Shocks in Nonlocal Media

We investigate the formation of collisionless shocks along the spatial profile of a Gaussian laser beam propagating in nonlocal nonlinear media. For defocusing nonlinearity the shock survives the smoothing effect of the nonlocal response, though its dynamics is qualitatively affected by the latter, whereas for focusing nonlinearity it dominates over filamentation. The patterns observed in a thermal defocusing medium are interpreted in the framework of our theory.

Figure 1

1D spatial profiles of phase chirp $u(x)$ (a),(b) and amplitude $|\psi (x)|=\sqrt{\rho (x)}$ (c),(d), as obtained from Eqs. (2, 3) with $\epsilon ={10}^{-3}$, $\alpha =0$, $\chi =-1$ (defocusing), and ${\psi }_0=\exp (-x^2)$, for different $z$ as indicated: (a),(c) local case, ${\sigma }^2=0$; (b),(d) nonlocal case, ${\sigma }^2=5$.

Figure 2

From Eq. (5): (a) $u(x)$ for different $z$ and $\chi =1$ (focusing), $\sigma =1$; (b) as in (a) for $\chi =-1$ (defocusing); (c) shock distance $z_s$ ($\chi =-1$ bold solid, $\chi =1$ thin solid) and position $x_s$ (defocusing case, dashed line).

Figure 3

Focusing case ($\chi =1$, $\epsilon =0.01$). Level plot of intensity: (a) nonlocal case (${\sigma }^2=25$); (b) chirp profiles for various $z$, ${\sigma }^2=25$; (c) quasilocal case (${\sigma }^2={10}^{-5}$).

Figure 4

2D evolution according to Eqs. (2, 3) with ${\sigma }^2=1$, $\alpha =1$: (a) radial phase chirp at different $z$, as indicated, showing steepening and shock formation for $\epsilon ={10}^{-2}$; (b) corresponding intensity profile $|\psi {|}^2$ (maximum scaled to unity) at $z=4.9$; (c) transverse intensity profile vs $x$ (at $y=0$) at $z=1/(6\epsilon )$ and different values of $\epsilon$ (${\alpha }_0=62{\mathrm{cm}}^{-1}$, $\sigma =0.3$).

Figure 5

Radial profiles of intensity observed in the thermal medium for different input powers. The insets show the corresponding 2D output patterns.