二維各向異性光子晶體之特性研究與有限元素法建模分析

Abstract

為了研究各向異性材料對於建構完整的能帶結構所造成之影響，我們針對在光子晶體中的電磁波模態可以退耦成橫向電場與橫向磁場模態之情況，先發展出一個以有限元素法為基礎所建立的廣義純量式特徵值演算法來分析二維各向異性光子晶體的能帶結構。我們分析了具有正方晶格與三角晶格的二維各向異性光子晶體，並且審查這兩種晶格結構的第一布里淵區中不同子區域間之關係。在分析各向異性光子晶體時，第一布里淵區中足夠多之必要子區域皆須納入考慮，因此用來為各向等性光子晶體建構完整能帶結構之概念須加以適當修正。 然後，在電磁波的傳播方向平行於週期性平面之條件下，一個以有限元素法為基礎的全向量式特徵值演算法更進一步被提出來針對具有三維各向異性材料的二維光子晶體進行能帶結構分析。我們嘗試同時考慮所有電場或磁場分量並推導出一個矩陣特徵值方程式來處理最廣泛之二維光子晶體問題。經由分析各向等性的光子晶體，我們闡述了此全向量式特徵值演算法與廣義純量式特徵值演算法之間的關係，而藉由探討一個各向異性光子晶體之特性，我們說明了此全向量式特徵值演算法的重要性。對於二維各向異性光子晶體更深刻的認識與了解皆可以利用此全向量式特徵值演算法來獲得。 最後，在電磁波的傳播方向不再平行於週期性平面之條件下，我們亦展示了一個以有限元素法為基礎的全向量式特徵值演算法，此演算法可以為具有三維各向異性材料的二維光子晶體建構出能帶邊界圖。使用此全向量式特徵值演算法，各種具有不同材料組成與幾何形狀定義的光子晶體之能帶邊界圖皆可以方便且正確地建構出來。因此，我們可以利用此全向量式特徵值演算法來獲得關於光子能隙光纖之分析與設計相當有用的指導方針。

A generalized scalar finite element method (FEM) based eigenvalue algorithm for analyzing the band structures of two-dimensional (2D) anisotropic photonic crystals (PCs) when the wave modes in the PCs can be decoupled into the transverse-electric (TE) and transverse-magnetic (TM) ones is developed first to investigate the intrinsic effect of anisotropic materials on constructing complete band structures. The 2D anisotropic PCs with square and triangular lattices are analyzed, and the relationships among the distinct sub-zones of the first Brillouin zones (BZs) for these two lattices are examined. The concept of constructing complete band structures for isotropic PCs should be modified for anisotropic PCs by taking into accounts enough necessary sub-zones in the first BZ. Then a full-vectorial FEM based eigenvalue algorithm under the in-plane wave propagation is further proposed for the band structure analysis of 2D PCs with arbitrary 3D anisotropy. We attempt an idea of considering all the electric or magnetic field components simultaneously and formulate a matrix eigenvalue equation to deal with the most general 2D PC problems. The relationship between this full-vectorial algorithm and the generalized scalar one is clarified from the analysis of isotropic PCs, and the significance of this full-vectorial algorithm is demonstrated through the examination of an example anisotropic PC. It is seen that a deeper insight about 2D anisotropic PCs can be obtained utilizing this full-vectorial algorithm. Finally a full-vectorial FEM based eigenvalue algorithm under the out-of-plane wave propagation for the band edge diagram construction of 2D PCs with arbitrary 3D anisotropy is also presented. Based on this full-vectorial algorithm, the band edge diagrams of various PCs with different material combination and geometry definition can be conveniently and correctly constructed. Consequently, quite valuable guidelines for the analysis and design of photonic band gap (PBG) fibers can be acquired with the help of this full-vectorial algorithm.