Why square antennas produce round beams

Why do square antennas produce circular main beams? An intuitive answer to this question comes from applying a series expansion to the expression for the far-field radiation pattern, which, in turn, is the Fourier transform of the aperture fields. This paper explains how filtering properties of the resulting Bessel functions cause rounding of the pattern corners near the beam center. The same expansion and filtering explain rounding of the beams of planar apertures of arbitrary shape.