C6對稱之二維全介電質光子晶體中的拓樸邊界態

Abstract

本研究利用二維全介電質光子晶體(photonic crystal)調製出拓樸邊界態(topological edge state)。藉著光子晶體的色散關係(dispersion relation)，系統能帶(band)結構可以被展示。根據參考文獻，光子能帶存在著值得分析的拓樸性質。為了分析系統的拓樸性質，我們關注能帶結構中的雙狄拉克錐(double Dirac cone)存在與否，以及簡併(degeneracy)能帶的交互關係。透過改變光子晶體的參數，整個晶體系統的拓樸性質可以被成功地改變。針對晶體參數作分析，可以發現某些參數下系統處在不同拓樸相(topological phase)的臨界狀態。藉著接合不同拓樸相的光子晶體，可以在不破壞時間反演對稱性(time-reversal symmetry)的情況之下，實現量子自旋霍爾效應(quantum spin Hall effect)，這造成在不同拓樸性質的晶體區域交界之間存在拓樸邊界態。 相較於其他研究以 對稱結構為模擬對象，本研究找到了一種 對稱結構，並在模擬結果中展示了拓樸邊界態。透過調整材料參數，無論在橫磁(transverse magnetic)、橫電模態(transverse electric mode)之下，拓樸邊界態都可以在本研究的結構中被展示。拓樸邊界態擁有很低的背向散射(backscattering)，並且能以銳轉折角為行徑路線。因此，透過改變幾何參數造成相變並達成拓樸邊界態，這樣的技術在光學應用上存在極大的潛力。

In this thesis, two-dimensional full-dielectric photonic crystals are used to modulate the topological edge states. With the dispersion relations of the photonic crystals, the band structures can be displayed. According to the references, there are many valuable topological properties in the photonic bands. In order to analyze the topological properties of the system, we focus on the existence of double Dirac cones in the band structure and the interaction of degenerate bands. By changing the parameters of the photonic crystals, the topological properties of the crystal system can be successfully changed. It can be found that some critical states between different topological phases by sweeping the parameters. By combining photonic crystals of different topological phases, the quantum spin Hall effect can be realized without breaking the time-reversal symmetry. Therefore, there are topological edge states at the boundary of two regions with the different topological properties. Comparing with the references that focus on the structures with C6v symmetry, we find a new structure with C6 symmetry that possesses the topological edge states. By exchanging the material parameters, the topological edge states can be shown in the same structure for both transverse-magnetic and transverse-electric modes in this thesis. Topological edge states can propagate along the sharp corners without backscattering. Therefore, changing the geometry parameters, which leads to the phase transition and modulates the topological edge states, is a technique with great potential in optical applications.