Analysis of Scattering Problem by an Imperfection of Finite Extent in a Plane Surface

1.Introduction In this paper, a new method based on the modematching method in the sense of least squares [1], [2] for analyzing the two dimensional scattering problem of plane wave incidence to the infinite plane surface with an arbitrary imperfection of finite extent. The semi-infinite upper and lower regions of that surface are a vacuum and a perfect conductor, respectively. For this problem, the Rayleigh hypothesis [3] assert that the scattered field may be expanded in terms of outward-going wave functions and represented by the ordinary Fourier transform whose spatial spectrum is not band-limited in the upper region of that surface including the boundary. However, it was shown that the Rayleigh hypothesis could not be applied without a limiting condition by van den Berg and Millar and so on [4], . Namely, the condition about the profile of the imperfection in a surface was derived under which the hypothesis is rigorously valid. For example, when the imperfection is sinusoidal periodic groove structure of finite extent and the length of a period is D and a half depth of the groove is A, the condition is 2pA/D< 0.448. Moreover, if the profile of grooves is rectangular, Rayleigh hypothesis is can not be applied. In this paper, a new method is presented for analyzing the above-mentioned scattering problem. This method is based on the mode-matching method in the sense of least squares. In this method, the approximate scattered wave is represented by the integral transform with band-limited spectrum of plane waves. This approximate wave function is determined in such way that the mean-square boundary residual is minimized. 2. Formulation of the problem