Scattering by an Electrically-Small Circular Aperture in a Conducting Screen using the Gegenbauer Polynomial Expansion

In this paper, we used the electric vector potential formulation, along with the equivalence principle and image theory, to calculate the scattered fields by a sub-wavelength circular slot in an infinite conducting ground plane. The formulation begins with the radiation integrals of the established magnetic current density in the aperture due to an incident plane wave of a given polarization and angle of incidence. The quasi-static magnetic current density in the aperture is based on the Bouwkamp model published in his monograph on diffraction theory in 1950. This paper provides significant added value to an earlier work published by the authors as it allows evaluation of the radiation integrals for observation points off the axis of the aperture using the Gegenbauer polynomial expansion of the Green's function terms of the form 1/R
^α
. Taylor-series expansion of the exponential exp (-jkR) is also used in this work in order to obtain accurate results for the scattered fields in the intermediate and far-field observation regions. The formulation presented herein is valid only for the case where the observation distance from the origin of the coordinate system is larger than the radius of the aperture; i.e., r > a, where a is the radius of the aperture. Comparisons between analytical results obtained using the proposed formulation and data obtained using numerical integration illustrate the accuracy of the underlined approach.