Polarization properties of longitudinally inhomogeneous dichroic medium

Polarization properties of longitudinal inhomogeneous medium with linear and elliptical dichroism based on differential Jones matrix are considered. Properties of privileged polarizations have been analyzed. The integral Jones and Mueller matrices of media are obtained as a solution of the vector tr...

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Дата:	2015
Автори:	Kolomiets, I.S., Savenkov, S.N., Oberemok, Ye.A.
Формат:	Стаття
Мова:	English
Опубліковано:	Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України 2015
Назва видання:	Semiconductor Physics Quantum Electronics & Optoelectronics
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Цитувати:	Polarization properties of longitudinally inhomogeneous dichroic medium / I.S. Kolomiets, S.N. Savenkov, Ye.A. Oberemok // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2015. — Т. 18, № 2. — С. 193-199. — Бібліогр.: 22 назв. — англ.

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spelling	irk-123456789-1218132017-06-19T03:02:54Z Polarization properties of longitudinally inhomogeneous dichroic medium Kolomiets, I.S. Savenkov, S.N. Oberemok, Ye.A. Polarization properties of longitudinal inhomogeneous medium with linear and elliptical dichroism based on differential Jones matrix are considered. Properties of privileged polarizations have been analyzed. The integral Jones and Mueller matrices of media are obtained as a solution of the vector transfer equation. Orthogonalization properties of these classes of media are studied. Evolution of totally polarized light in longitudinal inhomogeneous medium with linear and elliptical dichroism has been analyzed. 2015 Article Polarization properties of longitudinally inhomogeneous dichroic medium / I.S. Kolomiets, S.N. Savenkov, Ye.A. Oberemok // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2015. — Т. 18, № 2. — С. 193-199. — Бібліогр.: 22 назв. — англ. 1560-8034 DOI: 10.15407/spqeo18.02.193 PACS 78.15.+e, 78.20.Fm http://dspace.nbuv.gov.ua/handle/123456789/121813 en Semiconductor Physics Quantum Electronics & Optoelectronics Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	Polarization properties of longitudinal inhomogeneous medium with linear and elliptical dichroism based on differential Jones matrix are considered. Properties of privileged polarizations have been analyzed. The integral Jones and Mueller matrices of media are obtained as a solution of the vector transfer equation. Orthogonalization properties of these classes of media are studied. Evolution of totally polarized light in longitudinal inhomogeneous medium with linear and elliptical dichroism has been analyzed.
format	Article
author	Kolomiets, I.S. Savenkov, S.N. Oberemok, Ye.A.
spellingShingle	Kolomiets, I.S. Savenkov, S.N. Oberemok, Ye.A. Polarization properties of longitudinally inhomogeneous dichroic medium Semiconductor Physics Quantum Electronics & Optoelectronics
author_facet	Kolomiets, I.S. Savenkov, S.N. Oberemok, Ye.A.
author_sort	Kolomiets, I.S.
title	Polarization properties of longitudinally inhomogeneous dichroic medium
title_short	Polarization properties of longitudinally inhomogeneous dichroic medium
title_full	Polarization properties of longitudinally inhomogeneous dichroic medium
title_fullStr	Polarization properties of longitudinally inhomogeneous dichroic medium
title_full_unstemmed	Polarization properties of longitudinally inhomogeneous dichroic medium
title_sort	polarization properties of longitudinally inhomogeneous dichroic medium
publisher	Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
publishDate	2015
url	http://dspace.nbuv.gov.ua/handle/123456789/121813
citation_txt	Polarization properties of longitudinally inhomogeneous dichroic medium / I.S. Kolomiets, S.N. Savenkov, Ye.A. Oberemok // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2015. — Т. 18, № 2. — С. 193-199. — Бібліогр.: 22 назв. — англ.
series	Semiconductor Physics Quantum Electronics & Optoelectronics
work_keys_str_mv	AT kolomietsis polarizationpropertiesoflongitudinallyinhomogeneousdichroicmedium AT savenkovsn polarizationpropertiesoflongitudinallyinhomogeneousdichroicmedium AT oberemokyea polarizationpropertiesoflongitudinallyinhomogeneousdichroicmedium
first_indexed	2025-07-08T20:33:56Z
last_indexed	2025-07-08T20:33:56Z
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fulltext	Semiconductor Physics, Quantum Electronics & Optoelectronics, 2015. V. 18, N 2. P. 193-199. doi: 10.15407/spqeo18.02.193 © 2015, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine 193 PACS 78.15.+e, 78.20.Fm Polarization properties of longitudinally inhomogeneous dichroic medium I.S. Kolomiets, S.N. Savenkov, Ye.A. Oberemok Taras Shevchenko Kyiv National University, 4, Glushkov Ave, 03127 Kyiv, Ukraine Phone: +38(044)-526-0580, e-mail: kolomiets55@gmail.com Abstract. Polarization properties of longitudinal inhomogeneous medium with linear and elliptical dichroism based on differential Jones matrix are considered. Properties of privileged polarizations have been analyzed. The integral Jones and Mueller matrices of media are obtained as a solution of the vector transfer equation. Orthogonalization properties of these classes of media are studied. Evolution of totally polarized light in longitudinal inhomogeneous medium with linear and elliptical dichroism has been analyzed. Keywords: Jones matrix, Mueller matrix, dichroism, privileged polarization states, orthogonalization properties, evolution of polarization state. Manuscript received 02.12.14; revised version received 11.03.15; accepted for publication 27.05.15; published online 08.06.15. 1. Introduction Liquid crystals, which twenty years ago were quite exotic object of research [1-3], today have many important practical applications. In particular, the cholesteric and twisted nematic types of liquid crystals are widely used to create a variety of displays [4-6]. From the polarimetric viewpoint, these crystals are the class of nondepolarizing, longitudinally inhomogeneous media [7]. Jones and Mueller matrix methods [8, 9] to describe the polarization properties of such class of media were first used in the works [10, 11]. Description of media based on their equivalent representation by sequence of layers each of which is characterized by corresponding differential Jones or Mueller matrices [12]. Further analysis of properties inherent to these media is based on the study of evolution of polarized light [13, 14]. To date, we see growth of interest in studying of these media. In particular, recently a number of papers considering the different polarization properties of longitudinally inhomogeneous media with linear and elliptical birefringence have been published [15-17]. The case when the molecular plane in this medium is simultaneously characterized by linear birefringence and linear dichroism was also considered [18]. The main goal of this paper is to consider propagation of polarized light in longitudinally inhomogeneous medium characterized by linear and elliptical dichroism. 2. The equivalent representation of the medium The equivalent representation of longitudinally inhomogeneous medium with amplitude anisotropy is similar to the case of longitudinally inhomogeneous medium with phase anisotropy [13, 17]. However, in this case one monomolecular plane is thin (the thickness is Semiconductor Physics, Quantum Electronics & Optoelectronics, 2015. V. 18, N 2. P. 193-199. doi: 10.15407/spqeo18.02.193 © 2015, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine 194 much smaller than the longitudinal dimension of the medium) linear or elliptical partial polarizer. If each of these polarizers is linear, then they are characterized by the parameter )( 2 0 eo kk     . If the polarizers are elliptical, then they are also characterized by the parameter )( 2 0 rl kkr     . These parameters are specific relative absorption between linear and circular eigenpolarizations per unit length, ko, ke, kr, kl – absorption coefficients of ordinary, extraordinary, right and left circular eigenpolarizations respectively. In the direction of light propagation along the axis perpendicular to the input surfaces of polarizers (molecular planes) each subsequent polarizer is rotated relatively to the previous one at the equal angle θ0: p   2 0 , (1) where p is a step of helical structure of the medium, i.e., the smallest distance between the planes with the same orientation of molecules. Then, molecular plane orientation for the thickness z can be determined as: z0 . (2) The axis z coincides with the axis of twist in the layered medium (helical axis). It should be noted that, by analogy to the case of longitudinally inhomogeneous medium with birefringence, the equation (2) is valid for longitudinally inhomogeneous medium with dichroism under the following conditions: 1\|)sin(/)2tan(\| 00  – for linear dichroism and 1\|)sin(/)2tan(\| 0 2 0 2 0  r for elliptic dichroism. 3. Matrix models of medium under consideration Anisotropic properties of one molecular layer in this medium are described by the differential Jones matrix. This matrix can be obtained in a similar way to that presented in [13] for the case of longitudinally inhomogeneous medium with birefringence. In circular basis [19], the differential Jones matrices of longitudinally inhomogeneous medium with linear (LD) and elliptical (ED) dichroism are as follows:                0 2 1 2 1 0 0 0 2 0 2 0 zi zi LD ei ei N , (3)              0 2 0 2 00 0 0 2 1 re er N zi zi ED . (4) For the transition from circular to linear bases related with the Cartesian rectangular coordinate system, one can use the following relation [19]: 1 FFNN CirLin , (5) where F – transition matrix of the form: . 11         ii F (6) As a result, using Eqs. (3)-(6), one can obtain the following differential Jones matrices for classes of media under consideration in the linear basis: , )2cos( 2 1 )2sin( 2 1 )2sin( 2 1 )2cos( 2 1 0000 0000              zz zz N LD Lin (7) . )2cos( 2 1 ))2sin(( 2 1 ))2sin(( 2 1 )2cos( 2 1 00000 00000              zzir zirz N ED Lin (8) Using Ref. [11], the differential Mueller matrices corresponding to the Jones matrices described by Eqs. (7) and (8) can be calculated: , 0000 000)2sin( 000)2cos( 0)2sin()2cos(0 00 00 0000                 z z zz mLD Lin (9) , 000 000)2sin( 000)2cos( )2sin()2cos(0 0 00 00 00000                 r z z rzz mED Lin (10) The integral polarization properties of these classes of media with the microscopic thickness z can be described by the integral Jones matrix T. To calculate this matrix, we use the vector transfer equation [11]: ,NE dz dE  (11) where E is 21 Jones vector describing the polarization of light. Substituting the differential Jones matrices (7), (8) in Eq. (11) and solving them with initial conditions E1(0) = E01; E2(0) = E02, the elements of integral Jones matrices can be found using the equations [20]: ., ,, 002 2 22 001 2 21 002 1 12 001 1 11 0102 0102     EE EE E E T E E T E E T E E T (12) Then, for classes of media under consideration the integral Jones matrices are as follows: Semiconductor Physics, Quantum Electronics & Optoelectronics, 2015. V. 18, N 2. P. 193-199. doi: 10.15407/spqeo18.02.193 © 2015, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine 195 , 2 ))(2)(( 2 )( 2 ))(2)(()( 2 2 ))(2)(()( 2 2 ))(2)(( 2 )( 1 1000010 1000001 1000001 1000010                                                                  zA SzSzC zA CzAC zA SzCzSzS zA AC zA SzCzSzS zA AC zA SzSzC zA CzAC A T LD (13) . 2 ))()2()(( 2 )( 2 ))()()2(()( 2 2 ))()()2(()( 2 2 ))()2()(( 2 )( 1 10000010 10000001 10000001 10000010                                                                  zB SzSrizC zB CzBC zB SzSzCirzS zB AC zB SzSzCirizS zB BC zB SzSirzC zB CzBC B T ED (14) where ,4 2 0 2 0 A ,)2( 2 00 2 0  irB C(x) = cos(x), S(x) = sin(x), C1(x) = ch(x), S1(x) = sh(x). Similarly, using the Stokes vector transfer equation for differential Mueller matrices, see [11, 19], the corresponding integral Mueller matrices can determined. 4. The features of privileged polarizations In this section, to describe polarization we use the concept of a polarization complex variable  defined as the ratio of the Jones vector components  = Ey / Ex. To describe evolution of polarization, we use the differential transfer equation in the complex variable terms [19]: 211122 2 12 )(/ NNNNdzd  . (15) Substituting in Eq. (15) the elements of differential Jones matrices (in circular basis), Eqs. (3) and (4), one can obtain the corresponding equations of polarization evolution for longitudinally inhomogeneous medium with linear and elliptical dichroism: , 2 1 2 1 / 00 2 0 22 0 zizi eedzd         (16)  .1 2 1 / 242 00 00   zizi eerdzd (17) The equations (16) and (17) have the following partial solutions: ,02 2,12,1 zi eK   (18) ,02 4,34,3 zi eK   (19) respectively. Eqs. (18) and (19) describe evolution of polarizations characterized by complex variables K1,2 and K3,4 with the coordinate z propagating in longitudinally inhomogeneous media with linear and elliptical dichroism, respectively. The azimuths of these polarizations are changed linearly with the coordinate z (there is gradual rotation from one to another molecular layer). Whereas, the ellipticity remains unchanged, when light propagates in the medium. So, according to the classification presented in [13], [14] and [19] the polarizations described by complex variables K1,2 and K3,4 call privileged polarizations. They propagate in longitudinally inhomogeneous medium characterized by linear (K1,2) or elliptical (K3,4) dichroism as in optically active medium with the values of optical activity  = 1,2 (1,2 – azimuth of polarizations states variables K1,2 and K3,4). These complex variables are the solutions of the characteristic equations (16) and (17): ,1 22 2 0 0 0 0 2,1               ii K (20) .1 22 2 0 00 0 00 4,3               irir K (21) From the equations (20) and (21), it follows that if r0 = 0, then K1,2 = K3,4. To calculate the azimuths and ellipticity of polarizations in circular basis, the following equations can be used [19]: ),arg( 2 1 2,1 K (22) . 1\|\| 1\|\| arctan2,1          K K (23) As a result, azimuths and ellipticity for polarizations K1,2 are as follows: , 42 arg 2 1 0 2 0 2 00 2,1            i (24) . 1)/2()/2(1 1)/2()/2(1 arctan 00 2 00 00 2 00 2,1              i i (25) Analyzing the expressions (24), (25) and using the conditions of polarization’s orthogonality K1,2 * = –1, we should consider three cases: 1) if 2 0 2 04  , then K1,2 are two non-orthogonal linear polarizations; 2) if 2 0 2 04  , then K1 = K2 are two coincident linear polarizations with orientation 1 = 2 = 45. 3) if 2 0 2 04  , then K1,2 are two non-orthogonal elliptical polarizations with same orientation 1 = 2 = 45. In the case of elliptical dichroism, the azimuths and ellipticity for polarizations K3,4 differ from polarizations Semiconductor Physics, Quantum Electronics & Optoelectronics, 2015. V. 18, N 2. P. 193-199. doi: 10.15407/spqeo18.02.193 © 2015, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine 196 K1,2: in Eqs. (24) and (25), we have the sum 00 2  ir instead of the parameter 02 i . As a result, the class of media with elliptical dichroism is always characterized by elliptical non-orthogonal privileged polarizations. It should also be noted that, for both considered classes of media, the privileged polarizations are orthogonal only if 0 = 0. (i.e. when the medium is homogeneous). In this case, the privileged polarizations are eigenpolarizations of corresponding homogeneous medium. Comparing the features of the privileged polarizations for longitudinally inhomogeneous media with linear and elliptical dichroism with relevant properties of longitudinally inhomogeneous media with linear and elliptical birefringence [13, 17], some differences can be noted. In particular, unlike the case of birefringence for the case of dichroism the privileged polarizations are always non-orthogonal and its azimuths do not coincide with the azimuth of dichroism in the input molecular plane. 5. Orthogonalization properties Orthogonalization properties of the medium are the ability of the medium to transform some input polarization to orthogonal one [21]. As it was demon- strated in [21, 22], for existence of orthogonalization properties in the medium, the following two conditions have to be fulfilled:       0 0 2 0R F (26) where F and 2 0R – parameters which are the functions of integral Jones matrix elements Tij. The dependences of F and 2 0R on Tij are presented in [15, 17, 22]. Substituting the elements of Jones matrix Eq. (13) in these dependences, we get dependences of the value F on (a) the parameters of anisotropy ξ0, 0 and (b) parameter 0 at ξ0 = 1.8 (1/mm) for longitudinally inhomogeneous medium with linear dichroism. Results are presented in Fig. 1. As can be seen from Fig. 1, for given the values of anisotropy parameters ξ0 = 1.8 (1/mm), θ0 = 0.59 (rad/mm), this class of media will have orthogonalization properties for the thickness z =2 mm. In this case, 041.02 0 R . With account of the above values of anisotropy, to determine the polarizations that are orthogonalized by this class of media, we will analyze the product ),( inpinpoutinp ef   ( inp , out – complex variables describing input and output polarizations) defined basing on the approach presented in [15] and [17]. The results are shown in Fig. 2. It can be seen from Fig. 2 that in longitudinally inhomogeneous medium with linear dichroism two linear polarizations with azimuths θinp1 = –46.12°, θinp2 = –66.75° are orthogonalized. a b Fig. 1. Dependence of the function F on the anisotropy parameters: (a) ξ0, θ0; (b) θ0, ξ0 = 1.8 (1/mm) for medium with inhomogeneous linear amplitude anisotropy. The similar analysis is carried out for orthogonalization properties of longitudinally inhomogeneous media with elliptical dichroism. In particular, taking the values of anisotropy parameters ξ0 = 1.8 (1/mm), r0 = 0.7 (1/mm), θ0 = –0.6 (rad/mm), and thickness z =2 mm, this class of media orthogonalizes the elliptical polarizations with parameters θinp1 = –55.4° and einp = –16°. 6. Evolution of linear polarized light In the previous two sections, we examined the properties of two special types of polarizations: privileged and those orthogonalized in longitudinally inhomogeneous medium with amplitude anisotropy. Now consider what happens with other types of input polarizations, when they propagate along the helical axis in this class of media. Let us consider this problem for linear polarization with the azimuth θinp described by the Stokes vector [19]: T inpinpinpS )02sin2cos1(  . (27) For that, we use the Stokes vector transfer equation of the following form [11]: .mS dz dS  (28) Semiconductor Physics, Quantum Electronics & Optoelectronics, 2015. V. 18, N 2. P. 193-199. doi: 10.15407/spqeo18.02.193 © 2015, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine 197 a b c Fig. 2. Dependences of the product inp * out on the azimuth θinp and ellipticity εinp of input light for medium with parameters ξ0 = 1.8 (1/mm), θ0 = 0.59 (rad/mm), z = 2 mm: (a) dependence of Re (inp * out) on θinp and εinp; (b) dependence of Im(inp * out) on θinp and εinp, (c) dependence of Re (inp * out) on θinp at εinp = 0. Substituting the differential Mueller matrices, Eqs. (9) and (10), describing longitudinally inhomogeneous medium with linear and elliptical dichroism in Eq. (28), and solving it with initial conditions Eq. (27), we obtain: ,),,,()( 00 zfzS inpout  (29) ,),,,,()( 000 zrfzS inpout  (30) where Sout (z) is the output Stokes vector. To analyze evolution of polarization and the intensity of light with linear polarization (27), we use the equations: ,arctg 2 1 )( ,arctg 2 1 )( 2 )3( 2 )2( )4( )2( )3(                      outout out out out out out SS S z S S z (31) (1)( ) ,out outI z S (32) where θout, out and Iout are the azimuth, ellipticity and intensity (normalized by input intensity Sinp(1)) of light in the output of medium with the thickness z. The results are presented in Figs. 3 and 4. a b c Fig. 3. Evolution of linear polarization with azimuth θinp = –1.2 (rad/mm) in longitudinally inhomogeneous medium with linear amplitude anisotropy: (a) evolution of the azimuth for the case ξ0 = 1.22 (1/mm), θ0 = 0.5 (rad/mm); (b) evolution of the azimuth for the case ξ0 = 1.22 (1/mm), θ0 = 0.7 (rad/mm), and (c) evolution of the intensity for both cases. Semiconductor Physics, Quantum Electronics & Optoelectronics, 2015. V. 18, N 2. P. 193-199. doi: 10.15407/spqeo18.02.193 © 2015, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine 198 a b c Fig. 4. Evolution of linear polarization with the azimuth θinp = –0.8 (rad/mm) in longitudinally inhomogeneous medium possessing elliptical amplitude anisotropy with parameters ξ0 = 1.22 (1/mm), θ0 = 0.3 (rad/mm), r0 = 0.7 (1/mm), (a) evolution of the azimuth; (b) evolution of the ellipticity, and (c) evolution of the intensity. Fig. 3 shows that in longitudinally inhomogeneous medium with linear dichroism depending on the ratio between the anisotropy parameters ξ0 and 0, the azimuth of input linear polarization can be changed linearly or nonlinearly. Here, the ellipticity remains unchanged (equal to zero). Fig. 4 shows that, in the medium with elliptical dichroism, the azimuth changes linearly, i.e., there is rotation of the polarization plane, and ellipticity tends effectively to some constant non- zero value depending on anisotropy of the medium. The intensity of light in both cases of media (with linear and elliptical dichroism) decreases rapidly with the coordinate z. 7. Conclusions We show that privileged polarizations exist for both types of longitudinally inhomogeneous media: with linear and elliptical dichroism. However, unlike the case of inhomogeneous birefringence for the media with longitudinally inhomogeneous linear and elliptical dichroism, privileged polarizations are always non- orthogonal, and their azimuths do not coincide with orientation of the input molecular plane. Also, taking definite ratio between anisotropy parameters, in the case of linear dichroism the privileged polarizations can change from linear to elliptical and even become the same. Both classes of longitudinally inhomogeneous dichroic media demonstrate orthogonalization properties. However, unlike the case of inhomogeneous birefringent medium characterized by continuum of orthogonalized polarizations, longitudinally inhomo- geneous dichroic media have a finite number of orthogonalized polarizations. Note that, when arbitrary linear polarization propagates in inhomogeneous medium with linear dichroism, it remains linear, and the azimuth changes can be both linear and non-linear. 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Polarization properties of longitudinally inhomogeneous dichroic medium

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