本論文主要探討以蜂窩晶格組成之光子晶體的谷拓樸邊緣態 (the valley topological edge state)，藉由將多個單元晶胞組成之超晶胞 (supercell)，先使用平面波展開法計算其能帶結構與谷拓樸邊緣態，並以時域有限差分法 (FDTD) 模擬電磁波在材料中的傳播。此種拓樸邊緣態形成之原理是藉由打破蜂窩晶胞內兩圓柱之 C_3υ 對稱性使其約化為 C_3 對稱性。 在光子晶體的模擬中，發現谷拓樸邊緣態的手徵性 (chirality) 並不優秀，但在特定的邊界其電磁波傳播仍然有一定的手徵性。在計算能帶的貝瑞曲率 (Berry curvature) 後發現其藉由兩種不同排列方式所構成的邊界，在第一布理淵區 (the first Brillouin zone) 中 〖K、K〗^′ 點的局部陳數 (the local Chern number) 差異並非如理論所言等於 1，實際上差異是小於 1，這是導致其不良好手徵性之原因。 最後我們藉由改變單元晶胞中介電質柱之形狀或性質以以使谷邊緣態的實現具有更多的靈活度。我們也研究了三角晶格 (triangular lattice) 與可果美晶格 (Kagome lattice) 中的谷拓樸邊緣態。它們的存在性也是藉由打破晶胞內兩圓柱分佈之 C_3υ 對稱性使其約化為 C_3 對稱性而實現的。 ;In the thesis, we mainly discuss the valley topological edge states in photonic crystals which composed of honeycomb lattices of dielectric pillars. We use the supercell theory and the plane wave expansion method to calculate the band structure and valley topological edge states. Besides, we use the finite difference time domain method to simulate the propagation of electromagnetic waves in the material. The principle of the formation of this topological edge state is to reduce the C_3 symmetry of the distribution of the two cylinders in the honeycomb lattices to C_3υ symmetry. In the boundary formed by two different arrangements, according to the calculated Berry curvature of the photonic crystal energy band, it is found that the sum of the local Chern numbers belonging to the 〖K and K〗^′ points in the first Brillouin zone of the two photonic crystals is not as what the theory expected to be equal to 1, but a value less than 1. We believe this is the reason for its poor chirality. Finally, we consider more general structures to realize the valley edge states by changing the shapes or properties of the dielectric pillars in more flexible ways in a unit cell. We also study the valley topological edge states in the triangular lattice and the Kagome lattice photonic crystals. The existence of the edge states in these structures can also be explained by the reduction to C_3 symmetry through breaking the C_3υ symmetry of the distribution of the two cylinders in a unit cell.
