Exactly calculable field components of electric dipoles in planar boundary

The Sommerfeld integrals for the electromagnetic fields in the planar boundary between air and a homogeneous, isotropic medium, due to a horizontal and a vertical electric dipole each lying along the interface, are examined in detail. In the case of the horizontal dipole, the tangential electric field is given in terms of series that involve confluent hypergeometric functions, namely, the Fresnel and exponential integrals. A similar exposition is presented for the magnetic and vertical electric fields of the vertical dipole. When the index of refraction of the adjacent space is of a sufficiently large magnitude, the derived series converge rapidly and uniformly with the distance from the source. Specifically, their rates of convergence are shown to be independent of distance. It is pointed out that the corresponding formulas of King et al. are valid down to any distance close to the source, where they smoothly connect to known “quasi-static” approximations.