手性介質麥斯威爾方程在接近臨界點時的能帶結構變化

Variation of Band Structure Near Critical Point of Maxwell Equation with Chiral Medium

張芷菱(Chi-Lin Chang)

指導教授 ： 林文偉

國立交通大學/理學院/應用數學系暨研究所/碩士(2016年)

摘要

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。

關鍵字

馬克斯威爾方程 ； 手性係數 ； 能帶結構 ； 奇異值分解 ； 面心立方晶體

並列摘要

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.

並列關鍵字

Maxwell equation ； chirality parameter ； band structure ； SVD ； FCC

參考文獻

[1] R. L. Chern, H. E. Hsieh, T. M. Huang, W. W. Lin, and W. Wang. Singular

Value Decompositions for Single-Curl Operators in Three-Dimensional

[3] W. W. Hager. Updating the Inverse of a Matrix. SIAM Rev., 31:221–239, 1989.

[4] T. M. Huang, H. E. Hsieh, W. W. Lin, and W. Wang. Matrix Representation

of the Double-Curl Operator for Simulating Three Dimensional Photonic