1. Introduction

Mueller matrix polarimetry is a well-known method for materials characterizations [1–9]. A Mueller matrix polarimetry system included a polarization state generator (PSG) and polarization state analyzer (PSA) to characterize the complete state of polarized light reflected or transmitted after incident on sample. On the early stage of technology, the Mueller matrix polarimetry was only performed by a static measurement that the PSG was commonly built with a polarizer or compensator mounted on the mechanical stage to manually produce the polarization stage scanning [10,11]. These technique enable static measurements of the physical parameters of an optical sample to be reliable determined. However the time consuming to perform the measurement is high. To improve the performance of measuring, many similar systems have been built and proposed. Hall et al. [12] proposed a system by employing two photoelastic modulators (PEM) to build the PSA. Another dynamic Mueller matrix polarimetry measurement system by using liquid crystal retarders (LCV) was proposed as [13] or by combination of LCV and PEM was developed as [14]. The speed of these proposed systems were improved. Cross et al. [15] developed a real time measurement system by using two PEMs and a step scan interferometer for dynamically measuring the Mueller matrix of air and quartz crystal. The working frequency of PSA and PSG are 37 kHz and 50 kHz, respectively. Sanchez et al. [16] developed a system for dynamically measuring the Mueller matrix of samples with the frequency of PSG and PSA at 50 kHz by using four PMEs. Other real time measurement system using four PEMs to extract complete Mueller matrix of sample with the frequency of PSG at 60 kHz were proposed as [17,18]. The studies mentioned above achieved a remarkable high speed performance. The revolution for a high speed measurement system is still taken as a consideration from researchers, especially for application in determining the thickness of thin film growing, optical properties of liquid crystal cell with voltage changing, or the other on-line monitoring.

The TNLC has attracted many attentions by its interesting properties based on the precisely controlled realignment of liquid crystal molecules between different ordered molecular configurations under the action of an applied electric field. Many studies have employed dynamic Mueller matrix ellipsometry measurement system to characterize the limited properties of TNLC cell. Fukazawa et al. [19] presented a time-resolved polarization-modulated spectroscopic ellipsometry technique for examining the dynamic properties of nematic liquid crystals. Chao et al. [20] proposed a PEM method for obtaining real-time measurements of the twist angle and phase retardation of voltage-driven TNLC cells. Also, Patarroyo and Herrero [21] used a generalized transmission ellipsometry technique based on a standard rotating analyzer ellipsometer to determine the molecular tilt profile of a commercial voltage-driven nematic liquid crystal variable retarder. The methods described above provide an effective approach for dynamically measuring only certain properties of liquid crystal cells, and it is found that all physic parameters (ie. twist angle, tilt angle, cell gap and rubbing direction) cannot be extracted fully.

Accordingly, the present study proposes a new dynamic Mueller polarization scanning ellipsometry approach based on three electro-optic modulators (EO), and a GA is applied for extracting the physical properties and effective ellipsometric parameters of a TNLC cell. The validity of the proposed approach is demonstrated by obtaining dynamic measurements of the pretilt angle, twist angle and rubbing direction of a voltage-driven TNLC cell.

2. Extraction of effective ellipsometric parameters of TNLC cell using Stokes polarimetry method

TNLC cell is constructed as a model with three layers in arrangement of glass, LC, and glass based upon [22] and can be described by Stoke vector and Mueller matrix as

In previous studies by the present group [23], it was shown that the ellipsometric parameters of an optical sample can be obtained using four linear polarized lights and one circular polarized light. In traditional ellipsometry methods, the ellipsometric parameters describe the amplitude ratio and phase difference between two waves (i.e., the p-wave and the s-wave) orientated parallel and perpendicular to the incident (X-Y) plane, respectively. In this study, the orthogonal p- and s- waves are redefined as p’- and s’- waves, and are orientated at an arbitrary angle of θ relative to the X-Y coordinate system. (Note that full details of the P’-S’ coordinate system and the associated notations can be found in [23]). In practice, the laboratory coordinates are fixed and represent the X-Y coordinates. For any input polarization state corresponding to the X-Y coordinates, the output Stokes vector based on the P’-S’ coordinate system can be expressed as

In other words, the axes of the four linear input lights in the P’-S’ coordinate system must be rotated through an additional angle θ in order to express them in the X-Y coordinate frame. It is noted that in the particular case of θ = 0°, the P’-S’ coordinate system coincides with the X-Y coordinate system, and thus the effective ellipsometric parameters reduce to the traditional ellipsometric parameters.

In the present study, the effective ellipsometric parameters of the TNLC cell are extracted using four linearly polarized lights with the state of polarizations (SOPs) of 0°, 45°, 90° and 135°, respectively, Note that the depolarization effect is ignored, and hence additional circular polarized lights are not required. To obtain dynamic measurements of the TNLC cell properties, the four linear polarization states are scanned using an EO modulator. The corresponding Stokes vectors are as follows: Ŝ_0° = [1, 1, 0, 0]^T, Ŝ_45° = [1, 0, 1, 0]^T, Ŝ_90° = [1, −1, 0, 0]^T and Ŝ_135° = [1, 0, −1, 0]^T. The six effective ellipsometric parameters of an anisotropic are then obtained trigonometrically as follows:

For an anisotropic material, ∣r_p′p′/r_s′s′∣ = tan(Ѱ_p′p′), ∣r_p′s’/r_s’s′∣ = tan(Ѱ_p′s′) and ∣r_s′p′/r_s′s′∣ = tan(Ѱ_s′p′). Since ∣r_p′p′∣ ≥ 0, ∣r_p′s′∣ ≥ 0, ∣r_s′p′∣ ≥ 0 and ∣r_s′s′∣ ≥ 0, it follows that Ѱ_p′p′, Ѱ_p′s′ and Ѱ_s′p′ are located in the first quadrant. Furthermore, Δ_p′p′ = δ_p′p′ - δ_s′s′, Δ_p′s′ = δ_p′s′- δ_s′s′ and Δ_s′p′ = δ_s′p′ - δ_s′s′, where δ_p′p′, δ_p′s′, δ_s′p′ and δ_s′s′ are phase terms with values in the range of 0° ~360°.

3. Dynamic polarization scanning ellipsometry method

As described in the previous section, the effective ellipsometric parameters of the TNLC cell are extracted from the Stokes vectors corresponding to four linear polarization states with orientations in the range of θ = 0° ~180°. (Note that the SOPs obtained over the range of 180°~360° are symmetric). It is seen from Eq. (2) that the SOPs of the four linear polarization states at 0°, 45°, 90° and 135° in the P’-S’ coordinate system repeat as θ is scanned from 0° to 180°. Overall, Eq. (2) provides sufficient information to extract the effective ellipsometric parameters of the TNLC cell for any value of θ. Figure 1 illustrates the configuration of the proposed dynamic polarization scanning ellipsometry system. As shown, the light emitted from a He-Ne laser passes sequentially through a PSG, the TNLC cell, and a dynamic PSA. In traditional ellipsometry methods, the input SOPs are adjusted by rotating a polarizer using a mechanical stage. However, in the PSG used in the present study, the linear polarization states of the incident light are scanned in the range of 0°~180° by modulating the light emitted from the laser using a saw-tooth signal. Notably, this approach enables the linear SOPs to be scanned with a higher frequency. Furthermore, in accordance with the method described in [24], the output Stokes vectors are extracted dynamically using a PSA based on two EO modulators which are also driven by synchronized saw-tooth signals.

 

Fig. 1 Configuration of proposed dynamic polarization scanning ellipsometry system.

Download Full Size | PDF

3.1 Linear polarization scanner

In the linear polarization scanner, the principal axis of the EO modulator is adjusted to 45°, while that of the quarter-wave plate is adjusted to 0°. As a result, the Stokes vector of the light emerging from the PSG is given by

3.2 Polarization state analyzer

As shown in Fig. 1, the PSA comprises two EO modulators, an analyzer and a detector. The principal axes of the two modulators are set to 0° and 45°, respectively, while that of the analyzer is set to 0°. Consequently, the output Stokes vector has the form

4. Experimental setup and results

4.1 Experimental setup

As shown in Fig. 1, the linearly polarized lights were produced by passing the light emitted by a frequency-stable He-Ne laser (SL 02/2, SIOS Co., central wavelength 632.8 nm) through a polarizer (GTH5M, Thorlabs Co.), an EO modulator (ONESET Co., Model 350-50), and a quarter-wave plate (QWP0-633-04-4-R10, CVI Co.). The amplitude of the saw-tooth voltage signal used to drive the EO modulator was set as 0.635 V. As discussed in the previous section, the output Stokes vectors for each of the input lights were measured using a dynamic PSA consisting of two EO modulators (ONESET Co., Model 350-50), a polarizer (GTH5M, Thorlabs Co.) and a photo-detector (New Focus, Model 2001). In performing the experiments, the voltages applied to the three EO modulators and TNLC cell, respectively, and the intensity of the detected optical signal were recorded by a data acquisition (DAQ) card (NI USB-6366) with a maximal sampling rate of more than three million sampling points per second and a resolution of 16 bits in the range of ± 10V. Moreover, the sampling frequency of the DAQ card was set as 100 kHz such that the measured signals used to determine the Stokes vectors were each composed of 1000 sampling points. The dynamic output Stokes vectors were calculated on the basis of two hundred Stokes vector values per scanning period. In other words, the frequency of the PSA modulation signal was set to 200 times that of the linear polarization scanner. More specifically, the frequency of the saw-tooth voltage applied to the two EO modulators in the PSA was set to 100 Hz, while that of the voltage applied to the EO modulator in the PSG was set to 0.5 Hz. The pretilt angle, twist angle and rubbing direction of the TNLC cell sample are 5°, 90° and - 45°, respectively (reference information was provided by manufacturer).

In characterizing the TNLC cell, the azimuth angle of the input polarized light was scanned from 0° to 180° in each scanning period. Moreover, two hundred Stokes vector values were obtained per scanning period. Consequently, a high angular resolution of the cell parameters was obtained. In setting up the optical system shown in Fig. 1, a normal alignment of the optical components was achieved by placing a pinhole in front of each component and then adjusting the component such that the reflected laser beam passed through each pinhole in turn.

4.2 Experimental results

In the present study, GA was used to extract the pretilt angle, twist angle and rubbing direction of a TNLC cell with known optical parameters using the experimental values of the six effective ellipsometric parameters of the TNLC cell as a target function. The extracted values of the parameters were then used to calculate the corresponding effective ellipsometric parameters by means of Eqs. (3)-(16). Finally, the validity of the proposed model will be confirmed by comparing the simulated and experimental results of the effective ellipsometric parameters.

4.2.1 Characterization of TNLC cell without driving voltage (static measurements)

The feasibility of the proposed dynamic ellipsometry method was evaluated using a TNLC cell (Daxin Materials Corp.) with a known pretilt angle of 5°, twist angle of 90°, cell gap of 3.2 μm, and rubbing direction of −45°. Figure 2 shows the results obtained for the effective ellipsometric parameters of the TNLC cell (i.e., Ѱ_p’p’, Ѱ_p’s’, Ѱ_s’p’, Δ_p’p’, Δ_p’s’, and Δ_s’p’) given an incidence angle of 0°. Note that the blue and red lines represent the experimental results and the GA curve-fitting results, respectively. As described in the previous section, the experimental results comprise two hundred groups of effective ellipsometric parameters obtained from two hundred groups of measured Stokes vectors, where each group comprises the vector measurements obtained for linearly polarized input lights with azimuth angles of 𝜃 + 0°, 𝜃 + 45°, 𝜃 + 90° and 𝜃 + 135°, respectively. In addition, each measured Stokes vector is calculated on the basis of 1000 sampling points by means of discrete Fourier integration.

 

Fig. 2 Experimental results and GA simulation results for effective ellipsometric parameters of TNLC cell with no driving voltage

Download Full Size | PDF

Under a normal incidence condition, the optical characteristics of the TNLC cell are dependent on the cell gap, pretilt angle, twist angle, and rubbing direction. However, pretilt angle and cell gap are not independently but cooperatively expressed as a parameter [25], therefore the influences of cell gap and pretilt angle cannot be separated without a priori knowledge of cell gap or pretilt angle. Under an oblique incidence condition, the calculation of TNLC optical properties is very complicated and no simple equations as a function of tilt angle have been derived based upon Berreman’s 4x4 matrix or the extended Jones matrix [25]. Therefore, in the present study, the cell gap was set to the default value of 3.2 μm and the other parameters of the TNLC cell (i.e., the pretilt angle, twist angle and rubbing direction) were inversely extracted by the GA. In performing the GA extraction process, the error (fitness) function was formulated as

The GA generates an initial population of random candidate solutions of pretilt angle, twist angle and rubbing direction. Please note that the search spaces for the candidate solutions were specified as follows: 3° ≤ pretilt angle ≤ 7°, 70° ≤ twist angle ≤ 95°, and −50° ≤ rubbing direction ≤ −30°, respectively. The fitness value of pretilt angle, twist angle and rubbing direction was computed with a pre-defined value of error. The GA was terminated when the value of error less than this pre-define value and then the optimal fitness value was achieved.

As a result, the extracted value of the pretilt angle, twist angle and rubbing direction of the TNLC cell are 6.01°, 89.50° and - 44.72°, respectively. The deviation of the extracted values and the known value of the pretilt angle, twist angle and rubbing direction are just 1.01°, 0.5° and 0.28°, respectively. Figure 2 shows the simulation and experimental results of the effective ellipsometric parameters, Ѱ_p’p’, Ѱ_p’s’, Ѱ_s’p’, Δ_p’p’, Δ_p’s’, and Δ_s’p’, where the red and blue line represented for the simulated and experimental results. As shown, there is in a good agreement. In other words, the validity of the proposed approach for extracting the effective ellipsometric parameters of TNLC cells under static measurement conditions is confirmed.

4.2.2 Characterization of TNLC cell with driving voltage (dynamic measurements)

The dynamic polarization scanning ellipsometry system shown in Fig. 1 was used to extract the effective ellipsometric parameters of the TNLC cell as the driving voltage range from 0V to + 10V over a period of 12 seconds.

In practical applications, the relationships between the applied voltage and the distributions of the twist and tilt angles in the thickness direction (z) of the TNLC cell are more important than the actual values of the parameters themselves. In the present study, the distribution of the twist angle φ(z) was calculated in accordance with the following error function [26]

In characterizing the TNLC cell under dynamic modulation conditions, the cell gap, pretilt angle, twist angle and rubbing direction were set to the default values of 3.2 μm, 5°, 90° and −45°, respectively, and the GA commenced by generating an initial population of random solutions for system parameters A in Eq. (29) and B in Eq. (31). Note that system parameter C in Eq. (29) can be determined directly from system parameter A given boundary conditions for the total twist angle and maximal pretilt angle equal to 90° in both cases. When B is larger than 1.35, the maximal value of θ(z) may be larger than 𝜋/2 [26]. However, in most cases, the GA model is fitted to the experimental data with a value of B less than 1.35. Furthermore, if θ(z) becomes larger than 𝜋/2 during curve-fitting, it can simply be set to 𝜋/2 in the corresponding region(s) of the curve. Adopting such an approach, the distribution curve of the tilt angle may be clipped at 𝜋/2. Accordingly, in implementing the GA, the search spaces for the candidate solutions were set as 0 < A ≤ 100 and 0 ≤ B ≤ 2, respectively.

Figure 3 show the experimental and GA results for the effective ellipsometric parameters (Ѱ_p’p’, Ѱ_p’s’, Ѱ_s’p’, Δ_p’p’, Δ_p’s’, and Δ_s’p’). For applied voltages in the range of 0 ~2.54 V, the effective ellipsometric parameters in Fig. 3 are unchanged compared to results shown as Fig. 2. In other words, the Jones matrix of the TNLC cell remains constant under low driving voltages (<2.54V), and hence the optical properties are insensitive to the actual value of the voltage. However, for applied voltages in the range of 2.54 ~4.53 V, the electric energy is sufficient to overcome the elastic restoring energy, and thus the twist and tilt angles of the LC director increase with an increasing driving voltage. For applied voltages in the range of 4.53 ~10 V, the two hundred groups of effective ellipsometric parameters are once again the same. In general, the experimental results show that the TNLC cell changes from an anisotropic behavior to a quasi-isotropic behavior as the applied voltage increases.

 

Fig. 3 Experimental results and GA simulation results for effective ellipsometric parameters of TNLC cell for driving voltages. (a) 0 ~2.54 V. (b) 3.04V. (c) 4.03 V. (d) 4.53 ~10 V.

Download Full Size | PDF

In general, a good agreement exists between the experimental results and GA simulation results in Fig. 3. Thus, the validity of the proposed analytical model and ellipsometry system for determining the dynamic parameters of the TNLC cell are confirmed.

Figure 4 shows the distributions of the twist angle (blue line) and tilt angle (red line) across the width of the TNLC cell as a function of the applied voltage. The system parameters (i.e. A and B) were extracted from the effective ellipsometric parameters shown in Fig. 3 by the GA process. The results of the twist angle and tilt angle were then obtained by substituting A and B into Eqs. (29) and (31), respectively. The relationship between the twist angle and the TNLC position (z) becomes increasingly nonlinear as the applied voltage increases. Moreover, the tilt angle no longer remains constant at higher values of the driving voltage.

 

Fig. 4 Distributions of twist angle (blue line) and tilt angle (red line) of the LC director across width of TNLC cell as function of applied voltage.

Download Full Size | PDF

5. Conclusions and discussions

This study has presented a dynamic polarization scanning ellipsometry method based on the Mueller matrix formulation, Stokes polarimetry and a GA curve-fitting technique for extracting the physical properties and effective ellipsometric parameters of a TNLC cell. It has been shown that the extracted results for the pretilt angle, twist angle and rubbing direction deviate from the reference values by just 1.01°, 0.5° and 0.28°, respectively. Importantly, the proposed system uses an EO modulator to scan the SOPs of the input lights in the high speed performance. Consequently, the greater angular resolution of the measurement results is achieved. Overall, the experimental results confirm that the proposed dynamic polarization scanning elipsometry technique provides a straightforward and reliable means of extracting the effective ellipsometric parameters and physical properties of TNLC cells and other optical samples.