光子晶體是光電科技中最重要的材料之一，它是介電質模仿原子在晶體中的排列方式生成。光子晶體基本的光學特性就是具有光子能隙，尋找光子能隙必須使用數值方法去計算。電磁波在光子晶體中的傳播行為受到馬克斯威爾方程控制，在時間諧波的條件下，此控制方程變成與時間無關的頻域問題。數值計算的方法有許多種，經過比較後我們發現 Yee's scheme 是目前最好的離散方法，至今為止這套方法只能適用於簡單立方晶格與面心立方晶格，但三維空間中共有14種布拉非晶格，本論文的第一個重要的工作是將這套方法推廣到所有的布拉非晶格上。利用 Yee's scheme 將馬克斯威爾方程離散後，可得到一組廣義特徵值問題，我們將對此廣義特徵值問題進行分析。本論文的第二個重要工作，是尋找離散旋度算子的特徵分解，經過一系列繁雜的計算後，我們發現所有的晶格可歸納出兩種分解。我們有興趣的特徵值是最小的一些正實數，但此特徵值問題龐大的零空間嚴重影響了數值計算的收斂性，我們採用一種叫作零空間免除法的技巧，避開了這個困擾，但這個技巧使得我們的問題從稀疏矩陣便成了稠密矩陣，幸運的是我們得到的特徵分解與離散傅立業轉換有關，大幅提升了數值計算的效率。最後我們計算了光子晶體在各種晶格上的能帶結構，並在GPU叢集上實現了高效能的計算。

Photonic crystal is one of the most important materials in optoelectronics technology, it is made up of periodic dielectrics and imitates the arrangement of atoms in the crystal. The basic optical property of a photonic crystal is band gap, it is necessary to use the numerical method when computing the band gap. The propagation behavior of electromagnetic waves in photonic crystals is governed by Maxwell's equations, applying the time harmonic assumption, the governing equations will become a time-independent frequency domain problem. There are many numerical methods, after comparison, we believe that Yee's scheme is the best discrete method. So far, this method can only be applied to simple cubic lattice and face-centered cubic lattice, but there are 14 Bravais lattices in three-dimensional space. The first important work of this dissertation is to extend this method to all of the Bravais lattices.

Using the Yee's scheme to discretize the Maxwell's equations, we can get a general eigenvalue problem, and we will analyze this general eigenvalue problem. The second important task of this article is to find the eigen-decomposition of the discrete curl operator, after a series of complicated calculations, we find that all the lattices can be summed up into two kinds of decomposition. We are interested in finding the several few smallest real eigenvalues, but the large dimension of null space is 1/3 of all, this seriously affected the convergence of calculation, we use a technique which is called nullspace-free method to avoid this trouble. But this technique transforms the sparse matrix in our problem to a dense matrix, fortunately, the eigenvectors we found before are related to discrete Fourier transformation. The efficiency of calculation has been significantly improved by using the fast Fourier transformation. Finally, we calculate the band structures of photonic crystals on various lattices, and implement high performance calculations on the GPU.

1 Introduction 1
1.1 Photonic Crystal 1
1.2 Governing Equations 2
1.3 Numerical Methods 5
1.4 Yee’s Scheme 6
1.5 Notations and Overview 8
2 Background 10
2.1 Crystallographic Restriction Theorem 10
2.2 Bravais lattice 11
2.3 Primitive Translation Vectors 21
2.4 Reciprocal Lattice and Brillouin Zone 25
2.5 Point Groups and Space Groups 26
3 Explicit matrix representation of the curl operator 29
3.1 Matrix representation of curl operator for simple cases 30
3.2 Matrix representation of curl operator for general cases 32
3.2.1 Matrix representation of ∇×E = ιωμ0H 48
3.2.2 Matrix representation of ∇×H = -ιωεE 64
4 Eigen-decomposition of the partial derivative operators 74
4.1 Eigen-decomposition of the partial derivative operators for simple cases 74
4.2 Eigen-decomposition of the partial derivative operators for general cases 77
5 Singular Value Decomposition and the Fast Solver 89
5.1 Singular value decompositions for discrete curl operators 89
5.2 Null space free method 91
5.3 FFT based matrix-vector multiplication 93
6 Numerical Results 97
7 Conclusions 107
A Appendix 109

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