光子超穎材料(photonic metamaterials)的發展與研究，使得人類得以透過人造的材料來達到自然材料中所無法達到的特性，其特殊性質主要是來自於組成材料本身的特性以及其在空間中的幾何形狀與排列，透過週期性的排列能使超穎材料以自然材料所無法達到的方式來操控電磁波。光子超穎材料的分類有很多種，其中掌性結構對於圓偏振光(circularly polarized light)有特別的交互作用，因而發展出很多關於圓偏振光與掌性結構特性的研究，這種掌性結構也開啟許多設計圓偏振光相關光學儀器的可能性。在此論文中，我們主要研究的掌性結構為螺旋結構(helix)以及螺旋二十四面體(gyroid)，材料方面我們選擇的是介電質材料，相較於金屬的特殊的色散特性，介電質材料的色散性質在可見光波段相對比較單純，且擁有低吸收率的特性，利用此材料的週期排列，我們可以調整結構參數來控制其光學特性。為了能有效了解介電質掌性材料的特性，我們利用了有限時域差分法（FDTD）來進行研究，在空間中建立一個掌性結構的系統模型，來檢驗掌性結構與圓偏振光的交互作用。
在介電質螺旋陣列中，因其結構旋轉的構造以及幾何排列，能產生與結構同旋性的光子能隙外，也會產生與結構相反旋性的光子能隙，這種相反旋性的能隙是來自於幾個螺旋結構之間的排列產生出破碎的相反旋性結構而形成，我們期望透過調整結構參數及材料參數來調控兩個不同旋性能隙所產生的頻段。除此之外，透過系統性的參數研究結合其他文獻中的理論，我們發現不同頻段的能隙可能來自於不同的成因，這些研究結果對於應用介電質螺旋結構的光電元件之設計有相當程度的幫助。
在介電質螺旋二十四面體中，我們也發現光子能隙的存在，在能隙頻段附近的模態因為色散關係的快速變化，能夠導致一些異常的光學性質，其中包含負折射以及準直現象，我們透過建立色散曲面及等頻率線的方式來研究其色散關係，並觀察異常折射現象發生的條件，透過調整結構參數及材料參數也能調控其發生的頻段與角度，這些研究結果有助於應用介電質螺旋二十四面體結構的光學元件之設計。

The development of artificial materials, termed photonic metamaterials, has led to phenomena that do not exist in natural materials. The properties of metamaterials are derived both from the properties of their constituent materials and the geometrical arrangement. The constituent materials are usually arranged in periodic patterns, manipulating the waves in the ways that are unachievable with conventional materials. Such periodic structures with chiral morphology are capable of inducing chiroptical effects with respect to righthanded circularly polarized (RCP) light and left-handed circularly polarized (LCP) light, leading to new types of circular polarization-sensitive devices. In addition, by studying dispersion characteristics, some interesting optical properties can be discovered. In this work,
we present studies based on finite-difference time-domain (FDTD) method for analyzing the polarization-dependent properties and dispersion relation characteristics of dielectric helix and dielectric single gyroid (SG) structures. The corresponding band structures, circular
dichroism (CD) indices, coupling indices and reflectance spectra are examined to verify circular polarization-dependent properties. The dispersion surfaces and equi-frequency contours (EFCs) are applied to discover interesting wave guiding characteristics including negative refraction. Moreover, we also investigate how the frequency ranges of these optical properties
are tailored by varying the refractive index and structural parameters of the structures. These results are crucial for the design of functional devices at optical frequencies based on dielectric SG and helix structures.

Contents
Abstract i
Contents ii
1 Introduction 1
1.1 Introduction to metamaterials . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Photonic crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.3 Chiral structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3.1 Helix chiral structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3.1.1 Metallic helix chiral structure . . . . . . . . . . . . . . . . . 5
1.3.1.2 Dielectric helix chiral structure . . . . . . . . . . . . . . . . 6
1.3.2 Gyroid chiral structure . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3.2.1 Morphology . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3.2.2 Optical properties . . . . . . . . . . . . . . . . . . . . . . . 10
1.4 Hybridization gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.5 Negative refraction and collimation . . . . . . . . . . . . . . . . . . . . . . . 15
1.6 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2 Methods 19
2.1 Photonic band structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.1.1 Finite-difference time-domain method . . . . . . . . . . . . . . . . . . 19
2.1.2 The irreducible Brillouin zone . . . . . . . . . . . . . . . . . . . . . . 21
2.1.3 Dispersion surface and equi-frequency contours (EFCs) . . . . . . . . 22
2.2 Circular dichroism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.2.1 Directions of circular motion . . . . . . . . . . . . . . . . . . . . . . . 24
2.2.2 CD index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2.3 Coupling index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.3 Simulation setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.3.1 Helix chiral structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.3.2 Gyroid chiral structure . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3 Dual polarization-dependent band gap in helix chiral structure 29
3.1 Polarization-dependent properties in helix array . . . . . . . . . . . . . . . . 29
3.1.1 Band structure and reflectance spectra . . . . . . . . . . . . . . . . . 29
3.2 Effects of structural and material parameters on the polarization band gaps . 32
3.2.1 Gap shifting with structural parameters . . . . . . . . . . . . . . . . 33
3.3 Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.3.1 Bragg and hybridization gap . . . . . . . . . . . . . . . . . . . . . . . 35
3.3.2 Verification of Bragg gap . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.3.2.1 Decreasing pitch number . . . . . . . . . . . . . . . . . . . . 35
3.3.2.2 Varying pitch length . . . . . . . . . . . . . . . . . . . . . . 38
3.3.3 Verification of hybridization gap . . . . . . . . . . . . . . . . . . . . . 39
3.3.3.1 Resemblance between the coupling modes . . . . . . . . . . 39
3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.4.1 Gap map for RCP modes . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.4.2 Gap map for LCP modes . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.4.3 Gap map for complete gap and polarization gap . . . . . . . . . . . . 44
4 Extraordinary optical properties in single gyroid chiral structure 47
4.1 Dispersion relation and band gap in single gyroid . . . . . . . . . . . . . . . 47
4.1.1 Band structure and reflectance spectra . . . . . . . . . . . . . . . . . 47
4.1.2 Complete band gap and polarization-dependent band gap . . . . . . . 48
4.2 EFCs construction and analysis . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.2.1 Dispersion surfaces and EFCs . . . . . . . . . . . . . . . . . . . . . . 50
4.2.2 Positive and negative refraction . . . . . . . . . . . . . . . . . . . . . 51
4.2.3 Diverging behavior and collimation effect . . . . . . . . . . . . . . . . 53
4.3 Verification of extraordinary optical properties . . . . . . . . . . . . . . . . . 55
4.3.1 Positive and negative refraction . . . . . . . . . . . . . . . . . . . . . 55
4.3.2 Diverging behavior and collimation effect . . . . . . . . . . . . . . . . 56
4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.4.1 Effect of refractive indices . . . . . . . . . . . . . . . . . . . . . . . . 58
4.4.2 Structure parameters for the TiO2 SG structures . . . . . . . . . . . 59
5 Conclusions 61

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