手性介質麥斯威爾方程在接近臨界點時的能帶結構變化

Abstract

本研究主要目標為當折射率逐漸接近於零時，三維簡單立方與面心立方晶體的能帶結構之探討。我們使用Yee scheme去離散麥斯威爾方程，以及對離散化的式子做奇異值分解，而後發現其不變子空間使得我們能將零空間拔除進而縮減矩陣的大小，利用SIRA來解此特徵值問題，畫出能帶結構圖。然而我們也對當折射率小於零時的方程提出一點想法並呈現實驗結果在本文中。我們發現隨著手性係數愈大，簡單立方與面心立方的能帶結構圖逐漸被壓縮。

The main objective in this study is the band structure variance of simple cubic cubic(SC) and face center cubic(FCC) when the refractive index gradually approaches to zero. We discretize Maxwell equation by the Yee's scheme. After singular value decomposition(SVD), we find out its invariant subspace such that we can have the nullspace-free eigenvalue problem with small dimension. We solve this eigenvalue problem by Shift-Invert Residual Arnoldi method(SIRA) and then we are able to draw the band structure. We also have some ideas of the equation as the refractive index is smaller than zero and the experiment results are shown in this thesis. With the chirality parameter growing, the band structures of SC and FCC are compressing.