自從1980年被Klitzing發現凝聚態物理中之整數量子霍爾效應(integer quantum Hall effect)以及Haldane在1988年提出量子異常霍爾效應(quantum anomalous Hall effect)並將相關研究延伸至光學系統後，拓樸絕緣體(topological insulator)之概念引起了科學界的廣泛注目。拓樸絕緣體有一特性，當具有不同拓撲數之能帶的材料相互接觸時會產生一邊緣態，並且其邊緣態可以起到拓撲保護(topological protected)作用，產生背散射抑制(back scattering)和免疫缺陷(defect)的單向傳播(unidirectional propagating)行為，對於傳播效率提升具有相當大的助益。另一方面M‧V‧Berry在1984年對於週期性固體在電子動力學領域所展現之相位性質率先定義出拓樸相位，並由J‧Zak於1989年提出了一維Berry phase在固態物理中能帶之相關拓樸特性，並在2019年Hai-Xiao Wang等人提出了Wilson loop以便在光學模擬空間中求得拓樸相位之數值解，對於光子晶體所展現之能帶結構中的拓樸性質有了明確的描述及量化，為拓樸晶體之設計及特性有更明確的認知，激發了更多後續的理論及實驗研究。
本論文首先於FDTD模擬空間中建立了Wilson loop拓樸相位數值計算法，求得所建構之光子晶體在色散頻譜中所展現的拓樸相位性質，並且將其結果與能帶結構倒空間之第一布里淵區高對稱點之電磁場模態強度分布圖相結合，雙重驗證拓樸相位之準確性質，並且由於光子晶體可類比為無限多層法布立-佩羅共振腔系統(Fabry-Perot resonance cavity system)，本論文也採用了有限多層週期性法布立-佩羅共振腔系統及單層法布立-佩羅共振腔系統之穿透頻譜，觀察在不同空氣佔比下，能帶結構與穿透頻譜的關聯性，並且在研究成果中成功預測了能帶間拓樸相位性質，從而建立一套準則，並且將以上結果擴展至二維單向週期性光子晶體結構之介面模態進行討論。最後則是重現了相關領域論文之研究結果，以期對拓樸光學模擬的研究開拓道路。

Since Klitzing discovered the quantum Hall effect in 2D electron gas system in 1980, Haldane proposed the quantum anomalous Hall effect with no magnetic fields in 1988 and then proposed the concept to optical systems. Among these, the core concept of “topological insulators" has attracted widespread attention in recent decays. In the optical system, Wu proposed quantum spin Hall effect in 2015 and stimulated subsequent theoretical and experimental studies in photonic topological insulators.
From the other perspective, in 1984 M. V. Berry proposed the concept of geometrical phase, which is called the "Berry phase", exhibited in the periodic solid state physics using Quantum Electrodynamics. In 1989, J. Zak proposed the 1D Berry phase in 1D solid-state system and related it to the topological properties of the energy band structure and clearly quantified it. In 2019, Wang et al. proposed the "Wilson loop" to obtain the topological phase in optical lattices.
This work first established the Wilson loop topological phase calculation by FDTD simulation and obtain the topological phase properties exhibited in 1D photonic crystal band. We also correlated the nontrivial topological phase with symmetry inversion of mode pattern at  and X point. We also adopt the single-layer F-P resonator and multilayer Fabry-Perot resonance cavity systems to observe the correlation between the energy band structure and the transmission spectrum at different dielectric/air ratio, and successfully predicts the topological phase transition crossing energy bands. Finally, we calculate the topological properties in 2D photonic structure, in order to extend the Wilson loop FDTD approach for the photonic/plasmonic topological insulators.

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