摘要

本論文主要探討平面波展開法(plane-wave-expansion method, PWEM)和轉移矩陣法(transfer-matrix method, TMM)的收斂性問題，我們計算光子晶體的能帶結構，並將所求得的本徵值(eigenvalue)做多項式展開(power series expansion)研究其收斂速度。

本文主要可分成四個部份，第一個，討論不同填充率時，平面波展開法的收斂速度，結果發現在填充率為0.5時收斂速度最慢，增減填充率增加收斂速度都會變快。第二個，將原本dielectric function不連續的邊界分別做線性(linear)處理和改為raised cosine function結果收斂速度比原來邊界不連續的情況，收斂速度快很多，而raised cosine function效果要比linear 的效果較好，但是只在特定的情況下，所以知道收斂速度不只和Gibb’s phenomenon有關而已。第三個，平面波展開法對 展開和 展開的收斂性分析比較，結果發現對 展開收斂速度比較快，第四個，比較不同展開個數對轉移矩陣法收斂性的影響，結果發現如同平面波展開法，轉移矩陣法計算所得的穿透率隨著展開項數增多，逐漸收斂。

Abstract

In this thesis, we investigate the convergence of plane-wave-expansion method (PWEM) and transfer-matrix method (TMM). We calculate the bandstructure of photonic crystals and expand the eigenvalue of PWEM and TMM with power law method. The convergence is based on the x value of expansion.

This thesis consists of four parts. First, we focus on the convergence of PWEM on different filling fractions. The results show that the convergence is the slowest when filling fraction is 0.5. Second, the boundary changes from discontinuous to continuous. And then, we find that the convergence on continuous boundary is faster than that on discontinuous boundary. Third, the convergence on the expansion of and are investigated. The results show that the convergence on expansion of is faster than that of . Finally, TMM can converges with the limited number of series expansion.

參考文獻

[1] E. Yablonovitch, Phys. Rev. Lett. 58, 2059 (1987)
[2] S. Joho, Phys. Rev. Lett. 58, 2486 (1987)
[3] S. L. McCall, P. M. Platzman, R. Dalichaouch, D. Smith, and S. Schultz, Nature 354, 53 (1991)
[4] E. Özbay, G. Tuttle, R. Biswas, K.M. Ho, J. Bostak, and D. M. Bloom, Applied Physics Letters, 65, 1617 (1994)
[5] J. C. Knight, J. Broeng, T. A. Birks, and P. St. J. Russell, Science 282, 1476 (1998)
[6] Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoous, and E. L. Thomas, Science 282, 1679 (1998)
[7] S. Y. Lin, V. M. Hietala, L. Wang et al., Opt. Lett. 21, 1771 (1996)
[8] H. Kosaka et al., Phys. Rev. B58, R10096 (1998)
[9] E. R. Brown, “Millimeter-wave application photonic crystals”, presented at the Workshop on Photonic Bandgap structures, sponsored by U.S. Army Research Office, January 28-30, 1992, Park City Utah
[10] E. R. Brown, C. D. Parker, and E. Yablonovitch, J. Opt. Soc. Am. B 10 , 404 (1993)
[11] J. B. Pentry, A. Mackinnon, Phys. Rev. Lett. 69, 2772 (1992)
[12] K. S. Yee, IEEE Trans on Antenna and Propagation 14, 302 (1996)
[13] H. S. Sozuer , J. W. Haus, and R. Inguva, Phys. Rev. B 45,13962 (1992)
[14] Linfang Shen and Sailing He, J. Opt. Soc. Am. A 19 , 1021 (2002)
[15] B.Temelkuran, H Altug, and E.Ozbay, IEE Proc.: Optoelectron. 145, 409 (1998)
[16] Zhi-Yuan Li and Kai-Ming Ho, Phys. Rev. B 67, 165104 (2003)
[17] 林鳳瑜，兆赫頻段二維與三維光子晶體之製作與量測，清華大學物理系碩士論文，民92
[18] 林建宏，光子晶體的理論分析法，清華大學物理系碩士論文，民93