Exploring the dynamics of the Kelvin-Helmholtz instability in paraxial fluids of light

Paraxial fluids of light have recently emerged as promising analog physical simulators of quantum fluids using laser propagation inside nonlinear optical media. In particular, recent works have explored the versatility of such systems for the observation of two-dimensional quantum-like turbulence regimes, dominated by quantized vortex formation and interaction that results in distinctive kinetic energy power laws and inverse energy cascades. In this manuscript, we explore a regime analog to Kelvin-Helmholtz instability to examine in further detail the qualitative dynamics involved in the transition from smooth laminar flow to turbulence at the interface of two fluids with distinct velocities. Both numerical and experimental results reveal the formation of a vortex sheet as expected, with a quantized number of vortices determined by initial conditions. Using an effective length transformation scale we get a deeper insight into the vortex formation phase, observing the appearance of characteristic power laws in the incompressible kinetic energy spectrum that are related to the single vortex structures. The results enclosed demonstrate the versatility of paraxial fluids of light and may set the stage for the future observation of distinct classes of phenomena recently predicted to occur in these systems, such as radiant instability and superradiance.

Figure 1

Illustration of the typical experimental scenario for the observation of Kelvin-Helmholtz instability in paraxial fluids of light. (A) Simplified schematic of the experimental setup. (B) Evolution of the optical intensity along the crystal, obtained using numerical simulations for an initial state with $v\approx 4.4c_s$ under the conditions described in the main text. (C) State at the input (numerical simulation), with (C1) corresponding to the intensity profile and (C2) to the phase profile with its characteristic phase imprinted on the top half-plane using wavefront shaping techniques. (D) Typical state (numerical simulation) at the output, revealing the formation of a vortex sheet along the discontinuity interface.

Figure 2

Profiles obtained using numerical (Column 1) and experimental (Columns 2 and 3 for intensity field and phase distribution, respectively) methodologies for the output states with distinct velocities (A–C) as shown in the picture. The different values for the velocity lead to the generation of a different number of vortices at the interface.

Figure 3

Plot comparing the observed linear vortex density $n_v=N_{\mathrm{vortices}}/w$ (circles) and the theoretically predicted value given by equation $n_v=v_x/\left(2\pi \right)$ (dashed line) as described in the main text.

Figure 4

Numerical results obtained for a state with velocity $v_x=4.4c_s$, with (A–D) representing slices taken along the propagation direction, where the top corresponds to the intensity profiles and the bottom to the phase profiles. On the bottom, the stream plots illustrate the gradient of the phase, associated with the analog velocity of the paraxial fluid of light. The red arrows in (A2) represent the $y$ component of the velocity occurring due to the discontinuity at the half-plane interface.

Figure 5

Experimental results obtained for a state with velocity $v_x=4.4c_s$. with panels representing slices taken along the propagation direction, where the top corresponds to the intensity profiles and the middle to the phase profiles using the effective scale transformation technique described in the main text. On the bottom, (A3)–(C3) represent the incompressible part of the velocity related to the presence of vortex structures in the velocity field, revealing the appearance of a vortex at the expected point.

Figure 6

Plot showing the incompressible energy spectrum in arbitrary units obtained for distinct effective propagation lengths, confirming the presence of the expected power laws for the single vortex distribution, i.e., $k^{-1}$, towards large scales, in clear contrast with the $k^{-3}$ power law observed towards smaller scales as the instability develops into a vortex.