本論文主要探討兩種材料 (AB 週期層狀結構) 與三種材料 (ABC 週期層狀結構)
所構成的一維光子晶體系統的能帶拓樸相變 (topological transition) 與介面態
(interface state)。已知在 AB 週期層狀結構中，當設定晶胞具有反射對稱性
(inversion symmetry) 時，各能帶的札克相 (Zak phase) 只有兩個可能值： 0 與 π；
此時可以調整介電常數和每層的厚度來獲得所需的拓樸能帶。隨著以上參數調整並
利用一個晶胞內的光程差不變量，我們可以使一個能隙 (gap) 從打開變成閉合然後
又重新打開。此時計算能隙上下兩能帶的札克相，會發現出現了札克相互換的拓樸
相變 (topological transition)，也就是能帶反轉 (band inversion) 的現象。將拓樸相變
前後的兩種 AB 週期結構相接，會在能隙中發現透射率出現峰值，這代表著介面態
(interface state) 的產生。把以上的方法推廣到 ABC 層狀結構，會發現此結構利用光
程差不變量所調整出來的能隙會比 AB 週期層狀結構的還來的小許多，而透射率峰
值的頻寬變得更窄。這代表著介面態的局域性較差，所以需要在結構中使用更多的
晶胞數才能明顯看得出隨遠離介面距離而遞減的介面態。根據數值模擬的結果發現：
如果在 ABC 週期層狀結構當中將 B 的結構參數固定，只調整 A 和 C 的結構參數來
設計相變前的能隙，那麼直接將 A 與 C 層相互對調的結構就是設計上最簡單又可
使介面態具有較好局域效果的相變後結構。

This thesis mainly discusses the topological phase transition of the energy bands and
interface state of a one-dimensional photonic crystal system composed of two materials
(AB periodic layered structure) and three materials (ABC periodic layered structure). It is
known that in the AB periodic layered structure, when the unit cell is set to have inversion
symmetry, the Zak phase of each energy band can take only two possible values: 0 or π.
However, we can choose appropriate permittivity and thickness of each layer to design the
desired topological bands. With the adjustment of the above parameters and keeping the
optical path difference in a unit cell invariant, we can tune a bandgap from open to close
and reopen again. When calculating the Zak phase of the band above and below the energy
gap, it is found that there is a topological phase transition related to the band inversion
phenomenon. Connecting the two AB periodic structures before and after the topological
phase transition, a transmittance peak is found in the bandgap, which corresponds to the
existence of the interface state. Generalizing the above method and applying it to the ABC
layered structures, it is found that the bandgap designed in accordance with the invariance
of the optical path difference will be much smaller than that of the AB periodic layered
structures, and the bandwidth of the transmittance peak becomes narrower. This means that
the localization effect of the interface state in the ABC system is poorer, so a lot of unit
cells are necessary to clearly demonstrate the interface state that decreases exponentially
away from the interface. According to the results of the numerical simulation, it is found
that if the structural parameters of B are fixed in the ABC structure, and we adjust only the
parameters of A and C layers to design the bandgap, then the simplest way to get the new
structure as the structure after topological phase transition is just to exchange the A and C
layers in the original ABC structure. We also found that the connection of two ABC
structures formed by this way gives us the most obvious localized effect of the interface
state.

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