Optics Communications 281 (2008) 4560–4565

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Optics Communications

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

Modified total intensity ratio methods for measuring cell gap of twistednematic liquid crystal cells

Yu-Lung Lo *, Tsung-Chih Yu, Li-Shuan Su, Ya-Shan HuangDepartment of Mechanical Engineering, National Cheng Kung University, Tainan, Taiwan

a r t i c l e i n f o a b s t r a c t

Article history:Received 23 November 2006Received in revised form 26 May 2008Accepted 4 June 2008

Keywords:Twisted nematic liquid crystalCell gapPolarization modulation

0030-4018/$ - see front matter � 2008 Elsevier B.V. Adoi:10.1016/j.optcom.2008.06.021

* Corresponding author. Tel.: +886 6 2757575x621E-mail address: loyl@mail.ncku.edu.tw (Y.-L. Lo).

This paper modifies the total intensity ratio method (TIRM) used to measure the cell gap of twisted nema-tic liquid crystal (TNLC) cells. Compared to the conventional TIRM in which a mechanical mechanism isused to physically rotate the polarizer, the modified TIRM methods presented in this study measure thetotal intensity ratio using a polarization rotation modulator. Two modified measurement methods areintroduced. In the first, the saw-tooth signal applied to the electro-optic (EO) modulator is used as a ref-erence signal in order to determine the polarization state of the measured signal. In the second method, abeam splitter and an additional quarter wave plate are added to the optical configuration. The quarterwave plate is adjusted such that a phase-matching condition is obtained between the reference and mea-sured signals. The experimental results confirm that the modified TIRM approaches yield comparableaccuracy to the conventional TIRM. Furthermore, in the proposed approaches, sufficient intensity signalsto determine the cell gap can be obtained in just 2 s.

� 2008 Elsevier B.V. All rights reserved.

1. Introduction

The characteristics of liquid crystal displays (LCDs) have beenextensively investigated in the past literatures. The cell gap of anLCD panel has a fundamental influence on its response time.Hence, to improve the image quality, it is essential that thecell gap is accurately controlled during the LCD manufacturingprocess.

Various cell gap measurement techniques have been proposed[1–7]. For example, the phase compensation method [1] uses aphase compensator and calculates the cell gap on the basis ofthe phase difference. However, this method does not readily lenditself to automation because repeated phase difference measure-ments are required and it is necessary to use two sources withdifferent wavelengths to resolve the problem caused by multiplesolutions. Wu and Xu [2] measured the reflection spectra of a cellat specific orientations to extract the cell gap and twist angle. Bytaking the ratio of reflected light intensity at two different pola-rizer angles, Zhu et al. [3] measured the cell gap of the reflectiveTNLC cell. The spectroscopic ellipsometry method [4] varies theangles of both the polarizer and the analyzer in order to locatethe position of null transmission at specific wavelengths.Although this method has the advantage of a simple measure-ment setup, it cannot be applied to the measurement of small cell

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gaps. Duran et al. [5] measured cell thickness and twist angle byuse of single wavelength light source for the analysis of the out-put polarization state. The spectral total intensity ratio method(TIRM) [6] uses a spectroscope with a halogen light source torecord the transmittance in all the wavelength regions. Thismethod obtains accurate measurements for small cell gaps, butis readily affected by external light sources and involves a compli-cated integration of the intensity ratio of the transmitted light intwo arbitrary wavelength regions. Recently, the single-wave-length TIRM [7] has been proposed as a means of determiningthe cell gap by measuring the integrated intensity ratio of thetransmitted light in two arbitrary polarizer angle regions. Com-pared with the spectral TIRM technique, the single-wavelengthTIRM is more straightforward and provides precise results forboth large and small cell gaps. However, this method involvesrotating the polarizer using a step motor and is therefore a timeconsuming process. Furthermore, it is necessary to ensure anabsolutely precise positioning of the light spot on the rotatingpolarizer to eliminate fluctuations in the detected intensity signalcaused by alignment errors.

Accordingly, the present study develops two modified TIRM ap-proaches in which the rotating polarizer driven by a mechanicalstep motor is replaced by a polarization rotation modulator [8]based on two quarter-wave plates (QWP) and an E–O modulatordriven by a saw-tooth driving signal. The experimental results con-firm that the proposed methods not only have comparable accu-racy of cell gap measurement to the conventional TIRM, but alsoreduce the measurement time.

Fig. 2. Periodic variation of measured intensity signal as rotation angle of polarizerincreases from 0� to 360�.

Y.-L. Lo et al. / Optics Communications 281 (2008) 4560–4565 4561

2. Basic theory of single-wavelength TIRM

Fig. 1 presents a schematic illustration of the optical setupused in the conventional single-wavelength TIRM [7]. As shown,the polarizer is rotated from 0� to 180� while the rubbing direc-tion of the TNLC cell and the rotational position of the analyzerare fixed at 0� and 45�, respectively. As described below, thecell gap is then calculated by processing the variation in theintensity signal produced by the photodetector as the polarizeris rotated.

The cell gap is measured by integrating the total intensity ratioof the transmitted light in two arbitrary polarizer angle regions [7].The transmittance, T, of the configuration shown in Fig. 1 is givenby

T ¼ ðcos c sin cÞMTNLCð/Þcos usinu

� ���������

2

ð1Þ

where u is the polarizer angle, c is the analyzer angle, and MTNLC(/)is the Jones matrix of the twisted nematic LC (TNLC) cell. The opticaltransmittance of the TNLC cell can be expressed by a Jones matrixwith elements defined in terms of the twist angle /, cell thicknessd, and wavelength-dependent birefringence Dn of the LC material[9], i.e.,

MTNLCð/Þ ¼ Rð�/Þ �cos X� i � C � sin X

2X / � sin XX

�/ � sin XX cos Xþ i � C � sin X

2X

" #ð2Þ

where R(/) is the rotation matrix, given by

Rð/Þ ¼cos / sin /

� sin / cos /

� �ð3Þ

and

C ¼ 2pdDnk

; X ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi/2 þ ðC=2Þ2

qð4Þ

in which

Dn ¼ neffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ ðne

noÞ2 � 1

h isin2 h

r � no; ð5Þ

In Eq. (5), h is the pre-tilt angle of the TLNC cell and ne and no arethe extraordinary and ordinary refractive indices, respectively, ofthe LC material. As the polarizer is rotated, the intensity of the sig-nal produced by the photodetector varies periodically, as illus-trated in Fig. 2.

The total intensity of the transmitted light as the polarizer ro-tates from 0� to 180� is given by

Itotal ¼Z 180�

0�Tdu ð6Þ

He-Ne Laserλ=632.8nm

Photo Detector

RotatingPolarizer

TN cell Analyzer

)180°0°( → )0°( (45°)

y x

z

Fig. 1. Schematic illustration of conventional TIRM setup.

However, the absolute value of the measured total intensitymay be different from the calculated value because of the absorp-tion by the optical elements and the effects of external light. Inother words, the measured and calculated transmitted light inten-sities are related as follows:Z u2

u1

Texpdu 6¼Z u2

u1

Tcaldu ð7Þ

where Texp[Tcal] is the measured [calculated] transmitted lightintensity. However, as shown in Eq. (8), the measured value of thetotal intensity ratio, Rt, is in good agreement with the calculated va-lue since the absorption rate is a constant [7].

Rt ¼Ru4u3

TexpduRu2u1

Texpdu¼Ru4u3

TcalduRu2u1

Tcalduð8Þ

Therefore, as the experimental Rt is equal to the calculated Rt,the theoretical cell gap corresponding to the calculated Rt couldbe considered as the cell gap of the measured TNLC cell.

However, the single-wavelength TIRM described above has anumber of disadvantages, principal of which is the requirementto accurately align the light spot on the center of the rotating pola-rizer. If great care is not taken to ensure an accurate alignment, theintensity of the photodetector signal, which is used to calculate thegap size, will fluctuate, and hence the measurement performanceof the optical configuration will be reduced.

3. Modified optical configurations for TIRM

To minimize the effects of alignment errors while simulta-neously reducing the measurement time, this study proposes twomodified TIRM configurations for measuring the cell gap of TNLCcells. In both cases, the mechanical mechanism used in the conven-tional TIRM to rotate the polarizer is replaced by a polarizationrotation modulation system incorporating two QWPs and an EOmodulator.

In the first modified optical setup, shown schematically in Fig. 3,a saw-tooth signal produced by the function generator is used todrive the EO modulator. In the proposed setup, the direction ofthe laser light propagation is specified as the +z-direction, whilethe horizontal direction and the vertical direction are specified asthe x- and y-directions, respectively. As shown in Fig. 3, the polar-ization rotation modulator comprises QWP #1 fixed at 0�, an E–O

P.D.

PolarizerQWP #1

EO Modulator

QWP #2

LCsample Analyzer

FunctionGenerator

Polarization Rotation Modulator

PC with DAQ card

He-Ne Laserλ=632.8nm

)0°( )45°( (90°)

Y

ZX

Fig. 3. Schematic illustration of first modified TIRM setup.

Fig. 4. Variation of measured intensity signal over time for different cell gaps.

4562 Y.-L. Lo et al. / Optics Communications 281 (2008) 4560–4565

modulator fixed at 45�, and QWP #2 fixed at 90�. The E–O modu-lator is driven by a saw-tooth wave signal with an angularfrequency of x = 2p/T (T is the signal period). When the laser lightpasses through this polarization rotation modulator, a sinewave signal is output from the photodetector. The electric field,Et, of the light incident on the photo-detector can be expressedas

Et ¼ Að90�Þ � Rð�aÞ �MTNLCð/Þ � RðaÞ � PRðxtÞ � Pð0�Þ � Einput

¼0 00 1

� ��

cos a � sinasin a cos a

� ��MTNLCð/Þ �

cos a sina� sina cos a

� �

�cosðxt

2 Þ � sinðxt2 Þ

sinðxt2 Þ cosðxt

2 Þ

!�

10

� �

ð9Þ

where PR(xt) is the Jones matrix of the polarization rotation mod-ulator, x is the angular frequency of the saw-tooth driving signal,R(a) and R(�a) are the rotation matrices, and a is the rubbing direc-tion of the TNLC cell. The electric field can be derived as

Et ¼1

2X ½ðX � /Þ � sinð� xt2 þ /þ XÞ þ ð�X � /Þ � sinðxt

2 � /þ XÞ��C8X ½cosð�xt

2 � 2a� /þ XÞ � cosðxt2 þ 2aþ /þ XÞ�

!

ð10Þ

The intensity of the light received at the photodetector can beexpressed as

It ¼ jEtj2 ð11Þ

From Eq. (11), it can be inferred that the intensity of the lightincident on the photodetector varies periodically as a function oftime when the polarization rotation modulator is driven by asaw-tooth wave signal with an angular frequency of x. SpecifyingDn, a and the twist angle of the TNLC cell as 0.099, 45� and 90�,respectively, and assuming a driving signal frequency of 1 kHz.Fig. 4 presents the simulated variation of the intensity signal overtime for cell gaps of 3.7, 3.9 and 4.2 lm, respectively. It is observedthat both the phase and the amplitude of the intensity signal areaffected by the cell gap of the TNLC cell.

In the conventional TIRM, integrating regions of the detectedsignal were determined by the angles of the rotating polarizer.However, the corresponding polarizer angles could not be knownfrom the detected signal by using EO modulator. The time regionswere therefore used as the integrating regions instead of the anglesof the polarizer in the proposed method. Therefore, the intensityratio defined in Eq. (8) should be written as

Rt ¼R t4

t3ItdtR t2

t1Itdt

ð12Þ

The detected signals obtained with and without the TNLC cell,respectively, are recorded by photo-detector as illustrated inFig. 5. The trough of the detected signal without TNLC cell is con-sidered as the time origin in the experiment. Deciding time origin,a decision can be made as to the time regions over which the inten-sity of the measured signal should be integrated in order to calcu-late the cell gap using the method described in Section 2.

As mentioned above, the detected signal without TNLC in thesystem should be acquired for the set of time origin. This meansthe detected signals without and with TNLC cell have to be ac-quired sequentially in each measurement in the configuration ofFig. 3. Accordingly, the second TIRM configuration is developedfor simultaneously acquiring the detected signals without and withTNLC cell in which the optical setup of Fig. 3 is extended to the con-figuration shown in Fig. 6. As shown in Fig. 6, a beam splitter is in-serted after the QWP #2. The light transmitted by the beam splitterfollows the same optical elements as that shown in Fig. 3. How-ever, the reflected light ray passes through a quarter-wave plate(QWP #3), a second analyzer (Analyzer 2) set at 90� and is incidenton a second photodetector (PD #2). The output signal from P.D. #2is then supplied to the original lock-in amplifier to serve as the ref-erence signal. Note that the function of QWP #3 is to provide anadjustment facility with which to tune the phase of the reference

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 10-3

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3

Time (s)

Inte

nsity

The measured signal with TNLC cell.

The measured signal without TNLC cell.

The saw-tooth driving signal

Fig. 5. Variation of measured intensity signals with and without inserted TNLC celland saw-tooth wave reference signal.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

0.33

0.34

0.35

0.36

0.37

0.38

0.39

0.4

0.41

0.42X: 0.2603Y: 0.4225

dΔn [ μ m]

Tot

al in

tens

ity r

atio

X: 0.3663Y: 0.4225

Fig. 7. Total intensity ratio of two integrating time ranges, (0–0.6T) and (0.2–0.4T),for a 90� TNLC cell under the condition of a = 45�.

Y.-L. Lo et al. / Optics Communications 281 (2008) 4560–4565 4563

signal to that of the measured signal when the TNLC cell is not in-serted, i.e., QWP #3 compensates for the non-linear effect of thebeam splitter [10].

The electric field of the light incident on P.D. #2 is given by

Er ¼ A2ð90�Þ � Q :W:ðeÞ � PRðxtÞ � Pð0�Þ � Einput

¼0 00 1

� ��

cos e � sin e

sin e cos e

� �� e�ip4 0

0 eip4

!�

cos e sin e

� sin e cos e

� �

�cosðxt

2 Þ � sinðxt2 Þ

sinðxt2 Þ cosðxt

2 Þ

!�

10

� �

ð13Þ

where e is the principal axis angle of QWP #3. Furthermore, theintensity of the light incident on P.D. #2 is expressed as:

It2 ¼ jEt2j2 ¼12þ 1

4cosð2eÞ cosðxt � 2eÞ ð14Þ

PolarizerQWP #1

EO Modulator

FunctionGenerator

Polarization Rotation

He-Ne Laserλ=632.8nm

)0°( )45°(

Y

ZX

Fig. 6. Schematic illustration of s

From Eq. (14), it is apparent that the phase shift of the referencesignal is dependent on the angle of rotation of QWP #3. Therefore,phase matching between the reference signal and the measuredsignal without TNLC can be obtained by rotating the quarter waveplate.

In both optical setups described above, the time origin can bedetermined by the trough of the reference signal (without TNLC).Fig. 7 shows the total intensity ratio Rt ¼ ð

R 0:4T0:2T ItdtÞ=ð

R 0:6T0 ItdtÞ as

a function of Dn � d from 90� TNLC cell with the entrance directorangle a of 45�. As can be seen in Fig. 7, the Rt value of 0.4225 cor-responds to two extracted d � Dn of 0.2063 lm and 0.3663 lmwhen the measured range of d � Dn is from 0 to 0.5 lm. This ambi-guity can be solved by changing the integrating regions of Rt. Forexample, in Fig. 8 while the case that Rt ¼ ð

R 0:4T0:1T ItdtÞ=ð

R 0:5T0 ItdtÞ,

the ratio value of 0.6812 corresponds to two d � Dn values of0.3405 lm and 0.3663 lm. As a result, if the measured values oftotal intensity ratio in the two sets of integrating ranges of(0–0.6T, 0.2–0.4T) and (0–0.5T, 0.1–0.4T) are 0.4225 and 0.6812,

P.D. 1

QWP 2

L.C.sample

Analyzer 1.

Lock-inAmplifier

Modulator

Analyzer 2.

B.S.

QWP 3

PC

(90°)

(90°)

P.D. 2

econd modified TIRM setup.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

0.61

0.62

0.63

0.64

0.65

0.66

0.67

0.68

0.69

X: 0.3663Y: 0.6812

dΔn [μm]

Tot

al in

tens

ity r

atio

X: 0.3405Y: 0.6812

Fig. 8. Total intensity ratio of two integrating time ranges, (0–0.5T) and (0.1–0.4T),for a 90� TNLC cell under condition of a = 45�.

Table 1Experimental results obtained for cell gap using first modified system

Sample Time

1 2 3 4 5 6 Average cell

4564 Y.-L. Lo et al. / Optics Communications 281 (2008) 4560–4565

respectively, the retardation value of the sample can therefore beconsidered as 0.3663 lm. Furthermore, if we extend the d � Dnrange as illustrated in Fig. 9, one measured Rt may correspond totwo or more d � Dn values. Nevertheless, the variations of Rt inthe two sets of integrating ranges in Fig. 9 have apparent differ-ence. Therefore the ambiguity in extracting the thickness of TNLCcan be solved by changing the integrating ranges as mentionedabove even the thickness range is extended.

4. Experimental setup and results

To verify the accuracy of the modified TIRM approaches, anexperimental study was performed using a empty cell (EHC .Co.Ltd) with a twist angle of 90�, a pre-tilt angle of 1�, and cell gapof 10 lm. The empty cell filled with E7 liquid crystal (Merck Co.),was used as sample A for the measurement. Another TNLC cell withno = 1.483, ne = 1.569 at 632.8 nm wavelength was also used assample B in the test. The designed values of twist angle / = 90�, cellgap d = 3.7 lm, and pre-tilt angle h ffi 4� in sample B were obtainedfrom the Chi-Mei Optoelectronics Co., Taiwan. The schematic dia-gram of the system configurations are shown in Figs. 3 and 6.

0 0.5 1 1.5 2 2.5

0.1

0.2

0.3

0.4

0.5

0.6

dΔn [μm]

Rt

Fig. 9. Total intensity ratio of two sets of integrating time ranges, (dash line: 0–0.5T,0.1–0.4T) and (solid line: 0–0.5T, 0.1–0.4T).

The light source was a frequency stabilized He–Ne laser (Model:SIOS SL 02/2) operating at a wavelength of 632.8 nm. The EO mod-ulator (Conoptics INC. 370) was driven by a saw-tooth wave signalwith a frequency of 1 kHz. The detected signals from photo-detec-tors were acquired by DAQ card. Because of the high modulationfrequency of the proposed method, each measurement can beaccomplished in 2 s by the second modified configuration. To ver-ify the repeatability and stability of the proposed approaches, thecell gaps of the TNLC samples were measured six times in every10 min using each optical setup.

The experimental results using the first optical configurationare presented in Table 1. It is found that this TIRM measurementmethod have deviations of ±0.027 lm and ±0.016 lm for sampleA and B, respectively.

The experimental results obtained by the second optical setupare presented in Table 2.

We compared the results with the cell gap values measured byStokes parameters method [11] and interferometric method asshown in Table 3.

When actuating the E–O modulator using a driving frequency of1 kHz, the electrical signal from the function generator and thephotoelectric signal from the photodetector are approximatelyin-phase. Therefore, there is no more than a slight difference inthe experimental results presented in Tables 1 and 2. Though theoptical setup of the first measurement configuration is morestraightforward than that of the second one, the second one ishighly recommended for the high frequency dynamical measure-ment. The measurement error of the proposed method can beattributed to slight misalignments in the optical setup and manu-facturing flaws in the optical components. Hence the more accu-rate alignment of the optical system, the more precise value ofLC parameter would be achieved.

In experiment, the twist angle and the pretilt angle were as-sumed to be specific values. As can be seen in Fig. 10, the deviationof pretilt angle has little influence on the measured results. How-ever, the deviation of twist angle would have more apparent influ-ence on the measurement error than that of pretilt angle. Fig. 11shows there is slight deviation between the two Rt-d � Dn curves

gap (lm)

A 10.069 10.124 10.139 10.134 10.075 10.093 10.10 ± 0.027B 3.735 3.685 3.723 3.727 3.699 3.725 3.72 ± 0.016

Table 2Experimental results obtained for cell gap using second modified system

Sample Time

1 2 3 4 5 6 Average cellgap (lm)

A 10.139 10.088 10.068 10.096 10.121 10.113 10.10 ± 0.020B 3.717 3.735 3.729 3.741 3.713 3.683 3.72 ± 0.015

Table 3Comparison of measured cell gap data

Sample Designed cellgap value (lm)

Stokes parametersmethod (lm)

Interferometricmethod (lm)

Proposedmethod(lm)

A 10 9.70 9.98 10.10B 3.7 3.61 – 3.72

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

dΔn [μm]

Rt

Fig. 10. Total intensity ratio of two different pretilt angles, (dot: 8.4�, solid line:3.4�).

0 0.5 1 1.5 2 2.5

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

dΔn [μm]

Rt

Fig. 11. Total intensity ratio of two different twist angle, (dash line: 89�, solid line:90�).

Y.-L. Lo et al. / Optics Communications 281 (2008) 4560–4565 4565

with twist angles of 90� and 89�. As can be seen in the figure, thelarger value of retardation is, the more apparent deviation can beobserved. The maximum deviation of d � Dn which is about±0.025 lm in the large retardation ranges of 2–2.5 lm corresponds

to the deviation of cell gap about ±0.1 lm for sample A. In most LCdisplay application, however, the retardation value of LC panel isusually the smaller than that of sample A. As for sample B, theretardation value is in the range of 0.35–0.45 lm. The correspond-ing cell gap deviation is about ±0.05 lm for sample B as the inac-curacy of the twist angle is at the level of ±1�.Therefore, theproposed method has better performance for small cell gap mea-surement than large one.

5. Conclusions

This study has presented and verified the modified TIRM opticalconfigurations for TNLC cell gap measurement. Both configurationsreplace the mechanical mechanism used to rotate the polarizer inthe conventional TIRM with a polarization rotation modulationsystem incorporating two QWPs and an E–O modulator.

In the experimental results, the cell gap measurement deviationabout 0.015 lm is obtained. Also, in our investigation, the pro-posed system has better measurement ability for the TNLC cellwith small retardation than with large retardation. Furthermore,compared to the conventional TIRM, which requires several sec-onds to measure the cell gap, the modified TIRM techniques pre-sented in this paper need only 2 s to acquire the signal andderive the cell gap value. The results of this study indicate thatthe modified TIRM methods have the potential to carry out on-lineTNLC cell gap measurement in the industrial manufacturing ofLCDs.

Acknowledgements

The current authors would like to acknowledge the financialsupport of Chi-Mei Optoelectonics (CMO) and also the assistanceof Dr. T.S. Lin and Dr. I.L. Ho in the CMO without whom this studycould not have been achieved. This research is also partially sup-ported by the National Science Council, Taiwan, under grantNSC96-2628-E-006-005-MY3.

References

[1] A. Lien, H. Takano, J. Appl. Phys. 69 (1991) 1304.[2] S.T. Wu, G. Xu, IEEE Trans. Electron. Dev. 47 (2000) 2290.[3] X. Zhu, W.K. Choi, S.T. Wu, IEEE Trans. Electron. Dev. 49 (2002) 1863.[4] S.T. Tang, H.S. Kwok, J. Appl. Phys. 89 (2001) 80.[5] V. Duran, J. Lancis, E. Tajahuerce, J. Appl. Phys. 97 (2005) 043101.[6] J.S. Gwag, S.H. Lee, K.Y. Han, J.C. Kim, T.H. Yoon, J. Appl. Phys. 41 (2002) L79.[7] J.S. Gwag, K.H. Park, G.D. Lee, T.H. Yoon, J.C. Kim, J. Appl. Phys. 43 (2004) L30.[8] Y.L. Lo, T.C. Yu, Opt. Commun. 259 (2006) 40.[9] P. Yeh, C. Gu, Optics of Liquid Crystal Displays, 121, Wiley, New York, 2000.

[10] Y.L. Lo, J.F. Lin, S.Y. Lee, Appl. Opt. 43 (2004) 6248.[11] Y. Zhou, Z. He, S. Sato, J. Appl. Phys. 36 (1997) 2760.