本篇論文主要是採用數值模擬的方法，研究一維及二維光子晶體的光學特性及應用。對於一維結構的光子晶體，我們透過轉移矩陣法來計算由超導層、介電層相互交替排列的週期組成，並求得其透射及反射的光學頻譜。利用模擬的結果，我們分析不同的相對週期厚度及入射角度對能帶分佈的影響，進一步歸納出各個變量在整體的結構中，所可能扮演的角色及造成的效應。在處理二維組成的光子晶體時，我們利用平面波展開法探求其能帶結構；並應用時域有限差分法，以光學模擬上常被使用的FDTD套裝軟體，進一步透析其電場、磁場、及能量在光子晶體中的傳播方式，以及各個分量在行進中隨時間和空間的變化。最後，我們將探討光通訊系統中，由光子晶體組成的小尺寸光學元件，例如:二維光子晶體波導、耦合共振光學波導、以及指向耦合器，其特有的電磁特性與相關的光學應用。

In this thesis, we theoretically study the optical properties and their applications for one-dimensional and two-dimensional photonic crystals. In the one-dimensional photonic crystals (1DPCs) we use the transfer matrix method to calculate the transmittance and reflectance spectra for a superconductor-dielectric photonic crystal (SDPC). Based on the calculated results we investigate the photonic band structures as a function of film thickness and the angle of incidence as well. As for the two-dimensional photonic crystals (2DPCs), the optical properties will be explored by not only the plane wave expansion but FDTD method. In the final part, we design some useful two-dimensional photonic devices such as the photonic crystal waveguides (PCWs), coupled-resonator optical waveguides (CROWs), and director couplers (DCs).

1._E. Yablonovitch, “Inhibited Spontaneous Emission in Solid-State Physics and Electronics,” Phys. Rev. Lett. 58, 2059 (1987).
2._S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486 (1987).
3._M. Qiu, K. Azizi, A. Karlsson, M. Swillo, and B. Jaskorzynska, “Numerical studies of mode gaps and coupling efficiency for line-defect waveguides in two-dimensional photonic crystals,” Phys. Rev. B 64, 155113 (2001).
4._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, 10696 (2000).
5._Introduction to Photonic Crystals: Bloch’s Theorem, Band Diagrams, and Gaps(But No Defects)Steven G. Johnson and J. D. Joannopoulos, MIT 3rd (February 2003).
6._Alexei A. Erchak, Daniel J. Ripin, Shanhui Fan, Peter Rakich, John D. Joannopoulos, Erich P. Ippen, Gale S. Petrich and Leslie A. Kolodziejski, “Enhanced coupling to vertical radiation using a two-dimensional photonic crystal in a semiconductor light-emitting diode,” Appl. Phys. Lett. 78, 563 (2001).
7._J. C. Knight, J. Broeng, T. A. Birks, and P. St. J.Russell, “Photonic band gap-guidance in opticalfibers,” Sicience, 282, 1476 (1998).
8._Attila Mekis, J. C. Chen, I. Kurland, Shanhui Fan, Pierre R. Villeneuve, and J. D.Joannopoulos, “High Transmission through Sharp Bends in Photonic Crystal Waveguides,” Phys. Rev. Lett. 77, 3787 (1996).
9._M. Agio and C. M. Soukoulis, “Ministop bands in single-defect photonic crystal waveguides,” Phys. Rev. E 64, 055603R (2001).
10._P. G. Luan, and K. D. Chang, “Transmission characteristics of finite periodic dielectric waveguides,” Opt. Express 14, 3263 (2006).
11._欒丕綱、陳啓昌, 光子晶體—從蝴蝶翅膀到奈米光子學, 五南出版社 (2005).
12._Biró, L., P.; Bálint, Zs.; Kertész, K.; Vértesy, Z.; Márk, G., I.; Horváth, Z., E.; Balázs, J.; Méhn, D.; Kiricsi, I.; Lousse, V.; Vigneron, J.-P.: Role of photonic-crystal-type structures in the thermal regulation of a lycaenid butterfly sister species pair; Phys. Rev. E 67, 021907-1(2003).
13._C. Kittel, Introduction to Solid State Physics, 8th Ed., John Wiley & Sons, USA, (2005).
14._J. D. Joannopoulos, R. D. Meade, J. N. Winn, Photonic Crystals—Molding the Flow of Light, Princeton University Press (1995).
15._K. Sakoda, Optical Properties of Photonic Crystals, Springer, Berlin (2001).
16._A. Yariv, P. Yeh, Optical Waves in Crystals, John Wiley & Sons, New York, (1984).
17._A. Taflove, Computational Electrodynamics, Artech House, Boston, (1995).
18._D. K. Cheng, Field and Wave Electromagnetics 2nd, Addison-Wesley, New York, (1989).
19._George B. Arfken and Hans J. Weber, Mathematical Methods for Physicists, 5th Ed, Academic Press, (2000).
20._I. S. Gradshteyn, I. M. Ryzhik, Alan Jeffrey, Table of Integrals, Series, and Products, 5th Ed., Academic Press, (1994).
21._J. D. Jackson, Classical electrodynamics 3rd, John Wiley&Sons, New York, (1999).
22._L. Brillouin, Wave propagation in periodic structure 2nd, Mineola, New York, (1946).
23._J. D. Joannopoulos, R. D. Meade, J. N. Winn, Photonic Crystals—Molding the Flow of Light, Princeton University Press (1995).
24._Kazuaki Sakoda, Noriko Kawai, and Takunori Ito, “Photonic bands of metallic systems. I. Principle of calculation and accuracy,” Phys. Rev. B 64, 045116 (2001).
25._D. F. Kelley, and R. J. Luebbers, “Piecewise linear recursive convolution for dispersive media using FDTD,” IEEE Trans. Antennas and Propagation, 44, 792 (1996).
26._J. B. Judkins, and R.W. Ziolkowski, “Finite-difference time-domain modeling of nonperfectly conducting metallic thin-film gratings,” J. Opt. Soc. A 12, 1974 (1995).
27._K. S. Yee, :Numerical solution of initial boundary value problems involving Maxwell’s equation in isotropic media,” IEEE Trans. Antennas and Propagation, 14, 302 (1996).
28._F. Zheng, Z. Chen, and J. Zhang, “A finite-difference time-domain method without the Courant stability conditions,” IEEE Microwave Guided Wave Lett., 9, 441 (1999).
29._G. Mur, “Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic field equations,” IEEE Trans. Electromagnetic Compatibility, 23, 377 (1981).
30._J. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185 (1994).
31._S.D. Gedney, “An anisotropic perfectly matched layer absorbing media for the truncation of FDTD lattices,” IEEE Trans. Antennas and Propagation, 44, 1630 (1996).
32._Ting-Hang PEI* and Yang-Tung HUANG: Temperature Modulation of the Superprism Effect in Photonic Crystals Composed of the Copper Oxide High-Temperature Superconductor (Japanese Journal of Applied Physics Vol. 46, No. 24, 2007, pp. L593–L595).