所謂的光子晶體就是在空間中，將介電質材料週期性地排列的結構。受到週期結構的影響，電磁波的行為會呈現布洛赫波(Bloch wave)的傳播形態。從光子晶體的色散曲線中，我們可以探討能隙結構(Band structure)、缺陷態(Defect modes)、態密度(Density of states)及表面模態(Surface modes)的物理特性。常用的分析方法包括平面波展開法(Plane wave expansion method)、多重散射法(Multiple scattering method)、時域有限差分法(Finite difference time domain method)、傳遞矩陣法(Transfer matrix method)等；而本論文主要是以平面波展開法來分析二維光子晶體的光學特性。
本論文主要分為四章，第一章是介紹光子晶體的背景知識與由來，以及它的相關應用。第二章除了包括平面波展開法的理論公式推演以及一些物理特性回顧之外，再將平面波法延伸，結合超晶格法(Super cell method)、史托克定理近似(Stokes’s theorem approach)及微擾法(Perturbation method)，可針對特殊形狀介電質材料做能隙分析，及能帶結構預測。第三章針對新穎方法的收斂性、精確性做嚴謹的測試、檢驗以及討論。第四章總合本篇論文的各項重要特點下一個結論。

Photonic crystals is the structure to arrange dielectric medium in order. Electronic wave propagate in Bloch wave form because of periodic structure. We discuss band structure, defective mode, density of state and surface mode in Photonic crystals’ dispersion relation. The analytic methods include Plane wave expansion method, Multiple scattering method, Finite difference time domain method, Transfer matrix method and this paper major on Plane wave expansion method.
The paper divide into four chapter. First chapter introduce photonic crystals, including background and application. Second chapter besides we deduce the equation of Plane wave expansion method and review some physics character, combine super cell method, Stokes’s theorem approach, Perturbation method, we can analyze band gap and forecast band structure. Third chapter we test and verify our new method and at the last chapter we give a conclusion.

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