Axial correlation revivals and number factorization with structured random waves

TL;DR

This paper develops a theory linking structured random wave correlations with number theory and demonstrates an optical system capable of factoring numbers using wave interference patterns.

Contribution

It introduces a novel analytical relation connecting wave correlation functions with Gauss sums and implements an optical device for number factorization.

Findings

Derived an analytical relation between autocorrelation functions and Gauss sums.

Experimentally demonstrated number factorization using structured random waves.

Established a fundamental link between statistical optics and number theory.

Abstract

We advance a general theory of field correlation revivals of structured random wave packets, composed of superpositions of propagation-invariant modes, at pairs of planes transverse to the packet propagation direction. We derive an elegant analytical relation between the normalized intensity autocorrelation function of thus structured paraxial light fields at a pair of points on an optical axis of the system and a Gauss sum, thereby establishing a fundamental link between statistical optics and number theory. We propose and experimentally implement a simple, robust analog random wave computer that can efficiently decompose numbers into prime factors.