integrated so reducing the occupation area. Moreover, the semiconductor ring lasers offer new capabilities in the travelling-wave unidirectional oscillation. Unidirectional operation is desired because it offers the advantages of enhanced mode purity (high side- mode suppression ratio), reduced sensitivity to feedback and higher single beam power. Unidirectional ring lasers have been used for feedback laser diodes [39]. Other important applications of semiconductor ring lasers include multi-wavelength [40] and all-optical flip flop operation [41-42]. A number of experimental studies have been also carried out to analyse the operation regimes of the semiconductor ring lasers [43-45]. The main drawback of the semiconductor ring lasers could be represented by the instable regime of behaviour (switching between unidirectional and bidirectional). Thus, this physical situation needs an accurate physical model to individuate the design criteria in order to realise stable unidirectional regime. In this sense, recently it has been experimentally demonstrated that GaAs-AlGaAs ring lasers show either bidirectional or unidirectional regimes depending on the injection current [45]. In particular, bidirectional operation reveals that just above threshold the ring laser operate in a regime where the two counter-propagating modes are continuous waves. As the injected current is increased, a new regime appears where the intensities of the counter- propagating modes undergo alternate sinusoidal oscillations. Finally, for injection currents larger than a critical value, the unidirectional regime appears in a stable way. Therefore, the control of the unidirectional operation is of fundamental importance in order to use the semiconductor ring laser as an integrated source. In this chapter, we propose a very accurate physical model to analyse the operating regimes of a MQW semiconductor ring laser. Similarly, a physical model has been recently presented to analyse the operating regimes of any MQW semiconductor ring laser [46], as a strong generalization of the models previously proposed in literature. In fact, it can be applied to a generic MQW semiconductor structure, not only to a two level gas system [47]–[48] (for gas laser), and it considers the macroscopic effect of the backscattering non included in the model proposed in [47]–[48]. In addition, the Advances in Physical Modeling of Ring Lasers 171 model in [46], differently by [45], leads to calculate all the physical parameters without any semi-empirical approximation, taking into account the backscattering effect on the gain medium. It is based on four differential equations in total, two coupled equations for the counter-propagating modes, one rate equation for the carriers injected in the laser active region and one equation describing the phase difference dynamics between the two modes. Anyway, in the following section the theory of the physical model proposed in [46] is briefly summarized with the aim to introduce some recent advances of this numerical model. In particular we have investigated the influence of some technological parameters (ring sidewall roughness, ring radius) over the backscattering coefficient influencing the operation regime and performance of any MQW semiconductor ring laser.. Then, we present some numerical results and simulations applied to a GaAs-AlGaAs MQW semiconductor ring laser. In that section we will show as either unidirectional or bidirectional regime is related to the ratio between the injection current and the backscattering coefficient value, i.e. technological considerations have been summarized. Finally, simulation results relevant to an architecture of the ring laser with asymmetrical output coupler to obtain a stable unidirectional regime are presented. Theory The theory is based on the semi-classical interaction between radiation and matter. Then, the atomic systems are modelled as quantum phenomena while the electromagnetic (e.m.) field is classically described by the Maxwell’s equations. In particular, the electric dipole e r operator relates the system quantum-mechanical description with the polarization P of the medium used as a source of the e.m. field. Assuming a predominant single transverse mode as electric field inside the ring laser, we can write: ( ) ( ) ( ) ( ) ( ) E , F . . n n j t t n n t E t e c c ω φ + = + ∑ r r (1) where c.c. indicates the conjugate complex terms, ( ) n E t is the electric field amplitude, n ω is the angular pulsation of the optical mode inside the MQW ring cavity and ( ) n t φ is the time- dependent phase of the electric field. In general, the field function ( ) F r can be written as ( ) ( , ) ( ) F r G x y U z = , where ( , ) G x y and ( ) U z indicate the transverse and longitudinal profile of the electric field, z representing the propagation direction (curvilinear coordinate). The subscript n takes into account all possible longitudinal modes in the ring cavity. With this representation for the field, the polarization P induced by the gain medium is given by: ( ) ( ) ( ) ( ) ( ) P , F . n n j t t n n t P t e cc ω φ + = + ∑ r r (2) where ( ) n P t is the complex, slowly-varying component of the polarization of the n-th longitudinal mode. The wave equation for the time evolution of the electric field is given by: Vittorio M.N. Passaro and Francesco De Leonardis 172 2 2 2 0 0 2 2 2 1 t v t t μ μ ∂ ∂ ∂ −∇ + + = − ∂ ∂ ∂ J E P E (3) where v is the velocity of the electric wave in the ring resonator. The second term is included to take into account the cavity losses. In particular the current density is expressed in terms of the fictional conductivity σ = J E, where the conductivity σ is assumed as the sum of two contributions, 1 σ and 2 σ . The former includes all kind of optical losses (propagation loss, radiation loss, leakage loss, etc..) of the n-th longitudinal mode. It is related to the quality factor n Q of the cold cavity by means of the relationship ( ) 1 n n Q σ εω = , where ε is the permittivity of the cavity. The latter contribution is included to take into account the effect of the backscattering induced by the cavity sidewall roughness. According to [49], the backscattered wave induces a contribution to the fictional conductivity given by: 2 b σ ε = , where b is the backscattering rate. This rate coefficient is related to the amplitude reflectivity R due to the backscattering as ( ) eff eff b cR n R = π , being c the free-space light velocity, eff n the effective index of the optical wave in the ring cavity and eff R the ring effective radius. With the aim to study the dynamics of the two counter-propagating modes, we can particularise Eqs. (1)-(2) to only two modes, one clockwise (CW, as mode 1) and the other counter-clockwise (CCW, as mode 2). Under slowly-varying amplitude and phase approximations, extensively used in the laser dynamics modelling, the wave equation (3) produces the following set of equations: ( ) { } 1 2 1 1 1 2 2 1 1 1 1 1 cos Im 2 2 2 E E bE P Q ω ω ω ψ δ ω ε = − − + + (4) ( ) { } 2 1 2 2 2 1 1 2 2 2 1 1 cos Im 2 2 2 E E bE P Q ω ω ω ψ δ ω ε = − − − + + (5) { } ( ) 1 2 2 1 1 2 1 1 1 1 1 1 1 Re sin 2 2 E P b E E ω ω φ ψ δ ω ε ω = − − + + Ω − (6) { } ( ) 2 1 1 2 2 1 2 1 2 2 2 1 1 Re sin 2 2 E P b E E ω ω φ ψ δ ω ε ω = − − − + + Ω − (7) where 2 1 ψ φ φ = − and n Ω are the eigen-frequencies of the cold cavity eigen-modes. Eqs.(4)- (7) are the Lamb's self consistency equations and take into account the effects related to the gain medium and the macroscopic effect of the backscattering. The system (4)-(7) can be solved when the polarization vector is known. This model leads to evaluate the linear and nonlinear terms of P by performing quantum calculations applied to the MQW semiconductor structure. By this way, it is possible to calculate all physical parameters, involved in the ring laser dynamics, starting from the Advances in Physical Modeling of Ring Lasers 173 physical description of the MQW energy band without any semi-empirical approximation. To evaluate the P vector we use the density matrix formalism [50] to write: ( ) , Tr( ) ba ab ab ba b a P n M n M M ρ ρ ρ = = + ∑ (8) where Tr is the transposte of matrix and ρ is the density matrix operator, given by: aa ab ba bb ρ ρ ρ ρ ρ ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦ being aa ρ the probability to have an electron in a state of a BV (valence band) sub-band, bb ρ the probability to have an electron in b state of a BC (conduction band) sub-band, ab ρ and ba ρ the probabilities to have a transition between a and b levels or b and a levels, respectively (it holds * ba ab ρ ρ = ). M is the dipole moment operator (formed by the electron- hole pair relevant to two sublevels of the same order, one in BC and the other in BV) in the form of a 2x2 matrix as: 0 0 ab ba M M M ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦ ab M , ba M have been calculated as in [36]. In Eq. (8), n denotes the electron total density included in the conduction and valence sub-bands which must verify the following relationship for MQW semiconductor lasers: ( ) ( ) ( ) (0) (0) , g bb aa cv ba c ba v ba ba b a W n g W f W f W dW ρ ρ ∞ ⎡ ⎤ − = − ⎡ ⎤ ⎣ ⎦ ⎣ ⎦ ∑ ∫ (9) being ( ) cv ba g W the state density, W W W ba b a = − the energy difference between level b and a , f c ( f v ) the Fermi-Dirac distribution function at level b and a , respectively. By solving the continuity equation as proposed in our previous work [46], we derive the elements of the density matrix and, then, the polarization vector as a sum of one linear and one non linear terms. By substituting the components of the polarization vector obtained by the quantum-mechanical analysis in Eq. (4)-(5), considering only two modes, one clockwise (CW, 1) and the other counter-clockwise (CCW, 2), rearranging the equations in terms of the mode intensities 2 1 1 I E = and 2 2 2 I E = , and assuming 1 2 ω ω = , we obtain the rate equations for any semiconductor MQW ring laser: ( ) ( ) ( ) 1 1 1 1 1 12 2 1 2 1 1 12 2 1 2 2 2 cos tot dI I I I I I I I b I I dt α β θ ξ η δ = − − − + + Ψ + (10) Vittorio M.N. Passaro and Francesco De Leonardis 174 ( ) ( ) ( ) 2 2 2 2 1 21 1 1 2 2 2 21 1 1 2 2 2 cos tot dI I I I I I I I b I I dt α β θ ξ η δ = − − − + + Ψ − (11) with ( ) 2 1 1 2 tot ψ ψ π δ δ = − + − , ( ) 1 2 1 2 δ δ δ = + and: 4 2 i i iiii iiii M ω β ε = N I 4 2 i ij iijj iijj ijji ijji ijij ijij M ω θ ε ⎡ ⎤ = + + ⎣ ⎦ N I N I N I 4 2 i i iiij iiij iiji iiji ijii ijii M ω ξ ε ⎡ ⎤ = + + ⎣ ⎦ N I N I N I 4 2 i ij ijjj ijjj M ω η ε = N I 1, 2 i = and j i ≠ where ( ) ( ) ( ) { } Im nqkm g j nqkm cv ba c ba v ba qkm ba W g W f W f W je dW ∞ − Ψ = − ⎡ ⎤ ⎣ ⎦ ∫ I C , * * ( ) I n q k m V nqkm n F r F F F dr = ∫ N N is a real quantity (field overlapping integral, * ( ) ( ) n n n F r F r dr = ∫ N ), which is not zero just in the active region volume, and 2 n n n n g Q ω α = − is the net linear gain of MQW structure, being n g the gain of the MQW structure. The parameters nqkm Ψ and qkm C are defined in [31]. The introduction of the variable δ in Eq. (10)-(11) avoids the explicit dependence on the individual scattering phases 1 δ and 2 δ , and, then, all physical quantities will depend only on the average value δ . The equation system (10)-(11) has a stable solution as tot ψ = 0 when ψ π = and 1 2 δ δ = . The coefficients included in the model, i.e. i β , ij θ , i ξ and ij η , represent the self-saturation coefficient, the cross-saturation coefficient, the self interference coefficient and the cross interference coefficient, respectively. They are responsible of the mode competition phenomenon [46]. Then, by manipulating Eq. (6)-(7), we obtain: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 1 2 1 2 1 2 2 1 1 1 2 21 1 12 2 1 2 21 12 21 1 12 2 2 1 2 1 1 2 sin sin 2 tot tot tot d I I dt I I I I I I I I I I I I b I I ψ ω ω γ γ ν ν ν ν χ χ σ σ ψ δ ψ δ = Ω −Ω − − − − + − ⎛ ⎞ + − + − + − ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎛ ⎞ − + + − ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ (12) Advances in Physical Modeling of Ring Lasers 175 with 2 i i iiii iiii R ω ν ε = N 2 i ij iijj iijj ijji ijji ijij ijij R R R ω ν ε ⎡ ⎤ = + + ⎣ ⎦ N N N 2 i ij iiij iiij iiji iiji ijii ijii R R R ω χ ε ⎡ ⎤ = + + ⎣ ⎦ N N N 2 i ij ijjj ijjj R ω σ ε = N where 1 Ω and 2 Ω are the cold cavity frequencies for CW and CCW mode, respectively, 1 ω , 2 ω are the laser beam frequencies, 1 γ , 2 γ are the pulling effect coefficients, 1 ν , 2 ν are the self-pushing effect coefficients, 12 ν and 21 ν are the cross-pushing effect coefficients. The terms 12 χ , 21 χ , 12 σ , and 21 σ are nonlinear coefficients induced by the nonlinear contribution in the polarization vector. Moreover, it holds: ( ) ( ) ( ) { } 4 Re nqkm g j nqkm cv ba c ba v ba qkm ba W R M g W f W f W je dW ∞ − Ψ = − ⎡ ⎤ ⎣ ⎦ ∫ C and ( ) ( ) ( ) ( ) ( ) 2 2 2 2 ( ) 2 I g V i ba i i cv ba c ba v ba ba m W i ba in F r dr W W M g W f W f W dW h W W ω γ ε τ ∞ − = − ⎡ ⎤ ⎣ ⎦ ⎛ ⎞ / − + ⎜ ⎟ ⎝ ⎠ ∫ ∫ N being in τ the electron average relaxation time. In order to complete the ring laser model, we have introduced the classical rate equation for the injected carriers, in the form: ( ) ( ) ( ) ( ) 2 0 eff 2 3 1 1 1 1 12 2 1 2 1 1 12 2 1 2 0 eff 2 2 2 2 21 1 1 2 2 2 21 1 2 2ε n dN J = -AN-BN -CN - I g -β I -θ I - I I ξ I +η I dt ed hω 2ε n - I g -β I -θ I - I I ξ I +η I hω ⎡ ⎤ ⎣ ⎦ ⎡ ⎤ ⎣ ⎦ (13) where o ε is the vacuum dielectric permittivity, d is the active region thickness, J is the laser current density, n is the refractive index, A, B, C are the leakage recombination coefficient, the bimolecular recombination coefficient and the Auger coefficient, respectively. Eqs. (10)- (13) are the coupled differential equations for the two counter-propagating modes inside any MQW ring laser. Vittorio M.N. Passaro and Francesco De Leonardis 176 It is worth noting that the self and cross interference coefficients are equal to zero only in absence of any backscattering (i.e. R = 0) inside the laser cavity. In this ideal case Eqs (10)- (11) include only the self- and cross-saturation coefficients and, then, the only stable regime is the unidirectional one. In the following section we will show that the backscattering effect can be also responsible of a stable bidirectional regime, depending on R values. Since the backscattering effect depend on technological effects like the ring sidewall roughness, the goal of our model is also to evaluate the operating regime of the MQW ring laser related to the statistical information about this sidewall roughness. The sidewall roughness or boundary imperfections can have two significant effects: 1) energy scattering towards the radiation field, reducing the total quality factor of the optical mode; 2) power redistribution between the two counter-propagating waves influencing the operational regime of the ring laser. Differently from [45], where the backscattering effects are estimated in an empirical way, we calculate the R coefficient and the influence of the scattering losses by means of an analytic approach [52]. Thus, the sidewall imperfections can be described by a random function having a Gaussian distribution for its self-correlation function as: ( ) 2 ' 2 ' 2 ( ) exp / c c corr s s s s L σ ⎡ ⎤ − = − − ⎢ ⎥ ⎣ ⎦ (14) where s is the curvilinear coordinate, c L is the correlation length and c σ is the standard deviation of the roughness correlation function. We have analyzed the scattering loss due to sidewall imperfections by using the volume current method [52]. This method consists of the calculation of the current density associated to the sidewall roughness profile and, thus, solving the Maxwell’s equations in presence of this current source. Since the wave electric field has only components along the rˆ and φ ˆ directions in the plane of the ring (for TE polarization), we concentrate our attention on the dominant component E φ , because it has a peak at the ring edges. This component is influenced by the ring cross-section and radius. We have used a mixed numerical technique based on both effective index, conformal mapping and Wentzel-Kramers-Brillouin (WKB) methods to take into account these influences [52]. Thus, the electric field of the wave travelling along the ring excites an additional contribution to the current density in the regions as perturbed by the presence of roughness, and this contribution will induce the field radiation, calculated by the azimuth component of the vector potential φ A . The determination of vector potential has been executed by evaluating the free-space Green’s function. Then, we have determined the power density radiated by means of the radially directed Poynting vector. Both types of radiations have been considered, i.e. tunneling radiation and phase-matched radiation. However, for large resonator radii (>50 μm), we have found the phased-matched radiation as the only significant contribution to the scattering loss. Now, we have evaluated the scattering-related quality factor of the ring resonator, scatt Q . It is defined as the ratio between the stored energy and the power lost by scattering per each round trip, in the form: 2 exp( / 2) 1 exp( ) eff eff scatt eff stored scatt lost scatt eff n R R P Q P R π π α π λ α π − = = − − (15) Advances in Physical Modeling of Ring Lasers 177 by which the scattering coefficient scatt α has been calculated as: 2 1 2 sinh eff eff scatt eff scatt n R R Q π α π λ − ⎛ ⎞ = ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ (16) and the total quality factor of the ring cavity has been estimated as: 4 2 1 exp( / 2) 1 1 exp( ) eff eff total eff n total eff n R R Q R π π η α π λ η α π − − = − − − (17) where the parameter η is the coupling efficiency between the ring laser and an output bus waveguide and the total optical losses in the ring laser is given by total scatt prop bend leak α α α α α = + + + , being scatt α the sidewall roughness scattering loss, leak α the leakage loss to the substrate, prop α the propagation loss and bend α the curvature-induced bending loss. The leakage loss to the substrate is negligible by introducing in the ring laser structure a buffer layer. The bending loss coefficient can be considered negligible due to the strong confinement in the ring resonator and to its large radius (>50 µm). Thus the optical losses are mainly dominated by prop α and scatt α . Finally, the current induced by the sidewall roughness is a field source inducing a power transfer between the two counter-propagating waves with a backscattering amplitude reflectivity R given as [52]: ( ) ( ) 2 2 2 2 2 0 0 2 / 4 exp eff rib c eff c R R n t F k n L φ π ε ωδ π σ ⎡ ⎤ = − ⎢ ⎥ ⎣ ⎦ (18) where 2 n δ is the ring-air relative permittivity change, rib t is the ridge overall height of the ring cavity, F φ is the azimuth component of the normalized electric field travelling inside the ring cavity. Therefore, the backscattering coefficient is related by our model to the ring laser technological parameters, i.e. 2 n δ , c L , c σ , rib t , eff R . Numerical Results The numerical simulations have been performed by considering a standard GaAs-AlGaAs MQW ring laser structure. The active region is constituted from one (or more) GaAs wells sandwiched between two Al 0.2 Ga 0.8 As waveguide regions, each 100 nm thick. The p-type and n-type Al 0.4 Ga 0.6 As cladding layers are 1.0 and 1.5 μm thick, respectively. A top GaAs cap has been also included. A total optical loss of 25 cm -1 has been taken into account. We have assumed a ring radius of 200 μm (small ring), a rib width of 2 μm and considered the backscattering coefficient as a parameter. By this way we can show as the physical operation of the MQW ring laser depends on the relationship between the injection current and backscattering coefficient. Anyway, it is possible to calculate the backscattering coefficient Vittorio M.N. Passaro and Francesco De Leonardis 178 starting from the statistical information of the ring sidewall roughness taken from experimental data, as explained by Eq. (18). Analysing the stationary solution of Eqs. (10)-(11), the relationship 1 2 0, δ δ π + = must be satisfied. In particular 1 2 0 δ δ + = gives rise to a condition of minimum stimulated energy, whereas 1 2 δ δ π + = gives rise to an instable condition of maximum stimulated energy and can be discarded. We have investigated the impact of the output coupler configuration [46] on the operating characteristics of the semiconductor MQW ring laser. We have assumed an evanescent field coupler, a very common element in integrated optics. However, one of the main problems with these couplers is the sensitivity of the coupling efficiency η to the changes of coupler dimensions, particularly the coupling gap. Such variations make it difficult to accurately obtain a given coupling efficiency, a high reproducibility and good stabilization of the ring laser operating regime. Fig. 6 shows the stationary regimes of 1 I and 2 I versus the coupling efficiency. Figure 6. Intensities of both beams and phase difference versus the coupling efficiency. In this simulation we have assumed c L =0.07 µm, c σ = 0.012 µm and an injection current of I =100mA. The plot shows that the operating regime of the ring laser becomes bidirectional by increasing the coupling efficiency, starting from an unidirectional condition. In fact, for a coupling efficiency ranging from 5% to 16%, the quality factor of the ring resonator (see Eq. 17) assumes relatively large values, so inducing the current I to be well greater than the threshold, th I . A dominant beam in the ring cavity grows due to the mode Advances in Physical Modeling of Ring Lasers 179 competition effect, as induced by the self- and cross-saturation coefficients i β , ij θ . For a coupling efficiency larger than 16%, a significant part of the optical power leaves the ring resonator and this induces the threshold current to be close to I = 100mA. In this case, it is not possible for only one of the counter-propagating beams to be completely extinguished. In fact, the backscattering effect between the beams always induces radiation travelling in the opposite direction. Fig. 7 shows the stationary regimes of 1 I and 2 I versus the correlation length c L for different values of the standard deviation c σ , by assuming an injection current value of I =100mA (one well) and 1 2 0, δ δ π + = . As usual, mode 1 designates the CW solution and mode 2 the CCW one, respectively. Figure 7. Intensities of both beams versus the correlation length for various standard deviations of ring sidewall roughness function. The plot shows that for a value of c σ smaller than a critical value, depending of the injection current ( , c th σ =0.0047 µm in this case), the MQW ring laser works in an unidirectional regime without any dependence on c L . This means that for each value of c L the backscattering coefficient is too small to compensate the mode competition effect induced by the self- and cross- saturation coefficients i β , ij θ (see Eqs. (10)-(11)). Therefore, a dominant beam in the ring cavity grows while the backscattering effect will always maintain a very weak wave travelling in the opposite direction, being orders of magnitude below the dominant beam. The dominant beam can be randomly either the CW or CCW beam, Vittorio M.N. Passaro and Francesco De Leonardis 180 depending mathematically on the initial conditions or, physically, on the local optical losses inside the ring cavity. For values of roughness standard deviation c σ > , c th σ , there exists a range for c L where the MQW ring laser shows a bidirectional operating regime. In particular, this range increases with increasing c σ . Since the self-correlation function of the sidewall roughness is described as a Gaussian function (see Eq. (14)), this means that exists a range of c L , close to the peak of the Gaussian shape, where the backscattering coefficient compensates the mode competition effect. In the range of c L where the regime is bidirectional, it is possible to observe a maximum in the plot. This maximum occurs where the peak of the Gaussian self-correlation function is situated. The previous discussion is also confirmed in Fig. 8, which shows the intensities of CW and CCW beams versus the injection current in the stationary condition, for different values of the statistical parameters of the sidewall roughness. Figure 8. Intensities of both beams versus the laser current for various roughness correlation lengths. The plot again shows that the operation regime of the MQW semiconductor ring laser depends on the relationship between ( c L , c σ ) and the injection current. If the current is close to the threshold, i.e. th I I ≈ , the beams hold the same intensity (bidirectional condition). For currents well larger than th I , a dominant beam in the ring cavity grows inducing the unidirectional regime. In Fig.8 we have assumed c σ =12 nm and used different values of c L , larger than the position of the Gaussian peak. We can observe that the range of injection currents where the MQW ring laser works in a bidirectional regime increases by increasing Advances in Physical Modeling of Ring Lasers 181 the c L value. Figs. 7 and 8 are particularly important because they lead to an estimation of the MQW ring laser operation regime related to the etching step of the fabrication process. In fact, by performing a number of measurements of the etching profile of ring sidewalls obtained on different samples, it is possible to extract the statistical (Gaussian) information on the sidewall roughness function and, then, give theoretical predictions by our model on the laser working regime. Fig. 9 shows the intensities of CW and CCW beams versus the effective ring radius for different values of injection current in case of c L =0.07 µm and c σ = 0.012 µm. It is possible to observe that the operating regime of the MQW ring laser is influenced by the ring cavity sizes. In fact, for each value of the injection current the graph shows that the operational regime converts from unidirectional to bidirectional by increasing the ring radius. This behaviour depends on the circumstance that the density current decreases by increasing the ring radius and, therefore, the intensity of the dominant beam is reduced. In this condition the backscattering effect is not negligible and the beams hold the same intensity. The curves stopped for an appropriate value of the ring radius, depending on the injection current. For ring radii larger than this value, the MQW ring laser remains under the threshold. Figure 9. Intensities of both beams versus the effective ring radius for various laser currents. On the basis of our results, one additional component has to be included in the architecture of the MQW ring laser to favour only one circulating direction over the other, i.e. to achieve an unidirectional regime also for injection current values where the ring laser should be bidirectional. The solution consists of an output coupler including a grating [46]. Vittorio M.N. Passaro and Francesco De Leonardis 182 Then, we have investigated the influence of the grating reflectance g R and the output coupler reflection and transmission coefficients, c R and c T respectively, over the laser behaviour. It is clear that the presence of the grating strengthens the CW solution with respect to the CCW one. A number of parametric simulations by varying g R and c R show that, at constant injection current I =100 mA, correlation length c L =0.07 µm and standard deviation c σ = 12 nm (the value for bidirectional regime of the MQW ring laser without output coupler), it is possible to realise a purely unidirectional condition for g R > 0.9 and c R =10%. It is also possible to obtain an unidirectional condition with a value of g R < 0.9, but it still needs to increase c R . Conclusion In this chapter a short review on ring fiber lasers and recent advances of a highly detailed physical model of MQW semiconductor ring lasers is presented. In particular, solutions to multimode operation in fiber ring lasers and high performance in multi-wavelength erbium- doped fiber lasers have been described. Moreover, the operation regimes of GaAs-based semiconductor ring lasers have been demonstrated and related to the physical coefficients included in the model. Different behaviour of the ring laser can be observed as a function of injection current, ring radius and statistical information of the ring sidewall roughness as determining the backscattering coefficient. Results relevant to output coupling by a grating are discussed to guarantee a unidirectional regime in any condition. Thus, this chapter puts into evidence as integrated MQW ring lasers are promising candidates to realise sources with enhanced mode purity, reduced sensitivity to feedback and higher single beam power. However, in particular applications such as optical test and measurements, optical wavelength-division-multiplexing communications systems and sensing, the fiber ring lasers lead to obtain high performance essentially due to their wide tunable range and very narrow linewidth. References [1] Massicott, J. F., Armitage, J. R., Wyatt, R., Ainslie, R. J., and Craig-Ryan, S. P., High gain broadband 1.6 µm Er doped silica fiber amplifier, Electronics Letters, 1990, 26, 1645–1646. [2] Haber, T., Hsu, K., Miller, C. and Bao, Y., Tunable erbium-doped fiber ring laser precisely locked to the 50-GHz ITU frequency grid, IEEE Photonics Technology Letters, 2000, 12, 1456–1458. [3] Yu, Y., Tam, H. Y., Lui, L. F. and Chung, W. H., Fiber-laser-based wavelength division multiplexed fiber Bragg grating sensor system, IEEE Photonics Technology Letters, 2001, 13, 702–704. [4] Agrawal, G. P., Nonlinear Fiber Optics, Academic Press, 2001. [5] Jain, R. K., Lin, C., Stolen, R. H. and Ashkin, A., A tunable multiple Stokes cw fiber Raman oscillator, Applied Physics Letters, 1977, 31, 89-90. [6] Mears, R. J., Reekie, L., Juancey, I. M., Payne, D. N., Low-noise Erbium-doped fiber amplifier operating at 1.54 µm, Electronics Letters, 1987, 23, 1026-1028. Advances in Physical Modeling of Ring Lasers 183 [7] Pask, H. M., Carman, R. J., Hanna, D. C., Tropper, A. C., Mackechnie, C. J., Barber, P. R., Dawes, J. M., Ytterbium-doped silica fiber lasers: versatile sources for the 1-1.2 µm region, IEEE J. Selected Topics in Quantum Electronics, 1995, 1, 2-13. [8] Oron, R., Bunimovich, D., Nagli, L., Katzir, A., Hardy, A. A., Design considerations for rare earth doped silver halide fiber amplifiers, Optical Engineering, 2001, 8, 1516-1520. [9] Sun, Y., Zyskind, J. L. and Srivastava, A. K., Average inversion level, modeling, and physics of erbium-doped fiber amplifiers, IEEE J. Selected Topics in Quantum Electronics, 1997, 3, 991-1007. [10] Bellemare, A., Karásek, M., Riviere, C., Babin, F., He, G., Roy, V. and Schinn, G.W., A broadly tunable erbium-doped fiber ring laser: Experimentation and modeling, IEEE J. Selected Topics in Quantum Electronics, 2001, 7, 22–29. [11] Dragic, P. D., Analytical Model for Injection-Seeded Erbium-Doped Fiber Ring Lasers, IEEE Photonics Technology Letters, 2005, 17, 1629-1631. [12] Mollenauer, L. F., Mamyshev, P. V., Gripp, J., Neubelt, M. J., Mamysheva, N., Gruner- Nielsen, L. and Veng, T., Demonstration of massive wavelength-division multiplexing over transoceanic distances by use of dispersion-managed solitons, Optics Letters, 2000, 25, 704–706. [13] Tamura, K., Haus, H. A. and Ippen, E. P., Self-starting additive pulse mode-locked erbium fiber ring laser, Electronics Letters, 1992, 28, 2226–2227. [14] Haus, H. A., Ippen, E. P. and Tamura, K., Additive-pulse mode-locking in fiber lasers, IEEE J. Quantum Electronics, 1994, 30, 200–208. [15] Fermann, M., Andrejco, M. J., Silverberg, Y. and Stock, M. L., Passive mode locking by using nonlinear polarization evolution in a polarizing-maintaining erbium-doped fiber laser, Optics Letters, 1993, 18, 894–896. [16] Hofer, M., Fermann, M. E., Haberl, F., Ober, M. H. and Schmidt, A. J., Mode locking with cross-phase and self-phase modulation, Optics Letters, 1991, 16, 502–504. [17] Collings, B. C., Cundiff, S. T., Akhmediev, N. N., Soto-Crespo, T. J., Bergman, K. and Knox, W. H., Polarization-locked temporal vector solitons in a fiber lasers: experiment, J. Optical Society of America B, 2000, 17, 354–365. [18] Spaulding, K. M., Yong, D. H., Kim, A. D., Kutz, J. N., Nonlinear dynamics of mode- locking optical fiber ring lasers, J. Optical Society America B, 2002, 19, 1045-1054. [19] Zhang, J. L., Yue, C. Y., Schinn, G. W., Clements, W. R. L. and Lit, J. W. L., Stable single- mode compound-ring erbium-doped fiber laser, J. Lightwave Technology, 1996, 14,104–109. [20] Horowitz, M., Daisy, R., Fisher, B. and Zyskind, J., Narrow-linewidth, singlemode erbium- doped fiber laser with intracavity wave mixing in saturable absorber, Electronics Letters, 1994, 30, 648–649. [21] Kim, S. K., Stewart, G., Johnstone, W. and Culshaw, B., Mode-hop-free single-longitudinal- mode erbium doped fiber laser frequency scanned with a fiber ring resonator, Applied Optics, 1999, 38, 5154–5157. [22] Cheng, Y., Kringlebotn, J. T., Loh, W. H., Laming, R. I. and Payne, D. N., Stable single- frequency travelling wave fiber loop laser with integral saturable-absorber-based tracking narrow-band filter, Optics Letters, 1995, 20, 875–877. [23] Song, Y. W., Havstad, S. A., Starodubopv, D., Xie, Y., Willner, A. E. and Feinberg, J., 40- nm-wide tunable fiber ring laser with single-mode operation using a highly stretchable FBG, IEEE Photonics Technology Letters, 2001, 13, 1167–1169. [24] Libatique, N. J. C., Wang, L. and Jain, R. K., Single-longitudinal-mode tunable WDM- channel-selectable fiber laser, Optics Express, 2002, 10, 1503–1507. Vittorio M.N. Passaro and Francesco De Leonardis 184 [25] Yeh, C.-H., Huang, T.-T., Chien, H.-C., Ko, C.-H. and Chi, S., Tunable S-band erbium- doped triple-ring laser with single-longitudinal-mode operation”, Optics Express, 2007, 15, 382–387. [26] Yeh, C. H., Lee, C. C. and Chi, S., A Tunable S-Band Erbium-Doped Fiber Ring Laser, IEEE Photonics Technology Letters, 2003, 15, 1053–1054. [27] Liu, H. L., Tam, H. Y., Chung, W. H., Wai, P. K. A. and Sugimoto, N., Low Beat-Noise Polarized Tunable Fiber Ring Laser, IEEE Photonics Technology Letters, 2006, 18, 706-708. [28] Xu, L., Glesk, I., Rand, D., Baby, V. and Prucnal, P. R., Suppression of beating noise of narrow-linewidth erbium-doped fiber ring lasers by use of a semiconductor optical amplifier, Optics Letters, 2003, 28, 780–782. [29] Liu, H. L., Tam, H. Y., Chung, W. H., Wai, P. K. A. and Sugimito, N., La-codoped bismuth- based erbium-doped fiber ring laser with 106-nm tuning range, IEEE Photonics Technology Letters, 2005, 17, 297–299. [30] Wei, D., Li, T., Zhao, Y. and Jian, S., Multi-wavelength erbium-doped fiber ring lasers with overlap-written fiber Bragg gratings, Optics Letters, 2000, 25, 1150–1152. [31] Bellemare, A., Karasek, M., Rochette, M., LaRochelle, S. and Tetu, M., Room temperature multifrequency erbium- doped fiber lasers anchored on the ITU frequency grid, J. Lightwave Technology, 2000, 18, 825–831. [32] Graydon, O., Loh, W. H., Laming, R. I. and Dong, L., Triple-frequency operation of and Er- doped twin core fiber loop laser, IEEE Photonics Technology Letters, 1996, 8, 63–65. [33] Das G. and Lit, J. W. Y., L-band multiwavelength fiber laser using an elliptical fiber, IEEE Photonics Technology Letters, 2002, 14, 606–608. [34] Zhao, C.-L., Yang, X., Lu, C., Ng, J. H., Guo, X., Chaudhuri, P. R. and Dong, X., Switchable multi-wavelength erbium-doped fiber lasers by using cascaded fiber Bragg gratings written in high birefringence fiber, Optics Communications, 2004, 230, 313–317. [35] Poustie, A. J., Finlayson, N. and Harper, P., Multiwavelength fiber laser using a spatial mode beating filter, Optics Letters, 1994, 19, 716–718. [36] Mao, Q. and Lit, J. W. Y., Switchable multiwavelength erbium-doped fiber laser with cascaded fiber grating cavities, IEEE Photonics Technology Letters, 2002, 14, 612–614. [37] Shum, P., Tang, M., Gong, Y., Dong, X., Fu, S., Dong, H. and Yang, X., Nonlinearity enhances operation of fiber ring laser, SPIE, 2007, DOI: 10.1117/2.1200701.0562. [38] Liu, Y., Hill, M. T., Calabretta, N., de Waardt, H., Khoe, G. D. and Dorren, H. J. S., Three- state all-optical memory based on coupled ring lasers, IEEE Photonics Technology Letters, 2003, 15, 1461-1463. [39] Mahnkopf, S., Kamp, M., Marz, R. and Forchel, A., Unidirectional ring laser diode with gain-coupled distributed feeback, Electron. Letters, 2003, 39, 1055-1056. [40] Bente, E. A. J., Barbarin, Y., den Besten, J. H., Smit, M. K., Binsma, J. J. M., Wavelength selection in an integrated multiwavelength ring laser, IEEE J. Quantum Electronics, 2004, 40, 1208-1216. [41] Zhang, S., Liu, Y., Lenstra, D., Hill, M. T., Ju, H., Khoe, G.-D., and Dorren, H. J. S., Ring laser optical flip-flop memory with single active element, IEEE J. Selected Topics in Quantum Electronics, 2004, 10, 1093-1100. [42] Tangdiongga, E., Yang, X., Li, Z., Liu, Y., Lenstra, D., Khoe, G.-D., Dorren, H. J. S., Optical flip-flop based on two coupled mode-locked ring lasers, IEEE Photonics Technology Letters, 2005, 17, 208-210. [43] Sorel, M., Laybourn, P. J. R., Giuliani, G. and Donati, S., Unidirectional bistability in semiconductor waveguide ring lasers, Applied Physics Letters, 2002, 80, 3051-3053. Advances in Physical Modeling of Ring Lasers 185 [44] Sorel, M., Laybourn, P. J. R., Scirè, A., Balle, S., Giuliani, G., Miglierina, R., Donati, S., Alternate oscillations in semiconductor ring lasers, Optics Letters, 2002, 27, 1992-1994. [45] Sorel, M., Giuliani, G., Scirè, A., Miglierina, R., Donati, S., Laybourn, P. J. R., Operating regimes of GaAs-AlGaAs semiconductor ring lasers: experiment and model, IEEE J. Quantum Electronics, 2003, QE-39, 1187-1195. [46] De Leonardis, F. and Passaro, V. M. N., Accurate Physical Modeling of Multi Quantum Well Ring Lasers, Laser Physics Letters, 2005, 2, 59-70. [47] Shimosako, Y., and Numai, T., Semiclassical approach in analysis of ring laser. 1: Derivation of rate equations including backscattering and interference, Japanese J. Applied Physics Part 1, 2000, 39, 3983-3990. [48] Shimosako, Y., and Numai, T., Semiclassical approach in analysis of ring laser. 2: Mode coupling due to backscattering and interference, Japanese J. Applied Physics Part 1, 2000, 39, 3991-3996. [49] Haus, H. A., Statz, H., Smith, I. W., Frequency locking of modes in a ring laser, IEEE J. Quantum Electronics, 1985, QE-21, 78-85. [50] Yamada, M. and Suematsu, Y., Analysis of gain suppression in undoped injection lasers, J. Applied Physics, 1981, 52, 2653-2664. [51] Asada, M., Kameyama, A., Suematsu, Y., Gain and intervalence band absorption in quantum-well lasers, IEEE J. Quantum Electronics, 1984, QE-20, 745-753. [52] Armenise, M. N., Passaro, V. M. N., De Leonardis, F., Armenise, M., Modeling and Design of a Novel Miniaturized Integrated Optical Sensor for Gyroscope Applications, J. Lightwave Technology, 2001, 19, 1476-1494. In: Optical Fibers Research Advances ISBN: 1-60021-866-0 Editor: Jurgen C. Schlesinger, pp. 187-203 © 2007 Nova Science Publishers, Inc. Chapter 6 INVESTIGATION OF OPTICAL POWER BUDGET OF ERBIUM-DOPED FIBER Hideaki Hayashi a, b , Setsuhisa Tanabe a and Naoki Sugimoto b a Graduate School of Human and Environmental Studies, Kyoto University, Sakyo-ku, Kyoto 606-8501, Japan b Research Center, Asahi Glass Co.,Ltd., Kanagawa-ku, Yokohama 221-8755, Japan Abstract We investigated optical power budget of an erbium-doped fiber (EDF). In addition to the output signal and amplified spontaneous emission (ASE) powers from the fiber end, lateral spontaneous emissions and scattering laser powers in the EDF were measured quantitatively by using an integrating sphere. Compared with the signal and ASE powers, it was found that considerable powers were consumed by the laterally emitting lights. As an optically undetected loss which limits power conversion efficiency (PCE) of the fiber amplifier, the effect of nonradiative decay from the termination level of pump excited state absorption (pump ESA) was estimated from decay rate analyses of the relevant levels. The nonradiative loss was comparable to amplified signal power in the EDF when pumped with a 980 nm LD. Nonradiative decay following cooperative upconversion (CUP) process is also discussed using rate equations analysis. 1. Introduction With the spreading and popularization of Internet and broadband communication, larger data traffic and higher processing speed are required in optical telecommunication systems. To meet these demands, optical fiber communication networks have developed rapidly. By using transmission fibers, metropolitan area networks (MANs) in inner-city has been established for several years as well as long-haul networks [1]. There are two method of increasing the information capacity in a single fiber; one is time division multiplexing (TDM), and the other is wavelength division multiplexing (WDM) [2]. The TDM is a technology of increasing the bit rate. On the other hand, the WDM is a technology of coupling optical signals of different wavelengths in the same fiber. The transposable Hideaki Hayashi, Setsuhisa Tanabe and Naoki Sugimoto 188 bandwidths are represented as the TDM speed times the number of wavelengths in the WDM system. In the optical fiber networks, optical amplifiers are one of the key components. The optical signals decay due to the background losses of the transmission fibers (typically 0.2 dB/km at around 1550 nm). The insertion losses of optical add-drop multiplexer (OADM) components also decrease the signal intensity, thus the amplifications of the signals are necessary at every few tens kilometers. For the optical amplifiers in the WDM systems, it is required that as many optical signals with different wavelengths as possible are amplified at the same time. As a practical amplification medium, erbium doped fibers (EDFs) have been extensively studied due to their excellent gain operation around 1.5 μm in the loss minimum window of transmission silica fiber [3, 4]. Figure 1 shows loss spectrum of a transmission silica based fiber and amplification bandwidths of EDFs. Since the development of an efficient silica-based erbium doped fiber amplifier (EDFA) in 1987 by a research group of University of Southampton [5], considerable research efforts have been made to improve the efficiency and broaden the bandwidths. To extend the bandwidths from conventional C-band (1530-1565 nm) to L-band (1570-1610 nm) in the WDM system, several glass hosts for EDFA such as fluoride, tellurite, Bi 2 O 3 -based, and multi component antimony silicate (MCS) have been proposed since the later half of the 1990’s [6-9]. 1400 1500 1600 Wavelength (nm) T r a n s m i s s i o n l o s s ( d B / k m ) 0.2 0.3 0.4 0.5 C-band L-band Silica EDF for Bi 2 O 3 -based EDF for C-band: 1530-1565 L-band: 1580-1605 3 dB down bandwidth C-band: 1530-1565 C+L-band: 1535-1610 Extended L-band: 1545-1620 Figure 1. Loss spectrum of a transmission silica based fiber and the bandwidths of Silica EDFs and Bi 2 O 3 -based EDF. Here “bandwidth” means 3 dB down bandwidth. Silica based EDFs have been installed in the actual optical network system and practically played a critical role. However, their power conversion efficiency (PCE) is limited to 50-55% when pumped with 980 nm LD at present (i.e. ER-1090 amplifier by Sumitomo Electric Industries, Ltd. or HP980 amplifier by OFS). Since pump LD cost represents a significant proportion of the total amplifier cost, increasing the PCE is a concern for the amplifier development. Although the PCE is one of the most important factors for the amplifier design, it is not perfectly understood what limits the PCE in the EDFs. Except for Investigation of Optical Power Budget of Erbium-Doped Fiber 189 emissions or loss origins that can be evaluated at the output end of the fiber (i.e. amplified signal, amplified spontaneous emission (ASE) or splice loss), considerable optical power budget of the EDFs is not clear. For example, lateral spontaneous emission of 1550 nm band has not ever been evaluated quantitatively although there is a report that the lateral emission spectra have been measured to calculate the cross section [10]. In order to optimize the PCE and amplifier performance, understanding of overall optical power budget of the EDFs is essential. In this study, we constructed a novel evaluation system for measuring lateral emissions from the EDF by using an integrating sphere. We used a Bi 2 O 3 -based EDF (BIEDF) for the evaluation due to its potential for high performance amplifier [8, 11-13]. The lateral emissions such as spontaneous emission, upconversion emission, scattering light of laser diode (LD), and the scattering light of signal or ASE were measured quantitatively as well as the in-situ data results of the gain properties as a fiber amplifier. The variations of the lateral emissions with signal wavelength, signal power, or pump power were investigated. In addition, we estimated the effect of other nonradiative decay processes that follow pump excited state absorption (pump ESA) or cooperative upconversion (CUP). To investigate the nonradiative decay from the termination level of the pump ESA, the luminescence decay of the 550 nm band was measured. The effect of the CUP is then discussed theoretically using rate equations and optical propagation equations. Finally, we present the optical power budget of the BIEDF and clarify what decreases the PCE in the amplifier. 2. Background The configuration and principle of an EDFA is shown in Fig. 2. An EDFA is composed of pump lasers, WDM couplers that couple input signals with pumping lights, isolators that prevent the reflection of output signals, and an EDF as an amplification medium. The EDF can be operated as a laser for the signal wavelength ranging in the band around 1550 nm, by utilizing a pump beam of a LD at the wavelength of 980 nm or 1480 nm [14, 15]. The 4f energy diagram of Er 3+ ion and the main transitions involved in the three level laser operation are shown in Fig. 3. The ground state absorption (GSA) cross-section of the Er 3+ ion exhibits a peak at 980 nm, and the Er 3+ ions are excited from the ground 4 I 15/2 level to the 4 I 11/2 level. They decay to the metastable 4 I 13/2 level immediately, and the stimulated emission from the 4 I 13/2 level to the 4 I 15/2 level takes place. In addition to these transitions, the following transitions are accounted in this study: the quantum noise due to the ASE; the CUP via two photons in the first excited level of 4 I 13/2 . Energy transfers from one Er 3+ ion to other, and then the remaining excited Er 3+ ion rapidly decay back to the 4 I 13/2 level; 1550 nm-band spontaneous emission (1550 nm-SE); the pump ESA from the 4 I 11/2 level to the 4 F 7/2 level; the upconversion emission around 550 nm-band (550 nm-SE); Nonradiative transitions (NRs) between the 4 F 7/2 level and the 4 I 13/2 level. The PCE of optical amplifiers is calculated using following expression: PCE(%) = ( (P sOUT – P sIN ) / P pIN ) × 100, (1) Hideaki Hayashi, Setsuhisa Tanabe and Naoki Sugimoto 190 where P sOUT , P sIN , and P pIN are output signal power, input signal power, and launched pump power, respectively (Unit: W). 980nm 1480nm Pump LD Pump LD Er-doped fiber Signal hν WDM coupler Isolator E n e r g y Er 3+ ions at ground state Excited ions in active-center Figure 2. Configuration and principle of Er-doped fiber amplifier. 0 5 10 15 20 E n e r g y ( × 1 0 3 c m - 1 ) Er 3+ 4 I 13/2 4 I 15/2 4 I 11/2 4 I 9/2 4 S 3/2 4 F 9/2 2 H 11/2 4 F 7/2 980 nm Amp.Sig. NR CUP 1550nm- SE 550 nm-SE Signal ASE ESA GSA 0 5 10 15 20 E n e r g y ( × 1 0 3 c m - 1 ) Er 3+ 4 I 13/2 4 I 15/2 4 I 11/2 4 I 9/2 4 S 3/2 4 F 9/2 2 H 11/2 4 F 7/2 980 nm Amp.Sig. NR CUP 1550nm- SE 550 nm-SE Signal ASE ESA GSA Figure 3. 4f energy diagram of Er 3+ ion and the relevant transitions. 3. Preparation and Gain Characteristics of BIEDF The glass preform containing Bi 2 O 3 and SiO 2 as main constituents was prepared using a conventional melting method. For the fiber core composition, 0.5 mol% of Er 2 O 3 was added to the glass batch. Single mode EDF (cladding diameter of 125 μm) with plastic coatings was then fabricated. The core diameter of the BIEDF was 3.9 μm. The refractive index of the core and the numerical aperture (NA) of the fiber at 1550 nm were 2.03 and 0.20, Investigation of Optical Power Budget of Erbium-Doped Fiber 191 respectively. A BIEDF of 16 cm length was fusion-spliced to high NA fibers (Nufern 980HP) using a commercial fusion-splicer. The insertion loss of the spliced BIEDF at 1310 nm was 0.61 dB. By using a cutback method, the propagation loss of the BIEDF at 1310 nm was estimated to be 0.77 dB/m. Accordingly, the average splice loss per point was estimated to be 0.24 dB. Angled cleaving and splicing were applied to suppress the reflection due to the large difference of refractive induces between the BIEDF and the silica fibers [12]. It was confirmed that pig-tailed BIEDFs passed Bellcore (Telcordia) GR-1221 CORE qualification test [16]. Gain and noise figure profiles of the 16 cm BIEDF are shown in Fig. 4. The BIEDF was pumped with 140 mW by forward direction at 980 nm. The gain of the BIEDF reached 18.8 dB at 1535 nm in case the input signal power was –10 dBm. 1520 1540 1560 1580 1600 1620 0 5 10 15 20 25 Wavelength (nm) G a i n a n d N o i s e F i g u r e ( d B ) Gain NF Figure 4. Gain and NF profiles of the 16 cm BIEDF. Launched pump: 140 mW forward at 980 nm; Input signal: –10 dBm at 1535 nm. 4. Lateral Emission Properties of BIEDF 4.1. Lateral Emission Measurement Experimental setup for evaluating the lateral and fiber-propagating emission powers is shown in Fig. 5. The BIEDF of 16 cm length was coiled with 6 cm diameter and set in an integrating sphere (10inch: Model LMS-100s, Labsphere Inc.). The input and output end of high NA silica fibers were connected with instruments through small hole (5mm diameter) of the integrating sphere. The splice points were set just outside of the sphere. It was then pumped with a LD (FITEL) by forward direction at the wavelength of 980 nm. The pump power and temperature of the LD were controlled with a LD-driver (Model 525, Newport Corp.) and a temperature-controller (Model 325, Newport Corp.), respectively. A tunable laser (Model TLS210, Santec Corp.) was used for a single-channel signal source, and then the pump light and the signal light was coupled using a WDM coupler/Isolator (WDM/ISO). The spontaneous emissions and scattering lights laterally emitted from the BIEDF were detected with two kinds of fiber multi-channel CCDs with Si and InGaAs detectors. Each CCD Hideaki Hayashi, Setsuhisa Tanabe and Naoki Sugimoto 192 coupled proprietary spectrometer. The maximum wavelength range of the visible spectrometer (Model USB-2000, Ocean Optics Inc.) and the near-infrared spectrometer (Model NIR-512, Ocean Optics Inc.) were 350-1000 nm and 900-1700 nm, respectively. A premium-grade fiber with 1mm core (Model QP1000-2-VIS/NIR, Ocean Optics Inc.) was used to link the CCD and the output port of the sphere. For the spectral calibration, a standard halogen lamp (Model SCL-600, Labsphere Inc.) was used. The lamp was set at the center of the sphere and driven at 2.60A with a current-regulated DC stabilized power supply (Model PAN-5A, Kikusui Electronics Corp.). The absolute powers of total radiant flux of lateral emissions were then calculated. At the same time, the output spectra of fiber- propagating signal and ASE were detected with an optical spectrum analyzer (OSA: Model MS9780A, Anritsu Corp.) with 1 nm resolution. First, we measured the spectral power distribution of various emissions. The pump power, the input signal power, and the signal wavelength dependences of the emissions were then investigated. Integrating sphere CCD/ Spectrometer PC BIEDF OSA WDM /ISO Pump LD Device Under Test Splice point Signal source Figure 5. Experimental setup for evaluating the lateral and fiber-propagating emission powers of the BIEDF. Basically the splice points were set outside of the integrating sphere. 4.2. Spectral Power Distribution First, we show absolute power spectrum of lateral emissions and output emissions from the fiber end (Fig. 6). The ordinate represents spectral power distribution of radiant flux. The upconversion emission around 520 nm ( 2 H 11/2 → 4 I 15/2 ) and 550 nm ( 4 S 3/2 → 4 I 15/2 ), scattering light of LD around 980 nm, spontaneous emission of 1550 nm band, and ASE were detected by the two kind of multi-channel CCD which was connected with integrating sphere. We can also see weak emission around 660 nm that is related to the pump ESA process [17, 18]. The spectral shapes of the upconversion emission and 1550 nm band spontaneous emission were approximately identical with those in bulk glass [8]. When 100 mW of pump power and 0 dBm of signal power at 1530 nm were input, the optical powers of the upconversion emission, the LD scattering, and the 1550 nm band spontaneous emission, were 0.2 mW, 0.2 mW, and 3.1 mW, respectively. Here the splice points were set outside of the integrating sphere. In the case that the splice points were set inside of the sphere, the scattering of pump LD was increased to 4.3 mW, and 1.8 mW of the scattering light of the amplified signal was Investigation of Optical Power Budget of Erbium-Doped Fiber 193 detected by the CCD. The optical powers of amplified signal at 1530 nm and ASE band that detected by the OSA were 11.9 mW and 0.2 mW, respectively. When the splice points were set outside of the integrating sphere, the sum of emission powers detected by the OSA and the CCDs were 12.1 mW and 3.5 mW, respectively. On the other hand, when the splice points were set inside of the sphere, the optical powers of the LD and signal scattering lights increased. The differences should represent the scatterings at the splice points. That is, the LD and signal scattering lights at the splice points result in the power losses of 4.1 mW and 1.8 mW, respectively. 500 1000 1500 10 -2 10 -1 10 0 10 1 10 2 10 3 10 4 Wavelength (nm) S p e c t r a l p o w e r d i s t r i b u t i o n (μ W / n m ) 550nm -SE 980nm -Scat. ASE 1550nm -SE Amp.Si g. Figure 6. Spectral power distribution of various lateral emissions and amplified signal from the BIEDF. Launched pump and input signal power were 100 mW and 0 dBm, respectively. The splice points of the BIEDF were set outside of the integrating sphere. 550 nm-SE = 550 nm band spontaneous emission; 980 nm-Scat. = Scattering light of the LD at 980 nm; 1550 nm-SE = 1550 nm band spontaneous emission; Amp.Sig. = Amplified signal at 1530 nm; ASE = Amplified spontaneous emission. 4.3. Signal Wavelength Dependence Figure 7 shows the signal wavelength dependences of the optical powers of the lateral emissions and the fiber propagating emissions. The launched power of the pump LD at 980 nm was fixed to 100 mW. Input signal power was set to 0 dBm. The right axis in the figure shows the gain of the output signal (square plots, unit: dB). In the wavelength range from 1530 nm to 1560 nm that corresponds to the C-band, we can see that the signal gains more than 10 dB were obtained with the BIEDF of only 16 cm length. The optical power of the 1550 nm band spontaneous emission was larger than that of the ASE in the entire C-band region. As for the ASE, the spontaneous emission of 1550 nm band, and the scattering light of the 980 nm LD, the optical powers of their emissions showed negative correlations with that of the amplified signal at measured wavelengths. The correlation of the ASE was the strongest among these emissions. These results indicates that more powers are consumed for Hideaki Hayashi, Setsuhisa Tanabe and Naoki Sugimoto 194 the output signal power in the C-band by lowering the ASE and lateral powers in vain, which is desirable as a fiber amplifier. On the other hand, the optical power of the upconversion emission around 550 nm showed weak positive correlation. In other words, the upconversion emission power was large at the wavelength that the output signal power was large. This suggests that the upconversion emission is populated by the signal photons. In addition to the pump ESA, signal ESA using the signal photons can also occur. The initial level of upconversion emission, 4 S 3/2 , would be populated by the signal photons through pump ESA. It can be said from the above correlation that the effect of the input or output signal wavelength on the signal ESA process is smaller than that of the output signal power. 1500 1520 1540 1560 1580 10 2 10 3 10 4 Wavelength (nm) O p t i c a l p o w e r ( μ W ) Amp. Sig. ASE 0. 55u-SE 0. 98u-Scat. 1. 55u-SE S i g n a l g a i n ( d B ) 10 0 -10 Figure 7. Signal wavelength dependence of optical powers of various emissions in the BIEDF. Launched pump and input signal power were100 mW and 0 dBm, respectively. The splice points of the BIEDF were set outside of the integrating sphere. 550 nm-SE = 550 nm band spontaneous emission; 980 nm-Scat. = Scattering light of the LD at 980 nm; 1550 nm-SE = 1550 nm band spontaneous emission; Amp.Sig. = Amplified signal; ASE = Amplified spontaneous emission. 4.4. Signal Power Dependence The signal power dependences of the optical powers of various emissions are shown in Fig. 8. The launched pump power was set to 100 mW, and input signal wavelength was fixed to 1530 nm. The ASE, the spontaneous emission of 1550 nm band, and the scattering light of the 980 nm LD decreased with increasing the input signal power. Even in the small signal region, the lateral emission power was larger than the ASE at the same 1550 nm band. The lateral 1550 nm spontaneous emission was larger than the amplified signal when the input signal power was smaller than -20 dBm. On the other hand, the upconversion emission around 550 nm increased with the input signal power. This positive correlation also suggests the existence of the signal ESA process using the input and amplified signal photons, because the output signal power of a fiber amplifier increases with increasing the input signal power. Investigation of Optical Power Budget of Erbium-Doped Fiber 195 -30 -20 -10 0 10 2 10 3 10 4 Input signal power (dBm) O p t i c a l p o w e r ( μ W ) Amp. Sig. ASE 0. 55u-SE 0. 98u-Scat. 1. 55u-SE Figure 8. Input signal power dependence of optical powers of various emissions in the BIEDF. Launched pump power were 100 mW. The splice points of the BIEDF were set outside of the integrating sphere. 550 nm-SE = 550 nm band spontaneous emission; 980 nm-Scat. = Scattering light of the LD at 980 nm; 1550 nm-SE = 1550 nm band spontaneous emission; Amp.Sig. = Amplified signal; ASE = Amplified spontaneous emission. 4.5. Pump Power Dependence 10 1 10 2 10 0 10 1 10 2 10 3 10 4 Exci tati on power (mW) O p t i c a l p o w e r ( μ W ) Amp.Si g. ASE 0.55u-SE 0.98u-Scat. 1.55u-SE Figure 9. Pump power dependence of optical powers of various emissions in the BIEDF. Input signal power was 0 dBm. The splice points of the BIEDF were set outside of the integrating sphere. 550 nm- SE = 550 nm band spontaneous emission; 980 nm-Scat. = Scattering light of the LD at 980 nm; 1550 nm-SE = 1550 nm band spontaneous emission; Amp.Sig. = Amplified signal; ASE = Amplified spontaneous emission. Hideaki Hayashi, Setsuhisa Tanabe and Naoki Sugimoto 196 Figure 9 shows the pump power dependence of the optical powers of various emissions. All the emission species increased with the pump power, and the dependence of the upconversion emission around 550 nm was nearly 2 and obeying quadratic law. This means that the emission occurs as a result of the pump ESA or the CUP, each of which is due to a two- photon process. The pump power dependence of the lateral 1550 nm spontaneous emission was small and almost saturated under the pump power of larger than 60 mW. This indicates that there exist sufficient photons at the 4 I 13/2 level even when the excitation power is very small. 5. Nonradiative Loss Other than lateral emissions described above, various nonradiative decay processes can be considered; deactivation by hydroxyl group in glass, nonradiative decay which is related to the pump ESA, the decay which is related to the CUP, and the multiphonon relaxation from the 4 I 11/2 level. Among these origins, the effect of hydroxyl groups was neglected here because this BIEDF was sufficiently dehydrated during the fabrication [19, 20]. 5.1. Pump ESA Process 5.1.1. Lifetime Measurement of Er 3+ : 4 S 3/2 Level To analyze the effect of the nonradiative decay from the termination level of the pump ESA, luminescence decay of 550 nm band was measured, and then the quantum efficiency of the Er 3+ : 4 S 3/2 level was calculated from the measured lifetime [21]. Second harmonic of Nd: YVO 4 laser at 532 nm (Model J80-H10-532QW, Spectra Physics) was used as a pump source. The pump power was adjusted to 1 W, and the pump light that was modulated into pulses (Repetition: 15000 Hz; Pulse width: 13 ns) was incident on the optically polished Er-doped Bi 2 O 3 -based glass sample (18×15×3.5 mm in size). The luminescence of 550 nm band of the 0 1 2 3 4 [×10 -5 ] I n t e n s i t y ( a r b . u n i t ) Ti me(s) Excitation: 532 nm Power: 1 W Monit ering: 550 nm Sl it: 8 mm τ f =2.7 μs Figure 10. Luminescence decay curve of the Er 3+ : 4 S 3/2 level in the Bi 2 O 3 -based glass. Circle plots represent measured data, and solid line represents single exponential fitting of these data. Investigation of Optical Power Budget of Erbium-Doped Fiber 197 glass sample was monochromated (Model 1681B, Spex) and detected with a photomultiplier (Model 1424M, Spex) that 0.8 kV of voltage was applied. The signal was collected using a sampling oscilloscope (500 MHz; Model TDS520, Tectronix Corp.), and the lifetime was determined by least square fitting of the obtained decay curve with exponential functions. Measured luminescence decay curve of 550 nm band ( 4 S 3/2 → 4 I 15/2 ) is shown in Fig. 10. Sharp peak that can be observed near zero of the decay time must be the scattering of the pump LD, because the monitoring wavelength is relatively close to the pumping one. After excluding the effect of the LD scattering, the lifetime of the 4 S 3/2 level was determined to be 2.7 μs by using single exponential function. 5.1.2. Nonradiative Losses Following Pump ESA By using the obtained lifetime value, we discuss decay from the 4 S 3/2 level as a result of the pump ESA process. For simplicity, we assumed that all the photons that were excited to the 4 F 7/2 levels relax nonradiatively to the 4 S 3/2 level. This assumption will be valid because the energy gap between the 4 F 7/2 and the 4 S 3/2 is narrow (750 cm -1 ) [22]. Generally quantum efficiency of an emission, η, is written as follows: η = A × τ f = A / (A + W), (2) where A is spontaneous emission probability, W is nonradiative transition probability, and τ f is fluorescence lifetime. We calculated the A coefficient from the Judd-Ofelt analysis (3100 s -1 ) [23-25]. Accordingly, the quantum efficiency was estimated to be 0.8%. The nonradiative energy loss from the 4 S 3/2 level to the 4 I 11/2 level (unit: W), P NR ( 4 S 3/2 → 4 I 11/2 ), can be then expressed as follows: NR P ( 4 S 3/2 → 4 I 11/2 ) = { R P ( 4 S 3/2 → 4 I 15/2 ) / η} ×{ΔE ( 4 S 3/2 → 4 I 11/2 ) / ΔE ( 4 S 3/2 → 4 I 15/2 )}, (3) where P R ( 4 S 3/2 → 4 I 15/2 ) is upconversion emission power, ΔE is energy gap between two 4f levels. The nonradiative energy loss from the 4 I 11/2 level to the 4 I 13/2 level, P NR ( 4 I 11/2 → 4 I 13/2 ), can be considered separately. NR P ( 4 I 11/2 → 4 I 13/2 ) = [P L －{ R P ( 4 S 3/2 → 4 I 15/2 ) + NR P ( 4 S 3/2 → 4 I 11/2 ) }－P R ( 4 I 11/2 → 4 I 15/2 )] ×{ E Δ ( 4 I 11/2 → 4 I 13/2 ) / E Δ ( 4 I 11/2 → 4 I 15/2 )}. (4) Here P L is launched pump power, P R ( 4 I 11/2 → 4 I 15/2 ) is the optical power of 1000 nm emission band. By using these expressions described above, P NR ( 4 S 3/2 → 4 I 11/2 ) and P NR ( 4 I 11/2 → 4 I 13/2 ) were calculated to be 13 mW and 31 mW, respectively. Although the visible upconversion luminescence power was only 0.2 mW, we have to count the nonradiative decay from the 4 S 3/2 level due to low η. Hideaki Hayashi, Setsuhisa Tanabe and Naoki Sugimoto 198 5.2. Coorpelative Upconversion Process 5.2.1. Calculation of Cup Process We can estimate the effect of the CUP process using the rate equations analysis. Figure 11 shows the 4f energy level diagram of Er 3+ ions and transitions used for the analysis. When a BIEDF is pumped with a 980 nm LD by forward direction, the time dependence of populations can be expressed as follows [26-28]: ( ) ( ) 4 41 3 31 2 2 1 12 13 2 21 21 21 1 / N A N R CN N R R N W R A dt dN + + + + − + + = , (5) 2 2 3 32 1 12 2 21 21 21 2 2 ) ( / CN N W N R N W R A dt dN − + + + + − = , (6) ( ) 2 2 4 43 3 31 34 32 1 13 3 / CN N W N R R W N R dt dN + + + + − = , (7) 3 34 4 43 41 4 ) ( / N R N W A dt dN + + − = , (8) where N 1 , N 2 , N 3 , and N 4 represent the population of the 4 I 15/2 , 4 I 13/2 , 4 I 11/2 , and 4 F 7/2 levels, respectively. For simplicity, we neglected the intermediate levels between the 4 I 11/2 and the 4 F 7/2 levels, and assumed that all the photons pumped at the 4 F 7/2 level via the pump ESA transit nonradiatively to the 4 S 3/2 level. Total Er 3+ ion number density for the calculation was set to 1.54 × 10 26 m -3 , which corresponded to 0.5 mol% of Er 2 O 3 . R 21 , R 12 , R 31 , R 13 , and R 34 are radiation transition rate between these levels that are calculated from absorption and emission cross sections (σ s e , σ s a , σ p e , σ p a , and σ ESA , respectively). A 21 and A 41 represent spontaneous emission probabilities that are calculated by the Judd-Ofelt analysis [25]. Nonradiative transition probability, W 43 , is calculated in the way described in Section 5.1.2. W 21 and W 32 can be also estimated from the lifetime measurements of the bulk glasses in the same way as described in Section 5.1.1. The fiber length and the numerical aperture were set to 16 cm and 0.20, respectively. C represents cooperative upconversion coefficient. Here we assumed homogeneous distribution of Er 3+ ions in the glass and homogeneous upconversion process [29-31]. Mode field diameter at 980 nm and at 1530 nm were set to 4.2μm and 6.3 μm, respectively. The signal and pump lightwaves propagating along the fiber (I s and I p ) are expressed as the following set of ordinary differential equations [26-28]. dI s / dz = (σ s e N 2 – σ s a N 1 ) Γ s I s – α s I s (9) dI p / dz = – (σ p a N 1 – σ p e N 3 + σ ESA N 3 ) Γ p I p – α p I p (10) Γ s and Γ p are overlap factor at the signal wavelength and pumping wavelength, respectively. α s and α p are parameters that represent the intrinsic fiber background loss at the signal and pumping wavelength, respectively. Here we assumed that the α s were identical with α p , and treated them as fitting parameters. We applied the Quimby’s assumption that σ ESA is equal to 2σ p a [32]. Although spontaneous decay was accounted for, the ASE was neglected since the Investigation of Optical Power Budget of Erbium-Doped Fiber 199 input signal power was sufficiently large (0 dBm) and the fiber length was sufficiently short. Splice loss from the BIEDF to high-NA silica fiber was set to 0.24 dB/point. Assuming a steady state condition (the time derivatives to be zero), the set of differential equations were numerically integrated using the fourth order Runge-Kutta method with an initial condition at the input end of the fiber (z=0 m) [27]. The parameters used for numerical calculations are shown in Table 1. Input signal and launched pump powers were set to 1 mW (0 dBm) and 100 mW, respectively. By using these calculations, we obtained the relationship between the output signal power and the CUP coefficient. Table 1. Parameters used for numerical calculations. Parameter Symbol Value Unit Spontaneous emission rate A 21 250 s -1 A 41 3100 s -1 Nonradiative decay rate W 32 69 s -1 W 32 3.30×10 4 s -1 W 43 3.70×10 5 s -1 Signal emission cross section at 1530 nm β s e 7.41 ﾗ 10 -25 m 2 Signal absorption cross section at 1530 nm β s a 8.19 ﾗ 10 -25 m 2 Pump emission cross section at 980 nm β p e 3.06 ﾗ 10 -25 m 2 Pump absorption cross section at 980 nm β p a 2.36 ﾗ 10 -25 m 2 Overlap factor at 980 nm β s 0.82 Overlap factor at 1530 nm β p 0.52 Er 3+ ion density β 1.54 ﾗ 10 26 m -3 Cooperative upconversion coefficient C Fitting parameter m 3 /s Background loss β Fitting parameter N 1 0 5 10 15 20 E n e r g y ( × 1 0 3 c m - 1 ) Er 3+ 4 I 13/2 4 I 15/2 4 I 11/2 4 I 9/2 4 S 3/2 4 F 9/2 4 F 7/2 R 13 R 34 A 41 Amp.Sig. R 21 W 21 W 32 W 43 C N 2 N 3 N 4 A 21 R 12 R 31 Figure 11. 4f energy diagram of Er 3+ ion and the transitions used for the rate equations analysis. Hideaki Hayashi, Setsuhisa Tanabe and Naoki Sugimoto 200 5.2.2. Effect of Cooperative Upconversion Here we estimate the effect of the cooperative upconversion (CUP) using rate equations analysis as described in the former section. Figure 12 shows the variation of calculated output signal with the CUP coefficients. The difference between the output power at a given CUP coefficient and the output at zero of the coefficient (value at y-intercept) represents energy loss via the CUP process. The calculations were performed for three values of α. For any α, the output power decreased exponentially with increasing the CUP coefficient. Snoeks et al. reported that the value of the CUP coefficient was 3.2 ×10- 24 m 3 /s in a soda lime silicate glass that was doped with 1.4×10 -26 m -3 of Er 3+ ions [31]. When we assume that the CUP coefficient of the BIEDF (1.54×10 -26 m -3 of Er 3+ ion number density) is same as that of the soda lime silicate, the curve of α = 4 seems reasonable. In this case, the effect of the CUP process results in approximately 10 mW. If we decrease the Er concentration in glass, the CUP will be reduced because the CUP coefficient is a function of the Er 3+ ion density [33]. 0 0.5 1 1.5 2 [×10 -23 ] 0 5 10 15 20 25 30 35 40 O u t p u t p o w e r ( m W ) CUP coeffi cient (m 3 /s) α=0 α=4 α=8 Soda li me silicate Figure 12. Variation of signal output power with the CUP coefficient in the BIEDF doped with 1.54 × 10 26 m -3 of Er 3+ ions. Plots represent calculation data, and solid lines are exponential fitting of these data. Dashed line represents literature value for a soda lime silicate glass doped with 1.4 × 10 26 m -3 of Er 3+ ions (C = 3.2 × 10 -24 m 3 /s) [31]. 6. Energy Budget of BIEDF The optical power budget of the BIEDF that has been clarified in this study is shown in Table 2. Here the launched pump power and the input signal power were 100 mW and 1 mW (0 dBm), respectively. The insertion loss of 0.61 dB corresponds to 13.1 mW. The output signal power at 1530 nm and the sum of lateral emissions and scattering powers were 11.9 mW and 9.4 mW, respectively. It can be said that considerable powers were consumed by the lateral emissions and scatterings in the BIEDF. Investigation of Optical Power Budget of Erbium-Doped Fiber 201 Taking into account the output signal, the ASE, the lateral emissions, and the insertion loss, 65% of total power (65 mW) was not detected either by the CCDs or the OSA. The power of the nonradiative decay from the termination level of the pump ESA to the 4 I 11/2 level was estimated to be 13 mW. That from the 4 I 11/2 level to the 4 I 13/2 level was 31 mW. Approximately 10 mW can be attributed to the nonradiative decay following the CUP. We can say that nonradiative decays above also affect the decrease of the PCE in the BIEDF. Even counting all sources of loss described above, however, we could not identify approximately 11% of total launched power. A possible reason is that we underestimate nonradiative losses at present. For precise estimation of the pump ESA effect, high measurement accuracy of the very weak upconversion luminescence is necessary. For the CUP effect, we will have to consider the clustering of the Er 3+ ions and resulting pair induced quenching [15, 34, 35]. Table 2. Energy budget of the BIEDF when pumped with 100 mW of launched power. Emission species and source of loss mW Amplified signal 12 Insertion loss (splice loss+background loss) 13 980 nm LD scattering (at splice point) 4.1 980 nm LD scattering (w/o splice point) 0.2 Signal scattering (at splice point) 1.8 Amplified spontaneous emission 0.2 1550 nm band spontaneous emission 3.1 550 nm band upconversion emission 0.2 Nonradiative decay from the 4 S 3/2 to the 4 I 11/2 13 Nonradiative decay from the 4 I 11/2 to the 4 I 13/2 31 Nonradiative decay following CUP aprx.10 Unidentified aprx.11 Launched pump power: 100 mW Input signal power: 1 mW 7. Conclusion We have analyzed optical power budget of an erbium-doped amplifier (EDF). Lateral spontaneous emissions and scattering laser powers in a Bi 2 O 3 -based EDF (BIEDF) were evaluated quantitatively by using an integrating sphere. Comparing with amplified signal, it was clarified that considerable power was consumed by the laterally emitting lights. While the LD scattering, the signal scattering, and the 1550 nm band emission powers decreased with increasing input signal power, the lateral 550 nm emission power increased. In the same way, among the lateral emissions, only 550 nm band showed positive correlation with the spectrum of the output signal. These results suggested that the upconversion emission was promoted by the signal ESA. Hideaki Hayashi, Setsuhisa Tanabe and Naoki Sugimoto 202 As a result of decay rate analysis, it was revealed that the nonradiative power loss related to the pump excited state absorption (pump ESA) was comparable with the output signal power because the quantum efficiency of the initial level of the upconversion emission was only 0.8%. In addition, as a result of rate equations analysis, it was suggested that the effect of nonradiative decay following the cooperative upconversion (CUP) was not negligible when Er 3+ ion density was an order of 10 -26 m -3 . These analyses performed in this study can be applicable for not only a BIEDF but also commercial silica based EDF that the power conversion efficiency (PCE) is usually limited to 50-55%, other rare earth-doped amplifiers or lasers. The measurement system using an integrating sphere is also useful to analyze the lateral emissions from waveguide amplifiers in which precise control of their structures is necessary. References [1] Kartalopoulos, S. V. Introduction to DWDM Technology; IEEE Press: NJ, 1999, pp. 209-222. [2] Keiser, G. E. Opt. Fiber Technol. 1999, vol. 5, 3-39. [3] Sudo, S. Optical Fiber Amplifiers; Artech House: MA, 1997, pp. 1-53. [4] Tanabe, S.; Hanada, T. J. Non-Cryst. Solids 1999, vol. 196, 101-105. [5] Mears, R.J.; Reekie, L.; Jauncey, I. M.; Payne, D. N. Electron. Lett. 1987, vol. 23, 1026- 1028. [6] Yamada, M.; Ono, H.; Kanamori, T.; Sudo, S.; Ohishi, Y. Electron. Lett. 1997, vol. 33, 710-711. [7] Oishi, Y.; Mori, A.; Yamada, M.; Ono, H.; Nishida, Y.; Oikawa, K. Opt. Lett. 1998, vol. 23, 274-276. [8] Tanabe, S.; Sugimoto, N.; Ito, S.; Hanada, T. J. Lumin. 2000, vol. 87-89, 670-672. [9] Goforth, D. E.; Minelly, J. D.; Ellison, A. J. G.; Trentelman, J. P.; Samson, B. N. Technical Digest of Optical Amplifiers and their Applications 2000, Nara, 2000, pp. OTuA4-1. [10] Zech, H. IEEE Photon. Technol. Lett. 1995, vol. 7, 986-988. [11] Sugimoto, N. J. Am. Ceram. Soc. 2002, vol. 85, 1083-88. [12] Ohara, S.; Sugimoto, N.; Ochiai, K.; Hayashi, H.; Fukasawa, Y.; Hirose, T.; Nagasima, T.; Reyes, M. Opt. Fiber Technol. 2004, vol. 10, 283-295. [13] Hayashi, H.; Sugimoto, N.; Tanabe, S. Opt. Fiber Technol. 2006, vol. 12, 282-287. [14] Barnes, W. L.; Laming, R. I.; Tarbox, E. J.; Morkel, P. R. IEEE J. Quantum Electron. 1991, vol. 27, 1004-1010. [15] Prudenzano, F. J. Lightwave Technol. 2005, vol. 23, 330-340. [16] Bell Communications Research, Generic requirements GR-1221-CORE 1994, Issue 1. [17] Auzel, F. Chem. Rev. 2004, vol. 104, 139-173. [18] Sun, H.; Xu, S.; Dai, S.; Wen, L.; Zhang, J.; Hu, L.; Jiang, Z. J. Non-Cryst. Solids 2005, vol. 351, 288-292. [19] Hayashi, H.; Sugimoto, N.; Tanabe, S.; Ohara, S. J. Appl. Phys. 2006, vol. 99, 093105. [20] Hayashi, H.; Sugimoto, N.; Ochiai, K.; Ohara, S.; Fukasawa, Y.; Tanabe, S. Extended Abstract of International Congress on Glass XX, Kyoto, 2004, pp. O-14-028. [21] Tanabe, S.; Hayashi, H.; Hanada, T.; Onodera, N. Opt. Mat. 2002, vol. 19, 343-349. Investigation of Optical Power Budget of Erbium-Doped Fiber 203 [22] Miyakawa T.; Dexter, D.L. Phys. Rev. B 1970, vol. 1, 2961-2969. [23] Judd, B.R. Phys. Rev. 1962, vol. 127, 750-761. [24] Ofelt, J.S. J. Chem. Phys. 1962, vol. 37, 511-520. [25] Tanabe, S.; Photonics Based on Wavelength Integration and Manipulation; IPAP Books: Tokyo, 2005, vol. 2, pp. 101-112. [26] Becker, P. C.; Olsson, N. A.; Simpson, J. R. Erbium-Doped Fiber Amplifiers; Academic Press: NY, 1999, pp 153-195. [27] Komukai, T.; Yamamoto, T.; Sugawa, T.; Miyajima, Y. IEEE J. Quantum Electron. 1995, vol. 31, 1880-89. [28] Khoptyar, D.; Jaskorzynska, B. J. Opt. Soc. Am. B 2005, vol. 22. 2091-2098. [29] Myslinski, P.; Nguyen, D.; Chrostowski, J. J. Lightwave Technol. 1997, vol. 15,112-120. [30] Bilxt, P. IEEE Photon. Technol. Lett. 1991, vol. 3, 996-998. [31] Snoeks, E.; van den Hoven, G. N.; Polman, A.; Hendriksen, B.; Diemeer, M. B. J.; Priolo, F. J. Opt. Soc. Am. B 1995, vol. 12, 1468-74. [32] Quimby, R.S. Appl. Opt. 1991, vol. 30, 2546-52. [33] Gapontsev, V.P.; Platonov, N.S. Materials Science Forum 1989, vol. 50, 165-222. [34] Nilsson, J.; Bilxt, P.; Jaskorzynska, B.; Babonas, J. J. Lightwave Technol. 1995, vol. 13, 341-349. [35] Masuda, H.; Takada, A.; Aida, K. J. Lightwave Technol. 1992, vol. 10, 1789-99. In: Optical Fibers Research Advances ISBN: 1-60021-866-0 Editor: Jurgen C. Schlesinger, pp. 205-229 © 2007 Nova Science Publishers, Inc. Chapter 7 RECENT DEVELOPMENTS IN ALL-FIBRE DEVICES FOR OPTICAL NETWORKS Nawfel Azami Institut National des Postes et Télécommunications, Madinat Al Irfane, Rabat-Instituts, Rabat, Morocco. Suzanne Lacroix Ecole Polytechnique de Montréal, Laboratoire des fibres optiques, Montréal, Québec, Canada Abstract All-fibre components are essential components of optical networks systems. Development of such devices is of great importance to allow network functions to be performed in the glass of the optical fibre itself. Among of all fabrication techniques, the Fused Fibre Biconical Taper (FBT) technique allows optical devices with high performances. Although fibre devices are mainly based on the passive directional coupler basic structure, research is made to design components that perform complex functionalities in today optical networks systems. Recent developments on all-fibre devices in network systems are presented. Research is mainly focused on enhanced fabrication and stability of FBT fabrication technique, passive thermal compensation for stable interferometer optical structure, broadband spectral operation for multi-wavelength operations and new interferometer designs. An overview of recent fused fibre devices for optical telecommunications is presented to understand the main functionalities of these fibre devices. The limiting factors are explained to understand challenges on fibre devices development. Introduction The fibre is not only the choice transmitting medium for high speed long-haul telecommunication. It is also currently used in sensing networks applications and more recently in quantum information systems. Components are key elements of such networks. All-fibre devices and their full compatibility with the transmission medium make them particularly attractive to perform operations such as multiplexing, routing, or filtering with Nawfel Azami and Suzanne Lacroix 206 low insertion loss, low polarization mode dispersion, and ease of interconnection. In this chapter, new developments of all-fibre components for optical networks systems are presented. A number of techniques have been developed to fabricate all-fibre components. Among them, the fusion-tapering or, in short, the FBT technique is extensively used especially for the fabrication of 2x2 couplers. Because of their intrinsic low loss, they offer the possibility of high power handling (such as in all-fibre lasers), as well as individual photon manipulation (such as quantum information processing in quantum key distribution and quantum computing.) Most devices and components considered herein are however firstly designed for use in standard telecommunication networks. In the first part of this chapter, Fused Biconical Taper (FBT) fabrication technique is described as well as the basic designs structures of all-fibre devices. A basic optical fibre communication system is presented in Fig. 1. An electrical signal from the data source is fed into the optical transmitter, which is contains a laser or an LED. The modulated light from the transmitter is launched into the fibre and transmitted to the receiver via a demodulator. The receiver consists of a light detector with appropriate amplification and noise filtering. In a digital system a decision gate is also included. Optical fibres prove economic when good use can be made of the bandwidth that they offer. Optical Wavelength Division Multiplexing (WDM) and Dense WDM (DWDM) systems have been developed to perform multi channels propagation in a single optical fibre. Development of stable multiplexers/demultiplexers is of great importance to combine wavelength channels in the optical fibre. These types of multiplexers can also be used as demodulators when Differential Phase Shift keying modulation is used. Designs of all-fibre wavelength multiplexers/demuliplexers are usually complex since they require techniques for thermal compensation of the wavelength channel drift. Moreover the sinusoidal spectral response of basic structures such as tapered fibre couplers or Mach-Zehnder interferometers is not appropriate. A flattened spectral response is more appropriate since it allows minimizing insertion loss even when the carrier wavelength drifts. It also reduces crosstalk between adjacent channels. The second part of this chapter is dedicated to new developments on stable WDM/DWDM. In particular, passively temperature-independent all-fibre devices techniques and new design of flat top multiplexers are presented. During the last twenty years, interest in communicating by sending signals along optical fibres has grown enormously. This interest lies in the very high capacity of transmission in optical fibres, the very low attenuation of the signal during the propagation, as well as the high performances of Erbium doped fibre amplifiers (EDFA) and Raman amplifiers. Development of these amplifiers allows achievement of multi-channel lightwave systems with high bit rates performances. For silica fibres, the attenuation is quite small, particularly in the C-band, between 1525 nm and 1570 nm. Erbium doped optical fibres demonstrated high performances on amplification of signals with low noise. However, multi-channel systems need additive components when compared to single-channel communicating systems. As an example, Erbium gain non-uniformity causes power divergence of WDM channels, limiting the system performances. Gain flattening filters (GFF) and Dynamic gain equalizers modules are requested to flatten the amplifiers gain. Development of such devices using FBT fabrication technique is presented in the third part. Raman amplifiers have also proved their high capacity of achieving high gain with low noise. Distributed Raman amplification is very attractive since it allows amplification of Recent Developments in All-Fibre Devices for Optical Networks 207 signals in the transmission fibre. Because of the high pump power needed in Raman amplification, all-fibre components are of more benefit thanks to their high optical power handling. Multi-channel signal systems need multi-wavelength pump lasers when Raman amplification is used. For this reason, and for system reconfiguration agility, wideband devices are of particular interest for Raman amplifiers. In the last part, new developments on large bandwidth all-fibre devices for Raman amplification are presented. Figure 1. Schematic optical fibre network configuration. 1. Fused Biconical Taper Fibre optic couplers either split optical signals into multiple paths or combine multiple signals on one path. The number of input and output ports, expressed as an NxM configuration, characterizes a coupler, N representing the number of input fibres, and M the number of output fibres. Fused couplers can be made in any configuration, but the simplest is the 2x2 symmetric directional coupler, which is the equivalent in guided optics of a beam splitter in bulk optics. Although the most frequent components are 2x2 couplers, tapered single fibres are also a basic component of interest in themselves and, as such, are studied in the following. 1.1. Manufacturing The fusion-tapering manufacturing technique consists in fusing laterally two (or more) fibres together using, as an heat source, a micro-torch, an oven or a CO 2 laser. Depending on the fusion duration, one obtains a cross section with a degree of fusion ranging theoretically from zero (for unfused fibres) to 1 (corresponding to a circular cross section theoretically obtained after an infinite duration). From a practical point of view, the degrees of fusion usually range between 0.5 and 0.7. Example cross sections are shown in Fig. 2. 0.005 0.25 0.5 0.75 1 Figure 2. Cross sections of 2x2 symmetric fused fibre couplers. The degrees of fusion and indicated below each cross section. Note the deformation of the cores as the fusion increases. Nawfel Azami and Suzanne Lacroix 208 As shown schematically in Fig. 3, the fused structure is then stretched so as to create a biconical structure until the desired profile or the desired response is obtained. Tools and rules for the elaboration of recipes to design specific components are given in Ref. [1,2]. Each tapering recipe includes several tapering segments, usually no more than 3 or 4, except for complex concatenated structures, such as Mach-Zehnder interferometers. Apart from the fibre local temperature, each segment is characterized by four main parameters, namely, the pulling speed, the elongation, the flame position, and the effective width the flame or length of the hot zone during the process. Temperatures range is typically between 1450 ±50 °C but may reach 1700 °C. The ends of the tapered structure are usually pulled apart at equal and opposite speeds relative to the centre of the heat source. As a result, the tapered structure is symmetric so that the slopes of the down-taper transition and of the up-taper transition regions are identical. Pulling speeds, typically of the order of millimetres per minute, are usually constant for a given segment and depend on the fibre temperature. For a given temperature, it is adjusted so that the fibre neither breaks nor sags during the process. The final elongation of the component for a given segment determines the end of this particular segment. During the tapering process, diagnostics are made: the shape of the device is controlled through a binocular microscope; its optical transmissions (in both arms) are recorded at a given wavelength as a function of elongation or for a whole range of wavelengths using a broadband light source and an Optical Spectrum Analyzer (OSA) as a detector. Broadband light source Diagnostics (OSA) Heat source (flame, CO 2 laser, oven) Streching motors Figure 3. Manufacturing of a 2x2 coupler using the FBT technique. 1.2. Adiabaticity Concept The slopes of the longitudinal structure largely determine the behaviour of the component. The propagation along a tapered fibre is said to be adiabatic whenever the fibre transmission is not affected by the taper slope [2]. This is only possible for gentle slopes. In contrast, when the slopes are abrupt, transfer of power to higher mode may occur. This is, from a general point of view, undesirable for couplers as this causes power leakage. An adiabaticity criterion is derived for every particular structure, whether a single fibre, or a coupler made of two or more fused fibres. The fused fibres may be identical to create a symmetric structure or not, in the more general case of asymmetric couplers. The adiabaticity criterion provides the upper limit normalized slope that a structure may have for an adiabatic behaviour. Details of the calculation and graphical representation of adiabaticity criteria are given in Ref. [1,2]. For most structures made of standard 125 µm diameter fibres, the limit slope is of the order of 10 - 3 µm -1 . While adiabaticity is usually required for couplers, non-adiabaticity of tapered single Recent Developments in All-Fibre Devices for Optical Networks 209 fibres can be used to design a variety of all-fibre spectral filters (see Section 3.1). Their principle of operation is overviewed below. 1.3. Non-adiabatically Tapered Fibres When one tapers a fibre with steep slopes, e.g. heating over a zone of the order of a few millimetres, one observes oscillations in the transmitted power as a function of elongation, at a given wavelength. For a given elongation, these oscillations are also present as a function of wavelength. As explained in more details below, this is the result of the alternation of local modes coupling and beating effects along the tapered structure. In the downtaper region, as the fibre diameter decreases, the fundamental LP 01 core mode expands in the cladding. When the diameter is reduced by a factor of 2 or more, the fundamental mode becomes guided by the cladding-air interface. The mode is said to be “cut off” as a core mode: it becomes a cladding mode. If the slope is steep, some power is transferred to other cladding modes (LP 02 , LP 03, ...) by coupling effects. In the central region, where the slopes are small, the adiabaticity criterion is again obeyed and the excited LP 0m modes, all of them being cladding modes, accumulate phase differences through the beating effect. While arriving on the uptaper region, mode coupling again occurs before the power is finally recovered in the core. Depending on the relative phase of the excited LP 0m modes, (therefore on the wavelength and on the elongation) power may be partially or totally recovered in the core. All the power, which is not recovered in the core, is in the cladding modes and possibly trapped by the protective jacket of the fibre, thus lost. This process of coupling-beating-coupling thus confers to a tapered fibre an oscillatory behaviour according to the various parameters affecting the modal phase differences accumulated mostly in the beating region. The LP 01 and LP 02 modes are responsible for the main oscillation. Higher order modes (LP 03 , ...) possibly superimpose to it smaller amplitude and larger frequency oscillations. For a pair of modes, e.g. LP 01 and LP 02 , the wavelength response is essentially sinusoidal and it is exploited to design spectral filters, such as those to flatten the Erbium doped fibre gain described in Section 3.1. 1.4. Transfer Matrices of 2x2 FBT Symmetric Couplers In the following, the principle of operation of adiabatic 2x2 FBT symmetric couplers is overviewed. For a coupler made of individual guides in close proximity, the power transfer from branch to branch is usually analyzed in terms of coupling between the modes of the individual guides. However, in the case of FBT couplers, it is necessary to call for supermodes. For a 2x2 symmetric coupler (the only coupler considered herein for the sake of simplicity), these are referred to as SLP 01 and SLP 11 , respectively. They are the fundamental and the first asymmetric modes of the superstructure, i.e., the fused structure. Note that the adiabaticity criterion, which is supposed to be obeyed, refers to these supermodes. This concept of supermodes is essential for the following reasons. For the power transfer to occur, the fused structure is tapered down to a diameter such that the cores no more play their guiding roles. Nawfel Azami and Suzanne Lacroix 210 As the cores are reduced, the fields spread out of the cores and the guiding process is ensured by the cladding-air interface. As a result, individual guides can no more be identified in the fused and tapered region, where the power transfer occurs. The transfer of power is then described in terms of a beating phenomenon between the fist two supermodes SLP 01 and SLP 11 , which are equally excited at the entrance of the coupler when light is launched only one of the entrance branch. Due to their different propagation constants, they accumulate a phase difference along the structure. Whenever they are in phase, the power is retrieved in the main branch, while, whenever they are out of phase, the power is retrieved in the secondary branch. An intermediate phase difference value corresponds to a branching ratio between 0 and 1. More quantitatively, a coupler is characterised by its transfer matrix M α = e iα cosα i sinα i sinα cosα ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ (1) where α is an average common propagation phase, which is, for this reason often omitted. It is defined as 2α = ∫ 0 L (β 01 + β 11 )dz (2) and 2α is the accumulated phase difference between both supermodes along the length of the coupler 2α = ∫ 0 L (β 01 −β 11 )dz (3) β 01 and β 11 being, in these formulas, the propagation constants of the supermodes SLP 01 and SLP 11 , respectively. Note that these propagation constants are wavelength dependent, which confer the coupler a spectral dependence. The transfer matrix relates the amplitudes in the two exit branches to those in the entrance branches. For example, an excitation in a single branch corresponds to the entrance vector 1 0 ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ and thus to an exit vector e iα cosα i sinα i sinα cosα ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 1 0 ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = e iα cosα i sinα ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ Note the i factor, corresponding to a π/2 phase factor between both branches, which is unusual referring to the analogy between a fibre coupler and a beam splitter. Recent Developments in All-Fibre Devices for Optical Networks 211 The corresponding intensity transmissions in main and secondary branches, respectively labelled 1 and 2, are T 1 = cos 2 α = 1+ cos2α 2 T 2 = sin 2 α = 1−cos2α 2 (4) A 50%/50% splitter, also referred to as 3 dB coupler, is thus a coupler having α=π/4+pπ/2, with p integer (usually null, to ensure small spectral dependence). Due to the lack of circular symmetry of the guiding structure, the couplers are inherently polarisation dependent. The cross section has two symmetry axes x and y, which define the principal polarisation axes. The coupler transmissions must more generally be written as a superposition of transmissions in each polarisation T 1 = T 1x + T 1y =ηcos 2 α x + (1−η)cos 2 α y T 2 = T 2x + T 2y =ηsin 2 α x + (1−η)sin 2 α y (5) where η and (1-η) are the proportions of power launched in the x and y polarisations, respectively. However, strongly fused couplers are virtually polarisation insensitive inasmuch their waist is not too small. This is the case of most 3dB and other standard beam splitters, the polarisation dependence of which is ignored. The transfer matrices are very useful tools to predict the responses of more complex structures made of concatenation of several couplers. The simplest one is the Mach-Zehnder (MZ) interferometer made of two concatenated couplers, which may be different, thus characterised by α ≠α’. Such an interferometric structure is sketched in Fig. 4. Figure 4. All-fibre Mach-Zehnder structure. The phase difference between the arms ϕ=β 1 L 1 −β 2 L 2 is realised through a length difference L 1 −L 2 and/or a propagation constant difference β 1 −β 2 =2πν(n 1 −n 2 )/c. For a phase difference ϕ between the two MZ arms, the transmission column vector (containing individual guide amplitudes) may be calculated by using matrix products as follows Nawfel Azami and Suzanne Lacroix 212 ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − 0 1 cos sin sin cos . 0 0 . ' cos ' sin ' sin ' cos 2 2 ' α α α α α α α α α ϕ ϕ α i i e e e i i e i i i i (6) The intensity transmissions in each branch are then easily derived to be T 1 = 1 2 (1+ cos2α cos2α' −sin2α sin2α' cosϕ) T 2 = 1 2 (1− cos2α cos2α' +sin2α sin2α' cosϕ) (7) The different parameters α, α’, and ϕ give a flexibility to design a variety of different components with specific functionalities. As an example, the DWDM components are examined in Section 2. In this case, one has the couplers parameters α=α’=π/4, which are almost wavelength independent over the range of interest. 2. Stable Wavelength Division Multiplexer All-Fibre Devices WDM Mach-Zehnder Interferometers (MZI) are extensively used as multiplexer, demultiplexer, add-drop modules, and in many other applications. For most of these applications, a control of the thermal dependence of the refractive index is required. In order to simplify the description of the MZI transmission as a function of the optogeometrical parameters of the two fibres (Fig. 4), let us suppose an ideal MZI with no loss and an infinite isolation by using 3dB couplers. Using eq. 7, the transmittivity from port 1 to port 2 can be written as: ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − π = ) ( 2 cos ) ( 2 2 1 1 2 L n L n c T ν ν (8) where ν is the signal frequency, c is the light velocity, L 1 and L 2 are the lengths of fibres 1 and 2 in the central zone respectively, and n 1 and n 2 their effective indices. The Free Spectral Range (Δν) and the p th transmission peak frequencies ν p are given by: ( ) ( ) 2 2 1 1 2 2 1 1 2 . 2 L n L n c p L n L n c p − = − = Δ ν ν (9) Inter-channel spectral distance Δν is then induced by fibres with different refractive index profiles or/and different lengths. MZIs are known for their narrow band capabilities. For this purpose, they must be stable over a range of environmental conditions, such as temperature, within a defined range in case of temperature variations. However, the refractive indices or the optical path lengths of the two connecting fibres of the device between the two couplers Recent Developments in All-Fibre Devices for Optical Networks 213 usually vary with temperature. If the thermooptic coefficients (i.e. the temperature dependence of their refraction indices) of the two fibres are not equal or if the optical paths of the two fibres are not equal, the temperature variations cause variations in the differential phase shift. Consequently, the channel spacing of the device, defined as the wavelength separation between the transmission peaks of two adjacent channels, as well as the peak wavelengths and the pass-band, become unstable. This would cause significant problems for WDM applications, due to the small separation between channels. In the next section, results of temperature-independent all-fibre MZI manufactured with the FBT technology are presented. The thermal dependence of an optical fibre can be expressed with the aid of the thermo- optic coefficient, which describes the change of the index of refraction with the change of temperature dn/dT. If both arms in the central zone are equal (L 1 =L 2 =L), the thermal shift of the MZI transmission peaks is given by: dν p dT ν p − 1 Δn dn 1 dT dn 2 dT − ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ ⋅ 1 L dL dT ⋅ + ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ ⋅ (10) where Δn= n 1 – n 2 and L is the length of the central zone of the MZI. The thermal expansion coefficient for silica (L -1 .dL/dT) is about 5.10 -7 °C -1 . The contribution of the thermal expansion of silica fibre to the thermal shift of the MZI transmission peaks is nearly 0.75 pm/°C for a transmission peak at 1.55 μm. Thermal expansion of silica is usually neglected in Mach-Zehnder interferometer structures and is not an issue for thermal compensation. However, thermal expansion of silica is of great importance in tapered couplers design because of the impact on supermodes propagating index. 2.1. Passive Thermal Compensation Using UV Treatment It is well known that photosensitive fibre hydrogenation may produce large refractive index changes if the fibre is exposed to UV radiation [3,4]. This process has been extensively used for fabrication of Bragg gratings and balanced MZI. More recently, it has been shown that hydrogenation of an optical fibre followed by UV exposure can control the thermal dependence of the refractive index. This may be used in a device, such as an all-fibre MZI. In the following section, the process applied to one fibre-arm of the MZI is presented. This process, applied before the fabrication of the MZI, consists of hydrogenation and exposition to UV radiation. The optical fibre is put in a pressure chamber, filling the chamber with hydrogen at a suitable pressure (about 1800 psi) and left there for a period of time suitable to achieve the desired photosensitivity (about 12 hours). This process produces an increase in the index of refraction of the fibre, which becomes n+dn [3,5,6]. Thereafter, the photosensitive fibre is exposed to UV radiation. As is mentioned in ref. [6], such an exposure can lead to a further increase of the fibre refractive index. It has been recently found that one can control or adjust the thermal dependence of the optical fibre by controlling the UV exposure time of the photosensitive fibre [7]. Moreover, it has been discovered that the change of the thermal dependence provided by this method remains constant even though the index of refraction is further changed, for example by exposing the fibre to heat. Thus, by Nawfel Azami and Suzanne Lacroix 214 heating the fibre to a temperature greater than 800 °C, one can bring down the fibre index of refraction back to the value n, without affecting the adjusted thermal dependence. As a demonstration, all-fibre MZIs were fabricated using the treated fibre (fibre 1) and a dissimilar fibre (fibre 2) with different refractive indices (Fig. 5). Thermal dependences of the fibres are also different. Fig. 6 illustrates the change of the thermal dependence of the all fibre MZI as a function of the UV exposure time of the hydrogenated fibre (Corning SMF-28™). The thermal dependence of the MZI does not change during the first 5 minutes of exposure to UV radiation. The change in thermal dependence then starts to occur gradually and continues more steeply as shown in Fig. 6. Between 10 and 25 minutes of exposure, the thermal dependence change is essentially linear. As is shown in Fig. 6, the reproducibility of the thermal dependence is good. The small variability of the thermal dependence may be due to variations in the fabrication process of the MZI (for example small variations of the temperature from device to device) and also to variations in the final free spectral range of the MZI that have been tested (20nm ± 1nm). 3 dB 3 dB Fibre 1 Fibre 2 Figure 5. All-fibre Mach-Zehnder interferometer with different fibres. -20 -10 0 10 20 30 0 5 10 15 20 25 30 35 Time of exposure to UV (min) T h e r m a l d e p e n d e n c e ( p m / C ) Figure 6. Experimental thermal dependence of all fibre MZI as a function the UV exposure time of one of its fibre arms. Recent Developments in All-Fibre Devices for Optical Networks 215 2.2. Passive Thermal Compensation Using Specific Dopants in Fibres The cores of silica fibres are usually doped with Germanium to increase the refractive index with respect to the undoped cladding. However, many other dopants can be used to control the refractive index, such as Fluor or Phosphorus. Concentrations of such dopants in the fibre core or cladding have direct impact on the effective index of the fibre and on the temperature dependence of this effective index. The adjustment of the composition with dopants can take place in the core of the fibre or in the cladding or both. It has been demonstrated that the type of dopant used and its concentration can be selected to control the thermal wavelength drift of a MZI to about 1-2 pm/°C accuracy within a desired temperature range which is generally between about -35°C and +85°C [8]. 2.3. Flat-top WDM Devices WDM optical systems allow multi-channels communication in a single optical fibre. A channel is spectrally characterised by a wavelength and a width. Channel spacing in WDM systems is constantly decreasing and can be as low as 25 GHz. In many cases, sinusoidal spectral response of multiplexers/demultiplexers is not appropriate, especially in long haul optical networks where tight specifications on insertion loss, crosstalk and differential group delay are required. Moreover, fluctuations of the signal laser wavelength may induce loss fluctuations in Dense Wavelength Division Multiplexing (DWDM) systems when a sinusoidal transmission device is used. Flat-top spectral responses are preferred because they allow minimizing the crosstalk between adjacent channels, the fluctuations of channel loss, and the differential group delay. Thin films DWDM devices can be easily designed to meet Figure 7. Typical spectral response of flat top interleaver with three cascaded couplers. Nawfel Azami and Suzanne Lacroix 216 flat top channel specifications. However, all-fibre DWDM devices are still attractive because of their low polarization mode dispersion. In their usual two couplers Mach-Zehnder interferometers configuration, only sinusoidal spectrum can be achieved. Flat-top channel spacing multiplexer has been implemented by three cascaded couplers of different coupling ratios linked by two differential delays [9,10,11]. A non sinusoidal spectrum is obtained by using three couplers instead of the usual two, while the second differential delay is exactly twice the first one. Cascading wavelength insensitive couplers allow constant isolation and insertion loss over more than 100 nm. As an example, Fig. 7 shows typical optical spectra of a 100 GHz spacing DWDM interleaver consisting of three cascaded couplers. 2.4. All Fibre Optical Add Drop Module Optical Add-Drop Modules (OADMs) are key devices for optical networks. OADMs are the access points to the optical network and allow adding or droping wavelengths at different sites along the network. The most usual all-fibre design used a balanced MZI with two identical Fibre Bragg Gratings (FBGs) embedded in the two MZI arms [12]. Optical signals are launched into port 1 (Fig. 8). The 3dB coupler splits the input power evenly into the two MZ arms. Only those signals carried at the Bragg wavelength get reflected by the FBGs and return back into the first 3dB coupler. Whenever the optical paths of both reflected waves are balanced, all the wavelengths over the bandwidth of interest are phase-matched and all the optical energy is transferred into port 4 with little energy returning back to the bar path (see eq. 7 with α=π/4 and φ=0). The port 4 becomes the drop-port, at which signals at the Bragg wavelength of the FBGs get filtered out from other channels. Signals carried at wavelengths other than the Bragg wavelength transmit through the FBGs and merge into the second 3dB coupler. Similar to the reflected one, all the transmitted waves over the wavelength span of interest are phase-matched under a balanced MZ structure and most of the energy is carried into port 3. Port 3 then becomes the pass-port, through which signals outside of the FBG reflection band are transmitted. Port 2 can then be used as the add-port, into which other signals carried at the Bragg wavelength are launched. Those additional signals get reflected by the FBGs, carried through the cross path arm of the second 3dB coupler, and join port 3 without interfering with each other The most common fabrication method approach is that a MZI is made first and the FBG pair is then written on the established interferometer [13-15]. Another approach for which available FBGs are integrated into a MZ interferometer has also been demonstrated [16]. 3 dB 3 dB 1 4 λ g λ g λ g -dropp λ g -add IN OUT 2 3 Figure 8. Balanced all-fibre Optical Add Drop Module. Recent Developments in All-Fibre Devices for Optical Networks 217 2.5. All Fibre Differential Phase Shift Keying Demodulator The differential phase shifted keying (DPSK) modulation has been attracted great attention for its application for dense wavelength division multiplexing (DWDM) transmission, since DPSK with optical Mach-Zehnder interferometer (MZI) demodulation provides several advantages over the conventional intensity modulation detection [17]. Optical differential- phase shift keying (DPSK) is a modulation format that offers high receiver sensitivity, high tolerance to major nonlinear effects in high-speed transmissions, and high tolerance to coherent crosstalk [18,19]. In DPSK, data information is carried by the optical phase difference between successive symbols. As an example, a Conventional DPSK (CDPSK) uses phase difference in the set (0,π) [20]. Figure 9. DPSK demodulator. (a) successive symbols with π-phase difference. (b) successive symbols in phase. For direct detection of DPSK signal (by conventional intensity detectors), a DPSK demodulator is used to convert the phase-coded signal into an intensity-coded signal. Fig. 9 illustrates demodulation of a DPSK optical signal using 1-bit-time-unbalanced Mach-Zehnder interferometer designed with 3 dB couplers (also called delay line interferometer). The incoming differential phase-shift keying optical signal is first split into two equal-intensity beams in two arms of a Mach Zehnder, in which one beam is delayed by an optical path difference corresponding to 1-bit time delay. After recombination, the two beams interfere with each other constructively or destructively, depending on the optical phase difference between adjacent bits. Using eq. 7 with α=α’=π/4 (3 dB couplers) and ϕ the phase difference corresponding to 1 bit time, one can easily show that the resultant interference intensity of two adjacent bits in phase is directed in port 3 (output port), while the resultant interference intensity of two adjacent bits having π-phase difference is directed in port 2. The resultant interference intensity is the intensity-keyed signal in output port 3. The all-fibre Mach- Nawfel Azami and Suzanne Lacroix 218 Zehnder interferometer design demonstrated low insertion loss, low polarization dependent loss and low polarization dependent isolation over a wide spectral band by using 3dB wavelength insensitive couplers [21]. Tunable all fibre DPSK demodulators are also very attractive since they allow re-configuration of wavelength channels. In an all-fibre structure, phase tuning is typically achieved by applying an electrical voltage to one arm of the MZ that has been metallized to this aim. The refractive index of the metallized arm change because of the thermo-optic effect, which allows tuning of the phase difference ϕ. 3. Developments on All-Fibre Devices for Erbium Amplifiers Erbium-based optical amplifiers have been developed during the 1980s to replace the expensive and complex electronic repeaters. The advantage of Erbium-doped fibre amplifiers (EDFAs) lies in the practical issues related to coupling losses, polarization insensitivity, high gain, low noise, and capability to regenerate several channels simultaneously. However, EDFAs need components for their integration in optical networks. As an example, EDFAs usually incorporate a gain equalizer filter to flatten the gain spectrum. Because of their high performance Erbium amplifiers are also used in a two-stage configuration. The mid-stage allows incorporating many devices to optimize the network performance, such as a chromatic dispersion compensation fibre, a Polarization Mode Dispersion (PMD) compensation module, a dynamic gain compensator, a gain flattening filter, and add-drop modules…Fig. 10 illustrates a basic configuration of a two-stage EDFAs. Pump/Signal combiner EDFA Isolator EDFA Mid-stage GFF DGC pump pump Figure 10. Two-stage EDFA basic configuration. 3.1. Fibre Gain flattening Filters All-fibre amplifiers are commonly used in telecommunication networks to amplify signals on a wide bandwidth. Filters are required to flatten the non-uniform gain of EDFAs or Raman amplifiers. Fixed gain flattening filters (GFFs) flatten the gain profile of optical amplifiers by selectively removing excess power. These filters are often fabricated using short- or long- period fibre gratings. However, efficient gain flattening filters can also be achieved with a cascade of tapered fibres [22]. As discussed in Section 1.3, abruptly tapered fibres allow the coupling between the fundamental mode and several cladding modes. The controlled taper profile is used for tailoring the filter loss spectrum. Tapered fibres show a sinusoidal spectral Recent Developments in All-Fibre Devices for Optical Networks 219 response. They can thus be combined to create any spectral filter as in a Fourier series (Fig. 11). GFF are static filters used to flatten the amplifier spectrum gain for a given gain shape. Hence, GFFs are designed for given operating conditions, such as total signal power, pump power, number of channels, and temperature conditions. When these parameters vary, the gain shape of the amplifier also varies, and the GFF is no longer able to flatten the amplifier spectrum. Figure 11. Spectral response of a gain flattening filter for erbium-doped amplifiers made by concatenation of four tapered fibre filters. 3.2. Dynamic Gain Tilt Compensation In this section, we focus on recent development of the EDFA gain control using all-fibre devices. An optical amplifier may not always operate at the gain value for which the gain flatness is optimized. Many factors contribute to this sub optimal operating condition: span- loss variation, input channel count change, and spectral tilt due to stimulated Raman scattering. As a result, the amplifier gain is tilted, and such tilt can have significant impact on the system performance. Generally, spectral gain flatness of an EDFA due to change of operating conditions is characterized by the Dynamic Gain Tilt parameter (DGT). DGT (dB/dB) is defined as the gain variation at wavelength λ,when the gain variation at a reference wavelength λ 0 is 1 dB. ) ( ) ( ) ( 0 λ λ λ G G DGT Δ Δ = (11) The DGT is a characteristic function of erbium ions and do not depend on fabrication techniques or opto-geometrical parameters of the fibre. It is then an efficient parameter to characterize the variation of the gain in EDFAs. It can be easily shown that the DGT is a function of the absorption and emission Giles parameters: Nawfel Azami and Suzanne Lacroix 220 ) ( ) ( ) ( ) ( ) ( 0 0 λ α λ λ α λ λ s s g g DGT + + = (12) In the case of C-band EDFAs, the dynamic gain-tilt (DGT) is the main factor of the gain flatness deterioration, especially in the case of 980 nm pumping. Another important amplifier control function is to maintain the output signal level per channel. The dynamic control of the per-channel output power in the EDFA is important to avoid SNR degradation. Many designs have been proposed for dynamic gain tilt compensation. Variable optical attenuator (VOA) is the most common device used for gain tilt compensation [23]. However, the large insertion loss of the VOA deteriorates the signal to noise ratio and/or the power conversion efficiency of the amplifier. Automatic power control (APC) scheme and a variable attenuation slope compensator (VASC) demonstrate better performances than the VOA [24]. The APC is employed in the first EDFA stage and the VASC in the mid-stage does not change its insertion loss in spite of the attenuation slope change. In reference [25], an all-fibre Mach- Zehnder interferometer with appropriate couplers is presented. The design allows dynamic gain tilt compensation by only changing the isolation of the interferometer while the centre wavelength remains unchanged. The all-fibre structure allows high optical performance including low insertion loss, low polarization dependent loss and low polarization mode dispersion. The gain slope tuning is made using the thermo-optic effect while the device still passively insensitive to external temperature variations. For illustration, this all-fibre device is presented in the following. Figure 12. Mach-Zehnder interferometer with metallised fibre in one arm. A MZ can be used in a linear spectral region to compensate the gain tilt of an EDFA. The MZ couplers are identical and wavelength dependent such that 0 dB insertion loss is realized at λ 0 (1520 nm) and 3dB is realized at λ 1 (1580 nm). The MZ is then characterized by a minimum insertion loss at λ 0 and a maximum insertion loss at λ 1 . One of the two branches of the MZ is metallised to allow phase tuning between the two arms of the interferometer (Fig. 12). Applying an electric voltage allows to increase the temperature, and thus change the refractive index of the fibre. As a result, phase changes occur between the two arms via thermo-optic effect, allowing the change of the slope in the 1530-1570 nm spectral band. The phase change has only an impact on the isolation at λ 1 , while central wavelengths stay unchanged. Figure 13 shows the spectral response of the MZs for different phase differences induced between the two branches. Initial configuration is such that the isolation of the components is 0 dB. A flat transmission near 0 dB over the C-band is realised for no applied voltage. We focus on the 1540-1565nm spectral band where the EDFA has a linear DGT. The Recent Developments in All-Fibre Devices for Optical Networks 221 maximum insertion loss at 1540nm is 1 dB for –5 dB gain tilt. The deviation from linearity is less than ±0.25 dB. The polarization dependent loss (PDL) is less than 0.3dB. A maximum of 3 Volts allow –5 dB tilt between 1540 nm and 1565 nm. The response time, defined by the characteristic time allowing a change of the attenuation slope from 5 dB to 5/e dB was 210 ms. -25 -20 -15 -10 -5 0 1500 1550 1600 Wavelength (nm) d B -6 -4 -2 0 1540 1550 1560 Wavelength (nm) d B -25 -20 -15 -10 -5 0 1500 1550 1600 Wavelength (nm) d B -6 -4 -2 0 1540 1550 1560 Wavelength (nm) d B Figure 13. (a) Transmission of MZI for different applied voltage. V=0, 1, 2.25, 3 and 3.6 Volts. (b) Zoom of the 1540-1565 nm range. Two concatenated MZIs are required to allow positive and negative slope compensation as well as a constant average insertion loss for any slope. The first MZI has optical characteristics presented in the previous section. The second MZI has a complementary transmission (couplers with 0 dB at λ 1 and 3 dB at λ 0 ). ). A great advantage of this all-fibre DSC is that a very low electrical power is needed to compensate the gain tilt. A maximum total electrical power of 250 mW is needed due to the low thermal conductivity of silica. 4. Developments on All-Fibre Devices for Raman Amplifiers Raman Fibre Amplifiers (RFAs) are of great interest for the development of long distance, high capacity WDM systems. Their main advantages are their low noise, wide amplification bandwidth and saturation characteristics. RFAs have also the advantage that the optical amplification occurs in the transmission fibre transmission itself. RFAs differ in principle from EDFAs as they utilize the stimulated Raman scattering effect to create optical gain. However, RFA suffer from polarization dependent gain (PDG). A solution to reduce PDG is the use of pump laser with low degree of polarization (DOP). One can scramble the state of polarization of the pump with the aid of a depolarizer. Experimental and theoretical investigations have been reported on the statistical properties of PDG [26-30]. These reports show that the PDG is linked to the PMD of the fibre. Fig. 14 illustrates the basic design of an optical fibre system using Raman amplification. In a RFA, a strong pump laser provides gain to signals at longer wavelengths through stimulated Raman scattering. One of the major attractions of Raman amplification is that it can be used over a very wide wavelength range by multiplexing together different pumps wavelengths. Nawfel Azami and Suzanne Lacroix 222 Figure 14. Raman amplification configuration. Polarization beam combiners (PBC) are key component of diode-pumped Raman amplifiers. They allow combining the output of two pumps operating at the same wavelength but in different polarization modes into a single fibre, thereby doubling the effective power output. PBCs can also be used in EDFAs, typically to combine 1,480 nm pumps in the second stage. The combination of the two linear-orthogonal polarizations having the same power, allows in addition a complete depolarization of the output pump. Moreover, polarization combiners allow protecting the system from the failure of any single laser. RFAs may also be subject to dynamic reconfiguration of the pumps lasers wavelengths. Broadband polarization combiners can be necessary to ensure system reconfiguration. Optical depolarizers play also a key role by scrambling the state of polarization of wave pumps that are not doubled. 4.1. Broadband All-Fibre Polarization Combiners The fibre optic coupler made by the FBT technology has been one of the most widely used devices in optical fibre systems. Other than the most common function of optical power splitting, such couplers may have other applications. In particular, they can be designed to operate as optical polarization beam splitter/combiners [31,32]. For example, optical networks use optical polarization beam splitters in their PMD compensator modules, and pump depolarizers in Raman amplifiers. A large bandwidth is usually specified in case of multi-channel lightwave systems. Ideally, a bandwidth as large as the complete optical band (S+C+L) is suitable. However, all-fibre polarization beam splitters suffer from a narrow bandwidth, which limits the spectral operating wavelength range to a few nanometers. A spectral width of 17 nm for –15 dB extinction ratio has been reported [33]. A wider spectral width of 38 nm for –15 dB extinction ratio has been demonstrated using a weakly fused coupler design [34]. Recently, an all-fibre polarization beam splitter/combiner has been reported on a very wide band of more than 200 nm for –15 dB extinction ratio [35]. In the following section a brief description of this device is presented. An all-fibre MZI design is used for which specific couplers are designed. The first coupler is an all-fibre polarization beam combiner (PBC) and the second coupler is an all- Recent Developments in All-Fibre Devices for Optical Networks 223 fibre WDM coupler. The structure is schematized on Fig. 15. Input fibres (fibres 1 and 4) are polarization-maintaining fibres (PMF). The central zone of the MZI (fibres A and B) and outputs (fibres 2 and 3) are standard single mode fibres (SMF). In regard to Fig. 15, the x- polarization is coupled into port 4 and the y-polarization is coupled into port 1. Experimental transmissions from input ports 1 and 4 to output ports A and B are illustrated in Fig. 16 for both polarizations. By using transfer matrix formalism, the transmitivity of the MZI is given by ) ( ). ( ). ( , , λ φ λ Y X PBC MZ WDM Y X M M M M = (13) where the transfer matrix of the WDM and the PBC are given in section 1.4. Let us note α x,y (λ) and α w (λ) the phase difference between the symmetric and anti- symmetric super-modes of the fused fibre PBC for x- and y-polarizations, and the WDM coupler respectively. These parameters are defined in section 1.4 (eq. 2). The central zone of the interferometer structure is characterized by a φ-phase shift between the two branches of the MZI (fibres A and B). It has been shown that, If α w (λ)=α x (λ) and φ=π then the device transfer matrix for x- and y-polarizations becomes wavelength independent. The spectral dependence of the PBC can be counterbalanced by a π-phase between the two MZI arms while using a WDM coupler having the same spectral dependence transmitivity as the PBC. Figure 15. Wideband polarization combiner design. Fig. 17 shows typical experimental spectral transmissions of the interferometer structure. The non-perfect extinction ratio at the input ports (PM fibres 1 and 4) induces polarization beating and spectral ripples on the device transmission. The extinction ratio at the PM fibres inputs is estimated to be –30 dB. For such an extinction ratio, the ripples are only observed on the isolation ports. The insertion loss is lower than 0.2 dB in the 1440 nm-1550 nm- Nawfel Azami and Suzanne Lacroix 224 wavelength range. Insertion loss increases near 1440 nm because of the water absorption peak, which induces excess loss, and increases above 1550 nm because the spectral transmissions shift between the PBC and the WDM coupler degrades the isolation. More than 200 nm spectral width for –15 dB extinction ratio is achieved. Figure 16. Experimental transmissions of PBC. 1 and 4: input ports; A and B: output ports. -30 -25 -20 -15 -10 -5 0 1400 1450 1500 1550 1600 Wavelength (nm) d B X(4-3) Y(1-3) X(4-2), Y(1-2) -30 -25 -20 -15 -10 -5 0 1400 1450 1500 1550 1600 Wavelength (nm) d B X(4-3) Y(1-3) X(4-2), Y(1-2) 1 and 4: input ports; 2 and 3: output ports. Figure 17. Experimental spectral transmissions of the interferometer structure. 4.2. Stable All-Fibre Depolarizers Passive depolarizers are used to scramble the State Of Polarization (SOP) of an incoming light source, reducing the mutual coherence between the orthogonal polarization components of the light source. Highly birefringent (Hi-Bi) fibres are usually used to depolarize wide- band sources but are not suitable for narrow-band sources because of the long length required. Recently described passive devices are based on incoherent fibre ring structures, using a cascaded fibre ring design [36] or a dual fibre ring design [37]. These designs allow Recent Developments in All-Fibre Devices for Optical Networks 225 depolarizing any SOP by using directional couplers in a fibre ring scheme and by adjusting the ring birefringence. Recently a new technique for the single-mode fibre depolarizer based on polarization combiners using a linear design (as opposed to ring design) has been proposed. The use of a linear design allows a one-way propagation of the two orthogonal polarization components, which make the device very stable. This new design provides low loss, low polarization dependent loss and high depolarization of light for any input SOP. The assembly of such a device is made using power light detection that makes the integration device easier than the degree of polarization optimization technique. Figure 18. All-State of Polarization all-fibre depolarizer design. The all-SOP all-fibre depolariser linear design presented in ref [38] is a combination of two polarisation combiners (PC) and a 2x2 directional coupler (Fig. 18). An optical phase delay (delay1) is induced between the waves propagating in the two branches A and B. A polarisation rotator device is used to realise half-π rotation of the light wave SOP propagating in fibre B. Interference occurs at the coupler since the SOPs at the inputs of the coupler are parallel. The average intensities at the outputs of the 2x2 coupler are equal if the delay induced is much greater than the coherence length of the light source. A second phase delay (delay2) is induced between the waves propagating in A and B fibres. A half-π rotator device is used such that the SOP of the wavelength propagating in A and B fibres are orthogonal and aligned with the eigen axis of PC 2 . To increase the polarisation scrambling at the output of the depolariser, the condition on equal average intensities I X and I Y has to be satisfied. If the optical delay induced by delay1 is much greater than the coherence length of the source and if Nawfel Azami and Suzanne Lacroix 226 a 3 dB transmission coupler is used then the average intensities for x and y-polarisations at the input of PC 2 are equals for any SOP at the input. Also, the two orthogonal components x and y are completely uncorrelated when the delay lengths (ΔL 1 and ΔL 2 ) as well as the difference length (ΔL 1 -ΔL 2 ) are greater than the coherence length of the laser DOP less than 4.5% is obtained for coherence length less than 1 m. The linear design eliminates the re-circulations in PC or coupler fibre occurring in a fibre ring configuration. The loss is then given by the sum of the losses of each subcomponents. Low loss (1.5 dB), and low polarization dependent loss (0.5 dB) over 1400–1500 nm spectral range, for all SOPs, and for a 0–70°C temperature range, has been demonstrated with this type of design. By using a wide band PC, the depolariser presented is made achromatic over a 100 nm spectral band (Fig. 19). The one passage light propagation in symmetrical branches (identical fibres) makes it very stable. Although the PM fibre is often temperature sensitive the small length of half wave PM fibre used as a rotator device (1.8 mm) keeps the SOP thermally stable. The DOP variation obtained is +/-2% and loss variation is ±0.05 dB for a 0°C to 70°C temperature range. In addition, the all silica-fibre structure allows the depolarisation of any laser with a coherence length lower than the loop length and permits high power handling. By design, this depolariser allows 2 inputs and 2 outputs, for each input corresponds an output. Figure 19. Maximum degree of polarization (for any input SOP) versus wavelength. Conclusion Fused Biconical Taper fibre devices have shown great integration in today optical networks. From basic directional coupler to cascaded Mach-Zehnder interferometers designs, all fibre components are used to perform many functionalities in all parts of the optical network. In the transmission part, temperature-independent WDM and DWDM interleavers with flat-top spectrum are attractive because of their very low chromatic dispersion, differential group delay, and polarization dependent loss. In the amplification part, all-fibre devices based on fused biconical taper fabrication technique demonstrated high potential to multiplex pumps and signals. Cascaded tapered fibres and cascaded couplers demonstrated their capability to correct the amplifier gain non-uniformity being respectively used as Gain Flattening Filters Recent Developments in All-Fibre Devices for Optical Networks 227 and dynamic gain compensators. For Raman amplification, broadband polarization combiners and depolarizers have been developed to limit gain and channels power fluctuations while high power handling and wide spectral band are ensured, which allow broadband multichannel amplification and amplifiers re-configuration. Access to the network can be performed anywhere along the fibre by using all-fibre optical add-drop modules while polarization mode dispersion is minimized compared to circulators based design. Many other functionalities can be performed by all-fibre devices. References [1] S. Lacroix, N. Godbout, and X. Daxhelet, “Fused Biconical Components,” Chapter 6 of Optical Fiber Components: Design and Applications Editor Habib Hamam, Research Signpost (2006). [2] W. J. Stewart, and J. D. Love, “Design limitation on tapers and couplers in single-mode fibres,” in 5th Int. Conf. Integrated Opt. &Opt. Fibre Commun., 11 th European Conf. Opt. Commun., IOOC/ECOC'85, ed. Instituto Internazionale delle Comunicazioni, Venice, Italy, 1985, pp 559-562. [3] G. Meltz, W. W. Morey, and W. H. Glenn, “Formation of Bragg gratings in optical fibers by a transverse holographic method,” Opt. Lett., 1989, vol. 14, pp 823–825. [4] K. O. Hill, Y. Jujii, D. C. Johnson, and B. S. Kawasaki, “Photosensitivity in optical fiber waveguides: application to reflection filter fabrication,” Appl. Phys. Lett., 1978, vol. 32, pp 647 649. [5] M. Douay, W. X. Xie, T. Taunay, P. Bernage, P. Niay, P. Cordier, B. Poumellec, L. Dong, J. F. Bayon, H. Poignant, and E. Delevaque, “Densification involved in the UV- based photosensitivity of silica glasses and optical fibers,” J. Lightwave Technol., 1997, vol. 15, pp 1329–1342. [6] P. J. Lemaire, R. M. Atkins, V. Mizrahi, and W. A. Reed, “High pressure H2 loading as a technique for achieving ultrahigh UV photosensitivity and thermal sensitivity in GeO2 doped optical fibers,” IEE Electron. Lett., 1993, vol. 29, pp 1191–1193. [7] N. Azami, A. Villeneuve, F. Gonthier, “Method of adjusting thermal dependence of an optical fiber”, Optics Communications, 2005, Vol. 251, Issues 1-3, pp 6-9. [8] F. Gonthier, F. Seguin; Francois, N. Godbout, A. Villeneuve, “Passive thermal compensaton of all fibre Mach Zehnder interferometer” US Patent 6,850,654, 2005. [9] C. Chon, H. Luo, C. H. Huang, R. Huang, J. Chen, and J. R. Bautista, “Expandable 50- GHz and 100-GHz dense wavelength division multiplexers based on unbalanced and cascaded-fiber Mach-Zehnder architectures,” in Tech. Dig. National Fiber Optic Engineers Conference, NFOEC’99, 1999. [10] C. H. Huang, Y. Li, J.Chen, E. Sidick, J. Chon, K. G. Sullivan, and J. Bautista, “Low- loss flat-top 50-GHz DWDM and add/drop modules using all-fiber Fourier filters”, in Tech. Dig. National Fiber Optic Engineers Conference, NFOEC2000, 2000, pp 311-316. [11] J. C. Chon, C. H. Huang, and J. R. Bautista, “Ultra small dispersion, low loss, flat-top, and all-fiber DWDM and NWDM devices for high speed optical network applications”, European Conference on Optical Communication 2000, paper 11.3.1, 2000. Nawfel Azami and Suzanne Lacroix 228 [12] D.C Johnson, K.O. Hill, F. Bilodeau,,S. Faucher,. “New design concept for a narrowband wavelength selective optical tap and combiner”, IEE Electronics Letters, Vol.23, No.13, pp.668-669, 1987. [13] F. Bilodeau, D.C, Johnson, S. Theriault, B. Malo, J. Albert, K.O. Hill, “An all-fiber dense wavelength-division multiplexer/demultiplexer using photoimprinted Bragg Gratings”, IEEE Photonics Technology Letters, Vol.7, No.4, pp.388-390, 1995. [14] S.J. Madden, P.G. Jacob, J. Harradence, C. Robilliard, M.C. Ellas, S.J. Hewlett, K. Sugden, P.A. Otten, W.T. Holloway, M.D. Wallace, R.K. Selwood, “High performance modular DWDM implementation based on fiber Bragg Grating Mach Zehnder Interferometers”, in Tech. Dig. National Fiber Optic Engineers Conference, NFOEC’98, 1998, Vol.1, pp.453-463. [15] F. Bakhti, X. Daxhelet, P. Sansonetti, S. Lacroix, “Influence of Bragg grating location in fused 100% coupler for add and drop multiplexer realization”, OFC’98 Technical Digest, 1998, pp.333-334. [16] L.Y. Liu, J. Chen, A. Luo, J. Chon, K.Sullivin, J. Bautista, « Ultra low loss all fiber based FBG », in Tech. Dig. National Fiber Optic Engineers Conference, NFOEC’99, 1999. [17] A.H. Gnauck, "40-Gb/s RZ-differential phase shift keyed transmission", in Proc. Optical Fiber Communication Conf. OFC 2003, 2003, Paper ThE1, pp 450-451. [18] 18 X. Liu, “Nonlinear effects in phase shift keyed transmission”, in Proc. Optical Fiber Communication Conf. OFC 2004, 2004, Paper ThM4, pp 851-852. [19] X. Liu, C. Xu and X. Wei, “Performance analysis of time/polarization multiplexed40- Gb/s RZ-DPSK DWDM transmission”, 2004, IEEE Photon. Technol. Lett., Vol. 16, pp 302-304. [20] J. Wang, and J.M. Kahn, “Conventional DPSK Versus Symmetrical DPSK:Comparison of Dispersion Tolerances”, IEEE Photon. Technol. Lett., 2004, Vol. 16, pp 1585-1587. [21] F. Seguin and F. Gonthier, “Tuneable all-fiber, delay-line interferometer for DPSK demodulation”, 2005, OFC 2005, Vol. 5, pp 248 – 250. [22] X. Daxhelet and F. Gonthier, “Tapered fiber filters: Theory and applications,” in Proc. SPIE, Photonic East Boston, MA, vol. 4216A, no. 5, 2000, pp. 343-354. [23] M. Suzuki and S. Shikii, “The gain equalizing method of Erbium-doped fiber amplifiers for C-band and L-band”, IEEE/LEOS Annual Meeting, 1999, ThJ2. [24] H. Nakaji, M. Kakui, H. Hatayama, C. Hirose, H. Kurata, and M. Nishimura, “Superior noise performance and wide dynamic range Erbium-doped fiber amplifiers employing Variable Attenuation Slope Compensator”, IEICE Trans. Electron, 2001, Vol. E84-C, pp 598-604. [25] N. Azami, “All-Fiber Dynamic Gain Slope Compensator”, Optics Communications, 2003, Vol. 230/4-6 pp 325-329. [26] 26 P. Ebrahimi, M.C. Hauer, Q. Yu, R. Khosravani, D. Gurkan, D.W. Kim, D.W. Lee, and A.E. Willner, “Statistics of Polarization Dependent Gain in Raman Fiber Amplifiers due to PMD”, in Conference on Laser and Electro-Optics, 2001, Vol. 56 of OSA trends in Optics and Photonics Series, pp 143-144. [27] S. Popov, E. Vanin, And G. Jacobsen, “Influence of polarization mode dispersion value in dispersion-compensating fibers on the polarization dependence of Raman gain”, Optics Letters, 2001, Vol. 27, No 10, pp 848-850. Recent Developments in All-Fibre Devices for Optical Networks 229 [28] Q. Lin and G. P. Agrawal, “Statistics of Polarization-dependant gain in fiber-based Raman amplifiers”, Optics Letters, 2003, Vol. 28, No 4, pp 227-229. [29] E. Son, J. Lee, and Y. Chung, “Gain variation of Raman amplifier in birefringent fiber”, Optical Fiber Conference (OFC), 2003, Paper TuC5. [30] N. Azami, “Characterization of Polarization Dependent Gain in Raman fiber amplifiers”, Optics Communications, 2003, Vol. 230/1-3, pp 181-184. [31] M.S. Yataki, D.N. Payne, and M.P. Varnham, “All-fiber polarizing beamsplitter”, Electron. Lett., 1985, Vol. 21, pp 249-251. [32] T. Bricheno and V. Baker, “All-fiber polarization splitter/combiner”, IEE Electron. Lett, 1985, Vol. 21, pp 251-252. [33] M. Eisenhamm, E. Weidel, “Single mode fused biconical coupler optimized for polarization beam splitting”, J. of Lightwave Technol., 1991, Vol. 9, No 7, pp 853-858. [34] C.W. Wu, T.L. Wu, H.C. Chang, “A novel fabrication method for all-fiber, weakly fused, polarization beamsplitter”, IEEE Photonics Technol. Lett., 1995, Vol. 7, No. 7, pp 786-788. [35] N. Azami, A. Villeneuve, F. Gonthier, “All-Fiber Wide-Band Polarization Beam Combiner”, IEE Electron. Letters, 2003, Vol. 40, No 17, pp 1043-1044. [36] P. Shen, and J.C. Palais, ‘Passive single-mode fiber depolarizer’, Appl. Opt., 1999, Vol. 38, pp 1686–1691. [37] M. Martinelli, and J.C. Palais, ‘Dual fiber-ring depolarizer’, J. Lightwave. Technol., 2001, Vol. 19, pp 899–905. [38] N. Azami, E. Villeneuve, A. Villeneuve, F. Gonthier, “All SOP all fiber depolarizer design”, IEE Electron. Letters, 2003, Vol. 39, pp. 1573-1575. In: Optical Fibers Research Advances ISBN: 1-60021-866-0 Editor: Jurgen C. Schlesinger, pp. 231-256 © 2007 Nova Science Publishers, Inc. Chapter 8 ADVANCES IN OPTICAL DIFFERENTIAL PHASE SHIFT KEYING AND PROPOSAL FOR AN ALTERNATIVE RECEIVING SCHEME FOR OPTICAL DIFFERENTIAL OCTAL PHASE SHIFT KEYING M. Sathish Kumar 1a , Hosung Yoon 2b and Namkyoo Park 1c , 1 Optical Communication Systems Lab, School of EECS, Seoul National University, Seoul, Korea, 151-742, 2 Network Infra Laboratory, Korea Telecom, Daejeon, Korea, 305-811, Abstract Optical Differential Phase Shift Keying (oDPSK) with delay interferometer based direct detection receiver was proposed as an alternative for the conventional On-Off Keying (OOK) modulation schemes. Compared to OOK, oDPSK was predicted to have a 3dB improvement in performance due to its balanced detection receiver structure. It was also predicted that due to the optical signal occupying all the symbol slots, unlike in OOK, symbol pattern dependent fiber nonlinear effects will make less of an impact on long haul optical transmission schemes based on oDPSK. Subsequent successful demonstrations of these positive attributes of oDPSK resulted in active investigations into multilevel formats of oDPSK namely, optical Differential Quadrature Phase Shift Keying (oDQPSK) and optical Differential Octal Phase Shift Keying (oDOPSK). Significant developments in theoretical models of optically amplified lightwave communication systems based on the Karhunen-Loeve Series Expansion (KLSE) method assisted such investigations. In this chapter, we discuss some of the recent advances in oDPSK and its multilevel formats that have been achieved such as proposals for receiver schematics, theoretical analysis of receiver schematics, electronic techniques to counter polarization mode dispersion induced penalties, and application of coded modulation techniques. The chapter also proposes an alternative receiver schematic for oDOPSK which can separately detect the three constituent bits from an oDOPSK symbol. a E-mail address:
[email protected] b E-mail address:
[email protected] c E-mail address:
[email protected] M. Sathish Kumar, Hosung Yoon and Namkyoo Park 232 1. Introduction Single mode optical fibers with their enormous bandwidth of around 15THz and extremely low attenuation of 0.2dB/Km in the 1550nm window offer immense promise as a viable medium to realize high bit rate long distance data transmission systems. Over the years, there has been a remarkable growth in the data transmission rates achieved with one of the most recent experiments claiming 14Tbps over a distance of 160Km [1]. As the transmission distance and signaling rates increase, certain problems inherent to the optical fiber medium such as attenuation, chromatic dispersion, nonlinear effects, and Polarization Mode Dispersion (PMD) start to crop up which inhibits increments in the link distance and data transmission rates. Developments in the area of optical fiber amplifiers have resulted in mature technologies such as doped fiber amplifiers and fiber Raman amplifiers to overcome the attenuation limits. Also, developments in optical fiber device technology such as fiber gratings, both long period and Bragg, and high degree of control over refractive index profiles of core and cladding have helped in identifying feasible and effective methodologies to counter chromatic dispersion. Combating the ill effects of fiber nonlinearities and PMD still continue to be challenging problems primarily due to their statistical nature. More recently, alternate modulation techniques such as optical Differential Phase Shift Keying (oDPSK) [2] a bi-level version of optical differential phase modulation and optical duobinary signaling have been proposed and actively investigated upon. Optical duobinary schemes are based on the principle of introducing controlled inter symbol interference so that compared to On-Off Keying (OOK), for a given data transmission rate, the bandwidth of the optical signal propagating through the fiber is reduced. This obviously has an advantage over OOK in that spectral width dependent signal distorting mechanisms such as chromatic dispersion and PMD will have less of an impact. A tutorial discussion on duobinary signaling schemes could be found in [3]. In optical differential phase modulation, irrespective of whether it is bi-level or multilevel, the phase of the optical field during the current signaling interval is modulated relative to its phase in the previous signaling interval. The detection of the data at the receiver side at any particular signaling interval is hence dependent on the phase of the received optical signal in the previous interval. It is worth to note that optical differential phase modulation was under investigation during the late eighties and early nineties while coherent optical communication systems were aggressively explored [4]. However, the idea of optical differential phase modulation as most of the recent publications concentrate on is based mainly on an interferometric delay line based direct detection technique and will be the one discussed in this chapter. In comparison to OOK, oDPSK provides a 3dB performance improvement [2]. The 3dB improvement offered by oDPSK can easily be translated to an increase of approximately 15Km in the transmission length or a reduction in signal intensity dependent nonlinear effects such as stimulated Raman scattering, four wave mixing, cross phase modulation, etc. Moreover, since oDPSK has all the bit slots occupied by optical intensity, unlike in OOK, bit pattern dependent undesirable impacts of fiber nonlinear effects also get alleviated. While oDPSK transmits one bit per signaling interval, its multilevel versions, namely oDQPSK and oDOPSK, transmit two and three bits respectively per signaling interval. Obviously, for a given bit rate and pulse format (Return to Zero (RZ) or Non Return to Zero Advances in Optical Differential Phase Shift Keying… 233 (NRZ)), the oDQPSK and oDOPSK schemes can provide an increment in the spectral efficiency by factors of two and three respectively over OOK and oDPSK. This is significant in that as with optical duobinary transmissions, oDQPSK and oDOPSK due to their reduced bandwidth requirements will be more immune to spectral width dependent signal distorting mechanisms. Over and above these advantages, oDQPSK and oDOPSK carry with them more or less the same other advantages that oDPSK has over OOK. However, the price that needs to be paid for this improved spectral efficiency of the multilevel versions of oDPSK is an inferior error rate performance. This chapter aims at providing a review of some of the advances that have been achieved in the domain of optical differential phase modulation schemes such as proposals for receiver schematics, theoretical analysis of receiver schematics, electronic techniques such as equalizers to counter PMD induced penalties, and application of coded modulation techniques such as Trellis Coded Modulation (TCM) [5]. The chapter also proposes an alternative receiver schematic for oDOPSK which can separately detect the three constituent bits from an oDOPSK symbol. 2. Transmitter and Receiver Schematics for oDPSK and oDQPSK Figure (1) shows the transmitter and receiver schematics for oDPSK along with the resultant one dimensional signal space diagram and constellation in the inset. The coherent optical field emitted by the laser is phase modulated by a suitable optical phase modulator which is driven by differentially encoded NRZ data. The phase modulated output is passed through a pulse carver to obtain RZ optical pulses which are subsequently transmitted through the fiber. A pulse carver is essentially a Mach Zehnder Modulator (MZM) complimentarily driven by a sinusoidal clock [2] [6]. The phase modulator can be a simple optical phase modulator or a MZM [2]. At the receiver side, the optically amplified signal is first passed through an optical band pass filter and then through a delay line interferometer with one arm of interferometer introducing a time delay of T where T is the signaling interval. The constructive and destructive port outputs of the delay line interferometer are used to illuminate a pair of identical photo detectors to facilitate optoelectronic conversion. The difference of the two photo detector outputs is then passed through an electrical post detection filter whose output is sampled at appropriate time instants once every signaling interval to obtain y as shown in figure (1). The obtained sample y is compared with a threshold of zero to obtain the estimates of the transmitted binary data. It may be noted that the constructive port output (the top output in figure (1)) effectively feeds in duobinary modulated optical signal to the photo detector while the destructive port feeds in alternate mark inversion modulated optical signal [2] [7][8]. As mentioned in [2], it is this balanced detection using a pair of photo detectors that effectively gives a 3dB advantage for oDPSK over OOK. The concepts of multilevel optical differential phase modulation schemes such as oDQPSK and oDOPSK are in effect a two dimensional extension of oDPSK. In these multilevel versions of oDPSK, inphase and quadrature components of the optical carrier are phase modulated independently, combined and then transmitted. The inherent orthogonality between the inphase and quadrature components of the optical carrier enables unambiguous identification of the modulating data at the receiver side. M. Sathish Kumar, Hosung Yoon and Namkyoo Park 234 Figure 1. Transmitter and receiver schematics for oDPSK. The inset shows one dimensional signal space diagram. When it comes to receiver schematic for oDQPSK, there is no much of an option and more or less the same technique as the one used in electrical communication systems [3] is employed. However, oDOPSK opens up a range of options for detection. As such, we dedicate two separate sections later on for discussing the receiver schematics for oDOPSK. The rest of this section will discuss the transmitter and receiver schematic for oDQPSK. Figure (2) shows the transmitter and receiver schematic for oDQPSK. Comparing this schematic with that of oDPSK as given in figure (1), it can be readily observed that the transmitter schematic is in effect a parallel concatenation of two oDPSK transmitters. The incoming laser field is split into two equal parts in terms of power and passed through parallel phase modulators ensuring that the split optical fields have a relative phase shift of π/2 between them. This is to separate out the inphase and quadrature components of optical carrier. The inphase and quadrature phase modulated optical fields are then combined and guided through an optical fiber towards the oDQPSK receiver. At the receiver side, the received signal is split into two equal parts and passed through parallel concatenated delay interferometers. Compared to the delay interferometer setup depicted in figure (1) with regards to oDPSK, the difference here is that the arms of the delay interferometers introduce phase shifts that have to be such that the absolute value of the phase difference is π/2. More specifically 2 / 2 1 π θ θ = − (1) Advances in Optical Differential Phase Shift Keying… 235 Figure 2. Transmitter and receiver schematics for oDQPSK. The inset shows the signal space diagram with θ 1 and θ 2 as pπ/4 and qπ/4, such that p and q are odd integers satisfying the condition given in equation (1). The post detection electronic processing to extract the data bits can be simplified considerably by selecting θ 1 and θ 2 as pπ/4 and qπ/4, such that p and q are odd integers satisfying the condition given in equation (1), and using an electronic precoding circuit at the transmitter side as reported in [9]. The precoder is designed to satisfy the following Boolean expressions (for p=1, q=-1). ) ( ) ( ) ( . ) ( 1 1 1 1 1 2 1 1 − − − − − − ⊕ ⊕ + ⊕ ⊕ = k k k k k k k k k I b I Q I b I Q I (2.a) M. Sathish Kumar, Hosung Yoon and Namkyoo Park 236 ) )( ( ) ( . ) ( 1 2 1 1 1 1 1 1 − − − − − − ⊕ ⊕ + ⊕ ⊕ = k k k k k k k k k I b I Q I b I Q Q (2.b) Here I k and Q k are the NRZ data driving the top and bottom phase modulators respectively in figure (2) during the kth signaling interval and b 1k and b 2k are the two binary data bits which constitute the kth transmitted oDQPSK symbol and which are subsequently detected directly from the outputs y 1 and y 2 respectively. With the parameters selected as given in the above paragraph and with the appropriate precoder identified through equations (2.a) and (2.b), it is possible to obtain the information bit sequence carried by the inphase and quadrature components of the optical carrier from a bi-level detection of the outputs y 1 and y 2 respectively [9]. The resultant signal space diagram and constellation when θ 1 and θ 2 is pπ/4 and qπ/4 with p and q as discussed above is also shown in figure (2) in the inset. It may be noted that when one of the phase shifts of the delay interferometers of the receiver is set as |π/2| and the other as 0, the orientation of the signal vectors get rotated by π/4 from what it was when θ 1 and θ 2 were pπ/4 and qπ/4 with p and q as discussed above. This has an obvious disadvantage in that the outputs y 1 and y 2 will now be three valued as opposed to binary valued when θ 1 and θ 2 were pπ/4 and qπ/4 such that p and q are odd integers satisfying the condition given in equation (1). It can be inferred from the signal space diagram and the discussion above that the receiver schematic for oDQPSK with θ 1 and θ 2 as pπ/4 and qπ/4 in effect treats the inphase and quadrature components of the optical carrier as two separate independent oDPSK channels. 3. Transmitter and Receiver Schematics for oDOPSK Figure (3) depicts a possible transmitter schematic for oDOPSK [10]. The idea is based on the fact that an oDOPSK signal comprises essentially of two oDQPSK signals having a relative phase offset of π/4 between them [3]. The differentially encoded data b 1 and b 2 drive two parallel phase modulators as it was in the case of oDQPSK transmitters. However, the differentially encoded data b 3 brings about a phase shift of 0 or π/4 in the signal propagating through the last phase modulator. This effectively rotates the signal constellation by 0 or π/4 radians. Unlike in oDQPSK, wherein a direct mapping of the ideas from electrical communication systems was followed to arrive at possible receiver schematics, for oDOPSK, the major driving factors in identifying receivers have been optimized performance as well as ability to extract the three constituent bits directly through a bi-level detection of samples. This has led to suggestions of receiver schematics for oDOPSK which employ more than two delay interferometers. Before venturing into a review of such receiver schematics, it is worth to note that in principle it is possible to extract the eight distinct symbols that comprise the oDOPSK symbol set using a receiver schematic like the one used for oDQPSK. This should be quite elementary to understand since the signal space diagram of oDOPSK is two dimensional and to uniquely represent a point in a two dimensional space, only two coordinates are required. Those two coordinates could be obtained readily from a schematic exactly like the oDQPSK receiver by treating the outputs y 1 and y 2 as multilevel [11]. Advances in Optical Differential Phase Shift Keying… 237 Figure 3. Transmitter schematic for oDOPSK. Figure (4) depicts two receiver schematics proposed in [10] and [11] for oDOPSK. In case of figure (4.a), each interferometer output, after passing through the electrical post detection filter, is sampled once every signaling interval and treated as a bi-level sample. (a) (b) Figure 4. Receiver schematics for oDOPSK; (a) is after [11] and (b) after [10]. To be more specific, the absolute value of the samples obtained (y 1 to y 4 ) are disregarded and only their numerical signs are taken into consideration. From the fact that in oDOPSK the phase difference between successive symbols can take on only values which are integer multiples of π/4 mod 2π, table (1) can readily be formulated from the receiver schematic M. Sathish Kumar, Hosung Yoon and Namkyoo Park 238 shown in figure (4.a). Table (1) clearly shows that for each value of phase difference there is a unique combination of the numerical signs of y 1 to y 4 which easily enables the identification of the transmitted symbol and subsequent decoding of the symbol to its respective binary triplet. However, it should also be noted that out of 2 4 possible combinations, only 2 3 are made use of and the inevitable redundancy that exists throws up the option of error correction using maximum likelihood estimation techniques [11] [12]. Table 1. Relation between polarity of detected samples and phase of oDOPSK symbols for receiver shown in figure (4.a) Phase difference y 1 y 2 y 3 y 4 0 + + - - π/4 + - - - π/2 - - - - 3π/4 - - - + π - - + + 5π/4 - + + + 3π/2 + + + + 7π/4 + + + - Coming to the receiver schematic given in figure (4.b), the samples obtained from the delay interferometers outputs (y 1 , y 2 ) and the XOR logic block output (y 3 ) are treated as bi- level as was in the case of figure (4.a). We defer discussions on this receiver schematic for a later section wherein we discuss an alternative receiver schematic for oDOPSK. 4. Error Rate Performance Evaluation As is well known, optical amplifiers have become essential components of current state-of- the-art long haul fiber optic communication systems. Consequently, current fiber optic communication systems are essentially Amplified Spontaneous Emission (ASE) noise limited. As such, in performance evaluations, usually the sole noise source taken into account is the ASE. An accurate method to arrive at the Characteristic Function (CF) of the detected optically preamplified signal using the Karhunen Loeve Series Expansion (KLSE) method and subsequent evaluation of probability of error through saddle point integration was reported in [13] and modified later in [14] for computational efficiency. Though the KLSE method reported in [13] and [14] was implied for OOK systems, the method could easily be modified to use it for oDPSK as well as its multilevel versions as suggested in [2][15][16]. The major advantage of the KLSE method compared to others such as those developed in [17] is that the KLSE method could be used to evaluate the error rate performance for arbitrary pulse shapes and can account for the pre and post detection filter transfer functions along with other linear impairments of the fiber medium such as chromatic dispersion and PMD [14]. Advances in Optical Differential Phase Shift Keying… 239 In the following subsections we first discuss the essence of KLSE method for error rate evaluation of optically amplified lightwave communication systems and then the error rate performance. 4.1. Essence of the KLSE Method for Error Rate Evaluation Consider the general setup of a direct detection fiber optic communication receiver as shown in figure (5). Let the received optically amplified signal be ) ( ) ( ) ( t w t s t e + = (3) where s(t) and w(t) stand respectively for the desired signal and ASE noise sample function. The ASE noise is a complex Additive White Gaussian Noise (AWGN) process with a two sided Power Spectral Density of N O W/Hz. As the first step in deriving the KLSE method, it can be shown that [13] ∫ ∫ ∞ ∞ − ∞ ∞ − − = ' ) ) ' ( 2 exp( ) ( ) , ' ( ) ' ( 2 1 * dfdf t f f j f E f f K f E y k k π (4) where y k is the detected sample as shown in figure(5), E(f) is the Fourier transform of the received signal e(t), t k is the time instant at which the post detection filter output is sampled and K(f’,f) is as given below Figure 5. General setup of a direct detection fiber optic communication receiver. ) ' ( ) ' ( ) ( ) , ' ( * f H f f H f H f f K o e o − = (5) with H o (f) and H e (f) standing respectively for the transfer function of the optical and electrical filters. Rewriting equation (4) above as ∫ ∫ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − = ∞ → −∞ → b a b a k k b a k df df ft j f E f f K t f j f E Lt y ' ) 2 exp( ) ( ) , ' ( ) ' 2 exp( ) ' ( 2 1 * π π (6) M. Sathish Kumar, Hosung Yoon and Namkyoo Park 240 and by treating a and b as finite but sufficiently large in magnitude so as to cover several times the filter bandwidths [13], the inner integral in equation (6) becomes effectively the right hand side of a Fredholm integral equation of second kind which can be represented formally as [18] ∫ = b a m m m df f f f K f ) ( ) , ' ( ) ' ( φ λ φ (7) with φ m (f) and λ m being its eigenfunction and eigenvalue respectively and K(f’,f) acting as the Hermitian kernel. Based on the fact that the eigenfunctions form a complete set of orthonormal basis functions in the interval a<f<b, ) 2 exp( ) ( k ft j f E π can be expressed as a series expansion of the basis functions as ∑ + = m m m m k f N S ft j f E ) ( ) ( ) 2 exp( ) ( φ π (8) In equation (8) above the coefficients S m correspond to the desired signal and N m to the ASE noise. It can be shown that N m will have both its real and imaginary parts as Gaussian random variables with zero mean and variance N o /2 W/Hz each [13][14]. Substituting equation (8) into (6) and making use of the definition of the Fredholm integral equation as given in equation (7), it can readily be shown that ∑ + = m m m m k N S y λ 2 2 1 (9) Since the real and imaginary parts of N m are mutually uncorrelated and Gaussian distributed [13] [14], the CF of y k will be products of CF of non central chi square distributions each with single degree of freedom. This CF of y k will be as given below ∏ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − = Φ m o m m m o N j S j N j 2 2 exp 2 1 1 ) ( 2 ξ λ ξ λ ξ ξ (10) The probability density function (pdf) of y k can be obtained by inverting the CF. From the pdf so obtained, the probability of error can be evaluated. Since the CF given above cannot be inverted in a straightforward manner, saddle point integration method is used [14]. The alteration of the above discussed KLSE method so as to make it suitable for evaluating the error rate performance of oDPSK and its multilevel versions is rather straightforward in that all it needs to be done is to incorporate appropriately the transfer Advances in Optical Differential Phase Shift Keying… 241 functions of the delay line interferometers into the Hermitian kernel. Thus for oDPSK and its multilevel versions, each of the detected outputs (y k ) when expressed in the form of equation (6) will have its Hermitian kernel as ) ( ) ' ( 2 )) 2 ( exp( )) ' 2 ( exp( ) ' ( ) , ' ( * f H f H fT j T f j f f H f f K o o e ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + − + + − = θ π θ π (11) where θ is the phase shift introduced by the corresponding delay interferometer and T is as given earlier. Treating the signal s(t), i.e. the data sequence transmitted by the transmitter, as a periodic signal of sufficiently large period facilitates the usage of the FFT/IFFT algorithms which makes the computational process more efficient [14]. Also, by treating the data sequence as periodic and by representing it using appropriate pseudorandom sequence such as the de- Bruijn sequence [16], it is possible to estimate the impacts of fiber induced inter symbol interference caused due to chromatic dispersion and PMD. To incorporate these into the KLSE model, the fiber transfer function has to be included in the Hermitian kernel. The transfer function of the fiber which incorporates the chromatic dispersion is well known [14] and the principal states model of Poole [19] facilitates rather easy incorporation of the PMD effects into the fiber transfer function [16]. 4.2. Error Rate Performance Figure (6) shows the Bit Error Rate (BER) performance for oDPSK, oDQPSK and oDOPSK as a function of the Optical Signal to Noise Ratio (OSNR) in a back-to-back condition. The receiver schematics assumed in these evaluations are the ones discussed through figures (2) and (3) for oDPSK and oDQPSK respectively and the one discussed through figure (4.a) for oDOPSK. The optical filter was modeled as a first order Gaussian filter with its 3dB bandwidth as three times the baud rate and the electrical post detection filter was modeled as a fifth order Bessel filter with 3dB bandwidth as equal to the symbol rate. From here on, we refer to the 3dB bandwidth as just the bandwidth. The OSNR was evaluated with unpolarized ASE noise within a reference spectral width of 0.1nm. The bit rate was taken as 40Gbps for all three cases. The binary data sequence was modeled as pseudo random sequences of length 31, 63 and 511 respectively for oDPSK, oDQPSK and oDOPSK. These results as well as the others to be presented later in this chapter are for RZ pulse formats. We restrict ourselves to the RZ formats because in general as per the results in [11], [15] and [16], the RZ format gives a better error rate performance compared to the NRZ formats. Though the above results in back-to-back condition give a very good picture of the relative error rate performance of oDPSK, oDQPSK and oDOPSK schemes, it has to be noted that in actual long haul systems, signal distorting mechanisms like PMD will have to be dealt with. Electronic post detection equalization techniques such as Linear Equalizers (LE) [3] have been proposed as viable technologies to deal with PMD induced pulse broadening [20]. In the following we discuss the performance of the above discussed schemes in the presence of first order PMD with and without the use of LE. We begin with a very brief M. Sathish Kumar, Hosung Yoon and Namkyoo Park 242 review of LE schemes. More details could be found in standard digital communication text books such as [3]. Figure 6. Bit Error Rate performance for oDPSK, oDQPSK and oDOPSK as a function of the OSNR in back-to-back condition. Figure 7. A five tap Linear Equalizer. Advances in Optical Differential Phase Shift Keying… 243 The LE is essentially a transversal filter as shown in figure (7). The tap delay elements shown as square boxes in figure (7) introduce a delay of T s which can be equal to the signaling interval or a fraction of that. The tap weights c k with which each delay element output is multiplied is constantly updated using a tap updater algorithm called as the LMS algorithm. Details of the LMS algorithm are beyond the scope of this chapter and could be found in [3]. The sequence {v k } stands for the received distorted symbol sequence. The tap updater (the LMS algorithm) is in turn fed by an error signal e k as shown in figure (7). This error signal is derived from the difference between the equalized output k I ˆ and its nearest information symbol k I as estimated by the decision circuit. It should be easily understandable from this figure that the tap updater algorithm works towards minimizing the magnitude of this error signal which drives it. It may be noted that to estimate the kth information symbol, the LE discussed above has to depend not only on the kth received symbol v k but also on two before that and two after that. To incorporate the LE into the KLSE method of error rate evaluation so as to theoretically evaluate the performance improvement achievable with the use of LE in the presence of PMD induced pulse broadening, the LE is modeled as an electrical filter with transfer function [15] ∑ − = − = 2 2 ) 2 exp( ) ( k s k LE fkT j c f H π (12) where, it is assumed that the LE has five taps as in figure (7). Figure 8. OSNR penalty of equalized and unequalized oDQPSK and oDOPSK systems. M. Sathish Kumar, Hosung Yoon and Namkyoo Park 244 The improvement in performance in oDQPSK and oDOPSK systems by the use of a five tap LE in the presence of first order PMD effects is depicted in figure (8). The OSNR penalty to attain a reference BER of 10 -12 is plotted as a function of Differential Group Delay (DGD). The OSNR penalty for a particular DGD value was computed as the difference between the OSNR sensitivity in back-to-back condition (DGD=0ps) and the OSNR sensitivity at the DGD value of interest. The bit rate for all the considered systems was taken as 40Gbps and the tap delay T s of the LE was set as the symbol duration. The receiver optical and electrical filter bandwidths were optimized to provide maximum instantaneous DGD tolerances. It has been reported in [15] that the maximum instantaneous DGD tolerance was provided when the receiver optical and electrical filter bandwidths were optimized for a reference DGD of 25ps which turns out to be 1/2 and 1/3 times respectively of the symbol duration for oDQPSK and oDOPSK considered herein. What we mean by optimization of filter bandwidths for maximum DGD tolerance is as follows. When the filter bandwidths were optimized for maximum OSNR sensitivity at a DGD value of 25ps, the OSNR penalty of the system to instantaneous DGD values, especially around and beyond DGD values of 20ps, was lesser than what it was when the filter bandwidths were optimized for maximum OSNR sensitivity in a back-to-back condition [15]. The optimum filter bandwidths for a reference DGD of 25ps was found to be Bo = 3.5, Be = 0.8 times the baud rate for oDQPSK without equalizers and Bo = 3, Be = 0.6 times the baud rate for oDQPSK with LE where Bo and Be are the optical and electrical filter bandwidths respectively. For oDOPSK, Bo and Be were found to be 3.4 and 0.9 times the baud rate and 3.7 and 0.7 times the baud rate respectively for unequalized and equalized systems. Two important observations can be made from figure (8). The use of LE results in lower OSNR penalty for both the systems and the oDOPSK system has lower OSNR penalty compared to oDQPSK for a given bit rate. Reason for the latter is the fact that the oDOPSK symbols, at the same bit rate as supported by oDQPSK, are broader in time domain. Before moving on to the next section, we would like to mention that Decision Feedback Equalizers (DFE) could as well be used in place of the LE to electronically compensate for PMD induced pulse distortions. Results of OSNR penalty reduction in the presence of first order PMD by the usage of DFE is presented in [15]. A four tap DFE with two feed forward filter taps and one feed back filter tap was considered in [15]. The results in [15] show that improvements over LE occur only at relatively larger values of DGD that are above 50ps for the oDOPSK systems whereas for oDQPSK systems, there was practically no difference in the OSNR penalty between the LE and DFE systems. 5. Application of Coded Modulation Techniques As the demands on data transmission rate and distance increase, one of the prime candidates that can assist meeting these requirements is coded modulation technique. Coded modulation technique is the name given to such error control techniques which combine coding and modulation into a single operation unlike in conventional error control coding schemes wherein coding and modulation are treated as two different operations. One of the coded modulation techniques called as TCM has been successfully employed in electrical communication systems and will be discussed here from an application view point to oDQPSK systems. Advances in Optical Differential Phase Shift Keying… 245 (a) (c) (b) (d) Figure 9. The concept of TCM; (a) is a rate ½ convolutional encoder with one of the input bits driving it and the other connected directly to the output, (b) is the resultant trellis diagram, (c) is the partitioning of oDOPSK constellation and (d) is the trellis with the state transitions marked with the transmitted oDOPSK symbol selected based on the partitioning shown in (c). M. Sathish Kumar, Hosung Yoon and Namkyoo Park 246 One of the first reports of TCM applied to optical differential phase modulation systems was [21] wherein TCM was applied to oDPSK systems by incorporating a rate 1/2, constraint length 3 convolutional encoder into the modulation process. Later, in [22] the idea was extended to application of TCM to oDQPSK systems. TCM was essentially an invention of Ungerboeck [5] in the eighties and ever since has grown considerably and has found useful applications in electrical communication systems. For the benefit of readers who are not familiar with TCM techniques, we provide a brief overview. TCM is best explained with an example. Consider the setup depicted in figure (9.a). We have a rate 1/2 convolutional encoder [3] [5] whose input is driven by one of two parallel binary data streams. The other binary data stream appears as it is at the output of the system. At the final output of the system, there are three parallel binary data streams (two from the rate 1/2 convolutional encoder and one from the direct output). Thus, in a way, the combined setup can be viewed as a rate 2/3 convolutional encoder with its trellis structure as given in figure (9.b). During each signaling interval, depending on the combination of the three parallel binary data streams, one of 2 3 possible signal points from an octal signal set such as that of the oDOPSK is selected for transmission. The key to the success and effectiveness of TCM is the way the signal point to be transmitted is selected. A heuristic approach towards this, as explained in [5] [23], is to partition the constellation of the chosen modulation scheme progressively so that the minimum Euclidean distance after each level of partitioning is more than what it was before. The partitioning of the oDOPSK constellation based on this heuristic guideline is as shown in figure (9.c). The two outputs from the convolutional encoder elect that partitioned subset that has two signal points in it and the output that is directly connected to the input selects the final signal point to be transmitted. It should be obvious from the above discussion as to how the coding and modulation steps are interconnected and are inseparable in a TCM system. Figure (9.d) shows the trellis diagram of the TCM scheme discussed above with the transmitted signal points labeled on each transition. To complete this example on TCM, we explain briefly the demodulation/decoding operations at the receiver. Having received a noisy sequence of oDOPSK symbols of a particular length from a TCM based transmitter, the receiver computes the Euclidean distance between the received symbol sequence and all other possible symbol sequences of length the same as that of the received sequence. Due to the usage of a convoutional encoder to select the transmitted symbols, only certain particular symbol sequences would only be valid transmitted sequences. For example in the present system, with the initial state of the convolutional encoder as ‘00’, a look at the trellis structure shown in figure (9.d) readily tells that the sequence a-g-b cannot be a valid transmitted sequence of length three. It is this fact that arms the TCM system with its error correction capability. The sequence that is nearest in terms of Euclidean distance with the received noisy sequence is estimated as the actual transmitted sequence. This method of calculating the Euclidean distance and taking decisions based on the Euclidean distance is derived from the principle of maximum likelihood sequence estimation and from the assumption that the noise corrupting the transmitted symbols are statistically independent and Gaussian in distribution [3]. However, in the case of non Gaussian noise as well, if the noise corrupting the symbols is statistically independent and the statistical distribution of the noise is nearly Gaussian, Euclidean distance based estimation is justified. The maximum likelihood sequence estimation and the resultant Euclidean distance based estimation can be carried out efficiently using the Viterbi algorithm [3] [5]. An error in the Advances in Optical Differential Phase Shift Keying… 247 decoding stage occurs when the Viterbi algorithm selects a wrong path in the trellis. This can happen if the Euclidean distance between the received sequence and a valid sequence other than the actual transmitted sequence is less than the Euclidean distance between the received sequence and the actual transmitted sequence. Due to the manner in which the oDOPSK symbol sequence to be transmitted is selected, namely the partitioning of the constellation and the involvement of a convolutional code, the minimum Euclidean distance between any two given sequences will be equal to the maximum Euclidean distance (between diametrically opposite symbols) in the oDOPSK constellation [5]. Now since the dominant error event in the decoding stages of conventional error control coding scheme is free distance or minimum Hamming distance respectively for convolutional codes and block codes, it is understood that the dominant error event in the decoding of TCM will be the minimum Euclidean distance error which, in the present example, will be the event of erroneous estimation between diametrically opposite symbols in the oDOPSK constellation [3] [5]. Under high OSNR scenarios such as the ones often encountered in present day systems due to the usage of optical amplification technologies, the probability of such an error event will thus very well approximate and provide a lower bound on the Symbol Error Rate (SER) of TCM encoded systems. It is worth to note that though the basic modulation scheme involved in this example is oDOPSK, from the data transmission viewpoint the system is an oDQPSK system since there are only two information bits carried per signaling interval. In further discussions in this chapter, we refer to the above discussed TCM system as oDQPSK-TCM system. An important point that needs to be mentioned here with regards to the receiver for oDQPSK-TCM is that out of the two possible receiver schematics discussed in section 3 for oDOPSK, due to the fact that the multiple outputs from the receiver are not statistically uncorrelated and hence statistically dependent, they are not the ideal ones for TCM demodulation [24]. Due to this reason, we use a receiver scheme as was depicted in figure (2) with θ 1 = π/4 and θ 2 = 3π/4. The procedure for SER evaluation with such a receiver schematic under the assumption of the minimum Euclidean distance error event dominating significantly the error events in the decoding process is reported in [22] and references therein and is quite straightforward. Figure (10) depicts SER performance of oDPSK, oDQPSK and oDQPSK-TCM system as a function of OSNR. The OSNR was calculated as earlier with unpolarized ASE noise within a reference spectral width of 0.1nm. The optical and electrical filter parameters have been retained the same as those for the results depicted in figure (6) of the previous section. However, in these results, the signaling rate is taken as 20Gbaud for all the three cases whereby the oDPSK system supports a bit rate of 20Gbps and the oDQPSK and oDQPSK- TCM systems support a bit rate of 40Gbps. Also, a symbol error is said to have occurred in oDQPSK if any one or both the bits that constitute a symbol is received erroneously As can be seen, the application of TCM improves the performance of an uncoded oDQPSK system to that of the binary oDPSK system at identical OSNR values. More importantly, for a symbol rate of 20Gbaud, the oDQPSK-TCM supports an information bit rate of 40Gbps at the same performance as a binary oDPSK system that supports 20Gbps while consuming the same bandwidth. Thus the application of TCM effectively doubles the bandwidth efficiency without having to tradeoff SER. M. Sathish Kumar, Hosung Yoon and Namkyoo Park 248 Figure 10. SER performance of oDPSK, oDQPSK and oDQPSK-TCM system as a function of OSNR. Figure 11. OSNR penalty for increasing values of DGD for oDQPSK-TCM, uncoded oDQPSK without equalizer and uncoded oDQPSK which uses LE. Advances in Optical Differential Phase Shift Keying… 249 Figure (11) shows the OSNR penalty of uncoded oDQPSK with and without LE discussed in the previous section and that of the unequalized oDQPSK-TCM for increasing values of instantaneous Differential Group Delay (DGD). The results are referenced to the OSNR sensitivity of unequalized oDQPSK optimized in back-to-back condition for a SER of 10 -12 [15]. The equalized uncoded oDQPSK as well as the unequalized oDQPSK- TCM were optimized for a reference DGD of 25ps as was done in the last section with optical and electrical filter bandwidths as 3.5 and 0.8 times the baud rate respectively for unequalized oDQPSK and 3 and 0.8 times the baud rate respectively for oDQPSK-TCM. These results bring out the robustness of oDQPSK employing TCM in the presence of first order PMD. It can be observed that the coding gain due to TCM over uncoded, unequalized oDQPSK is retained throughout the range of DGD values considered in this figure. 6. Alternative Receiver Schematic for oDOPSK In this section, we discuss an alternative receiver schematic for oDOPSK which needs only two delay interferometers in contrast to the four required by the schematics depicted in figure (4). The receiver discussed in this section is very closely associated with an earlier one reported in [25]. The maximum likelihood detection principle [3] as applicable to oDOPSK can be readily explained with the help of figure (12) wherein we depict the receiver schematic discussed earlier through figure (2) with θ 1 = π/8 and θ 2 = 5π/8 along with the resultant oDOPSK constellation and signal space diagram. The term φ k in this diagram stands for the phase difference between the received oDOPSK symbols in the present and the previous intervals. This phase difference can take on discrete values ranging from 0 to 7π/4 in steps of π/4. Now having received y 1 and y 2 during a particular signaling interval, classical detection theory works on the principle of hypothesis testing as given below Figure 12. Receiver schematic with θ 1 = π/8 and θ 2 = 5π/8 and associated oDOPSK constellation and signal space diagram. M. Sathish Kumar, Hosung Yoon and Namkyoo Park 250 ) / ( ) / ( y k P s k y s P > < (13) where P(a/b) is the conditional probability of event a given event b and s , k , and y are respectively the vectors [s 1 s 2 ], [k 1 k 2 ] and [y 1 y 2 ] with the first entry within the parenthesis standing for the inphase component and the second entry for the quadrature component. The vectors s and k are two different signal points in the signal space diagram. It may be noted that what the above equation does in principle is it compares the probability of the transmitted symbol being s or k given that the vector y was received. This hypothesis test is conducted over all symbols and the transmitted symbol is estimated as the one that has the highest probability of being transmitted given that the vector y is received. Now if the noise that corrupts the transmitted signals is AWGN, and the two components of the received vector y namely y 1 and y 2 are statistically independent (which they will be if they are uncorrelated and have Gaussian distribution), the hypothesis test given in the last equation can be rewritten as 2 2 2 2 1 1 2 2 2 2 1 1 ) ( ) ( ) ( ) ( y k y k s k y s y s − + − < > − + − (14) which is in fact a Euclidean distance comparison in the signal space between the received vector y and the two signal points s and k . Since all the eight signal points in oDOPSK constellation are equidistant from the origin, equation (14) can be rewritten as 0 ) ( ) ( 2 2 2 1 1 1 s k k s y k s y > < − + − (15) The numerical sign of y 1 and y 2 can be readily made use of to identify the quadrant in which the transmitted symbol is most likely to be. Thus, while assigning binary triplets to the eight different symbols of the oDOPSK constellation, if two of the bits in these triplets are made to tally with the quadrant in the signal space diagram where the symbol lies, those two bits could be readily detected by detecting the numerical sign or in other words a bi-level detection of y 1 and y 2 . As per the signal space diagram and constellation shown in figure (12), starting from the point φ k =7π/4 in the first quadrant, the triplets could be assigned in an anticlockwise manner in the order (111), (110), (010), (011), (001), (000), (100) and (101). It Advances in Optical Differential Phase Shift Keying… 251 may be noted that this assignment is in agreement with gray encoding rules according to which closest points in the signal constellation have to differ by only one bit. To identify the third bit which separates the two signal points within a quadrant, we revert back to equation (15) and analyze as follows. Within the same quadrant, (s 1 -k 1 ) and (s 2 - k 2 ) would be of same magnitude and the only difference if any would be in the numerical signs. Thus, while conducting the test as given in equation (15) above to estimate the most probable transmitted symbol among two symbols from the same quadrant, what matters is not the magnitude of the differences (s 1 -k 1 ) and (s 2 -k 2 ) but their numerical signs. Thus equation (15) can be rewritten for symbols within the same quadrant as 0 2 2 1 1 s k y y > < Γ + Γ (16) with Γ 1 and Γ 2 being the numerical signs of (s 1 -k 1 ) and (s 2 -k 2 ) respectively. With reference to the signal space diagram shown in figure (12) and the concepts presented above, the following with regards to estimating the third bit can readily be arrived at Ist quadrant Γ 1 = +, Γ 2 = -, if s = 7π/4 and k = 3π/2; therefore, 0 1 0 2 1 > < − y y and 2 1 y y + = + IInd quadrant Γ 1 = -, Γ 2 = -, if s =π and k = 5π/4; therefore, 0 0 1 2 1 > < + y y and 2 1 y y − = - IIIrd quadrant Γ 1 = -, Γ 2 = +, if s = 3π/4 and k = π/2; therefore, 0 0 1 2 1 > < − y y and 2 1 y y + = - IVth quadrant Γ 1 = +, Γ 2 = +, if s = 0 and k = π/4; therefore, 0 1 0 2 1 > < + y y and 2 1 y y − = + M. Sathish Kumar, Hosung Yoon and Namkyoo Park 252 From the above we note that either (y 1 -y 2 ) or (y 1 +y 2 ) is the deciding factor and when one is the deciding factor the other has a constant numerical sign. From this observation, the four different decision rules given above can be combined into a single decision rule for estimating the third bit as given below. 0 1 0 ) )( ( 2 1 2 1 > < + − y y y y (17) The above equation suggests that a binary decision on the sum and difference of y 1 and y 2 followed by an XNOR operation on those decisions can readily provide an estimate of the third bit. In fact, this is the receiver schematic suggested in [25] with a minor variation in that the XNOR is replaced by the XOR apparently due to the swap in positions of 0s and 1s in the third bit of the triplet as compared to what it is herein. The receiver schematic depicted in figure (4.b) also works as per the same principle as discussed above. The two inputs to the XOR are effectively (y 1 -y 2 ) and (y 1 +y 2 ) [25]. Also, with an appropriate precoding of the binary data as given in [10], it is possible to directly obtain the three constituent data bits from the detected binary levels of y 1 , y 2 and the product (y 1 -y 2 )(y 1 +y 2 ). Further, if equation (17) is rewritten as 0 1 0 ) ( 2 2 2 1 > < − y y (18) it becomes obvious that the decisions can be taken depending solely on the difference of the absolute values of y 1 and y 2 . The conversion of the detected samples to their absolute values can be achieved in effect by considering the fact that the detected analog samples y 1 and y 2 are in fact dealt with in the receiver electronics in the digital domain through an analog to digital converter. More the resolution of the analog to digital converter better will be the resultant digital representation of the detected analog voltage. This is the methodology used in almost all the electrical soft decision decoding receivers [3]. In an analog to digital converter, it is possible in principle to identify the numerical sign of the digitally converted sample and as such it is possible to alter that numerical sign. Thus, if the detected sample y 1 or y 2 is negative, its numerical sign can be altered and the following decision rule can be applied to detect the third bit. 0 1 0 ) ( 2 1 > < − y y (19) Advances in Optical Differential Phase Shift Keying… 253 The advantage of this receiving scheme is that the dependence of the decision making variable on the sum as well as difference of y 1 and y 2 is removed and is now dependent only on the difference between y 1 and y 2 . This is of course at the cost of an additional electronic operation of changing the numerical sign of the detected samples. It may also be noted that a mere change in numerical sign does not alter the pdf of the detected samples. The complete schematic representation of this receiver is as given in figure (13). Figure 13. Schematic representation of an oDOPSK receiver which employs only two delay interferometers. The BER or probability of error for this receiver schematic can be readily arrived at as BER = ( P(y 1 >0/ b 1 = 0)+P(y 1 <0/b 1 = 1)+ P(y 2 >0/b 2 =0)+P(y 2 <0/b 2 =1)+P((y 1 -y 2 )>0/b 3 =0)+ P((y 1 -y 2 )<0/b 3 =1))/6 (20) where b 1 , b 2 and b 3 stand for the constituents of the binary triplets assigned to the eight symbols of the oDOPSK constellation and are assumed to take on logic levels 1s and 0s with equal probability. It has to be specially taken note of the fact that y 1 -y 2 as it appears in the above equation is after the numerical signs of both y 1 and y 2 are converted to positive. The above BER can be computed as discussed earlier in section (4) using the KLSE method. However, while computing the last two probabilities in equation (20) that involves the difference of y 1 and y 2 , the CF of y 1 -y 2 is required. This can be obtained by replacing the Hermitian kernel in the procedure outlined in section (4) by the difference of the Hermitian M. Sathish Kumar, Hosung Yoon and Namkyoo Park 254 kernels of the two arms of the delay interferometer after appropriately accounting for the power splitting at the front end of the receiver after the optical filter. (a) (b) Figure 14. BER performance for the oDOPSK receiver structure depicted in figure (13). Figure (14.a) and (14.b) shows the BER as a function of OSNR. Figure (14.b) zooms on to a portion of the BER curve. These figures show the individual BER for b 1 , b 2 and b 3 as well as the average BER. The optical and electrical filter bandwidths while evaluating these results were taken as 3 times and 1 time the baud rate respectively. The bit rate was set as 40Gbps. It can be noted that while the BER curves for b 1 and b 2 overlap as they should due to identical variances and same average means for y 1 and y 2 , the BER of b 3 is slightly more than that of b 1 and b 2 . This is obviously due to the fact that the variance of y 1 -y 2 is twice the variance of y 1 and y 2 . Before we conclude this chapter, it needs to be mentioned that we have not included a performance evaluation of the oDOPSK receiver schematic shown in figure (4.b). This is due to the fact that the BER evaluation of that receiver schematic using the accurate KLSE method becomes a cumbersome problem due to the fact that the XOR output is a function of both y 1 +y 2 as well as y 1 -y 2 . This renders the BER evaluation an exercise of double integration of the joint pdf of y 1 +y 2 and y 1 -y 2 . The BER performance of that receiver using other techniques of evaluation has appeared in [10] and its inclusion here is not suitable for a fair comparison with the results reported in this chapter since all the results in this chapter have been evaluated using the KLSE method. 7. Conclusion This chapter has presented a detailed review of optical differential phase modulation schemes touching upon oDPSK, oDQPSK and, oDOPSK. Various proposed receiver schematics for these modulation schemes were presented along with their error rate performance. An introduction to the KLSE method for error rate evaluation of the discussed systems was also Advances in Optical Differential Phase Shift Keying… 255 presented. Use of electronic equalization methods in the form of LE was discussed and the resultant improvement in OSNR penalty to achieve a target BER of 10 -12 was discussed. Coded modulation techniques in the form of TCM which can improve error rate performance of optical differential phase modulation schemes were introduced and a comparison was made between uncoded oDQPSK system and oDQPSK systems that use TCM. An alternative receiver schematic for oDOPSK which needs only two delay interferometers was presented along with its error rate performance. While oDOPSK, oDQPSK and, oDOPSK with electronic equalizers and/or TCM schemes hold a lot of promise for realizing future long haul optical transmission systems, the main stumbling block that can arise towards the successful application of post detection electronic processing based performance enhancers such as equalizers and TCM is the speed of the electronic hardware. In this regards, some developments in electronic hardware as reported in [26] and [27] holds some promise. References [1] Sano, A.; Masuda, H.; Kisaka, Y.;Asawa, S.;Yoshida, E.;Miyamoto, Y.;Koga, M.; Hagimoto, K.;Yamada,; Furuta, T.;Fukuyama, H. ECOC 2006. Cannes, (Post Deadline Paper) [2] Gnauck, A. H.; Winzer, P. J. J. Lightwave Tech. 2005, Vol. 23, 115-129 [3] Proakis, J. Digital Communication. McGraw Hill, 2001 [4] Betti, S.; Marchis, G. D.; Iannone, E. Coherent Optical Communication Systems. Wiley, New York, 1995. [5] Ungerboeck, G. IEEE Commun. Magazine. 1987, Vol.25, 5-11 [6] Winzer, P.J.; Dorrer, C.; Essiambre, R. J.; Kang, I. IEEE Photon.Technol.Lett. 2004, Vol 16, 1379-1381 [7] Penninckx, D.; Bissessur, H.; Brindle, P.; Gohin, E.; Bhakhti, F. Proc. ECOC 2001. Amsterdam, The Netherlands, 2001, 456-457 [8] Wei, X.; Liu, X.; Chandrashekar, S.; Gnauck, A. H.; Raybon, G.; Leuthold, J.; Winzer, P. J. Proc ECOC 2002. Copenhagen, Denmark, 2002, paper 9.6.3. [9] Griffin, R.A.; Carter, A.C. OFC 2002. 367-368. [10] Kim, C.; Li, G. Optics Express. 2004, Vol.12, 3415-3421. [11] Yoon, H.; Lee, D.; Park, N. Optics Express. Vol.13, 371-376 [12] Yoon, H.; Lee, D.; Park, N. Ninth Optoelectronics and Communications Conference and Third Conference on Optical Internet (IEEE Communication Society). 2004. paper 14C3-4. [13] Lee, J. S.; Shim, C.S. J. Lightwave. Tech. 1994, Vol.12, 1224-1228. [14] Forestieri, E. J. Lightwave Tech. 2000, Vol. 18, 1493-1503 [15] Yoon, H.; Kim, N.Y.; Park, N. IEEE Photons. Tech. Letters. 2005, Vol. 17, 2577-2579 [16] Wang, J.; Kahn, J.M. J. Lightwave Tech. 2004, Vol. 22, 362-371 [17] Marcuse, D. J. Lightwave Tech. 1990, Vol. 8, 1816-1823 [18] Cohen, H. Mathematics for Scientists and Engineers, Prentice Hall, 1992. [19] Poole, C.D.; Wagner, R.E. Electron. Lett. 1986, Vol. 22, 1029-1030 [20] Haunstein, H. F.; Wolfgang, S. G.; Dittrich, A.; Sticht, K.; Urbansky, R. J. Lightwave. Tech. 2004, Vol. 22, 1169-1182 M. Sathish Kumar, Hosung Yoon and Namkyoo Park 256 [21] Buelow, H.; Thielecke, G.; Buchali, F. Optical Fiber Communication (OFC) , Los Angeles, CA 2004, Paper WM5 [22] Kumar, M. S.; Yoon, H.; Park, N. IEEE Photons. Tech. Letters. 2007, Vol. 19, 1245-1247 [23] Ungerboeck, G. IEEE Commun. Magazine. 1987, Vol.25, 12-21 [24] Wang, Jin-Der.; Chung, H. Y. IEEE Trans. Commun. 1990, Vol. 38, 1549-1556 [25] Han, Y.; Kim, C.; Li, G. Electron. Lett, 2004, Vol. 40, 1372-1373 [26] Mizuochi, T.; Ouchi, K.; Kobayashi, T.; Miyata, Y.; Kuno, K.; Tagami, H.; Kubo, K.; Yoshida, H.; Akita, M.; Motoshima, K. OFC 2003, PD 21, Atlanta, 2003 [27] Dawid, H.; Fettweis, G.; Meyr, H. IEEE Trans. VLSI Systems. 1996, Vol. 4, 17-31 In: Optical Fibers Research Advances ISBN: 1-60021-866-0 Editor: Jurgen C. Schlesinger, pp. 257-277 © 2007 Nova Science Publishers, Inc. Chapter 9 A NEW GENERATION OF POLYMER OPTICAL FIBERS Rong-Jin Yu and Xiang-Jun Chen School of Information Science and Engineering, Yanshan University, Qinhuangdao 066004, China Abstract This chapter describes the background to the development of Polymer Optical fibers (POFs), discusses the optical and temperature resistant properties of polymers while emphasizing the intrinsic high attenuation of them. The first generation of POFs which consists of a solid-core surrounded by cladding and transmits light by total internal reflection, is puzzled by the difficulty of high attenuation. Then, the method of using a specific structure (i.e. hollow-core Bragg fiber) to solve the problem is presented. A new generation of POFs based on the hollow-core Bragg fibers with cobweb-structured cladding can guide light with low transmission loss and high bandwidth in the wavelength range of visible to terahertz (THz ) radiation. Efficient hollow-core guiding for delivery of power laser radiation and solar radiation can be achieved by replacing the traditional polymethylmethacrylate (PMMA) with heat-resistant polymers. Lastly, this chapter concludes with a discussion of applications in diverse areas. 1. Introduction The optical fiber is a circular dielectric waveguide that guides optical information and energy. The first theoretical study of wave propagation on circular cylindrical dielectric structures was made by Hondros and Debye [1]. It was not until 1966 that the proposed to use circular glass fibers as optical transmission lines was made [2]. Since then, the attenuation of more than / 1000dB km for silica fiber is reduced to about 0.2 / dB km by advanced purifying and manufacturing processes. The optical fiber is generally made from glasses, polymers or other materials. Compared to glass materials, polymeric materials have the shortcoming of intrinsic high attenuation while having the advantage of great flexibility. The elastic limits of polymeric materials are high. It is this high elasticity which allows the fabrication of tough and flexible POF with diameters of one millimeter and above. However, inorganic glasses with low elastic limits dictate that remarkable flexibility may only be observed in the optical Rong-Jin Yu and Xiang-Jun Chen 258 fibers of very small diameter, typically 125 m μ . The inherent brittleness of glass requires that a more elastic polymer coating be applied to the very fragile core-cladding structure to protect its surface and prevent the growth of Griffiths cracks and consequent fracture. For a long time in the past, the optical fibers consist of a solid-core surrounded by cladding, and transmit light by total internal reflection, with the addition of a few hollow waveguides with inner metallic or dielectric coatings to provide high reflectivity. The transmission losses of a solid-core fiber, including solid-core photonic crystal fiber, are larger than (at least equal to) the losses of the fiber-core material itself. Thus, first generation POFs which consist of a solid-core surrounded by cladding are facing the difficulty of high attenuation. One feasible way of solving the problem is to use a hollow-core fiber instead of the solid- core fiber. But the hollow-core fibers do not get any special attention and exploitation due to the very large success of low-loss high-index core (solid-core) silica fibers by total internal reflection and their extensive use in various fibers for transmitting light. Until a few years ago, the conventional wisdom in most books related to the fiber had been that confined and lossless propagation in fibers is accomplished by total reflection from the dielectric interface between the core and the cladding, or must make use of the concept of total internal reflection to save light inside the core of the optical fiber. In fact, very early in the developmental history of optical fibers, the idea of using Bragg reflection in a cylindrical fiber to obtain lossless confined propagation in a core with a refractive index lower than that of the cladding medium was proposed in the year 1978 [3]. Up to the late 1990s, the hollow-core Bragg fibers [4] and hollow-core photonic band gap fibers [5] were demonstrated. Both experimental and theoretical investigations confirm that transmission losses by using hollow- core Bragg fibers are dramatically lower than the losses of its constituent materials. It seems that hollow-core Bragg fibers are a suitable structure that can very effectively confine the transverse leakage of guided wave. Once adopting the hollow-core Bragg structures, surmounting the shortcoming of high attenuation and taking the advantage of great flexibility of POFs, a new generation of POFs can guide light with low transmission loss and high bandwidth in the wavelength range of visible to THz radiation. Efficient hollow-core guiding for delivery of power laser radiation and solar radiation can be achieved by replacing the traditional PMMA with heat-resistant polymers. We expect that a new generation of POFs will find many applications in diverse areas, and is irreplaceable for some applications. 2. Development History of POFs The first report of POF was in 1960’s, which was almost the same as the invention of silica- base optical fiber. In 1966, Du Pont invented the first POF named “Crofon” that was of step- index (SI) type composed of PMMA core surrounded by a partially-fluorinated polymer cladding. In 1975, Mitsubishi Rayon commercialized the first SI POF whose trade name was “Eska” and Asahi Chemical and Toray soon followed in 1970’s. The POF market was originally dominated by these three major companies who have been manufacturing SI type POFs composed of PMMA core [6]. In the more than forty years history of POFs, efforts were made to improve the performance of attenuation, bandwidth and temperature resistance of POFs. In the mean time, the work of Kaino (NTT’s Ibaraki Laboratories), Koike (Keio A New Generation of Polymer Optical Fibers 259 University) and their co-workers is very important in the advancement of technology of POFs in the area of the reduction of attenuation and the realization of high bandwidth. PMMA, which has been the most typical core material for POF, has large attenuation due to the intrinsic absorption loss of carbon-hydrogen stretching vibration. Theoretical attenuation limit of PMMA based POF is around 100 / dB km at 0.65 m μ wavelength, while the attenuation abruptly increases from near infrared to infrared region. The intrinsic absorption loss can dramatically decreased by using a polymer which has no carbon-hydrogen bonding in it. The work of Kaino and co-workers has brought about the reduction of the losses of PMMA-D8-core fibers to 20 / dB km at 680nm [7]. By substituting all hydrogen bonding in polymer molecules with fluorine, remarkable low attenuation of 10 / dB m was achieved by the perfluorinated (PF) polymer based graded-index (GI) POF. The first PF polymer based GI-POF was reported in 1994. In 2000, PF polymer based GI-POF named “Lucina” was commercialized from Asahi Glass Co., using a PF polymer named CYTOP [8]. However, the fibers reduce attenuation at the expense of increased cost. The deployment of both deuterated and fluorinated POFs is limited by the high cost of the fiber materials. Considering the transmission bandwidth of POFs, commercially available POFs have been of the SI type with a numerical aperture ( NA) of about 0.5, whose bandwidth of transmission is limited to about 5MHz Km ⋅ . For a SI-POF, quite a degree of improvement in its bandwidth can be achieved by reducing NA of the POF. For the low NA SI-POF ( NA= 0.31), its bandwidth is 160MHz at 100m ( 200MHz > at 50m ), which is currently commercialized. It meets the requirement of standardization of 156 / Mb s , transmission 50m approved by the ATM forum in May 1997. It is a common knowledge that the main limitation on the bandwidth of multimode optical fibers is modal dispersion, which means that different optical modes propagate at different velocities and the dispersion grows linearly with length. One way to overcome the modal dispersion is to use single mode (SM) POF. The first SMPOF was reported in 1991, which was successfully prepared by the interfacial-gel polymerization technique [9]. In the fiber, the core diameter was 3 15 m μ − and the attenuation of the transmission was about 200 / dB Km at 652nm wavelength. Another way to solve the problem for POFs with large cores is to use multi-layer step-index (ML-SI) POF [10], multi-core step-index (MC-SI) POF [11], or GI-POF [12]. In ML-SI POFs, the core region is composed of several layers with different refractive index. This concentric multi- layer structure decreases modal dispersion compared to conventional SI type POF and a data rate as high as 500 / Mb s for 50m transmission is achieved experimentally. MC-SI POF has a low numerical aperture (0.25) and a core region composed of 19 cores of small-core. By reducing the core diameter, not only modal dispersion but also bending loss is decreased. A data transmission at 500 / Mb s for 50m is also achieved by the MC-SI POF. For GI POF, the refractive index of the fiber core is graded parabola-like from a high index at the fiber core center to a low index in the outer core region. For the GI POF produced by the interfacial-gel polymerization method, its bandwidth measured is 3GHz for a fiber length of 100m. A low- loss PF polymer based GI POF has been developed and PF polymer based GI POF is able to transmit a data rate of 10 / Gb s or higher because of its material dispersion property [13]. For the temperature resistance of POFs, the high-temperature performance of a polymer is limited by its glass transition temperature ( g T ). For PMMA, g T is about 105℃. Maximum operating temperature for PMMA-core SI-POFs is 80℃. Ziemann et al. [14] had carried out a Rong-Jin Yu and Xiang-Jun Chen 260 accelerated aging test for the fibers from the three leading POF manufacturers. The results of the test show that for all fibers and wavelengths ( 650nm, 590nm, 525nm and 520nm) the estimated possible operating temperature for 20 years use is over 70℃. Some applications, such as in automobiles, aerospace environments and transmitted power demand performance at temperature in excess of 80℃. The g T of polycarbonate (PC) is around 170℃. The use of PC and partially fluorinated PC as core material enables temperatures of up to 115℃ and 145℃, respectively. The g T of polyethersulfone (PES) is about 225℃, maximum operating temperature of PES is 197℃. Polyimide material has even more high operating temperature (316℃). The attenuation of these high temperature resistant polymers is generally larger than that of PMMA, therefore making the polymers useless in fabricating the fibers, but using the polymers (such as polyimide) as the coating of high temperature resistant silica fiber. In a word, in the forty years development of POFs, there is no better position in both performance (especially attenuation) and cost comparing with silica glass fibers. Thus, first generation POFs have limited their penetration in important market-segments, and are only suited to ornament, illumination, sensors and short-distance data transmission applications. 3. Hollow-Core Fibers Hollow-core fibers reported to date in the literature can generally be classified into four types: (1) those in which the refractive index of the cladding is greater than that of the core, (2) those in which inner wall coating has high reflectivity, (3) hollow-core photonic bandgap fiber, and (4) hollow-core Bragg fiber. 3.1. Those in Which the Refractive Index of the Cladding Is Greater Than That of the Core As is known to all, waveguiding is achieved in conventional solid-core fibers due to the total internal reflection from the interface between the core with the refractive index core n and the cladding with the refractive index ( > ) clad core clad n n n . For the hollow-core fiber in which the refractive index of the core is lower than that of the cladding, the propagation of light is achieved by the regime of grazing incidence and is accompanied by radiation losses (leaky guide). In fact, this hollow-core fiber is a capillary tube, as shown in Fig.1. The coefficient of optical losses in the hollow fibers scales as 2 3 /a λ , where λ is the radiation wavelength and a is the core radius of the fiber. Thus, most of applications are performed by using the hollow-core fibers with large inner radii and short length. For example, a 10cm-long and 150 m μ -diameter hollow-core fiber filled with argon gas is used on extreme ultraviolet (EUV) light generated through the process of high-harmonic up-conversion of femtosecond laser [15]. A New Generation of Polymer Optical Fibers 261 Figure 1. Cross section of the hollow-core fiber with clad core n n > (leaky guide). 3.2. Those in Which Inner Wall Coating Has High Reflectivity Figure 2. Cross section of two hollow-core fibers with inner wall coatings. For hollow-core fiber in which inner wall coating has high reflectivity as shown in Fig.2, the fibers are metallic, glass or polymer tubes with inner metallic or (and) dielectric coatings provided with high reflectivity in the wavelength range of interest. Among them, the hollow- core fiber whose inner wall material has a refractive index less than one is referred to as an attenuated total reflectance (ATR) guide. In general, the 1 n < or ATR guides are made of sapphire, 2 GeO or some special 1 n < oxide glass. The idea of an 1 n < structure originated from Hidaka, et al. in 1981 [16]. To be useful for laser transmission, the ATR guides must have the region of anomalous dispersion, where n is less than 1, fall within some useful laser wavelength range. The first 1 n < guides studied by Hidaka, et al. focused on glass tubes made from lead and germanium doped silicates. By adding heavy ions to silica glass, he was able to shift the infrared edge to longer wavelengths so that the 1 n < region of anomalous dispersion occurred within the 2 CO laser wavelength band. Gregory and Harrington [17] pointed out that sapphire or 2 3 Al O has 1 n < from 10 to 16.7 m μ and, in addition, it has a very small k value of 0.05 at 10.6 m μ . This means that the theoretical loss is very low (less than 0.1 / dB m for a 1000 m μ -bore tube) for this material. But sapphire has a high modulus. Therefore, it cannot be bent to small diameters. These hollow-core fibers are an attractive alternative to solid-core infrared fibers. These fibers have losses as low as 0.1 / dB m at 10.6 m μ and may be bent to radii less than 5cm . For applications in high-power laser delivery, the fibers have been shown to be capable of transmitting up to 2.6kW of 2 CO laser power [18]. They also have usage in both temperature and chemical fiber sensor applications. Recently, hollow polycarbonate tubing with inner Cu coating is used on broadband THz hollow-core cladding(glass) hollow-core n<1 material structural tube hollow-core structural tube metallic film dielcetric film Rong-Jin Yu and Xiang-Jun Chen 262 transmission, and the lowest loss of 3.9 / dB m was obtained from a 3mm -bore fiber at 158.51 m μ [19]. 3.3. Hollow-Core Photonic Bandgap Fiber In 1999, the hollow-core photonic bandgap fiber have been experimentally demonstrated for the first time [5]. For hollow-core photonic bandgap fibers as shown in Fig.3, they are optical fibers with cladding made of fused silica or polymer incorporating arrays of air holes. The Figure 3. Cross section of hollow-core photonic bandgap fibers. core is formed by omitting several unit cells of material from the cladding. The “holey” cladding has a two-dimensional photonic bandgap that can confine light to the core for wavelengths around a minimum-loss wavelength. For example, the transmission losses of hollow-core plastic (PMMA) photonic bandgap fibers can be decreased by an order of magnitude with nine rings of air holes in comparison with conventional plastic fibers according to our numerical analysis by using multipole method [20]. Xu, et al. [21] analyzed the loss of an air-core silica glass photonic bandgap fiber and demonstrated that it is possible reduce the transmission loss to a level below / 0.01dB km , with eight rings of air holes at 1.53 m μ . Experimentally, the lowest loss reported in hollow-core photonic bandgap fibers with fused silica is 1.2 / dB km at 1620nm[22]. The ultimate limit to the attenuation of such fibers is determined by surface roughness due to frozen-in capillary waves. The attenuation of 1.2 / dB km at 1620nm already appears to be dominated by this mechanism. On the other hand, under consideration of fiber design, Roberts, et al. [22] pointed out that “the reduction in reported attenuation from 13 / dB km to 1.7 / dB km (and now 1.2 / dB km ) was partly from enlarging the core from 7 to 19 unit cells, reducing F by a factor of 3/ 2 (19 7) / 4.5 ≈ . F could be further reduced by enlarging the core to 37 cells for example. However, this would be accompanied by propagation of more higher-order core modes, increased bending loss and closer spectral packing of surface modes. Since these are already apparent in our 19-cell hollow-core photonic crystal fiber (HC-PCF), we expect 37-cell HC-PCFs to be of very limited practical value.” In view of this, it is very difficult for hollow-core photonic bandgap fibers to further reduce the attenuation and achieve single mode at the same time. The hollow- core fiber needs a cladding structure that can confine the transverse leakage of guided wave more effectively. Until now, it seems that hollow-core Bragg fiber is a suitable selection. A New Generation of Polymer Optical Fibers 263 3.4. Hollow-Core Bragg Fiber So far, there are three classes of hollow-core Bragg fibers: (a) “OmniGuide” fibers with very large cladding indices contrast [23], (b) ring-structured hollow-core fibers with a single material [24, 25], (c) cobweb-structured hollow-core fibers with a single material and a certain number of supporting strips [26, 27]. (a) OmniGuide fiber (b) Ring-structured fiber (c) Cobweb-structured fiber Figure 4. Cross section of hollow-core Bragg-fibers. In Bragg fibers, the hollow-core is surrounded by a 1-dimensional Bragg reflector consisting of alternating layers of high- and low-index materials. The cladding of “OmniGuide” hollow-core fiber consists of two (solid) materials with different indices as shown in Fig.4(a), that of ring-structured hollow-core fiber consists of a single material in which rings of holes are used to define the low-index layers as shown in Fig.4(b), and that of cobweb-structured hollow-core fiber consists of a single material in which air layers are used as the low-index layers as shown in Fig.4(c). These Bragg fibers made a breakthrough in fiber’s transmission losses less than the absorption losses of the material. The attenuation, which is lower than the material loss, has been observed in “OmniGuide” fibers and ring- structured fibers. The transmission losses of “OmniGuide” hollow-core fibers are dramatically lower than the losses of its constituent materials [28]: “Recent data show the losses of 0.65 / dB m for such fiber, when made of a material with losses of 30000 / dB m . Thus, its structure suppresses the losses of constituent materials by a factor of more than 45000.” But, it is difficult to find two materials they have larger indices contrast, similar thermal and mechanical properties, as well as compatible processing technique in realizing the structure. Therefore, only two material combinations have been demonstrated: Te ( n =4.6) in combination with a polymer ( n =1.59) [4], and 2 3 As Se ( n ~2.8) in combination with PES ( n ~1.55) [29]. For ring-structured hollow-core fibers, the attenuation, which is lower than the material (PMMA) loss, has been observed in the infrared ( 1120nm λ > ), specifically at 1390nm wavelength, at which the transmission loss is only 40 / dB m compared with the 420 / dB m material loss [30]. We proposed a modified cladding structure, i.e. a hollow-core Bragg fiber with cobweb- structured cladding [26]. The structure uses a single dielectric material and may solve the problem of structural support by using a certain number of supporting strips. The supporting strips are always symmetric in the cross-section and use the same dielectric material as alternating layers. Our research shows that the field profiles are slightly deformed due to the introduction of supporting structure. Although a small fraction of power is leaked out as a Rong-Jin Yu and Xiang-Jun Chen 264 result of the introduction of supporting structure, properly selected parameters of supporting structure will keep the loss at a low level, neglecting the presence of the supporting strips. The number and width of supporting strips should be as small as possible, generally, m (number of supporting strips) = 6~12 and s w (width of supporting strips) = 3 /3 ~ 0 / λ λ , where λ is the operating wavelength of the fibers. In comparison to “OmniGuide” fibers, the feasibility of cobweb-structured fibers is greatly improved. For ring-structured fibers, the refractive index of low-index layers in the cladding is between high-index (host material) and 1 (air). As a result, the cladding indices contrast of ring-structured fibers is smaller than that of cobweb-structured fibers. As far as the ability to confine the transverse leakage of guided wave is concerned, the ring-structured fibers are smaller compared to the cobweb-structured fibers. In order to compare the confinement losses of hollow-core ring-structured Bragg fiber with hollow-core cobweb- structured Bragg fiber, we make the design of analogous structure. Argyros et al. [25] have presented the design that supports a single-polarization, circularly symmetric nondegenerate mode in an air-core ring-structured Bragg fiber. The design presented has 0.403 i m μ Λ = , 0.578 e m μ Λ = , 0.355 d m μ = and core radius( o r ) =2.89 m μ , giving / 0.83 i d Λ = . The host material was assumed to be lossless with a refractive index of 1.49 (corresponding to PMMA material). When N (number of rings in cladding) = 9, the confinement losses of the 01 TE mode (lowest-loss mode) and 02 TE mode (second-lowest-loss mode) are about 0.83 / dB m and 57.14 / dB m, respectively. The ratio of the loss of the 02 TE mode to the loss of the 01 TE mode reaches approximately 70. In our design, the same parameters: 2 n (PMMA) = 1.49, co r (core radius) = 2.89 m μ , 2 d (thickness of high-index layers) = 0.243 m μ , 1 d (thickness of low-index layers) = 0.335 m μ and N (number of alternating layers in cladding) = 9, as well as 1 1 n = are used. The host material was also assumed to be lossless. The calculated results show that the least-loss wavelength of the 01 TE mode is located at 0.72 m μ . The confinement losses of the 01 TE mode and 02 TE mode at 0.72 m μ wavelength are 5 5.32 10 / dB m − × and 3 2.97 10 / dB m − × , respectively. The ratio of the loss of the 02 TE mode to the loss of the 01 TE mode reaches approximately 56. Thus it can be seen that the confinement loss of the 01 TE mode in the hollow-core cobweb-structured Bragg fiber is reduced by 15600 times in comparison to that of the air-core ring-structured Bragg fiber. These hollow-core Bragg fibers not only can reduce unwanted material properties, such as absorption, scattering, dispersion and nonlinearity to a large extent, but also can act as a modal filter [3]. Sterke et al. [31] found that such Bragg fibers can be guaranteed to be effectively single-moded. Johnson et al. [23] presented their work of “how the lowest-loss 01 TE mode can propagate in a single-mode fashion through even large-core fibers, with other modes eliminated asymptotically by their higher losses and poor coupling, analogous to hollow metallic microwave waveguides.” The single-mode operation of the Bragg fibers is achieved through asymptotic way during the transmission of guided waves, i.e. the number of modes in large-core Bragg fibers causes the change as follows, at the beginning, the transmission with multimode is followed by a few modes, and then the transmission becomes A New Generation of Polymer Optical Fibers 265 single moded at last. Thus the single mode is achieved in a certain length range of the fiber. Moreover, Bassett and Argyros [32] presented a method for calculating the single-mode length range: “The individual modes are characterized by two lengths, 1% L at which the transmitted power in that mode is reduced to 1%, and 0.01% 1% 2 L L = , at which the power is reduced to 0.01%. We characterize each fiber as a whole by two lengths, max 1% L L = for the best guided mode, and 0.01% sm L L = for the second best guided mode. We consider the usefully single moded for lengths between sm L and max L .” 4. Hollow-Core Bragg Fiber with Cobweb-Structured Claadding The refractive index profiles of hollow-core Bragg fiber with cobweb-structured cladding, together with those of ring-structured and “OmniGuide” hollow-core Bragg fibers are shown in Fig.5 for comparison. The parameters of the fiber with cobweb-structured cladding are co r (hollow-core radius), 1 n (=1, air), 2 n (high-index), 1 d (thickness of air layers), 2 d (thickness of high-index layers), 2 1 ( / ) d d η = , 1 2 ( ) d d Λ = + , N , m and s w , where N is the number of alternating layers in cladding, m and s w are the number and the width of the supporting strips, respectively. (a) cobweb-structured fiber (b) ring-structured fiber (c) “OmniGuide” fiber Figure 5. Profiles of refractive index for hollow-core Bragg fibers. In cylindrical waveguides, modes can be labeled by their ‘angular momentum’ integer m ; the ( , , ) z t ϕ dependence of the modes is given by ( ) j z t mf e β ω − + . In the hollow-core fiber with cobweb-structured cladding the modes will be affected by the supporting strip. Because the supporting strip is periodic in ϕ , the modes can be written as ( ) 2 / j z t m j n n e e β ω ϕ π ϕ φ − + ∑ , where n is integer, φ is the periodicity of supporting strip in ϕ direction. The effective wavevector / k m r ϕ = in the ϕ direction goes to zero for r → ∞ . So the bandgap of this structure is the same as “OmniGuide” Bragg fiber in Ref. [23] and purely depends on r k and β as long as s w is small enough. For designing hollow-core Bragg fiber with cobweb-structured cladding, some important structural parameters related to the permitted normalized frequency range of the 01 TE mode, and their varying rule were analyzed by using a plane wave expansion method [27]. The Rong-Jin Yu and Xiang-Jun Chen 266 lowest-loss mode in Bragg fiber is 01 TE mode. The simulated results for hollow-core Bragg fiber with cobweb-structured cladding show that leakage losses of 02 TE mode for the fibers with 0.05 η = , 2 0.25 d m μ = , 2 1.49 n = , 4 N = and different core radii ( 10 co r m μ = and 50 m μ ) at 0.65 m λ μ = are 6 5.4 10 × and 3 8.7 10 × times larger than those of 01 TE mode, respectively. Thus, the permitted frequency range of 01 TE mode is of especially interest. The most commonly used material in POF is PMMA, its refractive index is 1.49. Using PMMA as the high-index material of the Bragg reflection layers, the first two TE modes in the Bragg reflection layers are calculated with the plane wave expansion method [33]. Figure 6 shows the mode index of the first two TE modes in the Bragg reflection layers. 01 TE mode is the fundamental mode in the hollow core. Its mode index must be below 1 and approach to 1. The frequency range formed by two intersecting points ( P and Q) of the two TE mode curves and the air line ( 1 eff n = ) is approximately the permitted frequency range of 01 TE mode in hollow core. For 0.01 η = and 0.05, 1 n (air), 2 1.49 n = (PMMA), we can see from Fig.6 that this kind of structure can guide light in the hollow core over a wide frequency range. Different η have a strong effect on the permitted normalized frequency range of the 01 TE mode. For 0.01 η = , normalized frequency can achieve the range from 2.91 to 45.76, while for 0.05 η = , normalized frequency is within the range 1.34 to 9.5. The permitted normalized frequency range of 01 TE mode shrinks more than 5 times as η changes from 0.01 to 0.05. In order to figure out the influence of the structural parameter η of Bragg reflection layers on the permitted normalized frequency range of 01 TE mode, the permitted normalized frequency range of 01 TE mode with different η at a fixed 2 n (1.49) was calculated. The results are listed in Table 1. Figure 6. Permitted normalized frequency range of TE 01 mode for Bragg fiber with η =0.01, 0.05. The two curves indicate the first two TE modes in the Bragg reflection layers, and the solid line is the air line ( n =1) [27]. A New Generation of Polymer Optical Fibers 267 Table 1. The permitted normalized frequency range of 01 TE mode vs different η at a fixed 2 n (1.49) [27] η 0.5 0.4 0.3 0.2 0.1 0.08 0.06 0.05 0.04 0.02 0.01 Q point value 1.36 1.59 1.96 2.72 4.98 6.11 8.00 9.5 11.76 23.07 45.76 P point value 0.57 0.60 0.65 0.75 0.99 1.09 1.24 1.34 1.49 2.07 2.91 / λ Λ range ( ~ ) Q P 0.79 0.99 1.31 1.97 3.99 5.02 6.76 8.16 10.27 21.00 42.85 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 η n 2 =1.49 upper limit of d 2 ′ lower limit of d 2 ′ Figure 7. Range of normalized high index layer thickness ( 2 2 d d λ ′ = ) vs. η [27] Figure 8. Permitted normalized frequency range of TE 01 mode vs. 1 d [27] In regard to the range of allowed values of 2 d , we define 2 2 / d d λ ′ = as the normalized high-index-layer thickness, where λ is the operating wavelength of the fiber. The upper and lower limit of 2 d′ can be obtained by means of the Q and P point values for each η in Table 1. Take 0.05 η = as an example, the upper limit ( Q point value) of the permitted normalized frequency range of 01 TE mode is 9.5, which means 1 2 ( ) / 9.5 d d λ + = . Substituting 1 2 / 0.05 d d = into it, we can obtain 2 2 / 0.4524 d d λ ′ = = . The relationship between 2 d′ and η is shown in Fig.7. From Fig.7, we can see that the values of 2 d′ for the Rong-Jin Yu and Xiang-Jun Chen 268 upper line are approximately 0.45, indicating that the maximum thickness of 2 d cannot go beyond 0.45λ . In general, 2 d takes 0.4 ~ 0.3 λ λ . The minimum thickness of 2 d′ decreases when decreasing η . Figure 9. Permitted normalized frequency range of TE 01 mode as a function of 2 n for η = 0.05 [27]. In regard to the relationship between the 1 d and permitted normalized frequency range of 01 TE mode, the permitted normalized frequency range of 01 TE mode with 2 0.25 d m μ = and 2 1.49 n = at different 1 d is illustrated in Fig.8. One obvious feature of Fig.8 is that the permitted normalized frequency ranges of 01 TE mode and the corresponding thickness 1 d of air layer are approximately a linear relationship. Thus, so long as the thickness 1 d of air layer increases at a fixed 2 d , the normalized frequency range broadens. In regard to the relationship between 2 n and permitted normalized frequency range of 01 TE mode, a series of 2 n ranging from 1.45 to 5.8 at a fixed η (0.05) are calculated, as shown in Fig.9. The permitted normalized frequency range of 01 TE mode increases when 2 n decreases. Most of polymers have the refractive index smaller than 1.8. Therefore, they are advantageous as the materials of the fiber with a large transmission frequency range. In regard to the tolerance of the parameters, we take a dielectric material PMMA as an example. The design objective is a hollow-core fiber to use as optical fiber communication in the wavelength range from 0.65 m μ to 1.65 m μ . The design parameters are 0.05 η = , 2 0.25 d m μ = and 2 1.49 n = . Its normalized frequency / λ Λ is in the range from 5.25/0.65=8.1 to 5.25/1.65=3.2, all within the permitted normalized frequency range of 01 TE mode (9.5-1.34) as shown in Fig.6(b). If 2 d has a error of 2 20% d ± in the production process, this corresponds to 2 0.2 d m μ = and 0.3 m μ . For 2 0.2 d m μ = , the normalized frequency range is from 5.2/0.65=8 to 5.2/1.65=3.15. This is within the permitted normalized frequency range of 01 TE mode (11.76-1.49) as shown in Table 1 for 0.04 η = . For 2 0.3 d m μ = , the normalized frequency range is from 5.3/0.65=8.15 to 5.3/1.65=3.21. This is A New Generation of Polymer Optical Fibers 269 almost within the permitted normalized frequency range of 01 TE mode (8.00-1.24) as shown in Table 1 for 0.06 η = . If 1 d has a error of 1 25% d ± , this corresponds to 1 3.75 d m μ = and 6.25 m μ . For 1 3.75 d m μ = , the normalized frequency range is from 4/0.65=6.15 to 4/1.65=2.42. This is within the permitted normalized frequency range of 01 TE mode (7.21- 1.18) for 0.067 η = . For 1 6.25 d m μ = , the normalized frequency range is from 6.5/0.65=10 to 6.5/1.65=3.94. This is within the permitted normalized frequency range of 01 TE mode (11.76-1.49) for 0.04 η = . Finally, polymers are considered to have different refractive indices for the same material, due to different molecular weight or polymerization condition. If the index of PMMA has a variation of 2 0.02 n ± , which corresponds to 2 1.47 n = and 1.51, then the permitted normalized frequency range of 01 TE mode are (9.74-1.38) and (9.27-1.31), respectively. They are essentially consistent with the normalized frequency range (9.5-1.34) as originally designed for 2 1.49 n = and 0.05 η = . The confinement loss and transmission loss for hollow-core Bragg fiber with cobweb- structured cladding were modelled by using an asymptotic formalism [34]. Many results show that the fibers with only 3-4 alternating layers in cladding can achieve the low confinement loss and transmission loss, and the confinement and transmission losses decrease with increasing the hollow-core radius ( co r ). In order to achieve both low loss and wide wavelength range, fiber design should adopt smaller 2 d value and lager 1 d value, besides increasing co r and N . 5. Functional Exploiting of Hollow-Core Bragg Fiber with Cobweb - Structured Cladding With the appealing properties described above, the possibility of using hollow-core Bragg fiber with cobweb-structured cladding for transmitting the information and delivering the laser energy was analyzed. 5.1. Fibers for Use in Optical Communications from Visible to near Infrared Region Today, the capacity of optical fiber communications has expanded gigabits per second into terabits per second, enough to meet the current traffic demand due to the explosive growth of data transfer and internet services. Large-capacity and long-distance optical fiber communication trunk line has been installed in many countries. The next big step will be extending the network from fiber-to-the-curb into every building and home. In the area of fiber to the home (FTTH) or fiber to the premises (FTTP) application, passive optical networks (PON), especially ethernet passive optical networks (EPON) and gigabit ethernet passive optical networks (GEPON), are generally preferred for home fiber connections. Usually, the transmission bandwidth and transmission distance required for the Rong-Jin Yu and Xiang-Jun Chen 270 networks are 100MHz - 10GHz and 100 -10 m km, respectively. Therefore, the fibers with lower loss, higher bandwidth and cheaper cost are in demand. People have been trying to find materials and methods to meet those requirements and POF is one of the major approaches being explored in addition to silica glass single-mode fiber and multi-mode hard plastic clad fiber (HPCF). The simulated results for hollow-core Bragg fibers with cobweb-structured cladding had proved that depending on the modal-filtering effect, they may realize the transmission of 01 TE single-mode or a few modes, thus achieving the transmission of higher bandwidth ( GHz ) [35]. Figure 10. Absorption loss spectrum of PMMA [36]. A fiber design for use in optical communication from visible to near infrared region is presented. The fiber parameters are 2 1.49 n = (PMMA), 1 1 n = , 2 0.25 d m μ = , 1 5 d m μ = , 75 co r m μ = and 3 N = . According to absorption loss spectrum of PMMA [36] as shown in Fig.10, we calculate the transmission losses of the fiber. The absorption losses of PMMA at the wavelengths of 0.65 , 0.85 , 1.3 and 1.55 m μ are about 100 / dB km , 3 2.5 10 / dB km × , 4 2.5 10 / dB km × and 4 7.8 10 / dB km × , respectively. The transmission losses of 01 TE mode at these wavelengths are 4 3.9 10 / dB km − × , 3 4.3 10 / dB km − × , 0.13 / dB km and 0.80 / dB km , respectively. The results show that after inevitable factors (material purity, imperfection and nonuniformity of fiber structure and existence of supporting strips) being considered, the transmission losses of the fiber are still very low. Thus, by using an inexpensive material (PMMA), it allows the fibers to meet the needs of the transmission distance and bandwidth for EPON and GEPON, and to realize the wavelength division multiplexing (WDM). A New Generation of Polymer Optical Fibers 271 5.2. Fibers for Use in THz Waveguiding The THz radiation, whose frequency range is about 0.1 10THz − , has important applications in spectroscopy, imaging, space science and information transmission. To date, progress in THz wave generation and detection techniques has been enormous. However, most of the present THz systems rely on free space propagation due to the absence of low loss waveguides and transparent materials in the THz region. The waveguides constructed with some metals suitable for microwave guides or some dielectrics (such as silica) suitable to optical waveguiding have very high losses for THz wave. Even if for high-resistivity silicon, the most common material for the passive devices in the THz technology, its absorption coefficient is of the order of 1 0.04cm − . In recent years, THz waveguides have been fabricated from some dielectrics (such as sapphire, plastics) except from metals such as Cu, brass, and stainless steel. The loss coefficients of high-index core (solid-core) photonic crystal fibers using high-density polyethylene (HDPE) [37] and polytetrafluoroethylene (Teflon) [38] are less than 1 0.5cm − ( 0.1 3THz − ) and approximately 1 0.12cm − , respectively. Hollow polycarbonate waveguides with inner Cu coatings for broadband THz transmission have been reported [39]. The lowest loss 3.9 /m dB ( 1 0.00898cm − ) was obtained from a 3mm core diameter fiber at 158.51 m μ wavelength. Recently, a simple subwavelength-diameter ( 200 m μ ) plastic (polyethylene) wire, similar to an optical fiber for guiding a THz wave has been reported as well [40]. Its attenuation constant is reduced to less than 1 0.01cm − in the frequency range near 0.3THz . A fiber design for use in THz waveguiding is presented. The structural parameters of fibers (A, B, C) are as follows: 9 co r mm = , 2 1.52 n = , 1 1 n = , 2 25 d m μ = , 1 500 d m μ = and 3 N = (fiber A); 12 co r mm = , 2 1.52 n = , 1 1 n = , 2 70 d m μ = , 1 1050 d m μ = and 3 N = (fiber B); 16 co r mm = , 2 1.52 n = , 1 1 n = , 2 150 d m μ = , 1 2250 d m μ = and 3 N = (fiber C). The host material was assumed to be lossless with a refractive index of 1.52 (corresponding to HDPE material). The confinement loss as a function of wavelength for 01 TE , 02 TE , 01 TM and 02 TM modes is shown in Fig.11. The lowest-loss mode is 01 TE mode. The confinement loss of the 01 TE mode at the least-loss wavelength is 8 1.13 10 / dB km − × at 83.5 m μ (fiber A), 7 2.15 10 / dB km − × at 233 m μ (fiber B), and 7 5.05 10 / dB km − × at 500 m μ (fiber C). Figure 11. Confinement loss as a function of wavelength for 01 02 01 , , , TE TE TM and 02 TM modes. Rong-Jin Yu and Xiang-Jun Chen 272 Figure 12. Transmission loss as a function of wavelength for 01 02 01 , , , TE TE TM and 02 TM modes. Then, we attempt to take the calculation of losses further by including the material absorption. Based on the absorption spectra of HDPE in wavelength range 50 1200 m m μ μ − [41], the transmission losses of three hollow-core fibers (A, B, C) with cobweb cladding are calculated. The calculated results are shown in Fig.12. The data in Fig.12 show that the transmission losses of 01 TE mode for fiber A in the wavelength range of 2 6 0 5 0 m m μ μ − are below 5.5 / dB km. The lowest loss is 0.63 / dB km (corresponding to loss coefficient 6 1 1.45 10 cm − − × ) at 90 m μ . The transmission losses of 01 TE mode for fiber B in the wavelength range of 2 5 0 4 0 0 m m μ μ − are below 5.0 / dB km . The lowest loss is 2.0 / dB km at 270 m μ . The transmission losses of 01 TE mode for fiber C in the wavelength range of 1 4 0 20 00 m m μ μ − are below 5.6 / dB km. The lowest loss is 2.09 / dB km at 560 m μ . The above transmission losses were taken into account only the absorption spectra of the material (HDPE). In fact, certain spectral region in the THz waves may not be available for signal transmission due to the strong absorption of water present in the constituent materials and air-core for the polymer fibers [42]. Therefore, while using hollow-core polymer Bragg fiber with cobweb-structured cladding in transmitting light through air-core, it is very important to eliminate the water from the constituent material and avoid moist air in the environment during fabrication and storage. 5.3. Fibers for Infrared (IR) Applications IR optical fibers may be defined as fiber optics transmitting wavelengths greater than approximately 2 m μ . IR fibers can be useful for the medical, industrial, civil, and military arenas. For example, they are used in surgical applications by transmitting 2 CO laser radiation(10.6 ) m μ and : Er YAG laser radiation (2.94 ) m μ . When used as fiber sensors, IR fibers are generally used either to transmit blackbody radiation for temperature measurements or as an active or passive link for chemical sensing, achieving non-contact temperature monitoring and remote spectroscopic chemical sensing. The application in the industrial arena includes welding and cutting. Scanning near-field microscopy by using high-quality single- mode and multimode IR fiber-tapered tips can obtain 20-nm topographic resolution and about 200-nm optical resolution for a variety of samples. IR fibers are also used for military applications including anti-aircraft missile defense. The development of infrared fiber optics A New Generation of Polymer Optical Fibers 273 began in the year 1960. The first IR fibers were fabricated in the mid-1960’s using arsenic- sulphur glasses [43]. So far, there are four classes of infrared fibers: (i) fluoride, germanate, tellurite or chalcogenide glass based solid-core fibers; (ii) crystalline silver halide solid-core fibers; (iii) hollow-core fibers in which inner wall coatings have high reflectivity; and (iv) solid-core photonic crystal fibers and hollow-core photonic bandgap fibers. The optical-loss values of the sulfide based chalcogenide glass fibers at the Naval Research Laboratory have been reduced to only 0.1 to 0.2 / dB m in fiber lengths of about 500m by using improved chemical purification and better fiber fabrication techniques [44]. The optical losses of crystalline silver halide solid-core fibers by an extrusion process have been reduced to lower than 50 / dB km in a broad IR region from 9 to 14 m μ and lower than 1 /m dB in the region from 3 to 20 m μ [45]. The losses of rectangular hollow waveguides with 1- m -long and 1 1 mm mm × cross-section by first depositing thin-film coatings of 2 PbF on phosphor bronze strips and then soldering four of these phosphor bronze metal strips together are as low as 0.1 / dB m at 10.6 m μ [46]. Photonic crystal fibers for the middle infrared were fabricated by multiple extrusions of silver halide crystalline materials [47]. These fibers are composed of two solid materials: the core consists of pure AgBr (n=2.16) and the cladding includes AgCl (n=1.98) fiberoptic elements arranged in two concentric hexagonal rings around the core. IR transmissive As-S glass and As-Se glass triangular photonic band gap fiber structures were theoretically modeled [48]. From numerical simulations, Pottage et al. [49] discovered a new type of air-line bandgap that is of considerable importance in the design of practical hollow-core photonic bandgap fibers made from high-index glass (n≥2.0) for guidance in the mid/far-IR. A silica based hollow-core photonic bandgap fiber in which fiber-core diameter is 40 m μ (nineteen capillaries were omitted from the centre of the stack to form the core), the overall outside diameter is 150 m μ and the nearest-neighbor hole spacing is around 7 m μ , has been fabricated [50]. The peak of the bandgap is at 3.14 m μ with a typical attenuation of 2.6 / dB m . By further optimization of the structure, modeling suggests that a loss below 1 / dB m should be achievable. The design is a hollow-core Bragg fiber with cobweb-structured cladding for the mid-IR region. In the wavelength region between 100 m μ and 1 m μ , many longitudinal and rotational resonances of molecules are present in almost all substances, especially the long-chain polymers [2]. Polymers such as teflon and polyethylene show relatively strong absorption at 1 1000cm − (10 m μ ). The absorption coefficient α at 10 m μ wavelength is about 1 100cm − for teflon and about 1 50cm − for polyethylene [16]. A fiber design for use in infrared is presented. The structural parameters of fibers (A, B) are as follows: 1500 co r m μ = , 2 1.4 d m μ = , 1 30 d m μ = , 1 1 n = , 2 1.37 n = (teflon) and 3 N = (fiber A); 1200 co r m μ = , 2 2.8 d m μ = , 1 28 d m μ = , 1 1 n = , 2 1.55 n = (PES) and 3 N = (fiber B). The absorption coefficient of the host material (teflon) is 1 100cm − (corresponding to absorption loss 7 4.343 10 / dB km × ). The calculated results are shown in Fig.13(a). The data in Fig.13(a) show that the transmission Rong-Jin Yu and Xiang-Jun Chen 274 loss of 01 TE mode for fiber A in the wavelength range of 2.8 m μ to 10.6 m μ are below 39.5 / dB km . The lowest loss is 1.47 / dB km at 3.9 m μ . The absorption loss of the host material (PES) is 7 3 10 / dB km × . The calculated results are shown in Fig.13(b). The data in Fig. 13(b) show that the transmission loss of 01 TE mode for fiber B in the wavelength range of 8 m μ to 13 m μ are below 30.9 / dB km. The loss for the 10.6 m μ wavelength of 2 CO laser is 18.9 / dB km. Figure 13. Transmission losses of the mid-IR region for fibers (A, B). The numerical results show that despite the strong absorption of the polymers in the mid- IR region, the transmission losses of the fibers are lower by comparison with those of other IR fibers reported in the literature. And the polymer fibers have an advantage over other fibers in flexibility. 5.4. Circular-Polarization-Maintaining Single-Mode Fibers Standard single-mode fibers support two degenerate, orthogonally polarized modes ( 11 HE mode). Random imperfections in the fiber structure and external forces on the fibers can create asymmetries that break the polarization degeneracy, resulting in polarization mode dispersion and polarization fading in interferometers. Conventional polarization-maintaining fibers (highly birefringent fibers) and some single-polarization single-mode photonic crystal fibers supported a linear polarization mode. The fibers require accurate alignment of the birefringence axes of the two fibers when coupling, splicing and some sensing applications are considered. Therefore, in the year 1980, Jeunhomme and Monerie [51] have suggested the design of a circular-polarization-maintaining single-mode fiber cable . Recently, Argyros et al. [25] have presented the design that supports a single-polarization, circularly symmetric nondegenerate mode in an air-core ring-structured Bragg fiber. We presented the design that supports a circular-polarization-maintaining single mode in a hollow-core and cobweb-structured cladding Bragg fiber. The structural parameters of the fiber are 10 co r m μ = , 1 1 n = , 2 1.585 n = (PC), 2 0.21 d m μ = , 1 2.1 d m μ = and 3 N = . The intrinsic losses of the host material (PC) are 166 / dB km at 650 656nm − and 224 / dB km at 764nm[52]. The calculated results show that the transmission losses of 01 TE mode (lowest A New Generation of Polymer Optical Fibers 275 loss mode) are 0.226 / dB km at 650 656nm − and 0.170 / dB km at 764nm , those of 02 TE mode (second-lowest loss mode) are 3.513 / dB km at 650 656nm − and 1.848 / dB km at 764nm . The ratio of the loss of the 02 TE mode to the loss of the 01 TE mode is 15.54 ( 650 656nm − ) and 10.87 ( 764nm). In accordance with the research reported in Ref.32, the fiber is single moded for lengths between 11.4km and 88.5km( 650 656nm − ), and 21.7km and 117.6km ( 764nm). We expect that this type of hollow-core Bragg fibers with circular-polarization- maintaining single-mode and low-losses will find many applications, such as gyroscopes, current sensors and coherent communication systems. 6. Applications of Hollow-Core Bragg Fiber with Cobweb- Structured Cladding A new generation of POFs has the advantages of both low-cost and high-performance in terms of attenuation, bandwidth and flexibility. It will find many applications in diverse areas and increases market acceptance. As respects information transmissions, the new generation of POFs can guide the light of visible to terahertz radiation, and can be applied to optical fiber communications and optical fiber sensing, such as LANs, specially FTTH, THz wave fiber communications. It can also be used as an active or passive links for chemical sensing and remote spectroscopic chemical sensing, a variety of physical quantity sensing as well as medical diagnostics including noninvasive blood glucose monitoring and detection of tumors. As respects delivery of power laser radiation and solar radiation, hollow-core Bragg fibers with cobweb-structured cladding can deliver solar radiation into darkroom, be used for indoor illumination, replacing former guided light tube or solid-core polymer fiber. Efficient hollow-core guiding for delivery of power laser radiation ( 10.6 m μ 2 CO laser, 2.94 m μ : Er YAG laser, etc) can be achieved by replacing the traditional PMMA with heat-resistant polymers, and can be used for medical therapy and processing including micro-processing and material processing. By using gas-filled hollow-core Bragg fibers with cobweb-structured cladding, it is possible to obtain the EUV light generated through the process of high-harmonic up- conversion of femtosecond laser and ultrahigh efficiency laser wavelength conversion by pure stimulated rotational Raman scattering, as well as to use laser light to levitate and guide particles through the hollow-core fiber, etc. Circular-polarization-maintaining single-mode low-loss fibers and high-strength, flexibility and resistance to shock fibers will provide the possibilities for some new applications. These fibers will stimulate further progress, both in fiber and allied systems technologies. The new generation of POFs based on hollow-core Bragg fiber with cobweb-structured cladding will find many applications and is irreplaceable for some applications such as THz wave low-loss transmission. Rong-Jin Yu and Xiang-Jun Chen 276 References [1] D. Hondros and P. Debye. Ann. Physik. 1910, vol.32, 465-476 [2] K.C. Kao and G.A. Hockham. Proc. IEE 1966, vol.113(7), 1151-1158 [3] P. Yeh, A. Yariv, and E. Marom. J. Opt. Soc. Am. 1978, vol.68, 1196-1201 [4] Y. Fink, D.J. Ripin, S. Fan, C. Chen, J.D. Joannopoulos, and E.L. Thoms. J. Lightwave Technol. 1999, vol.17, 2039-2041 [5] R.F. Cregan, B.J. Mangan, J.C. Knight, T.A. Birks, P.st.J. Russell, P.J. Roberts, and D.C. Allan. Science 1999, vol.285, 1537-1539 [6] Y. Koike. Status of POF for “fiber to the display”. 15th Inter. Conf. on POF(POF 2006), 17-22, Seoul, Korea [7] T. Kaino, K. Jinguji, and S. Nara. Appl. Phys. Lett. 1983, vol.42, 567-569 [8] http://www.lucina.jp/eg_lucina/productseng f2.htm [9] Y. Koike and E. Nihei. Proc. 17th Euro. Conf. Opt. Comm.(ECOC’91), pp.41-44, Paris, France, 1991 [10] K. Nakamura, J. Okumura, K. Irie, and K. Shimada. Proc. Int’1 POF Conf.(POF’95), pp.114-116, Boston, USA, 1995 [11] H.Munekuni, S. Katsuta, and S. Teshima. Proc. Int’1 POF Conf.(POF’94), pp.148-151, Yokohama, Japan, 1994 [12] Y. Koike, T. Ishigure, and E. Nihei. J. Lightwave Technol. 1995, vol.13(7), 1475-1489 [13] T. Ishigure, Y. Koike, and J.W. Fleming. J. Lightwave Technol. 2000, vol.18(2),178-184 [14] O. Ziemann, W. Daum, A. Bräuer, J. Schlick, and W. Frank. Conf. Proc. POF 2000, 173-177, Cambridge, USA [15] R.A. Bartels, A. Paul, H. Green, H.C. Kapteyn, M.M. Murnane, S. Backus, I.P. Christov, Y. Liu, D. Attwood, and C. Jacobsen. Science 2002, vol.297, 376-378 [16] T. Hidaka, T. Morikawa, and J. Shimada. J. Appl. Phys. 1981, vol.52, 4467-4471 [17] C.C. Gregory and J.A. Harrington. Appl. Opt. 1993, vol. 32, 5302-5309 [18] A. Hongo, K. Morosawa, K. Matsumoto, T. Shiota, and T. Hashimoto. Appl. Opt. 1992, vol 31, 5114-5120 [19] J.A. Harrington, R. George, P. Pedersen, and E. Mueller. Opt. Express 2004, vol.12, 5263-5268 [20] T.S. Guan, M.Y. Chen, Z.L. Zhang, and R.J. Yu. Chin. Opt. Lett. 2005, vol.3, 313-315 [21] Y. Xu and A. Yariv. Opt. Lett. 2003, vol.28(20), 1885-1887 [22] P.J. Roberts, F. Couny, H. Sabert, B.J. Mangan, D.P. Williams, L. Farr, M.W. Mason, A. Tomlinson, T.A. Birks, J.C. Knight, and P.st.J. Russell. Opt. Express 2005, vol.13, 236- 244 [23] S.G. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, T.D. Engeness, M. Soljačić, S.A. Jacobs, J.D. Joannopoulos, and Y.Fink. Opt. Express 2001, vol.9 , 748-779 [24] A. Argyros, I.M. Bassett, M.A. van Eijkelenborg, M.C.J. Large, J. Zagari, N.A.P. Nicorovici, R.C. McPhedran, and C.M. de Sterke. Opt. Express 2001, vol.9, 813-820 [25] A. Argyros, N. Issa, I. Bassett, and M.A. van Eijkelenborg. Opt. Lett. 2004, vol.29, 20-22 [26] M.Y. Chen, R.J. Yu, Z.G. Tian, and X.Z. Bai. Chin. Opt. Lett. 2006, vol.4, 63-65 [27] L. Huo, R.J. Yu, B. Zhang, M.Y. Chen, and B.X. Li. Chin. Phys. Lett. 2006, vol.23, 2121-2124 A New Generation of Polymer Optical Fibers 277 [28] G. Dellemann, T.D. Engeness, M. Skorobogatiy, and U. Kolodny. Photonics Spectra June 2003, vol.37, 60-64 [29] B. Temelkuran, S.D. Hart, G. Benoit, J.D. Joannopoulos, and Y. Fink. Nature 2002, vol.420, 650-653. [30] A. Argyros, M.A. van Eijkelenborg, M.C.J. Large, and I.M. Bassett. Opt .Lett. 2006, vol.31, 172-174 [31] C.M. de Sterke, I.M. Bassett, and A.G. Street. J. Appl. Phys. 1994, vol.76, 680-688 [32] I.M. Bassett and A. Argyros. Opt. Express 2002, vol.10, 1342-1346 [33] S.G. Johnson and J.D. Joannopoulos. Opt. Express 2001, vol.8, 173-190 [34] Y. Xu, A. Yariv, J.G. Fleming, and S.-Y. Lin. Opt. Express 2003, vol.11, 1039-1049 [35] R.J. Yu, B. Zhang, M.Y. Chen, L. Huo, Z.G. Tian, and X.Z. Bai. Opt. Commun. 2006, vol.266, 536-540 [36] T. Kaino, M. Fujiki, and K. Jinguji. Rev. Elec. Commun. Lab. 1984, vol.32, 478-488 [37] H. Han, H. Park, M. Cho, and J.Kim. Appl. Phys. Lett. 2002, vol.80, 2634-2636 [38] M. Goto, A. Quema, H. Takahashi, S. Ono, and N. Sarukura. Jpn. J. Appl. Phys. 2004, vol.43(No. 2B), L317-L319 [39] J.A. Harrington, R. George, P. Pedersen, and E. Mueller. Opt. Express 2004, vol.12, 5263-5268 [40] L.-J. Chen, H.-W. Chen, T.-F. Kao, J.-Y. Lu, and C.-K. Sun. Opt. Lett. 2006, vol.31, 308-310 [41] G.W. Chantry, J.W. Fleming, P.M. Smith, M. Cudby, and H.A. Willis. Chem. Phys. Lett. 1971, vol.10, 473-477 [42] R.-J. Yu, B. Zhang, Y.-Q. Zhang. C.-Q Wu, Z.-G. Tian, and X.-Z. Bai. IEEE Photon. Technol. Lett. 2007, vol.19, 910-912 [43] N.S. Kapany and R.J. Simms. Infrared Phys. 1965, vol.5, 69-80 [44] J.S. Sanghera, I.D. Aggarwal, L.E. Busse, P.C. Pureza, V.Q. Nguyen, F.H. Kung, L.B. Shaw, and F. Chenard. Laser Focus World April 2005, vol.41, 83-87 [45] L.N. Butvina, E.M. Dianov, N.V. Lichkova, V.N. Zavgorodnev, an L. Kuepper. Proc. SPIE 2000, vol.4083, 238-253 [46] H. Machida, Y. Matsuura, H. Ishikawa, and M. Miyagi. Appl. Opt. 1992, vol.31, 7617- 7622 [47] E. Rave, P. Ephrat, M. Goldberg, E. Kedmi, and A. Katzir. Appl. Opt. 2004, vol.43(11), 2236-2241 [48] L.B. Shaw, J.S. Sanghera, I.D. Aggarwal, and F.H. Kung. Opt. Express 2003, vol.11, 3455-3460 [49] J.M. Pottage, D.M. Bird, T.D. Hedley, T.A. Birks, J.C. Knight, P.st.J. Russell, and P.J. Roberts. Opt. Express 2003, vol.11(22), 2854-2861 [50] J.D. Shephard, W.N. MacPherson, R.R.J. Maier, J.D.C. Jones, D.P. Hand, M. Mohebbi, A.K. George, P.J. Roberts, and J.C. Knight. Opt. Express 2005, vol.13(18), 7139-7144 [51] L. Jeunhomme and M. Monerie. Electron. Lett. 1980, vol.16, 921-922 [52] T. Yamashita and K. Kamada. Jpn. J. Appl. Phys. 1993, vol.32, 2681-2686 In: Optical Fibers Research Advances ISBN: 1-60021-866-0 Editor: Jurgen C. Schlesinger, pp. 279-300 © 2007 Nova Science Publishers, Inc. Chapter 10 DISSIPATIVE SOLITONS IN OPTICAL FIBER SYSTEMS Mário F.S. Ferreira and Sofia C.V. Latas Department of Physics, University of Aveiro, 3800-193 Aveiro, Portugal Abstract We introduce the concept of dissipative solitons, which emerge as a result of a double balance: between nonlinearity and dispersion and also between gain and loss. Such dissipative solitons have many unique properties which differ from those of their conservative counterparts and which make them similar to living things. We focus our discussion on dissipative solitons in optical fiber systems, which can be described by the cubic-quintic complex Ginzburg-Landau equation (CGLE). The conditions to have stable solutions of the CGLE are discussed using the perturbation theory. Several exact analytical solutions, namely in the form of fixed-amplitude and arbitrary-amplitude solitons, are presented. The numerical solutions of the quintic CGLE include plain pulses, flat-top pulses, and composite pulses, among others. The interaction between plain and composite pulses is analyzed using a two- dimensional phase space. Stable bound states of both plain and composite pulses are found when the phase difference between them is 2 / π ± . The possibility of constructing multisoliton solutions is also demonstrated. 1. Introduction Solitary waves have been the subject of intense theoretical and experimental studies in many different fields, including hydrodynamics, nonlinear optics, plasma physics, and biology [1]- [5]. In fact, the history of solitons dates back to 1834, the year in which James Scott Russell observed that a heap of water in a canal propagated undistorted over several kilometres [6]. However, the term “soliton” was coined only in 1965, to reflect the particle-like nature of solitary waves that remain intact even after mutual collisions [7]. Such waves correspond to localized solutions of integrable equations such as the Korteveg de Vries and nonlinear Schrödinger equations. In these circumstances, solitons were usually attributed only to integrable systems. However, the concept of soliton was subsequently broaden to include also the localized solutions of non-integrable systems. Mário F.S. Ferreira and Sofia C.V. Latas 280 Concerning the field of nonlinear optics, one can distinguish between temporal and spatial solitons [8]. Spatial optical solitons are beams of light in which nonlinearity counteracts diffraction, leading to a robust structure which propagates without change of form. Such structures will play a major role in the future in the field of all-optical processing and logic. Temporal solitons, on the other hand, represent shape invariant (or breathing) pulses, formed by a balance between nonlinearity and dispersion. It is believed that temporal solitons will play a major role in future all-optical high-capacity transmission systems [9] [10]. Until now, the main emphasis has been given to the well-known conservative soliton systems, where only the diffraction or dispersion needs to be balanced by the nonlinearity. However, a new field has emerged in the last few years concerning the formation of solitons in systems far from equilibrium [11]. These solitons are termed dissipative solitons or auto- solitons and they emerge as a result of a double balance: between nonlinearity and dispersion and also between gain and loss. Such dissipative solitons have many unique properties which differ from those of their conservative counterparts. For example, except for very few cases [5], they form zero-parameter families and their properties are completely determined by the external parameters of the optical system. They can exist indefinitely in time, as long as these parameters stay constant. However, they cease to exist when the source of energy or matter is switched off, or if the parameters of the system move outside the range which provides their existence. Even if it is a stationary object, a dissipative soliton shows non-trivial energy flows with the environment and between different parts of the pulse. Hence the dissipative soliton is an object which is far from equilibrium and which presents characteristics similar to a living thing. In fact, we can consider animal species in nature as elaborate forms of dissipative solitons. An animal is a localized and persistent “structure” which has material and energy inputs and outputs and complicated internal dynamics. Moreover, it exists only for a certain range of parameters (pressure, temperature, humidity, etc.) and dies if the supply of energy is switched off. The same analogy can be applied to individual organs within an animal, since each maintains its shape and function over time. Many non-equilibrium phenomena, such as convection instabilities, binary fluid convection and phase transitions, can be described by the complex Ginzburg-Landau equation (CGLE) [12]-[14]. In the field of nonlinear optics, the CGLE can describe various systems, namely optical parametric oscillators, free-electron laser oscillators, spatial and temporal soliton lasers, and all-optical transmission lines [9][15]-[27]. In these systems there are dispersive elements, linear and nonlinear gain, as well as losses. In some cases, the CGLE admits a multiplicity of solutions for the same range of system parameters. This reality again resembles the world of biology, where the number of species existing in the same environment is trully impressive. In this chapter we will discuss the cubic-quintic CGLE and the characteristics of some of its solutions. In Section 2 we present the CGLE and in Section 3 the perturbation approach to solve this equation is discussed. Some analytical and numerical solutions of the CGLE are presented in Sections 4 and 5, respectively. Finally, Section 6 summarizes the main conclusions. Dissipative Solitons in Optical Fiber Systems 281 2. The Complex Ginzburg-Landau Equation In one of the forms used in nonlinear optics, the cubic-quintic complex Ginzburg-Landau equation (CGLE) can be written as [5][19]-[27]: q q q q i q q q i T q i q i q q T q D Z q i 4 4 2 2 2 2 2 2 2 ν μ ε ∂ ∂ β δ ∂ ∂ ∂ ∂ − + + + = + + (1) where Z is the propagation distance or the normalized number of round trips, T is the retarded time, q is the normalized envelope of the electric field, β stands for spectral filtering ( β >0), δ is the linear gain or loss coefficient, ε accounts for nonlinear gain-absorption processes (for example, two-photon absorption), μ represents a higher order correction to the nonlinear gain-absorption, and ν is a higher order correction term to the nonlinear refractive index. The parameter D is the group velocity dispersion coefficient, with 1 ± = D , depending on whether the group velocity dispersion (GVD) is anomalous or normal, respectively. The CGLE is rather general, as it includes dispersive and nonlinear effects, in both conservative and dissipative forms. It is known in many branches of physics, including fluid dynamics, nonlinear optics and laser physics. Equation (1) becomes the standard nonlinear Schrödinger equation (NLSE) when the right-hand side is set to zero. When this does not happen, Eq. (1) is non-integrable, and only particular exact solutions can be obtained. In the case of the cubic CGLE ( 0 = =ν μ ), exact solutions can be obtained using a special ansatz [28], Horota bilinear method [29] or reduction to systems of linear PDEs [30]. Concerning the quintic CGLE, the existence of soliton-like solutions in the case 0 > ε has been demonstrated both analytically and numerically [5][20][26][31]. Exact solutions of the quintic CGLE, including solitons, sinks, fronts and sources, were obtained in [32], using Painlevé analysis and symbolic computations. It must be noted that Eq. (1) can not be used as it stands to describe the behaviour of femtosecond optical pulses. For such ultrashort pulses, some higher-order nonlinear and dispersive effects must be taken into account, which results in additional terms to be added to the right-hand side of Eq. (1) [33]-[38]. 3. Results from the Soliton Perturbation Theory Assuming that D=+1 and that all the other coefficients in the right-hand side of Eq. (1) are small, we can use the adiabatic soliton perturbation theory [9][34][39][40] to evaluate the dynamical evolution of the soliton parameters the amplitude η and the frequency κ , with which the one soliton solution is given by: [ ] { } [ ] ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ − − + − − + = σ κ η κ θ κ η η i Z Z Z i T Z i Z T Z h Z Z T q 2 2 ) ( ) ( 2 ) ( exp ) ( ) ( sec ) ( ) , ( (2) Mário F.S. Ferreira and Sofia C.V. Latas 282 Applying the perturbation procedure, we get the following set of ordinary-differential equations: 5 3 2 2 15 16 3 4 3 1 2 2 μη εη κ η βη δη η + + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + − = dZ d (3) κ βη κ 2 3 4 − = dZ d (4) As can be seen from Eq. (4), the soliton frequency approaches asymptotically to 0 = κ (stable fixed point) if 0 ≠ η . The stable fixed points for the soliton amplitude, on the other hand, are given by minimums of the potential function φ defined by: η φ η d d dZ d − = (5) Considering the Eq. (3), we have the following expression for the potential function: ( ) 6 4 2 45 8 2 6 1 ) ( μη η ε β δη η φ − − + − = (6) For the zero-amplitude state to be stable, the potential function must have a minimum at 0 = η , in addition to a minimum at 0 ≠ = s η η . These objectives can be achieved if the following conditions are verified [20]: 0 < δ , 0 < μ , 2 / β ε > , 4 8 15 s μη δ > (7) We can verify from the above conditions that the inclusion of the quintic term in Eq. (1) is necessary to have the double minimum potential. The stationary value for the soliton amplitude can be obtained from Eq. (6) and is given by: μ δμ ε ε ε ε η 8 5 / 24 ) ( 5 ) ( 5 2 2 − − − − − = s s s (8) where 2 / β ε = s for small values of β . However, the result given by Eq. (8) can be generalized for arbitrary values of β using s ε given by [20][26][41]: Dissipative Solitons in Optical Fiber Systems 283 2 2 9 2 1 4 1 3 2 β β β ε + − + = s (9) From Eq. (8) it can be seen that a stationary amplitude 1 = s η occurs when the coefficients satisfy the relation: 0 8 ) 2 ( 5 15 = + − + μ β ε δ (10) The discriminant in Eq. (8) must be greater than or equal to zero for the solution to exist. For given values of β , μ , and ε , the allowed values of δ to guarantee a stable pulse propagation must satisfy the condition 0 min ≤ ≤ δ δ , where ( ) μ ε ε δ 24 5 2 min s − = (11) When 0 = δ , the peak amplitude is found to achieve a maximum value: ( ) μ ε ε η s − − = 4 5 max (12) For 0 = μ and s ε ε = the peak amplitude becomes arbitrary. On the other hand, for given values of β , μ , and δ , the minimum value of allowedε becomes 5 / 24 min δμ ε ε + = s (13) Considering the last condition in Eq. (7) or, alternatively, from Eq.s (8) and (13) we find that there is a minimum value for the peak amplitude, given by: 4 min 8 15 μ δ η = (14) Fig. 1 shows the potential function given by Eq. (6) when the relation (10) is satisfied for 3 . 0 = β , 5 . 0 = ε , 25 . 0 − = μ (curve a), 34375 . 0 − = μ (curve b) and 5 . 0 − = μ (curve c). Curves a and b present a minimum at 1 = η and 0 = η since they satisfy the conditions (7), corresponding to negative values of the linear gain ( 05 . 0 − = δ and 1 . 0 − = δ , respectively). However, curve c has no minimum at 0 = η , since the corresponding value of δ is positive ( 033 . 0 = δ ). Mário F.S. Ferreira and Sofia C.V. Latas 284 Figure 1. Potential φ versus soliton amplitude when the relation (10) is satisfied for 3 . 0 = β , 5 . 0 = ε , 0 = ν , 5 . 0 − = μ (curve a), 34375 . 0 − = μ (curve b) and 25 . 0 − = μ (curve c). Figure 2. Phase portrait of Eq.s (3) and (4) corresponding (A) to curve c and (B) to curve b of Figure 1. Dissipative Solitons in Optical Fiber Systems 285 Fig. 2 illustrates the stability characteristics of the stationary solutions using the phase- plane formalism. Fig. 2a corresponds to curve c in Fig. 1, and we observe that, in this case, soliton propagation can be affected by background instability due to the amplification of small-amplitude waves. The steady-state solution shows a limited basin of attraction. For example, initial conditions with 7 . 0 = i η and 1 ± = i κ evolve toward the trivial solution 0 = s η of Eq.s (3) and (4). For these initial conditions, the nonlinearity is not sufficiently strong to balance dispersion, and the pulse disperses away. The dashed curves in Fig. 2a give approximate limits between different basins of attraction. From a perturbation analysis of Eq.s (3) and (4) around 0 = η , one can show that these curves cross the 0 = η axis at 33 . 0 ± = c κ . Thus, waves weak initial amplitudes grow up to 1 = s η if 33 . 0 < i κ . In this case, soliton propagation can be severely affected by the background instability. Fig. 2b corresponds to curve b in Fig. 1, and we can see that, in this case, the background instability is avoided, since the small-amplitude waves are attenuated, irrespective of their frequencyκ . Besides the stable stationary point at 1 = s η , we note, in this case, the existence of another stationary point at 5 . 0 ≈ s η , which is unstable. This simple approach shows that, in general, the CGLE has stationary soliton-like solutions, and that, for the same set of equation parameters, there may be two of them simultaneously (one stable and one unstable). Moreover, this approach shows that soliton parameters are fixed. 4. Exact Analytical Solutions Several types of exact analytical solutions of the CGLE have been obtained considering a particular ansatz [5][26]. However, due to restrictions imposed by the ansatz, these solutions do not cover the whole range of parameters. In the following, we will assume a stationary solution of Eq. (1) in the form: [ ] { } Z i T a id T a Z T q ω − = ) ( ln exp ) ( ) , ( (15) where a(T) is a real function and d, ω are real constants. 4.1. Solutions of the Cubic CGLE The cubic CGLE is given by Eq. (1) with 0 = =ν μ . Inserting Eq. (15) in this equation we obtain the following solution for a(T): a T A h BT ( ) sec ( ) = (16) where Mário F.S. Ferreira and Sofia C.V. Latas 286 2 2 2 3 2 ) 2 ( B d d B A β + − = (17) β β δ − + = d d B 2 (18) and d is given by ) 2 ( 2 ) 2 ( 8 ) 2 1 ( 9 ) 2 1 ( 3 2 2 β ε β ε εβ εβ − − + + ± + = d (19) On the other hand, we have ( ) ( ) 2 2 2 4 1 d d d d β β β δ ω + − + − − = (20) The solution (16)-(18) is known as the solution of Pereira and Stenflo [28]. Although the amplitude profile of the solution (16)-(18) is an hyperbolic secant as in the case of the NLSE solitons, two important differences exist between the CGLE and the NLSE solitons. First, for CGLE pulses the amplitude and width are independently fixed by the parameters of (1), whereas for NLSE solitons A=B. The second difference is that the CGLE solitons are chirped. The solution given by Eq.s (16)-(18) has a singularity at d d − + = β β 2 0 , which takes place on the line ) (β ε s in the plane ( , ) β ε defined by Eq. (9). For a given value of β , the denominator in the expression for B in Eq. (18) is positive for s ε ε < and negative for s ε ε > . Hence, for solution (16)-(18) to exist, the excess linear gain δ must be positive for s ε ε < and negative for s ε ε > . In the last case, both numerical simulations and the soliton perturbation theory show that the soliton is unstable relatively to any small amplitude fluctuations [20][26]. On the other hand, for 0 > δ and s ε ε < the solution (16)-(18) is stable, since after any small perturbation it approaches the stationary state. However, the background state is unstable in this case, since the positive excess gain also amplifies the linear waves coexistent with the soliton trains. The general conclusion is that either the soliton itself or the background state is unstable at any point in the plane ) , ( ε β , which means that the total solution is always unstable. The stationary value of the pulse width 1/B can be significantly reduced by a convenient choice of the system parameters [42]. In fact, it can be verified from Eq.s (11) and (12) that, for a given value of the filter strength β , as the nonlinear gain coefficient approaches the value s ε given by Eq. (9), the amplitude A increases to infinity and its width 1/B tends to Dissipative Solitons in Optical Fiber Systems 287 zero. This singularity can be used in soliton lasers to vary the pulse parameters by a small variation of the material parameters. If β and ε satisfy the Eq. (9) and δ = 0, a solution of the cubic CGLE with arbitrary amplitude exists, given by [5][26]: ) ( sec ) ( DT h C T a = (21) where C is an arbitrary positive parameter and C/D is given by: ( ) ( ) ( ) 1 4 1 3 2 1 4 1 4 1 9 2 2 2 2 2 2 − + − + + + = β β β β β D C (22) We have also d = + − 1 4 1 2 2 β β (23) 2 2 2 4 1 D d β β ω + − = Figure 3. Simultaneous propagation of four arbitrary-amplitude solitons with with amplitudes 2, 1.5, 1, and 0.5, for 0 = δ , 2 . 0 = β 0 = = μ ν and s ε ε = . Mário F.S. Ferreira and Sofia C.V. Latas 288 It can be verified that the cubic CGLE becomes invariant under the scale transformation Dq q → , DT T → , Z D Z 2 → when δ = 0. This is the reason for the existence of the arbitrary-amplitude solitons. On the ther hand, we can see that the limiting value of the amplitude-width product A/B for the fixed-amplitude solitons coincide with the value C/D on the line (22) [20]. This shows that arbitrary amplitude solitons can be considered as a limiting case of fixed amplitude solitons when δ →0. However, the arbitrary amplitude solitons have stability properties different from those for fixed amplitude solitons. In fact, arbitrary amplitude solitons are stable pulses, which propagate in a stable background because δ = 0. This feature is illustrated in Fig. 3, which shows the simultaneous propagation of four stable solitons with amplitudes 2, 1.5, 1, and 0.5, for 0 = δ , 2 . 0 = β and s ε ε = . 4.2. Solutions of the Quintic CGLE Considering the quintic CGLE and inserting Eq. (15) in Eq. (1), the following general solution can be obtained for 2 a f = [5][26]: ( ) T f f f f f f f f T f 2 1 2 1 2 1 2 1 2 cosh ) ( ) ( 2 ) ( α − − + = (24) where 2 2 3 d d β β μ α − − = (25) and d is given by Eq. (19). The parameters 1 f and 1 f are the roots of the equation: 0 ) 4 1 ( 3 ) 2 ( 2 3 8 2 2 2 2 2 = + − − + − + + − d d f d f d d β β δ β ε β β ν (26) and the coefficients are connected by the relation: 0 1 2 3 16 2 2 2 2 4 12 2 2 = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + − − − + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − − − + d d β ε β εβ μ β β ε β ε εβ ν (27) One of the roots of Eq. (26) must be positive for the solution (24) to exist, while the other can have either sign. When the two roots are both positive, the general solution given by Eq. (24) becomes wider and flatter as they approach each other. These flat-top solitons correspond to stable pulses, whereas the solution (24) is generally unstable for arbitray choice of parameters. If 2 1 f f = , the width of the flat-top soliton tends to infinity and the soliton splits into two Dissipative Solitons in Optical Fiber Systems 289 fronts. The formation and stable propagation of a flat-top soliton will be demonstrated numerically in Section 5. If β and ε satisfy the Eq. (9) and δ = 0, a solution of the quintic CGLE with arbitrary amplitude exists, given by: [ ] ( ) ( ) ( ) T P S P d T a T f 2 cosh 2 4 1 3 ) ( ) ( 2 2 + − + = = ε β β (28) where P is an arbitrary positive parameter and ( ) ( ) P d d d S 2 2 2 2 2 2 3 4 1 9 2 β β β μ ε β − − + + − = (29) β β 2 1 4 1 2 − + = d (30) P d β β ω 2 4 1 2 + − = When 0 → μ , the solution (28) transforms to the arbitrary-amplitude solution of the cubic CGLE, given by Eq. (21)-(22). 5. Numerical Solutions Due to restrictions imposed by the ansatz, the analytic solutions of the quintic CGLE presented above do not cover the whole range of parameters and almost all of them are unstable. To find stable solutions in other regions of the parameters, different approximate methods [41], a variational approach [43]-[45], or numerical techniques must be used. As shown by the perturbative analysis presented in Section 3, the parameter space where stable solitons exist has certain limitations. We must have 0 > β in order to stabilize the soliton in frequency domain. The linear gain coefficient δ must be zero or negative in order to avoid the background instability. The parameter μ must be negative in order to stabilize the soliton against collapse. Concerning the parameter ν , it can be positive or negative. Stable solitons can be found numerically from the propagation equation (1) taking as the initial condition a pulse of somewhat arbitrary profile. In fact, such profile appears to be of little importance. For example, Fig. 4 illustrates the formation of a fixed amplitude soliton of the cubic CGLE starting from an initial pulse with a rectangular profile. It must be noted that, in this case, the linear gain is positive but relatively small ( 003 . 0 = δ ) and the soliton propagation remains stable within the displayed distance. Mário F.S. Ferreira and Sofia C.V. Latas 290 In general, if the result of the numerical calculation converges to a stationary solution, it can be considered as a stable one, and the chosen set of parameters can be deemed to belong to the class of those which permit the existence of solitons. In the following we show some examples of stable soliton solutions found with this method. Figure 4. Formation of a fixed-amplitude soliton solution of the cubic CGLE starting from an initial pulse with a rectangular profile of amplitude Ao = 0.7 (a) and Ao = 1.0 (b), when 003 . 0 − = δ , 2 . 0 = β , and 09 . 0 = ε . Figure 5. (a) Evolution of the peak amplitude and (b) the final pulse profile when 01 . 0 − = δ , 15 0. = β , 2 0. = ε , 0 = ν , 1375 0. − = μ (dashed curves) or 4 0. = ε , 3875 0. − = μ (full curves), considering an input pulse ) T ( h sec ) T , ( q = 0 . Fig. 5 shows (a) the evolution of the peak amplitude and (b) the final pulse profile obtained numerically from Eq. (1), assuming an input pulse with a sech profile and considering the following parameter values: 01 . 0 − = δ , 15 0. = β , 0 = ν , 2 0. = ε , 1375 0. − = μ (dashed curves) or 4 0. = ε , 3875 0. − = μ (full curves). When inserted in Eq. (8), these values provide a stationary amplitude 1 = s η . This prediction of the Dissipative Solitons in Optical Fiber Systems 291 perturbation theory, as well as the stability of the stationary solution are confirmed by the numerical results of Fig. 5. For small values of the parameters in the right-hand side of Eq. (1) the stable soliton solutions of the CGLE have a sech profile, similar to the soliton solutions of the NLSE, and correspond to the so-called plain pulses (PPs). However, rather different pulse profiles can be obtained for non small values of those parameters. As an example, Fig. 6 illustrates the formation and stable propagation of a flat-top soliton, starting from an initial pulse with a sech profile. The following parameter values were considered: 1 . 0 − = δ , 5 . 0 = β , 66 . 0 = ε , 01 . 0 − = =ν μ . Figure 6. Formation and evolution of a flat-top soliton, considering an input pulse ) T ( h sec ) T , ( q = 0 , for 1 . 0 − = δ , 5 . 0 = β , 66 . 0 = ε , 01 . 0 − = =ν μ . Fig. 7 shows (a) the amplitude profiles and (b) the spectra of a plain pulse, as well as of two composite pulses (CPs). The following parameter values were considered: 01 . 0 − = δ , 5 . 0 = β , 03 . 0 − = μ , 0 = ν , 5 . 1 = ε (plain pulse), 0 2. = ε (narrow composite pulse) and 5 2. = ε (wide composite pulse). Fig. 7c illustrates the formation and propagation of the wide composite pulse starting from the plain pulse solution represented in a) and b). A composite pulse exhibits a dual-frequency but symmetric spectrum (Fig. 6b) and can be considered as a bound state of a plain pulse and two fronts attached to it from both sides [5]. The “hill” between the two fronts should be counted as a source, because it follows from the phase profile that energy flows from the centre to the CP wings. If one of the fronts of a CP is missing one has a moving soliton (MS) [5]. The MS always moves with a velocity smaller than the velocity of the front for the same set of parameters. In fact, the front tends to move with its own velocity but the soliton tends to be stationary, due to the spectral filtering. The resulting velocity of the MS is determined by competition between these two processes. Increasing slightly the nonlinear gain coefficient and keeping the values of the other parameters equal to those used in Fig. 7 the stationary wide composite pulse shown in Fig. 7c is lost and a non stationary expanding structure appear, as illustrated in Fig. 8. Mário F.S. Ferreira and Sofia C.V. Latas 292 Figure 7. (a) Amplitude profiles and (b) spectra of a plain pulse and of two composite pulses when 01 . 0 − = δ , 5 . 0 = β , 03 . 0 − = μ , 0 = ν , 5 . 1 = ε (plain pulse), 0 2. = ε (narrow composite pulse) and 5 2. = ε (wide composite pulse). Figure 7c illustrates the formation and propagation of the wide composite pulse, starting from the plain pulse solution. Figure 8. Nonstationary expanding structure obtained from an initial plain pulse when 01 . 0 − = δ , 5 . 0 = β , 03 . 0 − = μ , 0 = ν and 183 . 2 = ε . Dissipative Solitons in Optical Fiber Systems 293 Pulsating and exploding soliton solutions of the CGLE were also observed recently [46]. Pulsating solitons correspond to fixed solutions in the same way as the stationary pulses and can be found when the parameters of the CGLE are far enough from the NLSE limit. On the other hand, exploding solitons appear for a wide range of parameters of the CGLE and originate from soliton solutions which remain stationary only for a limited period of time. Following the explosion, there is a “cooling” period, after which the solution becomes “stationary” again. This is a periodic phenomenon, like other phenomena occurring in the nature. It can be verified that different stable stationary solutions of the quintic CGLE can exist simultaneously for the same set of parameters [5][19]. This can be understood considering that solitons, fronts and sources are elementary building units which can be combined to form more complicated structures. In more complex systems, the number of solutions may be very high. This reality again resembles the world of biology, where the number of species is trully impressive. 6. Soliton Bound States After finding the conditions for the existence of stable solitary-pulse solutions of the CGLE equation, the next natural step is to consider their interactions and, in particular, the possibility of the existence of bound states of these pulses [19][25][47]-[52]. In fact, the problem of soliton interaction is crucial for the transmission of information. In the case of Hamiltonian systems, the interaction between the pulses is inelastic. Energy exchange between the pulses is one of the mechanisms that makes the two-soliton solutions of these systems unstable, even when such stationary solutions do exist. The situation is rather different for dissipative systems. In this case, all solutions are a result of a double balance: between nonlinearity and dispersion and also between gain and loss. Moreover, the properties of dissipative solitons are completely determined by the external parameters of the optical system. For given values of the CGLE parameters, the amplitude and width of its soliton solutions are fixed. As a consequence, during the interaction of two solitons, basically only two parameters may change: their separation r and the phase difference, φ , between them. These two parameters provide a two-dimensional plane in which we may analyze of pulse interaction, namely the formation of bound states, their stability and their global dynamics [19][25][51][52]. This reduction in the number of degrees of freedom is a unique feature of systems with gain and loss. In the case of Hamiltonian systems, the amplitudes of the solitons can also change, which can affect the stability of the possible bound states. In order to analyze numerically the soliton interaction in the 2-D space provided by the separation, r, and phase difference, φ , between the two solitons, Eq. (1) can be solved with an initial condition ) exp( ) 2 / ( ) 2 / ( ) ( 0 0 φ i r T q r T q T q + + − = (31) Mário F.S. Ferreira and Sofia C.V. Latas 294 where 0 q is the stationary solution obtained numerically from Eq. (1) when the values of its parameters are specified. Initial condition (31) with arbitrary values for r and φ will result in a trajectory on the interaction plane. Bound states will be singular points of this plane. Figure 9. Trajectories on the interaction plane showing the evolution of two plain pulses for 01 . 0 − = δ , 5 . 0 = β , 5 . 1 = ε , 0 = ν , and 03 . 0 − = μ . We have X ) cos(φ r = and Y ) sin(φ r = . Fig. 9 shows an example of a numerical simulation of an interaction between the two solitons on the interaction plane, considering the following parameter values: 01 . 0 − = δ , 5 . 0 = β , 5 . 1 = ε , 0 = ν , 03 . 0 − = μ . This figure indicates that, for the given set of parameters, there are at least four singular points. The points 3 P and 4 P are saddles and correspond to unstable bound states. In these states, the phase difference between the solitons is zero or π . In addition, there are two symmetrically located stable foci (points 1 P and 2 P ), which correspond to stable bound states of two solitons with a phase difference 2 / π φ ± = between them. The stationary pulse separation in these bound states is 62 . 1 ≈ r . As a consequence of its asymmetric phase profile, the two-soliton solution corresponding to the stable bound states 1 P and 2 P in Fig. 9 moves with a constant velocity. The direction of motion depends on the sign of φ . An example of stable propagation of a two-soliton bound state with a phase difference of 2 / π between the pulses is given in Fig. 10. Dissipative Solitons in Optical Fiber Systems 295 Stable bound states of two CPs, with a phase difference 2 / π φ ± = between them, can also be observed. This is illustrated in Fig. 11, which shows the stable propagation of a bound state of two composites pulses with a phase difference 2 / π . The following parameter values were assumed: 01 . 0 − = δ , 5 . 0 = β , 0 . 2 = ε , 0 = ν , 3 . 0 − = μ . In contrast with the behaviour of the plain pulse bound state shown in Fig. 10, the CP bound state moves at the group velocity. Figure 10. Propagation of a bound state of two plain pulses with a phase difference of 2 / π between them. Figure 11. Propagation of a bound state of two composite pulses with a phase difference of 2 / π between them. Mário F.S. Ferreira and Sofia C.V. Latas 296 The two-soliton solution can be assumed as the building block to construct various multi- soliton solutions. An example is given in Fig. 12, corresponding to a four-plain pulse solution, with a phase difference of 2 / π between adjacent pulses. As observed in the case of the two-PP solution, multisoliton solutions formed by plain pulses move with a constant velocity along the T axis. Figure 12. Four-plain pulse solution with a phase difference of 2 / π between adjacent pulses. The dash-dotted (full) lines in (b) correspond to the initial (final) phase profiles. Figure 13. Five-plain pulse solution and the correspondent phase profiles. The dash-dotted (full) lines in (b) correspond to the initial (final) phase profiles of the pulses in (a). Multisoliton solutions formed by plain pulses with zero velocity can be obtained by choosing appropriately its phase profile. Fig. 13a illustrates the evolution of a five-soliton solution whose initial phase profile is given by the dash-dotted line in Fig. 13b. This phase profile evolves during the propagation, and achieves a final profile given by the full curve in Fig. 13b. In spite of some oscillations, this multisoliton bound state remains relatively stable Dissipative Solitons in Optical Fiber Systems 297 and propagates with zero velocity. The final phase profile shown in Fig. 13b corresponds indeed to a stationary stable solution. From Fig. 13 we can infer that a zero velocity multi- soliton solution formed by plain pulses must present a symmetric and concave phase profile, such that the temporal displacement of half of the structure is balanced by the opposite displacement of the other half. These solutions can be the basic building blocks for more complicated structures. 7. Conclusion The concept of dissipative solitons was explained in this chapter. In fact, this concept is wide- ranging and provides a new paradigm for the investigation of phenomena involving stable structures in nonlinear systems far from equilibrium. Here, we have considered the particular case of nonlinear optical fiber systems with gain and loss, which can be described by the cubic-quintic complex Ginzburg-Landau equation (CGLE). These include spatial and temporal soliton lasers, parametric amplifiers and optical transmission lines. However, the model can also be applied in other fields of physics. The conditions to have stable solutions of the CGLE were discussed using the perturbation theory. Several exact analytical solutions, namely in the form of fixed-amplitude and arbitrary-amplitude solitons, were presented. The numerical solutions of the quintic CGLE include plain pulses, flat-top pulses, and composite pulses, among others. We used the two-dimensional phase space (distance-phase difference) to analyze the dynamics of the two soliton system. We have found stable bound states of both plain pulses and composite pulses when the phase difference between them is 2 / π ± . Two-composite pulses bound states have zero velocity, which is in contrast with the behaviour of the bound states formed by plane pulses. As a consequence of the existence of two-soliton bound states, three-soliton and other multisoliton bound states also exist. In particular, we have shown the possibility of constructing stable bound states of multiple plain pulses with zero velocity by choosing appropriately the phase profile of the whole solution. References [1] Ablowitz, M. J.; Clarkson, P. A. (1991). Solitons, Nonlinear Evolution Equations, and Inverse Scattering, Cambridge University Press, New York. [2] Taylor, J. T., Editor (1992). Optical Solitons – Theory and Experiment, Cambridge University Press, New York. [3] Drazin, P. G. (1993). Solitons: An Introduction, Cambridge University Press, New York. [4] Gu, C. H. (1995). Soliton Theory and its Applications, Springer, New York. [5] Akhmediev, N. N., and Ankiewicz, A. A. (1997). Solitons: Nonlinear Pulses and Beams, Chapman and Hall, London. [6] Russell, J. S. (1844). Report of 14 th Meeting of the British Association for Advancement of Science, York, September, 311-390. [7] Zabusky, N. J., and Kruskal, M. D. (1965). Interactions of “solitons” in a collisionless plasma and the recurrence of initial statea, Phys. Rev. Lett., 15, 240-243. Mário F.S. Ferreira and Sofia C.V. Latas 298 [8] Kivshar, Y. A., and Agrawal, G. P. (2003). Optical Solitons. From Fibers to Photonic Crystals, Chapman and Hall, London. [9] Hasegawa, A., and Kodama, Y. (1995). Solitons in Optical Communications, Clarendon Press, Oxford. [10] Mollenauer, L. F., and Gordon, J. P. (2006). Solitons in Optical Fibers. Fundamentals and Applications. Elsevier Academic Press, San Diego. [11] Akhmediev, N. N., and Ankiewicz, A. A. (2005). Dissipative Solitons, Springer, Berlin. [12] Normand, C., and Pomeau, Y. (1977). Convective instability: a physicist’s approach. Rev. Mod. Phys., 49, 581-623. [13] Kolodner, P. (1991). Collisions between pulses of travelling-wave convection, Phys. Rev. A, 44, 6466-6479. [14] Graham, R. (1975). Fluctuations, Instabilities and Phase Transictions, Springer, Berlin. [15] Staliunas, K. (1993). Laser Ginzburg-Landau equation and laser hydrodynamics, Phys. Rev. A, 48, 1573-1581. [16] Jian, P.-S., Torruellas, W. E., Haelterman, M., Trillo, S., Peschel, U., and Lederer, F. (1999). Solitons of singly resonant optical parametric oscillators, Optics Letters, 24, 400-402. [17] Dunlop, A. M., Wright, E. M., and Firth, W. J. (1998). Spatial soliton laser, Optics Commun., 147, 393-401 [18] Ng, C. S., and Bhattacharjee, A. (1999). Ginzburg-Landau model and single-mode operation of a free-electron laser oscillator, Phys. Rev. Lett., 82, 2665-2668. [19] Akhmediev, N. N., Rodrigues, A., and Townes, G. (2001). Interaction of dual-frequency pulses in passively mode-locked lasers, Opt. Commun. 187, 419-426. [20] Ferreira, M. F., Facão, M. V., and Latas, S. V. (2000). Stable soliton propagation in a system with spectral filtering and nonlinear gain, Fiber Integrated Opt. 19, 31-41. [21] Ferreira, M. F., Facão, M. V., Latas, S. V., and Sousa, M. H. (2005). Optical solitons in fibers for Communication systems, Fiber Integrated Opt., 24, 287-314. [22] Ferreira, M. F., Facão, M. V., and Latas, S. V. (1999). Solitonlike pulses in a system with nonlinear gain, Photonics and Optoelect. 5, 147-153. [23] Matsumoto, M., Ikeda, H., Uda, T., and Hasegawa, A. (1995). Stable soliton transmission in the system with nonlinear gain, J. Lightwave Technol., 13, 658-665. [24] Latas, S. V., and Ferreira, M. F. (1999). Soliton stability and compression in a system with nonlinear gain, SPIE Proc., 3899, 396-405. [25] Akhmediev, N. N., Ankiewicz, A. A., and Soto-Crespo, J. M. (1998). Stable soliton pairs in optical transmission lines and fiber lasers, J. Opt. Soc. Am. B 15, 515-523. [26] Akhmediev, N. N., Afanasjev, V. V., and Soto-Crespo, J. M. (1996). Singularities and special soliton solutions of the cubic-quintic complex Ginzburg-Landau equation, Phys. Rev. E 53, 1190-1201. [27] Soto-Crespo, J. M., Akhmediev, N. N., and Afanasjev, V. V. (1996). Stability of the pulselike solutions of the quintic complex Ginzburg-Landau equation, J. Opt. Soc. Am. B 13, 1439-1448. [28] Pereira, N. R., and Stenflo, L. (1977). Nonlinear Schrödinger equation including growth and damping,” Phys. Fluids, 20, 1733-1734. [29] Nozaki, K., and Bekki, N. (1984). Exact solutions of the generalized Ginzburg-Landau equation, Phys. Soc. Japan, 53, 1581-1582. Dissipative Solitons in Optical Fiber Systems 299 [30] Conte, R., and Musette, M. (1993). Linearity inside nonlinearity: exact solutions to the complex Ginzburg-Landau equation, Physica, D, 69, 1-17. [31] Thual, O. and Fauvre, S. (1988). Localized structures generated by subcritical instabilities, J. Phys, 49, 1829-1833. [32] Marcq, P., Chaté, H., and Conte, R. (1994). Exact solutions of the one-dimensional quintic complex Ginzburg-Landau equation, Physica D, 73, 305-317. [33] Agrawal, G. P. (1989). Nonlinear Fiber Optics, Academic Press, San Diego. [34] Ferreira, M. F. (1994). Analysis of femtosecond optical soliton amplification in fiber amplifiers, Opt. Commun., 107, 365-368. [35] Latas, S. V., and Ferreira, M. F. (2005). Soliton propagation in the presence of intrapulse Raman scattering and nonlinear gain, Opt. Commun., 251, 415-422. [36] Facão, M. V. and Ferreira, M. F. (2001). Analysis of the timing jitter for ultrashort soliton communication systems using perturbation methods, J. Nonlinear Math. Phys., 8, 112-117 [37] Latas, S. V., and Ferreira, M. F. (2007). Stable soliton propagation with self-frequency shift, J. Mathematics and Computers in Simulation, 74, 370-387 [38] Ferreira, M. F. (1997). Ultrashort soliton stability in distributed fiber amplifiers with different pumping configurations. In Applications of Photonic Technology. Ed.s G. Lampropoulos and R. Lessard, Plenum Press, New York, 2, 249-254. [39] Karpman, V. I., and Maslov, E. M. (1977). Perturbation theory for solitons. Zh. Eksp. Teor. Fiz. 73, 537-559 (1977. Sov. Phys. JETP. 46, 281-291). [40] Essiambre, R. J., and Agrawal, G. P. (1997). Soliton communication systems. In Progress in Optics XXXVII. Ed. E. Wolf. Amsterdam: North Holland Physics and Elsevier Science. [41] Soto-Crespo, J. M., and Pesquera, L. (1977). Analytical approximation of the soliton solutions of the quintic complex Ginzburg-Landau equation, Phys. Rev. E 56 , 7288- 7295. [42] Ferreira, M. F., and Latas, S. V. (2002). Soliton stability and compression in a system with nonlinear gain, Optical Eng. 41, 1696-1703. [43] Anderson, D, (1983). Variational approach to nonlinear pulse propagation in optical fibers, Phys. Rev. A, 27, 3135-3145. [44] Kaup, D. J., and Malomed, B. A. (1995). The variational principle for nonlinear waves in dissipative systems, Physica D, 87, 155-159. [45] Ankiewicz, A. A., Akhmediev, N. N., and Devine, N. (2007). Dissipative solitons with a Lagrangian approach, Optical Fiber Technol., 13, 91-97. [46] Akhmediev, N., Soto-Crespo, J. M., and Town, G. (2001). Pulsating solitons, chaotic solitons, period doubling, and pulse coexistence in mode-locked lasers: complex Ginzburg-Landau equation, Phys. Rev. E 63, 056602. [47] Malomed, B. A. (1991). Bound solitons in the nonlinear Schrödinger-Ginzburg-Landau equation, Phys. Rev. A 44, 6954-6957. [48] Malomed, B. A. (1993). Bound states of envelope solitons, Phys. Rev. E 47, 2874-2880. [49] Afanasjev, V. V., and Akhmediev, N. N. (1996). Soliton interaction in nonequilibrium dynamical systems, Phys. Rev. E 53, 6471-6475. [50] Ferreira, M. F., and Latas, S. V. (2001). Interaction and bound states of pulses in the Gnizburg-Landau equation, SPIE Proc. 4271, 268-279. Mário F.S. Ferreira and Sofia C.V. Latas 300 [51] Ferreira, M. F., and S. V. Latas. 2002. Bound states of plain and composite pulses in optical transmission lines and fiber lasers. In Applications of Photonic Technology. Ed.s R. Lessard, G. Lampropoulos, and G. Schinn. SPIE. 4833:845-854. [52] Latas, S. V. , Ferreira, M. F., and Rodrigues, A. (2005). Bound states of plain and composite pulses: multi-soliton solutions, Optical Fiber Technol., 11, 292-305 In: Optical Fibers Research Advances Editor: Jurgen C. Schlesinger, pp. 301-313 ISBN 1-60021-866-0 c 2007 Nova Science Publishers, Inc. Chapter 11 BRIGHT - DARK AND DOUBLE - HUMPED PULSES IN AVERAGED, DISPERSION MANAGED OPTICAL FIBER SYSTEMS K.W. Chow † and K. Nakkeeran ‡ † Department of Mechanical Engineering University of Hong Kong, Pokfulam, Hong Kong ‡ School of Engineering, Fraser Noble Building, King’s college University of Aberdeen, Aberdeen AB24 3UE, UK Abstract The envelope of the axial electric field in a dispersion managed (DM) fiber sys- tem is governed by a nonlinear Schr¨ odinger model. The group velocity dispersion (GVD) varies periodically and thus realizes both the anomalous and normal dispersion regimes. Kerr nonlinearity is assumed and a loss / gain mechanism will be incorpo- rated. Due to the big changes in the GVD parameter, the correspondingly large vari- ation in the quadratic phase chirp of the DM soliton will be identified. An averaging procedure is applied. In many DM systems, an amplifier at the end of the dispersion map will compensate for the energy dissipated in that map. Here the case of gain not exactly compensating the loss is considered, in other words, a small residual ampli- fication / attenuation is permitted. The present model differs from other similar ones on variable coefficient NLS, as the inhomogeneous features involve both time and the spatial coordinate. The goal here is to extend the model further, by permitting coupled modes or additional degree of freedom. More precisely, the coupling of fiber loss and initial chirping is considered for a birefringent fiber. The corresponding dynamics is governed by variable coefficient, coupled NLS equations for the components of the orthogonal polarization of the pulse envelopes. When the self phase and cross phase modulation coefficients are identical for special angles, several new classes of wave patterns can be found. New stationary wave patterns which possess multiple peaks within each period are found, similar to those found for the classical Manakov model. For situations where the self- and cross-phase modulation coefficients are different, symbiotic solitary pulses are studied. A pair of bright-dark pulses exists, where either or both pulse(s) cannot propagate in that waveguide if coupling is absent. 302 K.W. Chow and K. Nakkeeran 1. Introduction Transmission of information (voice, video, and data) over distances (short, moderate, long, and ultra-long) is a common requirement in the past, present and future. Carrier communi- cation of information using the electromagnetic waves is the best technology for high-speed transmission. Out of different frequency bands in the electro-magnetic wave spectrum, op- tical regime has various advantages. Optical fibers are commonly used in optical commu- nication for channelling the light pulses for digital transmission. Both linear and nonlinear optical effects in fibers play vital roles in determining the dynamics of pulse propagation. The field of nonlinear optics has blossomed and is undergoing a new revolution in recent years. The nonlinear optical response is now a key element for new emerging technologies. This is particularly true for soliton and other types of nonlinear pulse transmission in opti- cal fibers/nonlinear materials, since this form of light propagation can be used to realize the long-held dream of very high capacity dispersion-free communications. In the recent past, it has been proved beyond doubt that solitons do exist not only in optics but also in many other areas of science. Solitons that exist in optics called “optical solitons” have been draw- ing a greater attention among the scientific community, as they seem to be right candidates for transferring information across the world through optical fibers. Nonlinear pulse propagation in a long-distance, high speed optical fiber transmission system can be described by the (perturbed) nonlinear Schr¨ odinger equation (NLSE). NLSE includes linear dynamics due to group velocity dispersion of the pulse, and nonlinear mech- anism due to the Kerr effect [1]. Much research efforts on the development of such a system have been made with the intention to overcome or control these effects [2, 3]. In this di- rection, recent numerical studies [4–6] and experiments [7] have shown that a periodic dispersion compensation seems to be the most effective way for improving the optical trans- mission system. The main purpose of dispersion management is to reduce several effects, such as radiation due to lumped amplifiers compensating the fiber loss [8, 9], resonant four- wave mixing [10, 11], modulational instability [12], jitters caused by the collisions between signals [13], and the Gordon-Haus effect resulting from the interaction with noise [14], also to decide a desired average value for the dispersion [12]. Basically, dispersion-management technique utilizes a transmission line with a periodic dispersion map, such that each period consists of two types of fiber, generally with different lengths and opposite group-velocity dispersion (GVD) [4]. Lakoba has proved the non- integrability of the system equation governing the pulse propagation in dispersion-managed (DM) fibers [15]. As analytical solution for DM solitons is not available, researchers have so far utilized the Lagrangian method to study the dynamics of DM solitons [4]. Very recently we have developed a complete collective variable theory for DM solitons which effectively includes the residual field due to soliton dressing and radiation [16]. Many works have reported on fitting a Hermite-Gaussian ansatz function for the oscillating tails of the numerical stationary solution (fixed point) of the DM solitons [4, 17–19]. From numerical studies [5, 6] of DM fiber line, the pulse is deformed from the ideal soliton, has a chirp and requires an enhanced power for the average dispersion. Meanwhile Kumar and Hasegawa [20] have obtained a new nonlinear pulse (quasi-soliton) by programming the dispersion profile such that the wave equation has a combination of the usual quadratic potential and the linear potential. Bright - Dark and Double - Humped Pulses... 303 The envelope of the axial electric field in a DM fiber system is governed by a NLS model. The GVD varies periodically and thus realizes both the anomalous and normal dispersion regimes. Kerr nonlinearity is assumed and a loss / gain mechanism will be incorporated. Due to the big changes in the GVD parameter, the correspondingly large variation in the quadratic phase chirp of the DM soliton will be identified. An averaging procedure is applied [21]. In many DM systems, an amplifier at the end of the dispersion map will compensate for the energy dissipated in that map. Here the case of gain not exactly compensating the loss is considered, in other words, a small residual amplification / attenuation is permitted. The present model differs from other similar ones on variable coefficient NLS [22], as the inhomogeneous features involve both time and the spatial coordinate. The goal here is to extend the model further, by permitting coupled modes or additional degree of freedom. More precisely, the coupling of fiber loss and initial chirping is considered for a birefrin- gent fiber [23]. The corresponding dynamics is governed by variable coefficient, coupled NLS equations for the components of the orthogonal polarization of the pulse envelopes. When the self phase and cross phase modulation coefficients are identical for special angles, several new classes of wave patterns can be found. The first result will be a stationary wave pattern which possesses multiple peaks within each period, similar to those found for the classical Manakov model [24]. Another new result is the family of symbiotic solitary pulses, and this novel finding is applicable to configuration where the self phase and cross phase modulation coefficients are different. Indeed the constraints imposed on these coefficients extend or generalize results obtained earlier in the literature. As a simple example, a pair of bright - dark pulses exists where each individual wave guide separately will only admit bright solitons. This coupling nonlinearity is truly remarkable. As a second example, bright or dark solitons are allowed to propagate in waveguides which would otherwise prohibit their existence. 2. Double-Hump Bright - Dark Periodic and Solitary Pulses We consider the averaged, dispersion management system for coupled waveguides: i ∂A ∂z + ∂ 2 A ∂t 2 + (AA ∗ + σBB ∗ )A + iβA + β 2 t 2 A = 0, (1) i ∂B ∂z + ∂ 2 B ∂t 2 + (σAA ∗ + BB ∗ )B + iβB + β 2 t 2 B = 0. (2) A, B are the complex envelopes of the axial electric fields, z is the distance and t is the retarded time. The quantity β measures the quadratic phase chirp, and the residual gain or loss is specifically selected to match this parameter. The parameter σ represents cross phase modulation coefficient arising from the coupling. The derivation of system (1, 2) from the first principle of averaging over a dispersion map can be found in our earlier work [21]. System (1, 2) can be solved exactly by several techniques, but we shall focus on the Hi- rota bilinear method. As the description has been given in our earlier work, the presentation here will be brief. The quadratic phase factor or chirp is first isolated as 304 K.W. Chow and K. Nakkeeran A = exp iβt 2 2 ϕ, B = exp iβt 2 2 ψ. (3) The reduced governing equations for the auxiliary variables ϕ and ψ will be free of quadratic terms in time. To search for the special modes of optical pulses, we further constrain the wave pattern to be expressed as ϕ = g exp(−iΩ 1 ) f , ψ = Gexp(−iΩ 2 ) f . (4) G, g and f are dependent variables for the bilinear operation with the restriction that f is real. Typically they are combinations of exponential functions for solitary pulses but elliptic functions for periodic patterns. The phase factors Ω 1 , Ω 2 are functions of the distance (z) only. They will have their derivatives determined in the bilinear equations, and hence they themselves are readily recovered by quadrature. The resulting bilinear equations are then f D 2 t + 2iβtD t + ∂Ω 1 ∂z + 2iβ g · f + g(−D 2 t f · f + gg ∗ + σGG ∗ ) = 0, (5) f D 2 t + 2iβtD t + ∂Ω 2 ∂z + 2iβ G · f + G(−D 2 t f · f + σgg ∗ + GG ∗ ) = 0. (6) They are solved by using rather straightforward differentiation formulas developed from first principles. D is the bilinear operator, with its definition and properties described more fully in Appendix A.. For periodic wave patterns, Hirota derivatives of theta functions can be simplified by identities involving products of theta functions (Appendix B.). As illustrative examples, the simplest periodic wave pattern will be given by the choice, g = A 0 θ 1 (t[h 1 (z)]), G = B 0 θ 3 (t[h 1 (z)]), f = θ 4 (t[h 1 (z)]). (7) The theta functions are Fourier series with exponentially decaying coefficients and the classical Jacobi functions can be expressed as ratios of theta functions. The amplitude parameters, A 0 , B 0 , isolated here for convenience will also be functions of z. The distance dependent wave number function h 1 (z) will render the period of the pattern to change with location, and the precise form is determined by forcing the odd Hirota derivatives to vanish. The loss / gain factor is not arbitrary as it has to match the precise forms of the functions A 0 , B 0 . Theta functions will be convenient in the intermediate calculations. However, the Jacobi elliptic functions are preferred in the final expressions, as they can be easily handled by most established routines in computer algebra. A summary on existing results will be instructive: 1. When the cross phase modulation coefficient, σ, is arbitrary, the wave system will permit periodic patterns in terms of a single elliptic function. The long wave limit will, not surprisingly, yield solitary bright or dark pulses. Bright - Dark and Double - Humped Pulses... 305 2. When σ is constrained to be unity, there are other varieties of solutions. In particular, one class of wave patterns can be expressed in terms of products of elliptic functions. The physical implication is that the intensity will display two, or perhaps more, peaks per period. The goal of this section is to derive still further new wave patterns by choosing products of elliptic functions as the starting point of these calculations, while still assuming the cross phase modulation coefficient, σ, to be one. The motivation comes from the choice of wave patterns for the case of coupled nonlinear Schr¨ odinger models with constant coefficients. Proceeding along the lines of reasoning just described will yield A = √ 6r √ 1 −k 2 1 −2c √ 1 −k 2 c − dn 2 (rte −2βz ) √ 1 −k 2 exp iβt 2 2 −2βz −iΩ 1 , (8) B = √ 6 rk sn(rte −2βz )dn(rte −2βz ) 1 −2c √ 1 −k 2 exp iβt 2 2 −2βz −iΩ 2 . (9) Ω 1 = r 2 exp(−4βz) 4β 6c 2 (1 −k 2 ) 1 −2c √ 1 −k 2 + 2 √ 1 −k 2 c , (10) Ω 2 = r 2 exp(−4βz) 4β 6c 2 (1 −k 2 ) 1 −2c √ 1 −k 2 −2(5 −4k 2 ) , (11) r is a free parameter and represents the wave number at the initial location (z = 0). The quantity c will be one of the roots of the quadratic equation 3c 2 −2c 1 −k 2 + 1 √ 1 −k 2 + 1 = 0, (12) k is the modulus of the elliptic function. Waveguide B will generally exhibit two peaks per period. Waveguide A will degenerate to a dark solitary pulse with multiple peaks in the long wave period. Figures 1a, 1b illustrate this pattern. 3. A Generalized Model with Different Dispersion Coefficients In many applications involving wave propagation along different channels or waveguides, the optical pulses will experience different measures of group velocity dispersion. Hence the coefficients of the second derivative terms of the coupled NLS equations will generally be different. Remarkably a special model system will still permit analytical progress, and we shall consider pairs of bright - dark solitons in this model. Generally bright (dark) soli- tons occur for the conventional NLS model in the anomalous (normal) dispersion regimes respectively. However, due to the special nonlinear effects in coupled NLS systems, these bright / dark solitons can occur in the appropriate waveguide which are otherwise prohib- ited in the single mode NLS. They have sometimes been termed ‘symbiotic solitons’ in the literature. In optical physics, such waves have indeed been studied for configurations associated with conventional NLS with Kerr nonlinearity [25], intra-pulse stimulated Raman scattering 306 K.W. Chow and K. Nakkeeran -10 -5 0 5 10 1.0 0.5 0.0 0 0.5 1 1.5 2 t [N o rm . U n it] z [ N o r m . U n i t ] | A | 2 [ N o r m . U n i t ] (a) -5 0 5 1.0 0.5 0.0 0 0.02 0.04 0.06 0.08 0.1 t [N o rm . U n it] z [ N o r m . U n i t ] | B | 2 [ N o r m . U n i t ] (b) Figure 1. Evolution of the periodic solution (8) and (9) for the parameters β = 0.05, r = 1, k = 0.9 and c = 1.61. [26], quasi-phase matched parametric oscillator [27], second harmonic generation [28], and three-wave solitons [29]. In other systems, symbiotic solitons also occur in phenomena connected with Bose - Einstein condensates [30], discrete systems [31], multi-dimensional NLS by separation of variables [32], and quadratic, nonlinear media with loss and gain [33]. More precisely, we shall consider i ∂A ∂z + δ ∂ 2 A ∂t 2 + (AA ∗ + σBB ∗ )A + iβA + β 2 t 2 A δ = 0, (13) i ∂B ∂z + ∂ 2 B ∂t 2 + (σAA ∗ + BB ∗ )B + iβB + β 2 t 2 B = 0. (14) Bright - Dark and Double - Humped Pulses... 307 Here A and B are again complex envelopes but the first waveguide is permitted to have a dispersion coefficient δ (positive or negative) relative to waveguide B. The chirp factors, however, must be modified to A = exp iβt 2 2δ ϕ, B = exp iβt 2 2 ψ. A periodic pattern is obtained earlier in the literature as A = rkQ 1 sn(rte −2βz ) exp −2βz + iβt 2 2δ − ir 2 e −4βz 4β (Q 2 1 +δ(1 −k 2 )) , (15) B = rQ 2 dn(rte −2βz ) exp −2βz + iβt 2 2 − ir 2 e −4βz 4β (Q 2 2 −k 2 ) , (16) Q 1 = 2(δ −σ) σ 2 −1 , Q 2 = 2(σδ −1) σ 2 −1 . (17) The restrictions are either δ > σ if σ > 1, (18) or δ < σ if σ < 1, (19) The long wave limit is A = rQ 1 tanh(rte −2βz ) exp −2βz + iβt 2 2δ − ir 2 e −4βz Q 2 1 4β , (20) B = rQ 2 sech(rte −2βz ) exp −2βz + iβt 2 2 − ir 2 e −4βz (Q 2 2 −1) 4β . (21) (20) represents a dark soliton, and propagates in the anomalous regime if δ is positive, while (21) remains a bright soliton in the anomalous dispersion regime. The contribution in the present work is to document another set of periodic / solitary wave pattern which relieves or compensates the constraints imposed by (18) and (19). Fur- thermore, for some parameter regimes, one can achieve a pair of symbiotic solitons with bright (dark) solitons propagating in the normal (anomalous) dispersion regimes respec- tively. Following reasoning similar to the previous sections, we can attain another set of wave profiles by exchanging the choice of elliptic functions in (15) and (16): A = rR 1 dn[rt exp(−2βz)] exp −2βz + iβt 2 2δ −iΩ 1 , (22) B = rkR 2 sn[rt exp(−2βz)] exp −2βz + iβt 2 2 −iΩ 2 , (23) 308 K.W. Chow and K. Nakkeeran where the parameters R 1 , R 2 are R 1 = 2(σ −δ) σ 2 −1 , R 2 = 2(1 −σδ) σ 2 −1 , (24) and the requirement of real square roots dictates that δ > 1 σ if σ < 1, (25) or δ < 1 σ if σ > 1, (26) (25) and (26) are different from (18) and (19). Ω 1 , Ω 2 are angular frequency parameters given by ∂Ω 1 ∂z = −r 2 exp(−4βz) δ(2 −k 2 ) + 2σ(1 −σδ) σ 2 −1 , (27) ∂Ω 2 ∂z = −r 2 exp(−4βz) 1 −k 2 + 2σ(1 −σδ) σ 2 −1 . (28) The long wave limit is even more instructive. On taking the limit k −→1, where in general (sn, cn, dn) become (tanh, sech, sech) respectively, one now has A = rR 1 sech[rt exp(−2βz)] exp −2βz + iβt 2 2δ −iΩ 10 , (29) B = rR 2 tanh[rt exp(−2βz)] exp −2βz + iβt 2 2 −iΩ 20 . (30) where Ω 10 , Ω 10 are the long wave (k −→1) limits of Ω 1 and Ω 2 respectively. For σ < 1, both waveguides are in the anomalous dispersion regimes (as δ > 1). For σ > 1, δ can be either positive or negative. In particular, negative values of δ here will imply that waveguide A is in normal dispersion regime. Remarkably, a bright (dark) soliton now propagates in the normal (anomalous) dispersion regime respectively. These phenomena are quite contrary to the well known results. 4. Conclusions A class of periodic and solitary waves has been studied for a system of coupled envelope equations. This systemcan model averaged, dispersion managed systems where the residual gain / loss in each cycle of the dispersion map has been carefully chosen. Waves with mul- tiple peaks per period or symbiotic pairs of solitary pulses are obtained analytically. They will enhance our capability in modeling such systems and strengthen our understanding in this and similar optical systems. Bright - Dark and Double - Humped Pulses... 309 Several aspects still allow rooms for future work and expansions. In particular, config- urations where both waveguides are in the normal dispersion regime have not been worked out in details yet, although the same physics is expected to hold true qualitatively. One issue which has not been addressed is the stability of these wave patterns. Numer- ical simulations of perturbed wave trains must be pursued. Recent works and experience have indicated that stability will probably still prevail in some parameter regimes. The precise elucidation will await further efforts. Acknowledgement Partial financial support has been provided by the Research Grants Council through the con- tract HKU7123/05E. KWCand KNwish to thank The Royal Society for their support in the form of an International Joint Project Grant. KWC and KN are very grateful to Prof. John Watson for his valuable support for this research collaboration. KN also wishes to thank the Nuffield Foundation for financial support through the Newly Appointed Lecturer Award. A. Hirota Bilinear Operator The Hirota operator for any two functions f and g is defined as [34, 35] D m x D n t g · f = ∂ ∂x − ∂ ∂x m ∂ ∂t − ∂ ∂t n g(x, t)f(x , t ) x=x ,t=t , (31) and the properties in association with differentiation of exponential functions are espe- cially striking (m, n are constants): D x [exp(imx)g · exp(inx)f] = [D x g · f + i(m − n)gf] exp[i(m + n)x], (32) D 2 x [exp(imx)g · exp(inx)f] = [D 2 x g · f + 2i(m − n)D x g.f − (m − n) 2 gf] ×exp[i(m + n)x]. (33) Most existing works on the Hirota operator focus on the case of constant wave number or frequencies. The important point in this work is to extend Hirota derivatives to the case of time or space dependent wavenumbers. The bilinear identities for Hirota derivatives, even for the case of variable wave number, can be obtained from simple, straightforward differentiation. Examples are: D z exp[tξ 1 (z) + ξ 2 (z)] · exp[tη 1 (z) + η 2 (z)] = {t[ξ 1 (z) − η 1 (z)] + ξ 2 (z) − η 2 (z)} · exp{t[ξ 1 (z) + η 1 (z)] + ξ 2 (z) + η 2 (z)}, (34) D z exp(a) · m(z) exp(b) = m D z exp(a) · exp(b) − 1 m ∂m ∂z exp(a + b) . (35) 310 K.W. Chow and K. Nakkeeran B. Theta Functions The theta functions θ n (x), n = 1, 2, 3, 4 in terms of the parameter q (the nome) are defined by [36–38]: θ 1 (x) = 2 ∞ n=0 (−1) n q (n+1/2) 2 sin[(2n + 1)x] , (36) θ 2 (x) = 2 ∞ n=0 q (n+1/2) 2 cos [(2n + 1)x] , (37) θ 3 (x) = 1 + 2 ∞ n=1 q n 2 cos (2nx) , (38) θ 4 (x) = 1 + 2 ∞ n=1 (−1) n q n 2 cos (2nx) , 0 < q < 1. (39) Basically they are Fourier series with exponentially decaying coefficients. Relationships between the theta and elliptic functions are: sn(u) = θ 3 (0)θ 1 (z) θ 2 (0)θ 4 (z) , cn(u) = θ 4 (0)θ 2 (z) θ 2 (0)θ 4 (z) , dn(u) = θ 4 (0)θ 3 (z) θ 3 (0)θ 4 (z) , (40) z = u θ 2 3 (0) , k = θ 2 2 (0) θ 2 3 (0) . (41) Arguments of the theta and elliptic functions are related by a scale factor. The modulus of the elliptic functions, k, is related to the theta constants by (41). Theta functions possess a huge number of identities involving addition and subtraction of arguments: θ 3 (x + y)θ 3 (x − y)θ 2 2 (0) = θ 2 4 (x)θ 2 1 (y) + θ 2 3 (x)θ 2 2 (y), (42) θ 4 (x + y)θ 4 (x − y)θ 2 2 (0) = θ 2 4 (x)θ 2 2 (y) + θ 2 3 (x)θ 2 1 (y), (43) Such identities can be proven by re-arranging terms of the multiple sums [37]. By con- sidering the leading and quadratic terms in the Taylor series of y in identities of the form (42-43), one obtains θ 4 (0) θ 4 (0) − θ 3 (0) θ 3 (0) = θ 4 2 (0), θ 4 (0) θ 4 (0) − θ 2 (0) θ 2 (0) = θ 4 3 (0), θ 3 (0) θ 3 (0) − θ 2 (0) θ 2 (0) = θ 4 4 (0). (44) D 2 x θ 3 (x) · θ 3 (x) = 2θ 2 (0)θ 2 3 (x) θ 2 (0) + 2θ 2 3 (0)θ 2 4 (0)θ 2 4 (x), (45) D 2 x θ 4 (x) · θ 4 (x) = 2θ 2 3 (0)θ 2 4 (0)θ 2 3 (x) + 2θ 2 (0)θ 2 4 (x) θ 2 (0) . (46) Hence formulas for D x θ m · θ n , D 2 x θ m · θ n can be developed for m, n integers using this line of reasoning. Bright - Dark and Double - Humped Pulses... 311 References [1] A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion,” Appl. Phys. Lett., 23, 142 (1973). [2] A. Hasegawa and Y. Kodama, Solitons in Optical Communication, (Oxford University Press, New York, 1995). [3] G. P. Agrawal, Nonlinear Fiber Optics, (Academic Press, San Diego, 1989). [4] V. E. Zakharov and S. Wabnitz, Optical Solitons: Theoretical Challenges and Indus- trial Perspectives, (Springer-Verlag, Heidelberg, 1998). [5] N. J. Smith, F. M. Knox, N. J. Doran, K. J. Blow and I. Bennion, “Enhanced power solitons in optical fibres with periodic dispersion management,” Electron. Lett., 32, 54 (1996). [6] T. Georges and B. Charbonnier, “Reduction of the dispersive wave in periodically amplified links with initially chirped solitons,” IEEE Photon. Technol. Lett., 9, 127 (1997). [7] M. Suzuki, I. Morita, N. Edagawa, S. Yamamoto, H. Toga and S. Akiba, “Reduction of Gordon-Haus timing jitter by periodic dispersion compensation in soliton transmis- sion,” Electron. Lett., 31, 2027 (1995). [8] W. Forysiak, F. M. Knox and N. J. Doran, “Stepwise dispersion profiling of periodi- cally amplified soliton systems,” J. Lightwave Technol., 12, 1330 (1994). [9] S. Kumar, A. Hasegawa and Y. Kodama, “Adiabatic soliton transmission in fibers with lumped amplifier: Analysis,” J. Opt. Soc. Am. B, 14, 888 (1997). [10] C. Kurtzke, “Suppression of fiber nonlinearities by appropriate dispersion manage- ment,” Photon. Technol. Lett., 5, 1250 (1993). [11] P. V. Mamyshev and L. F. Mollenauer, “Pseudo-phase-matched four-wave mixing in soliton wavelength-division multiplexing transmission,” Opt. Lett., 21, 396 (1996). [12] N. J. Doran, N. J. Smith, W. Forysiak and F. M. Knox, in Physics and Applications of Optical Solitons in Fibers ’95, (Kluwer Academic Press, 1996). [13] A. Hasegawa, S. Kumar and Y. Kodama, “Reduction of collision-induced time jitters in dispersion-managed soliton transmission systems,” Opt. Lett., 21, 39 (1996). [14] W. Forysiak, K. J. Blow and N. J. Doran, “Reduction of Gordon-Haus jitter by post- transmission dispersion compensation,” Electron. Lett., 29, 1225 (1993). [15] T. I. Lakoba, “Non-integrability of equations governing pulse propagation in dispersion-managed optical fibers,” Phys. Lett. A, 260, 68 (1999). 312 K.W. Chow and K. Nakkeeran [16] P. Tchofo Dinda, A. B. Moubissi and K. Nakkeeran, “A collective variable approach for dispersion-managed solitons,” J. Phys. A, 34, L103 (2001). [17] T. I. Lakoba and D. J. Kaup, “Hermite-Gaussian expansion for pulse propagation in strongly dispersion managed fibers,” Phys. Rev. E, 58, 1998 (1998). [18] S. K. Turitsyn, T. Sch¨ afer, K. H. Spatschek and V. K. Mezentsev, “Path-averaged chirped optical soliton in dispersion-managed fiber communication lines,” Opt. Com- mun., 163, 122 (1999). [19] P. Tchofo Dinda, K. Nakkeeran and A. B. Moubissi, “Optimized Hermite-gaussian ansatz functions for dispersion-managed solitons,” Opt. Commun., 187, 427 (2001). [20] S. Kumar and A. Hasegawa, “Quasi-soliton propagation in dispersion-managed opti- cal fibers,” Opt. Lett., 22, 372 (1997). [21] C. C. Mak, K. W. Chow and K. Nakkeeran, “Soliton Pulse Propagation in Averaged Dispersion-managed Optical Fiber System,” J. Phys. Soc. Japan, 74, 1449 (2005). [22] V. N. Serkin and A. Hasegawa, “Novel Soliton Solutions of the Nonlinear Schr¨ odinger Equation Model,” Phys. Rev. Lett., 85, 4502 (2000). [23] R. Ganapathy, V. C. Kuriakose and K. Porsezian, “Soliton propagation in a birefrin- gent optical fiber with fiber loss and frequency chirping,” Opt. Commun., 194, 299 (2001). [24] K. W. Chow and D. W. C. Lai, “Periodic solutions for systems of coupled nonlinear Schrdinger equations with five and six components,” Phys. Rev. E, 65, 026613 (2002). [25] M. Lisak, A. H¨ o¨ ok and D. Anderson, “Symbiotic solitary-wave pairs sustained by cross-phase modulation in optical fibers,” J. Opt. Soc. Am. B, 7, 810 (1990). [26] K. Hayata and M. Koshiba, “Bright-kink symbions resulting from the combined effect of self-trapping and intra-pulse stimulated Raman-scattering,” J. Opt. Soc. Am. B, 11, 61 (1994). [27] A. Picozzi and M. Haelterman, “Spontaneous formation of symbiotic solitary waves in a backward quasi-phase-matched parametric oscillator,” Opt. Lett., 23, 1808 (1998). [28] S. Trillo, “Bright and dark simultons in second-harmonic generation,” Opt. Lett., 21, 1111 (1996). [29] C. Durniak, C. Montes and M. Taki, “Temporal walk-off for self-structuration of three-wave solitons in CW-pumped backward optical parametric oscillators,” J. Opt. B: Quantum and Semi-Classical Optics, 6, S241 (2004). [30] V. M. Perez-Garcia and J. B. Beitia, “Symbiotic solitons in heteronuclear multicom- ponent Bose-Einstein condensates,” Phys. Rev. A, 72, 033620 (2005). Bright - Dark and Double - Humped Pulses... 313 [31] E. P. Fitrakis, P. G. Kevrekidis, B. A. Malomed and D. J. Frantzeskakis, “Discrete vector solitons in one-dimensional lattices in photorefractive media,” Phys. Rev. E, 74, 026605 (2006). [32] K. Hayata and M. Koshiba, “Bright-dark solitary-wave solutions of a multidimen- sional nonlinear Schrdinger equation,” Phys. Rev. E, 48, 2312 (1993). [33] S. Darmanyan, L. Crasovan and F. Lederer, “Double-hump solitary waves in quadrat- ically nonlinear media with loss and gain,” Phys. Rev. E, 61, 3267 (2000). [34] M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform, (SIAM, Philadelphia, 1981). [35] Y. Matsuno, The Bilinear Transformation Method, (Academic Press, New York, 1984). [36] M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, (Dover, New York, 1964). [37] D. F. Lawden, Elliptic Functions and Applications, (Springer, New York, 1989). [38] K. W. Chow, “A class of doubly periodic waves for nonlinear evolution equations,” Wave Motion, 35, 71–90. In: Optical Fibers Research Advances Editor: Jurgen C. Schlesinger, pp. 315-333 ISBN 1-60021-866-0 c 2007 Nova Science Publishers, Inc. Chapter 12 DYNAMICS AND INTERACTIONS OF GAP SOLITONS IN HOLLOW CORE PHOTONIC CRYSTAL FIBERS Javid Atai and D. Royston Neill School of Electrical and Information Engineering The University of Sydney, NSW 2006 Australia Abstract The existence and stability of gap solitons in a model of hollow core fiber in the zero dispersion regime are analyzed. The model is based on a recently introduced model where the coupling between the dispersionless core mode and nonlinear surface mode (in the presence of the third order dispersion) results in a bandgap. It is found that similar to the anomalous and normal dispersion regimes, the family of solitons fills up the entire bandgap. The family of gap solitons is found to be formally unstable but in a part of family the instability is very weak. Consequently, gap solitons belonging to that part of the family are virtually stable objects. The interactions and collisions of in-phase and theπ-out-of-phase quiescent solitons and moving solitons in different dispersion regimes are investigated and compared. 1. Introduction Gap solitons (GSs) were originally introduced in Ref. [1]. Recent years have witnessed an upsurge of research activity on gap solitons in various areas of physics such as nonlinear optics and Bose-Einstein condensation (BEC). In optics, a nonlinear dispersive medium whose spectrum contains one or more forbidden bands can support gap solitons. An ex- ample of such a system is a fiber Bragg grating (FBG). The periodic variation of linear dielectric constant in an FBG leads to a photonic band structure. The linear cross coupling between the counter-propagating waves results in a large effective dispersion (5 to 6 orders of magnitude larger than the dispersion of standard optical fiber) [2, 3]. For sufficiently high light intensities, the large Bragg grating induced dispersion may be counterbalanced by Kerr nonlinearity resulting in a gap soliton. Significant theoretical [3–6] and experimental [7–10] efforts have been directed to- ward understanding and characterizing GSs in periodic media. In particular, it has been 316 Javid Atai and D. Royston Neill shown that GSs in an FBG form a two-parameter family of solutions [4]. It has also been shown that approximately half of the soliton family is stable against oscillatory pertur- bations [11–13]. Experimental activities in this area have focused on generating zero- velocity (quiescent) GSs due to their potential applications in optical buffers and storage elements. To date, GSs with a velocity of 0.23 of the speed of light in the fiber have been observed [14]. GSs have been studied in more sophisticated systems such as in the pres- ence of higher order dispersion [15], quadratic nonlinearity [16], cubic-quintic nonlinear- ity [18], dual core fibers with FBGs [17] and in waveguide arrays [19] and photonic crystal fibers [20]. Since their demonstration in 1996 [21], photonic crystal fibers (PCFs) have been the subject of extensive research due their interesting and peculiar properties. PCFs are spe- cially designed optical fibers with many microstructured air holes running along the fiber’s length. They can be divided into two main categories depending on the mechanism of light guidance, namely the solid core and hollow core PCFs. Solid core PCFs are similar to conventional optical fibers in that they guide light through total internal reflection. On the other hand, in hollow core PCFs (HC-PCFs) the microstructured cladding surrounding the hollow core creates a photonic bandgap that guides the light [22–24]. Introduction of atomic or molecular gases into the core of HC-PCF results in efficient nonlinear optical interactions due to strong confinement of light in the core region. Some recent results include demonstrations of generation of stimulated Raman scattering (SRS) in hydrogen [25], and electromagnetically-induced transparency (EIT) [26, 27]. They have also been utilized in delivery of high energy pulses [28–30] and in soliton lasers [31]. In Ref. [34] a model for pulse propagation in HC-PCF based on experimental [32] and numerical [33] results was proposed. The model took into account the coupling of a linear dispersionless mode propagating in a gas-filled core with a nonlinear dispersive surface mode propagating in silica. In Ref. [35] a simpler model was considered where the second and third order group velocity dispersion terms were absent. The model contained a linear loss term which accounted for the power leakage from the core to the cladding. In both models a bandgap opens in the system’s spectrum. The models in Ref. [34, 35] belong to a general class of models that give rise to wavenumber bandgap [36, 37]. A wavenumber bandgap arises as a result of the coupling between the co-propagating waves (in this case the core and surface modes). On the other hand, a frequency bandgap (e.g. the above- mentioned bandgap structure in a FBG) is due to the coupling of counterpropgating waves. The stability of GSs in the model of Ref. [34] has been investigated in both anoma- lous [38] and normal [39] dispersion regimes. It is shown that, strictly speaking, GSs in both anomalous and normal dispersion regimes are unstable. However, due to the fact that instability is weak in a part of the soliton family, the GSs belonging to that part of the family are “virtually” stable objects. In addition, an important finding reported in Ref. [39] is that GSs in the normal dispersion are far more stable than their counterparts in the anomalous dispersion. In this article, we will first investigate the existence and stability of gap solitons in a HC-PCF in the special case when the second order group velocity dispersion is negligible. The model, which is based on the model of Ref. [34], and the characteristics of the bandgap and soliton solutions will be discussed in Section 2.. In Section 3. stability of quiescent and moving GSs will be presented and their stability will be compared with that of GSs in the Dynamics and Interactions of Gap Solitons... 317 anomalous and normal dispersion regimes. In Section 4. we will investigate and compare the interactions of quiescent GSs and collision dynamics of moving solitons in the normal, anomalous and zero dispersions. In particular we will analyze the effect of initial phase difference and separation on the outcome of collisions and interactions. The results are summarized in Section 5.. 2. The Model and Gap Soliton Solutions The system of equations governing the propagation of the above-mentioned surface and core modes in the zero GVD are based on the model introduced in Ref. [34]. In the normal- ized form it reads: iu z −icu τ + iγu τττ +|u| 2 u + v = 0, (1) iv z + icv τ + u = 0, (2) where u and v are the amplitudes of the surface and core waves, respectively, z and τ are the propagation distance and reduced time and c represents the group velocity mismatch between the modes. The coefficients of Kerr nonlinearity and the linear coupling between the modes have been scaled to unity. Therefore, there are two free parameters in the model namely γ and c. It should be noted that when γ = 0 Eqs. (1) and (2) reduce to the model of Ref. [35]. However, as was pointed out in Refs. [34,38,39], due to the small temporal width of solitons and that the carrier wavelength may be close or exactly equal to zero GVD point, the third order dispersion needs to be present. Undoing the rescalings and using a typical value of |β 3 | = 0.2 ps 3 /km the soliton’s width is found to be in the range of 100-300 fs. This value of β 3 corresponds to a normalized value of γ ≈ 0.3. Also, ∆z = 1 and ∆τ = 1 correspond to ranges 1-10 cm and 30-100 fs in physical units. In order to determine the linear spectrum of the system, we substitute (u, v) ∼ exp(ikz −iωτ) into the linearized Eqs. (1) and (2). This results in the following dispersion relation: 2k ± = −ω 3 γ ± (ω 3 γ + 2ωc) 2 + 4. (3) By definition we set c > 0 in which case the wavenumber bandgap exists for γ > 0. Straightforward analysis of Eq. (3) shows that the bandgap is −1 < k < +1 provided that c 2 ≥ 1 4 . The solid curves in Fig. 1 represent the branches of the dispersion relation (3) for c = 1 and γ = 0.3 with the bandgap being −1 < k < 1. In the gap, soliton solutions to (1) and (2) were sought in the form of {u(z, τ), v(z, τ)} = {U(τ), V (τ)} exp(ikz). Substituting this ansatz into (1) and (2) results in a set of equations for complex functions U(τ) and V (τ). These equations can be solved numerically using the relaxation method. It is found that, similar to the anomalous and normal dispersion regimes, the family of gap solitons completely fill the bandgap, and 318 Javid Atai and D. Royston Neill −5 −2.5 0 2.5 5 ω −10 −5 0 5 10 k Figure 1. Dispersion diagramcorresponding to c = 1 and γ = 0.3 for quiescent gap solitons (solid lines) and moving ones with δ = 0.7 (dashed lines). The bandgap for quiescent solitons is −1 < k < 1. The bandgap for moving solitons is −0.77 < k < 0.78. |U(τ)| and |V (τ)| are always single-humped. As is shown in Fig. 2, the real and imaginary parts of U(τ) and V (τ) are even and odd functions of τ, respectively. The GSs in the model of Eqs. (1) and (2), similar to their counterparts in the anomalous and normal dispersion regimes [34, 39], satisfy Vakhitov-Kolokolov criterion [40]. This criterion states that a necessary condition for the stability of solitons against nonoscillatory perturbations with purely real growth rates is dE dk > 0 where E is the energy of the soliton family and is given by: E(k) = +∞ −∞ |U(τ; k| 2 +|V (τ; k| 2 dτ. (4) However, the soliton family or part thereof may be unstable against oscillatory pertur- bations. A finding of Ref. [39] was that the energy of the soliton family in the normal dispersion regime is considerably lower than that of the anomalous dispersion regime. As is shown in Fig. 3, the energy of solitons in (1) and (2) is less than that of the anomalous case and greater than the normal dispersion case. Based on the results of Ref. [39] one may conjecture that the solitons in the zero dispersion regime are more stable than their counterparts in the anomalous dispersion and less so compared with the ones in the normal dispersion regime. This issue will be considered in the next section. Moving solitons can be obtained by rewriting Eqs. (1) and (2) in the boosted reference frame through the coordinate transform (z, τ) −→(z, τ −δz) where δ is the velocity shift. The dispersion relation of the transformed system of equations is given by: Dynamics and Interactions of Gap Solitons... 319 −20 −10 0 10 20 τ −2 −1 0 1 2 Im(U) Re(U) (a) −20 −10 0 10 20 τ −2 −1 0 1 2 3 4 Im(V) Re(V) (b) Figure 2. The real and imaginary parts of the U(τ) and V (τ) for a quiescent gap soliton with c = 1 and γ = 0.3 and k = 0. 2k ± = −(ω 3 γ + 2ωδ) ± (ω 3 γ + 2ωc) 2 + 4. (5) The bandgap defined by Eq. (5) varies with δ. The dashed curves in Fig. 1 display the branches of Eq. (5) for δ = 0.7. The bandgap for moving solitons exists in the range 320 Javid Atai and D. Royston Neill − 0.9 − 0.6 − 0.3 0 0.3 0.6 0.9 k 0 80 160 240 320 E n e r g y Anomalous Normal Zero Figure 3. The total energy of gap solitons with c = 1 and γ = 0.3 in the anomalous dispersion (see Ref. [34, 38]), the normal dispersion (see Ref. [39]) and the zero dispersion regime (Eqs. (1) and (2)). δ min < δ < c where δ min is negative and can be obtained numerically. It is also found that the bandgap defined by (5) is completely filled with soliton solutions all of which satisfy VK criterion. In addition, similar to the case of quiescent solitons, the energy of moving solitons in this model is found to be greater than that in the normal dispersion and less than that of moving GSs in the anomalous dispersion regime. 3. Stability of Solitons In this section we investigate the stability of GSs in this model by means of direct numer- ical simulations and linear stability analysis. Evolution of GSs were simulated by numeri- cally solving Eqs. (1) and (2) using the symmetrized split-step Fourier method. Absorbing boundary conditions were implemented in order to attenuate any radiation that reaches the boundaries of computational window. To seed any inherent instability in the system, the GSs found by the above-mentioned relaxation algorithm were initially perturbed asymmet- rically and then propagated. It is found that, the GSs in the model of (1) and (2), like their counterparts in the anomalous and normal dispersion regimes, are unstable against oscilla- tory perturbations. But, in a part of the GS family the instability is weak and as a result solitons may propagate for long distances before the instability is manifested. As a conse- quence, the GSs belonging to this part of the family can be considered as being “practically” stable A key result of Ref. [39] was that GSs in the normal dispersion regime are significantly more stable than their counterparts in the anomalous dispersion. Moreover, it was con- Dynamics and Interactions of Gap Solitons... 321 jectured that the higher degree of stability of GSs in the normal dispersion regime was, at least in part, due to the fact that their total energy was considerably smaller than GSs in the anomalous dispersion. Based on this conjecture and since the total energy of GSs in Eqs. (1) and (2) is greater (smaller) than those in the normal (anomalous) dispersion (see Fig. 3), one expects the GSs in the zero dispersion to be more stable than those in the anomalous and less stable compared to the GSs in the normal dispersion regime. Our simulations corrobo- rate this prediction. A comparison between the propagation of GSs in different dispersion regime is provided in Fig. 4. To quantify the degree of instability of GSs in this model, we have utilized a linear sta- bility analysis to calculate the instability growth rates for small perturbations. Substituting the following perturbed soliton solution {u(z, τ) , v (z, τ)} = {U δ (τ) + f(τ) e σz , V δ (τ) + g (τ) e σz } e kz (6) into the boosted equations (see Section 2.) and linearizing, we arrive at the following eigen- value problem: Ay = σy (7) where U δ (τ) and V δ (τ) are the soliton solutions corresponding to velocity δ and f (τ) and g (τ) are the eigenmodes of the small perturbations and σ is the corresponding complex eigenvalue. y = [f, f , g, g ] T , and A = −ik + 2i |U δ | 2 + D f iU 2 δ i 0 iU 2 δ ik − 2i |U δ | 2 + D f 0 −i i 0 −ik + D g 0 0 −i 0 ik + D g . with D f = D f = (c + δ) d dτ − γ d 3 dτ 3 , and D g = D g = −(c − δ) d dτ . In the above expressions, asterisk represents complex conjugate. The eigenvalue problem posed by Eq. (7) can be solved using standard numerical tech- niques. The results of the stability analysis are summarized in Fig. 5 as graphs of Re(σ) vs. k for c = 1, γ = 0.3 and δ = 0, 0.25 and 0.5. Since the instability growth rate for all the cases is positive the GSs are formally unstable. However, one observes that the growth rates for a part of the family, particularly toward the lower edge of the bandgap, are very small. In this case, the instability will only be observable after a long propagation distance. Solitons exhibiting this character are therefore “virtually stable” objects. We have adopted the defi- nition of Ref. [39] for virtual stability and quasi-stability. That is, for a soliton to be virtually stable it must remain stable for at least 300Z nonlin (where Z nonlin ∼ 1 |U δ | 2 ) and for it to be quasi-stable it must remain stable for propagation distances 50Z nonlin < z < 300Z nonlin . There are a number of noteworthy features in Fig. 5. Firstly, we note that the growth rates in the zero dispersion regime are greater than those in the normal dispersion region (c.f. Fig. 5 in Ref. [39]) and smaller than those in the anomalous dispersion regime (c.f. Figs. 3 and 4 in Ref. [38]). This is consistent with the results of the direct numerical simulations shown in Fig. 4. Secondly, increasing δ gives rise to larger growth rates, particularly for 322 Javid Atai and D. Royston Neill solitons near the upper edge of the bandgap. Nevertheless, varying δ does not have an appreciable effect on the border of stable and quasi-stable regions. The weak dependence of boundary of stable and unstable regions on the velocity of solitons has also been reported for GSs in a FBG [11, 13]. − 60 − 40 − 20 0 20 40 60 τ 48 (a) z 0 − 60 − 40 − 20 0 20 40 60 τ (b) z 0 250 − 60 − 40 − 20 0 20 40 60 τ 1600 0 (c) z Figure 4. Examples of propagation of asymmetrically perturbed quiescent gap soliton corre- sponding to k = −0.4, c = 1, and γ = 0.3 in (a) anomalous dispersion, (b) zero dispersion and (c) normal dispersion. In (c) the initial perturbation causes the soliton to acquire a small velocity. In this figure and all others below, only the u-component is shown. Dynamics and Interactions of Gap Solitons... 323 − 0.8 − 0.4 0 0.4 0.8 k 0 0.1 0.2 0.3 0.4 R e ( σ ) Stable Quasi-Stable (a) − 0.8 − 0.4 0 0.4 0.8 k 0 0.1 0.2 0.3 0.4 0.5 R e ( σ ) Stable Quasi-Stable (b) − 0.8 − 0.4 0 0.4 0.8 k 0 0.2 0.4 0.6 0.8 1 R e ( σ ) Stable Quasi-Stable U n s t a b l e (c) Figure 5. Instability growth rate of GSs in the model of Eqs. (1) and (2) with c = 1 and γ = 0.3 versus k for (a) quiescent gap solitons, (b) moving gap solitons with δ = 0.25 and (c) moving gap solitons with δ = 0.5. In the “Stable” region, the solitons propagate for long distances i.e. z 300Z nonlin without any conspicuous instability development. In the “Quasi-Stable” region, instability occurs in the range 50Z nonlin < z < 300Z nonlin . 324 Javid Atai and D. Royston Neill 4. Interactions and Collisions of Solitons In view of nonintegrability of the model, the collision dynamics and interactions between the solitons may be quite complex. In Refs. [38, 39], the collisions between in-phase GSs in the anomalous and normal dispersion regimes were considered. Moreover, in [39] the interaction of in-phase and π-out-of-phase quiescent GSs in the normal dispersion regime was investigated and it was shown that in the case of π-out-of-phase solitons the outcome of interaction depends on k and the initial separation of solitons. In this section we will investigate the interaction of quiescent solitons in the anomalous and zero dispersion regimes. In addition, the collisions of in-phase and π-out-of-phase moving GSs in anomalous, normal and zero dispersion regimes will also be studied. 4.1. Interactions of Quiescent Solitons in Anomalous and Zero Dispersions The interaction of GSs in zero and anomalous dispersion regimes was simulated by propa- gating two identical quiescent solitons belonging to the “Stable” regions with a time sepa- ration of ∆τ and a phase difference of ∆φ. It is found that, irrespective of dispersion regime, when ∆φ = 0 (i.e. GSs are in-phase), the solitons always repel each other. This behavior was reported for the GSs in the normal dispersion regime (see [39]) . In the case of ∆φ = π, the interaction of solitons becomes dependent on ∆τ and k.These interactions can be divided into three types, denoted here as Types A, B, and C. In the Type A interactions the pulses initially attract each other and collide without merging and then bounce back. An example of this type of interaction in the zero dispersion regime is shown in Fig. 6(a). In the Type B interactions the pulses attract and temporarily merge and form a “lump” which subsequently disintegrates into two separating solitons with different amplitudes and velocities. Fig. 6(b) displays an example this type of interaction. In the Type C interactions the pulses repel each other resulting in two separating pulses with different amplitudes and velocities (Fig. 6(c)). It should be noted that the velocity and amplitude of resulting solitons in the Type C interaction as well as the interaction between in-phase solitons depend on the degree of initial overlap of solitons. If the solitons are initially weakly overlapping then the difference between the velocities and amplitudes of the eventual moving solitons will be small (see for example Fig. 7 and Fig 8(b) in Ref. [39]). On the other hand, increasing the initial overlap between solitons (i.e. reducing ∆τ) leads to generation of solitons whose velocities and amplitudes differ considerably. Fig. 7 displays the regions in the plane of (∆τ,k) where the types A, B and C in- teractions occur in zero and anomalous dispersion regimes. A noteworthy feature in Fig. 7(a) is that in the zero dispersion the boundary between the types C and A is very weakly dependent on the initial separation of solitons. 4.2. Collisions of Moving Solitons In Refs. [38, 39] it was shown that in the anomalous and normal dispersion regimes the collisions between in-phase counterpropagating solitons belonging to the “Stable” region Dynamics and Interactions of Gap Solitons... 325 − 60 − 40 − 20 0 20 40 60 τ 50 0 (a) z − 60 − 40 − 20 0 20 40 60 τ 50 0 (b) z − 60 − 40 − 20 0 20 40 60 τ 50 0 (c) z Figure 6. Examples of interaction of quiescent gap solitons in the zero dispersion regime with c = 1, γ = 0.3, and ∆φ = π. (a) k = −0.64, ∆τ = 10; (b) k = −0.64, ∆τ = 8; (c) k = −0.9, ∆τ = 8. with δ = ±0.5 are always elastic. In particular, It was also found the relative collision induced loss of energy is ≈ 0.1%. In this section the effect of phase and velocity shift on the collisions will be considered. First, we consider the collisions between GSs with initial velocities ±0.25. As shown in Fig. 8, in-phase GSs in different dispersion regimes with δ = ±0.25 bounce off each other elastically. On the other hand, as is displayed in Fig. 9, the π-out-of-phase solitons with δ = ±0.25 collide and merge temporarily and form a “lump” which then quickly breaks 326 Javid Atai and D. Royston Neill 8 9 10 11 12 ∆τ − 0.9 − 0.8 − 0.7 − 0.6 − 0.5 k A A B C (a) 8 9 10 11 12 ∆τ − 0.9 − 0.8 − 0.7 − 0.6 k A B C (b) Figure 7. Regions of different types of interaction in the plane of (∆τ, k) for (a) the zero dispersion and (b) the anomalous dispersion. up into two separating solitons. In addition, the collisions do not generate any noticeable radiation. Figs. 10 and 11 show that the collisions of in-phase and π-out-of-phase solitons with δ = ±0.5 in different dispersion regimes . In this case, regardless of the initial phase Dynamics and Interactions of Gap Solitons... 327 − 60 − 40 − 20 0 20 40 60 τ 50 0 (a) z − 60 − 40 − 20 0 20 40 60 τ 50 0 (b) z − 60 − 40 − 20 0 20 40 60 τ 50 0 (c) z Figure 8. Examples of collisions of moving gap solitons with k = −0.6, c = 1, γ = 0.3, δ = ±0.25 and ∆φ = 0. (a) Anomalous dispersion; (b) zero dispersion; (c) normal dispersion. difference, the solitons collide and form a lump which breaks up into two solitons which travel at almost the same velocity as the initial solitons. The effect of the initial phase difference is that in the case of ∆φ = π the emerging solitons have different velocity shifts compared to those with ∆φ = 0. 328 Javid Atai and D. Royston Neill 5. Conclusion In this article, we have characterized the gap soliton solutions in a recently introduced model in the absence of the second order dispersion. Similar to the anomalous and normal disper- sion regimes, the family of GSs in this case is found to be formally unstable but in a part of the family the instability is very weak and the solitons belonging to that part of the family are therefore virtually stable. Interactions of quiescent solitons and collisions of moving − 60 − 40 − 20 0 20 40 60 τ 50 0 (a) z − 60 − 40 − 20 0 20 40 60 τ 50 0 (b) z − 60 − 40 − 20 0 20 40 60 τ 50 0 (c) z Figure 9. Examples of collisions of moving gap solitons with k = −0.6, c = 1, γ = 0.3, δ = ±0.25 and ∆φ = π. (a) Anomalous dispersion; (b) zero dispersion; (c) normal dispersion. Dynamics and Interactions of Gap Solitons... 329 − 60 − 40 − 20 0 20 40 60 τ 50 0 (a) z − 60 − 40 − 20 0 20 40 60 τ 50 0 (b) z − 60 − 40 − 20 0 20 40 60 τ 50 0 (c) z Figure 10. Examples of collisions of moving gap solitons with k = −0.6, c = 1, γ = 0.3, δ = ±0.5 and ∆φ = 0. (a) Anomalous dispersion; (b) zero dispersion; (c) normal dispersion. solitons in zero, anomalous and normal dispersion regimes are analyzed. Depending on the initial separation and the wavenumber, the solitons may either attract and bounce, attract and merge temporarily and break up into separating solitons, or repel each other. We also find that the outcome of the collisions of moving solitons depends on the initial phase and the velocity shift. In all dispersion regimes, when δ = 0.25, the in-phase solitons collide and bounce off each other elastically whereas the π-out-of-phase solitons collide and form a lump which subsequently disintegrates into two separating solitons. On the other hand, 330 Javid Atai and D. Royston Neill − 60 − 40 − 20 0 20 40 60 τ 50 0 (a) z − 60 − 40 − 20 0 20 40 60 τ 50 0 (b) z − 60 − 40 − 20 0 20 40 60 τ 50 0 (c) z Figure 11. Examples of collisions of moving gap solitons with k = −0.6, c = 1, γ = 0.3, δ = ±0.5 and ∆φ = π. (a) Anomalous dispersion; (b) zero dispersion; (c) normal dispersion. when δ = 0.5, the solitons always collide elastically and two separating solitons emerge. In this case, the velocity shifts of the emerging solitons depend on the initial phase difference. Dynamics and Interactions of Gap Solitons... 331 References [1] W. Chen, D.L. Mills, “Gap solitons and the nonlinear optical response of superlat- tices”, Phys. Rev. Lett. 58, 160-163 (1987). [2] P.St.J. Russel, “Bloch wave analysis of dispersion and pulse propagation in pure dis- tributed feedback structures”, J. Mod. Opt. 38, 1599-1619 (1991). [3] C.M. de Sterke, J.E. Sipe, “Gap solitons”, Prog. Opt. 33, 203-260 (1994). [4] A.B. Aceves, S. Wabnitz, “ Self-induced transparency solitons in nonlinear refractive periodic media”, Phys. Lett. A 141, 37-42 (1989). [5] J.E. Sipe, H.G. Winful, “Nonlinear Schrï¿ 1 2 inger soliton in a periodic structure”, Opt. Lett. 13, 132-133 (1988). [6] D.N. Christadoulides, R.I. Joseph, “Slow Bragg Solitons in Nonlinear Periodic Struc- tures”, Phys. Rev. Lett. 62, 1746-1749(1989). [7] B.J. Eggleton, R.E. Slusher, C.M. de Sterke, P.A. Krug, J.E. Sipe, “Bragg Grating Solitons”, Phys. Rev. Lett. 76, 1627-1630 (1996). [8] B.J. Eggleton, C.M. de Sterke, R.E. Slusher, “Nonlinear pulse propagation in Bragg gratings”, J. Opt. Soc. Am. B 14, 2980-2993 (1997). [9] C.M. de Sterke, B.J. Eggleton, P.A. Krug, “High-intensity pulse propagation in uni- form gratings and grating superstructures”, J. Lightwave Technol. 15, 1494-1502 (1997). [10] D. Taverner, N.G.R. Broderick, D.T. Richardson, R.I. Laming, M. Ibsen, “Nonlinear self-switching and multiple gap-soliton formation in a fiber Bragg grating”, Opt. Lett. 23, 328-330 (1998). [11] B.A. Malomed, R.S. Tasgal, “Vibration modes of a gap soliton in a nonlinear optical medium”, Phys. Rev. E 49, 5787-5796 (1994). [12] A. De Rossi, C. Conti, S. Trillo, “Stability, Multistabiity, and Wobbling of Optical Gap Solitons”, Phys. Rev. Lett. 81, 85-88 (1998). [13] I.V. Barashenkov, D.E. Pelinovski, E.V. Zemlyanaya, “Vibrations and oscillatory in- stabilities of gap solitons”, Phys. Rev. Lett. 80, 5117-5120 (1998). [14] J.T. Mok, C.M. de Sterke, I.C.M. Littler, B.J. Eggleton, “Dispersionless slow light using gap solitons”, Nature Phys. 2, 775-780 (2006). [15] A.R. Champneys, B. A. Malomed and M.J. Friedman, “Thirring Solitons in the Pres- ence of Dispersion”, Phys. Rev. Lett. 80, 4169-4172(1998). [16] W.C.K. Mak, B.A. Malomed and P.L. Chu, “Three-wave gap solitons in waveguides with quadratic nonlinearity”, Phys. Rev. E 58, 6708-6722(1998). 332 Javid Atai and D. Royston Neill [17] W.C.K. Mak, P.L. Chu and B.A. Malomed, Solitary waves in coupled nonlinear waveguides with Bragg gratings“”, J. Opt. Soc. Am. B 15, 1685-1692 (1998); J. Atai and B. A. Malomed, “Bragg-grating solitons in a semilinear dual-core system”, Phys. Rev. E 62, 8713-8718 (2000); J. Atai and B.A. Malomed, “Solitary waves in systems with separated Bragg grating and nonlinearity ”, Phys. Rev. E 64, 066617 [5 pages] (2001). [18] J. Atai and B.A. Malomed, “Families of Bragg-grating solitons in a cubic-quintic medium”, Phys. Lett. A 284, 247-252 (2001); J. Atai, “Interaction of Bragg grating solitons in a cubic-quintic medium”, J. Opt. B: Quant & Semiclass. Opt. 6, S177- S181 (2004) . [19] D. Mandelik, H.S. Eisenberg, Y. Silberberg, R. Morandotti and J.S. Aitchison, “Band- Gap Structure of Waveguide Arrays and Excitation of Floquet-Bloch Solitons”, Phys. Rev. Lett. 90, 053902 (2003); D. Mandelik, H.S. Eisenberg, Y. Silberberg, R. Moran- dotti and J.S. Aitchison, “Gap Solitons in Waveguide Arrays”, Phys. Rev. Lett. 92, 093904 (2003). [20] A. Ferrando, M. Zacares, P. Fernandez de Cordoba, D. Binosi and J. Monsoriu, “Spa- tial soliton formation in photonic crystal fibers”, Opt. Express 11, 452-459 (2003); D. Neshev, A.A. Sukhorukov, B. Hanna, W. Krolokowski and Y.S. Kisvshar, “Controlled Generation and Steering of Spatial Gap Solitons”, Phys. Rev. Lett. 92, 093904 (2004). [21] J.C. Knight, T.A. Birks, P.St.J. Russell, D.M. Atkin, “All-silica single-mode fiber with photonic crystal cladding”, Opt. Lett. 21, 1547-1549 (1996). [22] T.A. Birks, P.J. Roberts, P.St. J. Russell, D.M. Atkin, T.J. Shepherd, “Full 2-D pho- tonic bandgaps in silica/air structures”, Electron. Lett. 31, 1941-1943 (1995). [23] R.F. Cregan, B.J. Mangan, J.C. Knight, T.A. Birks, P.St. J. Russell, P.J. Roberts, D.C. Allan, “Single mode photonic band gap guidance of light in air”, Science 285, 1537- 1539 (1999). [24] P.St. J. Russell, “Photonic crystal fibers”, Science 299, 358-362. [25] F. Benabid, G. Bouwmans, J.C. Knight, P.St.J. Russell, F. Couny, “Ultra-high effi- ciency laser wavelength conversion in gas-filled hollow core photonic crystal fiber by pure stimulated Raman scattering in molecular hydrogen”, Phys. Rev. Lett. 93, 123903 (2004). [26] S. Ghosh, J. Sharping, D.G. Ouzounov, A.L. Gaeta, “Resonant optical interactions with molecules confined in photonic bandgap fibers”, Phys. Rev. Lett. 94, 093902 (2005). [27] F. Benabid, P.S. Light, F. Couny, P.St.J. Russell, “Electromagnetically-induced trans- parency grid in acetylene-filled hollow-core PCF”, Opt. Express 13, 5694 (2005). [28] D.G. Ouzounov, F.R. Ahmad, D. Mller, N. Venkataraman, M.T. Gallagher, M.G. Thomas, J. Silcox, K.W. Koch and A.L. Gaeta, “Generation of Megawatt optical soli- tons in hollow-core photonic band-gap fibers”, Science 301, 1702-1704 (2003); Dynamics and Interactions of Gap Solitons... 333 [29] F. Luan, J.C. Knight, P.St.J. Russell, S. Campbell, D. Xiao, D.T. Ried, B.J. Mangan, D.P. Williams, P.J. Robert, “Femtosecond soliton pulse delivery at 800 nm wavelength in hollow-core photonic bandgap fibers” Opt. Express 12, 835-840 (2004). [30] W. Gobel, A. Nimmerjahn, and F. Helmchen, “Distortion free delivery of nanojoule femtosecond pulses from Ti:sapphire laser through a hollow-core photonic crystal fiber” Opt. Lett. 29, 1285-1287 (2004). [31] H. Lim and F.W. Wise, “Control of dispersion in a femtosecond ytterbium laser by use of hollow-core photonic bandgap fiber”, Opt. Express 12, 2231-2235 (2004). [32] C.M. Smith, N. Venkataraman, M.T. Gallagher, D. Mller, J.A. West, N.F. Borrelli, D.C. Allan and K.W. Koch, “Low-loss hollow-core silica/air photonic bandgap fibre”, Nature 424, 657-659 (2003). [33] K. Saitoh, N. A. Mortensen, M. Koshiba, “Air-core photonic band-gap fibers: the impact of surface modes”, Opt. Express 12 (2004) 394. [34] I.M. Merhasin and B.A. Malomed, “Gap solitons in a model of a hollow optical fiber”, Opt. Lett. 30, 1105-1107 (2005). [35] D.V. Skryabin, “Coupled core-surface solitons in photonic crystal fibers”, Opt. Ex- press 12, 4841-4846 (2004). [36] S. Wabnitz, “Forward mode coupling in periodic nonlinear-optical fibers: modal dis- persion cancellation and resonance solitons”, Opt. Lett. 14, 1071-1073 (1989). [37] G. van Simaeys, S. Coen, M. Haelterman and S. Trillo, “Observation of Resonance Soliton Trapping due to a Photoinduced Gap in Wave Number”, Phys. Rev. Lett. 92, 223902 (2004). [38] J. Atai, B.A. Malomed, I.M. Merhasin, “Stability and collisions of gap solitons in a model of a hollow optical fiber”, Opt. Comm. 265, 342-348 (2006). [39] D.R. Neill, J. Atai, “Gap solitons in a hollow optical fiber in the normal dispersion regime”, Phys. Lett. A (in press). [40] M.G. Vakhitov, A.A. Kolokolov, “Stability of stationary solutions of nonlinear wave equations”, Radiophys. Quantum Electron. 16, 783 (1973). In: Optical Fibers Research Advances ISBN: 1-60021-866-0 Editor: Jurgen C. Schlesinger, pp. 335-353 © 2007 Nova Science Publishers, Inc. Chapter 13 MULTIWAVELENGTH OPTICAL FIBER LASERS AND SEMICONDUCTOR OPTICAL AMPLIFIER RING LASERS Byoungho Lee * and Ilyong Yoon School of Electrical Engineering, Seoul National University Gwanak-Gu Sinlim-Dong, Seoul 151-744, Korea Abstract We review various schemes for multiwavelength fiber lasers and semiconductor optical amplifier (SOA) ring lasers. Multiwavelength fiber lasers have applications in wavelength division multiplexing (WDM) optical communication systems, optical fiber sensors and optical spectroscopy. Erbium-doped fiber amplifiers (EDFAs), Raman amplifiers and SOAs are mainly used as gain media for multiwavelength fiber lasers. Because EDFAs are homogeneously broadened gain media, various methods have been researched to enable the multiwavelength generation. Due to the introduction of liquid nitrogen cooling, four-wave mixing, frequency shifted feedback, and so on, multiwavelength erbium-doped fiber lasers could become realized. On the other hand, because SOA and Raman amplifiers are gain media with inhomogeneous broadening, multiwavelength generation is relatively easy. The useful features of the multiwavelength lasers are mainly dependent on a comb filter. One of the most important features of multiwavelength lasers is tunability. The tunability of wavelengths and channel spacing is required for WDM optical communication systems. Much research has been conducted to enable implementation of tunable multiwavelength fiber lasers. Various comb filters such as Fabry-Perot filters, fiber Bragg gratings, and polarization-maintaining fiber loop mirrors can be used for multiwavelength fiber lasers. We review several schemes for multiwavelength SOA-fiber and Raman fiber lasers in this chapter. * E-mail address:
[email protected]. Tel: +82-2-880-7245, Fax: +82-2-873-9953 Byoungho Lee and Ilyong Yoon 336 1. Introduction The realization of the laser has made many applications possible. Among those applications, a light source for optical communication systems is one of the most important applications. As wavelength-division-multiplexing (WDM) optical communication systems have become more developed, multiwavelength light sources have also been widely researched. In the first stage, multiwavelength lasers could be made as a simple structure consisting of the array of lasers and a multiplexer [1, 2]. However, there have been difficulties with these lasers such as large insertion loss and bulky size. Therefore, multiwavelength fiber lasers using a single gain medium are desired. There are many possible gain media for optical communication such as erbium-doped fiber (EDF), semiconductor optical amplifier (SOA), and stimulated Raman scattering (SRS). In this chapter we review a wide variety of multiwavelength fiber lasers employing a single gain medium. Because EDF is a homogeneously broadened gain medium, a laser using EDF normally lases at a single wavelength. Various methods have been researched to enable the multiple wavelength generation, such as the introduction of liquid nitrogen cooling, four-wave mixing, frequency shifted feedback, and so on. For the multiwavelength EDF laser (EDFL), schemes to suppress mode competition are a main subject. On the other hand, Raman amplifiers and SOAs are inhomogeneously broadened gain media. Therefore, multiwavelength generation is relatively simple compared with an EDFL. Many methods have also been proposed to implement multiwavelength lasers using these technologies. One of the useful characteristics of multiwavelength fiber lasers is tuning capability. The tunability of lasing wavelengths and channel spacing is required for WDM optical communication systems. Therefore, much research has been conducted for the implementation of tunable multiwavelength fiber lasers. For multiwavelength SOA-fiber and Raman fiber lasers, the schemes for tuning and these lasers’ characteristics are main subjects. We classify and review many schemes for the design of tunable fiber lasers in this chapter. 2. Multiwavelength Fiber Lasers Using EDFA The most challenging difficulty of an EDF amplifier (EDFA) for a multiwavelength laser is that the EDFA is a homogeneously broadened medium. In a homogeneously broadened medium, all atoms in the excited state have the same gain spectrum. Therefore, when a laser employs a homogeneously broadened gain medium, only the wavelength which has the largest net gain (gain minus cavity loss) can survive. The other wavelengths decay due to loss. When a number of wavelengths are in a cavity, each channel experiences mode competition. However, because Er 3+ ions are surrounded by a glass host, the interaction with the silica and other dopants leads to some degree of inhomogeneous broadening contribution. Therefore, the schemes for the multiwavelength EDFL involve increasing of the competitiveness of weak wavelengths or decreasing of the homogeneously broadened linewidth of the EDFA. Multiwavelength Optical Fiber Lasers and Semiconductor Optical Amplifier… 337 2.1. Cavity Loss Balancing The first multiwavelength operation of an EDFL was demonstrated in 1992 [3]. The cavity losses of the lasing wavelengths are carefully controlled to suppress single channel lasing as shown in Fig. 1. The cavities of lasing wavelengths are separated and the losses of cavities are controlled independently so that many wavelengths can lase. This is equivalent to the net gain flattening. There is no dominant wavelength due to the flattened net gain. However, this method requires a careful control of cavity losses. Thus it can be easily expected that lasers employing this scheme are relatively unstable and sensitive to environmental conditions. Gain medium W D M PC FLM FLM λ 1 , λ 2 , …, λ 8 λ 1 λ 2 λ 8 (a) W D M λ 1 λ 8 W D M Gain medium Polarizer Isolator λ 1 , λ 2 , …, λ 8 (b) Figure 1. (a) An eight-channel laser configuration based on a linear cavity. (b) An eight-channel laser configuration based on a ring cavity (PC: polarization controller, FLM: fiber loop mirror, WDM: wavelength division multiplexer) [3]. Byoungho Lee and Ilyong Yoon 338 2.2. Liquid Nitrogen Cooling When an EDFA is cooled, the homogeneous linewidth of the EDFA is narrowed. Spectral hole burning and homogeneous linewidth were measured as a function of temperature in Ref. [4]. The homogeneous linewidth was measured as 1.3 nm at 61 K. It was shown that the homogeneous linewidth exceeded 11.5 nm at room temperature. For a multiwavelength application of an EDFA, there was other research to make an inhomogeneously broadened EDFA by liquid nitrogen cooling [5]. In Ref. [5], the main idea was the suppression of dynamic crosstalk between adjacent channels. The liquid nitrogen cooling made an 11 dB suppression of crosstalk. The channel spacing was 4 nm and the homogeneous linewidth was measured as ~1 nm. Comb filter Isolator Doped fiber 77K 75:25 coupler Output WDM coupler Pump Figure 2. A multiwavelength EDF ring laser configuration using a comb filter in the cavity [6]. The first multiwavelength EDFL by liquid nitrogen cooling (77 K) was presented in 1996 [6]. Figure 2 shows the configuration. 11 stable laser lines were demonstrated with 0.65 nm channel spacing around 1535 nm. Two types of comb filters were used in the experiment. Those were a chirped fiber Bragg grating (CFBG) Fabry-Perot filter and a sampled grating. 2.3. Four-Wave Mixing The self-stabilizing effect of four-wave mixing (FWM) can be used for a multiwavelength EDFL. The powers of lasing wavelengths are automatically balanced by several degenerated FWMs, 1 2 3 2ω ω ω = + , or nondegenerated FWMs, 1 2 3 4 ω ω ω ω + = + . For a phase matching condition, dispersion-shifted fiber (DSF) or photonic crystal fiber (PCF) are required. The self-stabilizing effect may be described as a “photonic Robin Hood.” This means that FWM takes the energy of a rich wavelength and gives it to a poor wavelength. Because power is transferred between wavelengths by the nonlinear process, this scheme can be thought of as automatic net gain equalization. Multiwavelength Optical Fiber Lasers and Semiconductor Optical Amplifier… 339 PC 1 PC 2 VOA 1 VOA 3 VOA 4 VOA 2 FBG 1 FBG 3 FBG 2 FBG 4 3-dB coupler 3 - d B c o u p l e r HN-PCF PC 3 Output 980nm Pump laser diode WDM coupler EDF Output F u s e d c o u p l e r 1 F u s e d c o u p l e r 2 Figure 3. An experimental setup for four-wavelength EDFL (HN-PCF: highly nonlinear photonic crystal fiber, VOA: variable optical attenuator, FBG: fiber Bragg grating, PC: polarization controller) [7]. Liu and Lu demonstrated a four-wavelength EDFL using a highly nonlinear PCF to suppress the mode competition at room temperature as shown in Fig. 3 [7]. Experimental results showed lasing wavelengths of 1540.28, 1543.58, 1546.79 and 1550.08 nm. EDFA Output 10 90 AWG(1xn) PC FBG 1 FBG n λ 1 λ 2 λ n Isolator DSF PC Figure 4. A schematic of the multiwavelength EDFL based on degenerate four-wave mixing in the DSF (EDFA: erbium-doped fiber amplifier, FBG: fiber Bragg grating, AWG: arrayed waveguide grating, PC: polarization controller) [8]. Byoungho Lee and Ilyong Yoon 340 Han et al. also presented similar multiwavelength EDFLs by using a DSF [8, 9]. Figure 4 shows a schematic diagram of the multiwavelength EDFL employing multiple fiber Bragg gratings (FBGs) and a 1 km DSF for 10 channels’ lasing with 0.8 nm channel spacing. In addition, they showed channel spacing tunability through the elimination of the effects of several FBGs. Output 10 90 Isolator DSF PC 1 980nm pump laser diode EDF PC 3 PC 4 PC 2 PMF 1 L 1 PMF 2 L 2 Lyot-Sagnac filter Figure 5. Schematics of the multiwavelength EDFL using DSF and Lyot-Sagnac filter (PMF: polarization-maintaining fiber, EDF: erbium-doped fiber, PC: polarization controller) [9]. A tunable multiwavelength EDFL was demonstrated in 2005 [9]. The tunability originated from a tunable Lyot-Sagnac filter as shown in Fig. 5. The wavelength spacing of the two-segment Lyot-Sagnac filter was [ ] 2 1 2 / ( ) n L L λ λ Δ = Δ ⋅ ± , where n Δ was the effective birefringence between two orthogonal polarization modes and 1 2 , L L were the lengths of the two polarization-maintaining fibers (PMFs) shown in Fig. 5. Thus, the channel spacing was switchable by polarization control. In the Lyot-Sagnac filter, clockwise and counterclockwise lights experienced optical path difference due to PMF segments. Therefore, the optical path difference between two lights led to comb-like filter characteristics. The birefringence of the two PMF segments may be summed or subtracted depending on the state of the polarization controllers (PCs). Experimental results showed 11 laser lines with 1 nm spacing and 17 laser lines with 0.8 nm spacing. Stable lasing characteristics and tuning capability were obtained due to FWM. Multiwavelength Optical Fiber Lasers and Semiconductor Optical Amplifier… 341 2.4. Frequency Shifting Technique Another scheme for a multiwavelength EDFL is a frequency shifted feedback scheme [10]. Figure 6 is a schematic diagram of the multiwavelength EDFL. In the scheme, an acousto- optic modulator (frequency shifter) shifts the frequency of light by 100 MHz for each round trip. This prevents single frequency lasing. Experimental results showed stable ~13 laser lines with 0.8 nm spacing. A Fabry-Perot etalon with a CFBG or sampled grating was used for the periodic filter. The experimental results showed good agreement with the simulation results. EDFA 1 EDFA 2 3-dB coupler Isolator Output Frequency shifter Periodic filter Figure 6. A schematic diagram of the multiwavelength EDFL employing a frequency shifted feedback scheme (EDFA: erbium-doped fiber amplifier) [10]. Table 1. Multiwavelength EDFLs Year First author [Reference] Comments Channel number Channel spacing (nm) Scheme 1992 N. Park [3] The first multiwavelength EDFL 6 4.8 Cavity loss balancing 1996 J. Chow [6] 11 0.65 Liquid nitrogen cooling 2000 A. Bellemare [10] ~13 0.8 Frequency shifting 2002 R. Slavik [11] High uniformity 18 0.8 Frequency shifting 2005 X. Liu [7] 4 3.3 Four-wave mixing 2005 Y.-G. Han [9] Channel spacing switching 17, 11 0.8, 1 Four-wave mixing 2006 Y.-G. Han [8] Channel spacing switching 10 0.8, 1.6, 2.4 Four-wave mixing Byoungho Lee and Ilyong Yoon 342 For the frequency shifting technique, a more in-depth study was published in 2002 [11]. These researchers improved the uniformity of the lasing wavelengths in the EDFL. Uniform 18 laser lines with 0.8 nm channel spacing were obtained in the experiments. Similar to the frequency shifted feedback scheme, the phase-modulation feedback scheme was also presented [12, 13]. A LiNbO 3 phase modulator was used for phase modulation. In Ref. [12], the sawtoothed and sinusoidal phase modulation of a few tens of kHz generated a multiwavelength operation. In Ref. [13], the authors reported that sinusoidal, sawtoothed, triangular and square waveforms are all suitable for multiwavelength lasing. They also indicated that the phase modulation of 500 Hz to a few tens of kHz is good. Important features of the above multiwavelength EDFLs are shown in Table 1 as a summary. 3. Multiwavelength Fiber-SOA and Fiber-Raman Lasers In a SOA, the gain medium is a semiconductor and not a single atom or ion. The recombination of electron-hole pairs makes spontaneous or stimulated emission. The SOA is electrically pumped. More electrons in the conduction band and more holes in the valance band lead to higher gain. The gain spectrum of the SOA depends on materials and structure. The intrinsic inhomogeneous broadening is an advantage of the SOA in its application to multiwavelength lasers. High gain per unit length and compact size are other advantages. On the other hand, the rectangular structure leads to a coupling loss for optical fiber and polarization-dependent gain. The fast carrier lifetime (~200 ps) leads to cross saturation and stronger nonlinear processes. SRS is an interaction between photon energy and molecular vibrational energy (optical phonon). The amplification is performed by the energy transfer from a pump beam to the signal beam (or light to lase). Unlike that of the EDFA and the SOA, Raman scattering does not require a population inversion for amplification. Very broad gain bandwidth is the main characteristic of the Raman amplification process. In the SRS, specific resonant frequency does not exist in contrast to the EDFA and the SOA. The wavelength of the pump beam determines the location of the gain spectrum which has a peak at 13.2 THz off the pump wavelength. It is a main advantage of a Raman amplifier that a specific gain medium is not required, i.e., amplification occurs in a common optical fiber. Therefore, lumped or distributed schemes are all possible. If several pump wavelengths are used properly, a flat gain over a wide bandwidth can be obtained [14]. Because Raman scattering is a weak effect, the SRS requires very high pump power (typically a few Watts) and a long length of fiber. Intrinsic inhomogeneous linewidth broadening is very attractive for a multiwavelength laser. In the SOA and Raman fiber lasers, multiwavelength generation is relatively easy because of their inhomogeneous broadening. Therefore, tunable capability has been a main subject of research involving these multiwavelength SOA and Raman fiber lasers. Tunability was even considered in the demonstration of the first multiwavelength Raman fiber laser [15, 16]. Thus, in this section, we focus on the tunable capability of SOAs and Raman fiber lasers. To avoid confusion, we use two different terminologies: switchability and tunability. Switchability and tunability denote discrete tuning and continuous tuning, respectively. Thus a wavelength switchable laser means a laser which can shift the spectral position of lasing wavelengths by some discrete steps. Multiwavelength Optical Fiber Lasers and Semiconductor Optical Amplifier… 343 3.1. Wavelength or Channel Spacing Switchability By using a sampled high-birefringence (Hi-Bi) fiber grating as a switchable comb filter, the wavelength switchable laser was demonstrated by Yu et al. as shown in Fig. 7 [17]. We can think of this as if two different sampled FBGs (SFBGs) are used due to the difference of the refractive indexes in the fast and slow axes of the Hi-Bi fiber. The control of a rotatable polarizer is equivalent to the selection of one of two SFBGs. In a SFBG, the center Bragg wavelength is 2 B eff n λ = Λ and the wavelength separation is 2 / 2 B eff n p λ λ Δ = , where eff n is the effective refractive index of the fiber core, Λ the individual grating pitch, and p is the sampling period. Therefore, while λ Δ is maintained at nearly the same value, it is possible to move only the center Bragg wavelength. Because the birefringence n Δ is of the order of 10 -4 , the channel spacing is hardly influenced by the choice of polarization axis. However, if we control the polarization of light incident on the sampled grating by using the rotatable polarizer, transmission peaks can be shifted by 2 n Δ Λ. The experimental result showed an interleaving characteristic. The laser output was shifted by 0.4 nm with the 0.8 nm channel spacing fixed. It had a disadvantage in that the number of switchable wavelength set was intrinsically limited to two. The amount of switchable wavelengths was determined by the choice of PMF. Thus, the maximum switchable range was limited by the birefringence of the PMF. PC SOA Isolator Isolator Polarizer Output Variable coupler SMF Sampled Hi-Bi fiber grating Figure 7. A schematic diagram of a wavelength switchable SOA-fiber ring laser employing sampled Hi- Bi FBG (SMF: single mode fiber) [17] Lee et al. presented wavelength switchable SOA fiber lasers employing two SFBGs [18] and a reflection type interleaver [19]. The former used two SFBGs connected to a polarization beam splitter (PBS) for waveband switching as shown in Fig. 8. A rotatable linear polarizer selected one of the two SFBGs. Contrary to the work by Yu et al. [17], the amount of waveband switching depended on the design of the SFBGs. In other words, the amount of wavelength shift was not limited by the choice of a PMF. The experimental result in Fig. 9 showed that 5 laser lines with 0.8 nm spacing could be switched by a spectral displacement of 10 nm. Byoungho Lee and Ilyong Yoon 344 SOA Isolator PC 75: 25 coupler Output Rotatable liner polarizer Light absorber Sampled fiber Bragg gratting 1 Sampled fiber Bragg gratting 2 PBS Figure 8. A schematic diagram of the waveband-switchable SOA-fiber laser using two SFBGs (PBS: polarization beam splitter, PC: polarization controller) [18]. 1535 1540 1545 1550 1555 1560 1565 -40 -30 -20 -10 0 Wavelength [nm] O p t i c a l p o w e r [ d B m ] Figure 9. Output optical spectra showing the waveband switching operation. Multiwavelength Optical Fiber Lasers and Semiconductor Optical Amplifier… 345 SOA Isolator Isolator 75: 25 coupler Output 50:50 coupler PC 1 PC 2 PMF 1 2 Figure 10. A schematic diagram of a SOA-fiber laser employing a reflective type interleaver (PMF: polarization maintaining fiber, PC: polarization controller) [19]. L1 L2 50:50 coupler λ/4 λ/4 λ/2 λ/2 λ/2 λ/2 a b c Tunable filter WDM 2 WDM 1 Raman fiber Pump laser Output Figure 11. An experimental setup for a tunable Raman fiber ring laser (WDM: wavelength division multiplexer) [21]. Another scheme using a reflective interleaver is shown in Fig. 10 [19]. The interleaver is composed of a PBS and a PMF loop mirror. A PC in the PMF loop mirror consists of two quarter-wave plates. The control of waveplates leads to an interleaving characteristic. The feature of this filter is that the transmission and reflection characteristics show interleaved sets of multiple wavelength peaks. Theoretically, for transmission, the filter has infinite Byoungho Lee and Ilyong Yoon 346 channel isolation and 3 dB insertion loss. On the other hand, for reflection, the filter shows 3 dB channel isolation and 0 insertion loss. 17 wavelengths were generated with 0.8 nm channel spacing. The laser lines could be shifted by 0.4 nm with channel spacing fixed. The PMF Lyot-Sagnac filter was also used for a multiwavelength SOA-fiber laser [20]. With the PMF Lyot-Sagnac filter, the SOA-fiber laser could have channel spacing switchability. In addition, the rotation of a quarter-wave plate made the lasing wavelength shift. Channel spacing switchability from 0.8 nm to 4.1 nm was demonstrated. Laser lines from 5 to 20 were observed. Continuous wavelength tuning was also shown. There have also been intensive research efforts for tunable multiwavelength Raman fiber lasers. Kim et al. demonstrated a multiwavelength Raman fiber ring laser with switchable channel spacing and a tunable lasing wavelength [21]. The multiwavelength source was composed of a Raman fiber and a Lyot-Sagnac filter as shown in Fig. 11. The experimental results showed a multiwavelength generation of up to 20 laser lines with 0.43 nm spacing. PMF 2 PMF 1 Lyot-Sagnac filter PC 2 (λ/2) PC 1 (λ/2) Fiber grating 97% Raman gain fiber Fiber grating 90% Pump combiner Output WDM coupler Pump Pump laser Figure 12. An experimental setup for a tunable multiwavelength Raman laser based on an FBG cavity incorporating PMF Lyot-Sagnac filter (PC: polarization controller, WDM: wavelength division multiplexer) [22]. Han et al. demonstrated a multiwavelength Raman laser with a similar filter as shown in Fig. 12 [22]. Although a similar PMF Lyot-Sagnac filter was used, there were two differences: a linear cavity structure employing FBGs and the use of PMFs with different birefringences. In the experimental result, the multiwavelength laser generated 7 channels with 0.6 nm spacing and 5 channels with 0.8 nm spacing. A phase modulator loop mirror filter (PM-LMF) could be used for wavelength switchability [23]. The PM-LMF is a sort of PMF loop mirror where a phase modulator is inserted. Because DC bias, RF power, or modulation frequency changes the birefringence of the phase modulator slightly, the spectral comb position can be shifted while the channel spacing is fixed. Experimentally, 21 laser lines with a 0.8 nm channel spacing were obtained. Multiwavelength Optical Fiber Lasers and Semiconductor Optical Amplifier… 347 Optical amplifier 10% Output Coupler Programmable Hi-Bi FLM Output Input Residual pump power DCF 6.9 km Pump laser WSC Isolator In Out Combiner 3 dB coupler PC 1 PC 2 PC n Hi-Bi, L 1 Hi-Bi, L 2 Hi-Bi, L n 2×2 switch Figure 13. A schematic of a multiwavelength laser and a programmable Hi-Bi fiber loop mirror (FLM) (DCF: dispersion compensation fiber, WSC: wavelength selection coupler) [24]. Chen demonstrated channel spacing switchable fiber lasers by using a programmable Hi- Bi fiber loop mirror as shown in Fig. 13 [24]. He employed a SOA or a Raman amplifier as an optical amplifier. Although many switchable sections are theoretically possible in Fig. 13, the two sections of the PMF were demonstrated. The use of 2×2 switches changes the combination of the PMF section more flexibly. The channel spacing expression is essentially equivalent to that of the PMF Lyot-Sagnac filter except that it has a more flexible birefringence combination. The experimental result showed 3.2 nm and 1.6 nm channel spacing switching. A tunable Raman fiber laser employing an electro-optical tuning scheme was presented in 2004 [25]. The comb filter uses an electro-optic polarization controller (EOPC) inserted in the PMF Sagnac loop filter. The PMF Sagnac loop filter is sometimes called a Lyot-Sagnac filter. Due to the PMF Sagnac loop filter, the channel spacing was switchable. When a driving voltage was applied to the EOPC, the additional birefringence was induced. This effect is equivalent to changing the length of the PMF slightly. Therefore, lasing wavelengths can be shifted as channel spacing is nearly fixed. This Raman laser had both channel spacing switchability and wavelength tunability. Experimental results indicated channel spacing switchability between 0.95 and 2.95 nm. Interleaved switching operation of 11 laser lines was also demonstrated with 0.88 nm channel spacing. 3.2. Wavelength Tunability In a SOA fiber laser, wavelength tuning possibility was presented in 2001 [26]. In fact, the work in Ref. [26] was about a multiplexed sensor. The sensor was a SOA ring laser using a transmission-type filter consisting of a circulator and multiple FBGs. Eight laser lines were Byoungho Lee and Ilyong Yoon 348 observed when 8 FBGs were used. The center wavelengths of the FBGs were 1534.44, 1543.68, 1546.32, 1549.38, 1552.5, 1554.06, 1556.28 and 1558.92 nm. The strain on each FBG changed each lasing wavelength. Therefore, the sensor was indeed a sort of a tunable laser although it was not clarified in the reference. Isolator Isolator PC GC- SOA PBS HWP2 HWP1 QWP PMF Port 2 Port 1 Output 75:25 coupler CW CCW Figure 14. The schematic diagram of a multiwavelength SOA-fiber ring laser employing a PDLC (GC- SOA: gain-clamped semiconductor optical amplifier, HWP: half-wave plate, QWP: quarter-wave plate, PMF: polarization-maintaining fiber, PBS: polarization beam splitter, PC: polarization controller, CW: clockwise, CCW: counterclockwise) [27]. 1545 1550 1555 1560 1565 1570 1575 -50 -40 -30 -20 -10 O p t i c a l p o w e r [ d B m ] Wavelength [nm] 1557 1558 1559 1560 1561 1562 1563 -50 -30 -10 O p t i c a l p o w e r [ d B m ] Wavelength [nm] 0 ° T h e a n g l e o f H W P 1 -50 -30 -10 2 0 ° -50 -30 -10 4 0 ° -50 -30 -10 6 0 ° -50 -30 -10 8 0 ° (a) (b) Figure 15. (a) The output spectrum of a multiwavelength laser (b) The tuning characteristic. Multiwavelength Optical Fiber Lasers and Semiconductor Optical Amplifier… 349 Yoon et al. published their research on the wavelength tunable SOA fiber ring laser [27]. They employed a polarization-diversity loop configuration (PDLC) comb filter as shown in Fig. 14. The PDLC comb filter consists of two half-wave plates, a quarter-wave plate, a polarization beam splitter, and a PMF. The PDLC filter can tune lasing wavelengths with the control of a half-wave plate alone. A single polarization which comes out of the PBS enters the PMF after passing through wave plates. The light experiences polarization change according to its wavelength by the PMF. When the light meets the PBS again, the transmittivity is determined by its polarization. The role of the wave plates is to make the polarization-change by the rotation of the first half-wave plate reproduce the polarization- change by the PMF. When we represent polarization change in the Poincare sphere, the trajectory of polarization change becomes a circle. Because the rotation of the first half-wave plate makes the trajectory rotate, the rotation of the first half-wave plate can shift the position of the filter comb. In the PDLC filter, clockwise and counterclockwise lights exist. Two counter propagating lights experience the same transmittivity and reflectivity. A 90° rotation of the angle of the first HWP corresponds to the sweep of the entire channel spacing. The channel spacing is determined by the length and birefringence of the PMF. In the experiment, 18 laser lines were observed with 0.8 nm channel spacing. Figs. 15 (a) and (b) show the output spectrum and tuning characteristics. The rotation of a half-wave plate can shift the position of lasing wavelengths linearly. Few-mode Bragg grating PC Raman fiber (SMF 50 km) Tunable chirped FBG Output Pump combiner 1425nm14351455 1465 Pump laser Pump (a) d Flexible metal plate Compression (Negative bending) Tension (Positive bending) (b) Figure 16. (a) An experimental setup for a multiwavelength Raman fiber laser based on few-mode FBGs (b) A tuning method based on the symmetrical bending of a flexible metal plate (FBG: fiber Bragg grating, PC: polarization controller) [28]. Byoungho Lee and Ilyong Yoon 350 For a wavelength-tunable Raman fiber laser, Han et al. demonstrated the few-mode FBG scheme as shown in Fig. 16 [28]. Because the few-mode FBG has multiple resonant wavelengths, a multiwavelength Raman fiber laser could be obtained without additional multichannel filters. The CFBG is used to form a linear cavity because of the broad reflection spectrum. Tuning of the CFBG is needed to match the reflection spectrum to that of a few-mode FBG. The lasing wavelength shift ( λ Δ ) can be defined as (1 ) p λ ρ ελ Δ = − , where p λ is the lasing wavelength of the Raman fiber laser, ρ is the photo-elastic coefficient, and ε is the strain induced by the bending of the fiber. The experimental results showed 3 laser lines with 3.5 nm spacing and the wavelength tuning characteristics. The number of laser lines was limited by the few-mode FBG. Han et al. also demonstrated temperature tuning of the lasing wavelength of a multiwavelength Raman laser using the few-mode FBG [29]. Three laser lines were also obtained and the temperature sensitivity was measured as 10.5 pm/°C. They also applied a similar structure to a temperature and strain sensor using a multiwavelength Raman laser with a phase-shifted FBG [30]. The experimental result showed that two lasing wavelengths could be shifted by strain and temperature with a fixed spacing. 3.3. Channel Spacing Tunability Dong et al. presented a multiwavelength SOA-fiber laser and Raman fiber laser employing a fiber Fabry-Perot filter based on a superimposed CFBG [31, 32]. Two super imposed CFBGs form the Fabry-Perot filter when the writing positions of the two CFBGs are slightly different. The tuning of the chirp rate changes the channel spacing. The SOA-fiber laser and Raman fiber laser could be implemented by using the same filter. The SOA-fiber laser generated 10~13 laser lines with 0.3~0.6 nm channel spacing and the Raman fiber laser generated 2~10 laser lines with 0.3~0.6 nm channel spacing. 3.4. Both Wavelength and Channel Spacing Tunability Roh et al. demonstrated a SOA-fiber laser with both wavelength and channel spacing tunability [33]. They employed a PDLC comb filter with a differential delay line (DDL) as shown in Fig. 17. The use of the DDL instead of a PMF could lead to spacing tunability as well as wavelength tunability. Both the channel spacing and lasing wavelength are continuously tunable. Channel spacing is tuned electrically and wavelength is tuned by the rotation of a half-wave plate. Because this laser adopted a PDLC comb filter, the wavelength tuning characteristics are totally equivalent to Ref. [27]. Experimental results showed channel spacing tunability of 0.4 ~ 1.6 nm with up to 23 laser lines. Multiwavelength Optical Fiber Lasers and Semiconductor Optical Amplifier… 351 PC Isolator SOA Isolator Output CW DDL QWP HWP1 HWP2 CCW 1 2 75:25 coupler S F θ h1 S F θ q S F θ D S F θ h2 PBS Figure 17. The schematic diagram of the channel spacing and wavelength tunable SOA-fiber ring laser (DDL: differential delay line, QWP: quarter-wave plate, HWP: half-wave plate, PBS: polarization beam splitter, PC: polarization controller, CW: clockwise, CCW: counterclockwise) [33] The features of the demonstrated SOA-fiber lasers and Raman fiber lasers are summarized in Tables 2 and 3. Table 2. Multiwavelength SOA-fiber lasers Year First author [Reference] Advantage Channel number Channel spacing (nm) Scheme 2001 S. Kim [26] 8 Fiber Bragg grating 2003 B.-A. Yu [17] Wavelength switching 4 0.8 Sampled fiber Bragg grating 2004 L. R. Chen [24] Spacing tuning 11, 6 1.6, 3.2 Programmable Hi- Bi FLM 2004 Y. W. Lee [19] Waveband tuning 17 0.8 Hi-Bi FLM 2005 M. P. Fok [23] Wavelength tuning 21 0.8 PM-LMF 2005 Y.-G. Han [20] Spacing and waveband switching 5~20 0.8~4.1 PMF Lyot-Sagnac filter 2005 X. Dong [31] Spacing tuning 13 0.4 Fabry-Perot 2006 I. Yoon [27] Wavelength tuning 18 0.8 PDLC 2006 S. Roh [33] Wavelength and channel spacing tuning 23 0.8 PDLC with DDL Byoungho Lee and Ilyong Yoon 352 Table 3. Multiwavelength Raman fiber lasers Year First author [Reference] Advantage Channel number Channel spacing (nm) Scheme 2001 F. Koch [15] Potential angle tuning 24 0.8 Fabry-Perot 2001 C. J. S. de Matos [16] Potential individual tuning 4 ~4.5 FBG 2003 C.-S. Kim [21] Channel spacing switching and wavelength tuning 20 0.4~3 Sagnac 2004 C.-S. Kim [25] Channel spacing switching and wavelength tuning 11 0.95, 0.88, 2.95 PMF Sagnac loop filter with electro-optic polarization controller 2004 Y.-G. Han [22] Channel spacing switching 7, 5 0.6, 0.8 PMF Lyot-Sagnac 2005 Y.-G. Han [28] Wavelength tuning 3 3.5 Few-mode fiber 2005 Y.-G. Han [30] Sensing (wavelength tuning) 2 1.4 Phase-shifted fiber 2006 X. Dong [32] Channel spacing tuning 2~10 0.3~0.6 Sample fiber Bragg grating 4. Conclusion As multiwavelength light sources become more important in WDM optical communication, there is an increasing amount of research on the multiwavelength fiber lasers. In this chapter we reviewed various schemes for a multiwavelength fiber laser to date. Feasible gain media are the EDFA, the SOA and the SRS. Each of these schemes has different challenging difficulties for multiwavelength generation. For the EDFA, the most difficult problem is its homogeneous broadening. The unstable lasing characteristic of the EDFA results from homogeneous broadening. As possible schemes to overcome homogeneous broadening, we reviewed the techniques of cavity loss balancing among wavelengths, self stabilization from FWM, liquid nitrogen cooling and frequency shifted feedback. On the other hand, inhomogeneous broadening of a SOA and Raman amplifier makes multiwavelength generation easy. The tunability in lasing wavelength and channel spacing becomes more important as optical communication systems become more flexible and efficient. We focused on the challenging issues of tuning multiwavelength SOAs and Raman fiber lasers. Practically, tunable channel spacing and tunable lasing wavelengths are required features in WDM optical communication systems. Tuning capabilities can be classified into switchability and tunability. Today’s tunable SOA-fiber and Raman fiber lasers were classified and reviewed. Because the tunable characteristic originates from characteristics of a filter, various filters for tunable lasers were introduced. Multiwavelength Optical Fiber Lasers and Semiconductor Optical Amplifier… 353 References [1] Hamakawa. A.; Kato, T.; Sasaki, G.; Shigehara, M. Conf. Opt. Fiber Comm. 1997, 297-298. [2] Takahashi, H.; Toba, H.; Inoue, Y. Electron. Lett. 1994, 30, 44-45. [3] Park, N.; Dawson, J. W.; Vahala, K. J. IEEE Photon. Technol. Lett. 1992, 4, 540-541. [4] Desurvire, E.; Zyskind, J. L.; Simpson, J. R. IEEE Photon. Technol. Lett. 1990, 2, 246 – 248. [5] Goldstein, E. L.; Eskildsen, L.; da Silva, K.; Andrejco, M.; Silberberg, Y. IEEE Photon. Technol. Lett. 1993, 5, 937 – 939. [6] Chow, J.; Town, G.; Eggleton, B.; Ibsen, M.; Sugden, K.; Bennion, I. IEEE Photon. Technol. Lett. 1996, 8, 60-62. [7] Liu, X.; Lu, C. IEEE Photon. Technol. Lett. 2005, 17, 2541-2543. [8] Han, Y.-G.; Tran, T. V. A.; Lee, S. B. Opt. Lett. 2006, 31, 697-699. [9] Han, Y.-G.; Lee, S. B. Opt. Express 2005, 13, 10134-10139. [10] Bellemare, A.; Karasek, M.; Rochette, M.; LaRochelle, S.; Têtu, M. J. Lightwave Technol. 2000, 18, 825-831. [11] Slavik, R.; LaRochelle, S.; Karasek, M. Opt. Commun. 2002, 206, 365-371. [12] Zhou, K.; Zhou, D.; Fengzhong, F.; Ngo, Q. N. Opt. Lett. 2003, 28, 893-895. [13] Ahmed, F.; Kishi, N.; Miki, T. IEEE Photon. Technol. Lett. 2005, 17, 753-755. [14] Liu, X.; Lee, B. IEEE Photon. Technol. Lett. 2004, 16, 428-430. [15] Koch, F.; Reeves-Hall, P. C.; Chernikov, S. V.; Taylor, J. R. Conf. Opt. Fiber Comm. 2001, 54, WDD7/1-WDD7/3. [16] De Matos, C. J. S.; Chestnut, D. A.; Reeves-Hall, P. C.; Koch, F.; Taylor, J. R. Electron. Lett. 2001, 37, 825-826. [17] Yu, B.-A.; Kwon, J.; Chung, S.; Seo, S.-W.; Lee, B. Electron. Lett. 2003, 39, 649-650. [18] Lee, Y. W.; Yu, B.-A.; Lee, B.; Jung, J. Opt. Eng., 2003, 42, 2786-2787. [19] Lee, Y. W.; Jung, J.; Lee, B. IEEE Photon. Technol. Lett. 2004, 16, 54-56. [20] Han, Y.-G.; Kim, G.; Lee, J. H.; Kim, S. H.; Lee, S. B. IEEE Photon. Technol. Lett. 2005, 17, 989-991. [21] Kim, C.-S.; Sova, R. M.; Kang, J. U. Opt. Commun. 2003, 218, 291-295. [22] Han, Y.-G.; Lee, J. H.; Kim, S. H.; Lee, S. B. Electron. Lett. 2004, 40, 1475-1476. [23] Fok, M. P.; Lee, K. L.; Shu, C. IEEE Photon. Technol. Lett. 2005, 17, 1393-1395. [24] Chen, L. R. IEEE Photon. Technol. Lett. 2004, 16, 410-412. [25] Kim, C.-S.; Kang, J. U. Appl. Opt. 2004, 43, 3151-3157. [26] Kim, S.; Kwon, J.; Kim, S.; Lee, B. IEEE Photon. Technol. Lett. 2001, 13, 350-351. [27] Yoon, I.; Lee, Y. W.; Jung, J.; Lee, B. J. Lightwave Technl. 2006, 24, 1805-1811. [28] Han, Y.-G.; Moon, D. S.; Chung, Y.; Lee, S. B. Opt. Express 2005, 13, 6330-6335. [29] Han, Y.-G.; Lee, S. B.; Moon, D. S.; Chung, Y. Opt. Lett. 2005, 30, 2200-2202. [30] Han, Y.-G.; Tran, T. V. A.; Kim, S.-H.; Lee, S. B. Opt. Lett. 2005, 30, 1114-1116. [31] Dong, X.; Shum, P.; Xu, Z.; Lu, C. IEEE LEOS Ann. Meeting 2005, 814 – 815. [32] Dong, X.; Shum, P.; Ngo, N. Q.; Chan, C. C. Opt. Express 2006, 14, 3288-3293. [33] Roh, S.; Chung, S.; Lee, Y. W.; Yoon, I.; Lee, B. IEEE Photon. Technol. Lett. 2006, 18, 2302-2304. In: Optical Fibers Research Advances ISBN: 1-60021-866-0 Editor: Jurgen C. Schlesinger, pp. 355-368 © 2007 Nova Science Publishers, Inc. Chapter 14 AGING AND RELIABILITY OF SINGLE-MODE SILICA OPTICAL FIBERS M. Poulain 1 , R. El Abdi 2 and I. Severin 3 1 UMR 6226, Université de Rennes1, F-35042 Rennes, France 2 LARMAUR, Fre-Cnrs 2717, Université de Rennes1, F-35042 Rennes, France 3 Universita Politechnica, Splaiul Independentei, IMST, 06042 Bucarest, Romania Abstract The optical fiber reliability in telecommunication networks has been still an issue, that’s why the question of how long an optical fibers might been used without a significant probability of failure isn’t out of interest. Much work was developed around this issue, but the optical fiber fatigue and aging process has not been yet fully understood. The reliability of the optical fibers depends on various parameters that have been identified: time, temperature, applied stress, initial fiber strength and environmental corrosion. The major and usually unique corrosion reagent is water, either in the liquid state or as atmospheric moisture. Glass surface contains numerous defects, either intrinsic, the so- called “Griffith’s flaws and extrinsic, in relation to fabrication process. Under permanent or transient stress, microcracks grow from these defects, and growth kinetics depend on temperature and humidity. Although polymeric coating efficiently protects glass surface from scratches, it does not prevent water to reach glass fiber. The work carried out during the last years made possible to apprehend in a more coherent way the problems of failure and rupture of fibers subjected to severe aging conditions. In the proposed chapter, some informations on the used characterization methodology for the silica optical fibers are given. In addition, Optical fibers analysis advantages, expected percussions and theoretical background are given to enlighten the potential concerned persons. The principal optical fiber test benches are described and some results are commented. Finally, final remarks are noted. M. Poulain, R. El Abdi and I. Severin 356 1. Introduction Terrestrial and submarine telecommunication networks depend critically on optical fibers. While main emphasis is put on transmission and signal characteristics [1], more basic features such as reliability and expected lifetime has appeared also as major concerns [2, 3]. However, these concerns become less popular with the deep crisis that occurred in the telecom market and the emergence of more advanced fibers: one may think that new fibers should replace the existing ones earlier than expected. In addition, until now, operators did not face serious problems in relation to fiber failure. Nevertheless the reliability issue remains more than ever a topical question for several reasons. Firstly, the impressive increase of the bit rate is accompanied by a power increase which is supported by the fiber core and can generate catastrophic failure phenomena and generate damage of the fiber ends or losses in the connectors. Secondly, current models include humidity, applied stress and temperature as major aging factors, but their accuracy for lifetime prediction is questionable. Aging of silica fibers is now rather well understood as numerous studies have been implemented in this area [2-17]. One must separate the case of the fatigue static behavior where fibers are subjected to a permanent strain, e.g. bended fibers, and the dynamic fatigue corresponding to an unexpected tensile stress arising from environmental changes. Failure mechanism involves surface phenomena, which raise fundamental questions. Surface defects, initiator for cracks grow have not yet been identified neither by Scanning Electron Microscopy (SEM) nor by Atomic Force Microscopy (AFM). The current random network model used actually to describe glass structure gives no explanation for the so-called Griffith’s flaws [18] and does not account for density fluctuations and inhomogeneities in glass. While other models, such as the vacancy model [14, 19], may provide a physical picture of these defects, they are still in an emerging state that limits their application. Water is also critical in fiber failure: fiber strength may increase by 100 % if water is missing, for example under vacuum, in a dry box or at liquid nitrogen temperature [3, 4, 16]. It is assumed that water molecules break the Si-O-Si chemical bonds of the vitreous network. This simple and logical model may be incomplete and ignore some aspects of the whole phenomenon. Polymeric coatings are largely used to inhibit surface flaws and proved to be efficient. However the reinforcement mechanism is not well understood. The general use of Weibull’s statistics in data processing may be inappropriate in some cases: it is widely observed that fiber strength value calculated from Weibull plots decreases as sample length increases, but Weibull formula is precisely expressed to be independent on fiber length. These questions, and others, have not only a fundamental interest, but still could have notable economic implications. The technology evolution and the research for low cost solutions lead to use new fibers and new components. Thus, polymeric fibers are being considered for the local distribution, while Bragg grating fiber components are now largely used in optical amplifiers. However, the reliability of these new components has still to be evaluated. A traditional stake is referred to the future fibers for the local distribution networks (Fiber To The Home, FTTH). These fibers will be submitted to notable permanent stresses, for Aging and Reliability of Single-Mode Silica Optical Fibers 357 example at door corners, thus exposure to temperature, humidity and sudden stress may be larger than in classical cables. The better understanding of the factors ruling aging and reliability of optical fibers should lead not only to scientific advances, but also to economical spin-offs. The telecom market will require light cables for local area networks, and the design of mini cables can be optimized on this basis. In addition various markets are likely to open in other fields where optical fiber components are key elements. This concerns optical fiber sensors, laser power delivery, fiber lasers, monitoring and control, remote spectroscopy, in line imaging, etc… Of particular interest is automotive industry in those case reliability and cost make essential points. On the fundamental level, the principal goal is to collect new information elements that could help answering recurrent questions. 2. Background Failure of fibers is rather well understood as it might be considered as a particular case of fragile material fracture [2-18]. While such materials exhibit a large resistance to compressive stress, they are much more sensitive to traction. Failure originates from surface flaws that may be described as microcracks. Their cracks growth under tensile stress as effective stress is amplified at the bottom of the crack. This growth is enhanced by water activity in most materials, including oxide glasses. It is generally assumed that water acts by breaking the chemical bonds between oxygen and silicium or other cations. For this reason the intrinsic strength K 1C of the material can be observed only in extremely dry conditions, e.g. vacuum or liquid nitrogen. In practice, optical fiber aging depends on various factors that may decrease effective fiber strength: residual applied stress, temperature and water. It is assumed that surface flaws are enlarged, consequently crack growth promotes. Maximum water activity is in aqueous solutions and it is expressed by the relative humidity (RH) in current atmosphere. Figure 1. Fracture morphology of silica optical fibre (see silica core – typical fragile surface fracture surrounded by the two layer epoxi-acrylate polymer coating). M. Poulain, R. El Abdi and I. Severin 358 As fiber surface has determined fracture to a large extent, external coating appears critical. This coating is polymeric in most cases, and modern optical fibers are coated by two different layers, a soft coating at glass surface and a hard coating at external surface (Fig. 1). The coating first makes a protection against scratches that occur in normal handling; it also fills the surface flaws gluing in some a way the two sides of the micro cracks and finally, it reduces water activity at glass surface. Ideally, coating should prevent any water molecule to reach glass surface. Unfortunately, polymeric coatings, including hydrophobic coatings, are very permeable to water. Only inorganic hermetic coating could make an efficient barrier against water. Polymeric coatings (e.g. epoxyacrylates) are preferred in practice because they are more efficient to inhibit surface defects [13, 20-24]. Various theoretical models are applied for mechanical characterization of optical fibers [25- 26], but the most common one is based on Weibull's statistics. The Weibull law expresses the failure probability F of a fiber with a length L subjected to an applied stress σ : [ ] ) ( ) ( ] 1 1 [ 1 o Ln Ln m F Ln L Ln σ σ − = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ − (1) where m is a size parameter and σ ο is a scale parameter. The evolution of ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ − ] 1 1 [ 1 F Ln L Ln in function of Ln (σ) is known as the Weibull plot. The values of m and σ ο are calculated from the slope of the curve and the intersection with the stress axis. The m parameter characterizes the defect size dispersion [26]. A high m value indicates that the distribution of the defect size is homogeneous while a low m value means that surface defects are varying in size. When the curve appears as a broken line with two distinct slopes – one small for low stress and the second one large, respectively – one has assumed two different families of defects, the first one corresponding to large extrinsic defects, and the second one relating to intrinsic flaws. Other plots encompass several straight lines relating to different groups of defects. The failure probability F is calculated from the relation: N i F i 5 . 0 − = (2) where i represents the rank of the measurement and N the total number of values. The σ ο parameter represents the stress corresponding to the fiber cumulative fracture probability F of is 50%. In the static fatigue measurements, the fiber is subject to a constant stress and one measures the time to failure. This time t f is ruled by the following relation: Aging and Reliability of Single-Mode Silica Optical Fibers 359 n a n i f S B t σ 2 − = (3) where B is a constant that depends on environment – typically water, S the initial inert strength of the fiber and n the stress corrosion parameter, and σ a the failure stress. The failure probability of F can be written as: F t L L t B S f f a n o n m n ( , ) exp . . = − − ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎛ ⎝ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ − − 1 2 2 σ (4) Or, in the logarithmic form: ( ) ( ) Ln t nLn LnB Ln S f a i n = − + + − σ 2 (5) The plot Ln t f ( ) as a function of Ln a ( ) σ describes the static fatigue behavior and gives access to the fatigue parameters n and BS i n−2 . 3. Experimental: Mechanical Measurements The mechanical strength of the fibers may be measured in different ways corresponding to static fatigue and dynamic fatigue [27, 28]. In the static fatigue tests, fibers are subject to a permanent stress, and the time to failure is recorded for a set a fibers. Then a statistical analysis gives values for the mean failure strength and the mean lifetime in the testing conditions: type of fiber, temperature, applied stress and water activity. The dynamic fatigue test consists in applying an increasing tensile strength until fiber breaks. From a convenient data processing one finds the mean fiber strength. Special equipments are used for these measurements, presented as follows. 3.1. Vertical Bench The static fatigue under axial tensile loading consists in subjecting a fiber sample to a uniform load as a suspending weight of known value. The two fiber ends are rolled up on a pulley provided with a system allowing to block the fiber sample ends and to avoid any slip (Fig. 2a). The higher pulley (noted 1) is fixed on a support, while the lower pulley (noted 2) is mobile and interdependent of a plate on which a chosen mass is applied. This set up leads to carry out static tensile tests on high length fiber samples (usually 4 m in length) for applied loads ranging between 5 and 50 N. A number of forty samples can simultaneously be tested (Fig. 2b). The measurement of the fiber fracture time for different M. Poulain, R. El Abdi and I. Severin 360 loads allows determining the static stress corrosion parameter. Thus, the fibers are submitted to different aging conditions and subsequently to mechanical tensile testing. Silica fiber Mass Beam of light Optical sensor Pulley2 Pulley1 Plate (a) (b) Figure 2. Vertical static tensile-test bench: (a) diagram of sample fiber under mass loading, (b) general view. 3.2. Static Fatigue under Permanent Curvature Another testing bench can be used for static testing [29]. Optical fibers, one meter in length, are subjected to bending stresses by winding around alumina mandrel with calibrated diameter sizes (Fig. 3a). The constant level of applied stress is adjusted by the proper choice of the mandrel size. The time to failure is measured, and this corresponds to the time required for the fiber strength to degrade until it equals the stress applied through winding round the mandrel. The time to failure is measured by optical detection when the ceramic mandrel moves out of the special holder. When fiber breaks, the mandrel rocks from its vertical static position and the time to failure is directly recorded with an accuracy of ±1 s. The testing setup consists of a large number of vats containing 16 holders each (Fig. 3b). The applied stress on the fiber depends on the mandrel diameter according to the Mallinder and Proctor relation [30] as follows: ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + = 2 1 ' 0 ε α ε σ E ; α α 4 3 ' = ; d d fiber glass + = φ ε (6) Aging and Reliability of Single-Mode Silica Optical Fibers 361 where σ is the applied stress (in GPa), E 0 is Young modulus (equal to 72 GPa for the silica), ε is the relative fiber deformation, α is the constant of the elastic nonlinearity (equal to 6), φ is the mandrel diameter (in μm), d glass is the glass fiber diameter and d fiber is the fiber diameter including the polymer coating. For example, for standard silica optical fibers used for telecommunication networks, d glass is equal to 125 μm and d fiber is equal to 250 μm; this leads to the corresponding stress of 3.92, 3.76, 3.34 and 3.22 GPa for the calibrated diameter mandrel of 2.3, 2.4, 2.7 and 2.8 mm respectively. The testing environmental conditions during static fatigue measurements (temperature and relative humidity) should be also taken into account. E R Clamping rings Wound fibre on calibrated mandrel Light beam (a) (b) Figure 3. Static bending test. 3.3. Vertical Dynamic Tensile Test Movable pulley Dynamometric cell Engine Device speed control Fixed pulley 5 0 0 m m Fiber Higher plate (a) (b) Figure 4. Schematic description of the dynamic tensile-test bench. M. Poulain, R. El Abdi and I. Severin 362 During a dynamic tensile test, the fiber is subjected to a deformation under a constant speed until the rupture. The two fiber ends are rolled up on pulleys, having 65 mm in diameter and covered with a powerful adhesive so as to prevent any fiber slip during the test (Fig. 4). The lower pulley is fixed while the higher pulley is mobile and its displacement velocity (v, mm/min) corresponds to the chosen deformation speed to carry out the test. Typical fiber length is 500 mm. During the test, the deformation and the tensile load are measured using a dynamometric cell while the fiber deformation is deduced from the displacement between the fixed lower pulley and the mobile higher plate. The test velocity has an important influence for the failure stresses as this might be seen in Fig. 5. High speeds lead to failure cracks with the same geometry (not curve slope variation for v=500 mm/min), while the low speeds lead to various crack forms. -4 -3 -2 -1 0 1 2 0,60 0,80 1,00 1,20 1,40 Failure stress (GPa) L n ( - l n ( 1 - F ) ) v- 50mm/min v- 150mm/min v-300mm/min v - 500mm/min (F represents the cumulative fracture probability) Figure 5. Evolution of failure stresses for different tensile test velocities v (mm/min). 3.4. Long Length Dynamic Tensile Bench This mechanical bench (Fig. 6) allows to carry out tensile tests on fibers with high lengths (from 0.5 m to 18 m) with broad speeds (ranging between 30 mm/min to 30 m/min with an accuracy of less than 2 per 1000) and under very diverse environmental conditions (temperature, aqueous solution...). Using the set up, one can obtain information on the defect size dispersion onto the fiber surface and can determine the dynamic stress corrosion parameter n. Indeed, this parameter is related to the velocity by the following relation: K A n I V = (7) where A is a parameter environment dependent, K I is the stress intensity factor and n is a parameter characterizing the material capacity to resist to a stress. Aging and Reliability of Single-Mode Silica Optical Fibers 363 3.5. Two Point Bending Bench Fibers can also be characterized by using a two point bending testing device (Fig. 7). Samples of 10 cm in length are bent and placed between the grooved faceplates of the testing apparatus, in order to avoid the fiber slipping during the faceplate displacements and to maintain the fiber ends in the same vertical plane. Figure 6. Thirty meters long dynamic tensile bench. Generally, a series of 30 samples are tested for different faceplate velocities (for example, 100, 200, 400 and 800 μm/s, respectively). The failure stress is calculated from the distance separating the faceplates, using the Proctor and Mallinder relation, improved by Griffioen [8]. Subsequently failure stress is obtained for each tested sample and tracing the classical Weibull plots one might calculate the statistical parameters. Due to a very short fiber sample part subjected under stress, this testing method is preferentially used to study the intrinsic defects or selected flaws. Fiber Fiber Faceplates Control Block Stepper motor Faceplates Computer Piezo Electric Sensor Fiber Figure 7. Dynamic two point bending bench. M. Poulain, R. El Abdi and I. Severin 364 4. Results The work carried out during the last years made possible to apprehend in a more coherent way the failure of the fibers subjected to severe aging conditions. An empirical relation defining the fiber lifetime t f according to the temperature, residual stress and water content was established. For this purpose, silica optical fibers were either immersed in hot water or heated in wet atmosphere with a controlled relative humidity (RH). The two relations derived from this set of measurements are the followings [14]: ( ) ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⋅ − ⋅ + = RT T E A t a f σ β φ 0 0 exp (8) ( ) ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⋅ − ⋅ + = − RT T E Z A t a f σ β φ δ 0 0 exp (9) The first relation (8) applies to fibers aged in liquid water, while the second one (9) concerns fibers exposed to humid atmosphere with variable RH (RH = Z), temperature T and applied stress σ. It is worth noting that the factor Φ corresponds to some kind of relaxation that decreases the effect of the applied stress. Its magnitude grows with temperature and applied stress. In a second set of measurements, fibers were immersed in hot water at 65°C or at 85°C for a long time (up to 24 and 27 months). Then they were characterized in static and dynamic fatigue. As one could expect, fiber strength in dynamic fatigue decreases as aging time increases. More surprisingly, the time to failure of aged fibers subjected to static fatigue increased enormously by comparison to non-aged fibers [31]. However this unexpected effect did not follow a regular evolution versus time, but a rather cyclic one (Fig. 8). (In legend calibrated mandrel diameter, in mm, in the case of the static fatigue testing set-up – Fig.3) Figure 8. Evolution of the fiber failure time in function of different aging conditions. Aging and Reliability of Single-Mode Silica Optical Fibers 365 The explanation for this lifetime increase can be found in the structural relaxation identified in the previous relations. The failure mechanism of the aged fibres involves surface phenomena, in relation to water activity. A layer of hydrated silica is likely to be formed at fibre surface [15]. This vitreous hydrated phase may relax under stress at room temperature, which partly compensates the external applied stress in static fatigue. The change of the glass surface was exemplified by the indentation behaviour that is different from that of normal silica [32]. New experiments are carried out as well on standard silica fibers as on new fibers [33]. Fibers with a hermetic coating, fibers before and after photo-printing, fibers of polymer, on average have diameters between 85 µm and 125 µm. The influence of temperature, water and various corrosive agents on the mechanical fiber strength is determined. The coating aging is also taken into account. Characterizations are also carried out on fibers belonging to different vitreous systems (fluorides, oxides, sulphides) to detect and analyze less visible phenomena when silica fibers are studied. For several silica fibers subjected to vertical static tensile testing (see Fig. 2) under various loadings, one can notice that more the suspended mass value is high; more the time of rupture is large (Fig. 9). For weak loads (15 N), two families of cracks exist (a slope break indicates the dispersion of the microcrack shapes). -5 -4 -3 -2 -1 0 1 2 0 2 4 6 8 ln (time to rupture) (h) l n ( - l n ( 1 - F ) ) 20N 15N 25N 30N Figure 9. Time to rupture evolution for different loadings (F represents the cumulative fracture probability). 5. Final Remarks The huge development of the telecommunication networks has been made possible by the availability of low cost and high quality silica optical fibers. As industrial production reaches millions of km, research rather focuses on networks and advanced components. Fiber reliability is not a critical issue at this time because few problems were encountered, most of them being accidental. However future fiber local loops will put fibers under large and M. Poulain, R. El Abdi and I. Severin 366 permanent stress. In addition, lighter and less expansive cables could be manufactured if transient or permanent stresses have no significant influence on fiber lifetime. Fiber aging has been the subject of numerous studies leading to theoretical models for lifetime assessment. While ground observations do not contradict these predictions, the accuracy of the models is questionable due to the complexity of the aging mechanism. In this respect, experiments implemented on a long time scale are likely to bring new information. There are some questions underlying the reliability studies. Aging parameters encompass time, temperature, applied stress and water activity. While the critical part of water in failure mechanism is well known, its real impact varies according to the physical state - liquid or vapor - and partial vapor pressure. The stress applied to the fiber may be temporary, for example during the proof test or network installation, or permanent when the fiber is bent in cable and connecting areas. The aging mechanism is assumed to enlarge or to extend the "Griffith flaws" which are spread at the fiber surface. These defects may be described as micro-cracks which grow under applied stress in wet environment. Although this mechanism is believed to be irreversible, water may also induce some curing effect which could correspond to the geometrical smoothing of the crack tip [34, 35]. There is a practical interest in collecting quantitative information on the aging of the commercial optical fibers over a long period of time. Such observations should allow a more accurate comparison between experimental and calculated strengths and make lifetime assessments more realistic as testing periods (> 2 years) become closer to the lifetime required by network users that is at least 20 years. Acknowledgments Authors express their gratitude to France Telecom for technical assistance and equipment supply and to Region Bretagne for financial support. References [1] Pal B. P., Fundamentals of fiber optics in telecommunication and sensor systems. (Wiley Eastern ltd, Delhi, 1992). [2] Olshansky R. and Maurer R. D., (1976). Tensile strength and fatigue of optical fibers. J. Appl. Phys. 47, 4497-4499. [3] Sakaguchi S., Kimura T., (1981). Influence of temperature and humidity on dynamic fatigue of optical fibers. J. Amer. Ceram. Soc. 64 [5], 259-262. [4] Duncan W. J., France P. W. and Craig S. P., The effect of environment on the strength of optical fiber. Pp. 309-328 in Strength of Inorganic Glass, Edited by C.R. Kurkjian, Plenum press, New York, 1985. [5] Matthewson M. J. and Kurkjian C. R., (1988). Environmental effects of the static fatigue of silica optical fiber. J. Amer. Ceram. Soc. 71 [3], 177-183. [6] Kurkjian C. R., Krause J. T. and Mathewson M. J., (1989). Strength and fatigue of silica optical fibers. J. ligthwave Tech. 7, 1360-1370. [7] Michalske T., Smith W., Bunker B., (1991). Fatigue mechanisms in high-strength silica- glass fibers. J. Am. Ceram. Soc. 74, [8], 1993-1996. Aging and Reliability of Single-Mode Silica Optical Fibers 367 [8] Griffioen, W. Optical fiber reliability. Thesis edited by Royal PTT, The Netherlands NV, PPT Research, Leidschendam, 1994. [9] Glaseman G. S., (1994). Assessing the long term reliability of optical fibers. Proc. National Fiber Optics Engineers Conference, 297. [10] Muraoka M., Ebata K., Abe H., (1993). Effect of humidity on small-crack growth in silica optical fibers. J. Am. Ceram. Soc. 76, [6], 1545-1550. [11] Volotinen T. T. – Water tests on optical fibers – Proc. SPIE 3848, 134-143, (1999). [12] Semjonov S. L., Kurkjian C. R., (2001). Strength of silica optical fibres with micron size flaws. J. Non-Cryst. Solids, 283, 220-224. [13] Armstrong J. M. and Matthewson M. J., (2000). Humidity dependence of fatigue of high-strength fused silica optical fibers. J. Am. Ceram. Soc. 83, [12], 3100-3108. [14] Poulain M., Evanno N., Gouronnec A. – Static fatigue of silica fibers – Optical fiber and fiber component mechanical reliability and testing II, M. J. Matthewson, C. R. Kurkjian, Editors, Proc. SPIE 4639, 64-74, (2002). [15] Berger S., Tomozawa M., (2003). Water diffusion into silica optical fiber. J. Non-Cryst. Solids, 324, 256-263. [16] Gougeon N., El Abdi R and Poulain M., (2004). Evolution of strength of silica fibers under various moisture conditions. Optical Materials, 27, 75-79. [17] Severin I., El Abdi R. and Poulain M., (2007). Strength measurements of silica optical fibers under severe environment. Optics & Laser Techn.. 39, [2], 435-441. [18] Griffith A. A., Phil. Trans. 221A, 163 (1920). [19] Poulain M., Vacancy model of ionic glasses. Proc Int. symp. Non Oxide Glasses, Part B, pp 22-26, Corning USA and “What is glass?”(briton langage) ΣKIANT, 1, 13-26, (1996). [20] Wei T., Skutnik J., (1988). Effect of coating on fatigue behavior of optical fiber. J. Non- Cryst. Solids, 102, 100-105. [21] Kurkjian C. R., Simpkins P. G., Inniss D., (1993). Strength, degradation and coating of silica lightguides. J. Am. Ceram. Soc. 76, [5], 1106-1112. [22] Shiue S. T., Ouyang H., (2001). Effect of polymeric coating on the static fatigue of double-coated optical fibers. J. App. Phy. 90, [11], 5759-5762. [23] Mrotek J. L., Matthewson M. J., Kurkjian C. R., (2001). Diffusion of Moisture through optical fiber coatings. Journal Light-wave Technol. 19, [7], 988-993. [24] Mrotek J. L., Matthewson M. J., Kurkjian C. R., (2003). Diffusion of Moisture through fatigue and aging-resistant polymer coatings on lightguide fibers. Journal Light-wave Technol. 21, [8], 1775-1778. [25] Schmitz G. K. and Metcalfe A. G., (1967). Testing of fibers. Mat. Res. Stand., 7 [4], 862-865. [26] Matthewson M. J., (1994). Optical fiber reliability models. Proc. SPIE, Critical Reviews, CR 50, 3-31. [27] Matthewson M. J., (1994). Optical fiber mechanical testing techniques. Proc. SPIE, Critical Reviews, CR 50, 31-59. [28] Severin I., El Abdi R., Poulain M. and Amza G., (2005). Fatigue testing of silica optical fibres. Journal of Optoelectronics and Advanced Materials, 7 [3], 1581-1588 . [29] International standard IEC 793-1-3, First Edition 1995-10 . [30] Mallinder, F.P., Proctor, B.A., (1964) Phys. Chem. Glass, 5, 91. [31] Gougeon N., El Abdi R. and Poulain M. (2003). Mechanical reliability of silica optical fibers. J. Non-Cryst. Solids, 316, 125-130. M. Poulain, R. El Abdi and I. Severin 368 [32] Gougeon N., Sangleboeuf J. C., El Abdi R., Poulain M. and Borda C. T., (2005). Indentation Behavior of Silica Optical Fibers Aged in Hot Water. Fiber and Integrated Optics. 24, [5], 491-500. [33] Severin I., Poulain M., ElAbdi R. (2005). Phenomena associated to aging of silica optical fibers. Photonic Applications in Devices & Communication Systems, P. Mascher, A. P. Knights, eds., Proc. SPIE 5970. [34] Hirao K., Tomozawa M. (1987). Kinetics of crack tip blunting of glasses. J. Am. Ceram. Soc. 70, [1], 43-48. [35] Hirao K., Tomozawa M., (1987). Dymanic fatigue of treated high-silica glass: Explanation by crack tip blunting. J. Am. Ceram. Soc. 70 [6], 377-382. INDEX A absorption spectra, 272 access, 53, 75, 112, 216, 359 accounting, 254 accuracy, 120, 149, 201, 215, 356, 360, 362, 366 acetone, 33 acetylene, 332 achievement, 206 acid, 31, 33, 36, 44, 104, 105 acrylate, 357 adaptability, 5 adjustment, 59, 215 adriamycin, 31, 48 adsorption, 40 aerospace, 5, 260 AFM, 356 agent, 31 aging, xii, 260, 355, 356, 357, 360, 364, 365, 366, 367, 368 aging process, xii, 355 albumin, 45 algorithm, 59, 60, 65, 68, 74, 77, 243, 246, 247, 320 alternative(s), x, 54, 65, 146, 231, 233, 238, 249, 255, 261 aluminum, 144 amplitude, x, 19, 78, 84, 85, 128, 131, 166, 171, 172, 177, 209, 279, 281, 282, 283, 284, 285, 286, 287, 288, 289, 290, 291, 293, 297, 304, 324 AN, 175, 231 annealing, 65 antibody, viii, 15, 28, 29, 30, 31, 36, 37, 38, 39, 40, 42, 43, 45, 46 antigen, 31, 38, 39, 40, 43 antimony, 188 APC, 220 apoptosis, 31 argon, 31, 260 arsenic, 273 assessment, 366 assignment, 251 assumptions, viii, 43, 52 asymmetry, 105, 152 atoms, ix, 119, 120, 122, 124, 128, 130, 131, 132, 133, 134, 136, 137, 138, 139, 140, 142, 143, 146, 147, 148, 149, 150, 151, 152, 153, 154, 336 attachment, 21, 38, 39, 40, 41, 42, 43, 44 attention, vii, ix, 3, 4, 5, 161, 163, 164, 176, 217, 258, 302 attenuated total reflectance, 261 Australia, 315 automobiles, 260 availability, 365 averaging, xi, 301, 303 avoidance, 108 B Bacillus, 22, 28, 29, 47 Bacillus subtilis, 22 backscattering, ix, 54, 70, 71, 72, 73, 161, 162, 164, 170, 171, 172, 176, 177, 179, 180, 181, 182, 185 bacteria, 4, 28, 41, 48 bandgap, xi, 258, 260, 262, 265, 273, 315, 316, 317, 318, 319, 320, 321, 322, 332, 333 bandwidth, viii, x, 51, 54, 62, 63, 65, 68, 69, 72, 74, 76, 77, 84, 114, 115, 168, 170, 188, 206, 207, 216, 218, 221, 222, 232, 233, 241, 247, 257, 258, 259, 269, 270, 275, 342 beams, 120, 132, 140, 146, 150, 151, 154, 178, 179, 180, 181, 217, 280 beef, 22, 28, 46 behavior, viii, 15, 18, 35, 40, 72, 78, 97, 102, 115, 324, 356, 359, 367 Beijing, 3 bending, 177, 259, 262, 349, 350, 360, 361, 363 Index 370 bias, 346 binary decision, 252 binding, 29, 31, 39, 40, 41, 42, 47 biomarkers, 30 biomechanics, 84 biomolecule(s), viii, 15, 46 biosensors, 4, 15, 35 biotin, 29 birefringence, ix, 83, 84, 87, 88, 89, 90, 91, 92, 94, 98, 101, 102, 104, 106, 107, 109, 110, 170, 184, 225, 274, 340, 343, 346, 347, 349 bismuth, 184 blackbody radiation, 272 blocks, 297 blood, 26, 31, 275 BN, 175 BNP, 25, 26 Boltzmann constant, 58 bonding, 36, 44, 259 bonds, 356, 357 boundary value problem, 59 Bragg grating, viii, xii, 4, 51, 54, 83, 84, 85, 92, 97, 98, 101, 104, 106, 115, 161, 169, 170, 182, 184, 213, 227, 228, 315, 331, 332, 335, 338, 339, 349, 351, 356 branching, 210 brass, 271 breast carcinoma, 31 breathing, 280 broadband, viii, x, 51, 53, 63, 64, 79, 85, 113, 182, 187, 205, 208, 227, 261, 271 buffer, 22, 23, 24, 25, 26, 27, 29, 30, 38, 39, 177 building blocks, 297 burning, 169, 338 C cabinets, 53 cables, vii, 53, 357, 366 calibration, 33, 94, 96, 98, 107, 192 Canada, 205 cancer screening, 31 candidates, 182, 244, 302 capillary, 120, 121, 127, 128, 131, 132, 133, 134, 135, 260, 262 carbohydrate, 37 carbon, 6, 8, 259 carboxylic groups, 36 carcinogenicity, 31 carcinogens, 31 carcinoma, 31, 48 cardiovascular disease, 30, 49 carrier, 4, 5, 114, 115, 165, 167, 206, 233, 234, 236, 317, 342 cDNA, 32 cell, 21, 26, 27, 28, 31, 35, 40, 44, 48, 148, 262, 361, 362 cell culture, 27, 31 cell growth, 28 ceramic, 360 CFBG, 338, 341, 350 CGLE, x, 279, 280, 281, 285, 286, 287, 288, 289, 290, 291, 293, 297 channels, 53, 54, 57, 63, 75, 76, 77, 84, 115, 169, 206, 213, 215, 216, 218, 219, 227, 236, 305, 338, 346 chemical bonds, 356, 357 chemical etching, 33, 98, 105, 106 chemical properties, 15 China, 3, 257 Chinese, 24 cladding, x, 6, 16, 17, 18, 19, 31, 88, 105, 106, 121, 122, 143, 144, 146, 167, 168, 177, 190, 209, 210, 215, 218, 232, 257, 258, 260, 261, 262, 263, 264, 265, 266, 269, 270, 272, 273, 274, 275, 316, 332 cladding layer, 177 classes, xi, 263, 273, 301, 303 cleaning, 36, 42 clustering, 201 CO2, 4, 207, 208, 261, 272, 274, 275 coagulation, 30, 48 coatings, 190, 258, 261, 271, 273, 356, 358, 367 codes, 112, 113, 114, 247 coding, 114, 244, 246, 247, 249 coherence, 89, 109, 151, 224, 225, 226 collaboration, 134, 309 collisions, xi, 149, 151, 279, 302, 315, 317, 324, 325, 326, 327, 328, 329, 330, 333 combined effect, 39, 312 communication, vii, viii, x, xi, xii, 51, 52, 55, 56, 79, 83, 84, 108, 169, 187, 206, 215, 231, 232, 234, 236, 238, 239, 242, 244, 246, 268, 269, 270, 275, 299, 302, 312, 335, 336, 352 communication systems, viii, x, xi, xii, 51, 52, 55, 56, 79, 108, 231, 232, 234, 238, 239, 244, 246, 275, 299, 335, 336, 352 community, 302 compatibility, 84, 205 compensation, x, 205, 206, 213, 218, 220, 221, 302, 311, 347 competition, 111, 169, 174, 179, 180, 291, 336, 339 competitiveness, 336 complementary DNA, 32 complexity, 165, 366 complications, 30 Index 371 components, ix, xi, 7, 18, 54, 57, 58, 68, 84, 87, 89, 90, 91, 93, 98, 104, 111, 122, 164, 166, 173, 176, 188, 205, 206, 207, 208, 212, 218, 220, 224, 225, 226, 233, 234, 236, 238, 250, 301, 303, 312, 356, 357, 365 composites, 295 composition, 190, 215 compounds, 4 computation, viii, 52, 61, 67 computing, 206, 253 concentration, 28, 29, 30, 31, 32, 33, 35, 37, 39, 40, 41, 42, 43, 44, 104, 105, 106, 107, 164, 168, 200, 215 condensation, 315 conduction, 173, 342 conductivity, 8, 172, 221 configuration, vii, 3, 4, 49, 57, 65, 72, 75, 77, 78, 135, 142, 164, 165, 166, 169, 178, 189, 207, 216, 218, 220, 222, 226, 227, 303, 337, 338, 349 confinement, vii, 16, 52, 177, 264, 269, 271, 316 confusion, 342 Congress, 202 conjecture, 318, 321 consolidation, viii, 51 constituent materials, 258, 263, 272 constraints, 303, 307 contaminant, 47 contamination, 7, 32, 48, 143 continuity, 122, 173 control, 5, 68, 71, 72, 81, 154, 166, 170, 202, 212, 213, 215, 219, 220, 232, 244, 247, 302, 337, 340, 343, 345, 349, 357, 361 convergence, 8, 59, 65, 73 conversion, ix, 4, 108, 164, 187, 188, 202, 220, 233, 252, 260, 275, 332 cooling, xi, 68, 97, 111, 148, 154, 293, 335, 336, 338, 341, 352 Copenhagen, 255 corn, 26 coronary heart disease, 30 correlation(s), 176, 179, 180, 182, 193, 194, 201 correlation function, 176, 180 corrosion, xii, 84, 355, 359, 360, 362 costs, 68, 84 couples, 55 coupling, xi, 7, 16, 18, 58, 85, 87, 114, 128, 134, 136, 146, 147, 177, 178, 179, 182, 185, 187, 209, 216, 218, 264, 274, 301, 303, 315, 316, 317, 333, 342 covalent bonding, 36, 44 coverage, 40, 41, 43 crack, 357, 362, 366, 367, 368 C-reactive protein, 25, 30 critical value, 170, 179 cross-phase modulation, xi, 166, 301, 312 CRP, 25 crystal growth, 154 crystalline, 59, 273 culture, 27, 28, 31 curing, 366 CVD, 30 cytochrome, 30, 31, 48 cytokines, 30, 31, 47 cytoplasm, 31 D damping, 298 data processing, 356, 359 data transfer, 269 decay, ix, 56, 120, 187, 188, 189, 196, 197, 199, 201, 202, 336 decibel, 62 decision making, 253 decisions, 246, 252 decoding, 114, 238, 246, 247, 252 decoupling, 108 defects, xii, 7, 164, 355, 356, 358, 363, 366 defense, 15, 272 deficiency, 30 definition, 78, 146, 240, 304, 317, 321 deformation, 84, 96, 99, 103, 207, 361, 362 degenerate, 71, 141, 274, 305, 339 degradation, 52, 54, 62, 220, 367 delivery, x, 6, 257, 258, 261, 275, 316, 333, 357 demand, 7, 52, 53, 260, 269, 270 Denmark, 255 density, vii, 3, 5, 6, 7, 9, 53, 56, 73, 154, 162, 163, 164, 172, 173, 175, 176, 181, 198, 199, 200, 202, 240, 271, 356 density fluctuations, 356 dependent variable, 59 depolarization, 222, 225 deposition, 45, 139, 154 derivatives, 199, 304, 309 destruction, 53 detection, viii, x, 4, 15, 21, 28, 29, 30, 37, 38, 40, 42, 44, 45, 46, 47, 48, 49, 131, 134, 217, 225, 231, 232, 233, 234, 235, 236, 237, 238, 239, 241, 249, 250, 255, 271, 275, 360 detection techniques, 271 detonation, 4 deviation, 68, 76, 77, 176, 179, 180, 182, 221 dielectric constant, 315 dielectric permittivity, 175 dielectrics, 271 Index 372 differential equations, 162, 165, 171, 175, 198 differentiation, 304, 309 diffraction, ix, 119, 120, 124, 125, 126, 143, 144, 152, 153, 154, 280 diffusion, 367 digital communication, 242 diode laser, 29, 133, 134, 165 diodes, 64, 65, 162, 165, 170 dipole, ix, 119, 120, 121, 128, 130, 134, 136, 146, 150, 151, 152, 154, 155, 171, 173 dipole moment, 173 dispersion, x, xi, 53, 58, 68, 71, 72, 110, 122, 123, 163, 165, 183, 206, 216, 218, 220, 226, 227, 228, 231, 232, 238, 241, 259, 261, 264, 274, 279, 280, 281, 285, 293, 301, 302, 303, 305, 307, 308, 309, 311, 312, 315, 316, 317, 318, 320, 321, 322, 324, 325, 326, 327, 328, 329, 330, 331, 333, 338, 347, 358, 362, 365 displacement, 297, 343, 362 distortions, 244 distribution, 5, 59, 69, 70, 120, 124, 126, 139, 140, 141, 142, 143, 144, 146, 150, 152, 153, 154, 173, 176, 192, 193, 198, 206, 246, 250, 356, 358 distribution function, 173 divergence, 206 diversity, 349 division, xi, 53, 112, 169, 182, 183, 187, 217, 227, 228, 270, 311, 335, 336, 337, 345, 346 DNA, viii, 4, 15, 21, 22, 27, 31, 32, 44, 48, 49 DOP, 221, 226 dopants, 215, 336 Doppler, 148 dream, 302 drinking water, 31 DRS, 72 DSC, 221 duration, 207, 244 dyes, 30 dynamic control, 220 dynamical systems, 299 E E. coli, viii, 15,23, 28, 29, 32, 35, 37, 38, 40, 41, 44 earth, 183, 202 EEA, 79 eigenvalue, 240, 321 Einstein, Albert, 154, 306, 312, 315 elaboration, 208 elasticity, 257 electric current, 34 electric energy, 4, 5 electric field, xi, 19, 121, 124, 127, 142, 171, 176, 177, 281, 301, 303 electrical power, 221 electrodes, 34 electromagnetic, viii, ix, 83, 84, 119, 120, 121, 124, 154, 171, 302 electromagnetic fields, ix, 120, 121, 154 electromagnetic waves, 302 electromagnetism, 5 electron(s), 55, 128, 131, 134, 144, 162, 173, 175, 280, 298, 342 ELISA, 31 elongation, 208, 209 emission, ix, 28, 30, 56, 57, 72, 94, 111, 165, 187, 189, 191, 192, 193, 194, 195, 196, 197, 198, 199, 201, 202, 219, 342 encoding, 114, 251 endotoxins, 29 endurance, vii, 3, 9 energetic materials, 4 energy, vii, xi, 3, 4, 5, 6, 7, 8, 55, 62, 77, 87, 134, 139, 151, 162, 164, 166, 173, 176, 178, 189, 190, 197, 198, 199, 200, 216, 257, 269, 280, 291, 301, 303, 316, 318, 320, 321, 325, 338, 342 energy density, 5 energy transfer, 342 enlargement, 54 environment, 5, 16, 30, 44, 272, 280, 359, 362, 366, 367 environmental change, 356 environmental conditions, 212, 337, 361, 362 enzyme(s), 4, 48 epoxy, 37 equilibrium, 39, 56, 280, 297 equipment, vii, 99, 100, 101, 104, 108, 366 erbium, ix, xi, 94, 111, 162, 164, 165, 166, 167, 168, 169, 182, 183, 184, 187, 188, 201, 219, 335, 336, 339, 340, 341 Escherichia coli, 22, 28, 45, 46 estimating, 251, 252 etching, 28, 33, 44, 98, 104, 105, 106, 107, 181 ethylene glycol, 38, 39 Euro, 276 European Union, 79 evanescent waves, 132, 133, 134, 151 evaporation, 143, 144 evidence, 31, 182 evolution, 4, 52, 54, 57, 59, 61, 62, 67, 68, 73, 74, 84, 89, 94, 96, 97, 105, 106, 110, 151, 162, 165, 171, 183, 281, 290, 291, 294, 296, 313, 356, 358, 364, 365 excitation, 29, 30, 32, 46, 127, 130, 131, 132, 134, 144, 196, 210 Index 373 exercise, 254 exploitation, 258 exponential functions, 197, 309 exposure, viii, 83, 85, 104, 106, 213, 214, 357 extinction, 169, 222, 223, 224 extrusion, 273 F fabrication, viii, ix, x, xii, 15, 33, 44, 181, 196, 205, 206, 213, 214, 216, 219, 226, 227, 229, 257, 272, 273, 355 Fabry-Perot filters, xii, 335 failure, xii, 84, 222, 355, 356, 358, 359, 360, 362, 363, 364, 365, 366 family, xi, 303, 315, 316, 317, 318, 320, 321, 328 fatigue, xii, 355, 356, 358, 359, 361, 364, 365, 366, 367, 368 feedback, xi, 54, 56, 161, 162, 163, 170, 182, 331, 335, 336, 341, 342, 352 FFT, 241 fiber aging, 357 fiber optics, 101, 104, 272, 366 fibers, vii, viii, ix, x, xii, 3, 4, 5, 6, 7, 8, 9, 15, 16, 17, 18, 21, 28, 29, 32, 34, 35, 37, 41, 42, 45, 51, 53, 55, 58, 59, 68, 72, 83, 84, 86, 87, 88, 89, 91, 92, 96, 97, 98, 104, 105, 106, 107, 108, 111, 119, 120, 121, 123, 124, 132, 133, 139, 154, 163, 167, 187, 188, 190, 191, 227, 228, 232, 257, 258, 259, 260, 261, 262, 263, 264, 265, 266, 269, 270, 271, 272, 273, 274, 275, 298, 299, 302, 311, 312, 316, 332, 333, 340, 355, 356, 357, 358, 359, 360, 361, 362, 364, 365, 366, 367, 368 fibre laser, 206 fibrinogen, 27, 30 film(s), 144, 215, 261, 273 filters, xii, 4, 109, 165, 166, 206, 209, 218, 219, 227, 228, 239, 335, 338, 350, 352 financial support, 309, 366 first generation, x, 257, 258 flame, 33, 34, 44, 208 flatness, 54, 69, 219, 220 flexibility, 15, 84, 110, 212, 257, 258, 274, 275 flight, 148 fluctuations, vii, 15, 55, 166, 215, 227, 286, 356 fluid, 280, 281 fluorescence, 26, 28, 29, 30, 31, 44, 48, 49, 56, 148, 150, 197 fluorine, 259 fluorophores, 48 focusing, 5, 119 food, 29, 44, 47 food poisoning, 29 Fourier, 125, 157, 219, 227, 239, 304, 310, 320 Fourier transformation, 125 four-wave mixing, xi, 169, 170, 311, 335, 336, 338, 339 France, 157, 161, 162, 164, 166, 168, 170, 172, 174, 176, 178, 180, 182, 184, 276, 355, 366 freedom, xi, 240, 293, 301, 303 FTTH, 269, 275, 356 function values, 65 functionalization, 37 fusion, 19, 33, 34, 35, 38, 44, 190, 206, 207 G gases, 4, 316 Gaussian, 127, 143, 144, 146, 147, 149, 153, 176, 180, 181, 239, 240, 241, 246, 250, 302, 312 gel, 45, 259 generalization, 165, 170 generation, ix, x, xi, xii, 54, 57, 108, 111, 113, 119, 120, 140, 141, 143, 152, 154, 257, 258, 260, 271, 275, 306, 312, 316, 324, 335, 336, 342, 346, 352 geometrical parameters, 219 germanium, 93, 122, 261 Germany, 81 Ginzburg-Landau equation, x, 279, 280, 297, 298, 299 Ginzburg-Landau equation (CGLE), x, 279, 297 glass(es), vii, ix, xii, 4, 5, 7, 8, 30, 88, 91, 120, 121, 122, 127, 131, 132, 133, 142, 188, 190, 192, 196, 197, 198, 200, 205, 227, 257, 258, 259, 260, 261, 262, 270, 273, 336, 355, 356, 357, 358, 360, 361, 365, 366, 367, 368 glass transition temperature, 259 global communications, 83 glucose, 35, 44, 275 glycine, 38 glycol, 38, 39 gold, 26, 37, 44, 48 graph, 41, 77, 96, 127, 181 gratings, viii, xii, 4, 54, 72, 83, 84, 91, 92, 98, 101, 102, 104, 108, 109, 114, 115, 161, 170, 184, 213, 218, 227, 232, 331, 332, 335, 340 gravity, 147, 151 grazing, 120, 132, 260 Green’s function, 176 Griffith’s flaws, xii, 355, 356 groups, 30, 36, 37, 55, 132, 163, 166, 196, 358 growth, xii, 28, 44, 46, 154, 232, 258, 269, 298, 318, 321, 323, 355, 357, 367 growth rate, 318, 321, 323 guidance, 17, 120, 127, 132, 133, 134, 136, 139, 146, 154, 155, 273, 316, 332 Index 374 H halogen, 192 Hamiltonian, 293 HD, 35 HDPE, 271, 272 HE, 122 healing, 31, 47 heart disease, 30 heat(ing), vii, x, 15, 17, 19, 33, 35, 38, 44, 97, 130, 136, 150, 151, 207, 208, 209, 213, 214, 257, 258, 275 height, 152, 177 helium, 133 hemoglobin, 30 high power density, 164 hip, 24, 26, 113 homeland security, 44 homogeneity, 111, 112 Hong Kong, 301 host, 264, 271, 273, 274, 336 house(ing), 33, 115, 202 humidity, xii, 4, 45, 280, 355, 356, 357, 361, 364, 366, 367 hybrid, 53, 65, 66, 68, 77 hybridization, 32, 44, 48 hydrocarbons, 45 hydrochloric acid, 36 hydrofluoric acid, 33, 44, 104 hydrogen, 45, 213, 259, 316, 332 hydrogenation, 213 hydrolysis, 36 hydroxide, 36 hydroxyl groups, 36, 196 hypothesis test, 249, 250 I identification, 233, 238 ignition energy, 8 IL-6, 27, 31 illumination, 260, 275 images, 4, 124, 142 imaging, 31, 120, 126, 143, 144, 146, 147, 150, 155, 271, 357 immobilization, viii, 15, 28, 29, 36, 37, 40, 43, 47 immunity, viii, 83, 84 impairments, 238 implementation, xii, 55, 57, 62, 66, 68, 112, 113, 115, 228, 335, 336 impurities, 55 in situ, 164 in vivo, 31, 48 incidence, 16, 120, 132, 260 inclusion, 254, 282 India, 55 indication, 31 indices, 6, 121, 122, 212, 213, 214, 263, 264, 269 indirect measure, 84 industrial production, 365 industry, 357 inelastic, 55, 293 infarction, 30 infinite, 121, 127, 207, 212, 345 information processing, 206 information technology, 83 inhibition, 48 initial state, 246, 297 inoculation, 23 insertion, viii, 83, 84, 108, 114, 165, 188, 190, 200, 201, 206, 215, 216, 218, 220, 221, 223, 336, 346 insight, 18 instability, xi, 111, 285, 289, 298, 302, 315, 316, 320, 321, 323, 328 instruments, 191 integrated optics, 178 integration, 59, 60, 218, 225, 226, 238, 240, 254 intensity, 4, 28, 30, 44, 53, 56, 102, 120, 122, 123, 124, 126, 127, 128, 130, 131, 132, 133, 134, 138, 139, 141, 142, 144, 145, 146, 147, 150, 152, 153, 154, 165, 180, 181, 188, 211, 212, 217, 232, 305, 331, 362 interaction(s), viii, xi, 17, 29, 43, 52, 53, 54, 55, 57, 64, 76, 77, 133, 136, 152, 166, 171, 279, 293, 294, 298, 299, 302, 315, 316, 317, 324, 325, 326, 332, 336, 342 interface, 16, 53, 124, 132, 142, 209, 210, 258, 260 interference, 5, 84, 108, 113, 114, 115, 152, 174, 176, 185, 217, 232, 241 internet, 80, 187, 255, 269 interpretation, 21 interval, 232, 233, 236, 237, 240, 243, 246, 247, 249 inversion, 63, 183, 233, 342 investment, 52 ionization, 134, 139 ions, 4, 128, 131, 189, 190, 198, 200, 201, 219, 261, 336 IR, 35, 42, 80, 272, 273, 274 Islam, 81 isolation, 212, 216, 218, 220, 223, 224, 346 isotope(s), 136, 137 Italy, 51, 81, 161, 227 iterative solution, 165 Index 375 J Japan, 33, 134, 187, 276, 298, 312 K Karhunen-Loeve Series Expansion (KLSE), x, 231, 238, 239, 240, 241, 243, 253, 254 kernel, 240, 241, 253 kinetics, xii, 32, 40, 48, 355 Korea, 119, 134, 231, 276, 335 L laser(s), vii, viii, ix, x, xi, xii, 3, 4, 5, 6, 7, 8, 9, 29, 31, 42, 51, 53, 54, 56, 64, 65, 68, 69, 70, 71, 72, 73, 74, 75, 81, 108, 111, 112, 115, 119, 120, 121, 123, 124, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 187, 189, 191, 196, 201, 202, 206, 207, 208, 215, 221, 222, 226, 233, 234, 257, 258, 260, 261, 269, 272, 274, 275, 280, 281, 287, 297, 298, 299, 300, 316, 332, 333, 335, 336, 337, 338, 339, 340, 341, 342, 343, 344, 345, 346, 347, 348, 349, 350, 351, 352, 357 laser radiation, x, 257, 258, 272, 275 lasing effect, 72, 73 lattices, 313 leakage, 53, 143, 172, 175, 177, 208, 258, 262, 264, 266, 316 lectin, 29 LED, 206 lens, 7, 120, 126, 147, 150, 155 leprosy, 31 lifetime, 84, 152, 196, 197, 198, 342, 356, 359, 364, 365, 366 light beam, 136, 146 light scattering, 136 light transmission, 18, 28, 39 likelihood, 238, 246, 249 limitation, 164, 227, 259 linear function, 109 linkage, 29, 38 links, vii, 53, 275, 311 liquid nitrogen, 336, 338, 352, 356, 357 liquid phase, 45 Listeria monocytogenes, 23, 28, 46, 47 literature, 16, 98, 164, 165, 166, 170, 200, 260, 274, 303, 305, 307 local area networks, 357 location, 18, 228, 304, 305, 342 long distance, 120, 221, 232, 320, 323 LPS, 29 luminescence, 189, 196, 197, 201 lysine, 28 M magnetic field, 150 malaria, 31 management, 52, 302, 303, 311 Manakov model, xi, 301, 303 manipulation, 119, 154, 206 manufacturer, 37 manufacturing, 207, 257, 258 mapping, 176, 236 market(s), 84, 258, 260, 275, 356, 357 Marx, 48 masking, 144 Massachusetts, 81 matrix, 59, 73, 81, 99, 100, 103, 108, 162, 173, 210, 211, 223 Maxwell's equations, 162 measurement, viii, 15, 32, 46, 47, 84, 98, 99, 100, 104, 108, 126, 148, 162, 169, 201, 202, 358, 359 measures, 60, 303, 305, 358 mechanical properties, vii, 5, 263 mechanical stress, 94 mechanical testing, 367 media, viii, xi, xii, 35, 47, 51, 169, 306, 313, 315, 331, 335, 336, 352 medical diagnostics, 44, 275 medicine, 84 melting, vii, 190 memory, 184 metals, 271 methanol, 26 microarray, 48 microcavity, 149 micrometer, 96 microorganisms, 28 microscope, 7, 31, 33, 105, 144, 146, 208 microscopy, 272 microspheres, 31 microwave, 108, 264, 271 military, 272 miniaturization, 5 mitochondria, 31 mixing, xi, 54, 169, 170, 183, 232, 302, 311, 335, 336, 338, 339, 341 model system, 305 modeling, 46, 59, 63, 68, 166, 183, 273, 308 Index 376 models, x, 162, 164, 165, 170, 231, 305, 316, 356, 358, 366, 367 modules, 52, 206, 212, 218, 222, 227 modulus, 261, 305, 310, 361 moisture, xii, 355, 367 molecular weight, 269 molecules, 15, 32, 43, 44, 49, 55, 259, 273, 332, 356 moon, 353 Morocco, 205 morphology, 357 motion, 131, 294 motivation, 305 multidimensional, 313 multiples, 54, 237 multiplicity, 280 multiplier, 128, 131, 134 muscles, 30 mutation, 65, 68 myocardial infarction, 30 myoglobin, 30, 47 N NADH, 24 nanometers, 222 National Science Foundation, 45 national security, 29 Nd, 4, 6, 164, 196 necrosis, 27, 31 neodymium, 164 Netherlands, 255, 367 network(ing), ix, x, 52, 114, 188, 205, 207, 216, 218, 226, 227, 269, 356, 366 neural networks, 65 New York, 79, 115, 155, 156, 157, 158, 255, 297, 299, 311, 313, 366 next generation, 108 NIR, 192 nitrogen, xi, 133, 335, 336, 338, 341, 352, 356, 357 nodes, 124, 146, 154 noise, viii, 51, 52, 54, 56, 57, 62, 63, 64, 75, 162, 165, 166, 167, 168, 169, 182, 184, 189, 191, 206, 218, 220, 221, 228, 238, 239, 240, 241, 246, 247, 250, 302 nonequilibrium, 299 nonlinear dynamics, 154 nonlinear optical response, 302, 331 nonlinear optics, 4, 279, 280, 281, 302 nonlinear Schrödinger model, xi nonlinear systems, 297 normalization constant, 127 numerical analysis, 123, 262 numerical aperture, 6, 7, 168, 190, 198, 259 numerical computations, 65 O observations, 244, 366 OFS, 188 oligomers, 48 one dimension, 233, 234 On-Off Keying (OOK), x, 231, 232, 233, 238 operator, 65, 171, 173, 304, 309 optical communications, ix, 4, 6, 76, 83, 91, 115, 120, 164 Optical Differential Phase Shift Keying (oDPSK), x, 231, 232, 233, 234, 235, 236, 237, 238, 239, 240, 241, 242, 243, 245, 246, 247, 248, 249, 251, 253, 254, 255 optical fiber, vii, viii, ix, x, xi, xii, 4, 5, 6, 7, 15, 17, 30, 31, 33, 34, 45, 46, 48, 51, 53, 54, 55, 57, 72, 76, 84, 85, 87, 102, 119, 120, 121, 123, 124, 127, 132, 136, 137, 138, 139, 146, 151, 154, 163, 164, 165, 183, 187, 188, 227, 232, 234, 257, 258, 259, 268, 269, 271, 272, 275, 279, 297, 302, 311, 312, 315, 316, 333, 335, 342, 355, 356, 357, 358, 361, 364, 365, 366, 367, 368 optical gain, 56, 221 optical polarization, 222 optical properties, 55, 56, 84 optical pulses, 163, 233, 281, 305, 311 optical solitons, 280, 302, 332 optical systems, 215, 308 optics, vii, ix, 4, 83, 101, 104, 109, 115, 119, 120, 126, 178, 207, 272, 279, 280, 281, 302, 315, 366 optimization, viii, ix, 52, 60, 64, 65, 68, 69, 76, 77, 83, 115, 225, 244, 273 optimization method, 65 ordinary differential equations, 198 orientation, 98, 236 orthogonality, 18, 19, 233 OSA, 69, 94, 167, 192, 193, 201, 208, 228 oscillation, 153, 162, 165, 166, 170, 209 oscillograph, 8 oxides, 365 oxygen, 357 P palladium, 45, 46 PAN, 192 parameter, xi, 6, 17, 41, 71, 177, 199, 219, 266, 280, 281, 287, 289, 290, 291, 294, 295, 301, 303, 305, 307, 309, 310, 316, 358, 359, 360, 362 Paris, 276 Index 377 particles, 154, 275 particulate matter, 21 partition, 246 passive, viii, ix, x, 54, 83, 84, 165, 166, 205, 224, 269, 271, 272, 275 pathogens, 28, 44, 48 PBC, 113, 222, 223, 224 PCF, 169, 170, 262, 316, 332, 338, 339 PDEs, 281 penalties, x, 70, 231, 233 performance, x, 7, 8, 62, 63, 65, 113, 115, 162, 165, 166, 169, 171, 182, 189, 218, 219, 220, 228, 231, 232, 233, 236, 238, 239, 240, 241, 242, 243, 244, 247, 248, 254, 255, 258, 259, 260, 275 periodicity, viii, 83, 89, 166, 265 permit, 290, 304, 305 permittivity, 172, 175, 177 pH, 38, 39, 42 phase shifts, 234, 236 phase transitions, 280 phonons, 55, 56, 63 phosphate, 28 photoelastic effect, 87, 110 photographs, 91, 105 photonic crystal fiber, vii, 258, 262, 273, 316, 332, 333, 338 photonic crystal fiber (PCF), 338 photons, 55, 56, 151, 189, 194, 196, 197, 198 photosensitivity, 84, 213, 227 physics, viii, 15, 44, 183, 279, 281, 297, 305, 309, 315 pitch, 343 plane of polarization, 141 plasma, 23, 26, 30, 279, 297 plasminogen, 30 plastics, 271 PM, 69, 92, 223, 226, 346, 351 PMMA, x, 257, 258, 259, 260, 262, 263, 264, 266, 268, 269, 270, 275 POFs, x, 257, 258, 259, 260, 275 polarity, 238 polarization, vii, ix, x, xi, xii, 57, 58, 64, 83, 84, 87, 88, 89, 90, 94, 96, 97, 98, 99, 100, 102, 103, 106, 108, 109, 110, 111, 112, 113, 114, 115, 140, 141, 148, 165, 166, 167, 169, 170, 171, 172, 173, 175, 176, 183, 206, 216, 218, 220, 221, 222, 223, 224, 225, 226, 227, 228, 229, 231, 264, 274, 275, 301, 303, 335, 337, 339, 340, 342, 343, 344, 345, 346, 347, 348, 349, 351, 352 polarized light, 87, 88 polycarbonate, 260, 261, 271 polyethylene, 271, 273 polyimide, 260 polymer(s), x, 257, 258, 259, 260, 261, 262, 263, 268, 269, 272, 273, 274, 275, 357, 361, 365, 367 polymer molecule, 259 polymeric materials, 257 polymerization, 259, 269 polymethylmethacrylate, x, 257 polystyrene, 22, 23, 29 poor, 35, 264, 338 population, 63, 65, 66, 68, 165, 198, 342 population size, 66 ports, 207, 223, 224 Portugal, 48, 51, 81, 83, 279 positive correlation, 194, 201 positive feedback, 56 power, vii, ix, x, 3, 4, 5, 6, 7, 8, 9, 19, 28, 30, 52, 53, 54, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 88, 111, 114, 115, 132, 134, 135, 138, 139, 142, 146, 147, 149, 150, 152, 153, 162, 163, 164, 165, 167, 169, 170, 176, 177, 179, 182, 187, 188, 189, 191, 192, 193, 194, 195, 196, 197, 199, 200, 201, 202, 206, 207, 208, 209, 210, 211, 216, 218, 219, 220, 221, 222, 225, 226, 227, 234, 254, 257, 258, 260, 261, 263, 265, 275, 302, 311, 316, 338, 342, 344, 346, 347, 348, 356, 357 prediction, 290, 321, 356 pressure, viii, 4, 15, 45, 68, 131, 149, 213, 227, 280, 366 prices, 68 probability, xii, 7, 173, 197, 198, 238, 240, 247, 250, 253, 355, 358, 359, 362, 365 probability density function, 240 probe, 27, 29, 32, 57, 61, 63, 64, 67, 68, 69, 70, 72, 75, 77, 148, 150 production, viii, 83, 84, 91, 108, 268, 365 production costs, 84 production technology, viii, 83, 108 progesterone, 31, 47 prognosis, 49 program(ming), 19, 35, 65, 79, 302 propagation, ix, 18, 19, 52, 61, 65, 74, 75, 83, 84, 85, 87, 89, 108, 109, 122, 127, 144, 146, 147, 165, 171, 172, 177, 189, 191, 206, 208, 210, 211, 225, 226, 257, 258, 260, 262, 271, 281, 283, 285, 287, 288, 289, 291, 292, 294, 295, 296, 298, 299, 302, 305, 311, 312, 316, 317, 321, 322, 331 propane, 33 protein(s), viii, 15, 24, 25, 26, 28, 29, 30, 31, 42, 44, 46, 47 protocol, 32, 37, 39 prototype, 54 PTT, 367 Index 378 pulse(s), x, xi, 6, 108, 128, 131, 163, 164, 165, 166, 169, 183, 196, 232, 233, 238, 241, 243, 244, 279, 280, 281, 283, 285, 286, 287, 288, 289, 290, 291, 292, 293, 294, 295, 296, 297, 298, 299, 300, 301, 302, 303, 304, 305, 308, 311, 312, 316, 324, 331, 333 pumps, viii, 51, 53, 54, 57, 59, 60, 61, 63, 64, 65, 67, 68, 69, 70, 74, 76, 77, 78, 79, 108, 112, 221, 222, 226 purification, 273 pyrene, 31 Q QED, 136, 139, 154 quantum phenomena, 171 quantum well, 161 quartz, 26 R radial distribution, 153 radiation, x, 4, 7, 80, 149, 166, 171, 172, 176, 179, 198, 213, 214, 257, 258, 260, 271, 272, 275, 302, 320, 326 radio, 114 radius, 6, 17, 18, 19, 33, 121, 127, 144, 145, 146, 147, 162, 171, 172, 176, 177, 181, 182, 260, 264, 265, 269 Raman, viii, xi, xii, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 163, 164, 165, 182, 206, 207, 218, 219, 221, 222, 227, 228, 229, 232, 275, 299, 305, 312, 316, 332, 335, 336, 342, 345, 346, 347, 349, 350, 351, 352 Raman and Brillouin scattering, 55 Raman spectroscopy, 55 Raman-scattering, 312 range, vii, x, 7, 28, 29, 30, 35, 41, 56, 58, 64, 65, 76, 84, 102, 108, 110, 132, 133, 162, 163, 166, 167, 168, 180, 182, 184, 191, 193, 207, 208, 212, 214, 215, 221, 222, 224, 226, 228, 234, 249, 257, 258, 261, 265, 266, 267, 268, 269, 271, 272, 274, 280, 285, 289, 293, 317, 319, 323, 343 reactant, 41 reactive sites, 41 reality, 56, 280, 293 reasoning, 305, 307, 310 reception, 76, 77 receptors, 48 recognition, 29, 32, 44 recombination, 175, 217, 342 recovery, 167 recurrence, 297 redistribution, 176 reduction, 40, 53, 54, 61, 131, 232, 244, 259, 262, 281, 293 redundancy, 238 reflection, vii, viii, ix, x, 7, 16, 70, 72, 83, 85, 90, 91, 93, 94, 96, 97, 98, 101, 102, 103, 105, 106, 108, 113, 136, 142, 168, 182, 189, 191, 216, 227, 257, 258, 260, 266, 316, 343, 345, 346, 350 reflectivity, 71, 85, 86, 90, 114, 172, 177, 258, 260, 261, 273, 349 refraction index, 85, 88, 89 refractive index(ices), viii, ix, 5, 6, 8, 15, 16, 17, 21, 35, 40, 42, 43, 44, 83, 84, 85, 87, 89, 121, 122, 132, 146, 168, 175, 190, 212, 213, 214, 215, 218, 220, 232, 258, 259, 260, 261, 264, 265, 266, 268, 271, 281, 343 regenerate, 36, 218 regeneration, 52 rehydration, 39 reinforcement, 356 relationship, 8, 9, 19, 146, 147, 150, 172, 173, 177, 178, 180, 199, 267, 268 relaxation, 59, 63, 175, 196, 317, 320, 364, 365 reliability, xii, 5, 355, 356, 357, 365, 366, 367 remote sensing, 44 reparation, viii, 15 resistance, 84, 258, 259, 275, 357 resolution, 61, 84, 119, 144, 162, 167, 192, 252, 272 resonator, 162, 163, 166, 172, 176, 177, 178, 179, 183 response time, 221 RF, 346 rings, viii, 51, 166, 262, 263, 264, 273, 361 RNA, 27, 32 robustness, 44, 249 Romania, 355 room temperature, 8, 36, 111, 338, 339, 365 root-mean-square, 139 roughness, ix, 161, 162, 171, 172, 176, 177, 178, 179, 180, 181, 182, 262 routines, 304 routing, 108, 205 Royal Society, 309 rubidium, 122, 134 Russia, 55 S SA, 43, 44, 51 safety, 5, 53 Salmonella, 22, 28, 46 Index 379 sample(ing), viii, 15, 16, 17, 23, 36, 37, 39, 40, 42, 43, 44, 94, 102, 105, 154, 196, 197, 233, 237, 239, 252, 343, 356, 359, 360, 363 sapphire, 8, 131, 147, 261, 271, 333 saturation, ix, 43, 44, 54, 120, 161, 162, 168, 174, 176, 179, 221, 342 scaling, 164 scattered light, 55, 70, 132, 143 scattering, viii, ix, 28, 39, 51, 53, 54, 55, 56, 57, 62, 79, 136, 143, 151, 153, 162, 164, 174, 176, 177, 187, 189, 191, 192, 193, 194, 197, 200, 201, 219, 221, 232, 264, 275, 299, 305, 312, 316, 332, 336, 342 schema, x, 231, 233, 234, 235, 236, 237, 241, 247, 249, 254 scholarship, 79 Schrödinger equation, 279, 281, 298 science, vii, 271, 302 scientific community, 302 search, 65, 166 security, 29, 44 seed(ing), 54, 74, 165, 320 selecting, 7, 63, 235 selectivity, 32, 40, 44 self-phase modulation, 183 semiconductor, ix, xi, 4, 54, 161, 162, 168, 169, 170, 171, 172, 173, 178, 180, 182, 184, 185, 335, 336, 342, 348 semiconductor lasers, 173 sensing, viii, ix, 15, 16, 21, 30, 31, 44, 45, 46, 83, 84, 98, 104, 115, 169, 182, 205, 272, 274, 275 sensitivity, 28, 29, 35, 41, 44, 94, 96, 97, 98, 114, 162, 170, 178, 182, 217, 227, 244, 249, 350 sensors, vii, viii, ix, xi, 4, 15, 16, 21, 41, 44, 45, 46, 47, 83, 84, 93, 98, 108, 111, 115, 260, 272, 275, 335, 357 separation, 93, 98, 136, 137, 213, 293, 294, 306, 317, 324, 329, 343 sepsis, 29 series, 134, 219, 240, 268, 304, 310, 363 serum, 23, 26, 31, 45 shape(ing), 17, 57, 88, 108, 127, 146, 151, 180, 208, 219, 280 sharing, 167 shock, 275 sign(s), 57, 120, 122, 237, 238, 250, 251, 252, 253, 288, 294 signaling, 49, 232, 233, 236, 237, 243, 246, 247, 249 signals, 5, 28, 53, 54, 57, 59, 61, 62, 63, 64, 67, 68, 69, 70, 73, 77, 98, 114, 130, 134, 148, 150, 164, 187, 188, 189, 206, 207, 216, 218, 221, 226, 236, 250, 302 signal-to-noise ratio, 54, 165 silane, 32 silica, xii, 5, 6, 16, 23, 24, 25, 26, 31, 32, 48, 87, 93, 122, 182, 183, 188, 191, 199, 202, 206, 213, 215, 221, 226, 227, 257, 258, 260, 261, 262, 270, 271, 273, 316, 332, 333, 336, 355, 356, 357, 361, 364, 365, 366, 367, 368 silicon, 154, 271 silver, 6, 183, 273 simulation, viii, 15, 18, 19, 20, 21, 64, 66, 67, 68, 69, 73, 74, 77, 81, 86, 90, 92, 110, 145, 149, 152, 162, 171, 178, 294, 341 SiO2, 190 sites, 7, 29, 41, 52, 71, 216 smoothing, 366 sodium, 36 sodium hydroxide, 36 software, 33, 34 solitons, x, xi, 183, 279, 280, 281, 286, 287, 288, 289, 290, 293, 294, 297, 298, 299, 302, 303, 305, 306, 307, 311, 312, 313, 315, 316, 317, 318, 319, 320, 322, 323, 324, 325, 326, 327, 328, 329, 330, 331, 332, 333 solvent, 37 species, 28, 29, 196, 201, 280, 293 specificity, 40 spectral component, 58 spectroscopy, xi, 30, 55, 162, 271, 335, 357 spectrum, ix, 35, 56, 58, 65, 69, 70, 71, 72, 74, 76, 77, 83, 91, 92, 93, 94, 98, 101, 102, 103, 106, 112, 131, 132, 136, 137, 140, 163, 166, 167, 188, 192, 201, 216, 218, 219, 226, 270, 291, 302, 315, 316, 317, 336, 342, 348, 349, 350 speed, 18, 65, 162, 187, 188, 205, 208, 217, 227, 255, 302, 316, 361, 362 speed of light, 18, 316 spin, 357 sprouting, 22, 46 stability, x, xi, 44, 59, 68, 73, 149, 165, 167, 205, 285, 288, 291, 293, 298, 299, 309, 315, 316, 318, 320, 321 stabilization, 72, 73, 74, 75, 178, 352 stages, 247 standard deviation, 179, 180, 182 standardization, 259 standards, 68, 149 staphylococcal, 47 Staphylococcus, 22, 29, 47 statistics, 356, 358 steel, 271 sterile, 37 stimulant, 29 stock, 39 storage, 272, 316 Index 380 strain, 40, 45, 84, 92, 93, 94, 96, 97, 98, 99, 100, 101, 102, 103, 104, 107, 108, 115, 348, 350, 356 strength, xii, 7, 17, 32, 34, 58, 132, 275, 286, 355, 356, 357, 359, 360, 364, 365, 366, 367 stress, xii, 7, 87, 88, 91, 93, 94, 104, 106, 107, 109, 110, 115, 355, 356, 357, 358, 359, 360, 361, 362, 363, 364, 365, 366 stress intensity factor, 362 stretching, 259 strong interaction, 29 structural relaxation, 365 subtraction, 310 sulphur, 273 sun, 116, 183, 202, 277 supply, 44, 192, 280, 366 suppression, 162, 167, 170, 185, 338 surface layer, 43 susceptibility, 56 suspensions, 35 switching, 108, 170, 331, 341, 343, 344, 347, 351, 352 symbols, 217, 236, 237, 238, 244, 246, 247, 249, 250, 251, 253 symmetry, 211 synchronization, 163 synthesis, 32 systems, viii, ix, x, xi, xii, 51, 52, 53, 54, 55, 56, 63, 69, 70, 75, 76, 77, 78, 79, 83, 84, 108, 114, 162, 164, 165, 169, 171, 182, 187, 188, 205, 206, 207, 215, 221, 222, 231, 232, 234, 236, 238, 239, 241, 243, 244, 246, 247, 254, 255, 271, 275, 279, 280, 281, 293, 297, 298, 299, 301, 303, 305, 306, 308, 311, 312, 316, 332, 335, 336, 352, 365, 366 T technical assistance, 366 technology, vii, viii, 3, 7, 51, 52, 55, 68, 83, 84, 108, 187, 213, 222, 232, 259, 271, 302, 356 teflon, 273 telecommunication networks, xii, 206, 218, 355, 356, 361, 365 telecommunications, x, 53, 56, 111, 162, 166, 205 temperature, viii, x, xii, 5, 7, 8, 15, 33, 36, 45, 57, 58, 84, 92, 93, 97, 98, 99, 100, 101, 102, 103, 104, 107, 108, 110, 111, 115, 140, 148, 150, 152, 166, 169, 184, 191, 206, 208, 212, 213, 214, 215, 219, 220, 226, 257, 258, 259, 260, 261, 272, 280, 338, 339, 350, 355, 356, 357, 359, 361, 362, 364, 365, 366 temperature dependence, 57, 97, 102 temperature gradient, 110 tensile strength, 359 tensile stress, 356, 357 tension, 33, 34 theory, vii, x, 3, 18, 85, 86, 121, 124, 125, 127, 144, 154, 162, 171, 249, 279, 281, 286, 291, 297, 299, 302 therapy, 31, 275 thermal expansion, 213 threat, 29, 44 threshold(s), vii, 3, 6, 8, 9, 53, 56, 72, 74, 132, 138, 139, 163, 164, 168, 170, 178, 179, 180, 181, 233 threshold level, 163 thrombin, 30, 31, 48 tics, x, 231, 233, 234, 235, 236, 237, 241, 247, 249, 254 time(ing), viii, xi, xii, 5, 8, 30, 32, 33, 37, 41, 42, 43, 45, 46, 47, 52, 57, 59, 61, 64, 65, 66, 67, 72, 84, 104, 105, 106, 108, 112, 113, 114, 115, 148, 150, 151, 152, 162, 163, 165, 170, 171, 175, 187, 188, 192, 197, 198, 199, 213, 214, 217, 221, 228, 233, 239, 244, 254, 258, 262, 280, 281, 293, 299, 301, 303, 309, 311, 317, 324, 355, 358, 359, 360, 364, 365, 366 tin, 175 TIR, 16 TNF-alpha, 27, 31 Tokyo, 158, 203 topology, 68 total energy, 320, 321 total internal reflection, vii, x, 16, 257, 258, 316 toxin, 23, 29 tracking, 183 trade, 258 traffic, 53, 187, 269 trajectory, 294, 349 transference, 81 transformation, 125, 288 transition(s), 130, 134, 138, 148, 150, 151, 164, 173, 189, 190, 197, 198, 199, 208, 245, 246, 259, 280 transition rate, 198 transition temperature, 259 translation, 93, 96 transmission, vii, viii, x, 6, 7, 16, 18, 19, 20, 21, 28, 33, 34, 35, 38, 39, 40, 41, 42, 43, 44, 45, 51, 52, 53, 54, 55, 57, 58, 63, 70, 71, 75, 76, 77, 78, 84, 114, 115, 136, 137, 138, 139, 182, 187, 188, 205, 206, 207, 208, 211, 212, 213, 215, 217, 220, 221, 223, 226, 228, 231, 232, 244, 246, 247, 255, 257, 258, 259, 260, 261, 262, 263, 264, 268, 269, 270, 271, 272, 273, 274, 275, 280, 293, 297, 298, 300, 302, 311, 343, 345, 347, 356 transmission path, 78 transmits, x, 8, 21, 232, 257 transparency, 316, 331, 332 Index 381 transport, 30, 48 transverse section, 91, 94, 105 trend, 43, 106, 130 triggers, 56 tuberculosis, 31 tumor necrosis factor, 27, 31 tumors, 275 tunneling, 176 typhoid, 31 U UK, 81, 301 ultraviolet light, viii, 83, 85 uniform, 16, 17, 18, 21, 22, 23, 68, 85, 86, 87, 110, 152, 218, 331, 359 urine, 23 users, 112, 113, 114, 366 UV, 213, 214, 227 UV radiation, 213, 214 V vacuum, 6, 36, 124, 128, 131, 132, 134, 142, 175, 356, 357 Vakhitov-Kolokolov criterion, 318 valence, 173 values, 19, 20, 21, 59, 60, 65, 67, 68, 70, 76, 87, 92, 96, 97, 99, 100, 101, 104, 108, 126, 139, 147, 162, 176, 178, 179, 180, 181, 200, 237, 244, 247, 248, 249, 252, 267, 273, 282, 283, 290, 291, 293, 294, 295, 308, 358, 359 vapor, 45, 131, 148, 366 variability, 214 variable(s), xi, 4, 59, 174, 220, 240, 253, 301, 302, 303, 306, 309, 312, 339, 364 variance, 240, 254 variation, xi, 92, 93, 94, 98, 142, 167, 200, 219, 226, 229, 252, 269, 287, 301, 303, 315, 362 vector, 127, 172, 173, 175, 176, 183, 210, 211, 250, 313 vehicles, 84 velocity, xi, 57, 87, 105, 133, 134, 139, 150, 163, 172, 212, 281, 291, 294, 295, 296, 297, 301, 302, 305, 316, 317, 318, 321, 322, 324, 325, 327, 329, 330, 362 vibration, 259 viscosity, 4 voice, 4, 302 W water absorption, 224 wave number, 6, 122, 132, 304, 305, 309 wave propagation, 257, 305 wavelengths, viii, xii, 42, 51, 53, 54, 63, 64, 68, 71, 72, 75, 76, 77, 87, 89, 97, 98, 106, 107, 109, 112, 113, 114, 163, 170, 188, 193, 208, 213, 216, 220, 221, 222, 260, 261, 262, 270, 272, 335, 336, 337, 338, 339, 342, 343, 346, 347, 348, 349, 350, 352 welding, 272 wells, 177 windows, 7, 8, 53 workers, 164, 259 wound healing, 31 writing, 350 X X-axis, 95, 97 Y Y-axis, 95, 97, 102 yield, 304, 305 ytterbium, 164, 333