Analysis of Second order Dispersion on Free Space Optical PropagationAnalysis of Second order Dispersion on Free Space Optical Propagation David F. W. Yap1, Y. C. Wong1, C. Wasli1, S. P. Koh2, S. K. Tiong2, M. A. E Mohd Tahir3 Faculty of Electronics and Computer Engineering, Universiti Teknikal Malaysia Melaka (UTeM) Hang Tuah Jaya, 76100, Durian Tunggal, Melaka, Malaysia. 1 College of Engineering, Universiti Tenaga Nasional (UNITEN) km 7, Kajang-Puchong Road, 43009 Kajang, Selangor Darul Ehsan, Malaysia. 2 Faculty of Engineering & Technology, Multimedia University, Jalan Ayer Keroh Lama, 75450 Melaka, Malaysia. 3
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[email protected] Abstract Free space optic (FSO) can be regarded as a potential and attractive option to fiber optic. FSO has the ability to go beyond the limit of fiber optics. Unfortunately, due to the dispersion effect in the atmosphere, FSO suffers from signal loss and attenuation. Thus, practical and detailed research is needed to improve the system. Simulation on FSO propagation using measured parameter values is important to gain better understanding and level of accuracy on the pulse behavior in free space. Using MATLAB as the simulation platform and with the help of experimental parameter values, an accurate model can be obtained and studied. This will allow some level of prediction on the behavior of the propagating light pulse in the atmosphere and subsequently the FSO performance can be further improved. Keywords: Atmospheric turbulence, Binary Pulse Position Modulation, Bit Error Rate, Simulation System. I. INTRODUCTION Laser communication in free space offers an attractive alternative for transferring high-bandwidth data when optical fiber cable is either impractical or not viable. Here, wireless optical connectivity can be used as the last mile to connect fiber backbone to end users, such as from ISSN: 2180 - 1843 Vol. 2 building to building, due to the cost and time-consumption on top of the impossibility and impracticality in laying down optic fibers [1]. Other advantages of adopting the optical wireless communication systems, also termed as free space optics (FSO) or lasercom (laser communications), includes [2]:a) no licensing or tariffs fees required for its utilization [3]; b) small, lightweight and compact; c) ease of installation and deployment (digging up of road is unnecessary); d) it offers very high data rates due to its large bandwidth; e) high security fears (the extremely directional, narrow beam optical link makes eavesdropping and jamming nearly impossible); f) it operates at low power consumption; g) there are no rf radiation hazards (eye-safe power levels are maintained). However, random fluctuations in the atmosphere’s refractive index can severely degrade the wave front of a signal-carrying laser beam, causing the receiver to suffer from intensity fading. No. 1 January - June 2010 59 Research related to pulse propagation in both fiber optic and FSO show the propagating pulse is affected by both linear and nonlinear elements. The linear effects include the group velocity dispersion (GVD) and third order dispersion (TOD). while the nonlinear effects comprise of self phase modulation (SPM). In this paper.1843 Vol. used to mathematically explain varying pulse envelope in Equation (2) is the linear part ofpropagating NLSE. dispersion coefficient respectively. suitable forof describing propagation in free space. Numerical solution for NLSE canEquation be obtained (3) by applying TOD and attenuation. NONLINEAR(1) SCHRÖDINGER EQUATION generalized form of NLSESchrödinger for complex envelope The Nonlinear Equation (NLSE) isA(z. 60 ISSN: 2180 .t). The β2 and are the quadratic and complex envelope A(z. NLSE is consists secondpulse order dispersion (SOD). E3 d 3 AtheDgeneralized 2form of NLSE for iE2 (1) dAEquation d 2 Arepresents envelope A(z.t). 2 Equation represents the II. it is important to have an accurate prediction model to estimate pulse behavior in the atmosphere.isTOD cubic dispersion respectively. β3 Equation (2) is the linear part of The It Nonlinear Schrödinger Equation (NLSE) usedand to NLSE. The nonlinear Schrödinger equation is briefly discussed in Section II while the type of pulses used in the simulation is shown in Section III. data reliability can be compromised and may lead to the increase in BER. Thus. Electronic and Computer Engineering This results in increased system bit error rates (BERs) particularly along horizontal propagation paths [4]. NLSE is the SPM. consists of secondcoefficient order dispersion (SOD). Thus. Thus. II.Journal of Telecommunication. Į is the attenuation factor nonlinear coefficient Numerical NLSE can and Ȗ is the solution nonlinearfor coefficient [5].t). Equation (3) is the nonlinear part of NLSE that αmedium is the attenuation factor and γ is the adenotes with linear and nonlinear elements. Numerical solution for NLSE can be obtained by applying split step Fourier (SSF) or beam propagation (BPM) method. Simulation results on the second order dispersion are presented in Section IV. . propagation in free space. Thus. Both the linear and nonlinear effects are responsible for pulse broadening as well as distortion [5]. mathematically explain varying pulse envelope propagating in attenuation. NLSE is suitable for describing pulse propagation in free space. In fiber optic the extent of these effects can be estimated and anticipated through numerous literatures and research. be obtained by applying split step Fourier (SSF) or beam propagation (BPM) method. is the split step Fourier (SSF) or beam propagation (BPM) method. The ȕ2 and ȕ3 are the quadratic and cubic suitable for describing pulse [5]. NONLINEAR SCHRÖDINGER EQUATION SPM. It a medium with linear and nonlinear elements. nonlinear part of NLSE that denotes the Equation (1) represents the generalized form of NLSE for II. Unfortunately for FSO the extent of these effects cannot be estimated easily due to the random nature of the atmosphere. The conclusion is given in Section V. A iJ(2)| Ais| the A linear part of complex Equation 2 dz 2 dt dt 2 order 2 dispersion (SOD). NONLINEAR SCHRÖDINGER EQUATION The Nonlinear Schrödinger Equation (NLSE) is used to mathematically explain varying pulse envelope propagating in a medium with linear and nonlinear elements. the simulation on FSO is carried out without the nonlinear effects. Based on the severity of these effects. The ȕ2 and ȕ3 are the quadratic and cubic dispersion coefficient w AL i w 2 A respectively. Equation (3) is the nonlinear part of NLSE that denotes the SPM. 1 w 3 A Į isDthe attenuation factor E2 E 3 [5]. NLSE. A and w zȖ is the 2nonlinear w T 2coefficient wT 3 6 2 . It consists of 6second TOD and attenuation. i E d 2 A E3 d 3 A D dA 2 A iJ | A |2 A wdzANL 2 dt 2 2 6 dt 2 2 . iJ | A | A wz . Pulse arrival. Pulse broadening generated by the GVD i phase of each spectral c that depends on the freq The generated frequenc frequencycomponents dependence c spectral broadening simplycompo beca velocity. ȕ i w2A 1 w3A D E3 E2 A 3 w T 2 III. frequency dependence broadening simply beca disperse during propaga [5]. Pulse broadeni linearly correlated with not rely on the sign of ȕ2 To observe the effec set to zero while GVD. Pulsebroadening broadeni generatedcorrelated by the GVD linearly withi phase each not relyofon the spectral sign of ȕc2 thatTo depends on the freq observe the effec Thetogenerated set zero whilefrequenc GVD.6 PULSE w T TYPE 2 2 . ȕ spectral components c velocity. Spectral compo compare to the trailing e arrival. Spectral disperse during propagae compare to the trailing [5]. They are III. Two types of pulses were used in the simulation. PULSE the chirped GaussianTYPE pulse and the chirped hyperbolic secant wANL [5] as shown2 in (4) and (5) respectively. pulse w AL wz iJ | A | A Two were used in the wz types of pulses ª 1 iC § t · 2 . T0 is the half-width at 1/e intensity point. DUEcarried TO SECONDout ORDER IV. SOD is governed by ȕ2. PULSE TYPE secant pulse§ [5] as shown in (4) and (5) Two types of pulses · simulation. at0 =1/e P0 isofthe initial peak (GVD). SIMULATION RESULTS DUE TO SECOND ORDER DISPERSION Second order dispersion (SOD) is a linear effect and the primary source No. pulse [5] as shown©ino(4) A0 A( z . GVD represents dispersion of group velocity that determines the broadening characteristic of the pulse. 2. Alland of the the primary ofcarried pulse broadening. All The power and C is thecharacteristic frequency chirp. P with the propagated d constant phase shift cau increase chirp affect Fig. while t is time 2 T © 0 ¹ ¼» ¬« period. Pulse propagation at z = unchipred Gaussian pulse and The broadening exp observed in Fig. P with the propagated d constant phase shift cau increase in chirp affect pulse spectral compo consequentially increase The magnitude of dela effects can be observed i . 0 = 1W andpoint. known as the group velocity dispersion (GVD). P0 = Second order dispersion (SOD) is a linear effect and the 1W and C = 0 (unchirped). º simulation.June 2010 Fig. wh effects caninbeFig. Second is a linear effect initial peakorder power is(SOD) the frequency chirp.values SOD of. t ) 2 (6) P0 ª 1 iC § t · º (4) A0 exp « ¨ ¸ » Equation 6 is the pulse initial amplitude. T0 = 2 ps. wh amount of broadening. primary source of pulse broadening. P0 is the 2 initial peak power§ and chirp. t ) A0Gaussian sec h ¨ pulse ¸ expand ¨ the 2chirped ¸ respectively.intensity Pby C =as0 (unchirped). The IV. All of the · t · C is§ the iCtfrequency A( z. are¨ the (4) « A( z . 1. From Eq. Eq. T0the isFrom the half-width the group velocity dispersion governed ȕ2. P0 is the Equation 6 dispersion isand theC pulse initial amplitude. simulations SIMULATION RESULTS of the were using DISPERSION the parameter values of. 2T0 ¹ T0 = 2 ps. SIMULATION RESULTS DUE TO SECOND ORDER Equation 6 is the pulse initial amplitude. the hyperbolic secant T 2T 0 ¹ ¹ and© (5) respectively. t ) A0 expThey ¸ »chirped Gaussian 2 T » 0 ¹ ¼ © chirped pulse and ¬« the hyperbolic III. 2. is simulations out using parameter while source t iswere time period. while t is time DISPERSION period. known T 2 ps. GVD represents dispersion group velocity that determines the broadening of the pulse. P0 = 1W© and A0 P0 (6) IV. observed i amount of broadening. Pulseinpropagation at z = pulse spectral compo unchipred Gaussian pulse and consequentially increase The broadening exp The magnitude of dela observed 1. ¸ expout ¨ using2 ¸the parameter values (5) To ¹C = ©0 (unchirped). 1. 1 January . They are t ·were §used iCtin2 the (5) A( zchirped . t ) A0 were sec h ¨carried simulations of. 1. T0 is the half-width at 1/e intensity point. From Eq. propagation (BPM) method. Pulse broadening occurs due to frequency chirps generated by the GVD induced phase shift. known as the group velocity dispersion (GVD). causes a delay on the pulse arrival. 2. arrival. GVD changes the phase of each spectral component of the pulse by an amount that depends on the frequency and the propagated distance [6]. This [5]. NLSE is faster generated by the GVD induced phase shift. This causes a delay on the pulse nonlinear part of NLSE that sign of β2. GVD represents dispersion of group velocity that determines the broadening characteristic of the pulse. ȕ3 and Ȗ in Eq. TOD and pulse compare to the trailing edges. Spectral QUATION of the group velocity leads to pulse broadening simply because different component of the pulse Equation (NLSE) is used to components at the leading edge travel ulse envelope propagating in disperse during propagation and do not arrive simultaneously compare to the edges. Pulse an be obtained by applying phase of each spectral component of the pulse by an amount is frequency dependent on the delay that depends on the and the propagated distance [6]. β2 = ps2/km [3]. Tonot observe the effect of SOD alone. Į is the attenuation factor linearly correlated with distance. areGVD. SOD is governed by β2. . The pulse broadening does rely on the sign of ȕ2. The frequency dependence of the group velocity leads to pulse broadening simply because different component of the pulse disperse during propagation and do not arrive simultaneously [5]. Pulse broadening is dependent on the delay and are the quadratic and cubic y. Spectral components at the not leadingrely edge travel r dispersion (SOD).Analysis of Second order Dispersion on Free Space Optical Propagation of pulse broadening. 1 are γ in Eq. β3 and To observe the effect of SOD alone. Thus. set ȕto zero while GVD. broadening The generated frequency chirps changes the velocity of each eneralized form of NLSE for and linearly correlated with distance. The generated frequency chirps changes the velocity of each spectral components causing them to frequency traveldependence in different velocity. The spectral components causing them to travel in different ion (2) is the linear part of broadening does on faster the velocity. [5]. set to zero1while 2 = 21 A iJ | A |2 A 21 ps2/km [3]. GVD changes the propagation in free space. Pulse broadening occurstrailing due to frequency chirps ear elements. 3 A D A T3 2 . . 3 km and 0. in comparison to hyperbolic secant pulse. P0 is the frequency chirp. From Figure 2(a) and 2(b). 1. 4 and hyperbolic secant pulse as in Eq. constant phase shift cause a constant increase in chirp.5 km but disappears as the pulse propagates. The broadening experienced by both pulses can be observed in Fig.June 2010 61 . As the pulse propagates. 1(a) and Fig.1843 Vol. Hyperbolic secant pulse shows lower broadening rate at about 34. E TYPE in the simulation. 1 January . 1. The change of velocity consequentially increases delay and cause further broadening. This can be observed as Gaussian pulse exhibits wider broadening and lower pulse amplitude as it propagates. There is one important attribute. while t is time 1/e intensity point. the waterfall plot for both pulses show similar characteristics in pulse broadening. secant where both pulses show a significant pulse Fig. This implies that both pulses have different effect to GVD. 5.0. 1(b). The pulse shape is defined by the pulse equation and both pulses manifest differently over the same parameters as can be seen in the Gaussian pulse which is presented by Eq.1. chirped). DUE TO SECOND ORDER SION D) is a linear effect and the ening. 1(b). From Eq. The magnitude of delay increases with the distance. Distortion can be seen between distances 0. 2(b). These effects can be observed in Fig. It is obvious SOD induced broadening increase linearly with propagating distance.5% compare to Gaussian pulse.5 km with β2 = 21 ps /km for (a) unchipred The broadening experienced by both pulses can be Gaussian pulse and (b) unchipred hyperbolic observed in Fig. Pulse propagation at z = 0. The increase in chirp affects the velocity and the arrival of the pulse spectral components. as can be observed in Fig.5. Nevertheless.1. Pulse broadening is linearly correlated with the propagated distance.1. hyperbolic secant pulse shows a faint distortion at both edges of its pulse. 1. The difference in broadening rate can be traced back to the difference in the pulse shape. 2 · º ¸ » ¹ »¼ (4) Ct 2 · ¸ T02 ¹ (5) (6) l amplitude. Broadening rate for both Gaussian and hyperbolic secant pulse can be observed in Figure 3. Hyperbolic secant pulse reveals a lower broadening rate compare to Gaussian. 2. These effects can be observed in Fig. They are e chirped hyperbolic secant respectively. 1. constant phase shift cause a constant increase in chirp. both pulses have displayed different broadening rates.1. Pulse broadening is linearly correlated with the propagated distance.5 km with ȕ2 = 21 ps2/km for (a) unchipred Gaussian pulse and (b) 2unchipred hyperbolic secant pulse amount of broadening. All of the ing the parameter values of. The Fig. These differences create variations and rare anomalies No. The magnitude of delay increases with the distance. 2 consequentially increases delay and cause further broadening.0. 1(a) and Fig. As the pulse propagates. Pulse propagation at z = 0. The change of velocity ISSN: 2180 . The increase in chirp affects the velocity and the arrival of the pulse spectral components. SOD is e group velocity dispersion sion of group velocity that acteristic of the pulse.5. where both pulses show a significant amount of broadening. Davidson.. variations and rare anomalies propagation in free space was simulated with the SODV. Opt. pg. Nonlinear Fiber Optics. P. Fiber Optic Communication Systems. Agrawal. result may serve as toa observe prediction model Simulations were done in order pulse behavior and response to linear Pulse propagation in free space that can be parameters. 4 and Fig. 2001. Pulse2. “Simulation of Propagation [2] in Nonlinear M. Opt.traced back to the difference in the pulse shape. and M. 2 No.Optical J. with ȕ2 = 21 ps /km. pg. and M. 2002. model that can be used to estimate or predict to an prediction extent the actual pulse behavior in free space. 2. Wiley & Sons. WIAS Report 2003. WIAS Report no.” CLEO Tech Digest. Ricklin and F. P. CA: Academic January . Bandelow. No.5 km can be observed in hyperbolic secant rbolic secant pulse 34. J. [5] G. V. Kim. C.” J. M. [3] M. [5] G. CA: Academic Press. 1 REFERENCES [1] G. Broadening factor for unchirped Gaussian and unchirped 2 hyperbolic secant pulse as in Eq. pulse over the distance. Agrawal. San Diego. Grant of Universiti Teknikal Malaysia Melaka under REFERENCES grant PJP/2009/FKEKK [1] G. Simulation result may serve as a space. 399. z = 1 km. pg.no. 23. Electronic and Computer Engineering Hyperbolic in the Gaussian pulse which is presented by Eq. Fiber Optics. May 2003. New York: John 2.Hyperbolic with ȕ2 = 21 ps2/km. Demircan. San Diego. Fiber Optic Communication Systems. Gaussian β2 = 21 (a)unchirped pulse and (b)unchirped hyperbolic secant pulse. Am. A. inc. Baltimore. Nonlinear Optics. Press. 5. 20. A. V. 5. used to estimate or predict to was with actual the SODpulse in orderbehavior to observe in the free pulse ansimulated extent the behavior in free space. 3rd edition. Potasek & S. ACKNOWLEDGMENT ACKNOWLEDGMENT This work is supported by the Internal Grant of Universiti Teknikal Malaysia Melaka under grant PJP/2009/FKEKK This work is supported by the Internal (20D) S614. Broadening factor for unchirped Fig. It is obvious y with propagating displayed different e reveals a lower can be observed as g and lower pulse o hyperbolic secant different effect to hyperbolic secant edges of its pulse. Pulse ACKNOWLEDGMENT 62 This work is supported by the Internal Grant of Universiti Teknikal Malaysia Melaka under grant PJP/2009/FKEKK (20D) S614. 3. 20. These differences create Secant pulse over the distance. [6] U.” CLEO Tech Digest. Bandelow. 21. Vol. Davidson. C. 181-182. 4 and e differences create [3] M. J. Agrawal. J. [6] U. 2001. Demircan. May 2003. CA: Academic (20D) S614. Pulse propagation free space Inresponse this paper dispersion effects inwere was simulated with the SOD in order to observe the pulse simulated individually in 1D and 2D behavior in free space. 2001. 181-182.propagation at z = 1 km and withat ȕ2 =z21 (a)unchirped Gaussian pulse and ps2/km (b)unchirpedfor hyperbolic secant pulse. with β2 = 21 ps2/km. “Spatiotemporal Effects in Nonlinear Dispersive and Diffractive Media Using Ultrashort Optical Pulses”. Chap. Fig. 1999. 5. z = 1 km. and P.June 2010 . Ricklin F. 2003. Agrawal. Simulation result may serve as a graphical representation. Agrawal. Potasek. Simulation individually in 1D and 2D graphical representation. “Spatiotemporal Effects in Nonlinear Dispersive and Diffractive Media Using Ultrashort Optical Pulses”. pg. 1999. Broadening factor for unchirped Gaussian and unchirped Hyperbolic Gaussian and unchirped Secant Secant pulse over the distance.Nonlinear “Atmospheric optical communication G. with a Gaussian Schell beam.inCONCLUSION order to observe the In this paper the dispersion effects were simulated pulse behavior in free space.” J. 399. Vol. MD. MD. 3. z = 1 km. Chap. 3. Baltimore. San Diego. “Atmospheric optical communication with a Gaussian Schell beam. 2002. M. Kesting. No. CONCLUSION In this paper the dispersion effects were simulated in 1D and 2D graphical representation. Potasek & S. Vol. invited paper. Fig. A. “Simulation of Propagation in Nonlinear Optical Fibers”. 2. 21. The pulse shape is defined by the pulse equation and both pulses manifest differently over the same parameters as can be seen Journal of Telecommunication. A. Kesting. 23. The pulse n and both pulses ters as can be seen ed by Eq. “Optical Processing in Free Space Using Ultrashort Pulses. P.. individually CONCLUSION Simulations were done in order to observe pulse behavior and to linearthe parameters.3 km and 0. Kim. ISSN: 2180 . invited paper. [4] J. Simulations wereto an prediction model that can be used to estimate or predict extentin theorder actual pulse in free space.1843 Vol. 0. “Optical REFERENCES Processing in Free Space Using Ultrashort Pulses. Am. Fibers”. Nonlinear Fiber Optics.5% compare to ening rate can be e shape. plot for both pulses ening. Soc. Pulse propagation = ps12/km kmforand with Fig. Soc. Vol. 3rd edition. New York: John Wiley & Sons. 2001. Nonlinear Optics.behavior done to behavior observe pulse and response to linear parameters. P. inc. Potasek. [2] M. [4][1] J.