Irradiance moments: their propagation and use for unique retrieval of phase

TLDR

Irradiance moments vary with distance from the pupil plane under Fresnel diffraction, and their relationship to pupil‑plane phase is generally nonlinear and not uniquely defined at a fixed image‑space point. By introducing analytic pupil functions, the authors show that all finite‑order irradiance moments exist even with arbitrary continuous phase aberrations, and that examining moment behavior near an axial image point allows unique, linear‑problem phase retrieval by measuring moments around either the pupil or image plane. The study demonstrates that for analytic pupil functions, all finite‑order irradiance moments exist even with arbitrary continuous phase aberrations, and that by measuring moments near an axial image point one can uniquely retrieve pupil‑plane phase via a linear problem, with examples provided.

Abstract

The functional dependence of irradiance moments with distance from the pupil plane is studied within the framework of Fresnel diffraction theory. The concept of analytic pupil function is introduced, and for such pupil functions it is shown that any finite-order irradiance moment exists, even in the presence of arbitrary continuous phase aberrations. The uniqueness of the relationship between pupil-plane phase and irradiance moments, when the moments are calculated over an orthogonal plane at a fixed point along the optical axis in image space, is obscure, and the relationship between phase and moments is generally nonlinear. However, by studying the behavior of irradiance moments throughout the neighborhood of a given axial point in image space, one may determine, for a large class of pupils, the pupil-plane phase uniquely (within an arbitrary additive constant), and only a linear problem need be solved for phase retrieval. In particular, unique phase retrieval may be accomplished by measuring moments in the neighborhood of either the pupil plane or the image plane. Examples of this technique are given.