On the efficient evaluation of integral arising in the sommerfeld halfspace problem

Abstract

An efficient approach is described for the numerical evaluation of the Sommerfeld integrals which arise in analyses of antennas radiating over a dielectric halfspace. The main features of this approach can be summarised as follows. The contour deformation principle is employed to express the original integral as a sum of the steepest descent-path (SDP) integral and the possible branch-cut-path (BCP) integral, which are then evaluated numerically. If the branch point is captured, the SDP is allowed to go to the bottom sheet of the Riemann surface, resulting in a continuous behaviour of the integrand. In the last case, the BCP is deformed to the SDP emanating from the branch point, on which the integrand decays rapidly. Also, the Zenneck pole is subtracted, resulting in a smooth behaviour of the integrand on the SDP.