分析光柵結構的有限差分法與傅立葉模態法

Abstract

本論文提出步階式數學形式之有限差分模態法來對光柵結構進行模擬並將其與傅立葉模態法(亦稱嚴格耦合波分析法)進行比較。我們的數值結果證實，對於橫電極化的所有情況以及橫磁極化在高導電性與無損金屬材料中，有限差分模態法比起嚴格耦合波分析法會有更好的收斂性和準確性。 對於有限差分模態法，我們考慮任意高階之邊界條件並將其與泰勒展開式結合。在不使所取格點數增加的情況下，我們亦將廣義道格拉斯(Douglas)方法套入使用來加速誤差收斂階數。使用前述技巧，可以對結構的每一層架構出稀疏矩陣並計算出存在於該層之模態所對應的場值分佈以及傳播常數。另外，我們也使用穆哈拉姆(Moharam)所提出之改良穿透矩陣方法來穩定多層光柵或甚至單層光柵層間的矩陣運算。 為了評估此種數值方法的可用性，我們將討論一些光柵的繞射特性，如入射角變化、厚度變化、佔空比變化所造成的影響，以及準確性、收斂性等等。另外，我們也使用結合週期性邊界和吸收邊界的二維有限差分法來與前述方法進行比較。

In this thesis, the finite-difference modal method (FDMM) with step-index formulation for simulating grating structures is proposed and compared with rigorous coupled-wave analysis (RCWA), also called Fourier modal method (FMM). It is verified that FDMM has better convergence and accuracy than RCWA for TE polarization in almost all cases and TM polarization for high conductive and lossless metallic materials. In the FDMM, the relation of interface conditions to arbitrary high orders is considered and combines with Taylor series expansion. The generalized Douglas (GD) scheme is also adopted to enhance the convergence order without considering more sampled points. With the techniques mentioned above, the sparse matrix of eigenvalue problem could be constructed to solve the fields and the propagation constants of modes inside each layer. In addition, the enhanced transmittance matrix approach proposed by Moharam emph{et al.} for RCWA is used to make matrix manipulation stable for multi-layer or even single layer gratings. The diffraction properties of gratings, such as accuracy, convergence, dependence of diffraction efficiencies on incident angle, thickness, duty cycle, etc, will be discussed for numerical assessment of FDMM. Moreover, two-dimensional finite-difference methods combined with periodic boundary conditions and absorbing boundary conditions will be executed for comparison.