近年來，介電質光波導在光通訊領域裡面是一個非常重要的通訊元件，在這邊我們提出了以向量解形式的正交基底展開法來分析二維的介電質光波導模態的問題。為了降低這種積體介電質波導的製作成本，如何將這些複雜的光波導元件不同模態的傳播常數計算的精準是一個相當重要的問題。
首先我們從耦合微分方程式用兩個橫向磁場的分量，滿足在介面不連續處連續性問題。然後以一維的例子來驗證此方法的精確度，又由於一維的例子有正解，因此可以做更多項基底的有效位數探討，接下來我們才可以用這個方法應用到更複雜的二維的例子上。利用二維簡單正交基底的線性組合，我們可以展二維介電質波導模態問題。藉由嚴謹的數值正解積分公式推導，我們可以化成矩陣的特徵值、特徵向量問題來求得基底的二維係數。
用我們的方法來求解二維矩形波導問題可以達到小數點後五位有效位數。因此我們可以確信我們的方法分析任何複雜的二維類矩形波導都是一個相當快速而且有效的方法。接下來我們又花了相當多的時間處理程式類矩形波導的問題，最後我們可以計算脊型波導結構的抗共振反射光波導，也能夠正確的求得場型，因此這個方法對於光波導模態的估算為一非常快速而且有效的方法。

We propose a vectorial basis-function expansion formulation for analyzing modal characteristics of the complex 2-D dielectric waveguides. To reduce costs and to shorten the product development cycle of integrated dielectric waveguides, it is crucial to be able to accurately compute the propagation constants ( ) as well as the electromagnetic field profiles of these complex optical devices so that the devices will perform as intended. Although waveguide propagation constants can be very accurately computed, accurate 2-D vector field solutions are harder to compute especially when there are degenerate modes with similar .
We first derive the coupled differential equations of the two transverse magnetic field components which satisfy the continuous boundary conditions across all material interfaces. Then we investigate and verify the accuracy of this method on 1-D rectangular waveguide so that we can apply the technique to those more complex 2-D waveguides. And the one dimension case has exact solution, so we can compare with our bases expansion method to verify the accuracy. By means of linear combination of simple 2-D orthogonal bases, we expand the mode of rectangular dielectric waveguide. Through rigorous mathematical closed-form integration, we obtain the equivalent matrix whose eigenvalues and associated eigenvectors become the mode propagation constants and mode field distribution functions of the underlying 2-D dielectric waveguide. We can reduce the size of the problem by choosing appropriate boundary conditions via particular mode polarization desired.
We examined optical fiber modes both the step-index profile and the graded-index profile to confirm the accuracy and feasibility of our method. We get at least five significant digits of propagation constant and detailed field description of the rectangular dielectric waveguide. Finally to do the rectangular-like case, even if the ridged ARROWs wavegude we can accurately get the fields. We believe that it is an effective method for modal analysis of 2-D complex dielectric-waveguides.

致謝 I
中文摘要 II
Abstract III
目錄 IV
第一章 導論 1
1.1 簡介 1
1.2 馬克斯威爾方程式 5
第二章 一維介電質波導與正交基底理論 7
2.1 一維波導結構理論分析 7
2.2 一維結構微分方程式推導 8
2.3 數值計算通式推導 12
2.4 導波基底理論分析推導 16
第三章 二維介電質波導理論分析 21
3.1 二維結構波導基本觀念 21
3.2 向量橫向耦合矩陣方程式(Vectorial transverse coupled Helmhotz equation) 22
3.3 波導模態張量積(tensor product)求解理論 28
3.4 磁場Helmhotz耦合矩陣方程式求模態解 32
第四章 一維波導數值計算模擬結果 38
4.1 一維平板波導 38
第五章 二維波導數值計算模擬結果 64
5.1 矩形介電質波導場型數值結果 64
5.2 二維介電質圓柱光纖波導場型數值結果 76
5.3 脊型波導波導場型數值結果 83
5.4 抗共振反射光波導(ARROWs, Antiresonant Reflecting Optical Waveguides)場型結果 86
第六章 結果與討論 88
6.1 結論 88
6.2 未來工作 91
參考文獻 95
中英對照表 98

1. J. E. Goell, "A circular-harmonic computer analysis of rectangular dielectric waveguides," Bell Syst. Tech. J. 48, 2133–2160 (1969)
2. Dietrich Marcuse, Fellow, IEEE, "Solution of the vector wave equation for general dielectric waveguide by the Galerkin method," IEEE J. Quantum Electronics, vol.28 NO. 2, p459-465, February 1992
3. Robert L. Gallawa, Member, IEEE, I. C. Goyal, Yinggang Tu, an Ajoy K. Ghatak, "Optical waveguide modes: An approximate solution using Galerkin's method with Hermite-Gauss basis functions," IEEE J. Quantum Electronics, vol. 27, NO. 3 p518-522, March 1991
4. Likarn Wang and N. Huang, "A New Numerical Method for Solving Planar Waveguide Problems with Arbitrary Index Profiles :TE Mode Solutions ," IEEE Journal of Quantum Electronics, Vol.35, No.9, September 1999
5. Likarn Wang and C. S. Hsiao, "A Matrix Method for Studying TM Modes of Optical Planar Waveguides With Arbitrary Index Profiles," IEEE Journal of Quantum Electronics, Vol.37, No. 12, December 2001
6. Chia-Chien Huang, Chia-Chih Huang, and Jaw-Yen Yang, "An Efficient Method for Computing Optical Waveguides With Discontinuous Refractive Index Profiles Using Spectral Collocation Method With Domain Decomposition," Journal of Lightwave Technology,Vol.21,No.10,October 2003
7. Chia-Chien Huang, Chia-Chih Huang, and Jaw-Yen Yang, "A Full-Vectorial Pseudospectral Modal Analysis of Dielectric Optical Waveguides With Stepped Refractive Index Profiles, "IEEE Journal of Selected Topics in Quantum Electronics, Vol. 11, No. 2, March/April 2005
8. Chia-Chien Huang and Chia-Chih Huang, "An Efficient and Accurate Semivectorial Spectral Collocation Method for Analyzing Polarized Modes of Rib Waveguides," Journal of Lightwave Technology, Vol. 23, No. 7, July 2005
9. Tso-Lun Wu and Hung-Wen Chang, "Vectorial modal analysis of dielectric waveguides based on a coupled transverse-mode integral equation. II. Numerical analysis," J. Opt. Soc. Am. A, Vol. 23, No. 6, June 2006
10. Yen-Chung Chiang, Yih-Peng Chiou, and Hung-Chun Chang, Senior Member, IEEE, "Improved Full-Vectorial Finite-Difference Mode Solver for Optical Waveguides With Step-Index Profiles," Journal of Lightwave Technology, Vol. 20, No. 8, August 2002
11. Day-Uei Li and Hung-Chun Chang, Senior Member, IEEE, "An Efficient Full-Vectorial Finite-Element Modal Analysis of Dielectric Waveguides Incorporating Inhomogeneous Elements Across Dielectric Discontinuities," IEEE Journal of Quantum Electronics, Vol. 36, No. 11, November 2000
12. Enrique Silvestre, Miguel Angel Abián, Benito Gimeno, Albert
Ferrando, Miguel V. Andrés, Member, IEEE, and Vicente E. Boria,
"Analysis of Inhomogeneously Filled Waveguides Using a
Bi-Orthonormal-Basis Method," IEEE Transactions on Microwave
Theory and Techniques, Vol. 48, No. 4, April 2000
13. Ishimaru,A., "Electromagnetic Wave Propagation, Radiation, and
Scattering ," Englewood Cliffs,N.J.:Prentice-Hall, 1991
14. 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 年 6 月
15. 張世明, "精簡正交基底之複雜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 年 6 月