臺灣博碩士論文加值系統

分析及模擬電場在光波導元件(optical waveguide device)中傳播的行為，傳統常用的方法為有限差分法(finite difference method)、有限元素法(finite element method)及光束傳播法(beam propagation method)等。而在本文中提出另一種方法，將波導元件的折射率(refractive index)分佈函數與電場以有限項傅立葉級數展開，代入原來基於馬克斯威爾波動方程式(Maxwell’s wave eq.)所描述的電場偏微分方程式，便可將原微分方程轉成一矩陣形式的特徵值問題(eigenvalue problem of matrix form)，再以線性代數的方法求得其解，得到符合原方程式的所有特徵模態(eigen modes)並以傅立葉餘弦的形式呈現；然而在波導元件中之電場只會以這些特徵模態的能量形式存在。基於模態疊加的理論(supermode theoty)，在波導元件中的電場可視為這些特徵模態之線性組合。
當電場往前傳播時，由於波導元件在不同位置幾何結構或折射率的改變，所以在不同位置所能存在的能量模態也不同。根據模態耦合法則(couple principle)可算出各模態的能量，若波導存在多個模態，則總電場可視為各別模態依其各自的傳播常數(propagation constant)然後疊加的結果，然而在電場的傅立葉餘弦級數的形式已求得的優勢，在進行模態耦合係數的積分計算時都可以求得解析解，不需以數值積分法來進似，所以計算極為精確有效率。所以這樣的計算方式可準確的模擬多個模態干涉的情形，如多模干涉元件(multimode interference device；MMI device)的行為，且會有高效率的表現。

第一章 簡介................................................4
第二章 理論依據 .........................................6
2-1、多層平面波導新數值解法 ...........................6
2-2、傳導模態(Guided-Mode)分析 .......................12
2-3、適當項數的決定與收歛性探討 ......................16
2-3.1 項數的決定 ...................................17
2-3.2 收歛性探討 ...................................22
2-4、平均功率的計算 ..................................25
第三章 元件的模型建立與分析結果 ........................27
3-1、錐形漸變結構(Taper) .............................27
3-2、樹狀分支結構(Y-Branch) ..........................34
3-3、多模干涉元件(Multi-Mode Interference Device) ....40
3-3.1 多模干涉元件之原理 ...........................40
3-3.2 多模干涉元件之模型建立與分析結果 .............43
第四章 結論 ............................................53
參考文獻 ................................................54

[1] C. C. Teng, “Traveling-wave polymeric optical intensity modulator with more than 40 GHz of 3-dB electrical bandwidth”. Appl. Phys. Lett., Vol. 60, No.13, pp. 1538-1540, March 1992.
[2] K. Al-hemyari, G. F. Doughty, C. D. W. Wilkinson, A. H. Kean, and C. R. Stanley, “Optical loss measurements on GaAs/GaAlAs single-mode Y-junctions and waveguide bends”, J. Lightwave Technol., Vol. 11, pp. 272-276, 1993.
[3] H. Hatami, P. L. Chu and M. J. Lederer, “A new low-loss wide-angle Y-branch configuration for optical dielectric slab waveguide”,IEEE Photonics Technol. Lett., Vol. 6, pp. 528-530.
[4] Jiri’ Ctyroky’ and Lars Thyle’n, “analysis of a directional coupler by coupled modes of a single waveguide”, Optics Letters, Vol. 19, No. 20, pp. 1621-1623, Oct. 1994.
[5] O. Bryngdahl, “Image formation using self-imaging techniques,” J. Opt. Soc. Amer., vol. 63, pp. 416-419, 1973.
[6] Lars Oddvar Lierstuen and Aasmund Sudb “8-Channel Wavelength Division Multiplexer Based on Multimode Interference Couplers” IEEE Photonics Technol. Lett., Vol. 7, No. 9, Set. 1995.
[7] Boston, Artech House“Numerical Techniques for Modeling Guided-Wave Photonic Devices” IEEE Journal of Selected Topics in Quantum Electronics, Vol. 6, No. 1, January/February 2000.
[8] Takanori Nakano, Kazutaka Baba, and Miltsunobu Miyagi, “Insertion loss and extinction ratio of a surface plasmon-polarition polarizer: theoretical analysis”, J. Opt. Soc. Am. B, Vol. 11, No. 10, pp. 2030-2035, Oct. 1994.
[9] 黃能添”A New Numerical Method for Slab Waveguide Problems” 國立清華大學碩士論文, 1998.
[10] J.H. Wilkinson ．C. Reinsch “ Linear Algebra in Numerical Analysis” Similarity Reduction of a General Matrix to Hessenberg Form , 339~358 Contribution Ⅱ/13 and Eigenvectors of Real and Complex Matrices by LR and QR triangularizations, 373~395 Contribution Ⅱ/15, 1971.
[11] Richard Syms John Cozens “Optical Guided Waves and Devices”International Editions pp.138-144.
[12] Bahaa E. A. Saleh Malvin Carl Teich “Fundamentals of Photonics” pp.248-252.
[13] Richard L. Burden J. Douglas Faires “Numerical Analysis” Sixth Edition pp.65-75.
[14] Kendall E. Atkinson lowa City, lowa “Elementary Numerical Analysis 2/e” pp. 425-432 August 1992.
[15] Lucas B. Soldano and Erik C. M. Pennings, Member, IEEE “Optical Multi-Mode Interference Device Based on Self-Imaging: Principles and Applications” Journal of Lightwave Tech. Vol. 13, No. 4, April 1995.
[16] N. S. Kapany and J. J. Burke, Optical Waveguides. New York: Academic, 1972.