採用余氏網格之向量有限差分 波束傳播法於光波導問題之研究

Abstract

本研究採用以余氏網格為基礎的全向量有限差分波束傳播法，以 分析各種光波導結構的模態特性。此方法採用余氏網格直接離散馬克 斯威爾方程式，並應用虛軸波束傳播法的技巧。此種直接傳播的方法 在計算時間與記憶需求方面具有效率，並可以有效地同時求得波導模 態傳播常數與場形分佈。此外，為能求解洩漏模態問題並計算其傳播 損耗，本研究採用完美匹配層以吸收傳向計算邊界的波能。余氏網格 的採用得以同時提供電場與磁場各分量，因而其特徵模態場形可直接 當作相關問題的初始場套用於同樣使用余氏網格的有限時域差分 法，以利後續的計算。本研究分別建立了顯式和隱式兩種不同離散方 式的虛軸波束傳播法，經由分析傳統光纖、波導與融燒式光纖耦合結 構以及光子晶體光纖，檢驗方法的精準度。數值結果顯示與正確分析 解或其他數值方法的結果有很好的吻合度。

Finite-difference full vectorial beam propagation methods using Yee’s mesh discretization scheme of both explicit form and implicit form are adopted in this thesis to analyze the modal characteristics of optical waveguide structures with various cross-sections. The method is derived from directly discretizing Maxwell’s equations with the use of Yee’s mesh and utilizing the technique of imaginary-distance beam propagation method (ID-BPM). Such a direct propagation method is efficient in terms of CPU time and memory requirements. The propagation constants and their corresponding field distributions can be obtained efficiently and simultaneously. In addition, for solving the leaky-mode problems and calculating the corresponding propagation loss, the perfectly matched layers (PMLs) are incorporated into our formulations. Because of the use of Yee’s mesh, the electric and magnetic field components can be simultaneously evaluated, and the obtained eigenmode profile can be utilized as an incident field of related problems to be analyzed by the finite-difference time-domain (FDTD) method which utilizes the same Yee’s mesh. We establish the explicit Yee’s mesh ID-BPM (EY-ID-BPM) and the implicit Yee’s mesh ID-BPM (IY-ID-BPM) and examine their accuracy through analyzing conventional fibers, waveguides, and fused fiber coupler structures as well as photonic crystal fibers (PCFs). The results are in close agreement with either analytical values or published results using different numerical methods.