Propagation of privileged waves in longitudinally inhomogeneous medium with linear birefringence and dichroism
Anisotropic properties of longitudinally inhomogeneous medium with linear birefringence and dichroism were considered. Differential Jones matrix model for this class of media was obtained. The equation for the polarization complex variable that describes the evolution of polarization state in thi...
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irk-123456789-1184272017-05-31T03:05:18Z Propagation of privileged waves in longitudinally inhomogeneous medium with linear birefringence and dichroism Koev, M.S. Kolomiets, I.S. Savenkov, S.N. Oberemok, Ye.A. Skoblya, Yu.A. Anisotropic properties of longitudinally inhomogeneous medium with linear birefringence and dichroism were considered. Differential Jones matrix model for this class of media was obtained. The equation for the polarization complex variable that describes the evolution of polarization state in this class of media was found. Partial solutions describing evolution of the privileged polarization states were obtained. The features of privileged polarization states and the conditions of their existence in this class of media were analyzed. 2014 Article Propagation of privileged waves in longitudinally inhomogeneous medium with linear birefringence and dichroism / M.S. Koev, I.S. Kolomiets, S.N. Savenkov, Ye.A. Oberemok, Yu.A. Skoblya // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2014. — Т. 17, № 4. — С. 403-407. — Бібліогр.: 11 назв. — англ. 1560-8034 PACS 78.15.+e, 78.20.Fm http://dspace.nbuv.gov.ua/handle/123456789/118427 en Semiconductor Physics Quantum Electronics & Optoelectronics Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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English |
| description |
Anisotropic properties of longitudinally inhomogeneous medium with linear
birefringence and dichroism were considered. Differential Jones matrix model for this
class of media was obtained. The equation for the polarization complex variable that
describes the evolution of polarization state in this class of media was found. Partial
solutions describing evolution of the privileged polarization states were obtained. The
features of privileged polarization states and the conditions of their existence in this class
of media were analyzed. |
| format |
Article |
| author |
Koev, M.S. Kolomiets, I.S. Savenkov, S.N. Oberemok, Ye.A. Skoblya, Yu.A. |
| spellingShingle |
Koev, M.S. Kolomiets, I.S. Savenkov, S.N. Oberemok, Ye.A. Skoblya, Yu.A. Propagation of privileged waves in longitudinally inhomogeneous medium with linear birefringence and dichroism Semiconductor Physics Quantum Electronics & Optoelectronics |
| author_facet |
Koev, M.S. Kolomiets, I.S. Savenkov, S.N. Oberemok, Ye.A. Skoblya, Yu.A. |
| author_sort |
Koev, M.S. |
| title |
Propagation of privileged waves in longitudinally inhomogeneous medium with linear birefringence and dichroism |
| title_short |
Propagation of privileged waves in longitudinally inhomogeneous medium with linear birefringence and dichroism |
| title_full |
Propagation of privileged waves in longitudinally inhomogeneous medium with linear birefringence and dichroism |
| title_fullStr |
Propagation of privileged waves in longitudinally inhomogeneous medium with linear birefringence and dichroism |
| title_full_unstemmed |
Propagation of privileged waves in longitudinally inhomogeneous medium with linear birefringence and dichroism |
| title_sort |
propagation of privileged waves in longitudinally inhomogeneous medium with linear birefringence and dichroism |
| publisher |
Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
| publishDate |
2014 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/118427 |
| citation_txt |
Propagation of privileged waves in longitudinally inhomogeneous
medium with linear birefringence and dichroism / M.S. Koev, I.S. Kolomiets, S.N. Savenkov, Ye.A. Oberemok, Yu.A. Skoblya // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2014. — Т. 17, № 4. — С. 403-407. — Бібліогр.: 11 назв. — англ. |
| series |
Semiconductor Physics Quantum Electronics & Optoelectronics |
| work_keys_str_mv |
AT koevms propagationofprivilegedwavesinlongitudinallyinhomogeneousmediumwithlinearbirefringenceanddichroism AT kolomietsis propagationofprivilegedwavesinlongitudinallyinhomogeneousmediumwithlinearbirefringenceanddichroism AT savenkovsn propagationofprivilegedwavesinlongitudinallyinhomogeneousmediumwithlinearbirefringenceanddichroism AT oberemokyea propagationofprivilegedwavesinlongitudinallyinhomogeneousmediumwithlinearbirefringenceanddichroism AT skoblyayua propagationofprivilegedwavesinlongitudinallyinhomogeneousmediumwithlinearbirefringenceanddichroism |
| first_indexed |
2025-07-08T13:57:57Z |
| last_indexed |
2025-07-08T13:57:57Z |
| _version_ |
1837087408161554432 |
| fulltext |
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2014. V. 17, N 4. P. 403-407.
© 2014, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
403
PACS 78.15.+e, 78.20.Fm
Propagation of privileged waves in longitudinally inhomogeneous
medium with linear birefringence and dichroism
M.S. Koev*, I.S. Kolomiets, S.N. Savenkov, Ye.A. Oberemok, Yu.A. Skoblya
Taras Shevchenko Kyiv National University,
4, Glushkov Ave, 03127 Kyiv, Ukraine
Phone: +38(044)526-0570, fax: +38(044) 526-1073
*E-mail: kolomiets55@gmail.com
Abstract. Anisotropic properties of longitudinally inhomogeneous medium with linear
birefringence and dichroism were considered. Differential Jones matrix model for this
class of media was obtained. The equation for the polarization complex variable that
describes the evolution of polarization state in this class of media was found. Partial
solutions describing evolution of the privileged polarization states were obtained. The
features of privileged polarization states and the conditions of their existence in this class
of media were analyzed.
Keywords: Jones matrix, birefringence, dichroism, privileged polarization states.
Manuscript received 02.04.14; revised version received 20.08.14; accepted for
publication 29.10.14; published online 10.11.14.
1. Introduction
The history of investigation of inhomogeneous non-
depolarizing media returns us to [1]. This problem was
solved using the Jones matrix methods [2]. In [3] at first
the Jones matrix for longitudinal inhomogeneous
anisotropic medium was presented. Next step was made
by R.M.A. Azzam in [4] where he considered the type of
longitudinal inhomogeneous medium with linear
birefringence. Considered in particular in [4] were
examples of these medium. It was cholesteric liquid
crystals and twisted nematics. Development of modern
display technology led to the fact that these types of
liquid crystals very widely used [5, 6]. Therefore, a
number of the next works [7-9] dealt with the features of
anisotropy formation in longitudinally inhomogeneous
medium with linear birefringence, and in the more
general case of elliptical birefringence. However, as we
know [10] for certain wavelengths in different types of
cholesteric liquid crystals and twisted nematics, besides
linear birefringence occurs as linear dichroism. One of
the main features of the anisotropy arising in
longitudinally inhomogeneous twisted media are waves
with privileged polarization states, which propagate in
this media as in the optically active media. To study the
features of this polarization in longitudinal
inhomogeneous media with linear birefringence and
dichroism is the main goal of this work.
2. The structure of medium
Non-depolarizing longitudinal inhomogeneous medium
with linear birefringence and dichroism can be
equivalently represented as a sequence of molecular
planes that consists of molecules oriented parallel to
each other [4]. Each this molecular plane can be
considered as a sufficiently thin layer (thickness of this
layer is considered to be much smaller than the size of
the medium in the direction of light propagation) with
linear birefringence and dichroism. In general, the axis
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2014. V. 17, N 4. P. 403-407.
© 2014, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
404
of birefringence and dichroism in these molecules can
differ. Thus, each molecule is characterized by two axes:
the fast axis (axis of birefringence) and the axis of least
absorption (dichroism axis) lying in the plane of each
molecular layer. In this media, at propagation in the
direction perpendicular to the molecular planes the
direction of anisotropy axis of each molecular layer
returns (twisted) compared to the previous one by some
angle. Consequently, the resulting sequence of molecular
planes is the object with linear birefringence and
dichroism that form longitudinally (along the direction
of light propagation, which coincides with the direction
around which twisting occurs) non-depolarizing
inhomogeneous medium with its linear birefringence and
dichroism. When considering propagation of polarized
light along the axis z of the Cartesian rectangular
coordinate system, which coincides with the axis of twist
of the layered medium (Fig. 1), the azimuth of
birefringence α and dichroism axis of molecular plane
at a distance z from the input molecular plane are
defined as:
,, 00 zz (1)
where α0, 0 are the specific values of orientation of the
birefringence and dichroism axis per unity thickness,
defined as:
,
2
,
2
00 pp
(2)
where p is the step of a helical structure of medium (the
smallest distance between the planes with the same
orientation of molecules).
3. Polarimetric models of media
Anisotropic properties of one molecular layer in this
medium are described by the differential Jones matrix
(in circular basis) that includes parameters of
birefringence and dichroism [3]:
0
2
1
2
1
0
00
00
2
0
2
0
2
0
2
0
zizi
zizi
ei
ei
N .
(3)
As it is known [11], the Jones vector contains
information about the state of light polarization, absolute
amplitude (intensity) and phase of the light wave. But in
the case when there is only relevant information about
the state of polarization, more convenient is a complex
variable χ, defined as:
,
x
y
E
E
(4)
where Ex, Ey are the projections of the Jones vector on
the x and y axes in the Cartesian rectangular coordinate
system, respectively. To describe evolution of the state
of polarization in anisotropic media, one can use the
scalar differential equation for the complex variable
[4, 11]:
211122
2
12 NNNN
dz
d
(5)
where Ni,j are elements of the differential Jones matrix
(3) that describes the properties of medium where light
propagates. Substituting the Jones matrix for a given
class of media (3) into Eq. (5), we get:
22
0
2
0
2
0
2
0
0000
2
1
zizizizi eiei
dz
d
.
(6)
As shown in [4, 11] in longitudinally inhomo-
geneous media, there can exist privileged waves that
propagate in it like to that in medium with optical
activity (azimuth changes linearly with the coordinate z
from one molecular layer to another, and the ellipticity
angle does not change). The complex variables K1,2 that
describe the state of polarization of these waves are a
particular solution of Eq. (5) and are sought in the form:
zieK 02
2,12,1
, (7)
where 2γ0 is the doubled specific angle of rotation of
polarization plane inherent to the privileged waves in
one molecular layer, χ1,2 are complex variables
describing evolution of the polarization states of
privileged waves in this class of media. Substituting
Eq. (7) into Eq. (6), we find that partial solutions in the
form of polarizations of privileged wave can exist in a
given class of media only if
α0 = 0. (8)
That is, orientations of the axes of birefringence
and dichroism in each molecular layer coincide. So, let
us consider the case (8). When performing the resulting
conditions for existence of the privileged wave, Fig. 1
becomes similar to that shown in Fig. 2.
Fig. 1. Molecular plane of longitudinally inhomogeneous
medium at a distance z from the input plane. The case 0 0.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2014. V. 17, N 4. P. 403-407.
© 2014, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
405
Fig. 2. Molecular plane of longitudinally inhomogeneous
medium at a distance z from the input plane. The case 0 0.
As a result, the axis of birefringence and dichroism
can be found using the second relation in Eq. (1), and
each molecule in the molecular plane has an elongated
shape, and there are appropriate mechanisms for linear
anisotropy in this class of media.
In the case α0 = 0, the anisotropic properties of one
molecular layer in this medium are described by the
differential Jones matrix (in circular basis):
0
2
1
2
1
0
0
0
2
00
2
00
zi
zi
ei
ei
N . (9)
4. Evolution of privileged waves
In the case α0 = 0, Eq. (6) describing evolution of
polarization states in anisotropic medium takes the
following form:
1
2
1 00 42
00
2
zizi eie
dz
d
. (10)
Eq. (10) has a partial solution in the form:
zieK 02
2,12,1
, (11)
where K1,2 are the roots of the characteristic equation for
(10) and describe the complex variables of polarization
of privileged waves:
1
22
2
0000
2,1
ii
K . (12)
To find the main characteristics of the polarization
states inherent to privileged waves, such as the azimuth
1,2 and angles of ellipticity e1,2, we used the
relation [11]:
2,12,1 Arg
2
1
K , (13)
1
1
Arctan
2
1
2,1
2,1
2,1
K
K
e . (14)
Substituting (12) in (13), (14), we get:
00
0
2
00
2
0
2,1
2
1
4
Arg
2
1
ii
, (15)
00
0
2
00
2
0
2,1
2
1
4
1
2
1Arctan
ii
e
.(16)
Being based on Eq. (15), it should be noted that in
contrast to the class of media in the absence of dichroism
[8, 9] the azimuths of polarization orientation of
privileged waves do not coincide with the respective
orientations of the molecular planes. Analyzing Eq. (16),
we find that polarizations of privileged waves are
elliptical. It is interesting to note that e1,2 = 0 only if
0 = 0, i.e. when this medium is homogeneous.
Now examine polarization of privileged waves on
orthogonality. To do it, we substitute (12) in the product
K1 K2
* (K1 and K2 are multiplied by z/z and moved from
the specific to the integral anisotropy parameters based
on the equations P = –ln(0z), = 0z, = 0z), which
should give –1 for the case of orthogonal corresponding
polarization states. Real and imaginary parts of the
obtained product K1 K2
* are shown in Fig. 3.
Fig. 3. The product K1 K2
* for longitudinally inhomogeneous
medium with linear birefringence and dichroism at θ0 =
10(deg/mm). The real (a) and imaginary (b) parts of the
product K1 K2
*.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2014. V. 17, N 4. P. 403-407.
© 2014, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
406
a b
c d
Fig. 4. Changes in the azimuth (a), (c) and angle of ellipticity (b), (d) with the coordinate z along the axis of light propagation in
longitudinally inhomogeneous medium with linear birefringence and dichroism and parameters ξ0 = 1.22(1/mm), δ0 =
83(deg/mm), θ0 = 29(deg/mm).
It implies from Fig. 3 that the polarization states of
privileged waves in general are not orthogonal, and
orthogonality ( 1Re 21 KK , 0Im 21 KK ) is only
achieved in the absence of linear dichroism (P = 0),
which is an obvious result [8, 9].
Equation (11) describes evolution of polarization of
privileged waves in this class of media. Substituting it
into Eqs (13) and (14) (K1,2 rather than substitute 1,2),
we obtained evolution of polarization parameters: the
azimuth and angle of ellipticity at the coordinate z
(Fig. 4).
Thus, Fig. 4 implies that these two elliptically
polarized privileged waves propagating along the axis of
the helical structure z of the inhomogeneous medium
“feel” only homogeneous and continuous rotation of the
azimuth with distance, and their ellipticity does not
change. That is to polarizes K1,2, the medium exhibits the
effective circular birefringence (optical activity) with the
angle of rotation φ = θ0 = α0(deg/mm) and shows no
linear birefringence and dichroism. The existence of
these waves is due to the features of the structure
inherent to the relevant class of media [10] and is the
result of converting polarizations of eigenwaves in
homogeneous medium with linear birefringence and
dichroism, which can be derived from this class of media
with θ0 = 0 (untwisted medium).
5. Conclusions
Ascertained in this work is that, in longitudinally
inhomogeneous medium with linear birefringence and
dichroism, the privileged waves can exist only if
coincidence between orientations of corresponding
anisotropy axes takes place. Our analysis of polarization
parameters of the privileged waves in this case shows
that they are non-orthogonal unlike the absence of linear
dichroism. Based on the results, we can formulate the
problem of synthesis of medium polarization-rotation
with settings of the initial ellipticity and azimuth.
References
1. A.S. Marathay, Matrix-operator description of the
propagation of polarized light through cholesteric
liquid crystals // J. Opt. Soc. Am. 61, p. 1363-1372
(1971).
2. H. Hurwitz, R.C. Jones, A new calculus for the
treatment of optical systems. II. Proof of the three
general equivalence theorems // J. Opt. Soc. Am.
31, p. 493-499 (1941).
3. R.C. Jones, A new calculus for the treatment of
optical systems. VII. Properties of the N-matrices //
J. Opt. Soc. Am. 38, p. 671-685 (1948).
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2014. V. 17, N 4. P. 403-407.
© 2014, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
407
4. R.M.A. Azzam, N.M. Bashara, Simplified
approach to the propagation of polarized light in
anisotropic media-application to liquid crystals // J.
Opt. Soc. Am. 62, p. 1252-1257 (1972).
5. B. Das, S. Vyas, J. Joseph et al., Transmission type
twisted nematic liquid crystal display for three
gray-level phase-modulated holographic data
storage systems // Opt. Las. in Eng. 47(11),
p. 1150-1159 (2009).
6. P. García-Martínez, M. del Mar Sánchez-López
et al., Accurate color predictability based on a
spectral retardance model of a twisted-nematic
liquid-crystal display // Opt. Communs. 284(10-11),
p. 2441-2447 (2011).
7. I.S. Kolomiets, S.N. Savenkov, Ye.A. Oberemok,
Studying the orthogonalization properties of
longitudinally inhomogeneous nondepolarizing
media // Registration, data storage and processing,
15(1), p. 23-30 (2013).
8. I.S. Kolomiets, S.N. Savenkov, Ye.A. Oberemok,
A.S. Klimov, The solution of the spectral problem
for longitudinally inhomogeneous nondepolarizing
media // Metallofizika i noveishie tehnologii, 35(9),
p. 1197-1208 (2013), in Ukrainian.
9. I.S. Kolomiets, S.N. Savenkov, Ye.A. Oberemok,
A.S. Klimov, Studying anisotropic properties of
longitudinal inhomogeneous nondepolarizing
media with elliptical phase anisotropy //
Semiconductor Physics, Quantum Electronics &
Optoelectronics, 16(4), p. 366-373 (2013).
10. S. Chandrasekhar, Liquid Crystals. Cambridge
University Press, 1980.
11. R.M.A. Azzam, N.M. Bashara, Elipsometry and
Polarized Light. New-York, 1977.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2014. V. 17, N 4. P. 403-407.
PACS 78.15.+e, 78.20.Fm
Propagation of privileged waves in longitudinally inhomogeneous medium with linear birefringence and dichroism
M.S. Koev*, I.S. Kolomiets, S.N. Savenkov, Ye.A. Oberemok, Yu.A. Skoblya
Taras Shevchenko Kyiv National University,
4, Glushkov Ave, 03127 Kyiv, Ukraine
Phone: +38(044)526-0570, fax: +38(044) 526-1073
*E-mail: kolomiets55@gmail.com
Abstract. Anisotropic properties of longitudinally inhomogeneous medium with linear birefringence and dichroism were considered. Differential Jones matrix model for this class of media was obtained. The equation for the polarization complex variable that describes the evolution of polarization state in this class of media was found. Partial solutions describing evolution of the privileged polarization states were obtained. The features of privileged polarization states and the conditions of their existence in this class of media were analyzed.
Keywords: Jones matrix, birefringence, dichroism, privileged polarization states.
Manuscript received 02.04.14; revised version received 20.08.14; accepted for publication 29.10.14; published online 10.11.14.
1. Introduction
The history of investigation of inhomogeneous non-depolarizing media returns us to [1]. This problem was solved using the Jones matrix methods [2]. In [3] at first the Jones matrix for longitudinal inhomogeneous anisotropic medium was presented. Next step was made by R.M.A. Azzam in [4] where he considered the type of longitudinal inhomogeneous medium with linear birefringence. Considered in particular in [4] were examples of these medium. It was cholesteric liquid crystals and twisted nematics. Development of modern display technology led to the fact that these types of liquid crystals very widely used [5, 6]. Therefore, a number of the next works [7-9] dealt with the features of anisotropy formation in longitudinally inhomogeneous medium with linear birefringence, and in the more general case of elliptical birefringence. However, as we know [10] for certain wavelengths in different types of cholesteric liquid crystals and twisted nematics, besides linear birefringence occurs as linear dichroism. One of the main features of the anisotropy arising in longitudinally inhomogeneous twisted media are waves with privileged polarization states, which propagate in this media as in the optically active media. To study the features of this polarization in longitudinal inhomogeneous media with linear birefringence and dichroism is the main goal of this work.
2. The structure of medium
Non-depolarizing longitudinal inhomogeneous medium with linear birefringence and dichroism can be equivalently represented as a sequence of molecular planes that consists of molecules oriented parallel to each other [4]. Each this molecular plane can be considered as a sufficiently thin layer (thickness of this layer is considered to be much smaller than the size of the medium in the direction of light propagation) with linear birefringence and dichroism. In general, the axis of birefringence and dichroism in these molecules can differ. Thus, each molecule is characterized by two axes: the fast axis (axis of birefringence) and the axis of least absorption (dichroism axis) lying in the plane of each molecular layer. In this media, at propagation in the direction perpendicular to the molecular planes the direction of anisotropy axis of each molecular layer returns (twisted) compared to the previous one by some angle. Consequently, the resulting sequence of molecular planes is the object with linear birefringence and dichroism that form longitudinally (along the direction of light propagation, which coincides with the direction around which twisting occurs) non-depolarizing inhomogeneous medium with its linear birefringence and dichroism. When considering propagation of polarized light along the axis z of the Cartesian rectangular coordinate system, which coincides with the axis of twist of the layered medium (Fig. 1), the azimuth of birefringence α and dichroism ( axis of molecular plane at a distance z from the input molecular plane are defined as:
,
,
0
0
z
z
q
=
q
a
=
a
(1)
where α0, (0 are the specific values of orientation of the birefringence and dichroism axis per unity thickness, defined as:
,
2
,
2
0
0
p
p
p
=
q
p
=
a
(2)
where
p
is the step of a helical structure of medium (the smallest distance between the planes with the same orientation of molecules).
3. Polarimetric models of media
Anisotropic properties of one molecular layer in this medium are described by the differential Jones matrix (in circular basis) that includes parameters of birefringence and dichroism [3]:
(
)
(
)
ú
ú
ú
û
ù
ê
ê
ê
ë
é
x
+
d
x
+
d
=
q
-
a
-
q
a
0
2
1
2
1
0
0
0
0
0
2
0
2
0
2
0
2
0
z
i
z
i
z
i
z
i
e
i
e
i
N
.
(3)
As it is known [11], the Jones vector contains information about the state of light polarization, absolute amplitude (intensity) and phase of the light wave. But in the case when there is only relevant information about the state of polarization, more convenient is a complex variable χ, defined as:
,
x
y
E
E
=
c
(4)
where Ex, Ey are the projections of the Jones vector on the x and y axes in the Cartesian rectangular coordinate system, respectively. To describe evolution of the state of polarization in anisotropic media, one can use the scalar differential equation for the complex variable [4, 11]:
(
)
21
11
22
2
12
N
N
N
N
dz
d
+
c
-
+
c
-
=
c
(5)
where Ni,j are elements of the differential Jones matrix (3) that describes the properties of medium where light propagates. Substituting the Jones matrix for a given class of media (3) into Eq. (5), we get:
(
)
(
)
2
2
0
2
0
2
0
2
0
0
0
0
0
2
1
c
x
+
d
-
x
+
d
=
c
q
a
q
-
a
-
z
i
z
i
z
i
z
i
e
i
e
i
dz
d
.
(6)
As shown in [4, 11] in longitudinally inhomo-geneous media, there can exist privileged waves that propagate in it like to that in medium with optical activity (azimuth changes linearly with the coordinate z from one molecular layer to another, and the ellipticity angle does not change). The complex variables K1,2 that describe the state of polarization of these waves are a particular solution of Eq. (5) and are sought in the form:
z
i
e
K
0
2
2
,
1
2
,
1
g
-
=
c
,
(7)
where 2γ0 is the doubled specific angle of rotation of polarization plane inherent to the privileged waves in one molecular layer, χ1,2 are complex variables describing evolution of the polarization states of privileged waves in this class of media. Substituting Eq. (7) into Eq. (6), we find that partial solutions in the form of polarizations of privileged wave can exist in a given class of media only if
α0 = (0.
(8)
That is, orientations of the axes of birefringence and dichroism in each molecular layer coincide. So, let us consider the case (8). When performing the resulting conditions for existence of the privileged wave, Fig. 1 becomes similar to that shown in Fig. 2.
Fig. 1. Molecular plane of longitudinally inhomogeneous medium at a distance z from the input plane. The case (0 ( (0.
Fig. 2. Molecular plane of longitudinally inhomogeneous medium at a distance z from the input plane. The case (0 ( (0.
As a result, the axis of birefringence and dichroism can be found using the second relation in Eq. (1), and each molecule in the molecular plane has an elongated shape, and there are appropriate mechanisms for linear anisotropy in this class of media.
In the case α0 = (0, the anisotropic properties of one molecular layer in this medium are described by the differential Jones matrix (in circular basis):
(
)
(
)
ú
ú
ú
û
ù
ê
ê
ê
ë
é
x
+
d
x
+
d
=
q
-
q
0
2
1
2
1
0
0
0
2
0
0
2
0
0
z
i
z
i
e
i
e
i
N
.
(9)
4. Evolution of privileged waves
In the case α0 = (0, Eq. (6) describing evolution of polarization states in anisotropic medium takes the following form:
(
)
(
)
1
2
1
0
0
4
2
0
0
2
-
c
x
+
d
=
c
q
q
z
i
z
i
e
i
e
dz
d
.
(10)
Eq. (10) has a partial solution in the form:
z
i
e
K
0
2
2
,
1
2
,
1
q
-
=
c
,
(11)
where K1,2 are the roots of the characteristic equation for (10) and describe the complex variables of polarization of privileged waves:
1
2
2
2
0
0
0
0
2
,
1
+
÷
÷
ø
ö
ç
ç
è
æ
x
-
d
q
x
-
d
q
=
i
i
K
m
.
(12)
To find the main characteristics of the polarization states inherent to privileged waves, such as the azimuth (1,2 and angles of ellipticity e1,2, we used the relation [11]:
(
)
2
,
1
2
,
1
Arg
2
1
K
=
q
,
(13)
÷
÷
ø
ö
ç
ç
è
æ
+
-
=
1
1
Arctan
2
1
2
,
1
2
,
1
2
,
1
K
K
e
.
(14)
Substituting (12) in (13), (14), we get:
(
)
÷
÷
÷
ø
ö
ç
ç
ç
è
æ
x
-
d
q
+
+
x
-
d
q
=
q
0
0
0
2
0
0
2
0
2
,
1
2
1
4
Arg
2
1
i
i
m
,
(15)
(
)
÷
÷
÷
÷
÷
÷
÷
÷
ø
ö
ç
ç
ç
ç
ç
ç
ç
ç
è
æ
x
-
d
q
+
x
-
d
q
+
-
=
0
0
0
2
0
0
2
0
2
,
1
2
1
4
1
2
1
Arctan
i
i
e
m
.(16)
Being based on Eq. (15), it should be noted that in contrast to the class of media in the absence of dichroism [8, 9] the azimuths of polarization orientation of privileged waves do not coincide with the respective orientations of the molecular planes. Analyzing Eq. (16), we find that polarizations of privileged waves are elliptical. It is interesting to note that e1,2 = 0 only if (0 = 0, i.e. when this medium is homogeneous.
Now examine polarization of privileged waves on orthogonality. To do it, we substitute (12) in the product K1 K2* (K1 and K2 are multiplied by z/z and moved from the specific to the integral anisotropy parameters based on the equations P = –ln((0z), ( = (0z, ( = (0z), which should give –1 for the case of orthogonal corresponding polarization states. Real and imaginary parts of the obtained product K1 K2* are shown in Fig. 3.
Fig. 3. The product K1 K2* for longitudinally inhomogeneous medium with linear birefringence and dichroism at θ0 = 10(deg/mm). The real (a) and imaginary (b) parts of the product K1 K2*.
It implies from Fig. 3 that the polarization states of privileged waves in general are not orthogonal, and orthogonality (
(
)
1
Re
2
1
-
=
*
K
K
,
(
)
0
Im
2
1
=
*
K
K
) is only achieved in the absence of linear dichroism (P = 0), which is an obvious result [8, 9].
Equation (11) describes evolution of polarization of privileged waves in this class of media. Substituting it into Eqs (13) and (14) (K1,2 rather than substitute (1,2), we obtained evolution of polarization parameters: the azimuth and angle of ellipticity at the coordinate z (Fig. 4).
Thus, Fig. 4 implies that these two elliptically polarized privileged waves propagating along the axis of the helical structure z of the inhomogeneous medium “feel” only homogeneous and continuous rotation of the azimuth with distance, and their ellipticity does not change. That is to polarizes K1,2, the medium exhibits the effective circular birefringence (optical activity) with the angle of rotation φ = θ0 = α0(deg/mm) and shows no linear birefringence and dichroism. The existence of these waves is due to the features of the structure inherent to the relevant class of media [10] and is the result of converting polarizations of eigenwaves in homogeneous medium with linear birefringence and dichroism, which can be derived from this class of media with θ0 = 0 (untwisted medium).
5. Conclusions
Ascertained in this work is that, in longitudinally inhomogeneous medium with linear birefringence and dichroism, the privileged waves can exist only if coincidence between orientations of corresponding anisotropy axes takes place. Our analysis of polarization parameters of the privileged waves in this case shows that they are non-orthogonal unlike the absence of linear dichroism. Based on the results, we can formulate the problem of synthesis of medium polarization-rotation with settings of the initial ellipticity and azimuth.
References
1.
A.S. Marathay, Matrix-operator description of the propagation of polarized light through cholesteric liquid crystals // J. Opt. Soc. Am. 61, p. 1363-1372 (1971).
2.
H. Hurwitz, R.C. Jones, A new calculus for the treatment of optical systems. II. Proof of the three general equivalence theorems // J. Opt. Soc. Am. 31, p. 493-499 (1941).
3.
R.C. Jones, A new calculus for the treatment of optical systems. VII. Properties of the N-matrices // J. Opt. Soc. Am. 38, p. 671-685 (1948).
4.
R.M.A. Azzam, N.M. Bashara, Simplified approach to the propagation of polarized light in anisotropic media-application to liquid crystals // J. Opt. Soc. Am. 62, p. 1252-1257 (1972).
5.
B. Das, S. Vyas, J. Joseph et al., Transmission type twisted nematic liquid crystal display for three gray-level phase-modulated holographic data storage systems // Opt. Las. in Eng. 47(11), p. 1150-1159 (2009).
6.
P. García-Martínez, M. del Mar Sánchez-López et al., Accurate color predictability based on a spectral retardance model of a twisted-nematic liquid-crystal display // Opt. Communs. 284(10-11), p. 2441-2447 (2011).
7.
I.S. Kolomiets, S.N. Savenkov, Ye.A. Oberemok, Studying the orthogonalization properties of longitudinally inhomogeneous nondepolarizing
media // Registration, data storage and processing, 15(1), p. 23-30 (2013).
8.
I.S. Kolomiets, S.N. Savenkov, Ye.A. Oberemok, A.S. Klimov, The solution of the spectral problem for longitudinally inhomogeneous nondepolarizing media // Metallofizika i noveishie tehnologii, 35(9), p. 1197-1208 (2013), in Ukrainian.
9.
I.S. Kolomiets, S.N. Savenkov, Ye.A. Oberemok, A.S. Klimov, Studying anisotropic properties of longitudinal inhomogeneous nondepolarizing media with elliptical phase anisotropy // Semiconductor Physics, Quantum Electronics & Optoelectronics, 16(4), p. 366-373 (2013).
10.
S. Chandrasekhar, Liquid Crystals. Cambridge University Press, 1980.
11.
R.M.A. Azzam, N.M. Bashara, Elipsometry and Polarized Light. New-York, 1977.
� �
a b
� �
c d
Fig. 4. Changes in the azimuth (a), (c) and angle of ellipticity (b), (d) with the coordinate z along the axis of light propagation in longitudinally inhomogeneous medium with linear birefringence and dichroism and parameters ξ0 = 1.22(1/mm), δ0 = 83(deg/mm), θ0 = 29(deg/mm).
© 2014, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
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