對於光電元件，我們主要關注的是元件的穩態行為，或窄頻特性。因此我們使用頻域的方法，如頻域有限差分法來計算赫姆霍茲方程(Helmholtz equation)之數值解。本實驗室近年發展計算赫姆霍茲方程式的相連局域場理論 (connected local fields, CLF)。二維度的相連局域場是由互相連接、彼此重疊的田字形的局部區域場所組成。CLF為半解析的數值方法。有點類似於有限差分法，但在本質上有所不同。CLF的概念是利用赫姆霍茲方程式的解析解在緊致網格 (compact stencil)中，分別展開的局域場相互連結所組成。相連局域場將複雜的赫姆霍茲方程式離散化，再透過矩陣方程式求解。
傳統的頻域有限差分法對於不連續介質係數的處理，不是非常理想，本論文針對二維水平介面網格的問題進行CLF的開發及研究。本論文主要分為兩個部分:
1. 回顧傳統頻域有限差分法在不連續介質上的處理:
在傳統有限差分法中，在維持緊致網格的前提下，有兩種方法，最容易的方法是透過界面連續條件，電或磁場的垂直微分連續，導出界面係數;令一種則是材料平均，類似有效折射率的概念，可導出界面係數。
2. 發展CLF 對於不連續介質的處理局部平面波展開法(Local Plane Wave Expansion, LPWE):
如上所述，CLF是利用赫姆霍茲方程式的通解，展開在緊致網格中的局域場，針對二維單一界面網格中，利用多道入射的平面波，加上反射、穿透波等正解。對此網格進行局域場展開。最後將此單一介面網格的赫姆霍茲方程式轉化為離散形式，得到一組CLF-LPWE的係數。
3.利用FBS(Fourier Bessel Series)，展開在二維水平界面緊致網格中的局域場，並以此方法推導的係數與CLF進行比較及驗證。
最後進行誤差分析，方法是將一平面波自任意角度入射、反射及穿透波之正解與上述幾種數值方法數值解進行比較及分析其絕對誤差。

For opto-electromagnetic passive devices, our primary concerns are in their steady-state behaviors or narrow-band characteristics. We use mainly the frequency-domain methods, like frequency- domain finite-difference (FD-FD) methods for the Helmholtz equation. In recent years, we proposed the method of Connected Local Fields (CLF) to solve Helmholtz equation problems. CLF method is the semi-analytical numerical method like FD method but with a whole new approach.
The concept of CLF method is to use analytic solutions of the Helmholtz equation for the local fields defined by a basic cell (defined inside the red perimeter) as illustrated in Figure 1 showing a basic cell with a horizontal dielectric interface. The classical FD-FD methods in handling media with dielectric interfaces are not really ideal. In this thesis, we use the concept of CLF to study how to obtain CLF-equivalent compact stencils for cells with dielectric interfaces.
This thesis is divided into two parts: First, we review methods in handling media with dielectric interfaces: To obtain a compact stencil the simplest method is the straight forward implementation of the interface condition (such as continuity of normal derivative) by finite-difference approximation of the. We could also apply “material averaging” method to modify the Helmholtz equation near a dielectric interface. In the second half of the thesis we will adopt the local plane wave expansion, (LPWE) approach to obtain CLF-worthy compact stencils. To derive compact stencils with dielectric interfaces, we use some linear combination of incident waves with their reflected and transmitted waves to expand the local fields near the interface. Finally we will also look into Fourier Bessel series (FBS) approach to construct these special compact CLF stencils.
We shall also present numerical performance comparisons of above methods. It is based on numerically exact, semi-analytical solutions of incident-reflected-transmitted plane wave triplet.

論文審定書頁 i
誌謝 ii
中文摘要 iii
英文摘要 v
目錄 vii
圖目錄 ix
第一章 導論 1
1-1研究動機 1
1-2研究方法 4
1-3 Helmholtz equation推導 6
第二章 數值方法簡介及回顧 10
2-1數值方法概念介紹 10
2-2 CLF簡介 15
2-3 回顧傳統非均勻區處理方法 18
第三章 LPWE方法及絕對誤差分析 26
3-1 LPWE方法 26
3-2 絕對誤差 28
3-3 誤差分析結果 31
第四章 FBS方法及絕對誤差分析 38
4-1 FBS方法 38
4-2 誤差分析結果 42
第五章 結論 47
參考文獻 49

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