# Propagation of waves from an arbitrary shaped surface -- a generalization of the Fresnel diffraction integral

## TL;DR

This paper develops a generalized method for calculating wave propagation from arbitrarily shaped surfaces using Laplace transforms, with applications in X-ray optics and phase retrieval.

## Contribution

It introduces a new approach to wave propagation from arbitrary surfaces, generalizing the Fresnel diffraction integral with solutions applicable to complex geometries.

## Key findings

- Derived a general wave propagation formula for arbitrary surfaces.
- Obtained an exact solution for a concave parabolic initial surface.
- Applicable to X-ray optics and phase retrieval problems.

## Abstract

Using the method of Laplace transform the field amplitude in the paraxial approximation is found in the two-dimensional free space using initial values of the amplitude specified on an arbitrary shaped monotonic curve. The obtained amplitude depends on one {\it a priori} unknown function, which can be found from a Volterra first kind integral equation. In a special case of field amplitude specified on a concave parabolic curve the exact solution is derived. Both solutions can be used to study the light propagation from arbitrary surfaces including grazing incidence X-ray mirrors. They can find applications in the analysis of coherent imaging problems of X-ray optics, in phase retrieval algorithms as well as in inverse problems in the cases when the initial field amplitude is sought on a curved surface.