本論文主要以蜂窩晶格 (honeycomb lattice) 排列之介電質圓柱所組成的光子晶體探討三種不同材料所產生谷拓樸邊緣態 (valley topological edge state)。我們先利用平面波展開法 (plane wave expansion method) 計算光子晶體的頻帶結構，再利用超晶胞 (supercell) 法計算谷拓樸邊緣態的模態色散曲線，並利用時域有限差分法 (finite difference time domain method, FDTD method) 法去模擬電磁波在光子晶體中的傳播行為。此種拓樸邊緣態是藉著使晶胞內兩圓柱的半徑不同，打破原來的蜂窩晶格的 C_3v 對稱性，將它約化成 C_3 對稱性而形成的。
在光子晶體中模擬拓樸邊緣態時，發現其手徵性 (chirality) 並不完美。我們在計算其頻帶的貝瑞曲率 (berry curvature) 後，發現其 K、K^' 點的局部陳數(local Chern number) 差異並非理論預測的等於1，而是小於1。這可能是導致手徵性不完美的原因。
最後我們也利用改變超晶胞當中介電質柱的形狀去對於不同材料產生谷拓樸邊緣態。此作法同樣也是利用打破晶胞內兩圓柱的 C_3v 對稱性將它約化成 C_3 對稱性而實現其谷拓樸邊緣態。

In this thesis, we study the topological valley-edge states propagating or localizing at the boundary between two photonic crystals consisting of dielectric cylinders arranged on a honeycomb lattice. Three different material parameters are considered. We first calculate the photonic band structures by using the plane wave expansion method. We then calculate the dispersion relation of the topological valley edge states by applying the supercell method. The detailed propagation behaviors of the electromagnetic waves of the edge states and their field patterns are obtained by implementing various FDTD simulations. The valley edge states are formed by breaking the C_3v symmetry of the original honeycomb lattice and reducing it to C_3 symmetry by making the radii of the two cylinders in the unit cell different.
When simulating the topological valley edge states, we found that the chirality of them is not perfect. According to the numerical calculation results, the difference between the local Chern numbers at the K and K’ points in the first Brillouin zone is not equal to the theoretical value of 1, but instead of value smaller than 1. This may be the cause of imperfect chirality.
Finally, we also discuss how to generate topological valley edge states by changing the shapes of the cylinders in a unit cell. The underlying mechanism for this kind of edge states is the same as before, namely reducing the original C_3v symmetry to C_3 symmetry by changing the shape of the cylinders in a unit cell of the photonic crystals.

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