A Numerical Methodology for Efficient Evaluation of 2D Sommerfeld Integrals in the Dielectric Half-Space Problem

The analysis of 2D scattering in the presence of a dielectric half-space by integral-equation formulations involves repeated evaluation of Sommerfeld integrals. Deformation of the contour to the steepest-descent path results in a well-behaved integrand, that can be readily integrated. A well-known drawback of this method is that an analytical expression for the path is available only for evaluation of the reflected fields, but not for the evaluation of the transmitted fields. A simple scheme for numerical determination of the steepest-descent path, valid for both cases, is presented. The computational cost of the numerical determination is comparable to that of evaluating the analytical expression for the steepest-descent path for reflected fields. When necessary, contributions from branch-cut integrals and a second saddle point are taken into account. Certain ranges of the input parameters, which result in integrands that vary rapidly in the neighborhood of the saddle point, require special treatment. Alternative paths and specialized Gaussian quadrature rules for these cases are also proposed. An implementation of the proposed numerically determined steepest-descent path (ND-SDP) method is freely available for download.