A Collective Propagation Law of Optical Vortex Constellations and Longitudinal Sensing

Abstract

Optical vortices are ubiquitous phenomena naturally appearing in wave physics, yet higher-order charges inherently split into constellations of lowest-order singularities at the smallest deviation from an ideal situation. While these constellations are common phenomena in real-world scenarios, characterizing their longitudinal evolution typically relies on exhaustive full-field descriptions or the ambiguous, sequential tracking of individual singularities. Here, we reveal and experimentally demonstrate a simple deterministic law describing the paraxial longitudinal propagation of an arbitrary constellation of optical vortices in standard Gaussian backgrounds. By mapping the constituting singularity coordinates to their elementary symmetric polynomials (ESPs), we capture the holistic evolution of the constellation during propagation, completely bypassing the practical need to sequentially track indistinguishable vortices. We further show that such complex ESPs provide a useful metrological tool for the estimation of longitudinal displacements. Our results reveal a previously unrecognized compact description of collective vortex dynamics, introducing a new route to longitudinal sensing through singularimetry.