Numerical investigation of coherent and turbulent structures of light via nonlinear integral mappings

TL;DR

This paper presents a numerical method using nonlinear integral mappings to model and analyze stable and turbulent light structures in nonlinear media, capturing phenomena like solitons, vortices, and optical lattices.

Contribution

It introduces a novel numerical approach that combines nonlinear integral transformations with stability analysis to simulate complex light structures in nonlinear optics.

Findings

Stable numerical schemes for nonlinear wave propagation.

Ability to model noise effects on wave dynamics.

Identification of phase-synchronized optical configurations.

Abstract

The propagation of stable coherent entities of an electromagnetic field in nonlinear media with parameters varying in space can be described in the framework of iterations of nonlinear integral transformations. It is shown that for a set of geometries relevant to typical problems of nonlinear optics, numerical modeling by reducing to dynamical systems with discrete time and continuous spatial variables to iterates of local nonlinear Feigenbaum and Ikeda mappings and nonlocal diffusion-dispersion linear integral transforms is equivalent to partial differential equations of the Ginzburg-Landau type in a fairly wide range of parameters. Such nonlocal mappings , which are the products of matrix operators in the numerical implementation, turn out to be stable numerical-difference schemes, provide fast convergence and an adequate approximation of solutions. The realism of this approach allows one to take into account the effect of noise on nonlinear dynamics by superimposing a spatial noise specified in the form of a multimode random process at each iteration and selecting the stable wave configurations. The nonlinear wave formations described by this method include optical phase singularities, spatial solitons, and turbulent states with fast decay of correlations. The particular interest is in the periodic configurations of the electromagnetic field obtained by this numerical method that arise as a result of phase synchronization, such as optical lattices and self-organized vortex clusters.