光學定理於時域有限差分法光學模擬之應用

Abstract

本篇論文主要是在討論如何將電磁理論裡的光學定理 (optical theorem) 使用在有限差分時域法 (finite-difference time-domain method) 之中，並且因為二維的有限差分時域法程式攥寫上較為容易所以在此只對二維的光學定理來做討論。首先是改寫光學定理原來的形式，使它能夠使用有限差分時域法裡習慣使用的符號來表示，並且介紹本論文裡為了使用光學定理而使用的額外功能，其中包含有分離散射場於總場的總場散射場技巧 (total-field / scattered-field technique)，還有使用近場資料來計算遠場場值的近場至遠場轉換 (near-to-far field transformation )，接著是一般使用上都不可或缺的吸收邊界，在這裡是選用完美吸收邊界 (perfectly matched layer absorbing boundary condition)。最後是用光學定理來計算二維無吸收圓柱的散射截面 (scattering cross section)，並且和解析解米氏理論 (Mie theory) 比較後觀察其誤差隨空間解析度的變化，其趨勢大體而言如有限差分時域法所擁有的二階準確度。同時也和其他用來計算散射截面 (scattering cross section) 的方法比較彼此的誤差差異，這些用來比較的方法有使用遠場的雷達截面 (radar cross section) 來計算，還有在近場直接使用散射截面 (scattering cross section) 定義還有消光截面 (extinction cross section) 定義這三種。最後得到的結果是使用光學定理來計算的誤差總是比較大一些，目前認為是因為有限差分時域法的格子點異方性 (anisotropy) 特性所造成的，並且如果針對零度角的光速來修正，可以得到光學定理能有最好的準確度。

In this thesis, we discuss how to apply the optical theorem to the finite-difference time-domain method (FDTD). Because it is easier to write the FDTD code in two dimensions, we only investigate the two-dimensional optical theorem. First, we rewrite the form of the optical theorem to conveniently put into execution in the FDTD simulation and introduce some techniques which are used in the FDTD method to practice the optical theorem. There are (1) scattered-field / total-field technique (SFTF), which can separate the scattered field from the total field, (2) the near-to-far field transformation (NTFF), which could calculate the far field from the near-field data, and (3) the perfectly matched layer absorbing boundary condition (PML), which could absorb the electromagnetic wave. Then, we apply the optical theorem to calculate the scattering cross section of the non-absorbing cylinder which has the analytical solution called the Mie theory. With the changes of the grid size, the inaccuracies have the trend approaching to the second order-accuracy of the FDTD method. We also compare the inaccuracy between the different methods which are used to calculate the scattering cross section. The first method is the sum of RCS method. The second and third methods are to apply the definition of the scattering cross section and the definition of the extinction cross section in the near-field region. Finally, we observe that the inaccuracy of the optical theorem is larger than the others and suspect that it is induced by the anisotropy of the FDTD square grids.