Deducing the Talbot Effect from Electrodynamics

TL;DR

This paper models the Talbot effect using Maxwell's equations, deriving conditions for its occurrence and convergence to ideal distributions, thus providing a fundamental electrodynamics perspective on this optical phenomenon.

Contribution

It introduces a Maxwell-based analytical model for the Talbot effect, connecting classical wave optics with fundamental electrodynamics and proving convergence to ideal distributions.

Findings

Analytical solution of Maxwell's equations for the Talbot effect

Derivation of the paraxial limit and rational Talbot effect

Proof of $L^2$ convergence to ideal distributions

Abstract

We propose a model based on Maxwells equations to describe the Talbot effect. After solving the model analytically, we prove that its solution is equivalent to the one amply found in the literature in the asymptotic limit. By considering the paraxial limit, we showcase how the rational Talbot effect arises, and derive an ideal paraxial distribution that would give such effects. We give a proof of $L^2$ convergence to these ideal distributions for any such grating. The universal nature of the Talbot effect makes our findings pertinent for a variety of physical systems and technological applications.