mueller–stokes polarimetric characterization of transmissive liquid crystal spatial light...

Optics and Lasers in Engineering 50 (2012) 599–607

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Optics and Lasers in Engineering

0143-81

doi:10.1

n Corr

E-m

journal homepage: www.elsevier.com/locate/optlaseng

Mueller–Stokes polarimetric characterization of transmissive liquid crystalspatial light modulator

Kapil Dev n, Anand Asundi

School of Mechanical and Aerospace Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798, Singapore

a r t i c l e i n f o

Available online 19 October 2011

Keywords:

Mueller matrix

Mueller–Stokes formalism

Liquid crystal spatial light modulator

Liquid crystal

Depolarization

66/$ - see front matter & 2011 Elsevier Ltd. A

016/j.optlaseng.2011.10.004

esponding author.

ail address: [email protected] (K. Dev).

a b s t r a c t

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 opticalelement device, which can modify optical function for real-timeapplications. SLMs have been used in many applications involvingoptical metrology, optical pulse shaping, display applications,optical information processing, adaptive optics, etc. Importantoptical properties associated with an optical wavefront such asamplitude, phase and polarization can be modulated using SLMs.SLMs based on liquid crystal materials have advantages such ashigh switching speed, high spatial resolution, large birefringencewith low voltage operation and can provide amplitude, phase andpolarization modulation for incident optical wavefront. In the TN-LCSLM, uniaxial anisotropic nematic liquid crystal material issandwiched between two inner ends of conductive glass plateswith liquid crystal molecules director gradually rotating from oneto other end giving twist orientation of 901. The distortion of twistuniformity and decrease in tilt angles of liquid crystal moleculesdirector on an application of electric field leads to the amplitudeand phase modulation of transmitted or reflected wavefront,respectively.

ll rights reserved.

The amplitude and phase modulation characterization of theTN-LCSLM is crucial to exploit its wavefront modulation proper-ties in different applications. In general, the amplitude and phasemodulations are coupled in the TN-LCSLM and thus characteriza-tion is necessary to use it as either amplitude-mostly modulatoror phase-mostly modulator. The TN-LCSLM can be modeled usingsimplified Jones matrix calculus and expression for the intensityand phase modulation can be evaluated. Jones matrix for theTN-LCSLM can be represented by 2�2 matrix depending on itsintrinsic physical parameters such as the twist angle, birefrin-gence and the orientation of LC director axis at input face of theTN-LCSLM. Jones calculus is simple and widely used for thephase-modulation characterization of TN-LCSLMs; however, char-acterization using Jones matrix calculus cannot represent scatter-ing of light or depolarization. Also, Jones matrices are onlyapplicable to the completely polarized state of light and cannotexpress partially or unpolarized light, which assures that Jonescalculus is applicable to non-depolarizing medium only.

When an optical beam interacts with matter whether throughtransmission, reflection or scattering, its polarization state isalways changed. One of the important reasons for this change inpolarization of incident optical beam is depolarization and isdefined as the coupling of completely polarized into partiallypolarized light. The TN-LCSLM modeled using Mueller matrix isrepresented by 4�4 array and provides detailed information

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K. Dev, A. Asundi / Optics and Lasers in Engineering 50 (2012) 599–607600

about polarimetric properties such as diattenuation, polarizance,depolarization and retardance. Also, polarization modulationcharacterization for the TN-LCSLM can be achieved using Muellercalculus, which is not possible using Jones calculus.

Many researchers have reported characterization of the LCSLMmeasuring Mueller matrices for different addressed gray scaleswithin its dynamic range. Pezzaniti et al. first described a methodto achieve phase-only modulation from the LCSLM minimizingpolarization modulation using its average eigenpolarizationstates [1]. These average eigenpolarization states were firstcomputed from modeled Mueller matrices and then certaindegree of depolarization was studied present within the LCSLM.Depolarization in LCSLM exists due to the bulk scattering, glassspacer balls and small high-frequency oscillations of the LCmolecules with response to applied voltage. Thus, unpolarizedlight produced due to existing depolarization affects the intensitymodulation from the LCSLM, which can be decisive factor for itsdisplay applications [2]. Since Mueller matrix calculi cannot aloneprovide phase modulation information, researchers have pre-sented different methods to characterize phase modulation andpolarization modulation simultaneously for the reflective liquidcrystal on silicon (LCOS) SLM utilizing the Jones matrix formalismtogether [3–7]. Mueller calculus cannot render information aboutthe LCSLM maximum phase modulation but it can predictheuristically incident state of polarization (SOP) for which outputSOP in terms of stoke parameters remains constant [5,6].

In this paper, polarimetric characterization of the transmissiveTN-LCSLM HOLOEYE LC2002 is carried out using Mueller matriximaging polarimeter. 17 different Mueller matrices at differentaddressed gray scale values for this TN-LCSLM are calculated at530 nm wavelength. Lu–Chipman polar decomposition for Muel-ler matrices is used to classify diattenuation, depolarization andretardance Mueller matrices for range of addressed gray scale.Later, Mueller–Stokes formulation is used to examine the effect ofmodeled Mueller matrices on 6 different completely polarizedstates. We have found that the transmissive TN-LCSLM producesmore than 30% depolarized light for incident circularly polarizedlight. Eigenpolarization states in terms of incident Stokes vectorare then calculated such that there is least variation of outputStokes vector on Poincare sphere. These eigenpolarization statesgive the phase-mostly modulation of the TN-LCSLM without anypolarization modulation.

2. Measurement of Mueller matrices

The TN-LCSLM is a polarization sensitive device and any incidentlight wavefront with unique polarization has different propagationthrough it in terms of amplitude, phase and polarization. The

QWP 1P

U

x

y

SLM

Fig. 1. Mueller matrix imaging polarimeter (MMIP) with HOLOEYE LC2002 TN-LCSLM s

A¼Analyzer, QWP¼Quarter waveplate, CCD¼Charge Coupling Device.)

amplitude and phase modulations using the TN-LCSLM have alwaysbeen in major focus but it can also be utilized for polarizationmodulation of incident light wavefront either transmitting orreflecting through its active area. Generally, polarization modula-tion can be only seen as change in contrast of light wavefrontcoming out of the TN-LCSLM and cannot be easily investigated asamplitude and phase modulation characterization using Jonescalculi. Also, polarization modulation characterization using Jonescalculi (applicable for completely polarized light) is difficult inpresence of depolarization in the TN-LCSLM. On the other hand,Mueller calculus provides an alternative approach to characterizepolarimetric properties of the TN-LCSLM. Mueller matrix repre-sented by 4�4 real value elements for an optical system or thepolarization sensitive sample describes the transformation of inci-dent polarized light and outgoing polarized light. In Muellercalculus, Stokes vector S describes the polarization state of lightbeam and Mueller matrix M describes the polarization alteringcharacteristics of sample. Mueller matrix imaging polarimeter(MMIP) is used to characterize polarimetric properties of polariza-tion altering optical sample. MMIP consists of important opticalcomponents such as polarization state generator (PSG), polarizationaltering sample and polarization state analyzer (PSA). Any state ofpolarization in MMIP can be generated using PSG having polarizerand quarter wave plate placed in order. Similarly, any state ofpolarization altered by sample can be detected or analyzed usingPSA with same components as in PSG but placed in opposite order.The polarization altering sample whose Mueller matrix is to becalculated is placed between PSG and PSA.

Different methods have been presented to calculate Muellermatrix for polarization altering sample. Azzam described methodto calculate all 16 elements of Mueller matrix from the calculatedFourier coefficients of transmitted light intensity. In his proposedmethodology, quarter wave plates placed in polarizing optics andanalyzer optics were synchronously rotated at angular speeds of oand 5o [8]. Based on similar idea, Pezzaniti et al. developed MMIP,which provides high precision measurements for the Muellermatrices at every pixel of an image captured using charge couplingdevice [9]. Also, there exists other method to achieve Muellermatrix for sample by simply capturing 16, 36 or 49 polarizationimages generated by MMIP [10].

In our experimental work, HOLOEYE LC2002 transmissiveTN-LCSLM polarimetric properties are characterized, which usesSony LCX016AL liquid crystal microdisplay. The TN-LCSLM activearea of 21�26 mm contains 832�624 square pixels with pixelpitch of 32 mm and has fill factor of 55%. HOLOEYE LC2002 hastwisted arrangement of nematic liquid crystal molecules betweeninner ends of two conductive glass plates. The experimental set-upthat uses MMIP is shown in Fig. 1. 17 Mueller matrices are modeledfor transmissive HOLOEYE LC2002 TN-LCSLM at 17 different

CCD

AQWP 2

andwiched between PSG and PSA components. (U¼Unpolarized light, P¼Polarizer,

+ + + + − − + − − + − −− + − − − + − − + − − +− + − − − + − − + − − +− + − − − + − − + − − +

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 50 (2012) 599–607 601

addressed gray scales between its dynamic range (0–255) usingMMIP at 530 nm wavelength. In this experiment, 6 different SOP(Horizontal (H), vertical (V), þ451 linear (P), –451 linear (M), rightcircularly polarized (R) and left circularly polarized (L)) generatedby PSG are allowed to pass through the TN-LCSLM active area andtransmitted polarized light is analyzed using PSA. These 6 differentSOP generated and analyzed by PSG and PSA, respectively, can befound out by fixing polarization analyzer unit at a time and rotatingthe other to get a null intensity without sample in between. AnIMAGINGSOURCEs charge coupling device (CCD) with 1280�960square pixels each having size of 4.65�4.65 mm is used to recordthe transmitted intensity after PSA. Mueller matrix for theTN-LCSLM at any particular addressed gray scale can be calculatedby recording 36 intensity images as depicted in Fig. 2. In this figure,the first letter denotes the input polarization state and the secondletter denotes the measurement polarization state. For example, the‘‘HH’’ element represents an intensity image acquired with incidenthorizontal polarization (H) and analyzer horizontal polarization (H).

The calibration of MMIP is essential before calculating preciseMueller matrices for the TN-LCSLM. In order to verify this,Mueller matrix for air is first calculated. Fig. 3 shows Muellermatrix for air calculated using MMIP, which is very close to 4�4unit matrix. The central region of 300�300 pixels is selectedfrom these recorded intensity images and average value is usedfor further calculations. All 16 elements of the Mueller matrix arenormalized with respect to first element m00. Typical value oferror in each element was found to lie between 1 and 5%. Afterobtaining satisfactory results from this standard measurement,the set-up was used to record Mueller matrices for the TN-LCSLMinvestigated in this study. In our earlier study, we have char-acterized phase modulation of the LC2002 TN-LCSLM usingDigital Holography method with different combination of con-trast and brightness values and we have found that the selectionof contrast and brightness value is not very critical for theTN-LCSLM LC2002 modulation curve [11]. Thus, in our experi-ment to measure the Mueller matrix of the TN-LCSLM using MMIPand to characterize all polarimetric quantities, it is operated atmaximum contrast and brightness settings of value ‘255’. Thedynamic range of the TN-LCSLM addressed gray scale value isequally divided into 16 intervals and 36 polarization images arerecorded at each of 17 addressed gray scale values using MMIP.Fig. 4 shows the spatial uniformity of the Mueller matrix elementscalculated at gray scale value ‘128’ addressed on the TN-LCSLMusing MMIP. Since the value of Mueller matrix elements over anarea of 300�300 pixels is uniform, the average value of Muellermatrix elements is taken for further measurements. The variation

in values of all 16 elements of Mueller matrix measured for theTN-LCSLM with respect to addressed gray scale within itsdynamic range is shown in Fig. 5. It is evident from this figurethat below gray scale value of ‘90’ there is much variation invalues of Mueller matrix elements and above this gray scale valueMueller matrix elements remain constant.

3. Lu–Chipman polar decomposition

After modeling the TN-LCSLM with Mueller matrices forseveral addressed gray scale values, it is important to characterizepolarimetric properties such as diattenuation, polarizance, retar-dance and depolarization associated with it. The informationabout these different polarimetric properties is embedded withinMueller matrices modeled for the TN-LCSLM and can be separatedfrom one another using the Lu–Chipman polar decompositionalgorithm. This polar decomposition of Mueller matrix allows oneto obtain sequence of three 4�4 matrices factors: a diattenuator,followed by a retarder, then followed by a depolarizer [12]. In ourwork, we have used the Lu–Chipman polar decomposition algo-rithm to separate out different polarimetric properties frommeasured 17 Mueller matrices modeled for the TN-LCSLMaddressed to different gray scales within its dynamic range.

The polarization state of an incident light passing throughnondepolarizing element can be changed by either change inamplitude 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 bechanged by depolarizing element as stated earlier. For polari-metric characterization of the TN-LCSLM, it is important tomeasure these polarimetric properties and their effect on incidentpolarized light. To analyze whether there exists any amount ofdepolarization within the TN-LCSLM, it is necessary to observe thediattenuation and polarizance present in the TN-LCSLM from themodeled Mueller matrices. In particular, a nondepolarizing Muel-ler matrix shows the equivalence of the magnitude of thediattenuation and the polarizance [12].

Diattenuation changes the intensity transmittance of theincident 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 calculatedusing the first row of Mueller matrix M, which determinesintensity transmittance:

D¼1

m00

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim2

01þm202þm2

03

qð1Þ

D¼1

m00

m01

m02

m03

0B@

1CA ð2Þ

Here, D denotes the magnitude of diattenuation vector D andmab (a,b¼0,1,2,3) represents an element of Mueller matrix.Another polarimetric quantity of interest is polarizance presentwithin the TN-LCSLM, which is the measure of degree of polar-ization in transmitted light when unpolarized or natural lightincident on it. Mathematically, polarizance value P with vector P

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.


can be calculated from first column of Mueller matrix M as

P¼1

m00

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim2

10þm220þm2

30

qð3Þ

P ¼1

m00

m10

m20

m30

0B@

1CA ð4Þ

Thus, the first column vector of Mueller matrix M gives themeasure of polarizance. It should be noted here that value ofthese two polarimetric quantities varies between 0 and 1. Fig. 6shows the diattenuation and polarizance variation calculatedusing Eqs. (1) and (3) from the Mueller matrices modeled forthe TN-LCSLM with respect to addressed gray scale. Since themagnitude of the diattenuation and polarizance values is differ-ent, it is apparent from this figure that there exists depolarizationwithin the TN-LCSLM below gray scale value ‘90’. Thus, the Lu–Chipman polar decomposition of Mueller matrices M modeled forthe TN-LCSLM must be carried out considering it as depolarizingelement as follows [12]:

M¼MDMRMD ¼1 DT

P m

" #ð5Þ

In Eq. (5), MD, MR and MD are depolarization, retarder anddiattenuator Mueller matrices deduced from total Mueller matrixM, respectively. Also, m is sub matrix of original Mueller matrixM modeled for the TN-LCSLM having no contribution from

diattenuation D and polarizance P. To separate all polarimetricproperties from Mueller matrix M, symmetric diattenuator matrixMD is first calculated using Eqs. (6) and (7) as follows:

MD ¼1 DT

D mD

!ð6Þ

mD ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffi1�D2

pIþð1�

ffiffiffiffiffiffiffiffiffiffiffiffiffi1�D2

pÞDD

Tð7Þ

Here, DT represents the transpose of diattenuation vector given byEq. (2), mDis sub matrix of diattenuator matrix MD given by Eq. (7)in terms of diattenuation unit vector D and 3�3 identity matrix I.Now a new Mueller matrix M0 is defined based on total Muellermatrix M as

M0 ¼MM�1D ð8Þ

It should be noted here that this new Mueller matrix M0has nocontribution from the diattenuation but consists of both retar-dance and depolarization. The new Mueller matrix M0 in terms ofdepolarization Mueller matrix MD and retardance Mueller matrixMRcan be rewritten as

M0 ¼MDMR ¼1 0T

PD mD

" #1 0T

0 mR

" #¼

1 0T

PD mDmR

" #¼

1 0T

PD m0

" #

ð9Þ

PD ¼P�mD

1�D2ð10Þ

-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

0 50 100 150 200 250

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

0 50 100 150 200 250

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

0 50 100 150 200 250

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 2500.00

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.


In Eq. (9), 0T represents the of zero column vector transposeand PD is a vector given by Eq. (10). Also, m0 represents sub matrixof Mueller matrix M0. Now, if l1, l2 and l3 are the eigenvalues of

m0ðm0ÞT , then mD, the sub matrix of depolarization Mueller matrixMD, can be obtained by

mD ¼ 7 ½m0ðm0ÞTþðffiffiffiffiffiffiffiffiffiffil1l2

pþ

ffiffiffiffiffiffiffiffiffiffil2l3

pþ

ffiffiffiffiffiffiffiffiffiffil3l1

pÞI��1

�½ðffiffiffiffiffil1

pþ

ffiffiffiffiffil2

pþ

ffiffiffiffiffil3

pÞm0ðm0ÞTþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffil1l2l3

pI� ð11Þ

In Eq. (11), negative sign is used with equation if determinantof m0 is negative. Otherwise, positive sign is applied. The depolar-ization conceded by the TN-LCSLM can be calculated usingEq. (11) and is defined as the coupling of completely polarizedlight into partially polarized light. Thus, higher value depolariza-tion is the degradation of completely polarized light and isconsidered as problem for modern displays. Finally, the retar-dance Mueller matrix can be calculated using

MR ¼M�1D M0 ¼

1 0T

0 mR

" #ð12Þ

In Eq. (12), mR represents the sub matrix of retardance Muellermatrix MR. This sub matrix has all information about retardancepresent in the TN-LCSLM and is defined as the phase differenceadded between two incident orthogonal polarizations. The polari-metric quantities such as retardance R and depolarization Dpresent in the TN-LCSLM can be calculated using relations givenby Eqs. (13) and (14), respectively:

R¼ cos�1 TrðMRÞ

2�1

� �ð13Þ

0 50 100 150 200 2500.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

Dep

olar

izan

ce (Δ

)


Δ

0 50 100 150 200 2500.5

1.0

1.5

2.0

2.5

3.0

3.5

Ret

arda

nce

(R)


R

Fig. 7. Retardance (R) and depolarization (D) measured with respect to addressed gray scale from modeled Mueller matrices for HOLOEYE LC2002 TN-LCSLM.


D¼ 1�9TrðmDÞ9

3ð14Þ

Here, Tr represents the trace of a matrix. The variation of thesepolarimetric quantities with respect to addressed gray scalecalculated using relations (13) and (14) is shown in Fig. 7. Itshould be noted here that retardance and depolarization shown inFig. 7 calculated from their respective Mueller matrices consist ofboth linear and circular constituents and may have differentvalues for different states of polarizations. Since depolarizationis important polarimetric property of the TN-LCSLM, its variationwith respect to different state of polarized light is discussed in thenext section using the Mueller–Stokes formalism.

4. Mueller–Stokes formulation

In the last section, Lu–Chipman polar decomposition is used toseparate three 4�4 Mueller matrices for different polarimetricquantities such as diattenuation, retardance and depolarizationfrom the modeled Mueller matrices for the TN-LCSLM evaluatedusing MMIP. The variation of all polarimetric quantities ismeasured with respect to addressed gray scales and it is foundthat there exists�15% depolarization within the TN-LCSM at alladdressed gray scales. However, it is crucial to study the effect ofthis existing depolarization within the TN-LCSLM on differentincident polarizations using the Mueller–Stokes formalism. TheMueller–Stokes formalism is generally used for experimentaldetermination of optical devices polarization behavior. In theMueller–Stokes combined formalism, incident polarized light ismathematically represented using 4�1 Stokes vector S and itsinteraction with optical device modeled using Mueller matrix M isanalyzed in terms of exiting Stokes vector S0 (¼M.S).

In our observation, six different incident SOPs in terms of Stokesvector (SH¼[1,1,0,0], SV¼[1,�1,0,0], SP¼[1,0,1,0], SM¼[1,0,�1,0],SR¼[1,0,0,1] and SL¼[1,0,0,�1]) are multiplied with differentmeasured Mueller matrices modeled for the TN-LCSLM at 17different addressed gray scales to evaluate exiting Stokes vector.The exiting Stokes vector bears the information about incidentpolarization altered by the TN-LCSLM and exiting light polarizationstate. Fig. 8 shows the variation in values of exiting Stokesparameters calculated for six different incident polarizationswith respect addressed gray scale. The variation in exiting Stokes

parameters for incident polarization clearly shows that theTN-LCSLM acts as polarization sensitive device, which alters anyincident polarization depending upon its polarization state andaddressed gray scale value. It is evident from Fig. 8(e) and (f) thatthere is least change in numerical values of exiting Stokes para-meters calculated for addressed gray scales on the TN-LCSLM whenincident polarization is circularly polarized light in comparison toexiting Stokes parameters calculated for other incident linearlypolarization states.

The degree of polarization (DoP) for exiting polarization statecan also be calculated with the help of the Mueller–Stokesformalism. The DoP for an electromagnetic wave is defined as aquantity to describe the portion of it, which is polarized and givenin terms of Stokes parameters S0, S1, S2 and S3 as

DoP¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiS2

1þS22þS2

3

qS0

ð15Þ

Fig. 9 shows the DoP measured for six different existing Stokesvectors calculated using the Mueller–Stokes formalism withrespect to addressed gray scale on the TN-LCSLM. It can be clearlyseen from Fig. 9 that incident right and left circularly polarizedlight are most affected due to depolarization existing within theTN-LCSLM. There is �30%–40% degradation in polarization ofincident circularly polarized light after passing through the TN-LCSLM. This observation suggests that incident circularly polar-ized light passes through transmissive TN-LCSLM with reduceddegree of polarization and having least change in its originalpolarization 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 theTN-LCSLM without any change in its polarization state for alladdressed gray scale values within its dynamic range. Eigenpo-larization states are important for phase-mostly modulation ofthe 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, opticaltweezers, 3D holographic display, adaptive optics, etc. From ourprevious observations using the Mueller–Stokes formalism anddifferent incident SOP, it has been found that incident circularlypolarized light (either right circularly or left circularly) fallsclose as an eigenpolarization state. Also it is reported by many

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

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1.0

SSSS

Sto

kes

Par

amet

ers

Addressed grey scale on SLM0 50 100 150 200 250 300

-1.0

-0.8

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0 50 100 150 200 250 300-1.0

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1.0 SSSS

Sto

kes

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amet

ers


-1.0

-0.8

-0.6

-0.4

-0.2

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Sto

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ers


SSSS

SSSS

Fig. 8. Measured exiting Stokes parameters when incident Stokes polarization vector is (a) horizontal, (b) vertical, (c) þ451 polarized, (d) �451 polarized, (e) right

circularly polarized and (f) left circularly polarized.


0 50 100 150 200 2500.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

HVPMRL

Deg

ree

of P

olar

izat

ion

(DoP

)


Fig. 9. Degree of polarization (DoP) calculated for six different exiting polarization

states through transmissive HOLOEYE LC2002 TN-LCSLM measured with respect

to addressed gray scale.

0-0.5

0.0

0.5

1.0

Sto

kes

para

met

er v

alue


SSSSSSSS

50 100 150 200 250

Fig. 10. Comparison of exiting Stokes parameters measured for Incident Stokes

vector SI¼[1,�0.08,�0.14,�0.28] using the Mueller–Stokes formulation and

experimental method (Solid line represents numerical results and dotted line

represents experimental results).


researchers that elliptically polarized light acts as eigenpolariza-tion states for LCSLMs [1,6]. With the help of the Mueller–Stokesformalism, eigenpolarization states can be mathematicallyderived using Mueller matrices modeled for the TN-LCSLM atdifferent addressed gray scales. Since eigenpolarization states donot change their polarization state after passing through the TN-LCSLM, this refers that exiting Stokes parameters should remainconstant for all addressed gray scales on it. Mathematically, theinput Stokes parameters of a vector are varied in such a way thatafter its interaction with modeled Mueller matrices of the TN-LCSLM measured for different addressed gray scales, it shouldgive rise to uniform exiting Stokes parameters.

In our observation, the input Stokes vector SI¼[1.0, �0.08,�0.14, �0.28] behaves as an eigenpolarization state since itproduces nearly uniform exiting Stokes parameters. This inputStokes vector has azimuth angle of 30.131 and 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 experimentallyusing quarter wave plate in front of polarizer to generate different

states of polarization and then transmit it through active area ofthe TN-LCSLM to record intensity images (IH, IV, IP, IM, IR and IL).These recorded intensity images for different addressed gray scalevalues are then processed to give exiting Stokes parameter [6].Another input Stokes vector SI¼[1.0, 0.08, 0.14, 0.28] havingequal and opposite ellipticity angle to first incident eigenpolar-ization vector also behaves as eigenpolarization giving constantexiting Stokes parameters for all addressed gray scale valuewithin its dynamic range. Fig. 10 shows that the exiting Stokesparameter calculated experimentally for different addressed grayscale on the TN-LCSLM are in good agreement with the Stokesparameter calculated heuristically using the Mueller–Stokesformalism.

5. Conclusion

In summary, this paper presents the polarimetric character-ization of the transmissive TN-LCSLM (HOLOEYE 2002) using theMueller matrix imaging polarimeter (MMIP). Mueller matrices forthe TN-LCSLM are calculated for 17 different addressed gray scalevalues within its dynamic range. Polarimetric quantities such asdiattenuation, retardance and depolarization are evaluated frommodeled Mueller matrices using Lu–Chipman polar decomposi-tion of Mueller matrices for the TN-LCSLM in terms of 4�4Mueller matrices and their variation with respect to addressedgray scale on the TN-LCSLM is studied. It has been found thatthere exists �15% depolarization within the TN-LCSLM for alladdressed gray scale values and the effect of this on series ofpolarizations is studied using the Mueller–Stokes formalism. Theexisting 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 lightpasses through transmissive TN-LCSLM with reduced degree ofpolarization and having least change in its original polarizationstate 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–Stokesformalism, which shows nearly uniform exiting Stokes parameterfor all addressed gray scale values on the TN-LCSLM. Thus, tooperate this TN-LCSLM in the phase-mostly modulation mode,quarter wave plate must be added after polarizer in order toincident elliptically polarized light on it. Also in the phase-mostlymodulation mode, the intensity modulation provided by the TN-LCSLM should be constant. Since Mueller matrix calculus does notgive any information regarding phase, numerical search of eigen-polarization states using the Mueller–Stokes formalism rendersessential information to employ TN-LCSLM in phase-mostlymodulation mode.

Acknowledgment

We thank the Optics and Photonics Society of Singapore andthe Nanyang Technological University for their support.

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