環型共振腔是一種光學上的元件，在特定的波長下會共振，特性有點類似Fabry-Perot共振腔，可以在特定的頻率做相位補償，使波數、相位改變，而在製程上會更為簡單，也容易積體化，所以會是一個重要的元件。本篇論文剛好是藉由執行中山大學四年期『卓越計劃』的機會，而能夠對環型共振腔這個結構做一個理論的推導與分析，期望能在這裡一窺究竟。
本篇論文中，我們先處理了一維的三種不同情況，理論由簡單而複雜，一維單向是最簡單的情況，而在這裡，我們有一個突破性的建設，我們考慮反射的發生，進行前人所沒有的新的理論推導，並且在數值模擬上發現了這樣的雙向理論是有其意義的，這是我們很高興能有突破性建設的一點；此外，我們也進行二維的研究，因為二維研究一定要考慮雙向問題，而現有的研究二維的方法並不能滿足我們的需求，所以決定發展一個新的方法，也就是我論文的主題，雙向多模解析法、Multi-Mode Propagation Method（MMPM），期望可以改善現有方法的缺失；最後在附錄中，我們以2002年光電研討會發表的一篇論文結尾，探討AWG的有效折射率的一些影響因素，並做一些分析，它與本論文的相關性在於所用的方法也是和MMPM類似，用耦合積分方程式解二維的模態，一氣呵成，期對本論文有畫龍點睛之效。

Micro ring-cavity, like the Fabry-Perot cavity, is an optical device that resonates at certain frequencies. It is used as a phase compensator, and filter. Easily fabricated, the micro ring-cavity can be mass-produced, the ring-cavity is becoming evermore important as integrated opto-electronic technology advances.

In this thesis, we begin with a novel one-dimensional theory that considers bi-directional traffic in the micro-ring cavity. By separating the device into easily manageable regions, and employing only fundamental modes in each of the sections, we obtain a closed-form formula for the transmission and reflection coefficient of this device. Under certain circumstances, when the directional coupler length is short but its coupling strength is strong, we observed a significant amount of reflection of optical energy at some frequencies. This phenomena is currently unknown to the opto-electronic industry.

To further study this, we developed a more rigorous multi-mode propagation method for two-dimensional bi-directional ring cavities. The problem at hand is first being sliced into regions of multi-layered sections. Within each section, we can express the fields in terms of the underlying waveguide modes of the structure. At the interfaces of these sections, we construct coupled integral equations, which are derived from the continuity requirement of the tangential fields. We have complete formulations for both TE and TM cases, down to the coupled matrix equation for the unknown modal coefficients at each junction.

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
第二章 一維環型共振腔的理論推導 5
2-1：一維單向的模型 5
2-2：一維雙向的模型 10
2-3：一維雙向對稱結構 23
第三章 一維數值計算結果 29
3-1：一維單向 29
3-2：一維雙向 30
3-3：一維雙向對稱結構 42
第四章 二維環型共振腔的理論推導 63
4-1 MMPM的前置作業及觀念 63
4-2：TM WAVE方程式推導 66
4-2.1推導聯立積分方程 66
4-2.2重疊積分的定義 81
4-3：TE WAVE方程式推導 114
4-3.1推導聯立積分方程 114
4-3.2重疊積分的定義 128
4-4：重點摘要 159
第五章 結果與討論 161
5-1：結果討論 161
5-2：學長的研究成果 163
5-3：未來工作 163
附錄A 極化與幾何形狀對AWG通道波導的
有效折射率改變的探討 165
A-1：簡介 165
A-2：會影響的因素及兩種不同波導結構的介紹 166
A-3：數值模擬 170
A-4：討論 177
A-5：結論 178
參考文獻 179
中英對照表 181

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