1. Introduction

Various techniques to evaluate liquid crystal (LC) cell parameters, such as the cell gap and pretilt angle techniques, have been demonstrated previously. These methods are based on intensity measurement by a conventional photometric ellipsometer, or polarimeter [1–4]. Recently, a multiwavelength light source was introduced into the optical setup that can successfully extend the system’s capabilities to measure the two-dimensional (2-D) distribution of cell parameters [5–7]; this involves a rotated polarizer and an analyzer that are arranged in conjunction with different compensators for different wavelengths. Kawamura et al. [7] used a broadband light source that was able to build up the 2-D pretilt angle and cell gap distributions of a twisted nematic LC (TNLC). Furthermore, Tang et al. [8] proposed a method that adopted spectroscopic ellipsometry for the measurement by the transmission of LC cell parameters. Another study by Ong [9] demonstrated measurement of the cell gap and the pretilt angle of a parallel-aligned LC (PALC) using oblique incidence; however, a compensator is required in this method in order to null the transmission of light intensity during the cell thickness measurement, while at the same time the variable oblique incidence angle is scanned so that the pretilt angle is obtained without using a compensator. As a result of this method, which uses a nulling technique for the transmission intensity, the detection sensitivity for the cell parameter measurement is improved. However, the insensitivity of the zero transmission intensity at the oblique incidence angle to the PALC causes some uncertainty in the cell parameter measurement and, in addition, the multiple reflections in the LC at small oblique incidence angles produce phase oscillation that affects the calculation of the cell parameters.

It is clear from the above that there are limitations to the measurement of cell parameters that result from adopting an oblique incidence technique. To circumvent this, an interferometric ellipsometer [10] using a range of large oblique-incidence angles (30°≤ϕ≤50°) at a single wavelength to measure the cell parameters of a PALC is proposed and experimentally demonstrated in this study. This approach belongs to the phase-sensitive detection method and focuses on measuring the phase retardation between the P and S waves in the PALC. The setup involves a common-path configuration for the interferometric ellipsometer, and this allows a common-phase noise rejection mode to be implemented that is immune to environmental disturbance and laser frequency noise. This ensures that the phase detection is better stabilized and that there is a high signal-to-noise ratio (SNR) for the measurements. At the same time, the detection sensitivity for the cell parameters at large oblique incidence angles is enhanced significantly and the effects of multireflection and spatial shifting of emerging beams from the PALC are avoided effectively by setting up a retro-reflected geometry in this interferometer. As a result, precise measurements of the cell parameters of the PALC are possible, and the experimental verification was demonstrated. Overall, there are a number of significant advantages of this phase-sensitive interferometric ellipsometer method compared with the conventional photometric polarimeter approach.

2. Principle and experiments

As shown in Fig. 1, a polarized common-path optical heterodyne interferometer is set up [11]. The mirror M₂ in the signal arm of the interferometer, which is able to reduce the spatial shifting of emerging beams, is arranged in order to retro-reflect the emerging beams of the extraordinary ray (E-ray) and ordinary ray (O-ray) from the PALC under test conditions. At the same time, the effects of the multiple reflections of the E-ray and the O-ray in the PALC at a large oblique incidence angle are reduced.

 

Fig. 1. Experimental setup: BS, beam splitter; AOM, acousto-optic modulator; M, mirror; A, polarizer; S, test sample on three-dimensional rotation stage; PBS, polarization beam splitter; D_p, D_s, photo detectors; BPF, band-pass filter; LIA, lock-in amplifier; DSC, digital stepping controller; PC, personal computer; PALC, parallel-aligned liquid crystal.

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Figure 2 shows the ray-tracing diagram for the tested PALC in this experiment, where the retro-reflected geometry of the PALC at a large oblique incidence angle is also shown. A thick (5 mm) BK-7 glass plate is attached to the PALC in order to reduce the multireflection effect. Additionally, two thin BK-7 glass plates of refractive index n=1.516 and thickness t≅600µm are used to confine the liquid crystal molecules. At the same time, two indium tin oxide (ITO) electrodes (n≅1.82,d≅0.1µm) are built into the two BK-7 glass plates. The PALC molecules are introduced between the two ITO electrodes as shown in Fig. 3.

 

Fig. 2. Retro-reflected geometry of the PALC at a large oblique incidence angle. A thick BK7 glass plate is attached to the PA-LC in order to reduce multiple reflections of the laser beam.

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Fig. 3. Nematic PALC device is equivalent to the phase retardation wave plate where the optical axis is pretilted at θ_pre to the surface.

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Finally, the P-polarized heterodyne signal I_P(Δωt) is generated by the P₁ wave from the reference beam, and the P₂ wave from the signal beam and can be expressed by

Similarly, the S-polarized heterodyne signal I_S(Δωt) is produced by the S₁ and S₂ waves coming from the reference and signal beams respectively:

where Δω=ω ₁-ω ₂ is the beat frequency. ${\delta }_p={\delta }_{p_2}-{\delta }_{p_1}$ and ${\delta }_s={\delta }_{s_2}-{\delta }_{s_1}$ are the phase differences between P₁ and P₂ waves and between S₁ and S₂ waves, respectively. ω ₁ and ω ₂ are the driving frequencies of the acousto-optic modulators (AOMs) in the reference and signal arms, respectively, as shown in Fig. 1. The parameters (I _p1,δ _p1), (I _p2,δ _p2), $(I_{s_1},{\delta }_{s_1})$, and $(I_{s_2},{\delta }_{s_2})$ are the intensities and phases of the P₁, P₂, S₁, and S₂ waves, accordingly. Therefore, the phase retardation is δ_s-δ_p≃δ _s2-δ _p2, where δ _s1-δ _p1≅0 is invariable from the reference arm and becomes twice the phase difference between the E-ray and the O-ray in the PALC because of the retro-reflected geometry. During measurement, the analyzers A₁ and A₂ in Fig. 1 are adjusted to 45° to the x-axis in order to initialize the interferometer for measurement [12]. Thus the input polarization state of the incidence linearly polarized beam becomes |X⁽ⁱ⁾|=1 and δ ⁽ⁱ⁾=0° theoretically, as described in a previous setup [12]. The eigen phase retardation of the PALC at normal incidence is

where 2mπ+Γ is the untwisted phase retardation of the PALC (0°≤Γ≤360°), m is the order number, and n_e(θ_pre) can be expressed by

After normally rotating the PALC, the optical axis lies on the x-z plane until δ ^(o)(β=β _min)=δ ^(o)=Γ so that Eq. (3) is satisfied. δ ^(o) is the phase retardation of the output polarization state and β is the rotation angle of the PALC. Figure 4 shows the ray tracing diagrams of the PALC at an oblique incidence angle in the different incidence planes. Figures 4(a) and 4(b) are scanned with respect to the variable oblique incidence angle in the y–z (tilted along x-axis) and x-z (tilted along y-axis) planes, respectively. Thus, the phase retardation at an oblique angle ϕ_tx tilted along the x-axis [see Fig. 4(a)] can be derived as

If θ_pre≤3°, then n_e(θ_pre)≅n_e is satisfied. When n_o and n_e are given, then δ_tx(ϕ_tx) becomes a function of the thickness d only. Thus, d can be obtained by using Eq. (5). Similarly, in Fig. 4(b), when the optical axis is perpendicular to the y-axis (lays on x-z plane), the phase retardation at ϕ_ty along y-axis becomes

where

n″_e is the refractive index observed by E″-ray in Fig. 4(b). Then θ_pre can be calculated by using Eqs. (6) and (7) at a large incidence angle.

 

Fig. 4. Schematic diagram of the nematic PALC at oblique incidence: (a) in y-z plane (tilted along x-axis), (b) in x-z plane (tilted along y-axis).

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In this experiment, a PALC with the refractive indices n _o=1.489, n _e=1.588 at 632.8 nm wavelength were used in the test. In Fig. 1, the PALC is placed between BS₂ and M₂ in order to compensate for spatial shifting and to avoid multiple reflections of the emerging E-ray and O-ray in the PALC. The oblique incidence angle is then scanned in a range from -50°≤ϕ_t≤-30° and a range from 30°≤ϕ_t≤50° at large incidence angles, both in the y-z plane (tilted along x-axis) and in the x-z plane (tilted along y-axis) separately. In order to offset the initial phase retardation at |ϕ_tx|=30°, which is due to the residual phase retardation of the optical components in the interferometer, δ_tx(-30°)≅0° and δ_tx(30°)≅0° are offset numerically. Then the phase retardation at the incidence angle in the range of -50°≤ϕ_tx≤-30° and 30°≤ϕ_tx≤50° are measured by use of a lock-in amplifier. Next, a thick BK-7 glass plate (5 mm) is attached to the PALC (see Fig. 2) in order to reduce the multiple reflections of the PALC at a large incidence angle. Figures 5(a) and 5(b) show the difference between the arrangements both with and without a thick glass plate attached to the PALC when a variable large incidence angle ϕ_tx is scanned in a range (30°≤ϕ_tx≤50°). It is obvious that the arrangement in Fig. 2 avoids multiple reflections.

 

Fig. 5. Phase retardation vs. oblique incidence angle: (a) without attaching the thick BK-7 glass plate, (b) after attaching the thick BK-7 glass plate.

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Moreover, Fig. 6(a) shows the results of the phase retardations of δ_tx at ϕ_tx in the ranges of -50°≤ϕ_tx≤-30°(red color) and of 30°≤ϕ_tx≤50° (blue color), where δ_tx(30°) and δ_tx(-30°) were offset numerically. Both phase retardations δ_tx and ϕ_tx seem to virtually overlap. This is in contrast with δ_ty, where there is a large deviation relative to the scanned angle ϕ_ty in the range of -50°≤ϕ_ty≤-30° (red color) and of 30°≤ϕ_ty≤50° (blue color) tilted along y-axis as shown in Fig. 6(b). According to Eq. (4), if θ_pre is less than 3°, then n_e(θ_pre)≅n_e is satisfied. The cell gap of the tested PALC at d=4.156 µm is then calculated by use of the relationship {[δ_tx(50°)-δ_tx(30°)]+[δ_tx(-50°)-δ_tx(-30°)]}/2=-44.472° in Fig. 6(a) where δ_tx(50°)-δ_tx(30°)≅δ_tx(-50°)-δ_tx(-30°). Similarly, when the PALC is scanned along the y-axis in the range of -50°≤ϕ_ty≤-30° and 30°≤ϕ_ty≤50° independently, then the pretilted angle θ_pre is obtained by use of the relation [δ_ty(50°)-δ_ty(30°)]-[δ_ty(-50°)-δ_ty(-30°)]=15.648° in Fig. 6(b) where δ_ty(30°) and δ_ty(-30°) are offset numerically, too. The measured cell gap of d=4.156 µm and the pretilt angle of θ_pre=2.59° are comparable to the data d≅4000 nm and θ_pre≅1~2° for the PALC device that was obtained from the Chi-Mei Optoelectronics Co., Taiwan; this comparison confirms the experimental verification of this method at large incidence angles.

 

Fig. 6. Phase retardation vs. oblique incidence angle: (a) in y-z plane (tilted along x-axis), (b) in x-z plane (tilted along y-axis). The error bar of the measurement is smaller than the line width of red and blue lines.

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In Fig. 7, the asymmetry of the phase retardation δ_ty for -50°≤ϕ_ty≤-30° and 30°≤ϕ_ty≤50° is seen experimentally. This is due to the nonzero pretilt angle of the PALC, which is tilted along the y-axis. This is in contrast to the symmetry of the phase retardation δtx for -50°≤ϕ_tx≤-30° and 30°≤ϕ_tx≤50° using the same measurement. According to Eq. (6), a computer simulation based on the values θ_pre=2.59° and d=4.156 µm obtained in this experiment shows a good fit between the measured data by this method where the initial phase retardations at |ϕ_ty|=30° are both compensated relative to the theoretical one. The deviations between the calculated and experimental data at 30°≤|ϕ_ty|≤50° in Fig. 7 might be caused by the residual effect of multiple reflections in the PALC.

 

Fig. 7. Comparison of the experimental (-50°≤ϕ_ty≤-30°, 30°≤ϕ_ty≤50°) and theoretical (-50°≤ϕ_ty≤50°) curves scanned in the x-z plane (y-axial tilt) where theoretical curve is calculated by (n_e,n_o)=(1.588,1.489), d=4.156µm, and θ_pre=2.59° at 632.8nm wavelength. The error bar is smaller than the line width of the measurements (red line).

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The conventional method used to characterize cell parameters of a nematic LC focuses on polarizer-sample-analyzer geometry by the photometric method. This is based on light intensity detection by which the phase retardation between the E-ray and O-ray are measured in terms of transmission intensity. There is limited sensitivity and dynamic range available for the intensity measurements used during cell parameters detection, and therefore we have proposed in this study to measure the cell parameters by phase-sensitive detection using a polarized common-path optical heterodyne interferometer. This not only enhances the sensitivity by the use of heterodyne signal detection and a wider dynamic range for phase measurement using a lock-in amplifier, but also provides larger oblique incidence angles on the PALC that are able to obtain high sensitivity for the phase retardation measurement. The result is high sensitivity for cell parameter detection compared to the conventional photometric method. In this setup, the multiple reflections and spatial shifting of the emerging E-ray and O-ray are avoided by the use of retro-reflected geometry in the interferometer. The experimental results demonstrate the utility of this method.

This method is also based on polarized common-path phase difference detection that enables the canceling of the common phase noise caused by both environmental disturbances and laser frequency noise. Therefore, a high SNR for the phase retardation detection is available. In this experiment, 0.3 °/hr for the phase stability was achieved, which implies a detection sensitivity of 0.3 nm for the cell thickness measurement of the PALC. Furthermore, 0.01°/min for the phase stability over a short testing period was obtained in these experiments, which implies 0.01° sensitivity for the pretilt angle detection according to Eq. (6) when using a lock-in amplifier. This contrasts with a pretilt angle measurement sensitivity of ±0.2° obtained by Kawamura et al. [7].

3. Discussion and conclusions

From the above results, it is clear that the proposed method of using a phase-sensitive polarized optical heterodyne interferometer at a large oblique incidence angle to characterize PALCs not only provides high sensitivity for the cell parameters measurements but also results in a wide dynamic range for the phase measurement compared to the conventional photometric method. It is based on a set of crossed polarizers that are used for optical geometry and intensity measurement. A large oblique incidence angle is scanned in this setup, and this is able to enhance the sensitivity of phase retardation measurement, while multiple reflections and the spatial shifting effect of the emerging beams in PALC are effectively avoided. Additionally, the scattering effect [13] in the PALC can be reduced too, because the heterodyne efficiency is critical to the alignment between the signal and reference beams. The scattered P₂ and S₂ waves of the signal beam in the PALC provide lower contributions compared with the non-scattered P₂ and S₂ waves to the heterodyne signals during measurement. The result is that the scattered P₂ and S₂ waves are rejected owing to the low heterodyne efficiencies of the P and S polarized heterodyne signals. As a consequence, the cell parameters of the PALC are able to be measured precisely using the developed phase-sensitive optical heterodyne interferometer at large oblique incidence angles. In this experiment it was possible to measure the sensitivity for the cell gap detection to 0.3 nm and for the pretilt angle to be 0.01° when a range of large tilted oblique incidence angles (30°<|ϕ_t|<50) on the PALC was scanned. The multiple reflections and spatial shifting effect were avoided by setting up a correct retro-reflected geometry in the interferometer. Moreover, this method would also seem to show potential for 2-D cell parameter measurement if the amplitude of the heterodyne signal is detected using a CCD camera [14]. As a result, to measure the phase and amplitude of the detected polarized heterodyne signal simultaneously in this setup, a 2-D highly sensitive characterization of the cell parameters of a PALC becomes applicable at a large oblique incidence angle.