在此研究中，我們利用轉移矩陣法來分別設計與分析作為窄帶和平台濾波器的一維光子晶體。以轉移矩陣法為基礎，討論了膜層的結構、膜層的厚度等，進而設計出一個相當良好的窄帶濾波器，並且具有比較大的電場增益。此外，再針對三種高斯型光子晶體(折射率高斯型、厚度高斯型、和雙高斯型的光子晶體)及其缺陷態，同樣採用轉移矩陣法，探討膜厚、折射率、及缺陷態等對透射率的影響。最後，利用傅立葉變換法，分析折射率高斯型和厚度高斯型等光子晶體，其傅立葉頻譜和透射率譜間之關係。

In this study, we designed and analyzed one-dimensional photonic crystals as a narrow-band and a flat filter by using the transfer matrix method. Based on the transfer matrix method, we discussed the thin whole layer structure and the distribution of the thickness of each layer to get a very good narrow-band filter with a nice gain of the electric field of the light wave. Furthermore, we also used the transfer matrix method to calculate their transmissive spectra for three types of Gaussian photonic crystals, such as Gaussian refractive type, and the Gaussian thickness type, and the twin Gaussian type photonic crystal, and we focused and discussed transmissive spectra affected by the material parameters of thickness and refractive index and the defect states appeared in the layer structure. Consequenctly, we analyzed the relation between the Fourier spectra of Gaussian refractive type and Gaussian thickness type photonic crystals and their transmissive spectra.

目 錄

書名頁…………………………………………………………………………i
中文摘要…………………………………………………………………………ii
英文摘要………………………………………………………………………iii
目錄……………………………………………………………………iv
圖錄……………………………………………………………………vi
第一章 序論………………………………………………………………… 1
1.1 光子晶體…………………………………………………………… 1
1.2 光子晶體的應用…………………………………………………… 2
1.3 光子晶體的理論方法……………………………………………… 4
1.4 研究動機…………………………………………………………… 6
1.5 研究目的和方法…………………………………………………… 6
1.6 論文架構…………………………………………………………… 8
第二章 轉移矩陣法…………………………………………………… 9
2.1 基本的Maxwel方程式…………………………………………… 9
2.2 轉移矩陣法………………………………………………………… 10
2.2.1 正向入射之單層膜矩陣…………………………………………… 11
2.2.2 斜向入射之膜矩陣………………………………………………… 15
2.2.3 多層膜矩陣………………………………………………………… 16
2.3 轉移矩陣程式基本說明…………………………………………… 18
2.4 對轉移矩陣法進行驗證…………………………………………… 18
第三章 折射率高斯型光子晶體平台濾波器……………………………… 32
3.1 典型的一維光子晶體平台濾波器………………………………… 32
3.2 高斯模型之平台濾波器…………………………………………… 32
3.3 折射率高斯型光子晶體…………………………………………… 33
3.4 傅立葉轉換………………………………………………………… 46
第四章 厚度高斯型和雙高斯型光子晶體平台濾波器…………………… 50
4.1 厚度高斯型光子晶體……………………………………………… 50
4.2 厚度高斯型光子晶體之傅立葉分析……………………………… 57
4.3 雙高斯型光子晶體………………………………………………… 59
第五章 結論………………………………………………………………… 62
參考文獻……………………………………………………………………… 64

參考文獻
[ 1 ] E. Yablonvitch, “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 Super-lattice”, Phys. Rev. Lett. 58, 2486 (1987).
[ 3 ] J. C. Knight, T. A. Birks, P. St. J. Russell and J. P. de Sandro, “Properties of photonic crystal fiber and the effective index model” , J. Opt. Soc. Am., 15, 748(1998).
[ 4 ] J. D. Joannopoulos, P. R. Villeneuve and S. Fan, “Photonic crystals: putting a new twist on light” , Nature, 386, 143(1997).
[ 5 ] S.-Y. Lin, E. Chow, V. Hietala, P. R. Villeneuve, and J. D. Joannopoulos, “Experimental Demonstration of Guiding and Bending of Electromagnetic Waves in a Photonic Crystal” , Science, 282, 274(1998).
[ 6 ] A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High Transmission through Sharp Bends in Photonic Crystal Waveguides” , Phys. Rev. Lett., 77, 3787(1996).
[ 7 ] R. Dalichaouch, J. P. Armstrong, S. Schultz, P. M. Platzman, and S. L. McCall “Microwave localization by two-dimensional random scattering”, Nature, 354, 53(1991).
[ 8 ] P. Halevi and F. Ramos-Mendieta, “Tunable Photonic Crystals with Semiconducting Constituents”, Phys. Rev. Lett., 85, 1875(2000).
[ 9 ] S. F. Mingaleev and Yu. S. Kivshar, “Self-Trapping and Stable Localized Modes in Nonlinear Photonic Crystals”, Phys. Rev. Lett., 86, 5474(2001).
[10] V. Berger, “Nonlinear Photonic Crystals”, Phys. Rev. Lett., 81, 4136(1998).
[11] N. G. R. Broderick, G. W. Ross, H. L. Offerhaus, D. J. Richardson, and D. C. Hanna, “Hexagonally Poled Lithium Niobate: A Two-Dimensional Nonlinear Photonic Crystal”, Phys. Rev. Lett., 84, 4345(2000).
[12] M. Borodiitsky, T. F. Krauss, R. Coccioli, R. Bhat, and E. Yablonvitch, “Light extraction from optically pumped light-emitting diode by thin-slab photonic crystals” , Appl. Phys. Lett., 75, 1036(1999).
[13] J. M. Drake and A. Z. Genack, “Observation of nonclassical optical diffusion”, Phys. Rev. Lett., 63, 259(1989).
[14] K. Sakoda. “Optical Properties of Photonic Crystals”, Springer-Vwelag, New York, 2001.
[15] K. M. Leung, and Y.F. Liu, “Photonic band structures: The Plane-Wave Method”, Phys. Rev. B, 41, 10188 (1990).
[16] E. Yablonvitch, “Photonic band-gap crystals”, J. Phys.: Condens. Matter, 5, 2443 (1993).
[17] V. Yannopapas, N. Stefanou and A. Modinos, “Theoretical analysis of the photonic band structure of face-centered cubic colloidal crystals”, J. Phys.: Condens. Matter, 9, 10261 (1997).
[18] S. A. Bulgakov and M. Nieto-Vesperinas, “Field distribution inside one-dimensional random photonic lattices”, J. Opt. Soc. Am., 15, 503 (1998).
[19] K. S. Yee, “Numerical Solution of Initial Boundary Value Problems Involving Maxwell’s Equations in Isotropic Media”, IEEE trans. Antennas and propagation, AP-14, 302 (1996).
[20] A. Taflove and K. Umashankar, “A Hybrid Moment/Finite-Difference Time-Domain Approach to Electromagnetic Coupling and Aperture Penetration into Complex Geometries”, IEEE trans. Antennas and propagation, AP-30, 617 (1982).
[21] R. W. Ziolkowski and M. Tanaka, “FDTD analysis of PBG waveguides, power splitters and switches”, Optical and Quantum Electronics, 31, 843 (1999).
[22] X. Wang, X.-G. Zhang, Q. Yu, and B. N. Harmon, “Multiple-scattering theory for electromagnetic waves”, Phys. Rev. B: Condens. Matter, 47, 4161(1993).
[23] N. Stefanou, V. Yannopapasa, and A. Modinosb, “Heterostructures of photonic crystals: frequency bands and transmission coefficient”, Computer Phys. Comm., 113, 49(1998).
[24] A. Cucinotta, S. Selleri, L. Vincetti, and M. Zoboli, “Perturbation Analysis of Dispersion Properties in Photonic Crystal Fibers through the Finite Element Method” , Journal of Lightwave Technology, 20, 1433(2002).
[25] Jian Zi, Jun Wan, and Chun Zhang, “Large frequency range of negligible transmission in one-dimensional photonic quantum well structures” , Appl. Phys. Lett., 73, 2084(1998).
[26] Feng Qiao, Chun Zhang, and Jun Wan, Jian Zi, “Photonic quantum-well structures: Multiple channeled filtering phenomena” , Appl. Phys. Lett., 77, 3698(2000).
[27] Xin Wang, Xinhua Hu, Yizhou Li, Wulin Jia, Chun Xu, Xiaohau Liu, and Jian Zi, “Enlargement of omnidirectional total reflection frequency range in one-dimensional photonic crystals by using photonic heterostructures ” , Appl. Phys. Lett., 80, 4291(2002).
[28] T. Nakagawa, N. Kawai, K. Ohta, and M. Kawashima, “Superlattices & Microstruct” , 1, 217(1985).
[29] I. Gomez, F. Dominguez-Adame, E. Diez, and V. Bellani, “Electron transport across a Gaussian superlattices” , J. Appl. Phys., 85, 3916(1996).
[30] H. H. Tung and C. P. Lee, “An Energy Band-Pass Filter Using Superlattice Structures ” , IEEE J. Quantum Electron, 32, 507(1996).
[31] 劉康懷，“光子晶體平台濾波器之設計與分析 ”，私立元智大學光電所碩士論文，民國94年。
[32] 歐陽徵標、楊琳玲、許桂雯、阮雙琛、李景鎮， “一維缺陷光子晶體的模式特性研究”，光電子激光, 15, 63(2005).
[33] 歐陽徵標、朱駿、李景鎮， “兩端有慢變結構的光子晶體的能帶特性研究”，光學學報, 22, 610(2002).