完美金屬平板次波長孔洞之電磁場散射解析解研究

Abstract

本論文研究當電磁波斜向入射至有限厚度且無限延伸完美導體平板(perfect conductor)上的次波長圓形穿孔時，所形成的散射場解析解。 本研究理論基礎建構在沒有外加電荷與電流下(source free)的馬克士威方程式(Maxwell equation)，藉由其所具有的電磁場雙重性，將電磁場用向量與純量位勢能函數(potential function)表示後求解。問題空間區分為考慮近場表現的內部區域與探討遠域輻射的外部區域，在長波極限假設下，入射場波數(wave number)與孔洞半徑乘積epsilon=ka是一微小參數，我們利用擬合展開法(matched asymptotic expansion)，同時配合Fabrikant所提出之混合邊界值理論，分別求解當TM極化入射與TE極化入射時孔洞所引發之散射場多極結構，並分析多極結構強度與孔洞深度以及入射角度之間的關係。孔洞所造成的遠域輻射場主次項，相當於垂直導體面之電偶極與平行導體面之磁偶極所產生之輻射電磁場，當導體板厚度無限薄時，本論文能與Bethe[1]之結果吻合。 當考慮有限厚度完美導體平板上圓孔散射問題時，本研究提供了基礎的理論幫助此類問題的處理。

The scattering of an oblique electromagnetic wave incident on a subwavelength circular aperture in an infinite perfect conductor plane with a finite thickness is investigated analytically. The theory is based on the duality property of source-free Maxwell equation and the resultant scattering fields are fully expressed in terms of the auxiliary scalar and vector potentials. Both of the TM and TE polarized incidence are considered. There are two regions defined by the analytical method: the near field inner region and the radiation outer region. We use the method by Fabrikant to solve the mixed boundary value problems. Because epsilon=ka which is the product of the incident wave number and the pore radius is a small parameter, we could use the method of matched asympototic expansion to find the multipole structure. The sophisticated three- dimensional interplay between the pore depth and the incident angle is also revealed. The leading term of scattering radiation field due to the hole can be considered as caused by an electric dipole perperdicular to the plane of the hole, and a magnetic dipole parallel to it. When the thickness of the plane goes to zero, the result can be matched with Bethe’s theory. The theory lays the foundation for the future extension to the sattering problem of structured perfect conductor planes.