Structured light entities, chaos and nonlocal maps

TL;DR

The paper develops nonlocal iterative maps with Green-function kernels $K(\vec{r}-\vec{r}')$ as an efficient numerical framework that is formally equivalent to nonlinear parabolic PDEs of the Ginzburg-Landau type, enabling accurate modeling of spatial chaos, solitons, vortices, and vortex lattices in optical resonators under multimode noise. By expressing propagation as convolution maps and relating them to slice-by-slice evolutions, it links discrete map dynamics to evolution equations such as diffusion, diffraction, and NLS-GLE, with explicit connections to the KPP equation and the Gross-Pitaevskii framework. The work demonstrates the emergence of localized wavetrains and solitons as fixed points of nonlocal maps, and shows how periodic transverse structures and vortex lattices arise in cavities with varying Fresnel numbers, including Talbot self-imaging phenomena and phase-locked lattices. Incorporating multimode noise yields realistic relaxation oscillations and spectra, illustrating the method’s ability to reproduce experimental patterns while maintaining rapid convergence via FFT-based computation. Overall, the nonlocal-map approach provides a flexible, boundary-aware, computationally efficient bridge between discrete dynamical systems and continuous PDE descriptions of pattern formation in nonlinear optical media.

Abstract

Spatial chaos as a phenomenon of ultimate complexity requires the efficient numerical algorithms. For this purpose iterative low-dimensional maps have demonstrated high efficiency. Natural generalization of Feigenbaum and Ikeda maps may include convolution integrals with kernel in a form of Green function of a relevant linear physical system. It is shown that such iterative $nonlocal$ $nonlinear$ $maps$ are equivalent to ubiquitous class of nonlinear partial differential equations of Ginzburg-Landau type. With a Green functions relevant to generic optical resonators these $nonlocal$ $maps$ emulate the basic spatiotemporal phenomena as spatial solitons, vortex eigenmodes breathing via relaxation oscillations mediated by noise, vortex-vortex and vortex-antivortex lattices with periodic location of vortex cores. The smooth multimode noise addition facilitates the selection of stable entities and elimination of numerical artifacts.

Figures (8)

• Figure 1: a) Bifurcation diagram of ring laser shows distribution of electric field amplitudes $E_n$ at gradually increased gain $G=G_{1,2...chaos}$, and histograms representing chaotic probability densities $P(E)$ for gain $G=10.595$ (b) and $G=39$ (c) after 5000 iterations.
• Figure 2: a) Layout of $unidirectional$ single transverse mode ring laser with nonlinear losses . Envelope of the laser pulse is modulated consecutively from one passage to another by hyperbolic tangent chaotic map. b) Layout of single transverse mode $unidirectional$ ring cavity described by Ikeda map. The phase lag between entrance field $E_0$ and intracavity field $E_n$ is proportional to light intensity $|E_n|^2$. c)Layout of confocal cavity of length $L_c=2F$ with saturable gain $G(E)$ at right mirror and saturable absorber $\alpha(E)$ at the opposite one. The fields on opposite mirrors $\tilde{E}_n(\vec{r}_{bot})$ and $E_n(\vec{r}_{bot})$ are linked via Fourier transform. Spatial soliton is formed by transverse modelocking. b) Layout of diode-pumped solid-state laser with slightly focusing output mirror where vortex-antivortex lattices appear due to transverse modelocking at high Fresnel numbers $N_f \sim 10^2 - 10^3$.
• Figure 3: a) Comparison of logistic map ($\bf 1$) $x \rightarrow \lambda x(1-x)$ and hyperbolic tangent map $x \rightarrow G x[1-tanh(x)]$ ($\bf 2,3$) for laser with nonlinear losses at ($G=6 , 9$), b) histogram for full chaos after period 3, representing chaotic probability density $P(E)$ obtained by 5000 iterations of logistic maps with $\lambda=3.9$, c) identical probability density $P(I)\sim 1/\sqrt{I(1-I)}$ for interference pattern for Michelson interferometer with independent phase-conjugating mirrors obtained by averaging over ensemble of 300 000 counts.
• Figure 4: Spatial soliton $E_n(\vec{r})$ excited in a given area of [512x512] computational mesh fitted for confocal cavity fig.2c of length $2F$ with saturable gain $G(E)$ at left mirror and saturable absorber $\alpha(E)$ at opposite mirror. The oscillatory rings around central part of soliton subjected to self-phase modulation occur due to interference with background.
• Figure 5: Location of negative instability increments (hatched area) for spatial soliton in confocal cavity of length $2F$ with saturable gain $G(E)$ at left mirror and saturable absorber $\alpha(E)$ at opposite mirror. The vertical axis is for number $n$ of spatial harmonic of excitation $\psi_{\zeta}$ .