Cell parameter measurement of a twisted nematic liquid crystal device using interferometric polarimeter under normal incidence

Meng-Han Liu; Wen-Chuan Kuo; Hsiang-Chun Wei; Chien-Chung Tsai; Chih-Jen Yu; Bau-Jy Liang; Chien Chou

doi:10.1364/OE.18.008759

1. Introduction

To characterize optically the cell parameters of a liquid crystal device, such as cell gap, pretilt angle, twisted angle, rubbing angle, and phase retardation as shown in Fig. 1 , assessing the performance of the liquid crystal display quantitatively is essential. Generally, there are two categories on cell parameter measurement: the crystal rotation method and the polarimetric method [1–4]. However, both methods can only partly measure the five cell parameters of a twisted nematic liquid crystal device (TNLCD). The crystal rotation method requires the tilting of the TNLCD to compensate for the pretilt angle when using a monochromatic light source [1,2], while the polarimetric method requires a multiple-wavelength illumination under normal incidence and an arrangement with an optical compensator in the polarizer-sample-compensator-analyzer (PSCA) configuration [3]. Therefore, a calibration of the compensator serving as a quarter wave phase retardation on different wavelengths is required during the measurement [4]. This causes not only a slow response but also a large uncertainty in measurement. The dispersion of the tested TNLCD also produces a phase error. Recently, a spectroscopic ellipsometer or an interferometric polarimeter has been proposed that is capable of properly detecting transparent liquid crystal cell parameters [5,6]. However, these methods can only partly measure the five cell parameters. Tsai et al. [7] proposed the phase-sensitive heterodyne interferometer to measure the cell parameters of TNLCD, in which a frequency stabilized He-Ne laser is used under normal incident while the tested TNLCD is rotated continuously along the z axis (laser beam direction) during the phase measurement. However, the cell parameter pretilt angle is not available because the transfer matrix of the TNLCD is based on Yeh and Gu’s transfer matrix [8], in which a small pretilt angle is assumed.

Fig. 1 The test TNLCD cell configuration. Φ: the twist angle; Γ: untwisted phase retardation; α: rubbing angle, where i is the director of rubbing in and o is the direction of rubbing out; d: cell gap of two glass substrates; θ: pretilt angle of TNLCD.

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Recently, Wei et al. [9] proposed an amplitude-sensitive heterodyne polarimeter (ASHP), which can obtain the cell parameters in terms of the ratio of the decoded amplitudes of detected heterodyne signals and a quarter wave plate (QWP) in the signal arm of the interferometer is set in front of tested TNLCD and is continuously rotated during measurement. This setup not only improves the speed of measurement but also opens the possibility of a two-dimensional (2-D) distribution of cell parameters at the same time. However, the pretilt angle is not available either because the ASHP is also based on Yeh and Gu’s transfer matrix of TNLCD. Potentially, the 2-D distribution of cell parameters is possible when a multiple-channel amplitude demodulator is available. The features of ASHP at normal incidence, single wavelength illumination, and high-speed measurement on cell parameters can be performed. In this study, we extend the ASHP capability to enable the measurement of all five cell parameters simultaneously by integrating Yeh and Gu’s transfer matrix with Lien’s transfer matrix of TNLCD under the condition of single wavelength illumination and normal incidence. In comparison with the conventional methods, this extended ASHP not only measures all the five cell parameters of TNLCD at the same time but also provides a simple setup and a fast response. In this paper, the working principle of the extended ASHP is described, and all five cell parameters of the TNLCD are determined by a single experiment wherein a rotating QWP introduces the polarization modulation. As the pretilt angle is not explicitly shown in Yeh and Gu’s transfer matrix, in which only the twist angle, rubbing angle, and untwisted phased retardation are available. However, using Lien’s transfer matrix simultaneously can provide the pretilt angle and cell gap explicitly at a range limited at 0° to 30° [10]. Once the rubbing angle and twist angle are obtained from Yeh and Gu’s transfer matrix, and the two transfer matrices are integrated together, then all five cell parameters of TNLCD can be obtained numerically and simultaneously. The experimental demonstration is shown in Section 3, which verifies the applicability of this method. Finally, 2-D distributions of all five cell parameters of the TNLCD are measured by translating the TNLCD precisely. The discussions and conclusions are provided in the last section.

2. Working principle

As we previously developed in ASHP [9], the p and s polarized heterodyne signals can be expressed as follows:

I_{p} = {| E_{p 1} \exp (i ω_{1} t) + E_{p 2}^{(o)} \exp [i (ω_{2} t + δ_{p 2}^{(o)})] |}^{2}

= E_{p 1}^{2} + {(E_{p 2}^{(o)})}^{2} + 2 E_{p 1} E_{p 2}^{(o)} \cos (Δ ω t + δ_{p 2}^{(o)}),

and

I_{s} = {| E_{s 1} \exp (i ω_{1} t) + E_{s 2}^{(o)} \exp [i (ω_{2} t + δ_{s 2}^{(o)})] |}^{2}

= E_{s 1}^{2} + {(E_{s 2}^{(o)})}^{2} + 2 E_{s 1} E_{s 2}^{(o)} \cos (Δ ω t + δ_{s 2}^{(o)}),

where

Δ ω = ω_{1} - ω_{2}

and we define X as

X \equiv \frac{E_{s}}{E_{p}} \exp [i (δ_{s} - δ_{p})] = | X | \exp (i δ),

where

| X | = E_{s} / E_{p}

, the ratio of amplitude of the s and p waves, and

δ = δ_{s} - δ_{p}

is their phase difference.

δ_{s}

and

δ_{p}

are the phases of s and p waves, respectively. Thus, the output state of polarization X ⁽ ^o ⁾ can be determined by the transfer matrix T, which is analytically associated with the input state of polarization X ⁽ ⁱ ⁾ by

X^{(o)} = | X^{(o)} | \exp (i δ^{(o)}) = \frac{t_{21} + t_{22} | X^{(i)} | \exp (i δ^{(i)})}{t_{11} + t_{12} | X^{(i)} | \exp (i δ^{(i)})},

T = M_{T N L C} M_{Q W P} = [t_{11} t_{12}; t_{21} t_{22}]

is the transmission transfer matrix of a tested module composed by a QWP and a TNLCD. From Eqs. (1) and (2), the amplitude ratio of the p and s polarized heterodyne signals can be expressed by

[\frac{I_{s a c}}{I_{p a c}}] = \frac{E_{s 1} E_{s 2}^{(o)}}{E_{p 1} E_{p 2}^{(o)}} = | X_{1} | | X_{2}^{(o)} |,

where I _sac and I _pac are defined as the intensities of s and p polarized heterodyne signals, respectively.

In the experiment, QWP is considered an elliptical wave plate, wherein the transmission transfer matrix M_QWP is described as

M_{Q W P} = [\begin{matrix} \cos^{2} β + \sin^{2} β e^{- i γ} & \sin β \cos β (1 - e^{- i γ}) e^{- i δ_{f}} \\ \sin β \cos β (1 - e^{- i γ}) e^{i δ_{f}} & \sin^{2} β + \cos^{2} β e^{- i γ} \end{matrix}],

where β is the fast axis angle of the QWP with respect to the x axis; γ refers to the phase retardation between fast and slow elliptical eigen-polarizations; and

δ_{f}

is the phase difference between x and y components in each elliptical eigen-polarization [11–13].

Generally, the transmission transfer matrix of TNLCD, M_TNLC , can be written as

M_{T N L C} = [\begin{matrix} A & B \\ - B * & A * \end{matrix}] = [\begin{matrix} a_{1} + i a_{2} & b_{1} + i b_{2} \\ - (b_{1} - i b_{2}) & a_{1} - i a_{2} \end{matrix}] .

When we use Yeh and Gu’s matrix, the pretilt angle of the TNLCD is implicit, whereas the transfer matrix can be expressed as [8]

M_{Y e h} = [\begin{matrix} a_{1} + i a_{2} & b_{1} + i b_{2} \\ - (b_{1} - i b_{2}) & a_{1} - i a_{2} \end{matrix}]

= [\begin{matrix} p \cos Φ + q r \sin Φ - i q s \cos (2 α + Φ) & - p \sin Φ + q r \cos Φ - i q s \sin (2 α + Φ) \\ p \sin Φ - q r \cos Φ - i q s \sin (2 α + Φ) & p \cos Φ + q r \sin Φ + i q s \cos (2 α + Φ) \end{matrix}],

where

p = \cos χ, q = \sin χ, r = Φ / χ, and s = Γ / 2 χ

are defined and

χ = {[Φ^{2} + {(Γ / 2)}^{2}]}^{1 / 2}

. Φ is the twist angle, Γ is the untwisted phase retardation, and α is the rubbing angle of the tested TNLCD. Therefore, the characteristic parameters (Φ, Γ, α) can be obtained from Eq. (8) [9].

Similarly, if we want to measure the pretilt angle and cell gap of the TNLCD, Lien’s model becomes appropriate, and the Jones matrix is expressed as [10],

M_{L i e n} = [\begin{matrix} \cos α & - \sin α \\ \sin α & \cos α \end{matrix}] [\begin{matrix} a_{1} + i a_{2} & b_{1} + i b_{2} \\ - (b_{1} - i b_{2}) & a_{1} - i a_{2} \end{matrix}] [\begin{matrix} \cos α & \sin α \\ - \sin α & \cos α \end{matrix}],

in which

a_{1} = \frac{1}{\sqrt{1 + u^{2}}} \sin Φ \sin (\sqrt{1 + u^{2}} Φ) + \cos Φ \cos (\sqrt{1 + u^{2}} Φ),

a_{2} = \frac{u}{\sqrt{1 + u^{2}}} \cos Φ \sin (\sqrt{1 + u^{2}} Φ),

b_{1} = \frac{1}{\sqrt{1 + u^{2}}} \cos Φ \sin (\sqrt{1 + u^{2}} Φ) - \sin Φ \cos (\sqrt{1 + u^{2}} Φ),

b_{2} = \frac{u}{\sqrt{1 + u^{2}}} \sin Φ \sin (\sqrt{1 + u^{2}} Φ),

where

u = (π d / λ Φ) [n_{e f f} (θ) - n_{o}]

and

n_{e f f} (θ) = {n_{e} / {1 + [{(n_{e} / n_{o})}^{2} - 1] \sin^{2} θ}}^{1 / 2}

. Φ is the twist angle, d is the cell gap, and θ is the pretilt angle of the TNLCD. The characteristic parameters (Φ, d, θ) can then be properly derived from Lien’s matrix.

Theoretically, all five cell parameters (Φ, Γ, α, d, θ) of the TNLCD can be obtained by analyzing the experimental data using ASHP in conjunction with Yeh and Gu’s transfer matrix and Lien’s transfer matrix simultaneously. Note that this method is theoretically limited by the condition of $n_{e f f} (θ) ≃ n_{e}$ .

3. Experimental setup and results

Figure 2 shows the optical setup of ASHP, in which the driving frequencies of the acousto-optic modulators are ω ₁ = 80.0329 MHz and ω ₂ = 80.000 MHz, respectively, while the beat frequency is Δω = 32.9 kHz. Two digital voltmeters were adopted to demodulate the amplitude of p and s polarized heterodyne signals simultaneously. In the measurement, the ratio of the decoded amplitudes at 0.33% in stability in 20 min was performed (see Fig. 3 ). In the first step of the measurement, a single QWP was inserted into the signal arm of the interferometer and was rotated continuously at 360° on its azimuth angle β along the z-axis, which coincides with the laser beam. As the QWP is generally treated precisely as an elliptical wave plate in which two orthogonal elliptical polarizations are considered, the elliptically birefringent parameters $(γ, δ_{f})$ of each eigen elliptical polarization state of the QWP are defined, where γ is the phase retardation of the fast or slow eigen elliptical polarizations, and $δ_{f}$ is the phase difference between p and s components in each elliptical eigen polarization. The rotation speed of QWP is 3.33°/sec in this experiment. It took 5 minutes to acquire a complete set (360°) of the amplitude ratio of the p and s polarized heterodyne signals $[I_{s a c} / I_{p a c}]$ for single tested point on TNLCD. Figure 4 shows the measured result of the calculation of the ratio of the demodulated amplitudes ${[I_{s a c} / I_{p a c}]}_{Q W P}$ of Eq. (5). The experimental data agree well with the theoretical calculation by using the least squares curve fitting method. Thus, the parameters of the tested QWP are $(γ, δ_{f}) = (89
.5438 °,
0
.4133 °)$ in the experiment. This is required in the calibration procedures to ensure the accuracy of the TNLCD cell parameters measurement.

Fig. 2 Experimental setup: BS, beam splitter; AOM, acousto-optic modulator; M, mirror; A, polarizer; QWP, quarter-wave plate; TN-LC, twisted nematic liquid crystal cell; PBS, polarization beam splitter; D, photodetector; DVM, digital voltmeter; DSM, digital stepping motor; PC, personal computer.

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Fig. 3 Plot of the stability in this experiment. The total measurement times were 1,200 s. The $[I_{s a c} / I_{p a c}]$ ratio was maintained at 1 before either QWP or TNLCD could be inserted into the sample arm.

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Fig. 4 Least squares fitting between the experimental data (dotted curve) and theoretical calculation (solid curve), in which the QWP of $(γ, δ_{f}) = (89
.5438 °,
0
.4133 °)$ was measured.

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Meanwhile, in the second step, the tested TNLCD was inserted into the signal arm and was located after the QWP (see Fig. 2). The laser beam was arranged at normal incidence. The QWP was continuously rotated along the z-axis, while the TNLCD was fixed during the measurement. In Eq. (5), the ratio ${[I_{s a c} / I_{p a c}]}_{Q T N L C}$ is calculated again and is then fit with Yeh and Gu’s and Lien’s transfer matrices sequentially. The cell parameters of the TNLCD, (Φ, Γ, α) are obtained through Yeh and Gu’s matrix. We then substitute the given data α into the Lien’s matrix; thus, the cell parameters of (Φ, d, θ) are calculated at the same time. By adding these two sets of cell parameters together, the five cell parameters (Φ, Γ, α, d, θ) of the tested TNLCD are precisely obtained. Figure 5 shows the good agreement of this experiment with Yeh and Gu’s transfer matrix and Lien’s matrix, respectively. Table 1 shows the consistency of the experimental data with the given data provided by Chi-Mei Electro-optical Co, Tainan, Taiwan [14]. These results verify the ability of the association of ASHP with Yeh and Gu’s transfer matrix and Lien’s transfer matrix to measure all the five cell parameters of the TNLCD precisely and simultaneously.

Fig. 5 Measurement of the least squares fitting among the experimental data (dot), theoretical calculation applying Lien’s theoretical model (circle), and theoretical calculation applying Yeh and Gu’s theoretical model (plus sign) for the tested TNLCD cell.

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Table 1. The comparison of experimental data versus given values.

View Table

The 2-D distribution of the cell parameters of TNLCD can be carried out as well because ASHP belongs to amplitude-sensitive detection [9]. Figure 6 demonstrates the 2-D distribution of the five cell parameters of TNLCD by precisely translating the TNLCD along the X and Y directions at 1 μm on displacement.

Fig. 6 2-D distribution of the TNLCD cell: (a) twist angle Φ, (b) untwisted phase retardation Γ, (c) rubbing angle α, (d) cell gap d, (e) pretilt angle θ.

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4. Discussion and conclusion

In this research, we demonstrate the possibility of determining the five cell parameters simultaneously under normal incidence and by using a single wavelength laser beam. The advantages of this proposed method overcomes the difficulty caused by the slow measurement speed in conventional methods either by tilting the TNLCD at a single wavelength or by using multiple-wavelength illumination at normal incidence. Aside from measuring all the five cell parameters simultaneously, this method also simplifies the optical setup and improves the measurement speed by integrating our previously developed ASHP with the two types transfer matrices. Furthermore, the heterodyne detection technique ensures high signal to noise ratio (SNR) of the detected signal, indicating high detection sensitivity on the cell parameters. To our knowledge, this is the first time that all the five cell parameters of the TNLCD were simultaneously measured particularly under normal incidence and by using a single wavelength laser beam. In Fig. 6b, since the phase retardation is proportional to its thickness, a relative larger deviation happened in untwisted phase retardation between measured and given value is resulted by a small cell gap difference between measured and given value accordingly. Moreover, because ASHP belongs to an amplitude-sensitive interferometer, it is potential to measure the cell parameters at high speed. Then the 2-D distribution of cell parameter measurement by this proposed method can be considered to characterize TNLCD in the spatial-temporal modulation detection [15].

Acknowledgement

This research was supported by National Science Council of Taiwan through Grant # NSC 96-2221-E-010-015-001 and NSC 98-2221-E-182-064-MY3.

References and links

1. F. Nakano, M. Isogai, and M. Sato, “Simple method of determining liquid crystal tilt-bias angle,” Jpn. J. Appl. Phys. 19(10), 2013–2014 (1980). [CrossRef]

2. H. L. Ong, “Cell thickness and surface pretilt angle measurements of a planar liquid-crystal cell with obliquely incidence light,” J. Appl. Phys. 71(1), 140–144 (1992). [CrossRef]

3. Y. Zhou, Z. He, and S. Sato, “A novel method determining the cell thickness and twist angle of a twisted nematic cell by Stokes parameter measurement,” Jpn. J. Appl. Phys. 36(Part 1, No. 5A), 2760–2764 (1997). [CrossRef]

4. M. Kawamura, Y. Goto, and S. Sato, “A two-dimensional pretilt angle distribution measurement of twisted nematic liquid crystal cells using Stokes parameters at plural wavelengths,” Jpn. J. Appl. Phys. 43(2), 709–714 (2004). [CrossRef]

5. S. T. Tang and H. S. Kwok, “Transmissive liquid crystal cell parameters measurement by spectroscopic ellipsometry,” J. Appl. Phys. 89(1), 80–85 (2001). [CrossRef]

6. T. C. Yu and Y. L. Lo, “A novel heterodyne polarimeter for the multiple-parameter measurements of twisted nematic liquid crystal cell using a genetic algorithm approach,” J. Lightwave Technol. 25(3), 946–951 (2007). [CrossRef]

7. C. C. Tsai, C. Chou, C. Y. Han, C. H. Hsieh, K. Y. Liao, and Y. F. Chao, “Determination of optical parameters of a twisted-nematic liquid crystal by phase-sensitive optical heterodyne interferometric ellipsometry,” Appl. Opt. 44(35), 7509–7514 (2005). [CrossRef] [PubMed]

8. P. Yeh, and C. Gu, Optics of Liquid Crystal Displays (Wiley Interscience, New York, 1999), pp. 119–136.

9. H. C. Wei, C. C. Tsai, L. P. Yu, T. E. Lin, C. J. Yu, M. H. Liu, and C. Chou, “Two-dimensional cell parameters of twisted nematic liquid crystal with an amplitude-sensitive heterodyne ellipsometer,” Appl. Opt. 48(9), 1628–1634 (2009). [CrossRef] [PubMed]

10. A. Lien, “The general and simplified Jones matrix representations for the high pretilt twisted nematic cell,” J. Appl. Phys. 67(6), 2853 (1990). [CrossRef]

11. I. Scierski and F. Ratajczyk, “The Jones matrix of the real dichroic elliptic object,” Optik (Stuttg.) 68, 121–125 (1984).

12. Y. C. Huang, M. Chang, and C. Chou, “Effect of elliptical birefringence on the measurement of the phase retardation of a quartz wave plate by an optical heterodyne polarimeter,” J. Opt. Soc. Am. A 14(6), 1367–1372 (1997). [CrossRef]

13. C. Chou, Y. C. Huang, and M. Chang, “Polarized common path optical heterodyne interferometer for measuring the elliptical birefringence of a quartz wave plate,” Jpn. J. Appl. Phys. 35(Part 1, No. 10), 5526–5529 (1996). [CrossRef]

14. The cell parameter of TNLCD was provided by Chi-Mei Optoelectronics Co., Tainan, Taiwan, where the refractive indices are n_e = 1.7426, n_o = 1.5216; twisted angle is – 90°; rubbing angle is – 45°; cell gap is 4 μm; and pretilt angle is 3.2°.

15. I. Moreno, J. A. Davis, K. G. D’Nelly, and D. B. Allison, “Transmission and phase measurement for polarization eigenvectors in twisted-nematic liquid crystal spatial light modulators,” Opt. Eng. 37(11), 3048–3052 (1998). [CrossRef]

Cell parameter measurement of a twisted nematic liquid crystal device using interferometric polarimeter under normal incidence

Abstract

1. Introduction

2. Working principle

3. Experimental setup and results

4. Discussion and conclusion

Acknowledgement

References and links

Cited By

Figures (6)

Tables (1)

Equations (16)

Optics Express

	given values	experimental data
twist angle	– 90°	– 90.16 ± 0.24°^a
twist angle	– 90°	– 90.48 ± 0.73°^b
untwisted phase retardation	≈503°	480.69 ± 2.62°^a
rubbing angle	– 45°	– 45.16 ± 0.24°^a
cell gap	4.00 μm	3.92 ± 0.01 μm^b
pretilt angle	≈3.2°	3.28 ± 0.01°^b