## Abstract

Two external-field-free methods are presented for measuring the azimuthal anchoring strength in twisted nematic liquid crystal (TNLC) cells. For asymmetrical TNLC samples, the twist angle is derived from the phase of the detected signal in a phase-sensitive heterodyne polarimeter and is then used to calculate the weak anchoring strength directly. The measurement resolution which is found to be about 0.01μJ/m^{2} makes the present method sensitive enough for the LC-based bio-sensing application. Using the proposed method, the weak azimuthal anchoring strength of a composite liquid crystal mixture (40% LCT-061153 + 60% MJO-42761) in contact with a plasma-alignment layer is found to be 7.19 μJ/m^{2}. For symmetrical TNLC samples, the liquid crystals are injected into a wedge cell, and the two-dimensional distributions of the twist angle and cell gap are extracted from the detected phase distribution using a genetic algorithm (GA). The azimuthal anchoring strength is then obtained by applying a fitting technique to the twist angle vs. cell gap curve. Utilizing the proposed approach, it is shown that the strong anchoring strength between a rubbed polyimide (PI) alignment layer and E7 liquid crystal is around 160 μJ/m^{2} while that between a rubbed PI alignment layer and MLC-7023 liquid crystal is approximately 32 μJ/m^{2}.

© 2010 OSA

## 1. Introduction

The physical behavior of liquid crystals (LCs) is largely determined by their surface properties. Of the various surface properties, the surface energy plays a particularly important role in governing the physics of the LCs, and therefore in determining their technical applications. Accordingly, the literature contains many proposals for estimating the anchoring energy of LCs. For example, it has been shown that the azimuthal anchoring energy of nematic LCs can be measured using a torque balance method [1–3]. However, in applying this method, it is necessary to measure the cell parameters (e.g. the cell thickness and the twist angle) with an extremely high degree of precision. Therefore, a requirement exists for more straightforward means of measuring the fundamental properties of LC samples.

In response to this requirement, the literature contains numerous proposals for measuring the azimuthal anchoring strength using either external-field or external-field-free methods. Akahane *et al.* [4] minimized the light transmitted through a TNLC cell by rotating both the cell and the analyzer, and then determined the twist angle and the optical retardation of the cell such that the surface azimuthal anchoring strength could be determined using an analytical formulation based on the Jones matrix. In 1997, Zhou *et al.* [5] presented a method for determining the cell thickness and twist angle of TNLC cells by measuring the Stokes parameters of the transmitted light. In a later study, the same group demonstrated the use of the Stokes parameter method (SPM) in determining the azimuthal anchoring strength in TNLC cells [6]. Fonseca and Galerne [7] proposed a simple method for measuring the azimuthal anchoring strength of nematic LCs by applying a fitting technique to the curve of the cell thickness vs. deviation angle of LC director in a wedge cell at the condition of waveguide regime. In a recent study, Govindaraju *et al.* [8] used Fonseca’s method to measure the anchoring energy of LC on surfaces in order to quantify different proteins captured by immobilized ligands.

The external-field-free methods described above provide a convenient means of determining the weak azimuthal anchoring strength in LC cells, but provide less accurate results when applied to LC cells with a strong azimuthal anchoring strength. Accordingly, various external-field methods have been proposed for measuring a wider range of anchoring strengths [9, 10]. However, these methods are theoretically complex and generally involve a time-consuming and complicated experimental procedure. In [11], the present group proposed a simple method for measuring the multiple parameters of a twisted TNLC sample by using a genetic algorithm (GA) to extract the twist angle and cell gap parameters inversely from the intensity ratio and phase of the signal obtained in a heterodyne polarimeter. In a more recent study [12], the same group proposed a hybrid approach based upon the Stokes parameter method and a GA for enabling the full characterization of a TNLC cell, including the cell thickness, the twist angle, the pre-tilt angle, and the azimuthal angle.

This study proposes two external-field-free methods for measuring the weak and strong anchoring strengths in asymmetrically-aligned and symmetrically-aligned TNLC cells, respectively, using a phase-sensitive heterodyne interferometer. In the case of the asymmetrical TNLC cell, the cell parameters are optimized in such a way that the detected phase signal is rendered sensitive to the twist angle but insensitive to the cell thickness, and the twist angle of the sample is then extracted from the phase of the detected heterodyne signal. The twist angle is then used to calculate the anchoring strength directly. The proposing method is more sensitive than the one utilized by Govindaraju *et al.* [8] for quantifying proteins captured on surfaces through interactions with immobilized ligands and therefore is suitable for label-free bio-sensing application. For the symmetrical sample, the twist angle and cell gap of a wedge cell incorporating the LCs of interest are extracted from the 2-D phase distribution of the sample using a GA, and the azimuthal anchoring strength is then derived from the twist angle vs. cell gap curve using a fitting technique. The validity of the proposed weak anchoring strength measurement method is demonstrated using an asymmetrically-aligned TNLC cell containing a composite LC mixture of 40% LCT-061153 and 60% MJO-42761. Meanwhile, the validity of the strong anchoring strength measurement method is demonstrated by measuring the anchoring strengths between a rubbed polyimide (PI) layer and E7 liquid crystal and MLC-7023 liquid crystal, respectively.

## 2. Basic theory

In TNLC display devices, a uniform director alignment is generally obtained using rubbed polymer films. When the rubbing directions on the two substrates in the cell are orientated at an angle to one another, the nematic LCs injected into the cell form a twisted structure as a result of the surface anchoring force. If the surface anchoring energy is weak (and no external field is applied), the twisted structure induces a deviation of the director at the cell surface from the easy axis of the LC alignment layer, and the deviation angle can then be used to estimate the magnitude of the anchoring energy. In practice, the deviation angle is determined by the balance between the elastic power of the twisted LC structure and the torsional effect of the anchoring energy. For the case of a TNLC cell containing nematic LCs doped with a chiral material and having substrates rubbed in anti-parallel or parallel directions, the free energy per unit area is given by the sum of the elastic energy *F*
_{b} and the surface anchoring energy *F*
_{s} [13], i.e.

*K*

_{22}is the twist elastic constant;

*d*is the cell thickness;

*ϕ*is the real twist angle;

_{t}*ϕ*

_{0}is the intrinsic pre-twist angle of the sample material and is equal to

*ϕ*

_{0}= 2π

*d*/

*p*where

*p*is the natural pitch of the material;

*w*is the azimuthal anchoring strength, and Δ

_{ϕ}*ϕ*is

*the deviation of the surface LC director from alignment (rubbing) direction. (Note that for a nematic LC with chiral pitch of*

*p*= ∞,*ϕ*_{0}= 0).In this study, two external-field-free methods are proposed for measuring the azimuthal anchoring strength in TNLC cells. In the first case, the two substrates of the TNLC cell are prepared in the same way such that the LCs are symmetrically aligned. As a result, an identical (strong) azimuthal anchoring strength is obtained at both substrates. In the second case, the two substrates are prepared using different methods such that a strong anchoring effect is obtained at the lower substrate while a weak anchoring effect is obtained at the upper substrate. Consequently, the LCs are asymmetrically aligned.

In the case of the symmetrical cell, the anchoring strength, *w _{ϕ}*, can be obtained by minimizing the free energy,

*F*, with respect to Δ

*ϕ*[13], i.e.

*ϕ*is the twist angle induced by the easy axis of the LC alignment layer. In the case of the asymmetrical cell, the anchorage effect at the lower substrate is so strong relative to that at the upper substrate that the value of

_{e}*w*at the lower substrate can be approximated as infinity. In other words, no torque-induced deviation of the LC director occurs at the lower substrate. Consequently, the azimuthal anchoring strength at the upper substrate can be obtained as

_{ϕ}Thus, by measuring the real twist angle *ϕ _{t}* and the cell thickness

*d*, Eqs. (4) and (5) enable the strong and weak azimuthal anchoring strengths in symmetrically-aligned and asymmetrically-aligned TNLC cells, respectively, to be obtained analytically given a knowledge of the twist elastic constant

*K*and the twist angle induced by the easy axis of the LC alignment layer

_{22}*ϕ*.

_{e}## 3. Azimuthal anchoring strength measurement for asymmetrical TNLC cells

#### 3.1 Basic measurement principle

As shown, the light source has the form of a He-Ne laser. The laser beam is passed through a Glan-Thompson polarizer rotated to 0° and is then modulated by an electro-optic modulator (EOM) driven by a saw-tooth waveform signal with an angular frequency *ω* and with a slow axis oriented at −45° to the x-axis. The light is then passed through the TNLC sample and is incident upon a quarter-wave plate (QWP) whose slow axis is set such that it forms an angle of 45° with that of the EOM. Finally, the light emerging from the QWP is passed through an analyzer (a Glan-Thompson polarizer rotated to −45°) and is incident upon a photo-detector (PD).

The light intensity at the PD, and the electric field vector emerging from the configuration shown in Fig. 1 , are given by

*α*is the azimuthal angle of the entrance LC director. The Jones matrix of an asymmetrically-aligned TNLC cell can be expressed as [14]

*ϕ*is the twist angle of the LC cell,

_{t}*Γ*is the linear retardation,

*d*is the cell gap, and $\Delta n$ is the birefringence of the LC material. Therefore, the intensity of the PD output signal is given by

In other words, the detected heterodyne signal has the form of a sinusoidal signal with the same angular frequency as the EOM driving voltage. As shown in Eq. (9), the detected signal consists of a DC term, *I _{DC}*, and a modulated term, $K\mathrm{sin}(\omega t+\sigma )$. The phase

*σ*of the sinusoidal term can be expressed as a function of the TNLC cell parameters$(\alpha ,{\varphi}_{t},d)$and can be extracted using the phase-lock method. Figures 2(a) and 2(b) illustrate the variation of the phase term in Eq. (9) for twist angles in the range −180° ~180° and retardation values in the range 0 ~2 μm. (Note that the azimuthal angle is assumed to be

*α*= 0°.) It can be seen that the phase,

*σ*, and the twist angle,

*ϕ*, are linearly related at retardation values of approximately 0.3 μm, 0.9 μm, and 1.5 μm. Furthermore, it is noted that the slope of the linearity at

_{t}*d*Δn = 0.3 μm is greater than that at the other retardation values. Thus, in extracting the twist angle from the phase of the heterodyne detected signal, a retardation of around 0.3 μm represents the optimal value in terms of enhancing the measurement resolution. Note that further simulations (results not shown here) reveal that there is no better value of the azimuthal angle than

*α*= 0° when extracting the twist angle from the phase of the detected signal.

Figures 3(a)
and 3(b) show the variation of *σ* as a function of the retardation *d*Δn in the range 0.30 μm to 0.38 μm for twist angles *ϕ _{t}* ranging from −180° ~180° and a constant azimuthal angle of

*α*= 0. It can be seen from Fig. 3(a) that for a given twist angle, the phase

*σ*is insensitive to the retardation

*d*Δn, i.e. the phase value varies only slightly as the retardation is increased from 0.30 μm to 0.38 μm. Meanwhile, Fig. 3(b), which presents the same results but from a different perspective, gives a clear illustration of the twist angle dependent phase

*σ*performance in the retardation

*d*Δn range 0.30 μm to 0.38 μm. It can be seen from Fig. 3(b) that the rate of change of the phase

*σ*with the twist angle is approximately constant for twist angles in the range −150° ~150°. From inspection, the average sensitivity (i.e. the average slope), $\Delta \sigma /\Delta {\varphi}_{t}$, is found to be around 1.2. In the heterodyne polarimeter shown in Fig. 1, the phase of the detected signal can be measured with a resolution of 0.1°, and thus it follows that the twist angle can be extracted with a precision of around 0.08°. Thus, the information in Fig. 3(b) provides not only an optimal design of retardation range in 0.3 μm ~0.38 μm and twist angle range in −150° ~150° for the azimuthal anchoring strength measurement but also the measurement performance of the proposing system.

For an asymmetrical TNLC cell with a strong anchoring substrate (input surface) and a weak anchoring substrate (output surface), the azimuthal anchoring strength of the weak alignment layer can be calculated from Eq. (5) in Section 2 given a knowledge of the actual twist angle *ϕ _{t}*, the cell thickness

*d*, the twist elastic constant

*K*and the twist angle induced by the easy axis of the LC alignment layer

_{22}*ϕ*. Figure 4 shows the simulation results obtained for the correlation between the real twist angle and the azimuthal anchoring strength of the weak alignment layer in a TNLC cell with parameters of

_{e}*ϕ*= 120°,

_{e}*K*

_{22}= 7.35 pN, and

*d*= 3

*μ*m. The results show that for a weak anchoring strength of around 7 μJ/m

^{2}, the measurement resolution of the twist angle (i.e. 0.08°) equates to an azimuthal anchoring strength resolution of approximately 0.01 μJ/m

^{2}, and the smaller the anchoring strength, the more sensitive the anchoring strength measurement.

#### 3.2 Experimental setup and results for asymmetrical cells

In setting up the heterodyne polarimeter shown in Fig. 1, a frequency stabilized He–Ne laser (Model: SIOS SL 02/2) with an output power of 3 mW was used as the light source and the EOM (Conoptics, Model 370) was driven by a saw-tooth signal with a frequency of 1 kHz. The phase of the detected signal was acquired using a lock-in amplifier (SR830, Stanford Research Systems Inc.,). The asymmetric TNLC sample was constructed by filling an empty test cell (twist angle *ϕ _{e}* = 120°, pre-tilt angle 3°, and cell gap 2.91

*μ*m) with a composite LC mixture (

*K*

_{22}= 7.35 pN) comprising 40% LCT-061153 and 60% MJO-42761. The LCs were aligned at the lower substrate via a PI rubbing layer and at the upper substrate by a plasma-alignment layer. In performing the measurement process, the azimuthal angle of the entrance LC director was set to

*α*= 0°. The retardation of the TNLC sample,

*d*Δn, was estimated to be 0.352 μm, i.e. within the optimal range of 0.3 μm ~0.38 μm. The weak azimuthal anchoring strength at the plasma-alignment layer was calculated from Eq. (5) using the method described in Subsection 3.1. Note that prior to the measurement process, the heterodyne signal detected without the TNLC sample in the system was calibrated in order to eliminate the phase effect induced by the EOM.

Figures 5(a)
~5(c) show the correlation between the phase *σ* and the twist angle for azimuthal angles of *α* = 0°, 1° and −1°, respectively. Note that the value of *α* = 0° represents the optimal azimuthal angle, while the values of *α* = + 1° and −1° are used to investigate the effect of small errors in the optimal azimuthal angle on the calculated value of the anchoring strength. The measured phase *σ* of the asymmetric TNLC sample is −67.5° ( = 292.5°). In Fig. 5(a), the real twist angle ${\varphi}_{t}$ is found to be 80.21° at a phase *σ* of 292.5°. Substituting this value of ${\varphi}_{t}$ into Eq. (5) together with the cell parameters (${\varphi}_{e},d,{K}_{22}$) provided by the manufacturer, the azimuthal anchoring strength of the weak alignment layer is found to be 7.19 μJ/m^{2}. This result is reasonable since the azimuthal anchoring strength of E7 liquid crystal at a plasma-alignment surface is known to be around 14.8 μJ/m^{2} [15]. Substituting the twist angle values extracted from Figs. 5(b) and 5(c) into Eq. (5) together with the cell parameters (${\varphi}_{e},d,{K}_{22}$), the azimuthal anchoring strengths of the weak alignment surface in the TNLC cell are found to be 7.13 μJ/m^{2} and 7.24 μJ/m^{2}, respectively. In other words, an error of 1° in the azimuthal angle results in a deviation of no more than 0.06 μJ/m^{2} in the calculated value of the weak anchoring strength. Thus, the proposed method provides a robust means of calculating the azimuthal anchoring strength in asymmetrical TNLC cells even when small errors exist in the azimuthal angle setting.

## 4. Azimuthal anchoring strength measurement for symmetrical TNLC cells

#### 4.1 Basic measurement principle

In symmetrical TNLC cells, the LC director deviates from the easy axis of both substrates, and thus the method described in Subsection 3.1 cannot be used to measure the anchoring strength since the value of *α* is unknown. Therefore, this section proposes an alternative method for measuring the azimuthal anchoring strength of symmetrical aligned LC samples using a TNLC wedge cell and a genetic algorithm (GA).

Figure 6
shows the simulation results obtained by Eq. (4) for the variation of the twist angle with the cell thickness for known azimuthal anchoring strengths of *w _{ϕ}* = 30~150 μJ/m

^{2}at both substrates with

*ϕ*= 60° and

_{e}*K*

_{22}= 6.5 pN in each case. Using the theoretical

*ϕ*-

_{t}*d*curve, the unknown azimuthal anchoring strength of a symmetrically-aligned LC sample can be obtained simply by fitting the experimental

*ϕ*-

_{t}*d*curve obtained by the GA. Note that the azimuthal strength obtained by fitting the multiple

*ϕ*-

_{t}*d*curves is expected to be more accurate than that obtained when using a sample with a uniform cell gap.

Figure 7
presents a schematic diagram of the measurement system utilized to measure 2-D parameters distributions from a TNLC cell. As shown, the configuration is modified from the one described in Section 3. A beam expander is used to extend the single-point methodology to full-field measurement and a QWP whose slow axis forms an angle of 45° with that of the EOM was placed at three positions, namely Position 1, Position 2, and Position 3 (see Fig. 7) sequentially in the measurement process. After passing the analyzer, the expanded beam is incident on a CCD triggered by a complex programmable logic device (CPLD) with a phase-shift driver. As the QWP is placed at Position 1, the form of the detected signals *I _{1}* consists of a DC term

*I*and a modulated term ${K}_{1}\mathrm{sin}(\omega t+{\sigma}_{1})$. Similarly, as the QWP was placed at Position 2 and Position 3, the detected signal can be expressed as ${I}_{2}={I}_{DC}+{K}_{2}\mathrm{sin}\left(\omega t+{\sigma}_{2}\right)$ and ${I}_{3}={I}_{DC}+{K}_{3}\mathrm{sin}\left(\omega t+{\sigma}_{3}\right)$, respectively. In accordance with the method presented in [16], the phase maps of (

_{DC}*σ*

_{1},

*σ*

_{2},

*σ*

_{3}) of heterodyne signals can be carried out as follows

The full-field distributions of the azimuthal angle *α*, cell thickness *d* and twist angle *ϕ _{t}* are extracted using a GA, and the strong azimuthal anchoring strength is then obtained by applying a fitting method to the experimental results for the twist angle vs. cell thickness.

#### 4.2 Experimental setup and results for symmetrical cells

In the measurement system shown in Fig. 7, empty polyimide-rubbed wedge-shaped cells (provided by LCD&PMR lab, NCTU, Taiwan) with pre-tilt angle of 3° were used as samples for the measurement. The twist angle induced by the easy axis of the LC alignment layer was equal to *ϕ _{e}* = 60°, and the cell was filled with either E7 liquid crystal (

*K*

_{22}= 6.5 pN, Merck Co.) or MLC-7023 liquid crystal (

*K*

_{22}= 8.25 pN, Merck Co.). The surfaces of the samples are separated by a 20

*μ*m spacer, and are not separated by a spacer at the other end. The illuminating light was provided by a frequency-stabilized He-Ne laser (Model: SIOS SL 02/2) with an output power of 3 mW, and the EOM (Conoptics, Model 370) was driven by a saw-tooth signal with a frequency of 1 kHz. In acquiring the intensity images, the exposure time of the 8-bit gray-level CCD (JAI, CV-A11) was set to a quarter period of 0.25 ms. Figure 8 illustrates the interface between the CPLD and the CCD [17]. The control clocks in the CPLD were processed using a cascaded sequence of frequency dividers in an embedded 2 MHz oscillator. Meanwhile, the frequencies of the external trigger signals provided to the function generator and the CCD were set to 1 kHz and 10 Hz, respectively. The phase of the CCD trigger signal was shifted sequentially using a bits / delay controller, and the exposure time, trigger mode and other settings of the CCD were controlled by a computer through IMAQ 1409 and RS-232 interfaces. The mean intensity values of the 30 frames acquired over a 3 second period were then used in computing the phase maps of (σ

_{1}, σ

_{2}, σ

_{3}).

Before inserting the sample into the experimental setup, the DC bias of the EOM was adjusted such that the detected signal and the driving signal were in phase (i.e., ${\sigma}_{initial}={0}^{\circ}$) in order to eliminate the effect of the phase effect induced by the EOM. Having calibrated the detected signal, the sample was introduced into the measurement system and the QWP was inserted at Position 1 (see Fig. 7). The 2-D phase distribution, *σ*
_{1}, was then obtained from the detected signal. The QWP was then moved to Position 2, and the new 2-D phase distribution, *σ*
_{2}, was obtained. Finally, the QWP was removed from the experimental system, and the corresponding 2-D phase distribution, *σ*
_{3}, was obtained.

Figure 9(a)
presents a typical image acquired by the CCD camera of the wedge cell filled with E7 liquid crystals. Note that the diffraction patterns in the image are the result of small particles on the lens surface and interference fringes introduced by the optical components within the interferometer. Figures 9(b)–10(d)
show the corresponding 2-D distributions of *σ*
_{1}, *σ*
_{2}, and *σ*
_{3}.

The measured 2-D phase distributions (σ_{1}, σ_{2}, σ_{3}) were substituted into the GA as the objective values for the error function, and the optimal values of *α*, ${\varphi}_{t}$, and *d* (i.e. the parameter values which minimized the error function) were extracted. In Eq. (9), it can be seen that the phase term of the detected signal varies as a function of cos2X. That is, as the cell gap *d* increases, the term $\text{X}={[{\varphi}_{t}{}^{2}+{(\Gamma /2)}^{2}]}^{1/2}$ also increases, and cos2X varies periodically between −1 and 1. Figures 10(a) and 10(b) show the variations of terms *A* and *B* in Eq. (9) as a function of *d*. The periodic properties of *A* and *B* lead to a corresponding periodicity in the measured phase distributions. This phenomenon is inevitable, and causes the GA to extract multiple solutions for the TNLC parameters if the search space for the cell gap parameter is assigned the full range of 0~20 μm. Therefore, in implementing the GA, the TNLC sample parameters ($\alpha ,{\varphi}_{t},d$) were randomly assigned within the ranges

Figures 11(a)
–11(c) present the cell parameter distributions obtained from the GA for the E7 wedge cell, while Figs. 12(a)
–12(c) present the equivalent results for the MLC-7023 wedge cell. Finally, Figs. 13(a)
and 13(b) show the azimuthal anchoring energies of the E7 and MLC-7023 wedge cells, respectively, obtained by fitting the cell thickness and twist angle data presented in Figs. 11 and 12. The anchoring strengths of the E7 and MLC-7023 cells are found to be 160 μJ/m^{2} and 32 μJ/m^{2}, respectively, and are therefore consistent with the values presented in previous studies [18,19]. For example, in [18], the azimuthal anchoring energy of E7 liquid crystal on a rubbed PI alignment layer was found to be 58 μJ/m^{2}, while in [19], the azimuthal anchoring energy of 5CB liquid crystal on a rubbed PI layer was found to be 330 μJ/m^{2}. Thus, even though the experimental values obtained for the azimuthal anchoring energies of E7 and MLC-7023 liquid crystal are different from the values presented in the literature, they are still within a reasonable range.

## 5. Conclusions

This study has demonstrated the use of a phase-sensitive heterodyne interferometer in determining the azimuthal anchoring strengths of asymmetrical and symmetrical TNLC cells. For the case of asymmetrical TNLC cells, the twist angle of the cell is extracted from the phase of the detected heterodyne signal and is then used to compute the weak azimuthal anchoring strength directly. It’s not necessary to rotate the optical elements during measurement, and the resolution 0.01μJ/m^{2} is better than the standard error 0.3μJ/m^{2} obtained by Govindaraju *et al.* [8] for quantifying proteins captured on surfaces through interactions with immobilized ligands. The present method thus may provide a more straightforward and more sensitive way for the future bio-sensing applications. For symmetrical TNLC samples, the LC is inserted into a wedge cell and the correlation between the twist angle and the cell gap is extracted from the recorded phase measurements using a genetic algorithm (GA). A fitting technique is then applied to the twist angle vs. cell gap measurements in order to determine the corresponding value of the strong azimuthal anchoring strength. Unlike the method presented in ref [7], this method can be used as the sample is not in the waveguide regime. The experimental results have shown that the weak azimuthal anchoring strength of a composite liquid crystal mixture (40% LCT-061153 + 60% MJO-42761) on a plasma-alignment layer is equal to 7.19 μJ/m^{2}. Meanwhile, it has been shown that the strong azimuthal anchoring strengths of E7 liquid crystal and MLC-7023 liquid crystal on rubbed polyimide alignment substrates are equal to 160 μJ/m^{2} and 32 μJ/m^{2}, respectively. The azimuthal anchoring strengths obtained in this study are in good general agreement with the results presented in the literature, and thus the validity of the proposed approach is confirmed.

The phase-sensitive measurement methods proposed in this study not only eliminate the effects of intensity variations, but also minimize the influence of energy absorption at the optical elements within the measurement system. Moreover, the common-path configuration and heterodyne scheme reduce the effects of environmental perturbations and improve the SNR of the detected signal. As a result, the resolution of the measurement results is significantly improved. Finally, the use of a wedge cell in determining the azimuthal anchoring strength of symmetrical TNLC samples yields a more accurate estimate of the anchoring strength than that obtained using a sample with a uniform cell gap.

## Acknowledgements

The authors gratefully acknowledge the financial support provided to this study by MIRDC (Metal Industries Research & Development Centre), Taiwan, and the National Science Council of Taiwan under Grant No. NSC96-2628-E-006-005-MY3.

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