Wave equations end transmission formulas for the output of a receiving antenna

The output of a linear receiving antenna satisfies the homogeneous wave equation and the outgoing wave condition in both the frequency and time domains. The homogeneous wave equations are satisfied throughout all space except where the minimum convex surface enclosing the receiving antenna (probe) intersects the minimum convex surface enclosing the transmitting antenna. It is assumed that the probe translates without rotation with respect to the transmitting antenna, and that multiple interactions between the probe and transmitting antenna are negligible (or gated out in the time domain). The frequency-domain wave equation (Helmholtz equation) has been used in the past to efficiently compute the coupling between co-sited antennas, and to suggest a simplified probe-corrected spherical near-field measurement method. In this paper we first give a rigorous derivation of these wave equations and their region of validity. We then use the wave equations to derive both the frequency-domain and time-domain probe-corrected plane-wave transmission formulas that determine the plane-wave spectrum (and thus the near and far fields) of the transmitting antenna in terms of the measured output and the plane-wave receiving characteristic of the probe