由於具光學雙折射性和介電各向異性的特性，液晶材料一直在各領域廣泛被研究，且為各種應用於許多光電元件之上。其中液晶透鏡與液晶透鏡陣列更是廣泛被應用在諸如成像系統，微型投影儀，二維/三維影像可切換顯示器等應用領域中。其中，圓孔型電極液晶透鏡由於其可電壓調控焦距、具有開/關切換的功能及製作簡單等特性，在應用上具有相當的優勢。在原理上，圓孔型電極液晶透鏡利用其圓孔型電極與整面電極在液晶層中產生非均勻性電場，使得液晶層形成適當的折射率梯度分佈，而達到匯聚或發散光線的效果。

本論文主要探討兩個主題：

其一是位於圓孔型電極中的共平面環形電極對圓孔型電極液晶透鏡在不同操作電壓、操作頻率及不同直徑的共平面環形電極，對透鏡性能的改善與探討，並以等效電路模型說明其影響的相關性，更進一步做為圓孔型電極液晶透鏡設計之考量。
其二是提出一種具有兩個正交均勻液晶層和自對位雙圓孔型陣列電極結構的偏振無關液晶透鏡陣列。其中，雙圓孔型陣列電極採用新提出的黃光微影製程，在中間層玻璃基板的雙面分別製作對準的圓孔型陣列電極，且不需額外的對位製程。所提出的液晶透鏡陣列具有電可調焦點，無不連續線和偏振無關的特性。
一般而言，圓孔型電極液晶透鏡需要一個介電層以確保電場可以穿透液晶透鏡中心的工作區域並產生徑向對稱且不均勻的電場，並避免不連續線的產生。產生的電場分佈提供合適的力來重新定向液晶分子方向，以實現理想的折射率梯度分佈。圓孔型電極液晶透鏡由於外加介電層的存在，在低於1 kHz的操作頻率下，由於直流阻絕效應的影響，往往需要較大的操作電壓與較高的能量損耗。
經由實驗結果發現，利用在圓孔型電極中心區域加入一個共平面的環形電極達到不同於單一圓孔型電極的電場分佈，在低於1 kHz的操作頻率下相較於傳統的圓孔型電極液晶透鏡有更低的操作電壓、更短的焦距與更好的成像品質。透過等效電路模型的提出，本研究分析了不同孔徑的共平面環形電極，與不同上基板厚度對圓孔型電極的電場分佈與其對應之透鏡能力的影響。
實驗結果顯示，在孔徑為1 mm，介電層(NOA65)厚度為82 μm的圓孔型電極液晶透鏡中，搭配外圈直徑為360 μm (線寬20 μm)的共平面環形電極，在低於1 kHz的操作頻率下能夠明顯地改善圓孔型電極液晶透鏡的最短焦距操作電壓、與透鏡的起始電壓。在100 Hz的操作頻率下共平面環形電極液晶透鏡相較於傳統圓孔型電極液晶透鏡在最短焦距的操作電壓下降了76% (從50 Vrms降至12 Vrms)。
此外，本論文提出的自對位雙圓孔型偏振無關液晶透鏡陣列，同樣在圖案化的電極表面上塗覆額外137 μm厚度的介電層(NOA65)來避免不連續線的發生。其介電層的厚度經過優化，相較於以往能夠減少等待不連續線消失的時間。本偏振無關液晶透鏡陣列在7 Vrms下由鋁和銦錫氧化物 (ITO)導電膜產生的圓孔型陣列電極顯示出可用的最小焦距，分別為12.82和13.35 mm。

Owing to the characteristics of birefringence and dielectric anisotropy, liquid crystal (LC) materials are extensively studied in various fields and applied to many optical-electronic devices. Among these materials, LC lens and LC lens arrays (LCLAs) are widely used in application fields, such as imaging systems, micro-projectors, and 2D/3D switchable displays. Among the aforementioned LCLAs, the hole-patterned electrode liquid crystal lens (HELCL) has considerable advantages in various applications due to its characteristics of electric tunable focal length, ON/OFF function, and simple fabrication. In principle, hole-patterned and entire surface electrodes generate a center-symmetrically non-uniform electric field in the LC layer. Therefore, LC molecules form an appropriate gradient refractive index (GRIN) distribution and achieve the effect of converging or diverging light.

This dissertation mainly discusses the following two themes.

In the first study, the performance improvements of an additional coplanar inner floating ring (CFR) electrode in the conventional HELCL with various experimental conditions, such as operating voltages, driving frequencies, and diametric sizes of CFR, have been investigated and discussed. An equivalent resistor–capacitor (RC) circuit model is proposed to illustrate the performance of CFR–HELCL considering various lens structures and further provide design consideration to fabricate the CFR–HELCL.
In the second study, a polarization-independent LCLA comprising two orthogonal homogeneously aligned LC layers with self-aligned dual hole-patterned array electrodes on both surfaces of middle glass substrates has been demonstrated and investigated. The positions of self-aligned dual hole-patterned electrodes can be exactly and consistently aligned via novel photolithography processes without further alignment steps. The proposed LCLA shows characteristics of electrically tunable focuses, absence of disclination line, and polarization-independence.
The dielectric layer in the HELCL generally not only facilitates the penetration of center-symmetrically non-uniform electric fields in the center of hole-patterned electrodes but also prevents the disclination line issues. If an ideal electric field distribution is available, then electrically driving molecular reorientations will be provided along with GRIN distributions to demonstrate the optical functions of LC lenses. However, the additional dielectric layer also increases operating voltages owing to direct current blocking effects.
Experimental results reveal that the CFR electrode in the HELCL mainly contributes better electric field distributions than that in the conventional HELCL. Therefore, the CFR–HELCL possesses superior lens performance, such as short focal length and improved imaging quality with lower operating voltages at 100 Hz, to that in the HPLCL at an driving frequency of 1 kHz. The RC equivalent circuit model is used to illustrate qualitatively the effect of CFR electrode with various diametric sizes and driving frequencies on the electric field distributions in the CFR–HELCL. The experimental results show that the CFR–HELCL with a 1 mm diametric hole-patterned electrode along with a 360 μm diametric CFR electrode (i.e., 320 μm inside diameter with 20 μm wide conductive line) and an 82 μm thick NOA65 dielectric layer achieves the shortest focal length, in which operating voltages and frequencies are significantly more improved than that in the conventional HELCL. The CFR–HELCL almost reduced 76% of operation voltages to achieve the shortest focal length in the CFR–HELCL at 100 Hz compared with the conventional HELCL at 1 kHz (reduced from 50 Vrms to12 Vrms).
In addition, the polarization-independent dual hole-patterned electrode LCLA (DHELCLA) shows that a 137 μm thick NOA65 dielectric layer coated Al (aluminum) and ITO (indium tin oxide) hole-patterned array electrodes successfully prevent disclination line issues with the least waiting time. Simultaneously, DHELCLA demonstrates that minimum focal lengths in individual LC layers are 12.82 and 13.35 mm at 7 Vrms respectively considering individual Al and ITO hole-patterned array electrodes.

[1] S. Sato, "Applications of liquid crystals to variable-focusing lenses," Opt Review. 6 (6), 471 (1999).
[2] H. C. Lin, M. S. Chen, Y. H. Lin, "A review of electrically tunable focusing liquid crystal lenses," Trans Electr Electron Mater. 12 (6), 234 (2011).
[3] C. J. Hsu, C. R. Sheu, "Using photopolymerization to achieve tunable liquid crystal lenses with coaxial bifocals," Opt Express. 20 (4), 4738 (2012).
[4] H. C. Lin, Y. H. Lin, "A fast response and large electrically tunable-focusing imaging system based on switching of two modes of a liquid crystal lens," Appl Phys Lett. 97 (6), 063505 (2010).
[5] H. C. Lin, Y. H. Lin, "An electrically tunable focusing pico-projector adopting a liquid crystal lens," Jpn J Appl Phys. 49, 102502 (2010).
[6] M. Ye, B. Wang, M. Kawamura, et al. "Image formation using liquid crystal lens," Jpn J Appl Phys. 46 (10A), 6776 (2007).
[7] D. Liang, J. Y. Luo, W. X. Zhao, et al. "2D/3D Switchable autostereoscopic display based on polymer-stabilized blue-phase liquid crystal lens," J Display Technol. 8, 609 (2012).
[8] M. Ye, B. Wang, T. Takahashi, et al. "Properties of variable-focus liquid crystal lens and its application in focusing system," Opt Review. 14 (4), 173(2007).
[9] Y. H. Lin, M. S. Chen, H. C. Lin, "An electrically tunable optical zoom system using two composite liquid crystal lenses with a large zoom ratio," Opt Express. 19(5), 4714(2011).
[10] M. Kawamura, M. Ye, S. Sato, "Optical trapping and manipulation system using liquid-crystal lens with focusing and deflection properties," Jpn J Appl Phys. 44(8), 6098(2005).
[11] M. Kawamura, M. Ye, S. Sato, "Optical tweezers system by using a liquid crystal optical device," Mol Cryst Liq Cryst. 478, 891(2007).
[12] Y. H. Hsu, W. Y. Lu, T. T. Wu, et al. "Switchable aperture of liquid crystal lens array fabricated with a hole-array patterned metal foil spacer," Mol Cryst Liq Cryst. 613, 143(2015).
[13] T. Nose, S. Sato, "A liquid crystal microlens obtained with a non-uniform electric field," Liq Cryst. 5 (5), 1425 (1989).
[14] M. Ye, B. Wang, S. Sato, "Liquid-crystal lens with a focal length that is variable in a wide range," Appl Opt. 43 (35), 6407 (2004).
[15] A. Mouquinho, K. Petrova, M. T. Barros, et al. New Polymers for Special Applications. Section 1.3. (IntechOpen, 2012).
[16] B. A. Averill, P. Eldredge, Chemistry: Principles, Patterns, and Applications. Section 11.8. (Pearson College Div, 2006).
[17] D. K. Yang, S. T. Wu, Fundamentals of liquid crystal devices. Chapter 1.3. (John Wiley & Sons, 2006).
[18] J. D. Jackson, Classical electrodynamics. Chapter 4.4. (Wiley, 1999).
[19] W. D. Jeu, P. Bordewijk, "Physical studies of nematic azoxybenzenes. II. Refractive indices and the internal field," J Chem Phys. 68 (1), 109 (1978).
[20] D. K. Yang, S. T. Wu, Fundamentals of liquid crystal devices. Chapter 1.5. (John Wiley & Sons, 2006).
[21] M. F. Vuks, "Determination of the optical anisotropy of aromatic molecules from the double refraction of crystals," Opt Spek. 20, 644 (1966).
[22] J. Li and S. T. Wu, "Extended Cauchy equations for the refractive indices of liquid crystals," J Appl Phys. 95 (3), 896 (2004).
[23] F. C. Frank, "I. Liquid crystals. On the theory of liquid crystals," Discussions of the Faraday Society 25, 19 (1958).
[24] I. C. Khoo, Liquid crystals: physical properties and nonlinear optical phenomena. Chapter 3.2. (John Wiley & Sons, 2007).
[25] H. W. Ren, D. W. Fox, B. Wu, et al. "Liquid crystal lens with large focal length tunability and low operating voltage, " Opt Express. 15, 11328 (2007).
[26] Y. J. Liu, X. W. Sun, Q. Wang, "A focus-switchable lens made of polymer-liquid crystal composite," J Cryst Growth. 288,192-164 (2006).
[27] A. Jakli, A. Saupe, One- and Two-Dimensional Fluids: Properties of Smectic, Lamellar, and Columnar Liquid Crystals. Chapter 5.1. (Taylor & Francis, 2006).
[28] C. H. Kuo, W. C. Chien, C. T. Hsieh, et al. "Influence of pretilt angle on disclination lines of liquid crystal lens," Appl Opt. 51,4269 (2012).
[29] O. Pishnyak, S. Sato, O. D. Lavrentovich, "Electrically tunable lens based on a dual-frequency nematic liquid crystal," Appl Opt. 45 (19), 4576 (2006).
[30] C. J. Hsu, C. R. Sheu, "Preventing occurrence of disclination lines in liquid crystal lenses with a large aperture by means of polymer stabilization," Opt Express. 19 (16), 14999 (2011).
[31] M. Ye, S. Sato, "Optical properties of liquid crystal lens of any size," Jpn J Appl Phys. 41 (5B), L571 (2002).
[32] X. Zhao, C. Liu, D. Zhang, et al. "Tunable liquid crystal microlens array using hole patterned electrode structure with ultrathin glass slab," Appl Opt. 51 (15), 3024 (2012).
[33] C. Y. Chien, C. H. Li, C. R. Sheu, "An electrically tunable liquid crystal lens with coaxial bi-focus and single focus switching modes," Crystals. 7 (7), 209 (2017).
[34] X. Zhao, D. Zhang, Y. Luo, et al. "Numerical analysis and design of patterned electrode liquid crystal microlens array with dielectric slab," Opt Laser Technol. 44 (6), 1834 (2012).
[35] S. Sato, "Liquid-crystal lens-cells with variable focal length," Jpn J Appl Phys. 18, 1679 (1979).
[36] J. F. Algorri, V. Urruchi, N. Bennis, et al. "Integral imaging capture system with tunable field of view based on liquid crystal microlenses," IEEE Photon Technol Lett. 28 (17), 1854 (2016).
[37] A. Hassanfiroozi, Y. P. Huang, B. Javidi, et al. "Hexagonal liquid crystal lens array for 3D endoscopy," Opt Express. 23 (2), 971 (2015).
[38] M. Ye, B. Wang, S. Sato, "Polarization-independent liquid crystal lens with four liquid crystal layers," IEEE Photon Technol Lett. 18 (3), 505 (2006).
[39] Y. H. Lin, H. S. Chen, "Electrically tunable-focusing and polarizer-free liquid crystal lenses for ophthalmic applications," Opt Express. 21 (8), 9428(2013).
[40] Y. H. Lin, H. S. Chen, H. C. Lin, et al. "Polarizer-free and fast response microlens arrays using polymer-stabilized blue phase liquid crystal,"Appl Phys Lett. 96, 113505 (2010).
[41] C. J. Hsu, C. H. Liao, B. L. Chen, et al. "Polarization-insensitive liquid crystal microlens array with dual focal modes," Opt Express. 22 (21), 25925 (2014).
[42] M. Estribeau, P. Magnan, " Fast MTF measurement of CMOS imagers using ISO 12233 slanted edge methodology," Proc SPIE. 5251, 243 (2004).
[43] I. A. Cunningham, A. Fenster, " A method for modulation transfer function determination from edge profiles with correction for finite element differentiation," Med Phys. 14 (4), 533 (1987).
[44] P. B. Greer, T. van Doorn, "Evaluation of an algorithm for the Assessment of the MTF using an edge method," Med Phys. 27 (9), 2048 (2000).
[45] A. F. Naumov, G. D. Love, M. Y. Loktev, et al. "Control optimization of spherical modal liquid crystal lenses," Opt Express. 4 (9), 344 (1999).
[46] J. Beeckman, T. H. Yang, I. Nys, et al. "Multi-electrode tunable liquid crystal lenses with one lithography step," Opt Lett. 43 (2), 271 (2018).
[47] C. J. Hsu, J. J. Jhang, C. Y. Huang, "Large aperture liquid crystal lens with an imbedded floating ring electrode," Opt Express. 24 (15), 16722 (2016).
[48] Y. H. Hsu, B. Y. Chen, C. R. Sheu, "Improvement of hole-patterned electrode liquid crystal lens by coplanar-inner ring electrode," IEEE Photonics Technol Lett. 31 (20), 1627 (2019).
[49] T. Scharf, P. Kipfer, M. Bouvier, et al. "Diffraction limited liquid crystal microlenses with planar alignment," Jpn J Appl Phys. Part 1(39), 6629 (2000).
[50] Y. H. Hsu, B. Y. Chen, C. R. Sheu, "Effect of the dimensions of coplanar-inner floating ring electrode on the performance of liquid crystal lenses," Crystals. 11(2), 200 (2021).
[51] Y. Y. Kao, P. C. Chao, C. W. Hsueh, "A new low-voltage-driven GRIN liquid crystal lens with multiple ring electrodes in unequal widths," Opt Express. 18(18), 18506 (2010).
[52] Y. H. Hsu, T. H. Kao, C. Y. Chien, et al. "Polarization-independent liquid crystal lens array with additional dielectric films over self-aligned dual hole pattern electrodes," Liq Cryst. 47 (9), 1312 (2020).