二維光子晶體的對稱性影響光子能隙之研究

Abstract

本篇論文以平面波的計算方法去探討結構對稱以及旋轉對稱於二維光子晶體光子能隙的影響。利用場分佈以及碰撞機制可以合理的瞭解能帶的特性。我們也設計多邊形與切邊兩種柱子結構去探討柱子的邊界形狀如何影響光子能隙於正方晶格中形成。從結果中可以瞭解多邊形柱子與圓形柱子於正方晶格中的能帶行為以及兩者的關係，同時對於各個偏極態的能隙特性將可以合理解釋。在切邊結構的計算上，則是利用能帶的觀念去探討能隙形成的決定因素，進而瞭解八邊形柱子無法在正方晶格中出現絕對光子能隙的原因。 在中空柱子於三角晶格的計算，則是設計三種變形結構與兩種旋轉結構去探討旋轉以及結構對稱性於光子能隙的影響。於柱子的變形計算中，磁場偏極的能隙主要是由柱子之間電場的作用所主導; 而於柱子的旋轉計算中，磁場偏極能隙的改變則是受旋轉對稱性的改變而影響。電場偏極的能隙不管是柱子變形或者是旋轉，該能隙都是由柱子的形狀來主導。我們也發現當中空柱子於旋轉的計算時，絕對光子能隙的形成主要是由磁場偏極的能隙來決定。反之，在中空柱子於變形的計算時，絕對光子能隙的形成則是由電場偏極的能隙來決定。

In this study, we investigate how structural and rotational symmetries affect the E-polarization and H-polarization band structures in the two-dimensional photonic crystal. The plane-wave method is employed to calculate band structures and field patterns. The features of band structures have been reasonably explained with field pattern and scattering mechanisms. To understand how the regular shape and boundary of rods affect the formation of PBG in square lattice, the N-polygonal and corner-cutting structures are considered. The effects of N-polygonal rods on the E-polarization and H-polarization band structures have been examined. Our calculations also show that the band structures of polygonal rod approach the same to that of circular rod while N is large. We use band-structure viewpoint to examine the formed PBG in the corner-cutting structure. The reason why the complete PBG of octagonal dielectric rods with square lattice is closed can be understand. Three deformed and two rotational structures are constructed to the formed PBG of hollow rods in the triangular lattice. The H-polarization band gaps are strongly affected by the interaction between the fields of the rods as the rods are deformed and affected by the reduction in the rotational symmetry as whole rods are rotated. Only the shapes of the rods affect the E-polarization band gaps as the rods are either deformed or rotated. Moreover, H-polarization modes determine the complete PBG width as the rods are rotated, whereas E-polarization modes determine the complete PBG width as the rods are deformed.