由於光子晶體能帶結構的特性可以被引用在各種物理及電機上的應用，所以近年來已經成為大家感興趣的一門研究。在這篇論文，我們主要致力於使用更有效率的方式，來呈現清楚的能帶結構的圖表；主要處理的問題是來自於一個二維光子晶體在橫向磁場模式上，在離散我們的問題之後，將會得到一個無窮維的特徵值問題，利用格林函數逼近法，又可以將我們問題轉化成一個非線性矩陣方程式，接下來我們引用保結構倍增演算法來解決這個非線性矩陣方程式得到這個問題的穩定解，而利用這個答案，我們可以呈現光子晶體能帶結構之圖形，並且成功地對這方面的問題引進更有效率之解決方法。

Full band gap structure is the most distinguished feature of photonic crystal and attracts extensive studies in its properties and applications. Our basic goal in this paper is to use an efficient way to illustrate the band structure of our problem and find the band gap as we can. First, we have a 2-D photonic crystals model in TM mode. After discretization, we can reduce our problem into an infinite general eigenvalue problem (GEP). Using Green’ s function approach, we can transform the infinite GEP into a finite order non-linear matrix equation (NME), whose solution can be used to illustrate the band structure figure. Next, we apply the structure-preserving doubling algorithm to solve this NME and get the solution with a nice efficiency by the special form of our matrices. Finally, we can successfully get the band structure figure and present an advanced way in the research of photonic crystals.

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