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Mueller–Stokes polarimetric characterization of transmissive liquid crystal
spatial light modulator
Kapil Dev
n
, Anand Asundi
School of Mechanical and Aerospace Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798, Singapore
article info
Keywords:
Mueller matrix
Mueller–Stokes formalism
Liquid crystal spatial light modulator
Liquid crystal
Depolarization
abstract
In the twisted nematic liquid crystal spatial light modulators (TN-LCSLM), distortion of uniform twist
and decrease in tilt angle of liquid crystal molecules on application of an electric field lead to amplitude
and phase modulations of the transmitted or reflected wavefront, respectively. The amplitude and
phase modulation characterization of TN-LCSLM using Jones calculi is simple and extensively used but
does not give any information about important polarimetric parameters such as diattenuation and
depolarizance. On the other hand, the characterization using Mueller calculi provides all information in
terms of polarimetric properties such as diattenuation, retardance (birefringence) and depolarization.
In this paper, polarimetric properties of the transmissive TN-LCSLM (HOLOEYE LC2002) are character-
ized measuring 17 different Mueller matrices at different addressed gray scale through Mueller Matrix
Imaging Polarimeter (MMIP) at 530 nm wavelength. Lu–Chipman polar decomposition for Mueller
matrix is utilized to separate out three independent Mueller matrices for diattenuation, depolarization
and retardance as a function of addressed gray scale. Further, Mueller–Stokes combined formulation is
used to examine the effect of depolarization present in the TN-LCSLM on six different states of
polarization and evaluation of eigenpolarization states for the TN-LCSLM has been presented.
&2011 Elsevier Ltd. All rights reserved.
1. Introduction
Spatial light modulators (SLMs) are used as switchable optical
element device, which can modify optical function for real-time
applications. SLMs have been used in many applications involving
optical metrology, optical pulse shaping, display applications,
optical information processing, adaptive optics, etc. Important
optical properties associated with an optical wavefront such as
amplitude, phase and polarization can be modulated using SLMs.
SLMs based on liquid crystal materials have advantages such as
high switching speed, high spatial resolution, large birefringence
with low voltage operation and can provide amplitude, phase and
polarization modulation for incident optical wavefront. In the TN-
LCSLM, uniaxial anisotropic nematic liquid crystal material is
sandwiched between two inner ends of conductive glass plates
with liquid crystal molecules director gradually rotating from one
to other end giving twist orientation of 901. The distortion of twist
uniformity and decrease in tilt angles of liquid crystal molecules
director on an application of electric field leads to the amplitude
and phase modulation of transmitted or reflected wavefront,
respectively.
The amplitude and phase modulation characterization of the
TN-LCSLM is crucial to exploit its wavefront modulation proper-
ties in different applications. In general, the amplitude and phase
modulations are coupled in the TN-LCSLM and thus characteriza-
tion is necessary to use it as either amplitude-mostly modulator
or phase-mostly modulator. The TN-LCSLM can be modeled using
simplified Jones matrix calculus and expression for the intensity
and phase modulation can be evaluated. Jones matrix for the
TN-LCSLM can be represented by 2 2 matrix depending on its
intrinsic physical parameters such as the twist angle, birefrin-
gence and the orientation of LC director axis at input face of the
TN-LCSLM. Jones calculus is simple and widely used for the
phase-modulation characterization of TN-LCSLMs; however, char-
acterization using Jones matrix calculus cannot represent scatter-
ing of light or depolarization. Also, Jones matrices are only
applicable to the completely polarized state of light and cannot
express partially or unpolarized light, which assures that Jones
calculus is applicable to non-depolarizing medium only.
When an optical beam interacts with matter whether through
transmission, reflection or scattering, its polarization state is
always changed. One of the important reasons for this change in
polarization of incident optical beam is depolarization and is
defined as the coupling of completely polarized into partially
polarized light. The TN-LCSLM modeled using Mueller matrix is
represented by 4 4 array and provides detailed information
Contents lists available at SciVerse ScienceDirect
journal homepage: www.elsevier.com/locate/optlaseng
Optics and Lasers in Engineering
0143-8166/$ - see front matter &2011 Elsevier Ltd. All rights reserved.
doi:10.1016/j.optlaseng.2011.10.004
n
Corresponding author.
E-mail address: kapi0001@e.ntu.edu.sg (K. Dev).
Please cite this article as: Dev K, Asundi A. Mueller–Stokes polarimetric characterization of transmissive liquid crystal spatial
light modulator. Opt Laser Eng (2011), doi:10.1016/j.optlaseng.2011.10.004
Optics and Lasers in Engineering ](]]]])]]]–]]]
about polarimetric properties such as diattenuation, polarizance,
depolarization and retardance. Also, polarization modulation
characterization for the TN-LCSLM can be achieved using Mueller
calculus, which is not possible using Jones calculus.
Many researchers have reported characterization of the LCSLM
measuring Mueller matrices for different addressed gray scales
within its dynamic range. Pezzaniti et al. first described a method
to achieve phase-only modulation from the LCSLM minimizing
polarization modulation using its average eigenpolarization
states [1]. These average eigenpolarization states were first
computed from modeled Mueller matrices and then certain
degree of depolarization was studied present within the LCSLM.
Depolarization in LCSLM exists due to the bulk scattering, glass
spacer balls and small high-frequency oscillations of the LC
molecules with response to applied voltage. Thus, unpolarized
light produced due to existing depolarization affects the intensity
modulation from the LCSLM, which can be decisive factor for its
display applications [2]. Since Mueller matrix calculi cannot alone
provide phase modulation information, researchers have pre-
sented different methods to characterize phase modulation and
polarization modulation simultaneously for the reflective liquid
crystal on silicon (LCOS) SLM utilizing the Jones matrix formalism
together [3–7]. Mueller calculus cannot render information about
the LCSLM maximum phase modulation but it can predict
heuristically incident state of polarization (SOP) for which output
SOP in terms of stoke parameters remains constant [5,6].
In this paper, polarimetric characterization of the transmissive
TN-LCSLM HOLOEYE LC2002 is carried out using Mueller matrix
imaging polarimeter. 17 different Mueller matrices at different
addressed gray scale values for this TN-LCSLM are calculated at
530 nm wavelength. Lu–Chipman polar decomposition for Muel-
ler matrices is used to classify diattenuation, depolarization and
retardance Mueller matrices for range of addressed gray scale.
Later, Mueller–Stokes formulation is used to examine the effect of
modeled Mueller matrices on 6 different completely polarized
states. We have found that the transmissive TN-LCSLM produces
more than 30% depolarized light for incident circularly polarized
light. Eigenpolarization states in terms of incident Stokes vector
are then calculated such that there is least variation of output
Stokes vector on Poincare
´sphere. These eigenpolarization states
give the phase-mostly modulation of the TN-LCSLM without any
polarization modulation.
2. Measurement of Mueller matrices
The TN-LCSLM is a polarization sensitive device and any incident
light wavefront with unique polarization has different propagation
through it in terms of amplitude, phase and polarization. The
amplitude and phase modulations using the TN-LCSLM have always
been in major focus but it can also be utilized for polarization
modulation of incident light wavefront either transmitting or
reflecting through its active area. Generally, polarization modula-
tion can be only seen as change in contrast of light wavefront
coming out of the TN-LCSLM and cannot be easily investigated as
amplitude and phase modulation characterization using Jones
calculi. Also, polarization modulation characterization using Jones
calculi (applicable for completely polarized light) is difficult in
presence of depolarization in the TN-LCSLM. On the other hand,
Mueller calculus provides an alternative approach to characterize
polarimetric properties of the TN-LCSLM. Mueller matrix repre-
sented by 4 4 real value elements for an optical system or the
polarization sensitive sample describes the transformation of inci-
dent polarized light and outgoing polarized light. In Mueller
calculus, Stokes vector Sdescribes the polarization state of light
beam and Mueller matrix Mdescribes the polarization altering
characteristics of sample. Mueller matrix imaging polarimeter
(MMIP) is used to characterize polarimetric properties of polariza-
tion altering optical sample. MMIP consists of important optical
components such as polarization state generator (PSG), polarization
altering sample and polarization state analyzer (PSA). Any state of
polarization in MMIP can be generated using PSG having polarizer
and quarter wave plate placed in order. Similarly, any state of
polarization altered by sample can be detected or analyzed using
PSA with same components as in PSG but placed in opposite order.
The polarization altering sample whose Mueller matrix is to be
calculated is placed between PSG and PSA.
Different methods have been presented to calculate Mueller
matrix for polarization altering sample. Azzam described method
to calculate all 16 elements of Mueller matrix from the calculated
Fourier coefficients of transmitted light intensity. In his proposed
methodology, quarter wave plates placed in polarizing optics and
analyzer optics were synchronously rotated at angular speeds of
o
and 5
o
[8]. Based on similar idea, Pezzaniti et al. developed MMIP,
which provides high precision measurements for the Mueller
matrices at every pixel of an image captured using charge coupling
device [9]. Also, there exists other method to achieve Mueller
matrix for sample by simply capturing 16, 36 or 49 polarization
images generated by MMIP [10].
In our experimental work, HOLOEYE LC2002 transmissive
TN-LCSLM polarimetric properties are characterized, which uses
Sony LCX016AL liquid crystal microdisplay. The TN-LCSLM active
area of 21 26 mm contains 832624 square pixels with pixel
pitch of 32
m
m and has fill factor of 55%. HOLOEYE LC2002 has
twisted arrangement of nematic liquid crystal molecules between
inner ends of two conductive glass plates. The experimental set-up
that uses MMIP is shown in Fig. 1. 17 Mueller matrices are modeled
for transmissive HOLOEYE LC2002 TN-LCSLM at 17 different
QWP 1
P
UCCD
x
y
A
QWP 2
SLM
Fig. 1. Mueller matrix imaging polarimeter (MMIP) with HOLOEYE LC2002 TN-LCSLM sandwiched between PSG and PSA components. (U ¼Unpolarized light, P¼Polarizer,
A¼Analyzer, QWP¼Quarter waveplate, CCD ¼Charge Coupling Device.)
K. Dev, A. Asundi / Optics and Lasers in Engineering ](]]]])]]]–]]]2
Please cite this article as: Dev K, Asundi A. Mueller–Stokes polarimetric characterization of transmissive liquid crystal spatial
light modulator. Opt Laser Eng (2011), doi:10.1016/j.optlaseng.2011.10.004
addressed gray scales between its dynamic range (0–255) using
MMIP at 530 nm wavelength. In this experiment, 6 different SOP
(Horizontal (H), vertical (V), þ451linear (P), –451linear (M), right
circularly polarized (R) and left circularly polarized (L)) generated
by PSG are allowed to pass through the TN-LCSLM active area and
transmitted polarized light is analyzed using PSA. These 6 different
SOP generated and analyzed by PSG and PSA, respectively, can be
found out by fixing polarization analyzer unit at a time and rotating
the other to get a null intensity without sample in between. An
IMAGINGSOURCE
s
charge coupling device (CCD) with 1280 960
square pix els each having size of 4.65 4.65
m
m is used to record
the transmitted intensity after PSA. Mueller matrix for the
TN-LCSLM at any particular addressed gray scale can be calculated
by recording 36 intensity images as depicted in Fig. 2.Inthisfigure,
the first letter denotes the input polarization state and the second
letter denotes the measurement polarization state. For example, the
‘‘HH’’ element represents an intensity image acquired with incident
horizontal polarization (H) and analyzer horizontal polarization (H).
The calibration of MMIP is essential before calculating precise
Mueller matrices for the TN-LCSLM. In order to verify this,
Mueller matrix for air is first calculated. Fig. 3 shows Mueller
matrix for air calculated using MMIP, which is very close to 4 4
unit matrix. The central region of 300 300 pixels is selected
from these recorded intensity images and average value is used
for further calculations. All 16 elements of the Mueller matrix are
normalized with respect to first element m
00
. Typical value of
error in each element was found to lie between 1 and 5%. After
obtaining satisfactory results from this standard measurement,
the set-up was used to record Mueller matrices for the TN-LCSLM
investigated in this study. In our earlier study, we have char-
acterized phase modulation of the LC2002 TN-LCSLM using
Digital Holography method with different combination of con-
trast and brightness values and we have found that the selection
of contrast and brightness value is not very critical for the
TN-LCSLM LC2002 modulation curve [11]. Thus, in our experi-
ment to measure the Mueller matrix of the TN-LCSLM using MMIP
and to characterize all polarimetric quantities, it is operated at
maximum contrast and brightness settings of value ‘255’. The
dynamic range of the TN-LCSLM addressed gray scale value is
equally divided into 16 intervals and 36 polarization images are
recorded at each of 17 addressed gray scale values using MMIP.
Fig. 4 shows the spatial uniformity of the Mueller matrix elements
calculated at gray scale value ‘128’ addressed on the TN-LCSLM
using MMIP. Since the value of Mueller matrix elements over an
area of 300 300 pixels is uniform, the average value of Mueller
matrix elements is taken for further measurements. The variation
in values of all 16 elements of Mueller matrix measured for the
TN-LCSLM with respect to addressed gray scale within its
dynamic range is shown in Fig. 5. It is evident from this figure
that below gray scale value of ‘90’ there is much variation in
values of Mueller matrix elements and above this gray scale value
Mueller matrix elements remain constant.
3. Lu–Chipman polar decomposition
After modeling the TN-LCSLM with Mueller matrices for
several addressed gray scale values, it is important to characterize
polarimetric properties such as diattenuation, polarizance, retar-
dance and depolarization associated with it. The information
about these different polarimetric properties is embedded within
Mueller matrices modeled for the TN-LCSLM and can be separated
from one another using the Lu–Chipman polar decomposition
algorithm. This polar decomposition of Mueller matrix allows one
to obtain sequence of three 4 4 matrices factors: a diattenuator,
followed by a retarder, then followed by a depolarizer [12]. In our
work, we have used the Lu–Chipman polar decomposition algo-
rithm to separate out different polarimetric properties from
measured 17 Mueller matrices modeled for the TN-LCSLM
addressed to different gray scales within its dynamic range.
The polarization state of an incident light passing through
nondepolarizing element can be changed by either change in
amplitude or change in phase of orthogonal field components.
These two kinds of nondepolarizing elements are called diatte-
nuator and retarder, respectively. Polarization of light can also be
changed by depolarizing element as stated earlier. For polari-
metric characterization of the TN-LCSLM, it is important to
measure these polarimetric properties and their effect on incident
polarized light. To analyze whether there exists any amount of
depolarization within the TN-LCSLM, it is necessary to observe the
diattenuation and polarizance present in the TN-LCSLM from the
modeled Mueller matrices. In particular, a nondepolarizing Muel-
ler matrix shows the equivalence of the magnitude of the
diattenuation and the polarizance [12].
Diattenuation changes the intensity transmittance of the
incident polarization and is measured as the difference in inten-
sity transmittance between two incident orthogonal polariza-
tions. Diattenuation present in the TN-LCSLM can be calculated
using the first row of Mueller matrix M, which determines
intensity transmittance:
D¼1
m
00
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
m
2
01
þm
2
02
þm
2
03
qð1Þ
D¼1
m
00
m
01
m
02
m
03
0
B
@1
C
Að2Þ
Here, Ddenotes the magnitude of diattenuation vector Dand
m
ab
(a,b¼0,1,2,3) represents an element of Mueller matrix.
Another polarimetric quantity of interest is polarizance present
within the TN-LCSLM, which is the measure of degree of polar-
ization in transmitted light when unpolarized or natural light
incident on it. Mathematically, polarizance value Pwith vector P
+++ +−− +− − +−−
−+− −−+ −− + −−+
−+− −−+ −−+ −−+
−+− −−+ −− + −−+
Fig. 2. Mueller matrix calculation for a sample from 36 polarization images with polarization states generated and analyzed by PSG and PSA, respectively.
Fig. 3. Mueller matrix for air calculated using 36 polarization images recorded
by MMIP.
K. Dev, A. Asundi / Optics and Lasers in Engineering ](]]]])]]]–]]] 3
Please cite this article as: Dev K, Asundi A. Mueller–Stokes polarimetric characterization of transmissive liquid crystal spatial
light modulator. Opt Laser Eng (2011), doi:10.1016/j.optlaseng.2011.10.004
can be calculated from first column of Mueller matrix Mas
P¼1
m
00
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
m
2
10
þm
2
20
þm
2
30
qð3Þ
P¼1
m
00
m
10
m
20
m
30
0
B
@1
C
Að4Þ
Thus, the first column vector of Mueller matrix Mgives the
measure of polarizance. It should be noted here that value of
these two polarimetric quantities varies between 0 and 1. Fig. 6
shows the diattenuation and polarizance variation calculated
using Eqs. (1) and (3) from the Mueller matrices modeled for
the TN-LCSLM with respect to addressed gray scale. Since the
magnitude of the diattenuation and polarizance values is differ-
ent, it is apparent from this figure that there exists depolarization
within the TN-LCSLM below gray scale value ‘90’. Thus, the Lu–
Chipman polar decomposition of Mueller matrices Mmodeled for
the TN-LCSLM must be carried out considering it as depolarizing
element as follows [12]:
M¼MDM
R
M
D
¼1D
T
Pm
"# ð5Þ
In Eq. (5), MD,M
R
and M
D
are depolarization, retarder and
diattenuator Mueller matrices deduced from total Mueller matrix
M, respectively. Also, mis sub matrix of original Mueller matrix
Mmodeled for the TN-LCSLM having no contribution from
diattenuation Dand polarizance P. To separate all polarimetric
properties from Mueller matrix M, symmetric diattenuator matrix
M
D
is first calculated using Eqs. (6) and (7) as follows:
M
D
¼1D
T
Dm
D
! ð6Þ
m
D
¼ffiffiffiffiffiffiffiffiffiffiffiffiffi
1D
2
pIþð1ffiffiffiffiffiffiffiffiffiffiffiffiffi
1D
2
pÞ^
D^
D
T
ð7Þ
Here, D
T
represents the transpose of diattenuation vector given by
Eq. (2), m
D
is sub matrix of diattenuator matrix M
D
given by Eq. (7)
in terms of diattenuation unit vector ^
Dand 3 3 identity matrix I.
Now a new Mueller matrix M
0
is defined based on total Mueller
matrix Mas
M
0
¼MM
1
D
ð8Þ
It should be noted here that this new Mueller matrix M
0
has no
contribution from the diattenuation but consists of both retar-
dance and depolarization. The new Mueller matrix M
0
in terms of
depolarization Mueller matrix M
D
and retardance Mueller matrix
M
R
can be rewritten as
M
0
¼MDM
R
¼10
T
PDmD
"#
10
T
0m
R
"#
¼10
T
PDmDm
R
"#
¼10
T
PDm
0
"#
ð9Þ
PD¼PmD
1D
2
ð10Þ
m00
100 200 300
100
200
300 -1
0
1m01
100 200 300
100
200
300 -1
0
1m02
100 200 300
100
200
300 -1
0
1m03
100 200 300
100
200
300 -1
0
1
m10
100 200 300
100
200
300 -1
0
1m11
100 200 300
100
200
300 -1
0
1m12
100 200 300
100
200
300 -1
0
1m13
100 200 300
100
200
300 -1
0
1
m20
100 200 300
100
200
300 -1
0
1m21
100 200 300
100
200
300 -1
0
1m22
100 200 300
100
200
300 -1
0
1m23
100 200 300
100
200
300 -1
0
1
m30
100 200 300
100
200
300 -1
0
1m31
100 200 300
100
200
300 -1
0
1m32
100 200 300
100
200
300 -1
0
1m33
100 200 300
100
200
300 -1
0
1
Fig. 4. Spatial uniformity of 16 different Mueller matrix elements measured by MMIP at ‘128’ addressed gray scale addressed on the TN-LCSLM.
K. Dev, A. Asundi / Optics and Lasers in Engineering ](]]]])]]]–]]]4
Please cite this article as: Dev K, Asundi A. Mueller–Stokes polarimetric characterization of transmissive liquid crystal spatial
light modulator. Opt Laser Eng (2011), doi:10.1016/j.optlaseng.2011.10.004
In Eq. (9), 0
T
represents the of zero column vector transpose
and PDis a vector given by Eq. (10). Also, m
0
represents sub matrix
of Mueller matrix M
0
. Now, if
l
1
,
l
2
and
l
3
are the eigenvalues of
m
0
ðm0Þ
T
, then mD, the sub matrix of depolarization Mueller matrix
MD, can be obtained by
mD¼7½m0ðm0Þ
T
þð ffiffiffiffiffiffiffiffiffiffi
l
1
l
2
pþffiffiffiffiffiffiffiffiffiffi
l
2
l
3
pþffiffiffiffiffiffiffiffiffiffi
l
3
l
1
pÞI
1
½ð ffiffiffiffiffi
l
1
pþffiffiffiffiffi
l
2
pþffiffiffiffiffi
l
3
pÞm
0
ðm
0
Þ
T
þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
l
1
l
2
l
3
pIð11Þ
In Eq. (11), negative sign is used with equation if determinant
of m
0
is negative. Otherwise, positive sign is applied. The depolar-
ization conceded by the TN-LCSLM can be calculated using
Eq. (11) and is defined as the coupling of completely polarized
light into partially polarized light. Thus, higher value depolariza-
tion is the degradation of completely polarized light and is
considered as problem for modern displays. Finally, the retar-
dance Mueller matrix can be calculated using
M
R
¼M
1
DM
0
¼10
T
0m
R
"# ð12Þ
In Eq. (12), m
R
represents the sub matrix of retardance Mueller
matrix M
R
. This sub matrix has all information about retardance
present in the TN-LCSLM and is defined as the phase difference
added between two incident orthogonal polarizations. The polari-
metric quantities such as retardance Rand depolarization
D
present in the TN-LCSLM can be calculated using relations given
by Eqs. (13) and (14), respectively:
R¼cos
1
TrðM
R
Þ
21
ð13Þ
-1
-0.5
0
0.5
1
0 50 100 150 200 250
m00
-1
-0.5
0
0.5
1
0 50 100 150 200 250
m01
-1
-0.5
0
0.5
1
0 50 100 150 200 250
m02
-1
-0.5
0
0.5
1
050100150200250
m03
-1
-0.5
0
0.5
1
0 50 100 150 200 250
m10
-1
-0.5
0
0.5
1
0 50 100 150 200 250
m11
-1
-0.5
0
0.5
1
0 50 100 150 200 250
m12
-1
-0.5
0
0.5
1
050100150200250
m13
-1
-0.5
0
0.5
1
0 50 100 150 200 250
m20
-1
-0.5
0
0.5
1
0 50 100 150 200 250
m21
-1
-0.5
0
0.5
1
0 50 100 150 200 250
m22
-1
-0.5
0
0.5
1
050100150200250
m23
-1
-0.5
0
0.5
1
0 50 100 150 200 250
m30
-1
-0.5
0
0.5
1
0 50 100 150 200 250
m31
-1
-0.5
0
0.5
1
0 50 100 150 200 250
m32
-1
-0.5
0
0.5
1
0 50 100 150 200 250
m33
Fig. 5. Variation in values of all 16 elements of Mueller matrix for HOLOEYE LC2002 TN-LCSLM measured with respect to addressed gray scale.
0 50 100 150 200 250
0.00
0.02
0.04
0.06
0.08
0.10
Addressed grey scale on SLM
Diattenuation (D)
Polarizance (P)
Fig. 6. Diattenuation and polarizance measured with respect to addressed gray
scale from modeled Mueller matrices for HOLOEYE LC2002 TN-LCSLM.
K. Dev, A. Asundi / Optics and Lasers in Engineering ](]]]])]]]–]]] 5
Please cite this article as: Dev K, Asundi A. Mueller–Stokes polarimetric characterization of transmissive liquid crystal spatial
light modulator. Opt Laser Eng (2011), doi:10.1016/j.optlaseng.2011.10.004
D
¼19TrðmDÞ9
3ð14Þ
Here, Tr represents the trace of a matrix. The variation of these
polarimetric quantities with respect to addressed gray scale
calculated using relations (13) and (14) is shown in Fig. 7.It
should be noted here that retardance and depolarization shown in
Fig. 7 calculated from their respective Mueller matrices consist of
both linear and circular constituents and may have different
values for different states of polarizations. Since depolarization
is important polarimetric property of the TN-LCSLM, its variation
with respect to different state of polarized light is discussed in the
next section using the Mueller–Stokes formalism.
4. Mueller–Stokes formulation
In the last section, Lu–Chipman polar decomposition is used to
separate three 44 Mueller matrices for different polarimetric
quantities such as diattenuation, retardance and depolarization
from the modeled Mueller matrices for the TN-LCSLM evaluated
using MMIP. The variation of all polarimetric quantities is
measured with respect to addressed gray scales and it is found
that there exists15% depolarization within the TN-LCSM at all
addressed gray scales. However, it is crucial to study the effect of
this existing depolarization within the TN-LCSLM on different
incident polarizations using the Mueller–Stokes formalism. The
Mueller–Stokes formalism is generally used for experimental
determination of optical devices polarization behavior. In the
Mueller–Stokes combined formalism, incident polarized light is
mathematically represented using 4 1 Stokes vector Sand its
interaction with optical device modeled using Mueller matrix Mis
analyzed in terms of exiting Stokes vector S
0
(¼M.S).
In our observation, six different incident SOPs in terms of Stokes
vector (S
H
¼[1,1,0,0], S
V
¼[1,1,0,0], S
P
¼[1,0,1,0], S
M
¼[1,0,1,0],
S
R
¼[1,0,0,1] and S
L
¼[1,0,0,1]) are multiplied with different
measured Mueller matrices modeled for the TN-LCSLM at 17
different addressed gray scales to evaluate exiting Stokes vector.
The exiting Stokes vector bears the information about incident
polarization altered by the TN-LCSLM and exiting light polarization
state. Fig. 8 shows the variation in values of exiting Stokes
parameters calculated for six different incident polarizations
with respect addressed gray scale. The variation in exiting Stokes
parameters for incident polarization clearly shows that the
TN-LCSLM acts as polarization sensitive device, which alters any
incident polarization depending upon its polarization state and
addressed gray scale value. It is evident from Fig. 8(e) and (f) that
there is least change in numerical values of exiting Stokes para-
meters calculated for addressed gray scales on the TN-LCSLM when
incident polarization is circularly polarized light in comparison to
exiting Stokes parameters calculated for other incident linearly
polarization states.
The degree of polarization (DoP) for exiting polarization state
can also be calculated with the help of the Mueller–Stokes
formalism. The DoP for an electromagnetic wave is defined as a
quantity to describe the portion of it, which is polarized and given
in terms of Stokes parameters S
0
,S
1
,S
2
and S
3
as
DoP ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
S
2
1
þS
2
2
þS
2
3
qS
0
ð15Þ
Fig. 9 shows the DoP measured for six different existing Stokes
vectors calculated using the Mueller–Stokes formalism with
respect to addressed gray scale on the TN-LCSLM. It can be clearly
seen from Fig. 9 that incident right and left circularly polarized
light are most affected due to depolarization existing within the
TN-LCSLM. There is 30%–40% degradation in polarization of
incident circularly polarized light after passing through the TN-
LCSLM. This observation suggests that incident circularly polar-
ized light passes through transmissive TN-LCSLM with reduced
degree of polarization and having least change in its original
polarization state in comparison to other incident polarizations.
The Mueller–Stokes formalism can also be helpful in evaluat-
ing the eigenpolarization states for the TN-LCSLM under opera-
tion. These incident eigenpolarization state traverses through the
TN-LCSLM without any change in its polarization state for all
addressed gray scale values within its dynamic range. Eigenpo-
larization states are important for phase-mostly modulation of
the TN-LCSLM with constant amplitude or intensity modulation.
Phase modulation property of the TN-LCSLM has potential appli-
cations in optical data processing, laser pulse shaping, optical
tweezers, 3D holographic display, adaptive optics, etc. From our
previous observations using the Mueller–Stokes formalism and
different incident SOP, it has been found that incident circularly
polarized light (either right circularly or left circularly) falls
close as an eigenpolarization state. Also it is reported by many
050100150200250
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
0.20
Depolarizance (Δ)
Addressed grey scale on SLM
Δ
0 50 100 150 200 250
0.5
1.0
1.5
2.0
2.5
3.0
3.5
Retardance (R)
Addressed grey scale on SLM
R
Fig. 7. Retardance (R) and depolarization (
D
) measured with respect to addressed gray scale from modeled Mueller matrices for HOLOEYE LC2002 TN-LCSLM.
K. Dev, A. Asundi / Optics and Lasers in Engineering ](]]]])]]]–]]]6
Please cite this article as: Dev K, Asundi A. Mueller–Stokes polarimetric characterization of transmissive liquid crystal spatial
light modulator. Opt Laser Eng (2011), doi:10.1016/j.optlaseng.2011.10.004
0 50 100 150 200 250 300
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
S
S
S
S
Stokes Parameters
Addressed grey scale on SLM
0 50 100 150 200 250 300
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
Stokes Parameters
Addressed grey scale on SLM
0 50 100 150 200 250 300
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
S
S
S
S
Stokes Parameters
Addressed grey scale on SLM
0 50 100 150 200 250 300
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
Stokes Parameters
Addressed grey scale on SLM
0 50 100 150 200 250 300
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0 S
S
S
S
Stokes Parameters
Addressed grey scale on SLM
0 50 100 150 200 250 300
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0 S
S
S
S
Stokes Parameters
Addressed grey scale on SLM
S
S
S
S
S
S
S
S
Fig. 8. Measured exiting Stokes parameters when incident Stokes polarization vector is (a) horizontal, (b) vertical, (c) þ451polarized, (d) 451polarized, (e) right
circularly polarized and (f) left circularly polarized.
K. Dev, A. Asundi / Optics and Lasers in Engineering ](]]]])]]]–]]] 7
Please cite this article as: Dev K, Asundi A. Mueller–Stokes polarimetric characterization of transmissive liquid crystal spatial
light modulator. Opt Laser Eng (2011), doi:10.1016/j.optlaseng.2011.10.004
researchers that elliptically polarized light acts as eigenpolariza-
tion states for LCSLMs [1,6]. With the help of the Mueller–Stokes
formalism, eigenpolarization states can be mathematically
derived using Mueller matrices modeled for the TN-LCSLM at
different addressed gray scales. Since eigenpolarization states do
not change their polarization state after passing through the TN-
LCSLM, this refers that exiting Stokes parameters should remain
constant for all addressed gray scales on it. Mathematically, the
input Stokes parameters of a vector are varied in such a way that
after its interaction with modeled Mueller matrices of the TN-
LCSLM measured for different addressed gray scales, it should
give rise to uniform exiting Stokes parameters.
In our observation, the input Stokes vector S
I
¼[1.0, 0.08,
0.14, 0.28] behaves as an eigenpolarization state since it
produces nearly uniform exiting Stokes parameters. This input
Stokes vector has azimuth angle of 30.131and ellipticity angle of
8.131.Fig. 10 shows the comparison of exiting Stokes para-
meters calculated heuristically using the Mueller–Stokes formal-
ism and the exiting Stokes parameters calculated experimentally
using quarter wave plate in front of polarizer to generate different
states of polarization and then transmit it through active area of
the TN-LCSLM to record intensity images (I
H
,I
V
,I
P
,I
M
,I
R
and I
L
).
These recorded intensity images for different addressed gray scale
values are then processed to give exiting Stokes parameter [6].
Another input Stokes vector S
I
¼[1.0, 0.08, 0.14, 0.28] having
equal and opposite ellipticity angle to first incident eigenpolar-
ization vector also behaves as eigenpolarization giving constant
exiting Stokes parameters for all addressed gray scale value
within its dynamic range. Fig. 10 shows that the exiting Stokes
parameter calculated experimentally for different addressed gray
scale on the TN-LCSLM are in good agreement with the Stokes
parameter calculated heuristically using the Mueller–Stokes
formalism.
5. Conclusion
In summary, this paper presents the polarimetric character-
ization of the transmissive TN-LCSLM (HOLOEYE 2002) using the
Mueller matrix imaging polarimeter (MMIP). Mueller matrices for
the TN-LCSLM are calculated for 17 different addressed gray scale
values within its dynamic range. Polarimetric quantities such as
diattenuation, retardance and depolarization are evaluated from
modeled Mueller matrices using Lu–Chipman polar decomposi-
tion of Mueller matrices for the TN-LCSLM in terms of 4 4
Mueller matrices and their variation with respect to addressed
gray scale on the TN-LCSLM is studied. It has been found that
there exists 15% depolarization within the TN-LCSLM for all
addressed gray scale values and the effect of this on series of
polarizations is studied using the Mueller–Stokes formalism. The
existing depolarization within the TN-LCSLM affects mostly cir-
cularly polarized light and degrades its polarization by 30%–
40%. This study suggests that incident circularly polarized light
passes through transmissive TN-LCSLM with reduced degree of
polarization and having least change in its original polarization
state in comparison to other incident linear polarization states.
Elliptically polarized light is heuristically calculated as eigenpo-
larization state for the TN-LCSLM using the Mueller–Stokes
formalism, which shows nearly uniform exiting Stokes parameter
for all addressed gray scale values on the TN-LCSLM. Thus, to
operate this TN-LCSLM in the phase-mostly modulation mode,
quarter wave plate must be added after polarizer in order to
incident elliptically polarized light on it. Also in the phase-mostly
modulation mode, the intensity modulation provided by the TN-
LCSLM should be constant. Since Mueller matrix calculus does not
give any information regarding phase, numerical search of eigen-
polarization states using the Mueller–Stokes formalism renders
essential information to employ TN-LCSLM in phase-mostly
modulation mode.
Acknowledgment
We thank the Optics and Photonics Society of Singapore and
the Nanyang Technological University for their support.
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0.0
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light modulator. Opt Laser Eng (2011), doi:10.1016/j.optlaseng.2011.10.004
