傳輸矩陣法及其在光子晶體上的應用

Abstract

在本論文中，我們以傳輸矩陣法進行數值模擬並探討數種二維光子晶體的特性。傳輸矩陣法常被用於處理光柵結構的電磁場傳播問題上。以平面波為基底展開後，亦可用於探討有限長�半無限長的二維光子晶體內的電磁波傳播特性，解決光子晶體波導與方向耦合器內的電磁場分佈問題。加上布拉格條件，亦可用來計算無窮大光子晶體的色散關係，並以之描繪能帶圖。傳輸矩陣法的各個分割區塊互不相同，故可處理非週期性之結構。因此我們可以此方法計算二維異質結構的光子晶體傳輸頻譜。由數值模擬而得的特性，可使我們進一步了解各種積體光路的元件結構，以期對各結構的尺寸規格進行最佳化。

In this thesis, we use a transfer-matrix method (TMM) to numerically investigate the transmission characteristics of two-dimensional photonic crystals. The transfer-matrix method has been routinely used to calculate the transmission and reflection spectra of electromagnetic waves propagating through conventional gratings. By adopting the plane-wave bases to expand the electromagnetic field components, we can deal with the electromagnetic wave propagation inside finite/semi-infinite 2D photonic crystals, and further solve the field distributions for related structures such as waveguides and directional couplers. By matching the Bloch condition, we can also calculate the dispersion relation for an infinite photonic crystal and plot the band diagram. Since the slicing scheme in the TMM requires no periodicity along the propagating direction, we can apply this method to solve the transmission spectrum for hetero photonic crystals. Through the properties learned from the simulations, we can better realize the functional elements in optical integrated circuits, and design optimal feature size for each element.