Two-dimensional linear and nonlinear Talbot effect from rogue waves

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. Differing 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.

Figure 1

Linear 2D Talbot effect. (a) Intensity isosurface plot of the propagating rogue-wave incidence, according to the linear Schrödinger equation. The beam intensities are displayed at the distances of $1/4,1/2$, and 1 Talbot lengths. (b) Corresponding intensity carpet in the $xz$ plane at $y=0$. (c) Intensity profiles along the $x/y$ axes, depicting the profiles at $z=0,z=z_T/4,z=z_T/2$, and $z=z_T$ from (a). Other parameters: $q=1/4,\mathcal{D}=\sqrt{2}\pi$, and $M=3+2\sqrt{2}$. (d) Relationships between $\mathcal{D}$ and $z_T,q$, and $M$.

Figure 2

Same as described in the caption to Fig. 1 but for the 2D nonlinear Talbot effect.

Figure 3

Talbot length dependence on the amplitude of the incident beam with fixed transverse periods.

Figure 4

(a) 2D Talbot effect coming from a product of two doubly periodic ABs with $k=1.2$ in the plane $x-y=0$. Dashed lines show the locations of half Talbot length and Talbot length, respectively. [(b)–(d)] Intensities at the initial place, half Talbot length, and Talbot length.

Figure 5

2D Talbot effect resulting from Eq. (11). Figure setup is as in Fig. 4.