利用修改之平面波展開法和頻域有限差分法來分析光子晶體

Abstract

藉由將馬克士威方程式推導為一個利用已知頻率來求波向量的特徵值方程式後，利用不同的數值方法來替代此特徵值方程式中的運算子，便可以利用不同的數值方法來計算光子晶體的頻帶結構，論文中將會介紹將有限差分法和平面波展開法帶入此特徵值方程式的優點及運算結果。利用修改過後的平面波展開法，可以分析以色散性材料所製成的光子晶體，同時可以有效率的計算等頻率曲線，突破傳統平面波展開法無法直接計算色散性材料的限制，因此可以利用等效折射率的概念來設計平板光子晶體。而修改後之有限差分法除了同樣可以計算以色散性材料所製成的光子晶體外，藉由觀察所計算出波向量的虛數部分可以推測電磁波在光子晶體中的衰減情況，在有限差分法中也較容易加入完美匹配邊界，便可以計算有限的週期會帶給光子晶體波導多少的衰減，可利用計算出來的結果設計光子晶體波導。

An eigenvalue equation was derived from Maxwell equations. By this equation, eigenfrequencies are calculated with known wavevectors. Different numerical methods are used to solve the eigenvalue equation. In this thesis, plane-wave expansion method and finite-difference method are used to solve this eigenvalue equation. Photonic crystals made by dispersive and absorptive material can be analyzed directly by these methods, while conventional plane wave expanison method can not. Isofrequency curves and isofequency surfaces are calculated efficiently by these methods. Finite-difference method is also used to analyze the finite size photonic crystals by using perfect match layer rather than periodic boundary conditions. The loss caused by leakage in photonic crystal waveguides is estimated.