On the aperture and pattern space factors for rectangular and circular apertures

A consequence of a recently discovered edge condition for planar apertures is that all planar aperture distributions are separable physically into a product of an edge factor and an aperture space factor, analogous to the way in which the radiation pattern separates into a product of an element factor and a pattern space factor. An exact relationship between these aperture and pattern space factors for physically realizable vector fields is derived here for rectangular and for circular apertures. For rectangular apertures it leads to a two-dimensional set of doubly orthogonal functions that are characteristic of the aperture geometry. Characteristic functions for circular apertures, however, are shown to exist only if the vector fields are circularly symmetric, although for scalar fields they exist for completely arbitrary aperture distributions with arbitrary edge taper. For rectangular apertures the characteristic functions consist of products of spheroidal functions and for circular apertures they are obtained from a generalization of the spheroidal functions. Some of the properties of these generalized spheroidal functions are developed here.