Exact image theory for the Sommerfeld half-space problem, part II: Vertical electrical dipole

The exact image theory, introduced earlier by the present authors for the Sommerfeld problem with a vertical magnetic dipole above a planar interface of two dielectric media, is now extended to the more complicated problem of a vertical electric dipole. The image source is seen to be a line current which lies in complex space for best convergence. The image current function is expressable as an integral expression, easily adapted for numerical calculations and unlike for the vertical magnetic dipole, its form is dependent on the dielectric factor ratio of the two half-spaces. Series expansions are given for an easy calculation of the image current. The reflection coefficient method and Sommerfeld surface-wave expressions are obtained as limit cases of the exact image theory. The method is tested for a large grid of field points and good accuracy for a modest calculation algorithm is observed.