From the analysis in Chapter 3, we know that the intrinsic anisotropies of NLC materials indeed play important roles on the optical pattern formation phenomena. In this Chapter, we present the experimental results according to the analysis in Chapter 3.
4-1 Sample preparation, experimental setup and measurements
Fig. 4.1 Experimental setup: B.S., beam splitter; d, cell thickness; L, feedback length; LC, liquid crystal; L’i, lens
The experimental setup is the same as we shown in Fig. 2.2 and we redraw it as in Fig. 4.1. The Sample used in our experiment is a nematic liquid-crystal cell prepared by sandwiching the nematic E7 between two indium tin oxide-coated (ITO) glass windows that had been treated with polyvinyl alcohol (PVA) and achieved the parallelly planar alignment by rubbing. The nematic E7 is a mixed liquid crystal; it has many components. The material parameters of E7 including the temperature
range of nematic phase, refractive index, and elastic constants are listed in Appendix I as a reference. Different values of the parameters for E7 are obtained from different published sources because of the variations of different measuring methods. The thickness of the liquid-crystal cell is about 68 µm which is controlled by the calibrated Myler spacer and measured by using a micrometer. The sample area is about 2x2 cm2 which is much larger than the laser-spot size of about 1.4 mm in diameter. The planar alignment quality is examined by microscopy and conoscopy to assure that there is no observable defects. The hyperbola conoscopic picture which shows the planar alignment is shown in Fig. 4.2.
Fig. 4.2 Conoscopic picture for the planar aligned NLC sample.
The applied external fields include a 1kHz electric field and an optical field. The 1kHz electric field is generated by a microcomputer’s waveform synthesizer (Quatech Inc., WSB-A12M) applied normally to the sample’s glass windows and its amplitude could be programmably controlled. The laser beam used is from a CW Ar-ion laser (Spectra-Physics, Model 2080-15s) with wave length at 514.5 nm and linear polarization at TEM00 mode. The original beam waist from the
Ar-ion laser is 1.9 mm and the input beam diameter is controlled by a pinhole of 1.4 mm in diameter. The pinhole is used to block the stray light in the low intensity wings of the beam and allows the high intensity region to pass through the sample. The reflectivity of our reflecting mirror is about 0.65. The distance between the sample and the feedback mirror is 1.9 cm. The lens L1, L2 are arranged to observe the near-field picture of the reflected beam impinging on the sample. A silicon photodetector system (UDT, Model S380), which is covered by an iris diaphragm to reduce the scattering noise recorded, is used to detect the input light power. The observed picture was recorded by a digital camera with its exposure time of 1/800 second. The ambient temperature is roughly controlled by using a mounted thermo-electric cooler for eliminating laser heating effect and keeping the liquid crystal in nematic phase.
4-2 Experimental observations
In the following, the experimental results will be separated as three parts to describe the anisotropic effects and the electrical-modulating effect.
4-2-1 Optical pattern formation in a parallelly planar-aligned NLC film
In considering the prediction by the results of the LSA of the governing diffusion-like equation, the optical pattern should be formed as the input
light intensity is above the threshold intensity. Although the distribution of the threshold intensity in our planar-aligned homogeneous NLC film is anisotropic and has the minimum at ϕ=900 and the maximum at ϕ=00 and 1800, we first fix the applied voltage at 1.117 Vrms then input a light beam with power 0.83 W. From Eq. (11), the average tilt angle θa at this external field conditions is about 0.787 radian and the maximum threshold power obtained by the product of Eq. (17) and the beam area is about 0.8W as shown in Fig. 3.1(b). The near-field picture observed on the screen is shown in Fig. 4.3 and the stable hexagon is obviously formed.
Fig. 4.3 Near-field pattern observed on the screen; input power=0.83 W, biased voltage=1.117 Vrms, d=68 mm, L=1.9 cm,
R=0.65, beam diameter=1.4 mm, and exposure time=1.25 ms.
4-2-2 Obtaining different optical patterns by the optical method
The optical pattern formation from our quasi-electric-field-biased planar-aligned NLC film has been confirmed as shown in Fig. 4.3.
Considering the theoretical analysis in Chapter 3, the NLC (E7) film is classified in the region C in Table 3.1. The anisotropy between the diffusion lengths is always positive and the anisotropic distribution of the threshold power appears, as shown in Fig. 3.1(b) for example.
Therefore, in order to study the effects of the anisotropic distribution of the threshold power, we fix the biasing voltage at 1.117 Vrms and change the input light power from 0.71 W to 0.98 W with a power step about 0.03 W. The input light power is measured after the beam passing through the pinhole and the beam splitter. In our experiments, the laser beam is blocked when we change the input light to the desired power. Therefore the pattern formations always start from the homogeneous state. We find that, as the input power is 0.74 W and 0.78 W the observed near-field pictures are the stable rolls as depicted in Fig. 4.4. Furthermore, as the input power is close to 0.8 W, the observed near-field picture are not stable and the patterns are appear as the roll and the hexagon alternately.
The competition of the patterns is shown in Figs. 4.5(a)-(c).
(a) (b)
Fig. 4.4 Stable roll patterns observed on the screen (a) input power=0.74 W (b) input power=0.78 W
(a) (b) (c)
Fig. 4.5 Pattern sequence showing the competition between the roll and the hexagon patterns; input power=0.8 W.
As the input light power is at 0.83 W the stable hexagon is obtained, however, if the input light power increases continuously to about 0.98 W the pattern becomes unstable and turns chaotic finally. The hexagons and the chaotic patterns are shown in Fig. 4.6.
(a) (b)
Fig. 4.6 Stable hexagon and chaotic patterns observed on the screen (a) input power= 0.83 W (b) input power= 0.98 W.
4-2-3 Obtaining different optical patterns by the electric method
In this subsection, we keep investigating the electric effect on the optical pattern formation phenomena. In the above experimental observations, we see that the different patterns can be obtained by changing the input light power. Here, we fix the input light power at 0.91 W and change the biasing voltage to see what patterns can be obtained.
Fig. 4.7 shows the pattern when the biasing voltage is zero. There is no structured pattern that can be identified. Obviously to create a significant optical pattern, a biasing voltage is required when a planar-aligned homogeneous NLC film is used.
Fig. 4.7 Observed picture when the biasing voltage is zero with input light power= 0.91 W
In our experiments, we find that the patterns we get are similar to that we get by changing the input light power. The stable hexagons are obtained when the voltage is at 1.117 Vrms and 1.114 Vrms and are shown in Fig.
4.8.
(a) (b)
Fig. 4.8 Stable hexagon patterns observed on the screen (a) V= 1.117 Vrms (b) V= 1.114 Vrms
The stable rolls are obtained when the voltage is at 1.032 Vrms and are illustrated in Fig. 4.9. The pattern competition between the rolls and the hexagons is also observed when the biasing voltage is at 1.096 Vrms. See Fig. 4.10.
Fig. 4.9 Stable roll patterns observed on the screen; with V= 1.032 Vrms
(a) (b) (c)
Fig. 4.10 Pattern sequence showing the competition between the roll and the hexagon patterns; V= 1.096 Vrms.
According to Santamato’s explanation in Ref. 11, as the anisotropy between the diffusion lengths exists the roll patterns are expected to be seen and once the anisotropy is cancelled by rotating the sample at a
suitable angle the hexagon patterns are obtained. However, in our experiments, the hexagon patterns are seen even without canceling the anisotropy between the diffusion lengths. Not only the stable hexagons can be obtained but also the stable rolls are obtained by either changing the input light power or the biasing voltage. The pattern competition between the rolls and the hexagons is also observed when the input light power or the biasing voltage is between the values to obtain the stable rolls and the stable hexagons.
Chapter 5 Discussion and Conclusions
In this chapter, we discuss the experimental results and compare them with the theoretical LSA results we get in Chapter 2. Finally, we make some conclusion about our work in this dissertation and we also suggest some future works which may be done in this topic about the optical pattern formation by using NLC films.
5-1 Discussion
From the experimental results we presented in Chapter 4, we observe the stable rolls, the stable hexagons and the competition between the rolls and hexagons. To investigate the relation between the patterns and the external fields, we plot the distribution of the threshold intensity (or power) to see the corresponding relations.
5-1-1 Discussion about the observed patterns obtained by optical method
In order to explain the experimental results described in Section 4-2-2, we substitute all values of the experimental parameters, including the cell parameters, the material parameters and the external fields, into Eq. (11) to obtain the average tile angle θa and into Eq. (17) to calculate the
threshold intensity distribution. Threshold power is calculated by the product of Eq. (17) and the beam area. When the biasing voltage is about 1.117 Vrms, the calculated average tilt angle θa is about 0.787 radian. We plot the calculated threshold power distribution versus the azimuthal angles in Fig. 5.1 to relate these experimental observations to our arguments from the theoretical results exhibited in Section 4-2-2. Even though the actual laser beam is a gaussian beam, for simplicity we calculate the threshold power from the threshold intensity shown in Fig.
3.1(b) by multiplying the beam area. Since only the light with high intensity passing through the pinhole and the sample the optical power with the peak intensity reaching the threshold intensity should be lower then that shown in Fig. 5.1. Nevertheless, it is clear that the stable rolls and stable hexagons exist and are obtained indeed in the power regions near the minimum and the maximum threshold power, respectively.
0. 73
Fig. 5.1 The calculated curve of the threshold power versus the azimuthal angle ϕ when the average tilt angle θa is fixed at
about 0.787 radian and the biased voltage is 1.117 Vrms.
According to the theoretical analysis and the experimental observations that we have shown above, one can see that the anisotropy indeed play an
important role in the formation of the rolls. Moreover, some more words have to be said about the input power issue. When the power is above the maximum threshold, the hexagons appear. However, chaotic patterns will be formed if we keep increasing the input power. One interesting question which may appear is that what patterns will be formed as the input power is between the minimum and the maximum threshold power. Actually our experimental observations at 0.77 W and 0.8 W indicate that the hexagons and the rolls may compete with each other and are not stable.
This implies an interesting suggestion that the hexagons may still be formed without the requirement that all the modes with different azimuthal angles are allowed to appear. The experimental observations reasonably agree with the theoretical predictions discussed in Section 3-1.
5-1-2 Discussion about the observed patterns obtained by the electrical method.
In order to explain the experimental results we get in Section 4-2-3, following the same procedures in the preceding section, we insert the experimental parameters, including the cell parameters, the material parameters and the external fields values, into Eq. (11) to obtain the average tile angle θa and into Eq. (17) to calculate the threshold intensity distribution then obtain the threshold power by the product of Eq. (17) and the beam area. We obtain the results as illustrated in Fig. 5.2.
0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15
0 0.5 1 1.5 2 2.5 3
azimuthal angle j (rad.)
th reshold powe r (w)
V=1.114Vrms V=1.032Vrms input power=0.91W
Fig. 5.2 The calculated curves of the threshold power versus the azimuthal angle φ, from the bottom to the top are biased voltage=1.114 Vrms and 1.032 Vrms, respectively; the horizontal line indicates the input light power=0.91 W; with beam diameter=1.4 mm, d=68 µm, L=1.9 cm, R= 0.65.
Fig. 5.2 suggests that the stable hexagons are formed when the input light power is above and near to the maximum threshold, and the stable rolls are formed when the input light power is above and near to the minimum threshold. The competition between the rolls and the hexagons appear when the input light power is at some value between the maximum and the minimum thresholds.
In our calculation, we set the spatial frequency q as q ≈ π / λ0
L
. The accuracy of such an approximation has been verified by measuring the relations between the feedback length and the pattern period. The pattern period for the feedback length L=1.9 cm is 198 µm and the theoretically predicted value is 197.74 µm. The experimental results agree with the theoretical predictions reasonably well.5-1-3 Overall discussion
From the theoretical analysis and the experimental observations, the anisotropic properties of NLC materials indeed play important roles in the optical pattern formation. The orientation-dependent birefringence makes the pattern formation possible. The anisotropy of the elastic constant results in the anisotropic distribution of the threshold intensity for the patterns to be formed and this anisotropic distribution allows the observation of stable rolls and stable hexagons without using any external Fourier filter. Furthermore, the dielectric anisotropy of NLC materials enables to control the nonlinearity of the system by a small biasing voltage.
The pattern competition behavior between the rolls and hexagons we observed in our experiments is an interesting phenomenon worth continuing investigation. This dynamic behavior may associate with the dynamic property of the NLC molecules and with the instability of the roll state. The theoretical model we present here is only the instability analysis with respect to the homogeneous plane-wave state and the static state of the NLC molecules. Although only with the LSA results, the optical pattern formation phenomena can be effectively extended when the basic intrinsic properties are included.
The influence of the Talbot effect in this one-feedback-mirror system is also proved by measuring the period of the formed patterns.
5-2 Conclusions and future works
In this dissertation, we present the theoretical model to obtain the governing diffusion-like equation for both the orientation of the NLC directors and the Kerr-induced phase variation from the continuum theory for NLC materials for the one-feedback-mirror system to observe the optical pattern formation phenomena. The threshold intensity for the patterns to be formed is obtained from the linear stability analysis of the diffusion-like equation. Furthermore, the influences of the anisotropic properties of the NLC materials are analyzed.
The elastic anisotropy results in the anisotropic distribution of the threshold intensity. The anisotropic distribution of the threshold intensity is the key factor for the system to yield the pattern of stable rolls without using a Fourier filter. However, this does not imply that the hexagon patterns can not be obtained when the anisotropy of the threshold intensity distribution exists. The stable hexagons can still be obtained when the input light power is larger than the maximum threshold.
The anisotropic dielectric property of the NLC materials makes the modulation of the nonlinearity of the system possible by controlling the orientational distribution electrically. This property also facilitates the modulation of the formed patterns without changing the input light power.
The influences of the relative values of the Frank elastic constants are analyzed theoretically. We believe that once the material satisfying the elastic-constant requirement mentioned in Section 3-2 is available, the
method to obtain the vertical rolls, the horizontal rolls and the hexagons can be realized by simply modulating the biasing voltage.
The experimental results in our work qualitatively agree with the theoretical results well. The suggested forming properties in our theoretical analysis can be reasonably proved.
This work presented in this dissertation can be viewed as an opening of the study of the optical pattern formation phenomena from the point of view of the liquid crystal. We establish the theoretical model by using the simple linear stability analysis. The perturbation is added to the homogeneous plane wave state. However, the pattern competition phenomena we observed in our experiments are indeed associated with interaction of the roll and the hexagon patterns. Therefore, the stability analysis to the roll state can be included in the future studies.
Not only the instability analysis can be extended but also the dynamic behavior of the NLC molecules can be considered and it may become rather complicated.
Generally speaking, the study of the optical pattern formation phenomena is very interesting and the simple and tunable properties of liquid crystals can effectively be applied on the pattern formation phenomena and obtain interesting patterns without using Fourier filters.
There still are many interesting topics which can be studied in the future, including both the theory and the experiments.
References
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Appendix I
Curriculum Vita
基本資料:
姓 名: 徐旭寬
現在住址: 苗栗縣苗栗市福安里 10 鄰中華路 592 號
實驗室地址: 新竹市國立交通大學光電工程研究所液態晶體實驗室 電 話: 037-261213
實驗室電話: 03-5712121 轉 56344 生 日: 民國 64 年 12 月 22 日 籍 貫: 台灣 苗栗
婚姻狀況: 未婚 學 歷:
學 士: 國立成功大學物理學系 民國 87 年 6 月
碩 士: 國立交通大學光電工程研究所 民國 88 年 8 月直升博士班 博 士: 國立交通大學光電工程研究所 民國 93 年 6 月
Publication List
Journal Papers:
1. Yuhren Shen, Hsu-Kuan Hsu and Shu-Hsia Chen, “Phase-conjugate
reflection and self-starting optical phase-conjugate oscillation in planar nematic liquid-crystal cells,” J. Opt. Soc. Am. B, Vol. 20, No. 1, pp. 65-72 (2003)
2. Hsu-Kuan Hsu, Shu-Hsia Chen and Yinchieh Lai, “Crucial effects of the anisotropy on optical field induced pattern formation in nematic liquid crystal films,” Opt. Express, Vol. 12, No. 7, pp. 1320-1328 (2004)
3. Hsu-Kuan Hsu, Yinchieh Lai and Shu-Hsia Chen, “Quasi-static
3. Hsu-Kuan Hsu, Yinchieh Lai and Shu-Hsia Chen, “Quasi-static