OPTO-ELECTRONICS REVIEW 3(), 7 76 7 th International Workshop on Nonlinear Optics Applications Polarization degree fading during propagation of partially coherent light through retarders A.W. DOMAÑSKI * Faculty of Physics, Warsaw University of Technology, 75 Koszykowa Str., 66 Warsaw, Poland In the paper, degree of polarization fading during transmission of partially coherent light through bulk birefringent optical materials is analysed. It is also shown that classical Mueller matrix formalism for such cases should be changed and a depolarisation matrix should be added. As an example influence of degree of polarization fading on optical retarders parameters as well as on accuracy of degree of polarization measurements are discussed and a result of polarization degree fading measurements of laser diode light passing through a lithium niobiate crystal are presented. Keywords: polarization, coherence, birefringence, Mueller matrix.. Introduction A degree of polarization (DOP) of partially coherent light diminishes during transmission through birefringent media. The phenomenon is well known in optical fibers even with small birefringence, for example for optical fiber rotation sensor based on idea of optical fiber Sagnac interferometer [] as well as for polarimetric fiber optic sensors []. In the paper, we deal with a problem of DOP changes of partially temporal coherent light during propagation through linear retarders. In general, a linear retarder resolves a light wave into two orthogonal linearly polarized components travelling with two different velocities due to birefringence of a retarder. As a result, a phase shift between both components arises and a state of polarization (SOP) changes. The most popular retarders are made as quarter- or half-wave plates and they can either transform linearly polarized light into circular or rotate linear polarization, respectively. An ideal retarder should not change the DOP, particularly that quarter-wave plate is used as element of DOP measurement set-up. In a reality, DOP for partially temporal coherent light changes due to transmission through retarders in a way presented in the paper.. Degree of polarization for temporal partially coherent light Temporal coherence of light may be described by complex degree of coherence [3] g = * e-mail: domanski@if.pw.edu.pl J xy, () Jxx Jyy where Jxx = Ex E * x, Jyy = EyE * * y, Jxy = Ex Ey are the coherence matrix elements; stands for temporal mean value, * means the complex conjugate value. The complex degree of coherence depends on a width of the spectral characteristic Dn of the light source and changes during transmission though a birefringent material (with birefringence Dn) in z direction depending on the type of an applied light source. For light sources with Lorentzian spectrum, like laser diodes (LDs), the absolute value of a complex degree of coherence, i.e., degree of coherence is as follows [4] g L n nz = é D D ù exp - ë c û, () where c is the speed of light in a vacuum and z is the propagation distance. For Gaussian spectrum, that is typical for gas lasers as well as for light emitting diodes (LEDs) and superluminescent diodes (SLDs), the degree of coherence changes during propagation through birefringence materials in more complicated way g G é n nz = - æ è ç D D ö ù exp. (3) c ø ë ln û Let us assume the spectral characteristics Dn dependent on the coherence length DL, a parameter which may be measured in classical Michelson interferometer as a difference path length by which a visibility of the interference pattern diminishes by /e times, as follows Dn = c DL, (4) Opto-Electron. Rev., 3, no., 5 A.W. Domañski 7
Polarization degree fading during propagation of partially coherent light through retarders what leads to gl nz nz = é ù é - gg ë L û = - æ D è ç D ö ù exp, exp. (5) D Lø ë ln D û On the other hand DOP may be expressed through coherence matrix elements [3] P = - 4det[ J] { Sp[ J]}, (6) where Sp[ J] = J xx + J yy and det[ J] = Jxx Jyy - JxyJyx = Jxx Jyy - Jxy (due to J * xy = J xy because the coherence matrix is a Hermitian type). Combination of Eqs. () and (6) leads to the formula [5] P = - 4 ( - g ) ( Ex Ey + Ey Ex), (7) where E x, E y are the amplitudes of electric field components of incident light (Fig. ). 3. Transmission of partially coherent light through birefringent media Both components of linear polarized light coupled into crystal have different phase velocity (Fig. ). This causes a phase shift between both components and in the same linearly polarized light is transformed into circular, elliptical, or linear depending on value of the phase shift. For totally coherent light the SOP changes but DOP is equal to even for long pass length of light in the birefringent material. For partially coherent light characterized by the coherence length DL, the phase shift may be so high that light outgoing from the crystal is almost totally unpolarized. Based on Eqs. (5) and (7), the degree of polarization for partially coherent light with Lorentzian and Gaussian spectrum transmitted through birefringent material of the length l can be written as follows PC = - 4 [ -exp(-h s)] ( Ex Ey + Ey Ex), (8) where h s is, so-called, the degree of length shift and it depends on light spectrum h sl for Lorentzian spectrum and h sg Dnl Ls = = DL D L, (9) nl Ls = æ è ç ö = æ D Lø è ç ö, () ln D ln DLø for Gaussian spectrum with L s = Dnl as so-called the length shift of electric vector components. For linear retarders like quarter-wave and half-wave plates, the length shift is equal to l Lsl = Dnl = + ml 4 4, () l Lsl = Dnl = + ml where m =,,,3, and means the order number of retarders. For polymer, mica, and liquid crystal electrically controlled retarders the order number m is very often equal to zero and then a length shift is much smaller than coherence length of commonly used light sources. It means that for such retarders h s» and since E x = E y, DOP is P». For quartz retarders, the situation is quite different. Thickness of multi-order quartz retarders is usually close to mm because of high aperture and mechanical stability problems with thin plates. Then, the length shift (L s = Dnl = µm) is the same order like coherence length for part of commonly used diode sources (Table ). For example, in superluminescent diode (SLD) the degree of length shift is equal to h sl = /4ln and for E x = E y the degree of polarization diminishes to.6976. Fig.. The length shift Dnl of partially coherent light passing through linear birefringent media. L is the length of a wave package of incoming light. 7 Opto-Electron. Rev., 3, no., 5 5 COSiW SEP, Warsaw
7 th International Workshop on Nonlinear Optics Applications Table. Degree of polarization fading due to transmission partially coherent light ( * Gaussian, ** Lorentzian spectra) through retarder with azimuth 45 (E x = E y ). Light source DL = l Dl L s = Dnl ( mm quartz, Dn =.) Ls D L h s P c Natural * ~ µm µm 5/ln ~ LED * ~ µm µm.5 /6ln.9 SLD * 5 µm µm µm /4ln /ln.4.7 LD ** mm m µm 7 7.99 He-Ne *. m m µm 7 4 4 8 /4ln ~ DOP of the natural light after passing through a linear polarizer is equal to but behind high thickness the retarder is P» (completely depolarized) since h sg = 5/ln. Producers of linear retarders design a compound zero-order quartz retarder in order to diminish a thermal instability. The compound retarder is made as combination of two multiple-order quartz waveplates glued in such a way that the slow axis of one plate is aligned with the fast axis of the other. Then, the birefringence of the retarder is almost totally compensated. It makes thick quartz retarder real zero-order one and it causes that the depolarization of partially coherent light passing through thick retarder may be neglected for almost all types of light sources. 4. Muller matrix formalism for partially coherent light Transmission of partially polarized light should be analysed based on classical Muller matrix formalism [6]. For partially polarized light, the parameters of Stokes vector of outgoing light [S ] may be described in simple matrix formula é '' S ù M M M M é 3 4 ùé ' S ù '' S M M M 3 M ' = 4 S, () '' S M M M M 3 3 33 34 ' S '' ë S 3û ëm4 M4 M43 M ' 44 ûë S3 û where M ij are the elements of Mueller matrix and [S ]isthe Stokes vector of incoming light. The Stokes vector parameters are described by the elements of coherence matrix S = J xx + J yy, S = J xx J yy, S = J xy J yx, S 3 = i(j yx J xy ). As an example of the Mueller matrix formalism application, an optical system shown in Fig. is taken into consideration out in pol [ S ] = [ MR][ MP][ S ] = [ MR][ S ] (4) where [M P ] is the Mueller matrix for linear polarizer and [M R ] is the Mueller matrix for retarder. Based on full matrix Eq. (4), with assumption that intensity of input light is normalized to and totally unpolarized and also assuming that our retarder is quarter-wave plate, we obtain pol in [ S ] [ MP] [ S ] éù é ùéù = and next ë û ë ûë û out [ S ] [ Ml / 4] [ S pol ] éù é ù éù - = ë û ë û ëû (5) what gives P =,P = and P = in the places shown in Fig.. For partially coherent light, DOP of outgoing light changes due to length shift caused by the retarder plate (P < ). It means that additional depolarization matrix should be introduced into matrix equation out in [ S ] = [ DC][ MP][ S ], (6) It may be concluded that a degree of polarization depends on the Stokes vector parameters as follows P = S + S + S3 S. (3) Fig.. Mueller matrices and Stokes vectors for light passing through linear polarizer and quarter-wave plate. Opto-Electron. Rev., 3, no., 5 A.W. Domañski 73
Polarization degree fading during propagation of partially coherent light through retarders where [D C ] is depolarization matrix due to partially coherent light used. Elements of [D C ] matrix are as follows é ù PC [ DC] =, (7) PC ë P Cû where P C is the degree of polarization with parameters depending on the coherence length DL of incoming light, the birefringence Dn, and the thickness l of retarder as shown Eq. (8). It should be marked that diagonal form of matrix is typical for depolarization [7]. Based on Eq. (8), the calculated degree of polarization fading is shown in Table. 5. Errors in Stokes vector measurements for partially coherent light A Stokes vector of light beam may be calculated based on measurements of six intensities I (i,j) by the use of linear analyser and quarter-wave plate according to the formula ésù S [ S] = = S ës 3û é I + I (, ) ( 9, ) ù m I - I (, ) ( 9, ), (8) e I - I (, 45 ) ( 35, ) I - I ( l 4, 45 ) ( l ë 4, 35 ) û where the first subscript index means lack or presence of quarter-wave plate and the second one gives azimuth of an analyser (Fig. 3). Mueller matrix equations for I (, ), I (,9 ), I (,45 ), I (,35 ) intensity measurements are as follows [ S ] = [ M ][ S], [ S ] = [ M ][ S], 9 [ S ] 3 = [ M ][ S], 45 [ S ] 4 = [ M ][ S], 35 and for I (l/4,45 ),I (l/4,35 ), respectively [ S ] 5 = [ M ][ D ][ M / ][ S], 45 C l 4 [ S ] 6 = [ M ][ DC ][ M 35 l / 4 ][ S], (9) () where D C is the depolarization matrix due to partially coherent light passing through the quarter-wave retarder. Fig. 3. Set-up for measurements of Stokes vector parameters. Next results are obtained directly from a set of matrix [Eq. (9)] m S = ( S + S) = I, e (, ) m S = ( S - S) = I, e ( 9, ) m S = ( S + S) = I, 3 e ( 45, ) S = ( S - S) = m I, 4 e ( 35, ) () and from Eq. () additional equations allowing to calculate S 3 m S = ( S + PC S3) = I, 5 e ( l/ 445, ) S = ( S - PC S3) = m I 6 e ( l / 4, 35 ). () Hence, all parameters of Stokes vector of light incoming to the detector may be calculated é S ù é I + I (, ) (, 9 ) ù S [ S ] = m I - I = (, ) ( 9, ) S e I - I (, 45 ) ( 35, ) ës I - I 3 û ( l 4, 45 ) ( l ë 4, 35 ) û é S + S ù. (3) é S ù S - S S = = ¹ [ S] S - S S 3 4 3 S + S ëpc S û ë 5 6 û We can see that for partially coherent light measured and calculated, the first three parameters of Stokes vector are the same as for totally coherent light. The fourth parameter of the Stokes vector is different and the difference depends on degree of a length shift and coherence length in a way pointed out by s S3 PC S3 4 -exp(- h ) = = - S 3, (4) æe E x y ö ç + èey E x ø where h s is shown in Eqs. (9) and (). Stokes vector parameters also allow to calculate the azimuth b, that is the angle which the major axis makes with OX axis and ellipticity b/a of polarized part of light beam according to formulae 74 Opto-Electron. Rev., 3, no., 5 5 COSiW SEP, Warsaw
7 th International Workshop on Nonlinear Optics Applications Fig. 4. Set-up for depolarization measurements of partially coherence light. and b a -æ S ö b = tan ç, (5) è S ø ì æ S öü ï - = ç 3 ï taní sin ç ý. (6) î ï è S + S + S3 øþï It means that the measured and real value of azimuth are the same but the measured ellipticity depends on P C and is different than a real value because æ b ö ç = è a ø ì æ P S öü ï -ç C 3 ï taní sin ç î ï è S + S + PC S3 ý ¹ æ è ç b ö ø. (7) a øþï real We concluded from Eq. (7) that in case of linearly polarized part of partially polarized light (S 3 = ) measured and real value of ellipticity are the same. The same we can say about the measured and real value of degree of polarization because S + S + PC S3 3 P = S. (8) S + S + PC S3 3 = Preal P real S + S + S3 3 Error in measurements of Stokes vector parameters diminishes when we apply zero order quarter-wave plate and also when measured light posses high temporal coherence. 6. Experimental verification of analysis results In order to test the results of the analysis, a simple experimental set-up was established (Fig. 4). As a light source, the Tosiba TOLD 93 laser diode (l @ 67 nm) was used with the driver (SENSOMED LD-5) allowing to change diode current from 8 ma till 4 ma that is from LED operation through threshold current ( 35 ma) up to single mode lasing. Light beam was polarized by the use of high-quality Glan-Thompson polarizer. As a retarder, a lithium niobiate single crystal was applied. Optical axis was placed in diagonal direction in a front plane of a cubic shape crystal ( mm). A length of coherence of light emitted by the diode was calculated basing on spectrum width measurements with applied a spectrometer. Due to low resolution of the used spectrometer, the width of a spectrum of laser diode working over the threshold current was not determined with a high accuracy. High-quality Glan-Thompson polarizer was used as an analyser in front of silicon photodiode working as a photodetector. Table. Results of measurements and calculations of depolarization of partially coherent light passing through birefringent crystal (LiNbO 3, Dn =.86, l=mm). Diode current (ma) Dl (nm) measured DL = l Dl calculated (µm) Dnl (µm) Dnl DL Azimuth q P c calculated * Gaussian ** Lorentzian spectra 4 5 8 86 48 ~ * P measured 45 * 9 5 8 86 48 ~ * 45 * 37.5 <.3 > 6 86 <.54 >.58 **.6 ±. 45 ** 4 <.5 > 3 86 <.7 >.76 **.8 ±. 45 ** Opto-Electron. Rev., 3, no., 5 A.W. Domañski 75
Polarization degree fading during propagation of partially coherent light through retarders The results of measurements and calculations are shown in Table. A degree of polarization was measured with the help of zero-order foil of quarter-wave plate type. As we can see, for linearly polarized light, passing through a lithium niobiate crystal with the azimuth of 45 ( E parallel to the optical axes), a degree of polarization is independent of coherence of light and is equal to. For the azimuth of (or 9 ), phase velocities of E components in directions of fast and slow axis are different. Hence depolarization of low coherence light (LED range of operation below threshold current (< 35 ma) is total. For the higher coherence of light (LD range of operation), depolarization took place only partially. 7. Conclusions Influence of partially coherent light on polarization degree fading takes place not only in retarders but in all linear and circular birefringent materials. The phenomenon is present also in materials with induced birefringence for example by external mechanical stresses or electric and magnetic fields. A lot of modern optoelectronic devices like modulators, isolators, circulators require high birefringence of optical medium and high degree of polarization of light passing through them in order to achieve the best parameters. On the other hand, the superluminescent diodes (SLDs), specially designed for optical coherence tomography as well as for optical fiber rotation sensors with their extremely low coherence length (5 µm), become popular for other applications. Hence the analysis of polarization degree fading for low temporal coherent light passing trough high birefringent materials should be done not only for optical fibers and bulk retarders but also for other optical and optoelectronic devices. References. W.K. Burns, Optical Fiber Rotation Sensing, Academic Press, San Diego, 994.. A.W. Domañski, M.A. Karpierz, A. Kujawski, and T.R. Woliñski, Polarimetric fiber sensors for partially polarized light source, Proc. th Intern. Congress Laser 95, 684 687 (996). 3. M. Born and E. Wolf, Principles of Optics, Pergamon Press, Oxford, 993. 4. J.W. Goodman, Statistical Optics, John Wiley&Sons, New York, 985. 5. J.I. Sakai, S. Machida, and T. Kimura, Degree of polarization in anisotropic single-mode optical fibers: Theory, IEEE J. Quantum Electron. OE-8, 488 495 (98). 6. W.A. Shurcliff, Polarized Light Production and Use, Harvard University Press, Cambridge, 966. 7. R.A. Chipman, Depolarization in the Mueller Calculus, Proc. SPIE 558, 84 9 (3). 76 Opto-Electron. Rev., 3, no., 5 5 COSiW SEP, Warsaw