本論文主要探討應用連續輻射模態於計算瞬間垂直位移板塊波導的散射的問題。 計算任意多層介電質波導的連續輻射模態問題已經被廣泛地探討。現有文獻多是以雙入射雙反射波的組解來求得多層介電質波導的連續輻射模態。推導方式雖然詳盡，但由於代數繁瑣、不夠直覺，並帶有部分爭議的物理涵義。本論文首先提出了一較接近直覺的方法來求解一任意非對稱多層介電質波導的連續輻射模態。我們將利用簡單、直接的說明，找出任意多層複雜結構下兩組相互獨立的連續輻射模態，並提出一個完整的程序使其正交並規一化。
第二，文章將現有的橫向耦合積分方程式經由特殊的修訂，選擇滿足週期邊界條件的板塊波導輻射模態。由於橫向耦合積分方程式所計算出的模態滿足完整及正交特性，所以可以應用於垂直位移板塊波導電磁場的精算。利用滿足週期邊界條件的波導模態，對於垂直位移介面兩區輻射模態間的重疊積分矩陣將可大大地簡化，因為不同空間頻率輻射模態彼此不相互影響、他們的重疊積分矩陣可形成位於主對角線上的二乘二的方塊矩陣群。遠離主對角線的積分值相對的小了兩個數量級以上。因此、當橫向耦合積分方程式上下邊界推至於無窮遠處時，橫向耦合積分方程式可得到正解的反衍形式，無須進行矩陣的數值反衍。當使用的輻射模態越多時，計算量減少的趨勢將越趨明顯！

In this thesis, we study the scattering problem of a vertically offset dielectric slab waveguide, using continuous radiation modes. The calculation of radiation modes of an arbitrarily layered waveguide has been thoroughly investigated in the literature. Most approaches were based on launching two incident waves: one from above and one from below, resulting in two transmitted waves and two reflected waves. Radiation modes were obtained by algebraic adjustments of each incident wave’s amplitude and phase. These radiation modes formed standing waves in both the substrates and superstrates. This implies that walls are located an infinite distance far from the first and the last interfaces. In addition to physical conflicts of simultaneous existence of the incident wave and the walls, the derivation details are complicated and non-intuitive. In our thesis, with a given propagation constant for an arbitrarily layered dielectric waveguide, we propose an intuitive method to obtain two independent radiation mode solutions. We also construct a specific procedure to orthogonalize and normalize these two radiation modes.
The second part of this thesis is focused on applying these radiation modes into a customized coupled transverse mode integral equation formulation (CTMIE), to the study of vertically offset slab waveguides. CTMIE requires two artificial boundaries placed in the substrate and superstrate. We choose to compute discretized radiation modes with the periodic boundary conditions. Under these circumstances, modes correspond to different spatial frequencies and thereby do not inter-couple. This means the matrix of the overlap integral between these two groups of modes (slightly vertically shifted) are block-diagonally dominated. The off-diagonal elements are two orders of magnitude smaller than the diagonal ones. As a result, when the two artificial boundaries are pushed towards infinity in the CTMIE formulation, we may obtain an exact inverse of the Greene’s matrix without relying on numerical inversion.

第一章 導論 1
1-1 研究背景 1
1-2研究動機 3
1-3 研究主題 6
第二章 傳統方法分析平行層波導垂直位移之探討 8
2-1平行層波導瞬間垂直位移的介紹 9
2-2平行層波導瞬間垂直位移析之文獻探討 10
2-3平行層波導輻射模態分析之文獻探討 11
2-4 文獻間對平行層波導瞬間垂直位移之分析與差異 14
2-5 橫向耦合積分方程式的介紹 19
2-6 CTMIE與PECAM對垂直位移板塊波導的數據分析結果 24
第三章 以一較為直覺方法求解任意多平行層介電質波導輻射模態- 29
3-1建立兩組具相同操作頻率且相互獨立的連續輻射模態 31
3-2重根輻射模態的存在性 33
3-3建立三層對稱波導結構的兩組獨立且連續輻射模態 36
3-4正交且規一化所得到的兩組獨立且連續輻射模態 39
3-5 結論 44
第四章 垂直位移板塊波導之特性分析 45
4-1解垂直位移板塊波導正解基底函數的選擇 46
4-2邊界牆特性與牆距對結構的影響 51
4-3上邊界條件與牆距對點源的理論分析 52
4-4 上下邊界條件對牆距選擇依據的分析 54
4-5相鄰模態因牆距選擇的差異性 57
4-6 數值結果驗證 62
第五章 連續輻射模態在垂直位移板塊波導之應用分析 64
5-1垂直位移板塊波導近似正解型式的推算 65
5-2以修正版橫向耦合積分方程式做修正上的應用 71
5-3 解垂直位移板塊波導正解基底函數的選擇 76
5-4解析解近似根值的估算 78
5-5 解析解近似根值特性的驗證 81
5-6 週期邊界條件Sturm-Liouville問題的探討 82
5-7週期邊界條件Sturm-Liouville在垂直位移波導中重疊積分的驗證 86
第六章 總結與未來工作 105
6-1總結 105
6-2未來工作 108
參考文獻 110

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