## Measurement of the pretilt angle and the cell gap of nematic liquid crystal cells by heterodyne interferometry

Optics Express, Vol. 17, Issue 16, pp. 14143-14149 (2009)

http://dx.doi.org/10.1364/OE.17.014143

Acrobat PDF (216 KB)

### 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 OSA

## 1. Introduction

1. T. J. Scheffer and J. Nehring, “Accurate determination of liquid crystal tilt bias angles,” J. Appl. Phys. **48**(5), 1783–1792 (1977). [CrossRef]

1. T. J. Scheffer and J. Nehring, “Accurate determination of liquid crystal tilt bias angles,” J. Appl. Phys. **48**(5), 1783–1792 (1977). [CrossRef]

2. B. L. Van Horn and H. H. Winter, “Analysis of the conoscopic measurement for uniaxial liquid-crystal tilt angles,” Appl. Opt. **40**(13), 2089–2094 (2001). [CrossRef]

*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

3. S. J. Hwang, S. T. Lin, and C. H. Lai, “A novel method to measure the cell gap and pretilt angle of a reflective liquid crystal display,” Opt. Commun. **260**(2), 614–620 (2006). [CrossRef]

4. I. Dahl, “How to measure the Mueller matrix of liquid-crystal cells,” Meas. Sci. Technol. **12**(11), 1938–1948 (2001). [CrossRef]

## 2. Principles

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

*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

1. T. J. Scheffer and J. Nehring, “Accurate determination of liquid crystal tilt bias angles,” J. Appl. Phys. **48**(5), 1783–1792 (1977). [CrossRef]

*α*is the pretilt angle,

*d*is the cell gap,

*λ*is the wavelength of incident light,

*φ*is the incident angle, and

*n*and

_{e}*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

*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

*I*is the test signal, and

_{t}*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}*I*and

_{t}*I*, are sent to the lock-in amplifier. Therefore, the phase retardation

_{r}*ϕ*can be accurately measured.

*φ*and -

*φ*can be written as

*α*can be obtained:

*α*can be estimated by numerical analysis when the extraordinary and ordinary refractive indices,

*n*and

_{e}*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

_{1}—a homogeneously aligned cell (with a spacer diameter of 5.7

*μ*m) containing E7 whose

*n*= 1.5216 and

_{o}*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

*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°.

*ϕ*, Δ

_{φ}*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.

*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

*n*and

_{e}*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

## References and links

1. | T. J. Scheffer and J. Nehring, “Accurate determination of liquid crystal tilt bias angles,” J. Appl. Phys. |

2. | B. L. Van Horn and H. H. Winter, “Analysis of the conoscopic measurement for uniaxial liquid-crystal tilt angles,” Appl. Opt. |

3. | S. J. Hwang, S. T. Lin, and C. H. Lai, “A novel method to measure the cell gap and pretilt angle of a reflective liquid crystal display,” Opt. Commun. |

4. | I. Dahl, “How to measure the Mueller matrix of liquid-crystal cells,” Meas. Sci. Technol. |

5. | M. H. Smith, “Method and apparatus for determining liquid crystal cell parameters from full Mueller matrix measurements,” US patent No. 7218398 (2007). |

**ToC Category:**

Optical Devices

**History**

Original Manuscript: May 18, 2009

Revised Manuscript: July 20, 2009

Manuscript Accepted: July 22, 2009

Published: August 3, 2009

**Citation**

Kun-Huang Chen, Wei-Yao Chang, and Jing-Heng Chen, "Measurement of the pretilt angle and the cell gap of nematic liquid crystal cells by heterodyne interferometry," Opt. Express **17**, 14143-14149 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-16-14143

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### References

- T. J. Scheffer and J. Nehring, “Accurate determination of liquid crystal tilt bias angles,” J. Appl. Phys. 48(5), 1783–1792 (1977). [CrossRef]
- B. L. Van Horn and H. H. Winter, “Analysis of the conoscopic measurement for uniaxial liquid-crystal tilt angles,” Appl. Opt. 40(13), 2089–2094 (2001). [CrossRef]
- S. J. Hwang, S. T. Lin, and C. H. Lai, “A novel method to measure the cell gap and pretilt angle of a reflective liquid crystal display,” Opt. Commun. 260(2), 614–620 (2006). [CrossRef]
- I. Dahl, “How to measure the Mueller matrix of liquid-crystal cells,” Meas. Sci. Technol. 12(11), 1938–1948 (2001). [CrossRef]
- M. H. Smith, “Method and apparatus for determining liquid crystal cell parameters from full Mueller matrix measurements,” US patent No. 7218398 (2007).

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