Optical Fresnel-Zone gain of a rectangular aperture

Abstract
An equation for the on-axis gain of a uniformly illuminated rectangular aperture is derived which is valid in the "Optical Fresnel Zone." This equation is formulated in terms of the ordinary radiation field gain multiplied by a correction factor which depends upon the aperture dimensions and the distance, R, from the aperture at which the gain is measured. A table of the function [C2(v)+S(v)]/v2 is given; C(v) and S(v) being the Fresnel integrals. The gain of a square aperture (LXL meters) in the Fresnel-zone region is compared with the gain of a circular aperture and it is shown that for apertures of equal Gsquare < Gcircle when (L2/λ) <R ≤(2L2 λ); it is also shown that Gsquare has minima, but not zeros at R=L2/7.30λ, R=L2/5.24λ. Furthermore it is shown that for RλL2/2λ the first order gain correction factor of a square aperture is equal to the gain correction factor of a circular aperture having a diameter equal to 1.208 times the sidelength of the square. An approximate formula for the Fresnel-zone gain of a long and narrow aperture is also given.

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