光子晶體是由週期性結構所組成的結構，假若經由適度的設計則具備改變光波傳遞的效果。而近幾年發現在其傳導區擁有獨特的色散性質，尤其是負折射的特性，使得該方面的研究成為最近學術界極度熱門的話題。
本文以平面波展開法求得二維光子晶體等頻圖，利用等頻圖來預測負折射角度，再使用時域有限插分法和有限元素法模擬光子晶體的異常折射現象。
為了調變改變光波傳遞的方向，將光子晶體的圓柱以橢圓柱取代，因為橢圓柱的幾何非等向性，使得等頻圖明顯產生變化，讓垂直入射的光波可以發生折射，分析觀察橢圓柱不同的旋轉角，來獲得不同的折射角度，將不同方向的晶格排列進而控制光波傳遞的方向，可以利用橢圓柱光子晶體的概念設計出光開關、可調式分光器。
我們選用液晶材料當可調變的材料，利用其液晶材料的非等向性和藉由電場控制液晶導軸角度的特性，將液晶材料加入光子晶體橢圓柱中，來觀察與分析不同液晶導軸角度和橢圓柱之幾何非等向性的關係，對折射角度的影響。發現負折射角度有較大的範圍，適合作為可調折射角之光學元件。最後，對於調至臨界角度之後的全反射，則提出可做為光開關之用途。還有改善其邊界條件，讓光開關的效果更好。

Photonic crystals (PhCs) are synthetic periodic structures that,when suitably designed, have the ability to change the propagation of light. Recently, it was proved that the diffraction effects of PhCs can produce the effective negative refraction or the negative index. Hence, the studying of PhCs is not limited in the band gap region. The anomalous refractive properties (especially negative refraction) of PhCs have become hot topics of scientific research over the past few years.
In this thesis, the plane wave expansion method is used to get the equifrequency surface. The anomalous refractive properties of photonic crystal are analyzed by using the equifrequency surface. Then we use the finite difference time domain method and the finite element method to simulate the light propagation in PhCs and compare the refractive angle with that predicted by the equifrequency surface.
In order to change the direction of the light propagation, the two-dimension columns within photonic crystals are replaced by elliptic rod. Due to the geometric anisotropy of elliptic rods, we can rotate the elliptic rods to obtain different structure factors. The anisotropic property can then be used to tune the refraction direction. The results can be applied to develop various photonic crystal devices, such as optical switches and tunable optical splitters.
The material properties of liquid crystals (LCs) can be altered easily by applying an external electric field. The LCs are infiltrated into the photonic crystal of elliptic rods and the relationship between the refractive angle and the angle of the director is studied. Owing to the large anisotropy consisting of elliptic rods, the range of negative refraction angle is superior to column. The direction of the negative refraction is controlled by changing the direction of the LCs director. The tunability was analysed at the specified frequency and a large tunable range of the negative refraction is achieved. The refraction of a tunable PhCs with nematic liquid crystals can be used to design an optical switch. Besides, the transmission of the optical switch can be improved by properly adjusting the boundary conditions.

[1] E. Yablonovitch,“Inhibited spontaneous emission in solid-state physics and electronics,”Physical Review Letters, Vol.58, No. 20, pp.2059-2062(1987).
[2] S. John, “Strong localization of photons in certain disorded dielectric superlattices,”Physical Review Letters, Vol.58, No.20, pp.2486-2489(1987).
[3] J. D. Jannopolous, R. D. Meade and J. N. Winn, “Photonic Crystal,” (Princeton U. Press, Princeton, N.J., 1995)
[4] E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou and C. M. Soukoulis,“Negative refraction by photonic crystals,” Nature423, No.5, pp.604 (2003).
[5] P.V. Parimi, W.T. Lu, P. Vodo, J. Sokoloff, J. S. Derov, and S. Sridhar, “I Negative Refraction and Left-Handed Electromagnetism in Microwave Photonic Crystals,” Phys. Rev. Lett. Vol.92, No.12, pp.127401 (2004).
[6] B. Momeni, J. Huang, M. Soltani, M. Askari, S. Mohammadi, M.Rakhshandehroo and A. Adibi, “Compact wavelength demultiplexing using focusing negative index photonic crystal superprisms”Opt. Exp. B Vol.14, No.6,pp.2413-2422 (2006).
[7] H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Self-collimating phenomena in photonic crystals,” Appl. Phys. Lett. 74, No.9, pp.1212 (1999).
[8] X. Yu, S. Fan, “Bends and splitters for self-collimated beams in hotonic crystals,” Appl. Phys. Lett., Vol.83, No.16, pp.3251-3253(2003)
[9] S. A. Ramakrishna, “Physics of negative refraction index materials,”Rep. Prog. Phys. 68, pp.449-521 (2005)
[10] J. B. Pendry, A. J. Holden, W. J. Stewart and I. Youngs, “Extremely Low Frequency Plasmons in Metallic Mesostructures,”Phys. Rev. Lett. 76 , No.25,pp.4773(1996)
[11] J. B. Pendry, A. J. Holden, D. J. Robbins and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,”IEEE Trans. Microwave Theory Tech. 47, Issu.11, pp.2075(1999)
[12] C.G. Parazzoli, et. al., “Experimental Verification and Simulation of Negative Index of Refraction Using Snell’s Law,”Phys. Rev. Lett. Vol.90,No.10,pp.107401(2003)
[13] J. B. Pendry, “Negative Refraction Makes a Perfect Lens,”Phys. Rev. Lett. 85,No.18, pp.3966-3369(2000)
[14] M. Notomi, “Theory of light propagation in strongly modulated photonic crystals:Refractionlike behavior in the vicinity of the photonic band gap,” Phys. Rev. B 62, No.16, pp.10696 (2000).
[15] B. Gralak, S. Enoch and G. Tayeb, “Anomalous refractive properties of photonic crystals,” J. Opt. Soc. Am. A 17, No.6, pp.1012-1020(2000).
[16] R. Gajie, R. Meisels, F. Kuchar, and K. Hingerl, “Refraction and rightness in Photonic crystals,” Opt. Express, Vol. 13, No. 21, pp.8596-8605(2005)
[17] Y.-F. Chen, P. Fisher, and F. W. Wise, “Sign of the refractive index in a gain medium with negative permittivity and permeability,”J. Opt. Soc. Am.B 23, No.1, pp.45-20(2006)
[18] K. H. Ho, C. T. Chan and C. M. Soukoulis, “Existence of photonic gap in periodic dielectrics structures,”Physical Review Letters, Vol.65, No.25, pp.3152-3155 (1990)
[19] M. Phihal, A. Shambrook, A. A. Maradudin and P. Sheng, “Two-dimensional photonic band structures,” Optics Communications, Vol.80 , pp.199-204 (1991)
[20] M. Phihal and A. A. Maradudin, “Photonic band structure of two-dimensional systems: The triangular lattice,” Physical Review B, Vol.44, No.16, pp.8565-8571 (1987)
[21] T. Hass, A. Hesse and T. Doll, “Omnidirectional two-dimensional hotonic crystal band gap structures,” Physical Review B, Vol.73,No.4, pp.045130 (2006)
[22] C. Y. Liu and L. W. Chen, “Tunable photonic-crystal waveguide Mach-Zehnder interferometer achieved by nematic liquid-crystal phase modulation,”Optics Express, Vol.12, No.12, pp.2616-2624 (2004)
[23] C. Y. Liu and L. W. Chen, “Tunable band gap in a photonic crystal modulated by a nematic liquid crystal,” Physical Review B, Vol.72, pp.045133 (2005)
[24] Y.Y. Wang and L. W. Chen, “Tunable negative refraction photonic crystals chieved by liquid crystals,” Opt. Express, Vol. 14, No.22, pp.10580-10587(2006)
[25] D. Scrymgeour, N. Malkova, S. Kim, and V. Gopalan, “Electro-Optic control ofthe superprism effect in photonic crystals,” Appl. Phys. Lett. 82,No.19,pp.3176(2003)
[26] S. Xiong and H. Fukshima, “Analysis of light propagation in index-tunable photonic crystals,” J. Appl. Phys. 94, No.2, pp.1286-1288(2003)
[27] W. Park and J. B. Lee, “Mechanically tunable photonic crystal structure,” Applied Physics Letters, Vol.85, No.21, pp4845-4847 (2004)
[28] J. Broeng, D. Mogilevstev, S. E. Barkou and A. Bjarklev, “Photonic crystal fibers: A new class of optical waveguides,” Optical Fiber Technology, Vol.5, pp.305-330 (1999)
[29] P. Russell, “Photonic crystal fibers,” Science, Vol.299, pp.358-362 (2003)
[30] V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ,”Sov. Phys. Usp, Vol.10, No.4, pp.509-514 (1968)
[31] R. A. Shelby, D. R. Smith and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292,No.6, pp.77(2001)
[32] S. Foteinopoulou and C. M. Soukoulis, “Negative refraction and left-handed behavior in two-dimensional photonic crystals,” Phys. Rev. B 67, 235107 (2003).
[33] S. Foteinopoulou, E. N. Economou and C. M. Soukoulis, “Refraction in media with a negative refraction index,” Phys. Rev. Lett. 90, 107402 (2003).
[34] S. Foteinopoulou1 and C. M. Soukoulis, “EM wave propagation in two-dimensional photonic crystals: a study of anomalous refractive effects”
[35] R. Gajić, R. Meisels, F. Kuchar, and K. Hingerl, “Refraction and rightness in photonic crystals” Opt. Express 13, 8596 (2005).
[36] D. Felbacq and R. Smaâli, “Bloch Modes Dressed by Evanescent Waves and the Generalized Goos-Hänchen Effect in Photonic Crystals,” Phys. Rev. Lett. 92, 193902-1 (2004).
[37] A. Martínez and J. Martí, “Negative refraction in two-dimensional photonic crystals : Role of lattice orientation and interface termination,” Phys. Rev. B 71, 235115 (2005)
[38] C. S. Kee, J. E. Kim, and H. Y. Park, “Absolute photonic band gap in a two-dimensional square lattice of square dielectric rods in air,” Phys. Rev. E, Vol.56, No.6, pp.R6291-6293 (1997)
[39] S. Feng, Z. Y. Li, Z. F. Feng, B. Y. Cheng, and D. Z. Zhang, “Imaging properties of an elliptical-rod photonic-crystal slab lens,” Phys. Rev. B, Vol.72, No.7,pp.075101-075101(2005)
[40] Z. Tang, R. Peng and D. Fan, “Absolute left-handed behaviors in a triangular elliptical-rod photonic crystal,” Opt. Express, Vol. 13, No.24,pp.9796-9803(2005)
[41] P. R. Villeneuve and M. Piche, “Photonic band gaps in two dimensional square and hexagonal lattices,”Phys. Rev. B, Vol. 46,No.8, pp.4969-4972(1992)
[42] N. Susa, “Large absolute and polarization-independent photonic band gaps for various lattice structures and rod shapes”,J. Appl. Phys., Vol. 91, Issue.6,pp.3501-3510(2002)
[43] H. Takeda and K. Yoshino, “Tunable refraction effects in two-dimensional photonic crystals utilizing liquid crystals,” Phys. Rev. E 67, 056607 (2003).
[44] Kurt Busch and Sajeev John, “Liquid-Crystal Photonic-Band-Gap Materials: The Tunable Electromagnetic Vacuum,” Phys. Rev. Lett 83, 967 (1999).
[45] S. W. Leonard, J. P. Mondia, H. M. van Driel, O. Toader, and S. John, “Tunable two-dimensional photonic crystals using liquid-crystal infiltration,” Phys. Rev. B 61,R2389 (2000).
[46] K. S. Yee, ”Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,”IEEE Transactions on Antennas and Propagation, Vol.14,No.3, pp.302-307(1996)
[47] A. Taflove and S. C. Hagness, Computational electrodynamics: The finite-difference time-domain method, Artech House, Boston (2000).
[48] J. Min, “The finite element method in electromagnetic,” John Wiley& Sons, Inc.,New York, 2002
[49] Alejandro Martínez and Javier Martí, “Negative refraction in two-dimensional Photonic crystals: Role of lattice orientation and interface termination,” Phys. Rev. B, Vol. 71, NO. 23, pp.235115(2005)
[50] C. Luo, S.G. Johnson, J.D. Joannopoulos, and J.B. Pendry, “All-angle negative refraction without negative effective index,” Physical Review B, Vol. 65, pp.201104 (2002).
[51] W. Belhadj, D. Gamra, F. AbdelMalek, S. Haxha and H. hriha,“Design of two-dimensional photonic crystal structurebased in all-angle negative refractive effect for application in focusing systems,” IET Optoelectron.,Vol. 1, No, 2, P91-95,April 2007
[52] Haifei Zhang, Linfang Shen, Lixin Ran, and Yu Yuan,” Layered superlensing in two-dimensional photonic crystals” OPTICS EXPRESS ,Vol. 14, No. 23, 11178 (2006)
[53] K. Ren, S. Feng, Z. F. Feng, Y. Sheng, Z.Y. Li, B. Y. Cheng and D. Z. Zhang, “ Imaging properties of triangular lattice photonic crystal at the lowest band,” PHYSICS LETTERS A, Vol.348, pp.405-409 (2006)
[54] X. Wang, Z. F. Ren and K. Kempa,” Unrestricted superlensing in a triangular twodimensionalphotonic crystal,” OPTICS EXPRESS, Vol. 12, No. 13, 28 June 2004
[55] Y. Y. Wang, J. Y. Chen, and L. W. Chen, “Optical switches based on partial band gap and anomalous refraction in photonic crystals modulated by liquid crystals”, OPTICS EXPRESS, Vol. 15, NO. 16, 10033 (2007)
[56] E. Hecht, “OPTICS”, 4th Edition(2001)