The field of an electric dipole in a half-space with a smoothly nonuniform perturbation of the boundary impedance

A vertical electric dipole embedded in a vacuum half-space with a planar boundary characterized by a smoothly nonuniform perturbation of the impedance is considered. The effect of the impedance perturbation on the field of the dipole is studied. The situation when the perturbation amplitude is comparable with the unperturbed impedance value is analyzed. The problem is solved using an auxiliary function satisfying the zero Dirichlet condition on the boundary. The expression of the Hertz potential is derived in the case of grazing electromagnetic wave propagation.