A photonic crystal (PC) is a structure that has periodic distribution of dielectric material, which has the most important feature of having a bandgap. Bandgap is a range of frequencies at which the propagation of light is forbidden. This feature is used to design a variety of photonic crystal devices. The key is to disrupt the periodic structure of photonic crystals and create defect modes. For this reason, calculation of defect modes in an effective and accurate approach is an extremely important topic.
Our research consists of developing an improved supercell method within full plane wave expansion. In view of the symmetry property of defect states, we construct the basis function which reflect the underlying lattice symmetry and develop an accurate and efficient computing method. By this method, we can compute the defect states of photonic crystal very accurately and fast. This allows us to focus on defect states of specific symmetry and calculate the defect states independently. By doing so, it helps the device designers to have a preliminary understanding of the states in the possible shortest time before experiments are carried out.

光子晶體是一種介電物質成週期性分佈的結構，其最大的特性即是擁有能隙，我們可以利用能隙的特性控制光的傳播。這樣的特性被用來設計為多種光子晶體元件，而最主要的方式即是利用破壞原本光子晶體的週期性，使其原本的能隙中出現缺陷態，以作為應用。由上述可知，如何有效、準確且快速的計算出光子晶體的缺陷態是很重要的課題。
我們的研究方法是以超晶格的全平面波展開法為理論基礎，加以改進。我們利用缺陷態擁有不同對稱性的特性，以重新建立基底的方式，發展了一個準確且快速的計算方法。利用此方法，可以非常準確且快速的計算出光子晶體的缺陷態。我們亦能針對不同的對稱性，獨立計算單一對稱性的缺陷態。如此，可讓元件設計者能在最短時間內對結構特性能有初步的瞭解，方便進行實驗上更進一步的模擬和研究。

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