Asymptotic expansion for spherical mode phase with application to near-field measurements

The exact-circuit representation of spherical modes allows the derivation of a convenient asymptotic relationship for the phase difference between a plane wave and a spherical wave as a function of the radial position and spherical mode index. One application of this asymptotic form is to allow the effects of range uncertainties on patterns measured in a near-field antenna range to be related simply to the more familiar finite range effects in a direct far-field measurement.