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 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 andinterface state of a one-dimensional photonic crystal system composed of two materials(AB periodic layered structure) and three materials (ABC periodic layered structure). It isknown that in the AB periodic layered structure, when the unit cell is set to have inversionsymmetry, 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 thedesired topological bands. With the adjustment of the above parameters and keeping theoptical path difference in a unit cell invariant, we can tune a bandgap from open to closeand reopen again. When calculating the Zak phase of the band above and below the energygap, it is found that there is a topological phase transition related to the band inversionphenomenon. Connecting the two AB periodic structures before and after the topologicalphase transition, a transmittance peak is found in the bandgap, which corresponds to theexistence of the interface state. Generalizing the above method and applying it to the ABClayered structures, it is found that the bandgap designed in accordance with the invarianceof the optical path difference will be much smaller than that of the AB periodic layeredstructures, and the bandwidth of the transmittance peak becomes narrower. This means thatthe localization effect of the interface state in the ABC system is poorer, so a lot of unitcells are necessary to clearly demonstrate the interface state that decreases exponentiallyaway from the interface. According to the results of the numerical simulation, it is foundthat if the structural parameters of B are fixed in the ABC structure, and we adjust only theparameters of A and C layers to design the bandgap, then the simplest way to get the newstructure as the structure after topological phase transition is just to exchange the A and Clayers in the original ABC structure. We also found that the connection of two ABCstructures formed by this way gives us the most obvious localized effect of the interfacestate.
 摘要 IAbstract II誌謝 III目錄 IV圖目錄 VI表目錄 VIII一、緒論 11-1光子晶體介紹 11-2 札克相(Zak phase) 3二、層狀結構的理論分析 52-1 傳遞矩陣 52-1-1 如何利用傳遞矩陣來求得能帶結構 102-1-2 AB週期結構中的穿透率和反射率 142-1-3 矩陣的相似變換 162-1-4布洛赫定理(Bloch Theorem) 172-2 討論在AB週期結構中能帶反轉的拓樸特性和介面態 172-2-1 AB週期結構能帶的反轉現象 182-2-2討論在AB周期結構中的介面態 19三、AB週期結構下的數據分析 213-1 AB週期結構能帶與電場分析 213-1-1 單一頻率情況下的電場 213-1-2多頻率情況下的電場 243-2 雙層週期結構下的札克相(Zak phase)數據分析 253-2-1 如何由電場分析札克相(Zak phase) 263-3 雙層週期結構下介面態分析 29四、在多層ABC周期結構下的能帶反轉和介面態 324-1 多層ABC週期結構的傳遞矩陣 324-2 多層ABC週期結構的札克相(Zak phase)分析 354-2-1 ABC結構場圖的數據分析 374-2-2 ABC結構札克相(Zak phase)的數據分析 394-3 在多層ABC週期結構下的介面態分析 414-3-1結構的多寡對於介面態透射率的影響 444-3-2參數的選擇對介面態的影響 45五、結論與未來展望 505-1 結論 505-2 未來展望 51參考文獻 52
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