本論文研究螺旋光子晶體的光子帶隙現象，內容主要偏重在改變螺旋結構之週期性變化的規則，進而分析其所造成之光子帶隙內的缺陷模，以及相關衍生的光學應用元件。本文中所利用的數值工具包含4×4傳遞矩陣法與有限元素法，並以膽固醇液晶為例當作所要研究的螺旋光子晶體。首先以較單純的等螺旋距結構介紹初步的物理光學現象，包含色散關係、光子群速與態密度、穿透與反射頻譜、缺陷模的基本現象等等。接著再進入論文核心所要討論之螺距梯度結構的光子帶隙現象，研究的缺陷型態主要有三種，分別為：扭旋缺陷、梯度落差缺陷與螺距落差缺陷。

在螺距梯度結構中引入扭旋缺陷會使圓偏振光在不同行進方向上出現不相同中心波長的缺陷模，利用缺陷模具單一方向的光傳輸性，可當作光學二極體使用，然而隨結構厚度增加，兩正交圓偏振光會出現不尋常的缺陷模交錯行為。在螺距梯度結構中引入梯度落差缺陷所產生的缺陷模，只存在於與螺旋結構旋向定義相同的圓偏振光上，且隨結構厚度增加而逐漸在穿透強度上衰減甚至消失，卻可在頻譜上形成更寬的反射帶，可作為超寬頻圓偏光反射鏡使用。在螺距梯度結構中引入螺距落差缺陷會產生多缺陷模，各缺陷模的中心波長關係以及自由頻譜範圍可與法布里-比洛干涉現象的結果類比，若再加上扭旋缺陷的綜合效應，所產生之缺陷模中心波長的位置變化會與扭旋角度的改變呈線性關係，此多缺陷模現象可作為具圓偏振光選擇性或非圓偏振光選擇性之光譜濾波器使用。

The properties of photonic band gaps in chiral photonic crystals are discussed in the thesis. The research is devoted to the photonic defect modes in the chiral photonic structures with gradient pitch length and the introduction of the different kinds of defects. In addition, various optical applications with respect to the defect modes are also demonstrated. Self-organized cholesteric liquid crystals (ChLCs) with spatially varying pitch are the examples. The finite element method and the 4×4 transfer matrix method are carried out for the study of the optical propagation simulations in the ChLC films. First, the optics of the cholesteric medium with constant pitch is talked over, including dispersion relations, the group velocity, the group delays, the electromagnetic density of modes, transmission and reflection spectra, the properties of the defect modes, etc. Then, the optics of the cholesterics with spatially varying pitch is focused on the photonic defect modes created by the following defects: twist defect (phase jump), gradient jump, and pitch jump.

The defect mode due to the introduction of a twist defect is considered in the chiral structures with spatially varying pitch. An unusual crossover behavior in reflection at the defect resonance wavelength of a single circularly polarized mode appears when the structure thickness increases over a specific value. Two different resonance wavelengths can be created by a twist defect in the identical ChLC composite film with linearly varying pitch. The behavior constitutes the operational mechanism for a passive optical diode.

The introduction of photonic defect modes in a ChLC can be achieved by a global deformation of the helix with a chiral anti-symmetry relative to the middle of the ChLC configuration. The defect modes can be excited only when the circularly polarized incident wave has the same handedness as the ChLC. The transmittance of the defect modes is greatly reduced when the structure thickness is large far away from a critical value, and even some defect modes will be completely suppressed. A wider forbidden band for polarization-selective reflection can be obtained. Such a ChLC film can be useful for the production of the ultra-wide-band reflective polarizer.

For normal and near-normal incidence of circularly polarized light with the same handedness as structure, the defect caused by a pitch jump in the cholesteric helix results in discrete peaks within a forbidden band in the transmission. The particular spectrum is similar to the feature of a Fabry-Perot interferometer. By introducing an additional phase jump, linear blue shifts of the defect modes in transmission spectra are correlated with an increase in the twist angle. Polarization-dependent or polarization-independent optical filters based on the special transmission properties of multiple photonic defect modes are described.

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