本論文主要目的在研究二維微共振腔之光學圖案，並利用有限時域差分法以及有限元素法來模擬設計二維微共振腔之光學圖案，利用改變微共振腔的形狀來調製出具備方向性放射的光學圖案，此外，可以使用非對稱的微共振腔以設計出具有高品質因子的微共振腔。

This dissertation addresses optical patterns in 2-dimensional micro-cavities. Finite difference time domain method and finite elements method are used to simulate the optical patterns with 2-dimensional micro-cavity. Unidirectional emission could be manipulated by tailoring cavity shape. In addition, cavity quality factor(Q) could be estimated in deformed micro-cavities for designing high Q resonators.

中文摘要 ……………………………………………………………………………………… i
英文摘要 ……………………………………………………………………………………… ii
誌謝 ……………………………………………………………………………………… iii
目錄 ……………………………………………………………………………………… iv
表目錄 ……………………………………………………………………………………… vi
圖目錄 ……………………………………………………………………………………… vii
一、 緒論 1
1.1 光學微共振腔的應用 1
1.2 光子晶體 3
1.3 橫向模態與方向性控制 4
1.4 研究動機 6
二、 模擬方法與光子晶體介紹 7
2.1 有限時域差分法(FDTD) 7
2.1.1 FDTD理論介紹 8
2.1.2 FDTD的邊界條件 17
2.1.3 平面波模擬的邊界條件設定 20
2.1.4 穩定條件 21
2.2 有限元素法 22
2.2.1 里茲法 22
2.2.2 有限元素法之求解 26
2.3 光子晶體共振腔設計 28
2.3.1 週期函數、傅立葉級數與倒晶格 29
2.3.2 布洛赫定理 30
2.3.3 布里淵區 33
2.3.4 光子晶體頻帶結構 36
2.4 共振腔Q值計算 37
三、 FDTD光學微共振腔模擬 39
3.1 FDTD模擬結構一 39
3.2 FDTD模擬結構二 42
四、 FEM電磁模擬 47
4.1 FEM模擬結構 47
4.2 FEM模擬結構二 54
4.3 FEM模擬結構三 60
五、 結論 65
參考文獻 66

[1] Kim, Se-Heon; Kim, Sun-Kyung; Lee, Yong-Hee,Physical Review B, vol. 73, Issue 23, id. 235117(2006)
[2] K. J. Vahala, Nature (London) 424, 839 (2003)
[3] M. Kneissl and N. M. Johnson,Appl. Phys. Lett. 86, 241110 (2005);
[4] 欒丕綱，陳啟昌，2005，光子晶體：從蝴蝶翅膀到奈米光子學，一版，五南圖書出版股份有限公司，台北市。
[5]R.Holland,”Thread: a Free-Field EMP Coupling and Scattering Code”, IEEE Trans. Nuclear Science, Vol.24, pp.2416-2421(1977)
[6] K.S.Yee,”Numerical Solution of Initial Boundary Value Problems Involving Maxwell’s Equations in Isotropic Media,”IEEE Trans. Antennas and Propagation, Vol.14, pp.302-307(1966)
[7] Sylvester, Peter P. Pelosi, Giuseppe, “Finite Elements for Wave Electromagnetic:
Methods and Techniques”, New York: Institute of Electrical and Electronics Engineers.
[8] G.Mur,”Absorbing Boundary Conditions for the Finite-Difference Approximation of the Time-Domain Electromagnetic-Field Equations ,”IEEE Trans. Electromagnetic Compatibility Vol.23,pp.377-382(1981)
[9] Jean-Pierre Berenger,”A Perfectly Matched Layer for the Absorption of Electromagnetic Waves”, J.Com.Physics Vol.114, pp. 185-200(1994)
[10] Optiwave, 2005, OptiFDTD Technical Background and Tutorial
[11] 王怡喬，2005，以有限差分時域（FDTD）法模擬一維金屬與介電層光子晶體之電磁波傳輸，國立清華大學，碩士論文
[12] Chern, G.D., et al.; “Unidirectional Lasing from InGaN Multiple-Quantum-Well Spiral-Shaped Micropillars”; Applied Physics Letters, vol. 83, No. 9; pp. 1710-1712; (Sep. 1, 2003).
[13] Boriskina, S. V., ‘’Directional Emission, Increased Free Spectral Range, and Mode Q-Factors in 2-D Wavelength-Scale Optical Microcavity Structures’’ ,Selected Topics in Quantum Electronics, IEEE Journal of Publication Date,vol.12,pp.1175-1182(2006)