Asymptotic formulas for coupling between two antennas in the Fresnel region

Abstract

The coefficient of coupling <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\eta</tex> , or the ratio of power transfer between two antennas closely spaced, is defined. A two-term asymptotic approximation for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\eta</tex> , valid for arbitrary aperture antennas, is derived. The asymptotic approximation is the leading part of an asymptotic series involving powers of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/Kd</tex> , where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Kd</tex> is the separation distance between transmitter and receiver measured in wavelengths. The first term of the asymptotic approximation for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\eta</tex> 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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\gamma</tex> 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.