Asymptotic formulas for coupling between two antennas in the Fresnel region

The coefficient of coupling , or the ratio of power transfer between two antennas closely spaced, is defined. A two-term asymptotic approximation for , valid for arbitrary aperture antennas, is derived. The asymptotic approximation is the leading part of an asymptotic series involving powers of , where is the separation distance between transmitter and receiver measured in wavelengths. The first term of the asymptotic approximation for is the usual Friis formula for power transfer between antennas spaced far apart. The second term in the approximation represents a correction term due to effects associated with the finite spacing between antennas. For test cases the on-axis Fresnel region fields of several aperture antennas are computed using the asymptotic approximation. In turn, the gain reduction factor which is the ratio of Fresnel region gain to far-field gain of an aperture, is computed for each example. These results are compared with exact expressions previously presented in the literature, and close agreement is noted. Also, a means of measuring the far field of a large-aperture antenna with a probe in its Fresnel region is suggested.