Beam Instabilities in the Scale-Free Regime

The instabilities arising in a one-dimensional beam sustained by the diffusive photorefractive nonlinearity in out-of-equilibrium ferroelectrics are theoretically and numerically investigated. In the “scale-free model,” in striking contrast with the well-known spatial modulational instability, two different beam instabilities dominate: a defocusing and a fragmenting process. Both are independent of the beam power and are not associated with any specific periodic pattern.

Figure 1

(a) Gain versus the ratio of the waist between the perturbation and the pump for $r>1$ (maximum gain attained at $k_y=0$). (b) Left scale: Gain versus $r$ for $r<1$; right scale: corresponding maximally amplified period.

Figure 2

Simulation of the defocusing instability after Eq. (3). (a) Output beam at $z=2$ for $\sigma =0.125$, with a striped beam with $w_0=1$ and $A_0=1$; (b) as in  (a) with $\sigma =0$ (linear propagation); (c) as in (a) with $w_0=0.5$. (d) Beam waist in the $x$ directions for (a) dotted line, (b) dashed line, and (c) continuous line.

Figure 3

Simulation of the fragmenting instability. The stripe solution is perturbed by a Gaussian noise $p$ with 10% amplitude ($|p/{\psi }_0|=0.1$): (a) input beam, (b) output beam at $z=2$ ($A_0=1$, $w_0=0.5$, $\sigma =0.125$. (c) Spectrum obtained after the numerical simulations (continuous thick line for $w_0=0.5$, thin line $w_0=4$), the colored region is after Eq. (12) showing the theoretically expected spectrum.