光子晶體藉由適當的設計，將具光子能隙的特性。二維光子晶體由於製造上較三維光子晶體容易，所以目前應用範圍較廣。本文研究三角形晶格空氣柱的二維光子晶體，基材為矽(介電常數為12.096)，利用時域有限差分法配合週期性邊界條件，找出其光子能帶結構與能隙。藉由改變空氣柱的截面形狀，討論截面形狀與能隙形成的關係，柱體形狀包括圓柱、方柱、第一型六角柱、第二型六角柱、第一型八角柱及第二型八角柱。

　　分析結果發現三角形晶格的空氣柱是良好的TE能隙光子晶體，各個結構的TE能隙至少都在40%以上，而在第二能帶與第三能帶之間的TM能隙將會決定完全能隙的大小。在六種結構中，以第二型六角柱的完全能隙寬度最大，較圓柱的完全能隙寬度更佳，能隙大小可達17.7%。

　　Photonic crystals could have photonic band gap properties by appropriate design. The fabrication of two-dimensional (2D) photonic crystals is easier than that of three- dimensional ones, so applications of 2D photonic cryastals at present are wider. 2D photonic crystals with triangular lattices of air cylinders are investigated in this thesis, and the substrate adopted is silicon (dielectric constant 12.096). With the periodic boundary condition, the finite difference time domain method is used to find photonic band structures and photonic band gaps. By changing the cross section shapes of air cylinders, the relation between the cross section shapes and band gaps is discussed. The shapes of cylinders are circle, square, the hexagon of the first type, the hexagon of the second type, the octagon of the first type, and the octagon of the second type.

　　The results show that triangular lattices of various air cylinders are good for the formation of TE band gap, and TE band gap of every structure is more than 40% at least. The complete band gaps would be decided by the TM band gaps between the second band gaps and the third band gaps. In the six structures, the complete band gap of the hexagon of the second type is the largest, up to 17.7% can be obtained.

【1】 http://home.earthlink.net/~jpdowling/pbgbib.html
【2】 G. Vogel, “Breakthrough of the Year”, Science 286, pp.2238-2239 (1999).
【3】 http://ab-initio.mit.edu/photons/tutorial/
【4】 T. Baba, “Photonic Crystals and Microdisk Cavities Based on GaInAsP–InP System”, IEEE Journal of Selected Topics in Quantum Electronics 3, pp. 808-830(1997).
【5】 http://www.blazephotonics.com/
【6】 M. Boroditsky, T. F. Krauss, R. Coccioli, R. Vrijen, R. Bhat, and E. Yablonovitch, “Light Extraction from Optically Pumped Light-Emitting Diode by Thin-Slab Photonic Crystals”, Appl. Phys. Lett. 75, pp. 1036-1038 (1999).
【7】 M. Loncar, T. Doll, J. Vuckovic, and A. Scherer, “Design and Fabrication of Silicon Photonic Crystal Optical Waveguides”, J. Lightwave Tech. 18, pp. 1402-1411 (2000).
【8】 E. Yablonovitch, “Inhibited Spontaneous Emission in Solid-State Physics and Electronics”, Phys. Rev. Lett. 58, pp. 2059-2602 (1987).
【9】S. John, “Strong Localization of Photons in Certain Disordered Dielectric Superlattice”, Phys. Rev. Lett. 58, pp. 2486-2489 (1987).
【10】 E. Yablonovitch and T. J. Gmitter, “Photonic Band Structure: The Face-Centered-Cubic Case”, Phys. Rev. Lett. 63, pp. 1950-1953 (1989).
【11】 K. M. Ho, C. T. Chart, and C. M. Soukoulis, “Existence of a Photonic Gap in Periodic Dielectric Structures”, Phys. Rev. Lett. 65, pp. 3152-3155 (1990).
【12】 E. Yablonovitch, T. J. Gmitter, and K. M. Leung, “Photonic Band Structure: The Face-Centered-Cubic Case Employing Nonspherical Atoms”, Phys. Rev. Lett. 67, pp. 2295-2298 (1991).
【13】 Z. Zhang and S. Satpathy, “Electromagnetic Wave Propagation in Periodic Structure: Bloch Wave Solution of Maxwell’s Equations”, Phys. Rev. Lett. 65, pp. 2650-2653 (1990).
【14】 K. M. Leung and Y. F. Liu, “Full Vector Wave Calculation of Photonic Band Structure in Face-Centered-Cubic Dielectric Media”, Phys. Rev. Lett. 65, pp. 2646-2649 (1990).
【15】 H. S. Sozuer, J. W. Haus, and R. Inguva, “Photonic Bands: Convergence Problem with the Plane-Wave Method”, Phys. Rev. B 45, pp. 13962-13972 (1992).
【16】 C. T. Chan, Q. L. Yu, and K. M. Ho, “Oder-N Spectral Method for Electromagnetic Wave”, Phys. Rev. B 51, pp. 16635-16642 (1995).
【17】 J. B. Pendry and A. MacKinnon, “Calculation of Photon Dispersion Relation”, Phys. Rev. Lett. 69, pp 2772-2775 (1992).
【18】C. M. Soukoulis, “The History and a Review of the Modelling and Fabrication of Photonic Crystals”, Nanotechnology 13, pp. 420-423(2002).
【19】H. Benisty, M. Rattier and S. Olivier, “Two-Dimensional Photonic Crystals: New Feasible Confined Optical Systems”, C. R. Physique 3, pp. 89-102 (2002).
【20】M. Plihal, A. Shambrook, A. A. Maradudin, and P. Sheng, “Two-Dimensional Photonic Band Structures”, Optics Communications 80, pp. 199-204(1991).
【21】M. Plihal and A. A. Maradudin, “Photonic Band Structures of Two-Dimensional Systems: The Triangular Lattice”, Phys. Rev. B 44, pp. 8565-8571(1991).
【22】P. R. Villeneuve and M. Piché, “Photonic Band Gaps in Two-Dimensional Square and Hexagonal Lattices”, Phys. Rev. B 46, pp. 4969-4972(1992).
【23】P. R. Villeneuve and M. Piché, “Photonic Band Gaps in Two-Dimensional Square Lattices: Square and Circular Rods”, Phys. Rev. B 46, pp. 4973-4975(1992).
【24】R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Existence of a Photonic Band Gap in Two Dimensions”, Appl. Phys. Lett. 61, pp. 495-497 (1992).
【25】D. Cassagne, C. Jouanin, and D. Bertho, “Photonic Band Gaps in Two-Dimensional Graphite Structure”, Phys. Rev. B 52, pp. R2217-R2220(1995).
【26】T. Baba and T. Matsuzaki, “Theoretical Calculation of Photonic Gap in Semiconductor 2-Dimensional Photonic Crystals with Various Shapes of Optical Atoms”, Jpn. J. Appl. Phys. 34, pp. 4496-4498 (1995).
【27】D. Cassagne, C. Jouanin, and D. Bertho, “ Hexagonal Photonic- Band-Gap Structures”, Phys. Rev. B 53, pp. 7134-7142(1996).
【28】C. M. Anderson and K. P. Giapis, “Larger Two-Dimensional Photonic Band Gaps”, Phys. Rev. Lett. 77, pp. 2949-2952(1996).
【29】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 56, pp. R6291-R6293(1997).
【30】S. Y. Li , B. Y. Gu, and G. Z. Yang, “Large Absolute Band Gap in 2D Anisotropic Photonic Crystals”, Phys. Rev. Lett. 81, pp. 2574-2577(1998).
【31】X. H. Wang, B. Y. Gu, Z. Y. Li, and G. Z. Yang, “Large Absolute Band Gaps Created by Rotating Noncircular Rods in Two-Dimensional Lattice”, Phys. Rev. B 60, pp. 11417-11421(1999).
【32】M. Qiu and S. He, “Optimal Design of a Two-Dimensional Photonic Crystal of Square Lattice with a Large Complete Two-Dimensional Bandgap”, J. Opt. Soc. Am. B 17, pp. 1027-1030(2000).
【33】L. Shen, Z. Ye, and S. He, “Design of a Two-Dimensional Photonic Crystals with Large Absolute Band Gaps Using a Genetic Algorithm”, Phy. Rev. B 68, pp. 035109-1~035109-5(2003).
【34】L. F. Marsal, T. Trifonov, A. Rodriguez, J. Pallares, and R. Alcubilla, “Large Absolute Photonic Band Gap in Two-Dimensional Air-Silicon Structures”, Physica E 19, pp. 580-585(2003).
【35】A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method, Artech House, Boston (1995).
【36】 J. D. Jaonnopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals, Princeton Unversity Press, Princeton, New Jersey (1995).