Direct and efficient solutions of integral equations for scattering from strips and slots

Abstract

Singular integral or integro-differential equations (SIE or SIDE) are often used for the analytical formulation of two-dimensional boundary-value problems. The methods for solving them depend primarily on the complexity of their kernel and on the kind (first or second) of the SIE itself. First-kind SIEs with a Laplacian kernel are characteristic in electrostatics. A successful method for solving them is a regularization approach based on the transformation of the SIE to an equivalent Fredholm regular integral equation of the second kind. Well-known inversion formulas are essential to this approach. In electromagnetics, a Hankel-type kernel complicates matters considerably; inversion formulas and regularization techniques end up as cumbersome indirect procedures making necessary the recourse to a more direct method. Such a method is developed in this paper in combination with a very suitable expansion of the Bessel function, that multiplies the logarithmic singularity of the Hankel kernel, into a series of Chebyshev polynomials of the first or second kind. It is essentially a direct analytical approach that requires fewer expansion functions per wavelength than the method of moments and whose matrix elements are not numerical integrals of singular functions, but quite concise and rapidly convergent series expansions. The efficiency of the method is shown by applying it to scattering of E-polarized waves from a strip conductor right on the interface between two different dielectric half-spaces and of E- or H-polarized waves from a slot in the presence of a uniaxially gyrotropic half-space. Asymptotic expressions for the far-scattered field are given in all these cases and numerical results are plotted and compared with existing similar ones in certain special situations.