Dispersion relations in Fraunhofer diffraction

Abstract

The Fraunhofer diffraction pattern of an aperture function, which can be described in terms of the superposition of a known aperture function with a half plane, has an amplitude distribution which is real along one direction and complex along the orthogonal direction. The real and imaginary parts of the amplitude distribution are related by Hilbert transforms (dispersion relations). These dispersion relations can be used to arrive at the diffraction patterns of a variety of apertures having symmetry properties. Theoretical results are presented to illustrate the diffraction pattern of various apertures.