光子晶體是由不同介電質透過週期性排列所組成的材料，且其具有光子能隙的特性，而光子能隙能夠阻斷特定頻率的光子。再者，光波屬於電磁波的一種，電磁波在光子晶體中的傳遞行為受到馬克士威爾方程組控制。藉由Yee's scheme將馬克士威爾方程組離散，便可獲得特徵值問題。針對此特徵值問題，經過奇異值分後發現，透過其不變子空間能夠將零空間拔除，其好處是由於原特徵值問題具有龐大的零空間，而我們所需要的特徵值是最接近零但非零的特徵值。透過拔除零空間，能夠避免其影響數值計算的收斂，我們將這個方法稱為零空間免除法。將零空間拔除後，矩陣維度大幅縮減，但也將原本的稀疏矩陣變成稠密矩陣，所幸發現，此稠密矩陣與傅立業轉換相關，對於數值計算效率提升。

近年來，物質的拓樸相變是凝聚物理領域最大的進展之一。陳數是一種拓樸不變量，計算量子在晶體的拓樸形態，而光子晶體出現相變的必要條件，是打破晶體的對稱性。若獲得非零陳數，則稱此光子晶體為拓樸光子晶體，拓樸光子晶體是非常特殊的材料，其表面可導電，而內部為絕緣。本論文的第一步則是參考文獻，探討陳數的計算，並試圖將陳數的計算套用在光子晶體上。本論文第二步是透過GPU叢集進行高效能計算，藉由改變手性係數，觀察14 種布拉非晶格的能帶結構變化，發現調整手性係數，能夠使低頻率的能隙出現。

A photonic crystal is a material composed of periodical arrangement of different dielectrics and it has the characteristics of the photonic band gap. The photonic band gap can block photons of a specific frequency. In addition, light waves are a kind of electromagnetic wave, and the Maxwell's equations control the transmission of electromagnetic waves in photonic crystals. With the Maxwell's equations, the eigenvalue problem can be obtained by using the Yee’s scheme to discretize the system of equations. After singular value decomposition, we find that the null space can be removed through the invariant subspace. The advantage is that the original eigenvalue problem has a large dimension of null space. The eigenvalues ​​we need are the closest to zero but non-zero eigenvalues. By removing the null space, we can avoid the convergence of numerical calculations, and we call this method is nullspace-free method. After removing the null space, the matrix dimension is greatly reduced, but the original sparse matrix is ​​also changed into a dense matrix. Fortunately, it has been found that this dense matrix is related to the Fourier transformation, and the efficiency of numerical calculation is improved.

In recent years, the topological phase transition of matter has been one of the biggest advances in condensed matter physics. Chern number is a topological invariant, calculating the topological form of the quantum in the crystal. Moreover, the necessary condition for the phase transition of the photonic crystal is to break the symmetry of the crystal. If a nonzero Chern number is obtained, the photonic crystal is called a topological insulator. The topological insulator is a very special material, the surface of which is conductive, and the interior is insulated.

The first step of this thesis is the reference literature, which discusses the calculation of the Chern number and tries to apply the calculation of Chern number to the photonic crystal. The second step of this thesis is to use GPU clusters to perform high-performance calculations. By changing the chirality parameter, we observe the band structure changes of 14 Bravais lattices, and find that adjusting the chirality parameter can enable low-frequency energy gaps to appear.

摘要---------------------------------------------------------i
Abstract----------------------------------------------------ii
致謝辭------------------------------------------------------iv
1、Introduction----------------------------------------------1
2、Background------------------------------------------------2
2.1、Source free Maxwell’s equations-------------------------2
2.2、Bravais lattice-----------------------------------------3
2.3、Brillouin zone-----------------------------------------13
2.4、Yee’s Scheme-------------------------------------------15
2.5、Explicit matrix representation of curl operator--------18
2.6、Eigen-decomposition of partial derivative operators----21
2.7、Singular value decomposition of single curl------------24
2.8、Eigenvalue problems of Maxwell’s equations-------------25
3、Photonic band structure and Chern number-----------------28
3.1、Chern number-------------------------------------------28
3.2、Chern number in discretized Brillouin zone-------------29
3.3、Method1: Chern number in the photon band problem-------33
3.4、Method2: Chern number in the photon band problem-------35
4、Numerical Results----------------------------------------37
4.1、Computational times of MATLAB and GPU------------------37
4.1、The band structure of adjusting gamma------------------38
4.1、Chern number in the Double Gyroid----------------------63
5、Conclusions----------------------------------------------64

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