圓柱介電質波導如光纖、光子晶體光纖(OF,PCF)是很重要的光通訊被動元件，目前在市面或是學術上有很多種套裝軟體或模擬分析的方法，來對介電質波導做模態分析與元件特性的探討，而在我們這篇論文中，我們提出四種方法去模擬分析軸對稱圓柱介電質波導。此四種方法分別是圓柱ABCD矩陣法、Runge-Kutta積分法、電磁軸場耦合聯立方程式及簡單基底展開法。其中前面兩種方法是改進相關文獻已知的方法並做一些修正，而後面兩者是本實驗室自行開發完成，這四種方法是目前最嚴謹、精準的方法。此四種方法的精髓簡述如下:
一.圓柱ABCD矩陣法使用四個連續的電磁場量為主要變數，推導出中間層的傳輸矩陣及其反矩陣。電磁場量即可使用矩陣相乘做為跨介面傳播的方式，把最內層與最外層的場量同時往一個介面拉，再加入連續條件即可求解模態的 值。
二. Runge-Kutta積分法常被使用在微分方程式的初值問題，在此我們利用Runge-Kutta積分法來求解四個連續電磁場量的一階微分方程式的做法，找出圓柱介電質波導的模態解。
三.電磁軸場耦合聯立方程式是利用軸向耦合電磁場求解，如同圓柱ABCD矩陣法的做法，但它使用的變數少只用到 和 ，而且這方法求解很穩定，可是公式推導和程式的撰寫會比較複雜。
四.簡單基底展開法是利用三角函數(正弦或餘弦)做為橫向耦合磁場的基底，做級數展開求解二階磁場的耦合方程式，而我們不選用橫向耦合電場求解，主要是考量電場垂直介面的分量不連續而磁場在介面處垂直和切線方向的分量皆連續的因素。
本論文中，我們會對四種方法做一些優缺點比較，藉由這些比較的結果，對這四種特殊數值方法能更清楚其特性及適用時機。

Cylindrical dielectric waveguides such as the optical fiber and photonic crystal fiber are very important passive devices in optical communication systems. There are many kinds of commercial software and methods of simulation at present. In this thesis, we proposed the following four methods to analyze arbitrarily profiled cylindrical dielectric waveguides: The first two methods are modified from published work while the last two methods are entirely developed by ourselves.
1. Cylindrical ABCD matrix method: We take the four continuous electromagnetic field components as main variables and derive the exact four-by-four matrix (with Bessel functions) to relate the four field vector within each homogeneous layer. The electromagnetic field components of the inner and outer layer can propagate toward one of the selected interface of our choice by using the method of ABCD matrix. We can then solve for the β-value of the waveguide mode with this nonlinear inhomogeneous matrix equation.
2. Runge-Kutta method: Runge-Kutta method is mostly used to solve the initial value problems of the differential equations. In this thesis, we introduce the Runge-Kutta method to solve the first-order four-by-four nonlinear differential equation of the electromagnetic field components and find the β-value of the cylindrical dielectric waveguides in a similar way depicted in method one.
3. Coupled Ez and Hz method: It uses the axial electromagnetic filed components to solve cylindrical dielectric waveguides. The formulation is similar to cylindrical ABCD matrix method, but it requires less variables then cylindrical ABCD matrix method. The numerical solution obtained from this method is most stable, but it is more complicated to derive harder to write the program.
4. Simple basis expansion method: The simple trigonometric functions (sine or cosine) are chosen as the bases of the horizontal coupled magnetic field equation derived from the second-order differential equation of the transverse magnetic field components. We do not select the horizontal coupling electric field because the normal component of the electric field is discontinuous on the interface. But the normal and tangential components of the magnetic field are continuous across the interfaces. The modal solution problem is converted to a linear matrix eigenvalue-eigenvector equation which is solved by the standard linear algebra routines.
We will compare these four numerical methods with one another. The characteristics and advantage as well as the disadvantage of each method will be studied and compared in detail.

誌謝 I
中文摘要 II
英文摘要 IV
目錄 VI
圖表目錄 VIII
第一章 導論 1
1-1 簡介 1
第二章 圓柱ABCD矩陣法求解光纖波導 6
2-1 光纖波導理論 6
2-2 光纖波導中電磁場的分析 12
2-3 TE及TM模態分析（ ） 19
2-4 結論 27
2-5 數值計算結果 39
2-5-1 step-index fiber 39

2-5-2 W-type fiber 52

2-5-3 Dielectric Tube Waveguide 58
第三章 Runge-Kutta積分法求解光纖波導 79
3-1 光纖波導理論分析 79
3-2 光纖波導公式推導 80
3-2-1 與參考文獻的公式推導比較 84
3-3 Runge-Kutta 的方法 87
3-4 數值結果 88
3-4-1 步階式與漸變式光纖波導數值結果 88
第四章 簡單基底展開法求解光纖波導 91
4-1 光纖波導理論分析 91
4-2 步階式(Step-index)光纖波導公式推導 92
4-3 漸變式(Graded-indexed)光纖波導公式推導 101
4-4 數值計算結果 107
4-4-1 步階式光纖波導模擬結果 107
4-4-2 W-type fiber模擬結果 112
4-4-3 Dielectric Tube Waveguide模擬結果 113
4-4-4 漸變式光纖波導模擬結果 118
第五章 總結 121
5-1 結果討論 121
5-2 求解光纖波導的建議 123
參考文獻 124
中英對照表 125

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