近年來光子晶體吸引了大家的注意，主要原因就是其具有光子頻帶結構，與電子能隙結構相當類似，而在光子帶隙中電磁波無法通過。利用光子晶體可以發展出許多光通訊元件，並且較傳統光元件性能優異，因此目前在製造方面之研究也大有人在。
本篇論文針對當光子晶體因製作過程中出現誤差，導致介電材質之形狀與尺寸產生偏差時，對其帶隙產生的影響進行模擬，並且與原設計系統之帶隙進行比對。主要是考量二維正方形、三角形及蜂巢形晶格結構，而介電柱形狀則為圓柱與方柱，分別對兩種不同之介電柱結構，介電常數分別為8.9和11.4，在圓柱半徑與方柱邊長以及形狀具有誤差下之狀況進行模擬分析。並且在介電質圓柱正方形晶格排列、介電質圓柱三角形晶格排列與介電質方柱三角形晶格排列之TM模態帶隙中出現與原設計系統不同之結果。

Periodic dielectric structures, which are called photonic crystals or photonic lattices, have attracted many interests in recent years. The existence of band gaps, which prohibits the propagation of electromagnetic waves in any direction, provides an opportunity to confine and control the propagation of electromagnetic waves.
This study aims at the influence of photonic band gaps which are brought by the errors because of the procedures of manufacturing. It is simulated that the crystal structures of 2-D square, triangular and honeycomb lattice, and the dielectric rods of circular and square whose size and form are little different from original design system. Consequently, it is found that the band gaps of 2-D photonic square lattice composed of cylinders, and triangular lattice composed of cylinders or square rods are different from original design system.

目錄.....................................................Ⅰ
表目錄...................................................Ⅳ
圖目錄...................................................Ⅶ
中文摘要.................................................Ⅹ
英文摘要................................................ⅩⅠ

第一章 序論............................................1

1.1. 光子晶體簡介....................................1
1.2. 研究目的........................................3
1.3. 文獻回顧........................................3
1.4. 論文結構........................................5

第二章 光子晶體理論基礎與對稱性..................6

2.1. 光子晶體理論基礎................................6
2.1.1. Maxwell方程式............................6
2.1.2. 晶格結構................................8
2.1.3. 布里淵區................................10
2.1.4. 光子晶體對稱性..........................13
2.1.5. 光子晶體的縮尺定律......................14
2.2. 二維光子晶體頻域分析...........................15
2.2.1. Bloch 理論..............................15
2.2.2. TE Mode.................................17
2.2.3. TM Mode.................................19
2.2.4. 介電質函數之求法........................20
2.3. 光子晶體對稱性分析.............................22
2.3.1. 二維正方形晶格結構......................22
2.3.2. 二維三角形晶格結構......................29
2.3.3. 對稱性分類之應用........................33

第三章 模擬結果......................................44

3.1. 介電質圓柱正方形排列............................44
3.2. 介電質圓柱正方形排列............................49
3.3. 介電質圓柱三角形排列............................54
3.4. 介電質方柱三角形排列...........................59
3.5. 介電質圓柱蜂巢形排列...........................64
3.6. 介電質方柱蜂巢形排列...........................70

第四章 結論與未來展望..............................78

4.1. 結論...........................................78
4.2. 未來展望.......................................81

參考文獻..................................................82









表目錄

表2-1 不同晶格排列之基底向量..............................11
表2-2 圓柱與方柱之 ......................................22
表2-3 之特徵標表.......................................23
表2-4 之特徵標表.......................................24
表2-5 之特徵標表.......................................24
表2-6 E與A模態在Γ、Δ點之對稱操作......................25
表2-7 二維正方形晶格 、 、 點對應 和 之模態...........26
表2-8 二維正方形晶格之最低三個 點之特徵標表示............27
表2-9 B1模態於 點出現之求解.............................27
表2-10 不同之 、 、 點所對應之膜態表示..................28
表2-11 之特徵標表.......................................31
表2-12 之特徵標表.......................................31
表2-13 二維三角形晶格 、 、 點對應 和 之模態..........32
表2-14 不同之 、 、 點所對應之膜態表示..................32
表2-15 蛋白石結構之布里淵區各點點群........................39
表3-1 介電質圓柱正方形排列誤差1%之TM模態模擬數值.........44
表3-2 介電質圓柱正方形排列誤差3%之TM模態模擬數值.........46
表3-3 介電質圓柱正方形排列誤差5%之TM模態模擬數值.........46
表3-4 介電質圓柱正方形排列誤差1%之TM模態模擬數值.........47
表3-5 介電質圓柱正方形排列誤差3%之TM模態模擬數值.........48
表3-6 介電質圓柱正方形排列誤差5%之TM模態模擬數值.........48
表3-7 介電質方柱正方形排列誤差1%之TM模擬數值.............50
表3-8 介電質方柱正方形排列誤差3%之TM模擬數值.............51
表3-9 介電質方柱正方形排列誤差5%之TM模擬數值.............51
表3-10 介電質方柱正方形排列誤差1%之TM模擬數值.............52
表3-11 介電質方柱正方形排列誤差3%之TM模擬數值.............53
表3-12 介電質方柱正方形排列誤差5%之TM模擬數值.............53
表3-13 介電質圓柱三角形排列誤差1%之TM模擬數值.............55
表3-14 介電質圓柱三角形排列誤差3%之TM模擬數值.............56
表3-15 介電質圓柱三角形排列誤差5%之TM模擬數值.............56
表3-16 介電質圓柱三角形排列誤差1%之TM模擬數值.............57
表3-17 介電質圓柱三角形排列誤差3%之TM模擬數值.............58
表3-18 介電質圓柱三角形排列誤差5%之TM模擬數值.............58
表3-19 介電質方柱三角形排列誤差1%之TM模擬數值.............60
表3-20 介電質方柱三角形排列誤差3%之TM模擬數值.............61
表3-21 介電質方柱三角形排列誤差5%之TM模擬數值.............61
表3-22 介電質方柱三角形排列誤差1%之TM模擬數值............62
表3-23 介電質方柱三角形排列誤差3%之TM模擬數值............63
表3-24 介電質方柱三角形排列誤差5%之TM模擬數值............63
表3-25介電質圓柱蜂巢形排列誤差5%內之TE模擬數值...........65
表3-26介電質圓柱蜂巢形排列誤差5%內之TM模擬數值...........66
表3-27 介電質圓柱蜂巢形排列誤差5%內之TE模擬數值..........68
表3-28介電質圓柱蜂巢形排列誤差5內%之TM模擬數值..........69
表3-29介電質方柱蜂巢形排列誤差5%內之TE模擬數值...........71
表3-30介電質方柱蜂巢形排列誤差5%內之TM模擬數值...........72
表3-31介電質方柱蜂巢形排列誤差5%內之TE模擬數值...........74
表3-32介電質方柱蜂巢形排列誤差5%內之TM模擬數值.........75-76
表4-1原設計系統(介電係數8.9)與誤差系統帶隙比對表...........79
表4-2原設計系統(介電係數11.4)與誤差系統帶隙比對表..........80






圖目錄

圖1-1 光子晶體示意圖.....................................2
圖2-1 二維的布拉維晶格示意圖.............................9
圖2-2 正方形晶格與倒晶格................................12
圖2-3 三角形晶格與倒晶格................................12
圖2-4 二維正方形晶格所具有之對稱性......................14
圖2-5 TE Mode示意圖.....................................18
圖2-6 TM Mode示意圖.....................................20
圖2-7 二維正方形晶格之第一布理淵區.......................23
圖2-8 二維正方形晶格於倒晶格空間中之延展.................26
圖2-9 二維正方形晶格TM Mode频帶關係圖...................29
圖2-10 三角形晶格所具有之對稱性..........................30
圖2-11 二維三角形晶格之第一布里淵區......................31
圖2-12 二維三角形晶格TM Mode频帶關係圖..................33
圖2-13 二維光子晶體COMITS實驗與理論比較圖...............34
圖2-14 電場平行介電柱之最低四能帶鏡射圖..................35
圖2-15 二維三角形晶格能帶與反射比圖......................37
圖2-16 二維三角形晶格能帶與反射比圖......................37
圖2-17 蛋白石結構之布里淵...............................38
圖2-18 三角形晶格具有一個缺陷柱之六種缺陷模態...........40
圖2-19 週期性擾動示意圖.................................41
圖2-20 具有兩種不同介電材質之E極化能帶圖...............42
圖2-21 週期性擾動示意圖.................................43
圖3-1 介電質圓柱正方形排列示意圖.......................45
圖3-2 介電係數8.9，誤差5%內樣本點TM帶隙..............47
圖3-3 介電係數11.4，誤差5%內樣本點TM帶隙.............49
圖3-4 介電質圓柱正方形排列示意圖.......................50
圖3-5 介電係數8.9，誤差5%內樣本點TM帶隙..............52
圖3-6 介電係數11.4，誤差5%內樣本點TM帶隙.............54
圖3-7 介電質圓柱三角形排列示意圖.......................55
圖3-8 介電係數8.9，誤差5%內樣本點TM帶隙..............57
圖3-9 介電係數11.4，誤差5%內樣本點TM帶隙.............59
圖3-10 介電質方柱三角形排列示意圖.......................60
圖3-11 介電係數8.9，誤差5%內樣本點TM帶隙..............62
圖3-12 介電係數11.4，誤差5%內樣本點TM帶隙.............64
圖3-13 介電質圓柱蜂巢形排列示意圖.......................65
圖3-14 介電係數8.9，誤差5%內樣本點TE帶隙..............67
圖3-15 介電係數8.9，誤差5%內樣本點TM帶隙...............67
圖3-16 介電係數11.4，誤差5%內樣本點TE帶隙..............70
圖3-17 介電係數11.4，誤差5%內樣本點TM帶隙..............70
圖3-18 介電質方柱蜂巢形排列示意圖........................71
圖3-19 介電係數8.9，誤差5%內樣本點TE帶隙...............73
圖3-20 介電係數8.9，誤差5%內樣本點TM帶隙...............73
圖3-21 介電係數11.4，誤差5%內樣本點TE帶隙..............76
圖3-22 介電係數11.4，誤差5%內樣本點TM帶隙..............77

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