以模擬馬克斯威爾方程的途徑，探討奈米光纖之間的耦合效率

Abstract

本論文是在交錯網格上發展一三維時域有限差分法(FDTD)，以求解馬克斯威爾方程。本文的方法是在時域內，在滿足電場和磁場零散度條件(亦即高斯定律)的架構下求解法拉第定律和安培定律。所提出的數值方法於時間離散方面使用具辛結構(Symplectic)之兩階數二階之Runge-Kutta方法，在經過長時間模擬後，解仍得以保持馬克思威爾方程能量守恆的性質； 另透過法拉第及安培旋度方程空間微分項的推導，以期求得在色散關係上相當準確的解。 為了達到具最佳數值色散的性質，本文所提出的數值方法能在時間上和空間上保有相當好的理論收斂斜率， 且能有效地減少實解相速度與數值相速度之間的誤差，而得以顯著地降低了因時域有限差分所造成的數值色散誤差以及各向異性誤差。本研究證實了所提出的數值方法在具辛結構與色散關係上皆具有良好的保持性，尤其在針對經長時間馬克斯威爾方程的數值模擬後，其效果尤為顯著。本文進而將此數值方法應用於探討消逝波之奈米光纖之間的耦合效率：給入HE11單模態，準確預估空氣中兩材料為二氧化矽之直徑為350 nm奈米光纖其間的耦合效率； 在兩光纖重疊的部分，隨不同x位置切yz平面及取y軸之1D場，了解電、磁場及波印廷向量之物理量在奈米光纖間的變化； 利用波印廷定理之能量守恆的性質說明此數值方法之準確性。此外，本文利用三維的模擬結果，了解HE11單模態之電、磁場在奈米光纖內部的流線走向，以及在兩光纖重疊範圍的煙線走向。

An explicit finite-difference scheme for solving the three-dimensional Maxwell’s equations in staggered grids is presented in time domain. The aim of this thesis is to solve the Faraday’s and Amp`ere’s equations in time domain within the discrete zero-divergence context for the electric and magnetic fields (or Gauss’s law). The local conservation laws in Maxwell’s equations are also numerically preserved all the time using proposed the explicit second-order accurate symplectic partitioned Runge-Kutta temporal scheme. Following the method of lines, the spatial derivative terms in the semi-discretized Faraday’s and Amp`ere’s equations are then properly discretized to get a phase very accurate solution. To achieve the goal of getting the best dispersive characteristics, this centered scheme minimizes the difference between the exact and numerical phase velocities. The significant dispersion and anisotropy errors manifested normally in finite difference time domain methods are therefore much reduced. The dual-preserving (symplecticity and dispersion relation equation) solver is numerically demonstrated to be efficient for use to get in particular a long-term accurate Maxwell’s solution. By applying the developed FDTD scheme, we aim to study the wave propagation issue on nanowires. The efficency of evanescent coupling between two air-clad silica nanowires with diameter D = 350nm in HE11 single-mode operation is addressed. In the overlapping region, we can see that the electric, magnetic and poynting fields vary between two nanowires by cutting 2D-planes and 1D-Y-axis with different a streamwise positions. From the property of conservation of energy of Poynting theory, we confirm the validity of the propoesd finite difference method. Based on the computed 3D result of solving the Maxwell’s equations, we can clearly predict not only the streamlines of eletric and magnetic fields of HE11 mode in two nanowires but can also reveal the streaklines of them in the overlapping region.