Propagation of dark stripe beams in nonlinear media: Snake instability and creation of optical vortices.

Abstract

We analyze the evolution of s111d dimensional dark stripe beams in bulk media with a photorefractive nonlinear response. These beams, including solitary wave solutions, are shown to be unstable with respect to symmetry breaking and formation of structure along the initially homogeneous coordinate. Experimental results show the complete sequence of events starting from self-focusing of the stripe, its bending due to the snake instability, and subsequent decay into a set of optical vortices. Focusing Kerr-type nonlinear media exhibit solitary wave solutions, where diffraction is compensated by the nonlinearity. These solutions are localized bright beams with zero field at infinity. Solitary solutions in defocusing media correspond to a dark region of small intensity with finite (and larger) intensity at infinity. Stripe s111d dark solitary solutions of the nonlinear Schrodinger equation in the context of a defocusing Kerr nonlinearity were analyzed in Ref. [1]. Later it became evident that these solutions are unstable in bulk media [2,3] due to the growth of perturbations along the initially homogeneous coordinate. Their breakup and subsequent spatial dynamics have been the subject of continuing interest since then [4]. Numerical analysis of spatial dynamics of light beams for a nonsaturable and saturable defocusing Kerr nonlinearity was undertaken in [5] and [6], respectively. Dark solitary wave solutions can only be investigated in approximate form experimentally, since they have finite energy at infinity. Experimental investigations of the propagation of dark stripe beams in bulk media have relied, therefore, on embedding a dark notch in a somewhat wider bright envelope [7‐11]. This has two immediate consequences. First, the bright envelope spreads in a defocusing medium, which limits the propagation distance over which a high contrast dark stripe can be maintained. Second, the growth rates for perturbations along the homogeneous coordinate turn out to depend on the relative width of the envelope to the dark notch. We have found, both theoretically and experimentally, that the instability growth rates decrease as the envelope of the bright background is narrowed. This may in part explain why the snake instability of dark stripe beams, observed here for the first time, has not been reported previously. We analyze below the breakup and subsequent spatial evolution of one-transverse-dimensional s111d dark stripe beams in bulk nonlinear media with a photorefractive nonlinear response. In bulk media these beams belong to a low-dimensional s111d subclass of higherdimensional s211d allowable solutions and are shown to be unstable due to breaking of the initially odd symmetry of the field and the appearance of spatial structure along the “hidden” homogeneous coordinate. We present theoretical and experimental data demonstrating all stages of this breakup and subsequent spatial evolution resulting in the formation of optical vortices (wave front dislocations) [12]. Formation of optical vortices has been predicted in laser cavities [13], and due to nonlinear propagation , in self-defocusing media [6,14]. Optical vortices are well known in linear optics [15], and were observed previously in self-defocusing optical media [9] by introducing strong perturbations at the entrance to the nonlinear medium. We demonstrate here that optical vortices are a natural consequence of nonlinear propagation of any dark stripe beam in a defocusing medium. The necessary seeds are provided by the natural noise in the system, and the characteristic scales are determined by the fastestgrowing modes of the transverse modulation instability. Propagation of an optical beam Bs$ rd in a defocusing photorefractive medium is governed by the equations [16]