掌性光子超穎材料為近年來應用於奈米光學元件領域裡之指標性結構，其強響應旋光性以及圓二色性的特性吸引了許多關注，雖然掌性結構能透過適當的週期排列以及幾何參數建構，但若要與旋性光交互作用，螺旋結構則為最理想的模型系統。近十年來，介電質螺旋結構已有許多特性研究，隨著微機電奈米製程的快速進步，除了一維或二維光子晶體的製作與能隙探討已有詳細的理論建立之外，亦開始有團隊以包括斜向沉積技術、直接雷射寫入技術、全像蝕刻術等展現其製造之三維結構與對應之能隙，但並沒有團隊全盤的分析其三維螺旋光子超穎材料能隙生成的理論機制。

因此，在本研究中，我們利用液晶能隙分析理論與有限時域差分法，研究三維螺旋光子超穎材料中布拉格共振和混成現象的出現、演變與其交互作用所產生的多種情況，並提供三維光子超穎材料中從布拉格能隙到混成能隙再到散射區間轉換機制為目標導向的有效搜索標準，達成透過幾何參數調整，操控兩種能隙之間的出現的頻率、頻寬以及相對強度的目標。除了不須使用高折射率介電質，亦較二維光子超穎材料多了另一個操控之自由度，也將為光子超穎材料與奈米光學元件設計開起新的設計途徑。

Chiral photonic nanostructures provide a variety of fascinating properties, such as strong optical activity and circular dichroism, which can be applied in many optoelectronic devices. Although chiral motifs can be constructed by a variety of arrangements, the helix geometry is still an ideal model system to elucidate the interaction of circularly polarized (CP) light with a chiral medium. Dielectric helix structures have been studied for decades and the experimental implementation has been realized on various platforms, such as glancing angle deposition (GLAD), direct laser writing (DLW), and holographic lithography. However, these studies are predominately focused on presenting the existence of gaps, without addressing the underlying mechanisms.
In this study, we use an analytical model adapted from liquid crystal and the numerical simulation based on FDTD to study the emergence and evolution of both Bragg resonance and hybridization phenomenon in a 3D dielectric helix structure. We show that the interplay of gaps with different mechanisms gives rise to versatile scenarios. We also provide a diagram that illustrates the scenarios with respect to different geometrical parameters. Such 3D configuration provides another degree of freedom to tailor the relative position between the two bands by geometrical parameters without the use of high-permittivity dielectrics, opening a new horizon to design optical devices.

第一章 緒論 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1
1.1 光子晶體與超穎材料介紹 . . . . . . . . . . . . . . . . . . . . 1
1.2 掌性材料介紹 . . . . . . . . . . . . . . . . . . . . . . . . .3
1.3 螺旋光子超穎材料 . . . . . . . . . . . . . . . . . . . . . . .3
1.3.1 螺旋光子超穎材料光學特性. . . . . . . . . . . . . . . . . . . 3
1.3.2 三維光子超穎材料製造 . . . . . . . . . . . . . . . . . . . . .6
1.4 混成能隙. . . . . . . . . . . . . . . . . . . . . . . . . . .7
1.5 動機 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
第二章 實驗方法 . . . . . . . . . . . . . . . . . . . . . . . . . . .10
2.1 數值方法 . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.1.1 時域有限差分法 . . . . . . . . . . . . . . . . . . . . . . . 10
2.1.2 平面波展開法 . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 能隙分析 . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2.1 約化布里淵區(The irreducible Brillouin Zone) . . . . . . . . 12
2.2.2 光子超穎材料色散關係 . . . . . . . . . . . . . . . . . . . . .13
2.2.3 圓二色性(Circular Dichroism, CD) . . . . . . . . . . . . . .13
2.3 模擬設定. . . . . . . . . . . . . . . . . . . . . . . . . . . .15
第三章 三維螺旋光子超穎材料能隙形成機制研究 . . . . . . . . . . . . . . 17
3.1 色散關係與光學反射頻譜 . . . . . . . . . . . . . . . . . . . . 17
3.2 三維螺旋光子超穎材料幾何構形與能隙變化探討 . . . . . . . . . . . 19
3.2.1 右旋三維反射頻譜分析與結構幾何參數 . . . . . . . . . . . . . 20
3.2.2 左旋三維反射頻譜分析與結構幾何參數 . . . . . . . . . . . . . 22
3.3 機制 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.3.1 布拉格能隙成因 . . . . . . . . . . . . . . . . . . . . . . . 23
3.3.2 非布拉格能隙成因 . . . . . . . . . . . . . . . . . . . . . . 25
3.4 三維反射頻譜分析 . . . . . . . . . . . . . . . . . . . . . . 30
3.4.1 橫向單位間距與頻率. . . . . . . . . . . . . . . . . . . . . .31
3.5 運用三維結構週期性操控能隙機制 . . . . . . . . . . . . . . .33
第四章 運用解析解近似三維螺旋光子超穎材料能隙 . . . . . . . . . . . 34
4.1 布拉格能隙與解析解 . . . . . . . . . . . . . . . . . . . . . 34
4.1.1 螺旋奈米雕刻薄膜解析解 . . . . . . . . . . . . . . . . . . . 34
4.1.2 運用代數變換與三維迭代近似法加快數值計算 . . . . . . . . . . 36
4.1.3 布拉格能隙能隙中心波長預測 . . . . . . . . . . . . . . . . . 37
4.1.4 結構修正 . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.1.5 結構修正後布拉格能隙能隙中心頻率預測 . . . . . . . . . . . . 39
4.1.6 布拉格能隙頻寬預測模型. . . . . . . . . . . . . . . . . . . .43
4.1.7 頻寬預測結果. . . . . . . . . . . . . . . . . . . . . . . . .44
4.1.8 體積分率與近似誤差. . . . . . . . . . . . . . . . . . . . . .45
4.2 混成能隙趨勢分析. . . . . . . . . . . . . . . . . . . . . . .46
4.2.1 本徵模態演變與橫向單位間距. . . . . . . . . . . . . . . . . .46
4.2.2 本徵模態演變與縱向週期性. . . . . . . . . . . . . . . . . . .47
4.3 機制相位圖 . . . . . . . . . . . . . . . . . . . . . . . . . 47
第五章 結果與未來展望 . . . . . . . . . . . . . . . . . . . . . . . 52
參考文獻 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

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