光波導為光通訊領域中非常重要的元件，而其中的山脊型波導可降低光波導之製作成本，但其計算卻比埋入式波導複雜。故發展一套適當的方法來模擬山脊型波導是非常重要的。
我們先由一維的模型出發，利用簡單正交基底展開的方法，將三層的介電質波導，依折射率分佈個別探討步階變化與漸變變化，並確定其精確度與可行性。接著進行二維模型分析，為了證明此方法之廣泛性以及探討其準確性，特別選擇圓柱狀之光纖。因光纖有精密之數值解可比較，且原本需利用貝索函數(Bessel functions)才能解的圓柱問題，僅需利用直角座標即可解決，如此可看出此方法的運用範圍很大。但若需極精密的解，還是得倚賴貝索函數才可以。而利用簡單基底，我們可算出其光纖的有效折射率的有效位數可達五位。
爲了加快計算速度以及增進精確度，我們將原本的簡單正交基底，改為利用導波模態作為正交基底，並用來解矩形波導之模態。我們利用導波基底的方法，其展開的基底項數約簡單正交基底的十分之一，即達到相同的精密度，確實加快了計算的速度。此方法只要適當修改各區長度參數與各區折射率參數，即可解山脊型波導的模態。

The dielectric ridge waveguide is an important passive component used in the optical communication system. Compared to its cousins—the buried ridge waveguides, it is less expensive to process but harder to design due to its inherent complex field structure (it has a larger index contrast between the core and the cladding). Therefore, it is crucial to develop an efficient and accurate method to analyze the modal characteristic of ridge waveguides.

We began with the expansion of one-dimensional three-layer dielectric slab waveguide using simple orthogonal basis functions. We examined both the step-index profile and the graded-index profile waveguides to confirm the feasibility of this method and to understand the level of accuracy our technique can reach.

We then proceeded to derive the vector formulation of two-dimension dielectric waveguides and compared our results against the exact numerical solutions of optical fiber modes using Bessel functions. Our 2-D Cartesian mode solver gave up to 6 significant digits of the fiber propagation constants.

Finally, for rectangular dielectric waveguides, we use the tensor product of 1-D guiding-mode bases as an improvement over the simple orthogonal bases to speed-up the numerical convergence and cut the storage requirement by a factor of ten without loss of accuracy which is around 4 to 5 digits over a wide range of parameters and mode types. We will use these bases to solve for the mode field distribution of ridge-waveguides and other complex structures.

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 簡介1
1-2 馬克思威爾方程式5
第二章 矩陣方程式7
2-1 一維矩陣方程式(TE) 7
2-2 二維矩陣方程式(H coupled) 9
第三章 簡單正交基底11
3-1 正交基底方法11
3-2 邊界條件之設定12
3-3 簡單正交基底14
3-4 導波模態基底17
第四章 以簡單基底分析圓柱波導(光纖) 27
4-1 二維圓柱之折射率為漸變分佈27
4-2 二維圓柱之折射率為步階分佈31
第五章 以導波模態基底分析山脊型波導35
5-1 山脊型波導結構35
5-2 山脊型波導之分析36
5-3 山脊型波導矩陣方程之係數38
5-4 矩形波導之特例45
第六章 數值模擬結果49
6-1 二維圓柱步階折射率49
6-2 二維圓柱漸變折射率56
6-3 二維矩形波導61
第七章 結論69
參考文獻71
中英對照表73

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