Two-dimensional linear and nonlinear Talbot effect from rogue waves

TL;DR

This paper introduces 2D linear and nonlinear Talbot effects generated by periodic diffraction patterns, with the nonlinear effect originating from rogue waves in a 3D nonlinear medium, revealing unique recurrence and phase shift phenomena.

Contribution

The study presents the first analysis of 2D nonlinear Talbot effects from rogue waves, linking them to self-Fourier transforms and exploring their dependence on period, intensity, and initial conditions.

Findings

Nonlinear Talbot effect arises from rogue waves in 3D media.

Recurrences occur at Talbot and half-Talbot lengths with a  phase shift.

Talbot length decreases with smaller initial wave periods.

Abstract

We introduce two-dimensional (2D) linear and nonlinear Talbot effects. They are produced by propagating periodic 2D diffraction patterns and can be visualized as 3D stacks of Talbot carpets. The nonlinear Talbot effect originates from 2D rogue waves and forms in a bulk 3D nonlinear medium. The recurrences of an input rogue wave are observed at the Talbot length and at the half-Talbot length, with a \pi phase shift; no other recurrences are observed. Different from the nonlinear Talbot effect, the linear effect displays the usual fractional Talbot images as well. We also find that the smaller the period of incident rogue waves, the shorter the Talbot length. Increasing the beam intensity increases the Talbot length, but above a threshold this leads to a catastrophic self-focusing phenomenon which destroys the effect. We also find that the Talbot recurrence can be viewed as a self-Fourier transform of the initial periodic beam that is automatically performed during propagation. In particular, linear Talbot effect can be viewed as a fractional self-Fourier transform, whereas the nonlinear Talbot effect can be viewed as the regular self-Fourier transform. Numerical simulations demonstrate that the rogue wave initial condition is sufficient but not necessary for the observation of the effect. It may also be observed from other periodic inputs, provided they are set on a finite background. The 2D effect may find utility in the production of 3D photonic crystals.