Fields of an infinitesimal dipole radiating near an impedance half-space by application of exact image theory

In this paper exact image theory is applied to the problem of an electric dipole, of arbitrary orientation, radiating above an impedance plane and the subtlety regarding the derivation of the explicit expressions of horizontal dipoles are discussed. First the Sommerfeld formulation for the diffracted electric fields is given followed by a discussion of the application of exact image theory to transform the Sommerfeld integrals into a form more conducive to numerical evaluation. Then explicit expressions for all electric field quantities, in terms of the exact image formulation, is shown. The field expressions for a horizontal dipole show a diverging exponential term not discussed by Lindell and Alanen. Also, it is shown that the expressions for the exact image can be rearranged to separate the space waves (direct+geometrical optics (GO)) from the surface wave and hence the behavior of the surface waves can be analyzed independent of the distance between source and observation points, and for very low values of impedance, where the saddle point is near the pole. Finally a comparison is made between the time required to evaluate the electric fields using the Sommerfeld formulation and using the exact image expressions.