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

In this paper, a new method based on the mode-matching method in the sense of least squares is presented for analyzing the two dimensional scattering problem of TE plane wave incidence to the infinite plane surface with an arbitrary imperfection of finite extent. The semiinfinite upper and lower regions of that surface are a vacuum and a perfect conductor, respectively. Therefore the discussion of this paper is developed about the Dirichlet boundary value problem. In this method, the approximate scattered wave is represented by the integral transform with bandlimited spectrum of plane waves. The boundary values of those scattered waves are described by only abscissa z and Fourier spectra are obtained by applying the ordinary Fourier transform. Moreover, new approximate functions are made by inverse Fourier transform of band-limited those spectra. Consequently, the integral equations of Fredholm type of second kind for spectra of approximate scattered wave functions are derived by matching those new functions to exact boundary value in the sense of least squares. Then it is shown analytically and numerically that the sequence of boundary values of approximate wave functions converges to the exact boundary value, namely, the boundary value of the exact scattered wave in the sense of least squares when the profile of imperfection part is described by continuous and piecewise smooth function at least. Moreover, it is shown that this sequence uniformly converges to exact boundary value in arbitrary finite region of the boundary and the sequence of approximate wave functions uniformly converges to the exact scattered field in arbitrary subdomain in the upper vacuum domain of the boundary in wider sense when the uniqueness of the solution of the Helmholtz equation is satisfied with regard to the profile of the imperfection parts of the boundary.