Axial correlation revivals and number factorization with structured random waves

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.