以有限元素法分析光波導傳播特性之研究

Abstract

本研究採用以曲線式混合基底元素之全向量有限元素模態解析法來分析光波導模態，並在此架構上實現非線性波束傳播法探討非線性光波傳播特性。本論文藉由模態解析法搭配完美匹配邊界層作為邊界條件以吸收超出數值空間之電磁波，可準確的計算損耗波導的能量散逸，並建立非線性光波導模態解析,以與非線性波束傳播法搭配研究非線性現象。對於完美金屬導體與完美磁性導體，本研究也提出精確的邊界設定演算法。另外，為了分析二維傳播方向週期性排列的線性與非線性光波導，本論文採用二階三角曲線式元素搭配週期性邊界條件發展另ㄧ模態解析法。本論文模擬分析結構包括圓柱型光纖、三維抗諧振反射光波導、不同空氣孔洞數目的多孔光纖、非線性方向性耦合器以及二維線性與非線性光子晶體波導，並針對損耗波導的橫截面方向發展出能量流場圖。

In this research, we improve an optical waveguide mode solver based on the fi- nite element method (FEM) and curvilineal hybrid edge/nodal elements, and implement a nonlinear beam propagation method (BPM) numerical model based on the related FEM scheme. The FEM mode solver is incorporated into it the perfectly matched layer (PML) absorbing boundary condition and can solve the leaky waveguide mode very accurately. We refine the algorithms of the mode solver related to rigorous boundary setting involving perfect electric conductor (PEC) and perfect magnetic conductor (PMC) and numerical implementation of PMLs. The mode solver is further generalized to the analysis of nonlinear waveguide modes for working together with the nonlinear BPM model. Another FEM based mode solver for two-dimensional (2-D) linear and nonlinear periodic optical waveguides is also implemented with second order triangular elements. Periodic boundary conditions are properly imposed in the propagation direction for efficient analysis. Numerical examples considered in this research include circular waveguide, 3-D antiresonant reflecting optical waveguide (ARROW), holey fibers of various numbers of air holes, nonlinear directional coupler, and 2-D linear and nonlinear photonic crystal waveguides. We in particular develop a scheme to present power flow diagrams in the cross-sectional plane for leaky modes.