On inverse problem with phase retrieval for an inclined line in the parabolic approximation

TL;DR

This paper addresses the inverse problem of phase retrieval for an inclined line in a 2D parabolic equation framework, proposing a numerical iterative method with regularization for stable solutions.

Contribution

It introduces a novel approach reducing the phase retrieval problem to a singular Cauchy integral equation and develops an iterative numerical solution with regularization.

Findings

Successful numerical experiments demonstrating the method's effectiveness.

Stable solutions achieved through regularization procedures.

The iterative method converges reliably for the tested scenarios.

Abstract

The inverse problem of amplitude reconstruction on an inclined line based on the values of amplitude or its module as recorded on semi-infinite line orthogonal to the beam propagation direction is considered within the framework of 2D parabolic equation. It is demonstrated that this inverse problem, in case when the complex image plane amplitude is known, can be reduced to a singular Cauchy type integral equation. The existence of its solutions requires that certain conditions be met but if a solution exists it is necessary unique. The obtained integral equation is then approximated piece-wisely and the resulting linear algebraic system is solved numerically while applying necessary regularization procedures to enhance the stability of its solutions. Finally, an iterative method of phase retrieval is developed and a set of numerical experiments is performed.