Radial and asymptotic closed form representation of the spatial microstrip dyadic Green's function

Abstract

Sommerfeld-type integrals are frequently encountered when deriving the fields due to current elements radiating over a lossless or lossy grounded slab. In this paper, the problem of evaluating the electromagnetic field sustained by an electric source embedded in stratified struetures with planar boundaries is completely solved in the spectral domain. The Sommerfeld integral representation of the microstrip Green's function converges rapidly for source and field points which are vertically separated with respect to the interface plane and becomes very poorly convergent when the source and the observation points are radially or laterally separated. For the case of source and the observation points vertically separated, we improve the Chebyshev decomposition method by evaluating the Chebyshev coefficients via the Fast Fourier Transform. As the lateral separation among source and observation points increases the usual Sommerfeld integrand starts to blow up and to oscillate rapidly. In order to evaluate the otherwise divergent and highly oscillatory Sommerfeld integrals, two different deformations from the real axis to the complex plane of the original Sommerfeld contour of integration are presented. The first one is a deformation over the lower-half of the complex plane, taking account for residues and branch cuts. The second one is an integration over the first quadrant of the complex plane. An extensive discussion of the saddle point methods of integration necessary for approximated closed form representation of the electromagnetic field together with applications of the theory to the explicit determination of the electromagnetic field radiated by sources in isotropically stratified planar regions with piecewise constant properties along y are, then, presented. Finally, numerical evaluations of the modulus of the electric field, sustained by an horizontal electric point-source over the planar interface and a comparison between the integration of the radial propagation representation and the closed asymptotic expansion are shown.