Topologically crafted spatiotemporal vortices in acoustics

Abstract

Vortices in fluids and gases have piqued the human interest for centuries. Development of classical-wave physics and quantum mechanics highlighted wave vortices characterized by phase singularities and topological charges. In particular, vortex beams have found numerous applications in modern optics and other areas. Recently, optical spatiotemporal vortex states exhibiting the phase singularity both in space and time have been described. Here, we report the topologically robust generation of acoustic spatiotemporal vortex pulses. We utilize an acoustic meta-grating with broken mirror symmetry which exhibits a topological phase transition with a pair of phase singularities with opposite topological charges emerging in the momentum-frequency domain. We show that these vortices are topologically robust against structural perturbations of the meta-grating and can be employed for the generation of spatiotemporal vortex pulses. Our work paves the way for studies and applications of spatiotemporal structured waves in acoustics and other wave systems.

Fig. 1. Acoustic meta-grating for the generation of acoustic spatiotemporal vortex pulses.

a Schematics of the meta-grating generating a spatiotemporal vortex pulse for airborne sound by breaking the mirror symmetry. The period of the meta-grating is $p=$ 33.34 mm. b The structure of the meta-grating, where the solid material is shown in gray, the position of the yellow block (air) is fixed, while the four white blocks (air) marked can be x-shifted to the left or right. The displacements of these blocks, $\delta x_i$, break the mirror symmetry with respect to the x = 0 plane, and this asymmetry is quantified by the dimensionless parameter $\eta$ (see explanations in the text). c–e Numerical simulations for the phase (top) and the amplitude (bottom) of the transmission spectrum function $T\left(k_x,\omega \right)$ for different values of the asymmetry parameter $\eta$. Phase singularities (vortices) with the winding numbers (topological charges) +1 and −1 are indicated by the white and black arrows, respectively. f The pair of vortices in the panels (d) and (e) corresponds to a single phase-singularity (nodal) line in the 3D space (k_x,ω,$\eta$) extended by the asymmetry parameter $\eta$. The vortex pair emerges at the critical values of ${\eta }_c=0.40$. This nodal line and the separated vortices are topologically protected against small perturbations in the system.

Fig. 2. Transmission spectrum function for different values of the asymmetry parameter η.

Experimentally measured phase (a–c) and amplitude (d–f) of transmission spectrum function for different values of the asymmetry parameter η. Vortices with the winding numbers of +1 and −1 are indicated by white and black arrows, respectively.

Fig. 3. Topologically protected generation of acoustic spatiotemporal vortex pulses.

a Schematics of the experimental setup with the incident acoustic Gaussian pulse and transmitted acoustic STVP. b Experimental sample of the acoustic meta-grating with the asymmetry parameter $\eta =1.0$. c The perturbed meta-grating where 16 particles of different shapes (sphere, pyramid, cube and ring) are randomly placed, and additional regular small cuts are introduced. d–g Numerical simulations of the transmitted pulse envelopes $S_{out}\left(x,t\right)$ in the space-time domain at different z-positions separated by the ${\lambda }_0/3$ intervals (${\lambda }_0$ is the central wavelength of the pulse). h–k Experimental measurements of the transmitted pulse envelopes corresponding to the numerical simulations in (d–g). l–o Experimental measurements of the topologically protected STVP generation using the perturbed meta-grating shown in (c).

Fig. 4. Propagation of acoustic spatiotemporal vortex pulse in real space.

a–f Spatial distributions of the pressure field amplitude, ${\mid P}_{out}\left(x,z\right)\mid$ (colormap), and the wave momentum density $\mathit{\Pi }\left(x,z\right)$ (arrows) in the transmitted STVP at different instants of time t. One can see the front edge of the pulse (a, b), the nodal point in the center (c, d), and the rear edge of the pulse (e, f). The pulse shape is deformed due to the diffraction.