各向異性光波導及光子晶體與週期性電漿子結構之有限差分頻域特徵模態分析方法之發展

Abstract

本論文提出以全向量有限差分頻域法推得標準特徵方程式，以之分析具各向異性材料的光波導及光子晶體。利用全向量式有限差分頻域法之波導模態求解模型，得以簡單而有效率地計算出含任意導電率張量的各向異性光波導的導波模態，例如鈮酸鋰光波導和液晶光波導。本研究引入適用各向異性材料的完美匹配吸收層於計算區域的外圍，因此得以分析計算洩漏式光波導。 針對於含各向同性或平面各向異性材料的二維光子晶體的能帶分析，在波傳播方向為平行於週期性平面而橫向電場與橫向磁場波模不互相耦合的情況下，本研究以純量有限差分頻域法推導特徵方程式做有效率的計算。本研究進而推導同時考慮電場與磁場分量的全向量特徵方程式以分析具任意三維各向異性的二維光子晶體，此時橫向電場與橫向磁場波模通常呈耦合狀態。此全向量特徵方程式可進一步推展以分析具任意導電率張量的三維光子晶體，本研究以之計算探討三維各向同性簡單立方光子晶體及具各向同性或各向異性材料的光子晶體平板結構的能帶特性。 除了各向異性波導模態與光子晶體能帶的計算外，本研究亦推導標準特徵方程式以分析計算具一維週期與忽略損耗的實際金屬的二維三維表面電漿子波導，得以有效率地獲得二維及三維週期性波導的色散能帶特性與導波模態。

We propose full-vectorial finite-difference frequency-domain (FDFD) method based standard eigenvalue algorithms for analyzing anisotropic optical waveguides and photonic crystals (PCs). Using the established full-vectorial waveguide mode solver, we can easily and efficiently calculate allowed guided modes on anisotropic optical waveguides with an arbitrary permittivity tensor, such as lithium niobate and liq- uid crystal (LC) optical waveguides. We also incorporate perfectly matched layers (PMLs) for anisotropic media into our formulation as the absorbing boundary condi- tion at the outer boundaries of the computational domain so that leaky waveguides can be treated. For band diagram analysis of 2-D PCs with isotropic or in-plane anisotropic ma- terials under the in-plane wave propagation for which the waves can be decoupled into transverse-electric (TE) and transverse-magnetic (TM) modes, we formulate the scalar FDFD method based eigenvalue algorithm. We then develop a full-vectorial FDFD based eigenvalue algorithm to analyze band diagrams of 2-D PCs with ar- bitrary 3-D anisotropy, in which the TE and TM modes are coupled. The band diagram analysis algorithm is further generalized to investigate 3-D PCs with an arbitrary permittivity tensor. The band diagrams of 3-D simple cubic PCs or PC slab structures with isotopic and anisotropic media are examined and investigated. In addition to the algorithms for anisotropic waveguide modes and PC band diagrams, we derive a standard eigenvalue algorithm for analyzing 2-D and 3-D surface plasmonic waveguides with 1-D periodicity and involving real metals ignoring the losses. We are able to efficiently obtain dispersion band characteristics and guided mode patterns for 2-D and 3-D periodic waveguides.