Observation of a Gradient Catastrophe Generating Solitons

We investigate the propagation of a dark beam in a defocusing medium in the strong nonlinear regime. We observe for the first time a shock fan filled with noninteracting one-dimensional gray solitons that emanates from a gradient catastrophe developing around a null of the optical intensity. This scenario turns out to be very robust, persisting also when the material nonlocal response averages the nonlinearity over dimensions much larger than the emerging soliton filaments.

Figure 1

Sketch of the experimental setup.

Figure 2

Transverse distribution of the intensity along the cell ($x\mathrm{\text{−}}z$ plane), as observed from top scattered light for four different input powers. Superimposed light curves (yellow or light gray) are retrieved intensity profiles at $Z=0.6\mathrm{mm}$, while in (d) the right dark (blue or dark gray) curve is relative to $Z=2.25\mathrm{mm}$. The diffraction fringes aside the central dark notch are due to the focusing system before the sample and reflect the slight convex wave front of the bright background.

Figure 3

Far-field intensity distribution in the $X\mathrm{\text{−}}Y$ transverse plane at the sample output for different input powers: the intensity is collected after about 1 m of free-air propagation.

Figure 4

Numerical simulations of Eqs. (1, 2) with $\sigma =\alpha =0$ and ${\psi }_0=\tanh (x)$: (a),(c) Level plots of $|\psi {|}^2$ for $\epsilon =0.05$ (a) and $\epsilon =0.02$ (c); (b) snapshot of case (a) at $z=4$; (d) snapshot of case (c) at the catastrophe point $z=0.75$: frequency $u$ (solid red (medium gray) curve and intensity (black solid curve) compared with the input (blue dashed curve).

Figure 5

Evolution in the nonlocal and lossy case: (a) Level plot of intensity (the color scale is adapted to compensate for losses); (b) snapshots at $z=7$, of intensity (black solid curve) and relative index change $\delta n$ (red solid curve). The input is the dashed blue curve. Here $\epsilon =0.05$ ($P\simeq 0.6\mathrm{W}$ in our experiment), ${\sigma }^2=1$, $\alpha =0.3$, yielding a physical breaking (catastrophe) distance $Z_b=L{z}_b=0.8\mathrm{mm}$.

Figure 6

Breaking distance $Z_b$ vs input power $P_{\mathrm{in}}$ (log-log scale): (a) local case, numerical results (circles) vs behavior $Z_b\propto P_{\mathrm{in}}^{-1/2}$ (solid line), characteristic of the hydrodynamic limit; (b) nonlocal case, experimental results (circles) vs data extrapolated from numerical simulations (dashed line) performed for different values of powers $P_{\mathrm{in}}$ ($\epsilon$). Here $\sigma =0.3$ was fixed to find the best agreement with the data.