Propagation of light beams in anisotropic nonlinear media: From symmetry breaking to spatial turbulence.

Propagation and spatial evolution of light beams in nonlinear media is a central topic of nonlinear optical dynamics. There has been continuing interest in this topic since the initial investigations of self-trapping of light beams in nonlinear media @1,2#. Analysis of the stability and nonlinear evolution of solitary-wave solutions of nonlinear propagation equations is one of the most crucial parts of the problem of self-trapping of optical beams. Exact ~111!-dimensional solitary-wave solutions to the nonlinear Schrödinger equation were found more than 20 years ago @3,4#. It was shown shortly thereafter @5# that these solutions are unstable in the case of a ~111!-dimensional stripe solitary-wave propagating in a ~211!-dimensional bulk nonlinear medium. The symmetry-breaking instability, which is due to the growth of perturbations along the initially homogeneous coordinate parallel to the stripe, is known as a transverse modulation instability and goes back to papers by Bespalov and Talanov @6# and Benjamin and Feir @7#, who discussed its manifestations for a homogeneous ~plane-wave! ground state. Although much theoretical work has been devoted to studying the instability of stripe solitary waves in both focusing and defocusing bulk media @8–12#, the transverse modulation instability has only recently been observed experimentally @13–15#. Our topic here is a theoretical and experimental study of this instability in bulk photorefractive media with an anisotropic focusing or defocusing nonlinear response. Bright stripe solitary solutions in focusing media decay into a line of bright filaments, while dark stripe solitary solutions in defocusing media are subject to a snake instability @5# and decay into a line of optical vortices @10,11#. When the initial conditions are not solitary solutions the beam cannot propagate intact in the nonlinear medium. Wider stripe beams radiate and decay into multiple stripes before the onset of the instability seen for solitary stripes. For beams that are not too wide the stripes interact with each other and decay into a partially ordered pattern of filamentation. A related instability in planar photorefractive waveguides has also been observed @16#. Circular beams with full ~211!-dimensional symmetry decay into a spatially disordered pattern @17#. Experiments with input speckle beams demonstrate the difference in the spatial statistics of the output field due to focusing or defocusing nonlinearities. We present results illustrating all stages of spatial evolution from symmetry breaking and decay of a solitary stripe to generation of a turbulent array of cylindrical filaments. Photorefractive crystals turn out to be very convenient for experimental study of these instabilities. Several groups have demonstrated self-focusing and self-defocusing in photorefractive media @18–21#. A large steady-state nonlinear response can be obtained with low-power visible lasers. Furthermore, the magnitude and sign of the nonlinearity ~focusing or defocusing! are easily controlled with an external voltage. The physics of light propagation in photorefractive media is considerably different than in Kerr media that are described by the nonlinear Schrödinger equation. The photorefractive nonlinearity is due to the action of a static electric field that is generated by the optical beam. The electric field is found by solving a particular form of Poisson’s equation for the electrostatic potential f , with a source charge distribution due to light-induced charge transport. The photorefractive nonlinearity is thus nonlocal. Given the electrostatic potential, the perturbation to the refractive index is dni j;ri jk]f/]xk , where r is the electro-optic tensor. The anisotropic nature of r results in a highly anisotropic nonlinear response. These differences result in some instability signatures that are not seen in Kerr media. In particular, the pronounced striped filamentation seen at intermediate stages of the nonlinear decay is unique to anisotropic media. Nonetheless, the transverse modulation instability and the final decay into bright and dark filaments are universal.