Time-diffracting 2D wave vortices

TL;DR

This work identifies a new class of time-diffracting 2D vortices that are localized in the (x,y) plane and diffract only in time, carrying well-defined transverse OAM. It develops a general integral framework $\Psi({\bf r},t) \propto e^{i\ell\varphi} \int_0^{\infty} f(k) J_{|\ell|}(kr) e^{-i \omega(k) t} k\, dk$, and provides an intuitive geometrical-optics ray model, a Gaussian-wavepacket model, and an exact solution for a specific spectrum to describe their temporal diffraction and localization. A temporal Gouy phase is shown to induce a net azimuthal rotation $\Delta\varphi = \pi \mathrm{sgn}(\ell)$ between $t\to-\infty$ and $t\to\infty$, linking phase evolution to the vortex’s OAM dynamics. The results predict strong spatiotemporal energy and OAM concentration at sub-wavelength scales and point to realizations in surface polaritons and planar waveguides with potential applications in nonlinear optics, vortex-laser technologies, and light–matter interactions.

Abstract

Wave vortices constitute a large family of wave entities, closely related to phase singularities and orbital angular momentum (OAM). So far, two main classes of localized wave vortices have been explored: (i) transversely-localized monochromatic vortex beams that carry well-defined longitudinal OAM and propagate/diffract along the longitudinal $z$-axis in space, and (ii) 2D-localized spatiotemporal vortex pulses that carry the more elusive transverse (or tilted) OAM and propagate/diffract along both the $z$-axis and time. Here we introduce another class of wave vortices which are localized in a 2D $(x,y)$ plane, do not propagate in space (apart from uniform radial deformations), and instead propagate/diffract solely along time. These vortices possess well-defined transverse OAM and can naturally appear in 2D wave systems, such as surface polaritons or water waves. We provide a general integral expression for time-diffracting 2D wave vortices, their underlying ray model, and examples of approximate and exact wave solutions. We also analyze the temporal Gouy phase closely related to the rotational evolution in such vortices. Finally, we show that time-diffracting 2D vortices can provide strong spatiotemporal concentration of energy and OAM at sub-wavelength and oscillation-period scales.

Figures (6)

• Figure 1: Schematics of (a) monochromatic spatial vortex beams propagating and diffracting along the $z$-axis; (b) spatiotemporal vortices propagating and diffracting along the $z$-axis and time $t$; and (c) 2D vortices propagating and diffracting in time $t$, which are the focus of this work.
• Figure 2: Examples of 2D wave vortices \ref{['eq:vortex']} for $\ell=2$ and wavenumber distributions $f(k) = \exp\left[-(k-k_0)^2/\Delta^2 \right]$ with different values of $\Delta$. Left column: plane-wave spectra in the ${\bf k}$-plane, where saturation and colors represent the amplitude and phase of the plane-wave components. Right column: real-space distributions of $\Psi({\bf r},t=0)$, where brightness and colors represent the amplitude and phase of the wavefunction. The limit $\Delta\to 0$, shown in (a), corresponds to the monochromatic Bessel vortex \ref{['eq:Bessel']} with multiple intensity rings. For $\Delta \sim k_0$, shown in (c), the vortex becomes practically localized within a single intensity ring.
• Figure 3: (a) Temporal evolution of the vortex from Fig. \ref{['fig:vortex']}(c) for waves with linear dispersion $\omega = kc$. In each panel, the intensity is normalized by its maximum value (for the evolution of intensity with time see Fig. \ref{['fig:radial']}). (b) Geometrical-optics rays \ref{['eq:rays']} and the corresponding moving point particles underlying the temporal diffraction shown in (a).
• Figure 4: Temporal evolution of the vortex described by a simple analytical model \ref{['eq:model']} with $\ell=2$ and $k_0L =3$.
• Figure 5: Temporal evolution of the vortex described by the exact solution \ref{['eq:exact']} with $\ell=2$.