The Talbot effect as the fundamental solution to the free Schr\"odinger equation

TL;DR

This paper rigorously demonstrates that the Talbot effect, typically modeled heuristically via the Schrödinger equation, can be exactly described as its fundamental solution in the sense of distributions, providing a solid mathematical foundation.

Contribution

It mathematically justifies the Talbot effect as the fundamental solution to the free Schrödinger equation, moving beyond heuristic and paraxial approximations.

Findings

The Talbot effect is exactly modeled by the Schrödinger equation in the sense of distributions.

The heuristic paraxial approximation is rigorously justified mathematically.

The work provides a distributional framework for understanding the Talbot effect.

Abstract

The Talbot effect is usually modeled using the Helmholtz equation, but its main experimental features are captured by the solution to the free Schr\"odinger equation with the Dirac comb as initial datum. This simplified description is a consequence of the paraxial approximation in geometric optics. However, it is a heuristic approximation that is not mathematically well justified, so K. I. Oskolkov raised the problem of "mathematizing" it. We show that it holds exactly in the sense of distributions.