Experimental Apparatus for Inverse Problem Retrieval

In chapter 4, we developed a practical inverse problem procedure to retrieve the LC director profile from simulated data. The theoretical investigation indicates our method to be authentic even when the data have been contaminated with noise as high as 10%. In this chapter, we plan to use the inverse problem retrieval technique to recover the LC director profile from real measured data. The apparatus is to be described in details below.

5.1 Experimental Apparatus for Inverse Problem Retrieval

Figure 5-1. The experimental setup used to measure the optical transmittance data for inverse problem retrieval of LC director profile.

Polarizer 

The experimental setup used to measure optical transmittance data for inverse problem retrieval of LC director profile is shown in Figure 5-1. A He-Ne laser is used for the light source. The optical transmittance is resolved into T and _x with an analyzer and detected with a CCD. As noted in Section 4.3, the more independent data are collected, the higher accurate solution be retrieved. Therefore, we combine a polarizer and a quarter wave plate to collect more data by setting the incident light at four different polarization states: three linear ( , and 112.5 to the laboratory x-axis) and one left circularly polarized light. A rotation stage is used to adjust the incident angle into the LC cell. We should let the covering range of incident angle as wide as possible. However, due to the experimental constrain, the widest range of incident angle with this apparatus only covers from - to 50 .

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22.5^D 67.5^D D

50^{D D}

The samples we measured are a 4 mμ -thick LC cell in a splay alignment mode (OCB) and a 4 mμ -thick LC cell with hybrid alignment. The measurement procedure of the two samples is repeated for three times by using different applied voltages of 0V, 2.5V, 5V.

5.2 Experimental Results and Discussion

The polarization-resolved optical transmittance measurement results of the LC

cell with hybrid alignment are presented in Figure 5-2 to Figure 5-4, with three different applied voltages of 0V, 2.5V, 5V, respectively. Similar measurement results of the OCB cell are presented in Figure 5-5 to Figure 5-7. Each figure comprises of

T - and x -resolved optical transmittance plots by using an input light wave at four different polarization states. In each figure, the optical transmittance data (cross symbols) and the simulation curve (open squares) are compared. Deviations between the two curves can be observed and are used to adjust the calculated LC director profile during the inverse problem retrieval process. Figure 5-8 shows the retrieved LC director profiles for the hybrid cell with different applied voltages of 0V, 2.5V, 5V.

Figure 5-9 presents the similar results for the OCB cell. In the Figures, two LC director profiles are plotted. One profile (blue symbols) is obtained from the FEM simulation on the LC cell with Q-tensor approach, and the other profile (red symbols) is retrieved from the measured optical transmittance data with our inverse problem retrieval technique. We also note that our inverse problem retrieval technique always converge to an almost identical result even the initial input profile is quite different.

Although small oscillations remain on the retrieved profile at nonzero applied voltages, the overall agreement between the retrieved and the simulated profiles are excellent, indicating our inverse problem retrieval technique is fairly reliable and accurate for practical applications. The same conclusion can also be drawn for the

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OCB cell presented in Figure 5-9.

Figure 5-2. The polarization-resolved optical transmittance measurement results

T and x of the LC cell with hybrid alignment are presented by using four different input polarization states (22.5^o, 67.5^o, 112.5^o, and CP). The LC cell was applied with 0V. Two curves are included for comparison: red open squares:

simulated curve with Berreman matrix technique, and blue cross symbols: the measured transmittance as a function of optical incident angle.

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Figure 5-3. The polarization-resolved optical transmittance measurement results

T and x of the LC cell with hybrid alignment are presented by using four different input polarization states (22.5^o, 67.5^o, 112.5^o, and CP). The LC cell was applied with 2.5V. Two curves are included for comparison: red open squares:

simulated curve with Berreman matrix technique, and blue cross symbols: the measured transmittance as a function of optical incident angle.

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Figure 5-4. The polarization-resolved optical transmittance measurement results

T and x of the LC cell with hybrid alignment are presented by using four different input polarization states (22.5^o, 67.5^o, 112.5^o, and CP). The LC cell was applied with 5V. Two curves are included for comparison: red open squares:

simulated curve with Berreman matrix technique, and blue cross symbols: the measured transmittance as a function of optical incident angle.

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Figure 5-5. The polarization-resolved optical transmittance measurement results

T and x the OCB cell with bend-splay alignment are presented by using four different input polarization states (22.5^o, 67.5^o, 112.5^o, and CP). The LC cell was applied with 0V. Two curves are included for comparison: red open squares:

simulated curve with Berreman matrix technique, and blue cross symbols: the measured transmittance as a function of optical incident angle.

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Figure 5-6. Figure 5-5. The polarization-resolved optical transmittance measurement results T and _x the OCB cell with bend-splay alignment are

presented by using four different input polarization states (22.5^o, 67.5^o, 112.5^o, and CP). The LC cell was applied with 2.5V. Two curves are included for comparison: red open squares: simulated curve with Berreman matrix technique, and blue cross symbols: the measured transmittance as a function of optical incident angle.

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Figure 5-7. Figure 5-5. The polarization-resolved optical transmittance measurement results T and _x the OCB cell with bend-splay alignment are

presented by using four different input polarization states (22.5^o, 67.5^o, 112.5^o, and CP). The LC cell was applied with 5V. Two curves are included for comparison: red open squares: simulated curve with Berreman matrix technique, and blue cross symbols: the measured transmittance as a function of optical incident angle.

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(a) 

(b) 

Figure 5-8. The retrieval director profiles of the hybrid cell by inverse problem method. (a) The coordinate system used to present the LC director profiles. (b) The retrieved director profiles of the hybrid cell biased at 0V, 2.5V, and 5V. Two profiles are included for comparison: red squares: retrieved profile, and blue symbols: the simulated profile calculated by the FEM with Q-tensor approach.

(a) 

(b) 

Figure 5-9. The retrieval director profiles of the OCB cell by inverse problem method. (a) The coordinate system used to present the LC director profiles. (b) The retrieved director profiles of the OCB cell biased at 0V, 2.5V, and 5V. Two profiles are included for comparison: red squares: retrieved profile, and blue symbols: the simulated profile calculated by the FEM with Q-tensor approach.

Chapter 6

Conclusions and Future Prospect of This Thesis Study

In conclusion of this thesis research, we have accomplished a finite element method (FEM) simulation of nematics liquid crystal by using Q-tensor approach and the optical response is also demonstrated by using Berreman matrix method. We use this simulation technique for comparing the result with the measurement data of the modified OCB cell. We find that including line-patterned hybrid alignment configuration into an OCB cell can effectively eliminate the transition time between the splay to the bend configuration. For the LC inverse problem, we have the detail analysis of the effect of data regularization, the noise influence, and the finite range of incident angle. We use our approach method for the data regularization of the inverse problem to reduce the oscillating behavior in the LC inverse problem retrieval procedure. The director profiles of an OCB and a hybrid cell with and without applied voltage have been successfully retrieved from measured optical transmittance data with our inverse problem retrieval technique.

Further improvements could be done in the future:

(1) Using a perturbation concept to modify the Berreman matrix method for analyzing three-dimensional LC devices.

The optical response for a three-dimensional LC device can only be calculated by the finite difference time domain (FDTD) method. The FDTD method takes a lot of computing time and needs to setup the whole equation system quite precisely. Although the Berreman matrix method can only calculate the optical response caused by azimuthal or pretilt angle, but the computing time is less. We can consult the concept in the reference [10] that using a perturbation to extend the Berreman matrix method for the change of the azimuthal and pretilt angle at the same time.

(2) Changing the line-patterned hybrid alignment configuration for the line-patterned homeotropic alignment configuration into an OCB cell for the symmetry structure.

Because the hybrid alignment configuration has the un-symmetry cell structure, so the relaxation time is not fast as the symmetry OCB cell. The homeotropic alignment configuration has the symmetry cell structure and also it can provide the vertical force as the hybrid alignment configuration.

(3) Replace Tikhonov regularization method with iterative regularization method [21,

The prior probability can be realized very well by neural network. We can offer the prior information about the liquid crystal to help training the neural network system. The iterative regularization method can be expressed as followed which compared to Tikhonov regularization

2

where is the iteration number. Figure 6-1 gives an example of the iterative regularization method. When the iteration number increases, the more singular values we can keep. If it can be well designed, the convergence would be fast and reliable.

k

fi 

σi

Figure 6-1. An example of the iterative regularization method with different iteration number.

(4) Use an optical microscope with high NA objective to simplify the data taking

procedure.

With one shot of image exposure, the resulting optical image contains the information with varying incident angles needed for the inverse problem retrieval with a schematic illustrated in Figure 6-2 (a). We can do a simple estimation about the numerical aperture of the objective lens which is around 0.76 for our incident angle range as in Figure 6-2 (b).

Lens 

SAMPLE 

Light Objective (a)  

(b)

50^D Objective Lens

( )

sin 50 0.76

= ≈

NA ^D

Figure 6-2. Optical microscope with high NA objective can be used to simplify the data taking procedure for inverse problem retrieval. (a) The experiment setup. (b) The NA value of the objective lens for our incident angle range.

References

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336-362, 1998

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http://www6.cityu.edu.hk/ma/people/caponnetto/pub_files/lip_nips(CP1).pdf

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http://kavaro.com/mediawiki/index.php/Flips

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