中文摘要
光纖模態的問題，從1970年後就有許多相關的分析，近二十年來光纖的結構及波導參數(如載波波長、光纖色散特性)經歷許多改變。而未來所採用的光纖結構也有可能會變化。本篇論文提供一種新的分析理論，推導出一個二階的橫向磁場聯立常微分方程式及其相關的邊界條件。接著我們使用有限差分近似的方式將其轉換成矩陣特徵值、特徵向量的問題，即可找出所分析光纖的傳播常數與場型分佈。
本篇論文中，我們可以處理階梯變化或連續變化的折射率，不需使用特殊函數，計算量也不會因為折射率的複雜變化而增加。磁場的耦合方程式除了採用有限差分的近似外，也可以用基礎函數展開的方式去分析模擬。本篇論文計算目前市場上所採用的光纖，如8/125、9/125、50/125、62.5/125；藉由本篇論文的分析並跟step index正解比較後除了可以清楚地看到光纖內磁場變化的情形，也可發現，在使用有限差分去分析時，取點愈密光纖間距Δx愈小，所得到的模態解就愈逼近正解。最後從分析結果得知，傳播常數與正解的差距在Δx< 時可以小於0.1%。

Abstract

There have been many studies on model characteristics of optical fibers since 1970. In the past twenty years we have seen changes in the index profiles and operating parameters (such as the carrier wavelength and fiber dispersion) of the fibers. We shall expect some change in the future as well. In this thesis we propose a 2nd-order coupled transverse magnetic field ordinary differential equations and associated boundary conditions for analyzing fiber modes with complex index profiles. Using finite-difference approximation, we convert the equations into matrix eigenvalue-eigenvector equations. These are then numerical solved (suing many commercial numerical softwares such as Matlab) to obtain propagation constants and field distribution inside the fibers.

We shall show that our method do not require the use of cylindrical functions nor will the computation increase due to the increase of complexity in the index profile. In addition to finite-difference approximation, we may expand the model solutions in terms of elementary functions. In this thesis we computed many commercially available optical fibers such as 8/125, 9/125 step index fibers and 50/125, 62.5/125 graded index fibers. We compared our results against exact solution (the step index case) and found that we have converging solutions as we reduce the fiber sampling grid size Δx. The difference in propagation constant is less than 0.1% when Δx is < λ/6 .

誌謝 I
中文摘要 II
英文摘要 III
目錄 IV
圖目錄 VI
表目錄 IX
第一章、導論 1
1-1：簡介 1
1-2：研究動機 1
1-3：有限差分解法 4
第二章、理論與公式之推導 5
2-1：馬克思威爾方程式與向量基本公式 5
2-2：不均勻材料之電場磁場方程式 7
2-3：磁場之耦合方程式 9
第三章、數值分析與計算 17
3-1：前言 17
3-2：單模光纖及單模的條件 17
3-3：磁場之邊界條件 19
3-4：步階型光纖(step-index fiber)在m=0時之計算 22
3-5：步階型光纖(step-index fiber)在m=1時之計算 35
第四章、結果與討論 47
參考文獻 53
中英對照表 54

參考文獻
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