Fresnel gain of aperture aerials

When a transmitting aperture A0 and a receiving aperture B0 are coupled in the Fraunhofer region, the ratio of received to transmitted power is given by the Friis formula. It is well known that the Fraunhofer gains Ga0 and Gb0 of A0 and B0 are constant and are determined independently. In the paper, the product of the gains GaGb in the Fresnel region is determined as an integral over both apertures, assuming the gains Ga and Gb for A0 and B0. It is pointed out that GaGb cannot generally be separated into the individual factors, but a formula similar to that of Friis still holds in the Fresnel region. The author proposes to define the `gain-product reduction factor¿, ¿a¿b, as the ratio GaGbGa0Gb0. The individual gain Ga or Gb may be determined only when A0 and B0 are identical and the illuminations of both apertures are the same. Then the `gain reduction factor¿, ¿, is defined as the ratio Ga/Ga0. Assuming uniform amplitude and phase of illumination over A0 and B0, ¿a¿b and ¿ are determined for circular and rectangular apertures. For circular apertures, the same factors are determined for a set of illuminations which approximate to commonly used tapers, and the effect of defocusing the primary feed in lenses and dishes is also discussed.