光子晶體是週期性的介電質材料。它的有趣與奇特的電磁特性使其成為最近十年內，奈米光學研究領域內最熱門的研究主題之一。在本論文中我們首先介紹光子晶體的基本物理特性，以及常用的數值演算法；藉由數值計算及模擬，我們可分析(金屬性)光子晶體及其元件的特性。在分析光子晶體波導中，我們會先討論微帶隙的成因；接著針對光子晶體波導在導波時所受到的幾何限制因素(晶格排列)，提出一個新的改善方法，並設計出可任意彎曲的介電質柱波導。延伸此波導架構，我們也提出一個可用於對稱及非對稱分波形態的分波器。另外，我們也設計了一種可針對不同極化方向產生正負折射的極化分波器。而在本論文的最後，我們也探討了均向及非均向性超常材料(metamaterial)的特性以及相關的應用。

Photonic crystals are periodic structures made of dielectric materials. Its fascinating and amazing electromagnetic properties have made it one of the most extensively studied topics in the nano-optics related research fields in the last decade. In this thesis, we introduce the properties of photonic crystal and the numerical algorithms for calculating these properties. By utilizing these numerical methods, we can analyze the characteristics of the (metallic) photonic crystals and design various devices based on them. In the research of photonic crystal waveguide, we study the formation of the mini-stopband. For releasing the geometric restrictions of the photonic crystal waveguide, we propose an improved method, and design periodic dielectric cylinder waveguide which can be bent arbitrarily without losing its wave-guiding ability. Based on this structure, we also propose the symmetrical and asymmetrical configurations of a new type of beam splitter. Besides, we design a polarization beam splitter utilizing both negative and positive refraction. In the final section, we also discuss the optical characteristics of the homogeneous and inhomogeneous metamaterial and their applications.

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