Three necessary conditions for the validity of the Fresnel phase approximation for the near-field beam pattern of an aperture

A simple, straightforward derivation of three necessary conditions that define the region of validity for the near-field directivity function (beam pattern) of an aperture (array) is presented. The derivation of all three criteria is based on determining what conditions must be satisfied in order to obtain a valid Fresnel approximation of the time-independent free-space Green's function. Comparisons with other near-field conditions in the literature are made.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">></ETX>