Algebraic Formulation of Diffraction Applied to Self Imaging

Abstract

One of Rayleigh’s diffraction integrals is generalized as to the nature of the functions on which it operates. These functions comprise the well-known Hilbert space ℒ2 of square-integrable functions. The algebraic properties are apparent in the existence of a left inverse for the diffraction operator, and in other places where the ordering of a sequence of operations is important. The conditions for self imaging are completely characterized by a certain part of the mathematical spectrum of the diffraction operator. The contribution of the evanescent waves to this spectrum is clearly shown. Both the infinite- and finite-aperture cases of diffraction are treated. In the infinite-aperture case we can employ a vector theory in which we obtain a complete characterization of self imaging. This self imaging is of an approximate nature. For the finite-aperture case we employ a scalar theory and obtain some partial results. However, in contrast to the infinite aperture we prove, for the case of an arbitrary finite aperture in a black screen, that there exist fields which image themselves exactly through the diffraction process. These fields are the eigenfunctions of a diffraction operator which corresponds to the sequential operations of inserting a screen into the incident field, discarding the evanescent-wave contribution, and diffracting through a distance z. The operator which corresponds to just the first two processes of inserting the screen and discarding the evanescent waves is of the type which generates the so-called generalized prolate spheroidal functions. These functions correspond to fields which have the interesting optical property of giving the same diffraction pattern whether the screen is applied or not (i.e., to within a constant positive factor). A conjecture is put forth concerning the evanescent-wave contribution to measurements that have been taken near the aperture. If the conjecture proves correct, it may be possible to show a consistency between the first Rayleigh diffraction integral and near-aperture measurements.