在光學積體電路之中所使用到具有連續變化的Y型分叉器、耦合器等彎曲結構，而這些彎曲結構在越作越小的晶片底下將會有越大的轉彎角度，因此本篇論文將以理論模擬及分析轉彎波導作為探討的方向。
一些商業軟體使用的波束傳播理論(BPM)作為軟體設計之基礎，然而此理論對於我們所要探討的急遽轉彎波導、斜角轉彎波導和U型轉彎波導的反射/穿透現象等問題有其先天上的限制。因此本論文中會先建立二維結構來模擬需要計算的轉彎波導結構，結合耦合橫向模態積分方程(CTMIE)與模態匹配理論來計算，將整個問題簡化成一個對角矩陣的等式。對於轉彎波導而言，電磁波有兩個主要的行進方向，因此我們將利用對稱性解法來解決行進方向改變的問題，合併這兩種結果便是真正結構的解。利用這種嚴謹的數值方法，我們將可以模擬出電磁波場量在波導內的傳播和分佈情形並求出能量的反射與穿透率等數據。此外我們考慮在轉彎處製造一斜切面，藉由全反射原理使光路轉彎以減低轉彎損耗的影響，並探討了一些轉彎波導的結構對於反射係數的影響，例如比較直角彎曲波導在轉角處切出45度角的全反射面與0度角之間的差別。

Many waveguide components in the integrated optics are built with bending structures, such as Y-branches, couplers, tapered waveguides, etc. The bending angles are getting larger and larger in order to fill into a smaller integrated optical circuit. The influences of wide bending angles are no longer ignorable.
Commercially available beam-propagation method (BPM) design tools are inadequate for simulating and optimizing the problem we consider. These include tightly curved waveguide sections, reflection/transmission from slanted end facets and U-turn reflectors. In this thesis, we applied the coupled transverse-mode integral-equation (CTMIE) formulation and mode matching method to study the field distribution in a 2-dimentional rectangular waveguide structure with perfect boundary conditions. The problem is first separated into parts and then converted into a block-diagonal matrix equation.
By considering the symmetry of the bending structures, the original problem is broken down to two smaller problems each with it’s own boundary conditions. The combined solutions provide the desired results.

目 錄
頁次
誌謝 I
摘要 II
目錄 IV
圖目錄 VI
表目錄 X
第一章、導論 1
1.1：簡介 1
1.2：對稱性求解法 5
第二章、 斜邊近似法解均勻電介質彎曲波導 8
2.1：均勻電介質彎曲波導結構 8
2.2：斜邊近似法的理論模型與TE模態理論推導 10
2.3：數值分析結果 21
2.4：對斜邊近似法理論之修正 26
2.5：數值結果分析 32
第三章、階梯近似法解均勻電介質彎曲波導 35
3.1：階梯近似法理論模型的建立與TE模態的理論推導 35
3.2：數值分析結果 45
第四章、結果與討論 51
4.1：90度與180度轉彎波導的數值分析 51
4.2：其他轉彎波導的數值分析 64
第五章、結論 71
參考文獻 73
中英對照表 75
附錄 76

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