應用於光波導系統之頻域有限差分法架構

Abstract

在本篇論文中，我們基於頻域有限差分法開發了一個能夠在光波導系統中準確且有效率地求得電磁場的穩態分佈和波導截面模態分佈的自製模擬器。在建構此模擬器的過程中如下：首先將模擬空間離散，並且將電場與磁場排列在所對應的楊氏網格裡。接著，透過設置完美匹配層，將無限空間變成有限模擬空間並且防止在模擬邊界上所造成的數值反射；除此之外，模擬區域被劃分成總場區和散射場區等兩個區域。其次利用楊氏網格和有限差分法將電磁波動方程式近似離散成一連串的線性方程組，藉由求解Ax=b的線性方程組以獲得電場和磁場在模擬空間裡分佈。另外，為了應用在光波導系統中，求解波導模態和傳播常數的模態解析器是非常重要的。為此，我們利用此頻域有限差分法開發出模態解析器，藉由求解特徵值/特徵方程問題求出相對應的模態分佈以及傳播常數。最後，我們藉由二維結構的表面電漿帶止濾波器和三維結構的絕緣層覆矽交錯波導兩個例子來驗證此模擬器的效能。模擬結果顯示由頻域有限差分法與商用軟體兩者所計算獲得的結果具有相當的一致性，清楚地說明了我們所研發的模擬器是一可應用在光波導系統下準確且有效率解析器。

In this thesis, we develop a homemade simulator based on the finite-difference frequency-domain (FDFD) method to acquire the steady-state electromagnetic field distributions and the modal distributions in optical waveguide systems efficiently and accurately. To implement this simulator, first we discretize the concerned space and arrange the electric and magnetic fields on the Yee grid. The concerned space was truncated by using perfectly matched layers to restrict the computation in a finite domain and to prevent numerical reflections at the boundaries. In addition, the computational domain is divided into two regions: the total field domain and the scattering field domain. Next, the electromagnetic wave equations are approximately discretized into a sequence of linear equations by using finite-difference method on the Yee grids. The distributions of the electric and magnetic fields are obtained by solving these linear equations Ax=b. In addition, in order to apply to the optical waveguide systems, the mode solver being able to realize the modal distributions and propagation constants of the concerned waveguides is very important. Then, we use this method to develop the corresponding mode solver by solving the eigenvalue/eigenfunction problems. At last, we use two examples of a two-dimensional plasmonic bandstop filter and a three-dimensional crossing waveguide based on the silicon on insulator substrate to verify these simulators. The simulated results obtained by our FDFD method are well agreement with those attained by the commercial software, manifesting that our developed simulator is an accurate and efficient solver for the optical waveguide systems.