直彎曲(straight bending)光波導在以往的分析中是一個困難的問題，傳統上波導計算方式是光束傳播法(beam propagation method, BPM），但是因為BPM使用單向傳播與近光軸近似，對電質彎曲波導無法進行反射計算。
直彎曲波導的斜面左右兩側的座標系統不匹配，造成各種計算方法上的困難。本論文中引入解析連續法去處理介電質波導直彎曲的所導致的斜面邊界電磁場必須連續的邊界條件。此法可以處理斜面兩側座標系統的不匹配，計算量與誤差可以大幅降低，由電磁場連續條件可推得兩組耦合積分方程式。再利用Galerkin 最小平方誤差(lease squared error) 法推導出矩陣方程。利用波導的對稱結構的特性，可再簡化矩陣計算量。
本論文主要的結果是求得小角度彎曲光波導的穿透係數和反射係數近似公式。再利用精確數值解的結果，推導出直彎曲波導的散射矩陣（scattering matrix），應用在兩段式彎曲波導（two-corner bends）的不同變化上。

Analysis of dielectric straight bending waveguides has been a difficult problem in the past. Traditionally the task for computing bending of optical waveguides is carried out by the beam propagation method (BPM). However, due its assumptions on one-way propagation and paraxial approximation, BPM is unable to consider the reflection of dielectric straight-bent waveguides when the bending angles are large.
In a straight-bent waveguide, two coordinate systems are needed to fully describe the ongoing complex scattering process in the transition region of the waveguide. It is extremely hard to analyze such an unbounded problems with two incompatible coordinate systems even for those general-purpose methods like the finite-difference, finite-element. In this thesis, we use the analytic continuity method (ACM) to deal with the boundary conditions that both the tangential electromagnetic field components must be continuous across the bending line. This method can handle the mismatch of two coordinate systems and decrease the amount of calculation and error for small bending angles. From the two coupled integral equation we can derive matrix equation via Galerkin least squared error method.
The main part of this thesis contains the derivation of the approximate formula of the transmission and reflection matrices (scattering matrices) for a micro-bent waveguide. We show numerical results of various two-corner bends using cascading of these scattering matrices.

第一章 導論 1
1.1 簡介 1
第二章 小角度直彎曲介電質波導理論分析 4
2.1 解析連續法的理論架構 4
2.2 對稱結構的理論架構 5
2.3 彎曲對稱結構的分析 9
第三章 小角度直彎曲介電質波導數值計算結果 21
3.1 近似解與數值解之比較 22
3.2 不同折射率的場型及能量圖之比較 28
3.3 單模的數值模擬 33
第四章 兩段式彎曲波導理論分析 38
4.1 串接散射矩陣 39
4.2 順向兩段式彎曲波導的理論分析 43
4.3 反向兩段式彎曲波導的理論分析 47
第五章 兩段式彎曲波導數值計算結果 52
5.1 順向兩段式彎曲波導數值模擬 52
5.2 反向兩段式彎曲波導數值模擬 63
5.3 兩段式彎曲波導數值結果討論 77
第六章 結論與未來研究 78
參考文獻 80
附錄A 82
附錄B 85
中英對照表 87

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[7]吳宙秦Chou-Chin Wu ;”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,June 2004

[8]許峻源Jiun-yuan Shiu ;”利用解析連續法分析小角度彎曲波導的反射係數與場型”,國立中山大學光電工程研究所碩士畢業論文June 2004

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