## Abstract

This work proposes a simple method, based on the crystal rotation technique and heterodyne interferometry, to simultaneously determine the pretilt angle and cell gap of nematic liquid crystal cells. When heterodyne light passes through a nematic liquid crystal cell, the phase retardation given by the characteristic parameters of the cell can be measured accurately by heterodyne interferometry. This phase retardation relates to the pretilt angle, cell gap, and angle of incidence on the cell. By using the measured phase retardations at two incident angles, the pretilt angle and cell gap of the nematic liquid crystal cell can be estimated by numerical analysis. This method is feasible, requiring only two incident angles and prior knowledge of two characteristic parameters—extraordinary and ordinary refractive indices of the liquid crystal. It is characterized by the advantages of simplicity of installation, ease of operation, high stability, high accuracy, and high resolution.

©2009 Optical Society of America

## 1. Introduction

Nematic liquid crystal cells are widely applied as electro-optical devices, such as liquid crystal displays (LCDs), spatial light modulators (SLMs) and liquid crystal lenses. The performance of the devices is dominated by the characteristic parameters of the nematic liquid crystal cells, including the pretilt angle, cell gap, twist angle, and birefringence. For example, both the pretilt angle and cell gap in LCDs affect the operation voltage and the response time, leading to a great impact on dynamic display. An uneven cell gap that causes defects of Mura not only degrades the image quality but also reduces the viewing angle. Therefore, the pretilt angle and cell gap should be monitored during the fabrication process to ensure the quality and performance of the products.

Several methods have been proposed for measuring the pretilt angle and cell gap of nematic liquid crystal cells [1–5]. The conventional crystal rotation method determines the pretilt angle according to the relationship between the light transmittance ratios and incident angles [1]. The conoscopic approach determines the pretilt angle by analyzing the displacement of interference fringes with a conoscope [2]. In these two methods, the stability of light source and the quality of fringe images characterize the accuracy and resolution of measurements. A limitation is that they can only measure the pretilt angles over a small range. Hwang *et al*. proposed a measuring method to determine the pretilt angle and cell gap by the voltage-dependent change in phase retardation of a nematic liquid crystal cell [3], which is particularly suitable for the measurement of cell gap. However, an additional measurement technique has to be adopted in advance when determining a large pretilt angle. A common disadvantage of both the crystal rotation method and Hwang’s method is that they are time-consuming when applied to multi-point examination of a large test area. Although the Mueller matrix polarimeter method is a mature technique for cell parameter measurement [4,5], it is still affected by the stability of the light source.

This work presents a simple method to simultaneously determine the pretilt angle and cell gap of nematic liquid crystal cells. It is based on the crystal rotation method and heterodyne interferometry. With only two incident angles and two material properties; i.e., the extraordinary and ordinary refractive indices, it can be applied more easily and rapidly. Furthermore, the introduction of common-path heterodyne interferometry ensures this method to be simple for implementation, highly stable, highly accurate, and of high resolution.

## 2. Principles

#### 2.1 Phase retardation of a nematic liquid crystal cell

In the crystal rotation method, a nematic liquid crystal cell is treated as a uniaxial crystal. Figure 1 depicts a beam of light transmitted through a nematic liquid crystal whose *p*- and *s*-polarized components are parallel and perpendicular to the incident plane (*x-z* plane), respectively. According to the law of refraction and the index ellipsoid equation, the phase retardation *ϕ* between the *p*- and *s*- polarizations of the transmitted light caused by a nematic liquid crystal can be written as [1]

where *α* is the pretilt angle, *d* is the cell gap, *λ* is the wavelength of incident light, *φ* is the incident angle, and *n _{e}* and

*n*are the extraordinary and ordinary refractive indices of the liquid crystal material, respectively. From Eq. (1), the phase retardation

_{o}*ϕ*is a function of

*α*,

*d*,

*n*,

_{e}*n*,

_{o}*φ*, and

*λ*.

#### 2.2 Measuring the pretilt angle and cell gap by heterodyne interferometry

Figure 2 shows the setup for measuring the pretilt angle and cell gap of a nematic liquid crystal cell. For convenience, the + *z*-axis is set as the direction of light propagation and the *x*-axis goes perpendicularly into the plane of the paper. A horizontally polarized (*x*-polarized) laser beam sequentially passes through a 45° polarizer (P) with respect to the *x*-axis and an electro-optic modulator (EO) (with its fast axis at 0° with respect to the *x*-axis) driven by a sawtooth wave with a frequency *ω*. The transmitted light, acting as a heterodyne light source with a frequency difference *ω* between the *p*- and *s*-polarized light, then passes through the nematic liquid crystal cell, mounted on a rotation stage with its rotation axis parallel to the *y*-direction, and a 45° analyzer (AN) with respect to the *x*-axis, and finally enters a photodetector (D). Due to the introduction of the 45° analyzer, the intensity of interference of the original *p*- and *s*-polarized light measured at D can be written as

where *I _{t}* is the test signal, and

*ϕ*is the phase retardation from the light that passes through the cell. On the other hand, the electric signal generated by the function generator (FG) is applied as the reference signal

*I*in the following form:

_{r}Both of these sinusoidal signals, *I _{t}* and

*I*, are sent to the lock-in amplifier. Therefore, the phase retardation

_{r}*ϕ*can be accurately measured.

According to Eq. (1), the difference in phase retardation between the incident angles of + *φ* and -*φ* can be written as

Furthermore, in accordance with Eqs. (1) and (4), a single variable function of pretilt angle *α* can be obtained:

Accordingly, the pretilt angle *α* can be estimated by numerical analysis when the extraordinary and ordinary refractive indices, *n _{e}* and

*n*, and incident angles ±

_{o}*φ*are known. In addition, the cell gap

*d*can be determined by substituting the obtained pretilt angle

*α*into Eq. (1).

## 3. Experimental results

To demonstrate the feasibility of the proposed method, three nematic liquid crystal cells with alignment layers rubbed in anti-parallel were measured. The specifications of these samples are as follows: (1) sample S_{1}—a homogeneously aligned cell (with a spacer diameter of 5.7 *μ*m) containing E7 whose *n _{o}* = 1.5216 and

*n*= 1.7426 @632.8 nm; (2) sample S

_{e}_{2}—a vertical aligned cell (with a spacer diameter of 6.0

*μ*m) consisting of MLC-6608 with

*n*= 1.4738 and

_{o}*n*= 1.5560 @632.8 nm; (3) sample S

_{e}_{3}—a liquid crystal cell with an intermediate pretilt angle in the range of 30–60° and a spacer diameter of 7.6

*μ*m, which was filled with MLC-2048 with

*n*= 1.495 and

_{o}*n*= 1.705 @632.8 nm. Note that all of the liquid crystals were obtained from Merck. A heterodyne light source with a 1 kHz frequency difference between

_{e}*p*- and

*s*-polarizations was used. The measurement system consisted of a 632.8 nm He-Ne laser, a 45° polarizer, and an electro-optic modulator (Model 4002, New Focus, Inc.) driven by a function generator (FG) and a high-voltage amplifier (HVA), as shown in Fig. 2. A high-resolution motorized rotation stage (Model SGSP-60-YAW-0B, Sigma Koki, Inc) with an angular resolution of 0.0025° was employed to regulate the orientation of the sample with its rotation axis parallel to the

*y*-direction. A lock-in amplifier with a phase resolution of 0.01° (Model SR830, Stanford Research System) was used for the measurement of the phase retardation, and a personal computer was applied to analyze the captured data. For convenience, the phase retardation was measured at

*φ*= -45° and

*φ*= 45°. Table 1 presents the experimental phase retardations as well as the estimated pretilt angle and cell gap. The reference data in Table 1 were obtained using a commercial Mueller matrix polarimeter (AxoScan™, Axometrics, Inc). The measured results are in good agreement with the reference data, demonstrating the capability of the proposed method.

## 4. Discussion

Considering the phase unwrapping, the phase retardation is closely associated with the birefringence Δ*n*, cell gap *d*, pretilt angle *α*, and incident angle *φ*. As such, when the diameter of ball spacers, type of liquid crystal, method of alignment and the direction of alignment of a liquid crystal cell are given, the relationships between the phase retardation and the incident angle for different pretilt angles can be obtained by using Eq. (1). For example, consider the liquid crystal cell S_{1}. Figure 3 plots the curves of phase retardation versus incident angle at various pretilt angles. Since sample S_{1} is a homogeneously aligned cell with a pretilt angle of about 1°~3°, the actual phase retardation could be obtained by adding 360° to the captured phase retardation recorded by the lock-in amplifier. Accordingly, based on the same principle, the proposed method is not pretilt angle-limited and can be used to measure cell gaps and pretilt angles from 0° to 90°.

From Eqs. (1) and (5), the errors of the pretilt angle |Δ*α*| and cell gap |Δ*d*| are

and

where Δ*ϕ _{φ}*, Δ

*ϕ*

_{± φ}and Δ

*φ*are the errors of the phase retardations of

*ϕ*and

_{φ}*ϕ*

_{± φ}, and the incident angle

*φ*, respectively. Considering the angular resolution of the lock-in amplifier, the second harmonic error and the polarization-mixing error in the experiment, |Δ

*ϕ*| = |Δ

_{φ}*ϕ*

_{± φ}|≅0.03°, can be estimated. Substituting these data and the experimental results into Eqs. (6)a) and (6b) yields the errors of pretilt angle

*α*and cell gap

*d*. Undoubtedly, these values vary with the experimental conditions. In the measurements taken from S

_{1}, S

_{2}, and S

_{3}, the errors of the pretilt angle |Δ

*α*| are calculated to be 1.2 × 10

_{-3}, 1.4 × 10

^{-3}, and 2.5 × 10

^{-3}°, respectively; the errors of the cell gap |Δ

*d*| are calculated to be 0.0073, 0.0070, and 0.0079

*μ*m, respectively.

From Eq. (6)a), Fig. 4 plots curves of error of pretilt angle |Δ*α*| versus pretilt angle from 0 to 90° for incident angles *φ* = ±45°, cell gap *d* = 5.7 *μ*m, and the liquid crystal E7. In these curves, |Δ*α*| is smaller than 0.008° at pretilt angles from 0° to 90°, except near 10°. Altering the incident angle while remaining other conditions unchanged, as shown in Fig. 5, the error of the pretilt angle |Δ*α*| can be smaller than 0.002° near *α* = 10° when the incident angle is set to *φ* = ± 14°. Additionally, the cell gap can also influence the error of |Δ*α*|. Figure 6 plots the curve of error of pretilt angle |Δ*α*| as a function of the cell gap with incident angles *φ* = ± 45° and a fixed pretilt angle *α* = 6.73° (tolerance of |Δ*d*| = 0.01 *μ*m at *α* = 6.73°) for the liquid crystal E7. The error of the pretilt angle |Δ*α*| can be kept less than 0.003° as long as the cell gap *d* exceeds 4 *μ*m. Accordingly, this method not only has an unrestricted measurement range of pretilt angles, but also enables sufficiently accurate results to be obtained with appropriate choices of incident angles.

## 5. Conclusions

This work demonstrates a simple method for simultaneously measuring the pretilt angle and cell gap of nematic liquid crystal cells, based on the crystal rotation method and heterodyne interferometry. The measurements can be effectively achieved by using only two angles of incidence with prior knowledge of only two material properties—*n _{e}* and

*n*. This method has no restriction on the range of pretilt angle; it can be implemented more easily and rapidly with the advantages of common-path heterodyne interferometry.

_{o}## Acknowledgements

The authors gratefully acknowledge financial support from the National Science Council of the Republic of China, Taiwan under Contract No. NSC 96-2221-E-035-049-MY2. They would also like to extend their gratitude to Professor Wei Lee of Chung Yuan Christian University for his editorial assistance in the revised manuscript.

## References and links

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