以近遠場轉換初探模擬解析度對遠場數值計算的影響

Abstract

繞射是波的基本特性，並且能用Huygens–Fresnel principle 跟波疊加原理加以描述每一個向前傳遞的波都可將之前的波前視為新的波源。繞射現象是一種真實的物理情形，然而對於有些過於複雜無法透過直接解析的情況，透過電腦數值計算的方式可以得到複雜邊界條件下的數值解。一般而言會將計算電磁學分成以下: full-wave法跟high-frequency法，在full-wave法種類下方程式又可以細分成偏微分或是積分方程形式，其中時域有限差分法(FDTD)是在時域下以偏微分方程的形式解Maxwell’s方程式的一種方法。 當我們使用FDTD去解析電磁波的問題時，數值相位誤差扮演著影響結果的重要腳色，在數值計算中，誤差的來源是用光在數值網格上行進的速度對於真實世界光在真空中的比值來定義誤差。因此我們能透過改變計算中數值網格的大小，去讓光在數值網格上行進的速度非常的接近真實的光速，因此提高答案的精確度，在此研究中，我們使用不同的網格大小λ/20到 λ/80，去探討解析度的差異對於遠場繞射情形的影響並且將所有的結果都跟單狹縫遠場繞射的解析解去做比對驗證，最終得到在在不同解析度的情況下，每個角度的計算數值對於解析解的差異。

Diffraction is a basic characteristic of wave which can be described by the Huygens–Fresnel principle and the principle of superposition of waves. The propagation of a wave can be visualized by considering every particle of the transmitted medium on a wavefront as a point source for a secondary spherical wave. Diffraction phenomenon is one the real-world physics problems, however some of them are not analytically calculable. Computational numerical techniques can overcome the inability to obtain the solutions of governing equations under complex structure and boundary conditions. The techniques of computational electromagnetics into two major categories: full-wave methods and high-frequency methods, under the category of full-wave methods, techniques can be classified as partial differential equation (PDE) based or integral equation (IE) based. FDTD is under the category of full-wave methods which is used to solve Maxwell’s curl equations at points on space grids in the time domain. When one uses FDTD methods to study electromagnetic wave problems, the numerical dispersion errors play a key role to determine the accuracy of the numerical results. In the numerical calculations, the wave propagating through the grid causes a phase error that directly affects the numerical results. In other words, the numerical dispersion error could be reduced by choosing a finer grid or higher resolution. In this research, we interested in the effect of grid resolution on the far-field pattern, so we compared the four different far-field pattern under the selected resolution which is from λ/20 to λ/60. The validation is done by comparing the results to the analytical solutions of single-slit diffraction.