Regular and irregular revivals of quasi-periodic random waves

Abstract

Paraxial wave packets with discrete spatial, temporal, or spatiotemporal spectra are known to undergo periodic axial revivals on propagation in either free space or linear transparent, weakly dispersive media. Such spectacular revivals, ubiquitously encountered in physics, from optics and acoustics to condensed matter physics, are distinguished by their strict periodicity. We show theoretically and verify experimentally that ensembles of quasi-periodic random wave packets exhibit a unique revival network composed of regular (periodic) and irregular (aperiodic) revivals. Moreover, individual realizations of a statistical ensemble self-reconstruct, in general, at different propagation distances than do ensemble averages. Our results shed new light on the fundamental physics of self-reconstruction of random wave packets with structured correlations.

Figures (4)

• Figure 1: (a) & (c) Talbot carpet of an ensemble averaged intensity corresponding to (a) $q=1$ and $p=2$, as well as (c) $q=1$ and $p=3$. The inverted black and red triangles mark the intensity at the source and its perfect replicas at multiple integers of $\overline{L}_{\mathrm{SI}}$. The upright blue triangles mark the replicas of the source shifted by half the period $T_p$. (b) & (d) Intensity profile at the source (solid black) and replicas of the source intensity at $z=\overline{L}_{\mathrm{SI}}/2$ (solid blue) and at $z=\overline{L}_{\mathrm{SI}}$ (red dash). We employ a (normalized) Gaussian mode power spectrum $\nu_m=e^{-2\xi_c m^2}$ with $\xi_c=0.1$.
• Figure 2: (a) Rational approximation of $\alpha=T_c/T_p=1/\sqrt{2}$ as a function of $q$. (b) Talbot carpet of the average intensity for $\alpha=1/\sqrt{2}$ as a function of the scaled propagation distance $z/\overline{L}_{\mathrm{SI}}^{(0)}$. Acceptable candidates for complete revivals are marked by colored inverted triangles, while shifted acceptable revivals are denoted by colored upright triangles. We use a (normalized) Gaussian mode power spectrum $\nu_m=e^{-2\xi_c m^2}$ with $\xi_c=0.1$.
• Figure 3: Experimentally generated Talbot carpets for (a): $q=1$, $p=2$, and (c): $q=1$, $p=3$. The inverted black and red triangles mark the intensity at the source and its replica, at the self-imaging distance. The upright blue triangles indicates a half-period shifted replica at half the self-imaging distance. (b) & (d) Intensity profile at the source (solid black) and replicas of the source intensity at half- (solid blue) and full (red dash) self-imaging distances. We employ a (normalized) Gaussian mode power spectrum $\nu_m=e^{-2\xi_c m^2}$ with $\xi_c=0.1$.
• Figure 4: (a) Experimentally realized Talbot subnetwork that includes revivals associated with rational approximations $3/4$ (green) and $5/7$ (blue) of $1/\sqrt2$. Upright green and blue triangles mark shifted images, while the inverted blue triangle denotes an image of the source. (b) Intensity profiles of the source and those of images of variable order corresponding to the same rational approximations as in panel (a). We use a (normalized) Gaussian mode power spectrum $\nu_m=e^{-2\xi_c m^2}$ with $\xi_c=0.1$.