Studying anisotropic properties of longitudinal inhomogeneous nondepolarizing media with elliptical phase anisotropy
In this paper, basing on the anisotropic properties of longitudinal
 inhomogeneous nondepolarizing media with linear phase anisotropy, more general type
 of media with elliptical phase anisotropy was studied. The features of propagation of
 light with privileged states of pol...
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
2013
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| Cite this: | Studying anisotropic properties of longitudinal inhomogeneous
 nondepolarizing media with elliptical phase anisotropy / I.S. Kolomiets, S.N. Savenkov, Ye.A. Oberemok, A.S. Klimov // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2013. — Т. 16, № 4. — С. 366-373. — Бібліогр.: 18 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860087863155097600 |
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| author | Kolomiets, I.S. Savenkov, S.N. Oberemok, Ye.A. Klimov, A.S. |
| author_facet | Kolomiets, I.S. Savenkov, S.N. Oberemok, Ye.A. Klimov, A.S. |
| citation_txt | Studying anisotropic properties of longitudinal inhomogeneous
 nondepolarizing media with elliptical phase anisotropy / I.S. Kolomiets, S.N. Savenkov, Ye.A. Oberemok, A.S. Klimov // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2013. — Т. 16, № 4. — С. 366-373. — Бібліогр.: 18 назв. — англ. |
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| container_title | Semiconductor Physics Quantum Electronics & Optoelectronics |
| description | In this paper, basing on the anisotropic properties of longitudinal
inhomogeneous nondepolarizing media with linear phase anisotropy, more general type
of media with elliptical phase anisotropy was studied. The features of propagation of
light with privileged states of polarization were observed. Transformation of polarization
states of eigenwaves along z axes of light propagation was studied. The orthogonalization
properties inherent to this type of medium were obtained. Evolution of linear polarized
light in this media was presented and discussed.
|
| first_indexed | 2025-12-07T17:21:05Z |
| format | Article |
| fulltext |
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2013. V. 16, N 4. P. 366-373.
© 2013, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
366
PACS 78.20.Fm, 78.20.Ek
Studying anisotropic properties of longitudinal inhomogeneous
nondepolarizing media with elliptical phase anisotropy
I.S. Kolomiets, S.N. Savenkov, Ye.A. Oberemok, A.S. Klimov
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. In this paper, basing on the anisotropic properties of longitudinal
inhomogeneous nondepolarizing media with linear phase anisotropy, more general type
of media with elliptical phase anisotropy was studied. The features of propagation of
light with privileged states of polarization were observed. Transformation of polarization
states of eigenwaves along z axes of light propagation was studied. The orthogonalization
properties inherent to this type of medium were obtained. Evolution of linear polarized
light in this media was presented and discussed.
Keywords: Jones method, birefringence, spectral problem, longitudinally
inhomogeneous medium.
Manuscript received 18.07.13; revised version received 05.09.13; accepted for
publication 23.10.13; published online 16.12.13.
1. Introduction
The history of investigation of longitudinal
inhomogeneous nondepolarizing media returns us to
Ref. [1]. The study of anisotropic properties of this type
of media in time of publication [1] was difficult as
caused by the complexity of the mathematical apparatus
used. This problem was solved using the Jones and
Mueller matrix methods [2, 3]. For the first time, the
Jones matrix of longitudinal inhomogeneous anisotropic
medium was presented in [4]. R. Azzam made the next
step in [5], where he considered the type of longitudinal
inhomogeneous medium with linear phase anisotropy. In
particular, in [6-8] the examples of these media were
considered: cholesteric and twisted nematic liquid
crystals. However, the studies of anisotropic properties
of this type of media were made only partially because
of complexity of these objects. Specifically, such
properties as the solution of spectral problem and
orthogonalization properties were missed.
Development of the modern display technology
results in the fact that these types of liquid crystals are
very widely used [9-11]. Optical anisotropy of these
media is in the basis of using various liquid crystals in
computer displays and indicator devices. Therefore, the
study of anisotropic properties of longitudinal
inhomogeneous anisotropic medium is very important
problem for display technology. The case of longitudinal
inhomogeneous nondepolarizing media with linear phase
anisotropy, which in particular includes cholesterics and
twisted nematics, was solved in [12, 13]. This paper
generalizes the results obtained in [12, 13] for the case
of the media characterized by both linear and circular
phase anisotropy (elliptic phase anisotropy).
2. Polarimetric models of media
Nondepolarizing longitudinal inhomogeneous medium
with elliptical phase anisotropy can be equivalently
represented as a sequence of molecular planes that
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2013. V. 16, N 4. P. 366-373.
© 2013, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
367
consist of elongated molecules, oriented parallel to each
other. At the same time, such molecules possess inherent
chirality. Each of these molecular planes can be present
as a thin phase plate with elliptical phase anisotropy
(elliptical birefringence). The fast (slow) axis of the
plate is parallel (perpendicular) to the direction along
which directed is the plane of the molecules that is
considered, and lies entirely in this plane. In these
media, the axis of each following molecular layer is
rotated relative to the previous one by some angle 0 [7]:
p
2
0 , (1)
where p is the step of helical structure in this medium
(the shortest distance between the planes with the same
orientation of molecules). Then, the molecular
orientation of the plane in distance z from the input can
be defined as:
z0 . (2)
Anisotropic properties of one molecular layer in
this medium are described by the differential Jones
matrix (in circular basis) [6]:
0
2
0
2
00
0
0
2
1
2
1
iei
eii
N
zi
zi
. (3)
And the anisotropic properties of this longitudinally
inhomogeneous medium consisting of a sequence of
molecular planes with a longitudinal size z are described
by the integral Jones matrix (in linear basis):
.
2
2
2
2
2
2
2
2
2
2
2
21
000000
000000
000000
000000
zA
SzSzCi
zA
CzAC
zA
SzCzSizS
zA
AC
zA
SzCzSizS
zA
AC
zA
SzSzCi
zA
CzAC
A
T
(4)
In Eqs (3) and (4), 0 0, are magnitude of linear
and circular birefringence per unit thickness in the
direction of light propagation, respectively, and the
following replacement is used:
200
2
0 4 A , xxC cos , xxS sin .
Note that the integral matrix (4) can be obtained from
the differential (3) by using the vector transfer equation
and technique presented in [14].
3. Propagation of radiation with privileged states of
polarization
One of the anisotropic properties of longitudinally
inhomogeneous media that does not exist in a
longitudinally homogeneous media is availability of the
privileged polarization states. The term “privileged” was
at first introduced in [7]. It means the state of
polarization of eigenwaves in longitudinal homogeneous
(untwisted) media of this type.
Evolution of polarization states along the z axis of
light propagation in anisotropic medium can be
described by the first order differential equation for
polarization complex variable [6]. For the case of
longitudinally inhomogeneous medium with elliptical
phase anisotropy, this equation is:
zizi eiieizdd 00 2
00
22
0 2
1
2
2
1
/χ , (5)
where is the complex variable that determines the
light polarization state. Eq. (5) has a partial solution in
the form:
zieK 02
2,12,1
, (6)
where 2,1K are privileged polarization states of light.
Azimuth of orientation coincides with orientation of
input molecular plane, and the angle of ellipticity of
privileged states is determined as:
.
1/21/2
1/21/2
arctan
2
000000
2
000000
2,1
e
(7)
These two privileged polarization states 2,1K are
orthogonal and elliptical in a general case, and
transformation into two elliptical eigenpolarization for
longitudinally inhomogeneous medium with elliptical
phase anisotropy takes place when 0, 00 .
Eq. (6) describes the change of privileged
polarization states of light propagation in the medium
along the z axis. Since Eq. (5) is written for the
differential Jones matrix in the circular basis (3), the
azimuth and angle of ellipticity of polarization 2,1p
can be determined as:
2,1arg
2
1
pp ,
1
1
arctan
2,1
2,1
p
p
pe . (8)
Graphical representation of (8) is shown in Fig. 1.
From Eq. (6) and Fig. 1, we can see that the
azimuths of privileged polarization states are changed by
the law θp(1,2) = α0, and for each molecular plane they
coincide with their principal axes, while the angle of
ellipticity remains unchanged and is defined by Eq. (7).
As a result of propagation of light with the privileged
state of polarization along the z axis, this type of
medium is characterized by efficiency of circular phase
anisotropy with the magnitude of the relative angle φ0 =
α0 (rad/mm).
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2013. V. 16, N 4. P. 366-373.
© 2013, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
368
a
b
c
d
Fig. 1. Evolution of the azimuth and angle of ellipticity in
longitudinally inhomogeneous medium with elliptical phase
anisotropy and parameters: δ0 = 1.22 rad/mm, α0 = 0.3 rad/mm,
φ0 = 0.7 rad/mm; (a), (b) azimuth, (c), (d) angle of ellipticity
for polarizations p1, p2, respectively.
4. The solution of the spectral problem
To analyze the basic anisotropic properties for the
medium of this type, we need to find properties of their
eigenwaves, i.e. to solve the spectral problem. For this,
we use the relations represented in [14], which gives the
solutions of the spectral problem for the integral (3) and
differential (4) Jones matrices. As a result,
eigenpolarization and eigenvalues for these matrices are
as follows:
;4
2
1
,
2
0
2
02,1
2
332
2
4
2,1
iV
A
ARRRAR
V
Ne
Te
(9)
;
2sin2
2cos4
,
000
00
2
0
2
0
2,1
2
332
2
1
2,1
zi
zi
Q
ARRRAR
Ne
Te
(10)
where the following substitutions are used:
.
2
2
2
,
2
22
2
,
2
4
,1
2
,
2
00000
0
000
0
4
2
2
0
2
003
2
2
02
001
zA
SzSizC
zS
zA
ACQ
zA
SzS
zA
CzAC
R
zA
SzSR
zA
CzCR
zA
SzCiR
(11)
It follows from relations (9) that, as in the case of
longitudinally inhomogeneous medium with linear phase
anisotropy [13], the relation zeN
V
eT eV (the relation
occurs in homogeneous medium [14]) is no longer
satisfied. Also, there is the dependence of the
eigenvalues of the integral matrix on the angle 0 .
However, as in the case of longitudinally homogeneous
medium, these eigenvalues are the phase factors, and
therefore only change the absolute phase of eigenwaves
(their amplitude remains unchanged).
It follows from relation (10) that, like to the case of
longitudinally inhomogeneous medium with linear phase
anisotropy, the eigenpolarization depends on the value of
z. Unlike homogeneous media, the eigenpolarization of
the integral and differential Jones matrices are not equal.
This leads to the fact that those values change with the
coordinate z when light propagates from one to another
molecular plane. Changes in the azimuth and angle of
ellipticity for eigenpolarizations (basing on equation for
azimuth and angle of ellipticity that are presented in [6])
with z values are shown in Fig. 2.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2013. V. 16, N 4. P. 366-373.
© 2013, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
369
a
b
c
d
Fig. 2. Changes in the azimuth and angle of ellipticity with
coordinate z along the axis of light propagation in
longitudinally inhomogeneous medium with elliptical phase
anisotropy and parameters δ0 = 1.22 rad/mm, α0 = 0.3 rad/mm,
φ0 = 0.7 rad/mm for (a), (b) e1T and (c), (d) e2T –
eigenpolarizations, respectively.
As it follows from Figs 2a and 2c, the azimuths of
eigenpolarizations are linear functions of z that are
described by the equations:
2
0
1
z
e
,
22
0
2
z
e . These linear relationships are broken
in two cases. 1) When the angle of ellipticity of
eigenpolarization takes the values 4/ that correspond
to the circular polarization for which the concept of the
azimuth is degenerated. 2) A jump of azimuth from the
maximum (minimum) to the minimum (maximum)
value, which corresponds to the mathematical properties
of functions arctg and has no physical meaning.
Dependences of the angles of ellipticity for both
eigenpolarizations on z are not linear and periodic. It
should be also noted that there are z values, at which the
angle of ellipticity corresponds to linearly polarized
waves. As a result, the integral Jones matrix of
inhomogeneous medium coincides with the matrix of the
linear phase plate with the value of the linear
birefringence [15]: 44arccos M , and orientation of
the axis of birefringence: 4342arctan MM ,
which coincides with the azimuth of one of
eigenpolarizations. In addition, there are the values for
which the angle of ellipticity corresponds to circularly
polarized waves. That is, the integral Jones matrix
coincides with the matrix of circular phase plate with a
value of circular birefringence: z0 .
From the condition 0arccos 44 M [15] (which
ensures the absence of linear phase anisotropy), we
obtain that in this type of medium eigenpolarization
becomes circular in the carrying value:
.0,,
2
2
0
2
00
kZk
k
z (12)
Similarly, the condition 0arctan
2
1
3322
3223
MM
MM
(absence of circular phase anisotropy) yields the
following relation that provides linear eigenpolarization:
,0sin2cos4
2sincos
10100
012
2
01
zBzB
zBBB
(13)
where 2
0
2
001 4 B , 2
0
2
002 8 B .
So, compared to longitudinally homogeneous media, the
content of eigenvectors of Jones matrices in
longitudinally inhomogeneous case changes. In
particular, they describe the polarization of light waves
in the input of medium that are reproduced at its output,
if it’s specific thickness. As a result, these waves cannot
be considered as eigenvectors in generally acceptable
understanding. But specific monomolecular plane is
homogeneous medium, so the eigenvectors of Jones
matrices, of course, describe the state of polarization of
its eigenwaves.
It also follows from Fig. 2 that eigenpolarization in
this type of medium is always orthogonal, which can be
checked by direct substitution of Eq. (10) into the
condition 1 that is transformed into an identity
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2013. V. 16, N 4. P. 366-373.
© 2013, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
370
at any values of parameters: 0, 0, 0, z. When checking
the possibility of degeneration of anisotropy in this type
of inhomogeneous medium and substituting elements of
the Jones matrix (4) into the condition
04 2112
2
1122 TTTT , we find that there is no
parameter values (0, 0, 0, z), in which the anisotropy
becomes degenerate.
5. Orthogonalization properties
Using the notion “orthogonalization properties of
media”, we mean that for given polarization of input
light the polarization of output one is orthogonal.
Implementation of orthogonalization properties of
nondepolarizing media that are described by some Jones
matrix was considered in [16]. Further, in work [17] it
was obtained the relation on anisotropic parameters that
provide existence of orthogonalization properties in
specific types of media. As a result, for the existence of
orthogonalization properties the inequality F > 0 should
be valid, where F is as follows:
222121
2
2
2
21
2
1
2
1 24 cbbaabacbaF . (14)
And
.
,
,
,
,
,
112211222
2112222112222
2112222112222
2
22
2
22
22112211
1
2
22
2
22
211222122122
1
2
22
2
22
211222211222
1
XYYXc
YYYXXXb
XXYYYXa
YX
YYXX
c
YX
XXYYYX
b
YX
YYYXXX
a
(15)
In Eqs. (15), Xij, Yij are real and imaginary parts of
the Jones matrix elements. If F = 0, then, for the
existence of orthogonalization properties, an additional
condition should be satisfied in elements of the Jones
matrix. This condition has the form:
0
4 1
2
1
2
12
0
c
ba
R . (16)
But in the case when 02
1 R , the intensity of light,
the polarization state of which in output becomes
orthogonal, equals zero. So, we have to check this
feature.
Substituting the elements of Jones matrix (4) for
this type of longitudinally inhomogeneous media with
elliptical phase anisotropy, as a result we get the value of
F as a function of the anisotropy parameters 0, 0, 0
and thickness z. This dependence is shown in Fig. 3a for
the thickness z = 2 mm and value of parameter of
anisotropy φ0 = π/2 rad/mm. Presented in Fig. 3b is the
dependence of F on the parameter of anisotropy 0 for
some values of parameters 0, 0.
a
b
Fig. 3. Dependence of the function F on the anisotropy
parameters (a) δ0, α0 and curve F = 0, (b) 0 at δ0 = π rad/mm
for a medium with inhomogeneous elliptical phase anisotropy
and parameters: φ0 = π/2 rad/mm, z = 2 mm.
Like to the case of an inhomogeneous medium with
linear phase anisotropy, in this type of medium with
parameters: φ0 = π/2 rad/mm, z = 2 mm, there exist
several parameters of anisotropy 0 , 0 for which the
medium might have orthogonalization properties. As an
example, Fig. 3b demonstrates the section of curves in
Fig. 3b at δ0 = π rad/mm, where we can find such value
φ0 = π/2 rad/mm that gives 0F . Substituting the
above values of anisotropy 0 , 0 , 0 and values z =
2 mm in Eq. (16), we obtain 3.832
1 R , and, therefore,
at a given set of parameters 0 , 0 , 0 , z, the
corresponding type of medium characterized by
orthogonalization properties.
We can find these states of polarization. To fulfil it,
we shall consider the most general case when the input
state of polarization is elliptical and described by the
Jones vector [6]:
inpinpinpinp
inpinpinpinp
inp eie
eie
E
sincoscossin
sinsincoscos
, (17)
where inp , inpe are the azimuth and angle of ellipticity
for the polarization ellipse of input light. Now, apply the
basic equation of the Jones matrix method:
inpout TEE , (18)
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2013. V. 16, N 4. P. 366-373.
© 2013, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
371
where outE are the Jones vectors of output light. Also,
pass from Jones vectors to corresponding complex
variables based on the relations )()( / inpxinpyinp EE ,
)()( / outxoutyout EE . As a result, we obtain the
equation in the form:
inpinpoutinp ezf ,,,, 00 , (19)
where orthogonalization properties sold at
1
outinp . (20)
Substituting the above values of parameters of
anisotropy and thickness of the medium in Eqs (17) to
(20), we find that according to the classification
presented in [14] this type of medium in a general case is
characterized by continuum of polarization states that
are orthogonalized. Examples of these polarizations
(solutions of Eq. (5)) are elliptical and linear polarization
states with parameters: 81.16inp , 04.4inpe , and
7.7inp , respectively. It should be noted that these
states of polarization coincide with the corresponding
polarization states for longitudinal inhomogeneous
medium with linear phase anisotropy that was described
in [12].
6. Evolution of the states of polarization
Now we study the evolution of linear state of
polarization along the z axis in a longitudinally
inhomogeneous medium with elliptic phase anisotropy.
In this case, we use the Mueller matrix method. In this
method, an arbitrary linear state of polarization with
azimuth inp is described by the Stokes vectors:
TinpinpinpS 02sin2cos1 . (21)
At the same time, the anisotropic properties of the
medium are described by the integral Mueller matrix
Eq. (22), which is obtained from the Jones matrix (4)
using the method presented in [6].
Anisotropic properties of molecular layer in this
method are described by the differential Mueller matrix:
02cos2sin0
2cos020
2sin200
0000
0000
000
000
zz
z
z
m
. (23)
Matrix (23) can be obtained from the differentiated
integral matrix (22) by using the equation
1
M
dz
dM
m or the relation between the differential
Jones matrix (3) and the differential Mueller matrix [5].
It should be noted that the differential matrix (23) differs
from the corresponding matrix for a homogeneous
medium with elliptical phase anisotropy [18] only by its
dependence on z. The state of polarization of output
light can be found using the equation:
out inpS MS . (24)
Substituting (21) and (22) into (24), we get:
zfzS inpout ,,,, 000 (25)
This equation describes the evolution of Stokes
vectors, which present the input state of polarization
with azimuth inp along the axis z in a medium with
anisotropy parameters 000 ,, . If polarized light
propagates in the medium, it is very informative to
consider evolution of the azimuth O and angle of
ellipticity Oe with z coordinates in direction of light
propagation, which are defined as follows:
.arctan
2
1
,arctan
2
1
2
)3(
2
)2(
)4(
)2(
)3(
outout
out
O
out
out
O
SS
S
e
S
S
(26)
2
2
0
2
000
2
0000
0
000
0
00
2
2
0000
000
0
2
000
3
0
2
000
2
00
2
0
2
000
2
0
2
00
2
0
2
00
2
0
2
0
4
2
22
2
2
2
2
22
2
2
2
2
24
22
2
00
12
0
2224
0
2244
0
01
A
zAC
A
zAS
A
zA
SzS
A
zA
CzC
zA
S
A
zSzAS
zCzAC
A
zA
SzS
A
zA
SzC
A
zCzAS
zSzAC
A
zAC
A
zASzCAzSzACA
A
zASzASzSzACzACzC
M
(22)
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2013. V. 16, N 4. P. 366-373.
© 2013, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
372
a
b
Fig. 4. Evolution of linearly polarized light with the azimuth
inp = –0.8 rad in longitudinally inhomogeneous medium with
elliptical phase anisotropy and parameter: δ0 = 1.22 rad/mm,
α0 = 0.3 rad/mm, φ0 = 0.4 rad/mm. a) Evolution of azimuth,
b) evolution of angle of ellipticity.
Fig. 4 presents the evolution of polarization
parameters O and eO for the cases of longitudinally
inhomogeneous medium with elliptical phase anisotropy.
From Fig. 4, we have that in this case the azimuth
is aperiodic function of z coordinates in contrast to that
in homogeneous medium of this type that have been
studied in [17]. The angle of ellipticity is a periodic
function of z coordinates as in the case of homogeneous
media of this type [17]. The period of ellipticity angle in
this type of media should be equal to the thickness z of
the medium in which eigenpolarization becomes circular
(see Eq. (12)) at k = 1. Similar results were obtained in
[5, 6] for longitudinally inhomogeneous medium with
linear phase anisotropy.
7. Conclusions
Based on the Jones and Mueller matrix methods, the
anisotropic properties of longitudinally inhomogeneous
nondepolarizing media with elliptical phase anisotropy
have been analyzed.
We have shown that the vector transfer equation for
this type of media contains the partial solutions that
describe evolution of privileged states of polarizations.
In particular, it appears that these privileged states of
polarization propagate in this type of media in such a
way like that in the medium characterized by circular
phase anisotropy. Namely, azimuths of polarization
coincide with the principal axes of the specific molecular
plane and the angles of ellipticity remain unchanged.
By solving the spectral problem, we have found
that the exponential relationship between the eigenvalues
of differential and integral Jones matrices derived in [4]
is no longer held. However, the eigenvalues in this case
remain phase factors. This results in the fact that the
concept of eigenpolarizations in the case of
longitudinally inhomogeneous nondepolarizing media is
transformed in comparison with the case of
homogeneous media. This is particularly evident in their
dependence on z coordinates. This means that the
traditional concept of eigenpolarizations is valid only for
definite thickness of the medium.
Next, we have shown that the infinite numbers of
input states of polarization can be orthogonalized by the
medium of this class.
Studying evolution of linear polarization in this
type of medium, it is shown that in comparison with the
same type of homogeneous media the azimuth of
polarization becomes a nonperiodic function of z
coordinates. However, the angle of ellipticity is periodic
with a definite period.
Summarizing, it has been found that the anisotropic
properties of this type of media are qualitatively
equivalent to the similar properties of longitudinally
inhomogeneous media with linear phase anisotropy. This
is due to the fact that the circular phase anisotropy is
always a longitudinally homogeneous. In fact, this is
natural generalization of the first equivalence theorem
[2] for the case of longitudinally inhomogeneous media.
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. R.C. Jones, A new calculus for the treatment of
optical systems II. Proof of three general
equivalence theorems // J. Opt. Soc. Am. 31,
p. 493-499 (1941).
3. H. Mueller, The foundation of optics // J. Opt. Soc.
Am. 38, p. 661 (1948).
4. 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).
5. R.M.A. Azzam, Propagation of partially polarized
light through anisotropic media with or without
depolarization. A differential 4×4 matrix calculus //
J. Opt. Soc. Am. 68, p. 1756-1767 (1979).
6. R.M.A. Azzam, and N.M. Bashara, Elipsometry
and Polarized Light. Elsevier, 1987.
7. R.M.A. Azzam, and 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).
8. R.M.A. Azzam, N.M. Bashara, and B.E. Merrill,
Trajectories describing the evolution of polarized
light in homogeneous anisotropic media and liquid
crystals // J. Appl. Opt. 12, p. 764-771 (1973).
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9. T.Z. Kosc, K.L. Marshall, A. Trajkovska-Petkoska
et al. Progress in the development of polymer
cholesteric liquid crystal flakes for display
applications // Displays, 25(5), p. 171-176 (2004).
10. 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. and Las. in Eng. 47(11),
p. 1150-1159 (2009).
11. J.L. Martínez, P. García-Martínez, M. del Mar
Sánchez-López et al. Accurate color predictability
based on a spectral retardance model of a twisted-
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PACS 78.20.Fm, 78.20.Ek
Studying anisotropic properties of longitudinal inhomogeneous nondepolarizing media with elliptical phase anisotropy
I.S. Kolomiets, S.N. Savenkov, Ye.A. Oberemok, A.S. Klimov
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. In this paper, basing on the anisotropic properties of longitudinal inhomogeneous nondepolarizing media with linear phase anisotropy, more general type of media with elliptical phase anisotropy was studied. The features of propagation of light with privileged states of polarization were observed. Transformation of polarization states of eigenwaves along z axes of light propagation was studied. The orthogonalization properties inherent to this type of medium were obtained. Evolution of linear polarized light in this media was presented and discussed.
Keywords: Jones method, birefringence, spectral problem, longitudinally inhomogeneous medium.
Manuscript received 18.07.13; revised version received 05.09.13; accepted for publication 23.10.13; published online 16.12.13.
1. Introduction
The history of investigation of longitudinal inhomogeneous nondepolarizing media returns us to Ref. [1]. The study of anisotropic properties of this type of media in time of publication [1] was difficult as caused by the complexity of the mathematical apparatus used. This problem was solved using the Jones and Mueller matrix methods [2, 3]. For the first time, the Jones matrix of longitudinal inhomogeneous anisotropic medium was presented in [4]. R. Azzam made the next step in [5], where he considered the type of longitudinal inhomogeneous medium with linear phase anisotropy. In particular, in [6-8] the examples of these media were considered: cholesteric and twisted nematic liquid crystals. However, the studies of anisotropic properties of this type of media were made only partially because of complexity of these objects. Specifically, such properties as the solution of spectral problem and orthogonalization properties were missed.
Development of the modern display technology results in the fact that these types of liquid crystals are very widely used [9-11]. Optical anisotropy of these media is in the basis of using various liquid crystals in computer displays and indicator devices. Therefore, the study of anisotropic properties of longitudinal inhomogeneous anisotropic medium is very important problem for display technology. The case of longitudinal inhomogeneous nondepolarizing media with linear phase anisotropy, which in particular includes cholesterics and twisted nematics, was solved in [12, 13]. This paper generalizes the results obtained in [12, 13] for the case of the media characterized by both linear and circular phase anisotropy (elliptic phase anisotropy).
2. Polarimetric models of media
Nondepolarizing longitudinal inhomogeneous medium with elliptical phase anisotropy can be equivalently represented as a sequence of molecular planes that consist of elongated molecules, oriented parallel to each other. At the same time, such molecules possess inherent chirality. Each of these molecular planes can be present as a thin phase plate with elliptical phase anisotropy (elliptical birefringence). The fast (slow) axis of the plate is parallel (perpendicular) to the direction along which directed is the plane of the molecules that is considered, and lies entirely in this plane. In these media, the axis of each following molecular layer is rotated relative to the previous one by some angle (0 [7]:
p
p
=
a
2
0
,
(1)
where
p
is the step of helical structure in this medium (the shortest distance between the planes with the same orientation of molecules). Then, the molecular orientation of the plane in distance z from the input can be defined as:
z
0
a
=
a
.
(2)
Anisotropic properties of one molecular layer in this medium are described by the differential Jones matrix (in circular basis) [6]:
ú
ú
ú
û
ù
ê
ê
ê
ë
é
j
d
d
j
-
=
a
-
a
0
2
0
2
0
0
0
0
2
1
2
1
i
e
i
e
i
i
N
z
i
z
i
.
(3)
And the anisotropic properties of this longitudinally inhomogeneous medium consisting of a sequence of molecular planes with a longitudinal size z are described by the integral Jones matrix (in linear basis):
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
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(
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(
)
(
)
(
)
(
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(
)
(
)
(
)
(
)
(
)
.
2
2
2
2
2
2
2
2
2
2
2
2
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
ú
ú
ú
ú
û
ù
÷
ø
ö
ç
è
æ
a
j
+
a
+
a
d
-
+
÷
ø
ö
ç
è
æ
a
÷
ø
ö
ç
è
æ
a
j
+
a
+
a
d
+
a
÷
ø
ö
ç
è
æ
-
ê
ê
ê
ê
ë
é
÷
ø
ö
ç
è
æ
a
j
+
a
-
a
d
+
a
÷
ø
ö
ç
è
æ
÷
ø
ö
ç
è
æ
a
j
+
a
+
a
d
+
÷
ø
ö
ç
è
æ
a
=
zA
S
z
S
z
C
i
zA
C
z
AC
zA
S
z
C
z
S
i
z
S
zA
AC
zA
S
z
C
z
S
i
z
S
zA
AC
zA
S
z
S
z
C
i
zA
C
z
AC
A
T
(4)
In Eqs (3) and (4),
00
,
dj
are magnitude of linear and circular birefringence per unit thickness in the direction of light propagation, respectively, and the following replacement is used:
(
)
2
0
0
2
0
4
j
+
a
+
d
=
A
,
(
)
(
)
x
x
C
cos
=
,
(
)
(
)
x
x
S
sin
=
. Note that the integral matrix (4) can be obtained from the differential (3) by using the vector transfer equation and technique presented in [14].
3. Propagation of radiation with privileged states of polarization
One of the anisotropic properties of longitudinally inhomogeneous media that does not exist in a longitudinally homogeneous media is availability of the privileged polarization states. The term “privileged” was at first introduced in [7]. It means the state of polarization of eigenwaves in longitudinal homogeneous (untwisted) media of this type.
Evolution of polarization states along the z axis of light propagation in anisotropic medium can be described by the first order differential equation for polarization complex variable [6]. For the case of longitudinally inhomogeneous medium with elliptical phase anisotropy, this equation is:
z
i
z
i
e
i
i
e
i
z
d
d
0
0
2
0
0
2
2
0
2
1
2
2
1
/
χ
a
-
a
d
+
c
j
+
c
d
-
=
,
(5)
where
c
is the complex variable that determines the light polarization state. Eq. (5) has a partial solution in the form:
z
i
e
K
0
2
2
,
1
2
,
1
a
-
=
c
,
(6)
where
2
,
1
K
are privileged polarization states of light. Azimuth of orientation coincides with orientation of input molecular plane, and the angle of ellipticity of privileged states is determined as:
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
.
1
/
2
1
/
2
1
/
2
1
/
2
arctan
2
0
0
0
0
0
0
2
0
0
0
0
0
0
2
,
1
÷
÷
÷
ø
ö
ç
ç
ç
è
æ
+
d
j
+
a
+
+
d
j
+
a
±
+
d
j
+
a
+
-
d
j
+
a
±
=
=
e
(7)
These two privileged polarization states
2
,
1
K
are orthogonal and elliptical in a general case, and transformation into two elliptical eigenpolarization for longitudinally inhomogeneous medium with elliptical phase anisotropy takes place when
0
,
0
0
=
j
a
.
Eq. (6) describes the change of privileged polarization states of light propagation in the medium along the z axis. Since Eq. (5) is written for the differential Jones matrix in the circular basis (3), the azimuth and angle of ellipticity of polarization
(
)
2
,
1
p
c
can be determined as:
(
)
(
)
2
,
1
arg
2
1
p
p
c
=
q
,
(
)
(
)
÷
÷
ø
ö
ç
ç
è
æ
+
c
-
c
=
1
1
arctan
2
,
1
2
,
1
p
p
p
e
.
(8)
Graphical representation of (8) is shown in Fig. 1.
From Eq. (6) and Fig. 1, we can see that the azimuths of privileged polarization states are changed by the law θp(1,2) = α0, and for each molecular plane they coincide with their principal axes, while the angle of ellipticity remains unchanged and is defined by Eq. (7). As a result of propagation of light with the privileged state of polarization along the z axis, this type of medium is characterized by efficiency of circular phase anisotropy with the magnitude of the relative angle φ0 = α0 (rad/mm).
a
b
c
d
Fig. 1. Evolution of the azimuth and angle of ellipticity in longitudinally inhomogeneous medium with elliptical phase anisotropy and parameters: δ0 = 1.22 rad/mm, α0 = 0.3 rad/mm, φ0 = 0.7 rad/mm; (a), (b) azimuth, (c), (d) angle of ellipticity for polarizations (p1, (p2, respectively.
4. The solution of the spectral problem
To analyze the basic anisotropic properties for the medium of this type, we need to find properties of their eigenwaves, i.e. to solve the spectral problem. For this, we use the relations represented in [14], which gives the solutions of the spectral problem for the integral (3) and differential (4) Jones matrices. As a result, eigenpolarization and eigenvalues for these matrices are as follows:
;
4
2
1
,
2
0
2
0
2
,
1
2
3
3
2
2
4
2
,
1
÷
ø
ö
ç
è
æ
j
+
d
=
+
+
=
i
V
A
AR
R
R
A
R
V
N
e
T
e
m
m
(9)
(
)
(
)
;
2
sin
2
2
cos
4
,
0
0
0
0
0
2
0
2
0
2
,
1
2
3
3
2
2
1
2
,
1
z
i
z
i
Q
AR
R
R
A
R
N
e
T
e
a
d
+
j
a
d
+
j
-
d
-
±
=
c
+
+
±
=
c
(10)
where the following substitutions are used:
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
.
2
2
2
,
2
2
2
2
,
2
4
,
1
2
,
2
0
0
0
0
0
0
0
0
0
0
4
2
2
0
2
0
0
3
2
2
0
2
0
0
1
÷
ø
ö
ç
è
æ
a
d
-
a
j
+
a
+
+
a
÷
ø
ö
ç
è
æ
-
=
÷
÷
÷
÷
÷
ø
ö
ç
ç
ç
ç
ç
è
æ
÷
ø
ö
ç
è
æ
a
j
+
a
+
+
÷
ø
ö
ç
è
æ
a
=
÷
ø
ö
ç
è
æ
a
j
+
a
=
÷
÷
ø
ö
ç
ç
è
æ
-
÷
ø
ö
ç
è
æ
a
=
÷
ø
ö
ç
è
æ
a
d
-
=
zA
S
z
S
i
z
C
z
S
zA
AC
Q
zA
S
z
S
zA
C
z
AC
R
zA
S
z
S
R
zA
C
z
C
R
zA
S
z
C
i
R
(11)
It follows from relations (9) that, as in the case of longitudinally inhomogeneous medium with linear phase anisotropy [13], the relation
z
eN
V
eT
e
V
=
(the relation occurs in homogeneous medium [14]) is no longer satisfied. Also, there is the dependence of the eigenvalues of the integral matrix on the angle
0
a
. However, as in the case of longitudinally homogeneous medium, these eigenvalues are the phase factors, and therefore only change the absolute phase of eigenwaves (their amplitude remains unchanged).
It follows from relation (10) that, like to the case of longitudinally inhomogeneous medium with linear phase anisotropy, the eigenpolarization depends on the value of z. Unlike homogeneous media, the eigenpolarization of the integral and differential Jones matrices are not equal. This leads to the fact that those values change with the coordinate z when light propagates from one to another molecular plane. Changes in the azimuth and angle of ellipticity for eigenpolarizations (basing on equation for azimuth and angle of ellipticity that are presented in [6]) with z values are shown in Fig. 2.
a
b
c
d
Fig. 2. Changes in the azimuth and angle of ellipticity with coordinate z along the axis of light propagation in longitudinally inhomogeneous medium with elliptical phase anisotropy and parameters δ0 = 1.22 rad/mm, α0 = 0.3 rad/mm, φ0 = 0.7 rad/mm for (a), (b) (e1T and (c), (d) (e2T – eigenpolarizations, respectively.
As it follows from Figs 2a and 2c, the azimuths of eigenpolarizations are linear functions of z that are described by the equations:
2
0
1
z
e
a
-
=
q
,
2
2
0
2
p
+
a
-
=
q
z
e
. These linear relationships are broken in two cases. 1) When the angle of ellipticity of eigenpolarization takes the values
4
/
p
±
that correspond to the circular polarization for which the concept of the azimuth is degenerated. 2) A jump of azimuth from the maximum (minimum) to the minimum (maximum) value, which corresponds to the mathematical properties of functions arctg and has no physical meaning.
Dependences of the angles of ellipticity for both eigenpolarizations on z are not linear and periodic. It should be also noted that there are z values, at which the angle of ellipticity corresponds to linearly polarized waves. As a result, the integral Jones matrix of inhomogeneous medium coincides with the matrix of the linear phase plate with the value of the linear birefringence [15]:
(
)
44
arccos
M
=
d
, and orientation of the axis of birefringence:
(
)
43
42
arctan
M
M
-
=
a
, which coincides with the azimuth of one of eigenpolarizations. In addition, there are the values for which the angle of ellipticity corresponds to circularly polarized waves. That is, the integral Jones matrix coincides with the matrix of circular phase plate with a value of circular birefringence:
z
0
a
=
j
.
From the condition
(
)
0
arccos
44
=
M
[15] (which ensures the absence of linear phase anisotropy), we obtain that in this type of medium eigenpolarization becomes circular in the carrying value:
(
)
.
0
,
,
2
2
0
2
0
0
¹
Î
d
+
j
+
a
p
=
k
Z
k
k
z
(12)
Similarly, the condition
0
arctan
2
1
33
22
32
23
=
÷
÷
ø
ö
ç
ç
è
æ
+
-
M
M
M
M
(absence of circular phase anisotropy) yields the following relation that provides linear eigenpolarization:
(
)
(
)
(
)
(
)
(
)
,
0
sin
2
cos
4
2
sin
cos
1
0
1
0
0
0
1
2
2
0
1
=
a
j
+
a
+
+
a
+
d
-
z
B
z
B
z
B
B
B
(13)
where
(
)
2
0
2
0
0
1
4
d
+
j
+
a
=
B
,
(
)
2
0
2
0
0
2
8
d
+
j
+
a
=
B
. So, compared to longitudinally homogeneous media, the content of eigenvectors of Jones matrices in longitudinally inhomogeneous case changes. In particular, they describe the polarization of light waves in the input of medium that are reproduced at its output, if it’s specific thickness. As a result, these waves cannot be considered as eigenvectors in generally acceptable understanding. But specific monomolecular plane is homogeneous medium, so the eigenvectors of Jones matrices, of course, describe the state of polarization of its eigenwaves.
It also follows from Fig. 2 that eigenpolarization in this type of medium is always orthogonal, which can be checked by direct substitution of Eq. (10) into the condition
1
-
=
cc
*
that is transformed into an identity at any values of parameters: (0, (0, (0, z. When checking the possibility of degeneration of anisotropy in this type of inhomogeneous medium and substituting elements of the Jones matrix (4) into the condition
(
)
0
4
21
12
2
11
22
=
+
-
T
T
T
T
, we find that there is no parameter values ((0, (0, (0, z), in which the anisotropy becomes degenerate.
5. Orthogonalization properties
Using the notion “orthogonalization properties of media”, we mean that for given polarization of input light the polarization of output one is orthogonal.
Implementation of orthogonalization properties of nondepolarizing media that are described by some Jones matrix was considered in [16]. Further, in work [17] it was obtained the relation on anisotropic parameters that provide existence of orthogonalization properties in specific types of media. As a result, for the existence of orthogonalization properties the inequality F > 0 should be valid, where F is as follows:
(
)
(
)
(
)
2
2
2
1
2
1
2
2
2
2
1
2
1
2
1
2
4
c
b
b
a
a
b
a
c
b
a
F
-
+
-
+
-
+
=
.
(14)
And
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
.
,
,
,
,
,
11
22
11
22
2
21
12
22
21
12
22
2
21
12
22
21
12
22
2
2
22
2
22
22
11
22
11
1
2
22
2
22
21
12
22
12
21
22
1
2
22
2
22
21
12
22
21
12
22
1
X
Y
Y
X
c
Y
Y
Y
X
X
X
b
X
X
Y
Y
Y
X
a
Y
X
Y
Y
X
X
c
Y
X
X
X
Y
Y
Y
X
b
Y
X
Y
Y
Y
X
X
X
a
-
=
-
+
-
=
+
-
+
=
+
+
=
+
-
+
-
=
+
+
+
+
=
(15)
In Eqs. (15), Xij, Yij are real and imaginary parts of the Jones matrix elements. If F = 0, then, for the existence of orthogonalization properties, an additional condition should be satisfied in elements of the Jones matrix. This condition has the form:
0
4
1
2
1
2
1
2
0
³
-
+
=
c
b
a
R
.
(16)
But in the case when
0
2
1
=
R
, the intensity of light, the polarization state of which in output becomes orthogonal, equals zero. So, we have to check this feature.
Substituting the elements of Jones matrix (4) for this type of longitudinally inhomogeneous media with elliptical phase anisotropy, as a result we get the value of F as a function of the anisotropy parameters (0, (0, (0 and thickness z. This dependence is shown in Fig. 3a for the thickness z = 2 mm and value of parameter of anisotropy φ0 = π/2 rad/mm. Presented in Fig. 3b is the dependence of F on the parameter of anisotropy
0
a
for some values of parameters (0, (0.
a
b
Fig. 3. Dependence of the function F on the anisotropy parameters (a) δ0, α0 and curve F = 0, (b) (0 at δ0 = π rad/mm for a medium with inhomogeneous elliptical phase anisotropy and parameters: φ0 = π/2 rad/mm, z = 2 mm.
Like to the case of an inhomogeneous medium with linear phase anisotropy, in this type of medium with parameters: φ0 = π/2 rad/mm, z = 2 mm, there exist several parameters of anisotropy
0
d
,
0
a
for which the medium might have orthogonalization properties. As an example, Fig. 3b demonstrates the section of curves in Fig. 3b at δ0 = π rad/mm, where we can find such value φ0 = π/2 rad/mm that gives
0
F
=
. Substituting the above values of anisotropy
0
d
,
0
a
,
0
j
and values z = 2 mm in Eq. (16), we obtain
3
.
83
2
1
=
R
, and, therefore, at a given set of parameters
0
d
,
0
a
,
0
j
, z, the corresponding type of medium characterized by orthogonalization properties.
We can find these states of polarization. To fulfil it, we shall consider the most general case when the input state of polarization is elliptical and described by the Jones vector [6]:
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
ú
û
ù
ê
ë
é
q
+
q
q
-
q
=
inp
inp
inp
inp
inp
inp
inp
inp
inp
e
i
e
e
i
e
E
sin
cos
cos
sin
sin
sin
cos
cos
,
(17)
where
inp
q
,
inp
e
are the azimuth and angle of ellipticity for the polarization ellipse of input light. Now, apply the basic equation of the Jones matrix method:
inp
out
TE
E
=
,
(18)
where
out
E
are the Jones vectors of output light. Also, pass from Jones vectors to corresponding complex variables based on the relations
)
(
)
(
/
inp
x
inp
y
inp
E
E
=
c
,
)
(
)
(
/
out
x
out
y
out
E
E
=
c
. As a result, we obtain the equation in the form:
(
)
inp
inp
out
inp
e
z
f
,
,
,
,
0
0
q
a
d
=
c
c
*
,
(19)
where orthogonalization properties sold at
1
-
=
c
c
*
out
inp
.
(20)
Substituting the above values of parameters of anisotropy and thickness of the medium in Eqs (17) to (20), we find that according to the classification presented in [14] this type of medium in a general case is characterized by continuum of polarization states that are orthogonalized. Examples of these polarizations (solutions of Eq. (5)) are elliptical and linear polarization states with parameters:
°
=
q
81
.
16
inp
,
°
=
04
.
4
inp
e
, and
°
=
q
7
.
7
inp
, respectively. It should be noted that these states of polarization coincide with the corresponding polarization states for longitudinal inhomogeneous medium with linear phase anisotropy that was described in [12].
6. Evolution of the states of polarization
Now we study the evolution of linear state of polarization along the z axis in a longitudinally inhomogeneous medium with elliptic phase anisotropy. In this case, we use the Mueller matrix method. In this method, an arbitrary linear state of polarization with azimuth
inp
q
is described by the Stokes vectors:
(
)
T
inp
inp
inp
S
0
2
sin
2
cos
1
q
q
=
.
(21)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
ú
ú
ú
ú
ú
ú
ú
ú
ú
ú
ú
ú
ú
û
ù
d
+
j
+
a
d
-
÷
÷
÷
÷
÷
ø
ö
ç
ç
ç
ç
ç
è
æ
÷
ø
ö
ç
è
æ
a
j
+
a
+
÷
ø
ö
ç
è
æ
a
÷
ø
ö
ç
è
æ
d
a
j
+
a
+
a
÷
ø
ö
ç
è
æ
a
d
-
÷
ø
ö
ç
è
æ
a
d
j
+
a
a
j
+
a
+
a
-
ê
ê
ê
ê
ê
ê
ê
ê
ë
é
-
d
j
+
a
-
a
j
+
a
-
a
j
+
a
+
d
a
j
+
a
+
a
j
+
a
+
d
-
j
+
a
+
d
a
=
2
2
0
2
0
0
0
2
0
0
0
0
0
0
0
0
0
0
0
2
2
0
0
0
0
0
0
0
0
2
0
0
0
3
0
2
0
0
0
2
0
0
2
0
2
0
0
0
2
0
2
0
0
2
0
2
0
0
2
0
2
0
4
2
2
2
2
2
2
2
2
2
2
2
2
2
2
4
2
2
2
0
0
1
2
0
2
2
2
4
0
2
2
4
4
0
0
1
A
zA
C
A
zA
S
A
zA
S
z
S
A
zA
C
z
C
zA
S
A
z
S
zA
S
z
C
zA
C
A
zA
S
z
S
A
zA
S
z
C
A
z
C
zA
S
z
S
zA
C
A
zA
C
A
zA
S
z
C
A
z
S
zA
C
A
A
zA
S
z
AS
z
S
zA
C
zA
C
z
C
M
At the same time, the anisotropic properties of the medium are described by the integral Mueller matrix Eq. (22), which is obtained from the Jones matrix (4) using the method presented in [6].
Anisotropic properties of molecular layer in this method are described by the differential Mueller matrix:
(
)
(
)
(
)
(
)
ú
ú
ú
ú
û
ù
ê
ê
ê
ê
ë
é
a
d
-
a
d
a
d
j
-
a
d
-
j
=
0
2
cos
2
sin
0
2
cos
0
2
0
2
sin
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
z
z
z
z
m
.
(23)
Matrix (23) can be obtained from the differentiated integral matrix (22) by using the equation
1
-
÷
ø
ö
ç
è
æ
=
M
dz
dM
m
or the relation between the differential Jones matrix (3) and the differential Mueller matrix [5]. It should be noted that the differential matrix (23) differs from the corresponding matrix for a homogeneous medium with elliptical phase anisotropy [18] only by its dependence
a
on z. The state of polarization of output light can be found using the equation:
outinp
SMS
=
.
(24)
Substituting (21) and (22) into (24), we get:
(
)
(
)
z
f
z
S
inp
out
,
,
,
,
0
0
0
q
j
a
d
=
(25)
This equation describes the evolution of Stokes vectors, which present the input state of polarization with azimuth
inp
q
along the axis z in a medium with anisotropy parameters
0
0
0
,
,
j
a
d
. If polarized light propagates in the medium, it is very informative to consider evolution of the azimuth
O
q
and angle of ellipticity
O
e
with z coordinates in direction of light propagation, which are defined as follows:
.
arctan
2
1
,
arctan
2
1
2
)
3
(
2
)
2
(
)
4
(
)
2
(
)
3
(
ú
ú
ú
û
ù
ê
ê
ê
ë
é
+
=
ú
ú
û
ù
ê
ê
ë
é
=
q
out
out
out
O
out
out
O
S
S
S
e
S
S
(26)
a
b
Fig. 4. Evolution of linearly polarized light with the azimuth (inp = –0.8 rad in longitudinally inhomogeneous medium with elliptical phase anisotropy and parameter: δ0 = 1.22 rad/mm, α0 = 0.3 rad/mm, φ0 = 0.4 rad/mm. a) Evolution of azimuth, b) evolution of angle of ellipticity.
Fig. 4 presents the evolution of polarization parameters (O and eO for the cases of longitudinally inhomogeneous medium with elliptical phase anisotropy.
From Fig. 4, we have that in this case the azimuth is aperiodic function of z coordinates in contrast to that in homogeneous medium of this type that have been studied in [17]. The angle of ellipticity is a periodic function of z coordinates as in the case of homogeneous media of this type [17]. The period of ellipticity angle in this type of media should be equal to the thickness z of the medium in which eigenpolarization becomes circular (see Eq. (12)) at k = 1. Similar results were obtained in [5, 6] for longitudinally inhomogeneous medium with linear phase anisotropy.
7. Conclusions
Based on the Jones and Mueller matrix methods, the anisotropic properties of longitudinally inhomogeneous nondepolarizing media with elliptical phase anisotropy have been analyzed.
We have shown that the vector transfer equation for this type of media contains the partial solutions that describe evolution of privileged states of polarizations. In particular, it appears that these privileged states of polarization propagate in this type of media in such a way like that in the medium characterized by circular phase anisotropy. Namely, azimuths of polarization coincide with the principal axes of the specific molecular plane and the angles of ellipticity remain unchanged.
By solving the spectral problem, we have found that the exponential relationship between the eigenvalues of differential and integral Jones matrices derived in [4] is no longer held. However, the eigenvalues in this case remain phase factors. This results in the fact that the concept of eigenpolarizations in the case of longitudinally inhomogeneous nondepolarizing media is transformed in comparison with the case of homogeneous media. This is particularly evident in their dependence on z coordinates. This means that the traditional concept of eigenpolarizations is valid only for definite thickness of the medium.
Next, we have shown that the infinite numbers of input states of polarization can be orthogonalized by the medium of this class.
Studying evolution of linear polarization in this type of medium, it is shown that in comparison with the same type of homogeneous media the azimuth of polarization becomes a nonperiodic function of z coordinates. However, the angle of ellipticity is periodic with a definite period.
Summarizing, it has been found that the anisotropic properties of this type of media are qualitatively equivalent to the similar properties of longitudinally inhomogeneous media with linear phase anisotropy. This is due to the fact that the circular phase anisotropy is always a longitudinally homogeneous. In fact, this is natural generalization of the first equivalence theorem [2] for the case of longitudinally inhomogeneous media.
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. R.C. Jones, A new calculus for the treatment of optical systems II. Proof of three general equivalence theorems // J. Opt. Soc. Am. 31, p. 493-499 (1941).
3. H. Mueller, The foundation of optics // J. Opt. Soc. Am. 38, p. 661 (1948).
4. 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).
5. R.M.A. Azzam, Propagation of partially polarized light through anisotropic media with or without depolarization. A differential 4×4 matrix calculus // J. Opt. Soc. Am. 68, p. 1756-1767 (1979).
6. R.M.A. Azzam, and N.M. Bashara, Elipsometry and Polarized Light. Elsevier, 1987.
7. R.M.A. Azzam, and 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).
8. R.M.A. Azzam, N.M. Bashara, and B.E. Merrill, Trajectories describing the evolution of polarized light in homogeneous anisotropic media and liquid crystals // J. Appl. Opt. 12, p. 764-771 (1973).
9. T.Z. Kosc, K.L. Marshall, A. Trajkovska-Petkoska et al. Progress in the development of polymer cholesteric liquid crystal flakes for display applications // Displays, 25(5), p. 171-176 (2004).
10. 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. and Las. in Eng. 47(11), p. 1150-1159 (2009).
11. J.L. Martínez, 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).
12. I.S. Kolomiets, S.N. Savenkov, Ye.A. Oberemok, Studing orthogonalization properties of longitudinally inhomogeneous nondepolarizing medium // Registration, data storage and processing, 15, №1, p. 23-30 (2013), in Russian.
13. 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 tekhnologii, 35, № 9, p. 1001-1011 (2013), in Russian.
14. I.S. Kolomiets, S.N. Savenkov, Ye.A. Oberemok. Eigenpolarizations orthogonality conditions for the first and second Jones equivalence theorems in approximation of homogeneous and layered media // Metallofizika i noveishie tekhnologii, 33, p. 493-502 (2011).
15. S.N. Savenkov, V.V. Marienko, E.A. Oberemok, and O.I. Sydoruk, Generalized matrix equivalence theorem for polarization theory // Phys. Rev. E, 74, 056607 (2006).
16. R.M.A. Azzam, Polarization orthogonalization properties of optical systems // Appl. Phys. A, 13, p. 281-285 (1977).
17. S.N. Savenkov, and Y.V. Aulin, Orthogonal properties of homogeneous anisotropy medium // Proc. SPIE, 6536, p. 65360D (2007).
18. I.S. Kolomiets, S.N. Savenkov, Exploration evolution of polarization state in homogeneous anisotropic medium based on differential Mueller matrix model // Intern. Sci. and Practic. Conf., 2007, p. 99-100.
� EMBED Equation.3 ��� (22)
© 2013, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
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|
| id | nasplib_isofts_kiev_ua-123456789-117821 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1560-8034 |
| language | English |
| last_indexed | 2025-12-07T17:21:05Z |
| publishDate | 2013 |
| publisher | Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
| record_format | dspace |
| spelling | Kolomiets, I.S. Savenkov, S.N. Oberemok, Ye.A. Klimov, A.S. 2017-05-26T19:01:37Z 2017-05-26T19:01:37Z 2013 Studying anisotropic properties of longitudinal inhomogeneous
 nondepolarizing media with elliptical phase anisotropy / I.S. Kolomiets, S.N. Savenkov, Ye.A. Oberemok, A.S. Klimov // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2013. — Т. 16, № 4. — С. 366-373. — Бібліогр.: 18 назв. — англ. 1560-8034 PACS 78.20.Fm, 78.20.Ek https://nasplib.isofts.kiev.ua/handle/123456789/117821 In this paper, basing on the anisotropic properties of longitudinal
 inhomogeneous nondepolarizing media with linear phase anisotropy, more general type
 of media with elliptical phase anisotropy was studied. The features of propagation of
 light with privileged states of polarization were observed. Transformation of polarization
 states of eigenwaves along z axes of light propagation was studied. The orthogonalization
 properties inherent to this type of medium were obtained. Evolution of linear polarized
 light in this media was presented and discussed. en Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України Semiconductor Physics Quantum Electronics & Optoelectronics Studying anisotropic properties of longitudinal inhomogeneous nondepolarizing media with elliptical phase anisotropy Article published earlier |
| spellingShingle | Studying anisotropic properties of longitudinal inhomogeneous nondepolarizing media with elliptical phase anisotropy Kolomiets, I.S. Savenkov, S.N. Oberemok, Ye.A. Klimov, A.S. |
| title | Studying anisotropic properties of longitudinal inhomogeneous nondepolarizing media with elliptical phase anisotropy |
| title_full | Studying anisotropic properties of longitudinal inhomogeneous nondepolarizing media with elliptical phase anisotropy |
| title_fullStr | Studying anisotropic properties of longitudinal inhomogeneous nondepolarizing media with elliptical phase anisotropy |
| title_full_unstemmed | Studying anisotropic properties of longitudinal inhomogeneous nondepolarizing media with elliptical phase anisotropy |
| title_short | Studying anisotropic properties of longitudinal inhomogeneous nondepolarizing media with elliptical phase anisotropy |
| title_sort | studying anisotropic properties of longitudinal inhomogeneous nondepolarizing media with elliptical phase anisotropy |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/117821 |
| work_keys_str_mv | AT kolomietsis studyinganisotropicpropertiesoflongitudinalinhomogeneousnondepolarizingmediawithellipticalphaseanisotropy AT savenkovsn studyinganisotropicpropertiesoflongitudinalinhomogeneousnondepolarizingmediawithellipticalphaseanisotropy AT oberemokyea studyinganisotropicpropertiesoflongitudinalinhomogeneousnondepolarizingmediawithellipticalphaseanisotropy AT klimovas studyinganisotropicpropertiesoflongitudinalinhomogeneousnondepolarizingmediawithellipticalphaseanisotropy |