In a coherent imaging system, the complex amplitude and phase of the images is necessary to perform any linear processing or wavefront reconstruction. Using the image intensities measured in two defocused planes, it is possible to reconstruct the lost phase by iterative techniques. A merit function is defined as the sum over the second image plane of the squared differences between the known modulus in that plane and the modulus calculated by defocusing the image in the first plane with a phase estimate. This merit function is used to evaluate the convergence properties of two types of iterative schemes:the Misell (Gerchberg-Saxton) algorithm, and a gradient searching steepest descent method. The convergence is studied as a function of the amount of defocus between the images, complexity of the image, and noise present in the measured intensities. Variations and improvements of the methods are discussed. For example, for small amounts of defocus, the Misell algorithm has difficulty converging; the application of alternating constraints in Fourier space may help convergence. For the steepest descent and other gradient related methods, convergence of the phase depends on the value of the modulus at that point. Results of computer experiments using simulated images and pupil distributions are shown.
R. H. Boucher, R. H. Boucher,
"Convergence Of Algorithms For Phase Retrieval From Two Intensity Distributions", Proc. SPIE 0231, 1980 Intl Optical Computing Conf I, (22 August 1980); doi: 10.1117/12.958840; https://doi.org/10.1117/12.958840