Optics Communications 269 (2007) 8–13 www.elsevier.com/locate/optcom Direct measurement of dispersion of the group refractive indices of quartz crystal by white-light spectral interferometry ´ Petr Hlubina *, Dalibor Ciprian, Lenka Knyblova Department of Physics, Technical University Ostrava, 17. listopadu 15, 708 33 Ostrava-Poruba, Czech Republic Received 26 April 2006; received in revised form 21 July 2006; accepted 23 July 2006 Abstract We present a white-light spectral interferometric technique employing a low-resolution spectrometer for a direct measurement of the dispersion of the ordinary and extraordinary group refractive indices of a quartz crystal over the wavelength range approximately from 480 to 860 nm. The technique utilizes a dispersive Michelson interferometer with the quartz crystal of known thickness to record a series of spectral interferograms and to measure the equalization wavelength as a function of the displacement of the interferometer mirror from the reference position, which corresponds to a balanced non-dispersive Michelson interferometer. We confirm that the measured group dispersion agrees well with that described by the dispersion equation proposed by Ghosh. We also show that the measured mirror displacement depends, in accordance with the theory, linearly on the theoretical group refractive index and that the slope of the corresponding straight line gives precisely the thickness of the quartz crystal. Ó 2006 Elsevier B.V. All rights reserved. PACS: 42.25.Hz; 42.25.Lc; 42.81.Gs; 78.20.Fm Keywords: White light; Spectral interferometry; Low-resolution spectrometer; Michelson interferometer; Group refractive index; Birefringence; Quartz 1. Introduction The refractive indices and their dispersion, that is, the wavelength dependence, are the fundamental parameters and characteristics of both isotropic and anisotropic optical materials. White-light interferometry based on the use of a white-light source in combination with a standard Michelson or Mach–Zehnder interferometer, is considered as one of the best tools to measure the refractive index dispersion for different optical materials. White-light interferometric methods enable high-accuracy measurements of group dispersion as well as higher-order dispersion of various optical elements over a broad spectral range. White-light interferometry usually utilizes either of two methods, that is, a temporal method or a spectral method, depending on whether interference is observed in the time * Corresponding author. Tel.: +420 597 323 134; fax: +420 597 323 139. E-mail address: [email protected] (P. Hlubina). domain or in the spectral domain, respectively. The temporal method involves measurement of the time of flight of optical pulses through a sample. A method for measuring the group delay introduced by an optical material consists in placing the sample in one arm of the interferometer and evaluating the temporal shift of the peak of the cross-correlation interferogram. As the central wavelength is varied, the relative group delay of the different frequency components is observed directly [1]. Alternatively, the spectral phase over the full bandwidth of the white-light source can be obtained in a single measurement by a Fourier transform of the cross-correlation interferogram [2–5]. The dispersion characteristics of the sample under study can be obtained by simply differentiating the measured spectral phase. The spectral method is based on the observation of spectrally resolved interference fringes (channeled spectrum) [6–10] and involves measurement of the period of the spectral fringes in the vicinity of a stationary-phase point [6] that appears in the recorded spectral interferogram when 0030-4018/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2006.07.049 / Optics Communications 269 (2007) 8–13 9 the group optical path difference (OPD) between two beams in the interferometer is close to zero. ð1Þ where l and L are the optical path lengths of the beam in the air in the first and in the second arm of the interferometer. Let us consider next that a sample of thickness t and refractive index n(k) is inserted into the first arm of the Michelson interferometer as is shown in Fig. (2) in which the OPD DM is replaced by the group OPD Dg ðkÞ given by M Dg ðkÞ ¼ 2ðL À lÞ À 2t½N ðkÞ À 1.12] and repeats the measurement. the measurement of the group refractive index dispersion of a given material is still possible in the vicinity of the stationary-phase point if one moves it in successive steps to different wavelengths [11. 1. The smaller the OPD adjusted in the interferometer the larger the period of spectral modulation. a spectral interferogram can be recorded at the output of the interferometer with the period of spectral modulation K(k) given by: KðkÞ ¼ k2 =DM . The OPD DM(k) between beams of the dispersive interferometer is given in this case by DM ðkÞ ¼ 2ðL À lÞ À 2t½nðkÞ À 1: ð3Þ The corresponding period of spectral modulation is given by Eq. . respectively. Experimental setup with a Michelson interferometer and a low-resolution spectrometer to measure the group refractive indices for ordinary and extraordinary polarizations in a uniaxial crystal. When a white-light source is used.P. Hlubina et al. let us consider a configuration of a non-dispersive Michelson interferometer. which is given by the slope of linear dependence of the measured mirror displacement on the theoretical group refractive index. 2. 1. Experimental method First. We confirm that the group refractive indices measured with a precision of 3 · 10À4 agree well with those obtained from the dispersion equation proposed by Ghosh [15]. The main limitation of the method is reached for thick or strongly dispersive materials because under such conditions the spectral interference fringes that are far from the stationary-phase point become difficult to resolve. The wavelengths are also referred to as the equalization wavelengths [12]. The OPD DM between beams at the output of the interferometer is given by: DM ¼ 2ðL À lÞ. We present a white-light spectral interferometric technique employing a low-resolution spectrometer for measurement of the dispersion of the group refractive indices of a quartz crystal over the wavelength range approximately from 480 to 860 nm. The modification of the technique with a tandem configuration of a Michelson interferometer and a birefringent optical element has been used in measurement of the group birefringence dispersion of an optical fiber of known length [13] and of a calcite crystal of known thickness [14]. Fortunately. ð2Þ where k is the wavelength and we consider DM P 0. we determine precisely the thickness of the quartz crystal. M where N(k) is the group refractive index defined as ð4Þ Mirror 1 Light source Optical fiber Sample t Micropositioner HL 2000 Collimator Beamsplitter Polarizer Objective Optical table Spectrometer PC S2000 Read optical fiber Micropositioners Mirror 2 Fig. which is infinite for the balanced interferometer with L = L0 = l. The technique utilizes a dispersive Michelson interferometer with the quartz crystal of known thickness to record a series of spectral interferograms and to measure the equalization wavelength as a function of the displacement of the interferometer mirror from the reference position corresponding to a balanced non-dispersive Michelson interferometer. This measurement gives the wavelength dependence of the group refractive indices for the ordinary and extraordinary polarizations in the quartz crystal. The aim of this paper is to extend the use of a white-light spectral interferometric technique presented in a previous paper [12] for measuring directly the dispersion of the ordinary and extraordinary group refractive indices of a quartz crystal of known thickness. Furthermore. 2 0. Be = 1.15662475 and Fe = 100.e k À F o.09509924.e þ þ : no.e Bo. no(k) and ne(k). n2 ðkÞ ¼ Ao. the equalization wavelength [12]. the maximum thickness tmax % 50 mm. in the quartz crystal is represented in the Sellmeier-like form proposed by Ghosh [15]: 0. Hlubina et al. the minimum thickness tmin is given by the minimum displacement change DLmin ¼ ½DLðk0 min Þ À DLðk0 max Þmin with which the group dispersion curve is measured tmin ¼ DLmin =½N ðk0 min Þ À N ðk0 max Þ: ð9Þ Similarly. The corresponding group refractive indices (5) are represented as N o. The dispersion relation for refractive indices of the ordinary and extraordinary polarizations. if the dispersion of a birefringent crystal is known.) When the case of thick or strongly dispersive materials is considered.e ðkÞ ( ) k2 Bo. Theoretical spectral interferogram for the thickness t = 20 000 lm of the quartz crystal and the mirror displacement DL = 11 400 lm demonstrating discrimination between the equalization wavelengths ko and ke. 2 that shows the theoretical spectral interferogram [16] for a quartz crystal of the thickness t = 2 cm and the mirror displacement DL = 11 400 lm. Ce = 1.) or DLe(ke) on the known group refractive index No(ko) or Ne(ke).e F o.e ðkÞ ¼ no. Ae = 1. We can estimate the minimum and maximum thicknesses of the sample whose group dispersion can be measured by the method. This method can be extended for measuring the dispersion of the group refractive indices for both the ordinary and extraordinary polarizations [15] in birefringent uniaxial crystals such as quartz and calcite. an optical fiber. If the measurement is restricted to the wavelength range from k0min to k0max. Similarly.e ðkÞ ðk2 À C o.e Þ2 ð12Þ Fig. U. a collimating lens.e Þ2 ðk2 À F o.10202242 and Fo = 100. the maximum thickness tmax is preferably given by the maximum displacement DLmax which can be adjusted in the interferometer: tmax ¼ DLmax =½N ðk0 min Þ À 1: ð10Þ Taking into account the material of known dispersion such as fused silica [12] and the spectral range restricted to the wavelengths k0min = 450 nm and k0max = 900 nm. Experimental setup The experimental setup used in the application of whitelight spectral interferometry to measure the dispersion of the group refractive indices for the ordinary and extraordinary polarizations in a quartz crystal is shown in Fig.e þ o. 2 demonstrates easy discrimination between the equalization wavelengths ko and ke satisfying Eq. This fact is demonstrated in Fig.28851804. Bo = 1. displacement DLo(ko) or DLe(ke) measured as a function of the equalization wavelength ke or ke gives directly the dispersion of the group refractive index No(ko) or Ne(ke) for a birefringent crystal of known thickness.e k2 þ 2 . k2 À C o. (8) says that the thickness of the birefringent crystal can be determined precisely from the slope of linear dependence of the measured mirror displacement DLo(ko. which corresponds to a standard travel of a translation stage. which corresponds to 10 measurements with a 10 lm step.8 Eq. 2. we obtain for DLmin ¼ 100 lm. satisfying the relation Dg ðk0 Þ ¼ 2ðL À lÞ À 2t½N ðk0 Þ À 1 ¼ 0: M ð6Þ 1. Generally speaking. Co = 1.6 1.28604141. 3.07044083. the spectral interference fringes have the largest period in the vicinity of a stationary-phase point [6] for which the group OPD is zero at one specific wavelength k0.02101864 · 10À2. Similarly we obtain for DLmax = 25 mm. the minimum thickness tmin % 3 mm.00585997 · 10À2. It consists of a white-light source: 7 W halogen lamp HL 2000 with launching optics. (6) gives for the mirror position L = L(k0) for which the equalization wavelength k0 is resolved in the recorded spectral interferogram the relation: Lðk0 Þ ¼ l þ t½N ðk0 Þ À 1: ð7Þ If we introduce the mirror displacement DL(k0) = L(k0) À L0 as the displacement of the second mirror of the dispersive interferometer from the mirror position of the balanced non-dispersive Michelson interferometer.4 ↑ λ o ↑ λ 600 650 e 0 500 550 700 750 800 Wavelength (nm) Fig.e k2 Do. we obtain the simple relation N ðk0 Þ ¼ 1 þ DLðk0 Þ=t.e ð11Þ where k is wavelength in micrometers and the dispersion coefficients at room temperature are as follows: Ao = 1. a . ð8Þ enabling to measure directly the group refractive index N(k0) as a function of the equalization wavelength k0 for a sample of known thickness t.10 P.e Do.e C o. Do = 1. De = 1. 1. Eq. / Optics Communications 269 (2007) 8–13 N ðkÞ ¼ nðkÞ À k dnðkÞ : dk ð5Þ 2 Spectral Intensity (A. (8). The plate is placed into the first arm of the interferometer in such a way that the collimated beam is incident on the surfaces perpendicularly and the orientation of the optical axis is shown in Fig. These functions are represented in Fig. The spectrometer resolution is in our case given by the effective width of the light beam from a core of the read optical fiber: we used the read optical fiber of a 50 lm core diameter to which a Gaussian response function corresponds. a 2048-element linear CCD-array detector with a Schott glass longpass filter. has spectral operation range from 350 to 1000 nm and includes a diffraction grating with 600 lines per millimeter. a collection lens and a read optical fiber. 5 by the circles and they are shown together with the theoretical functions given . it can be seen that the equalization wavelengths ke for the extraordinary polarization can be resolved in the spectral range from 489 to 860 nm and that the corresponding displacement DLe varies from 12 428 to 11 768 lm. We recorded one interferogram with high-visibility interference fringes. 4 shows the displacement DLo as a function of the equalization wavelength ko and the displacement DLe as a function of the equalization wavelength ke. 1. a microscope objective. the extraordinary polarization is transmitted. U.56 nm. Knowledge of the sample thickness t and the measured dependences of the displacement DLo on the equalization wavelength ko and the displacement DLe on the equalization wavelength ke enable us to evaluate directly the group refractive indices No(ko) and Ne(ke) of the quartz crystal as a function of the equalization wavelengths ko and ke. 3 shows an example of the spectral interferogram recorded for the displacement DL = 1 1908 lm. Fig. Example of the spectral interferogram recorded for the mirror displacement DL = 11 908 lm with the equalization wavelengths ko = 585. The wavelength of the spectrometer is calibrated so that a third-order polynomial relation between pixel number and wavelength is used. parameters of which are presented above. Hlubina et al. First type of interference fringes is located in a range of shorter wavelengths and corresponds to the ordinary polarization and the second type of the interference fringes is located in a range of longer wavelengths and corresponds to the extraordinary polarization.32 nm and ke = 703. 4. 3. 2). The recorded spectral interferograms have revealed that the equalization wavelengths ko for the ordinary polarization can be resolved in the spectral range from 483 to 851 nm and that the corresponding displacement DLo varies from 12238 to 11578 lm. Discrimination between both types of interference fringes can be easily achieved by means of a polarizer.32 nm and ke = 703. The thickness of the plate is t = (20 950 ± 10) lm. Experimental results and discussion First. in the first arm of the interferometer (see Fig. 3 also clearly demonstrates 600 Spectral Intensity (A. we determined the position L0 of mirror 2 for which the non-dispersive Michelson interferometer is balanced.). which was determined in this way with a precision better then 1 lm. The equalization wavelengths are determined by autoconvolution method [17] with an error of 0. the effect of the limiting resolving power of the spectrometer on the visibility of the spectral interference fringes identified only over a narrow spectral range in the vicinities of the equalization wavelengths ko = 585. The fiber-optic spectrometer S2000 of an asymmetric crossed Czerny-Turner design with the input and output focal lengths of 42 and 68 mm. We displaced mirror 2 manually by using the micropositioner with a constant step of 10 lm and performed recording of the corresponding spectral interferograms. parallel to the optical axis of the crystal with a precision of 15 arcmin. Fig. Fig. micropositioners.P.32 nm corresponding to the wavelength difference for adjacent pixels. a micropositioner connected to mirror 2 of the interferometer. Inc. The quartz plate under test consists of two polished surfaces.56 nm. / Optics Communications 269 (2007) 8–13 11 bulk-optic Michelson interferometer. the ordinary polarization is transmitted and for the polarizer oriented parallel to the optical axis. respectively. an A/D converter and a personal computer. a fiber-optic spectrometer S2000 (Ocean Optics. (2) to evaluate the corresponding OPD DM. a polarizer. Spectrometer sensitivity is at given light conditions affected by the spectrometer integration time: it can easily be varied under software control. Then we inserted a quartz crystal. In this spectral interferogram are resolved two types of interference fringes.) 400 200 ↑ λ 0 450 500 550 o ↑ λ 650 e 600 700 750 800 850 900 Wavelength (nm) Fig. determined the period K(k) for a given wavelength k and used Eq. For the polarizer oriented perpendicularly to the optical axis. Half of this OPD is equal to the displacement DL of mirror 2 from the position L0. It is clearly seen that the larger the adjusted displacement the shorter the equalization wavelength. We measured in this way the dependence of the adjusted displacement of mirror 2 on the equalization wavelengths corresponding to the ordinary or extraordinary polarizations. 1) and displaced mirror 2 to such a position to resolve spectral interference fringes in accordance with theory (see Fig. Similarly. 6 for the group refractive index No(ko) given by Eq.25 x 10 4 1.58 1.57 1.12 x 10 4 P.2 lm is obtained. Solid lines correspond to theory. By using this procedure we obtained the quartz crystal thicknesses to = te = 20949.e 2 : t t In our case.e ðko.5 lm with a standard deviation of 0. (8) by using the measured o.e ðko.59 1. Solid line is a linear fit. 1.55 1. / Optics Communications 269 (2007) 8–13 1. given by Eq. Hlubina et al. If the displacements DLo. 4.e Þ dðtÞ ð13Þ dðN o.ei Þ ð14Þ SDo.e) and the thickness of the birefringent crystal is known with a precision of d(t). (12) when the quartz crystal thickness to of 20949.e so that the standard deviations sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pm 2 e t i¼1 ½N o. The measured mirror displacement as a function of the wavelength for both polarizations and the thickness t = (20 950 ± 10) lm of the quartz crystal.e Þ evaluated from Eq.e measured as a function of the equalization wavelengths ko.23 1. 7 shows the linear dependence for the group refractive index Ne(ke) given by Eq. Parameter m is the number of the equalization wavelengths ko.5 lm with a standard deviation of 0. The measured mirror displacement as a function of the theoretical group refractive index for the ordinary polarization in the quartz crystal (circles). Second.6 Group Refractive Index Fig. the thickness of which is known with higher precision and/or is larger.e displacements DLo. (12) when the quartz crystal thickness te of 20949. Similarly.ei Þ À N o.e adjusted in the interferometer are known with precisions of d(DLo.2 lm is obtained.21 Ne 1.56 No 1.17 ΔL o 1. The theoretical wavelength dependences of the group refractive indices N to.e are linearly dependent on the group refractive indices No. from which it results that the displacements DLo.5 lm corresponding to the minimal values of the standard deviations SDo = 3. are compared with the wavelength dependences for the group refractive indices N e ðko. the group refractive indices No.55 450 500 550 600 650 700 750 800 850 900 Wavelength (nm) Fig.15 450 500 550 600 650 700 750 800 850 900 Wavelength (nm) Fig.e(ko. The measured group refractive index as a function of the wavelength (circles) for both polarizations and the thickness t = (20 950 ± 10) lm of the quartz crystal. 5.e) in obtaining the group refractive indices are 3 · 10À4.57 1.23 1.6 · 10À5.15 1.e Þ ¼ þ DLo. Higher measure- 1.56 1. the precisions d(DLo.21 ΔLe 1. There are also possibilities to compare the known quartz crystal thickness with that resulting from the comparison of the measured group refractive index dispersions with theory.e are obtained with precisions given by the following formula: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 dðDLo.e ðko.8 · 10À5 and SDe = 3. by the dispersion relation (12). We can estimate a precision of the group refractive index measurements. .ei resolved in the measured spectral range (in our case 67 values). (12).6 1.e ¼ mÀ1 are minimal.17 ΔLo 1.19 1.e Þ.e and such a choice of the quartz crystal thicknesses to.58 Mirror Displacement (μm) 1.19 1.e.e) with slopes giving the crystal thicknesses to. (8).e) are 1 lm and the precision d(t) is 10 lm so that the precisions d(No. the thicknesses to. ment precisions can be achieved using a quartz crystal. Fig.59 Group Refractive Index 1. we can utilize Eq. This fact is illustrated first in Fig. 6.e of the quartz crystal can be obtained by using a least-squares procedure. First.25 Mirror Displacement (μm) 1. [12] P. Commun. B 70 (2000) 45. Urbanczyk. J. Medhat. it allows to measure precisely the thickness of a uniaxial birefringent crystal.25 Mirror Displacement (μm) 1. Conclusions We used a white-light spectral interferometric technique employing a low-resolution spectrometer to measure the dispersion of the group refractive indices of a quartz crystal over a wide spectral range (480–860 nm).A. C.58 1. 11 (2003) 2793. Walmsley. [3] K. Technol. 1. Yamada. Technol. 193 (2001) 1. Am. Martynkien. Opt. Sci. Opt. and by an ˇ internal Grant of TU Ostrava (IGS HGF VSB-TUO).59 1. G. I. T. Opt. Commun.6 Group Refractive Index Fig. R.A. Naganuma. Hirlimann. Gurov. El-Zaiat. Acknowledgements This research was partially supported by the Grant Agency of the Czech Republic (Projects 102/06/0284. 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