Measuring the Phase of an Optical Field from Two Intensity Measurements: Analysis of a Simple Theoretical Model

Abstract

Under the scalar paraxial approximation, an optical wavefield is considered to be complex function dependent on position; i.e., at a given location in space the optical field is a complex value with an intensity and phase. The optical wavefield propagates through space and can be modeled using the Fresnel transform. Lenses, apertures, and other optical elements can be used to control and manipulate the wavefield and to perform different types of signal processing operations. Often these optical systems are described theoretically in terms of linear systems theory leading to a commonly used Fourier optics framework. This is the theoretical framework that we will assume in this manuscript. The problem which we consider is how to recover the phase of an optical wavefield over a plane in space. While today it is relatively straightforward to measure the intensity of the optical wavefield over a plane using CMOS or CCD sensors, recovering the phase information is more complicated. Here we specifically examine a variant of the problem of phase retrieval using two intensity measurements. The intensity of the optical wavefield is recorded in both the image plane and the Fourier plane. To make the analysis simpler, we make a series of important theoretical assumptions and describe how in principle the phase information can be recovered. Then, a deterministic but iterative algorithm is derived and we examine the characteristics and properties of this algorithm. Finally, we examine some of the theoretical assumptions we have made and how valid these assumptions are in practice. We then conclude with a brief discussion of the results.