近年來，由於波導傳輸，頻寬不斷地提高，圓柱介電質波導在些許橢圓變形後，極化色散的問題將變得相當地重要。本文針對橢圓介電質波導的問題，提出一可得精確解的方法，並討論其極化的特性。
一般處理此問題，常使用網格狀的方法(Mesh-dependent method)，例如：有限差分法(Finite difference method)或有限元素法(Finite element method)來計算，但這類方法所得的解並非嚴謹地滿足 Helmholtz equation，所得的解為近似解。一般而言，其基模(Fundamental mode)有效位數只能到3~4位。而非網格狀的方法(Mesh-independent method)，常使用橢圓座標系，因此在數值計算上，常需花很多的計算量來處理特殊函數(modified Mathieu functions)；而我們所提的用『柱座標解析連續』的方法，所得的解均滿足Helmholtz equation，有效位數可達6位以上，並且在計算上僅需處理貝索函數(Bessel function)。另外，我們提出來的方法，利用了對稱的現象，在邊界設了電牆及磁牆，使矩陣內所需的項數大量地減少。
本論文在介面處理上，利用模態匹配(mode matching)的觀念，使用權函數(weighting function)的技巧來減少矩陣的運算量。本法亦可應用在矩形介電質波導管、正六角形介電質波導管…等正多邊形介電質波導結構。

Dielectric waveguides are important passive devices in optical communication systems. Circular-core fibers with slight ellipticity may lead to polarization-mode dispersion. A clear understanding of the propagation characteristics of the elliptical fibers thus becomes important for theoretical as well as practical purposes.
Although mesh-dependent methods such as the finite-element method or finite-difference method, can be used to study such a complex structure, its computational task is very high. Strictly speaking, mesh-based solution does satisfy the Helmholtz equation and the solution only provided four to five significant digits. On the other hand, the highly accurate solution based on solving the Helmholtz equation of the elliptical coordinate system spend most its computational resources on computing the functional value and the zeros of the modified Mathieu functions of the first kind.
Our method is based on linear combination of the exact mode-field solutions of the dielectric optical fiber. We apply the analytical continuity principle to obtain the simultaneous equation of the expansion coefficient vector. Since each basis solution satisfies the Helmholtz equation exactly, the overall solutions are very accurate and provide more than six significant digits for fibers with small elliptical eccentricity. In addition, only the Bessel functions are needed in our computation. Using cylindrical coordinate and symmetry, together with ACM principle, we simplify the problem of modal analysis of dielectric elliptical waveguides. This method also can be applied to some regular polygonal dielectric waveguides such as the large area VCESL.

誌謝 Ⅰ
中文摘要 Ⅱ
英文摘要 Ⅲ
目錄 Ⅳ
圖表目錄 Ⅵ
第一章 導論 1
1-1 研究動機 1
1-2 簡介 1
1-3 基本公式推導 4
第二章 梯形介電質波導 10
2-1 梯形介電質波導模型 10
2-2 梯形介電質波導理論推導 10
2-3 梯形介電質波導數值結果 20
2-4 梯形介電質波導之討論 25
第三章 橢圓介電質波導 26
3-1 橢圓介電質波導模型 26
3-2 橢圓的向量分析 28
3-3 橢圓介電質波導理論推導 31
3-4 橢圓介電質波導數值結果 40
3-5 橢圓介電質波導之討論 51
第四章 正多邊形介電質波導與數值結果 54
4-1 矩形介電質波導模型 54
4-2 矩形的向量分析 56
4-3 矩形介電質波導數值結果 59
4-4 正六邊形介電質波導模型 61
4-5 正六邊形的向量分析 63
4-6 正六邊形介電質波導數值結果 67
4-7 正多邊形介電質波導之討論 70
第五章 結論 71
參考文獻 72

[1] L.J. Chu, “Electromagnetics in elliptic hollow pipes of metal,” J. Appl Phys., 9, 1938 , 583.
[2] C. Yeh, “Elliptical Dielectric Waveguides,” J. Appl. Phys., 33, Nov. 1962, 3235–3243.
[3] S.J. Zhang and Y.C. Shen, “Eigenmode sequence for an elliptical waveguide with arbitrary ellipticity,” IEEE Trans. Microwave Theory Tech., 45, 1995 , 227-230.
[4] C. Shu, “Analysis of elliptical waveguides by differential quadrature method,” IEEE Trans. Microwave Theory Tech., 48, 2000, 319-322.
[5] P.L. Jiang, S.Q. Li, and C.H. Chan, “Analysis of elliptical waveguides by a meshless collocation method with the wendland radial basis functions,” Microwave Opt. Technol. Lett., 32, 2002, 162–165.
[6] D.L.Young, S.P. Hu, C.W. Chen, C.M. Fan, and K. Murugesan, “Analysis of elliptical waveguides by the method of fundamental solution,” Microwave Opt. Technol. Lett., 44, 2005, 552–558.
[7] J.S.Goell, “A circular-harmonic computer analysis of rectangular dielectric waveguides”, Bell Syst. Tech. J., Sep. 1969, 2133-2160.
[8] 吳宙秦,“具有小角度斜切角終端波導之特性分析”, 國立中山大學光電工程研究所碩士畢業論文,2004
[9] 許峻源,“利用解析連續法分析小角度彎曲波導的反射係數與場形”,國立中山大學光電工程研究所碩士畢業論文,2004
[10] 林明崇,“圓柱座標多層介質波導之模態分析”,國立中山大學光電工程研究所碩士畢業論文,2003
[11] 曹碩芳,“簡單正交基底之二維向量介質波導模態解,” 國立中山大學光電工程研究所碩士畢業論文,2004
[12] Ishimaru,A., “Electromagnetic Wave Propagation ,Radiation ,and Scattering ”, Englewood Cliffs, N.J.: Prentice-Hall, 1991.
[13] Amnon Yariv, “Optical Electronics in Modern Communications”, Oxford, New York, 1997.
[14] Tso-Lun Wu and Hung-Wen Chang, “Guiding mode expansion of a TE and TM transverse-mode integral equation for dielectric slab waveguides with an abrupt termination,” J. Opt. Soc. Am. A, 2001.
[15] Tso-Lun Wu and Hung-Wen Chang, “Analysis of TE to x and TM to x Modes for Dielectric Slab Waveguides,” Opt., pp. 146-148 (2001).