Random motion theory of an optical vortex in nonlinear birefringent media

TL;DR

This paper develops a theoretical framework describing the stochastic motion of optical vortices in nonlinear birefringent media, linking vortex dynamics to polarization fluctuations and providing analytical solutions relevant for experimental studies.

Contribution

It introduces a novel stochastic model for optical vortex dynamics in nonlinear birefringent media based on the nonlinear Schrödinger equation and Langevin formalism.

Findings

Derived the Langevin and Fokker-Planck equations for vortex motion.

Found the relaxation distance expressed by a universal constant and vortex size.

Provided analytical solutions for vortex distribution evolution.

Abstract

A theoretical study is presented for the random aspect of an optical vortex inherent in the nonlinear birefringent Kerr effect, which is called the optical spin vortex. We start with the two-component nonlinear Schr\"{o}dinger equation. The vortex is inherent in the spin texture caused by an anisotropy of the dielectric tensor, for which the role of spin is played by the Stokes vector (or pseudospin). The evolutional equation is derived for the vortex center coordinate using the effective Lagrangian of the pseudospin field. This is converted to the Langevin equation in the presence of the fluctuation together with the dissipation. The corresponding Fokker-Planck equation is derived and analytically solved for a particular form of the birefringence inspired from the Faraday effect. The main consequence is that the relaxation distance for the distribution function is expressed by the universal constant in the Faraday effect and the size of optical vortex. The result would provide a possible clue for future experimental study in polarization optics from a stochastic aspect.