二維光子晶體色散關係的能帶結構之數值研究

Abstract

本論文主要在研究電磁波傳導進入不同材質的光子晶體所發生的色散現象 (dispersion relation)。光子晶體具有一維、二維、三維等形式，本論文主要討論電磁波在二維光子晶體中的傳播，並且討論時間頻率以及布洛赫波向量之間的關係，我們稱這種關係為色散關係(dispersion~relation)。我們研究的目標是找出時間頻率以及布洛赫波向量所形成的關係圖，由此關係圖可找到電磁波在光子晶體中傳導的頻帶結構，進而分析頻帶結構中是否存在全光子能隙 (full photonic band gap)。要得到此頻帶結構我們必須對馬克士威方程式 (Maxwell’s equation)所推導出的特徵值問題求解特徵值，因此特徵值問題的數值運算是本論文所要研究的課題。

In this paper, we study the electromagnetic wave propagation through different material photonic crystals. Photonic crystals were classified as 1D, 2D, 3D, etc. We focus on the 2D photonic crystals, and we want to discuss the relation between the time frequency and the Bloch wave vector. We also say that the relation is dispersion relation. Our goal is to find the band structures of photonic crystals from the relation of Bloch wave vector and the time frequency. And we transform the non-linear eigenvalue problem for a Lorentz material in air into a non-Hermitian linear eigenvalue problem. Thus, the main work of this paper is to solve the eigenvalue problems derived from Maxwell equations.