光纖模態的問題[1] [2]，從1970年後就有許多相關的分析，近二十年來光纖的結構及參數(如載波波長、光纖色散特性)經歷許多改變。而未來所設計的光纖結構也會有許多的變化。本文利用耦合z分量電磁場（coupled ， ）方法，推導出一N階橫向介面電磁場耦合聯立方程式，來分析圓柱波導及雷射與光纖間的耦合問題[3]。再將其轉換成矩陣特徵值、特徵向量的問題，即可找出光纖（圓柱波導）的傳播常數與場型分佈。
本篇論文中，在數學模型中，我們可以處理任意層數層結構的Step-Index Fiber、W-Type Fiber及Dielectric Tube等階梯變化的折射率，由於變數的減少、實數對稱矩陣的優點，及利用Wronskians行列式的簡化，將只有在對角線的元素才有貝索函數（Bessel function）的存在，因此我們將可以大大減少推導的複雜度及計算量。在數值計算結果方面，我們計算了目前市面上較普遍所採用的光纖型態，如Step-Index Fiber；及其他型態如W-Type Fibers及Dielectric Tube。藉由本篇論文的分析並跟傳統教科書[4]中的Step-Index Fiber推導正解比較後除了可以清楚地看到光纖內電磁場變化的情形，也可發現，使用本論文的分析方法不僅較快速且容易。
由參考文獻[5] 中發現，本篇論文除了計算出數值結果驗證數學模型之外，不僅推導過程容易簡單而且可推展至任意多層；而不需要從新推導數學式。

Since 1970, there have been many analytic theories studying waveguide modes in optical fibers. As the years progressed, the structure of optical fiber and its characteristics have undergone many changes. In recent years, the methods of analysis have also evolved into a more numerical style, such as the finite element and finite difference approach. In this thesis, we propose a semi-analytic, coupled Ez and Hz method for solving a multiple layered piecewise constant cylindrical dielectric waveguides.

In our mathematical model, we are able to handle any arbitrary layered structure, in particular, the step-index fiber, W-type fiber, and dielectric tubes. Within each layer, we express the azimuthal field components in terms of Bessel functions whose coefficients are determined by two pairs of Ez and Hz components that define the layer. By equating the transverse fields (above) on either side of each layer interface the coupled field equations are derived. The field components are either real, or purely imaginary, this allows us to formulate our matrix in real arithmetic. Further simplification is possible by using the Wronskians of the Bessel functions. The resulting matrices are both real and symmetric, which is consistent with the reciprocity principle. Compared to the traditional formulation from the early 1970s, we have reduced the variables by half and extended the formulation to include any arbitrary number of layers.

Numerous numerical results are presented in this thesis for all three types of fiber previously mentioned. Both lower-order modes as well as higher-order modes including TE, TM, HE, and EH modes are presented and discussed. Our formulations are compared to that of textbook formulas for the simple two-layered step index fiber, and are found to be identical.

目錄
頁次
誌謝……………………………………………………………I
中英文摘要………………………………………………….II
內容目錄……………………………………………………. IV
圖表目錄…………………………………………………….VI
第一章、導論…………………………………….……………1
1-1：簡介………………………………………………………1
1-2：馬克思威爾方程式………………………………………5
第二章、任意多層電介質波導……………………………….8
2-1：任意多層電介質波導理論……………..………….…….8
2-2：多層不連續介質波導之介面電磁場耦合方程式…...…16
2-3：TE及TM模態分析…………………………….………30
2-4：結論……………………………………………..……….38
第三章、數值分析與計算……………………………………50
3-1：前言……………………………………………...………50
3-2：Fiber mode的條件………………………….……….…50
3-3：Step-Index Fiber…………………………….……..……52
3-4：W-Type Fiber………..……………………………..……68
3-5：Dielectric Tubes…………………………………..….…76
第四章、結果與討論…………………………………………96
參考文獻……………………………………………………98
中英對照表……… ………………………… .………….…100

參考文獻
[1] Snitzer, E., ”Cylindrical Dielectric Waveguide Modes,” J. Opt. Soc. Am. 51:491, 1961
[2] Gloge D, “Weakly guiding fibers”, Appl.Opt.10:2252,1971
[3]李順天, ”Investigation of coupling between laser diodes and tapered fibers, 2-D case”, 國立中山大學光電工程研究所碩士畢業論文,2000
[4] Amon Yariv, ”Optical electronics in modern communications”, vol.3, pp.77-83, 1996
[5] Achint Kapoor G.S.Singh, ”Mode Classification in Cylindrical Dielectric Waveguides”, Journal of Lightwave Technology, vol.18 No.5, May 2000
[6] 張銘仁, 孫迺翔, “Numerical analysis for optical fibers”, 義
守大學電機工程研究所碩士畢業論文,2001
[7] Pochi Yeh, “Optical waves in layered media”, vol.1, pp. 8-9,
1988
[8] Akira Ishimaru, “Electromagnetic wave propagation,
radiation,and scattering”, vol.4, pp.91-97,1991
[9] 房景威, “Theoretical and Numerical Study of Fiber Modes with Arbitrary Axially Symmetric Index Profiles”, 國立中山大學光電工程研究所碩士畢業論文,2002

[10] Bahaa E.A.Saleh,and Malvin Carl Teich, “Fundamentals
of photonics”, vol.8 pp.273-305,1991
[11] Li T,“Structures,parameters,and transmission properties of
optical fibers”, Proc.IEEE 68:1175,1980