Break-up of two-dimensional bright spatial solitons due to transverse modulation instability

– We present the theory and the experimental observation of break-up of twodimensional bright spatial solitons propagating in a three-dimensional bulk photorefractive nonlinear medium due to transverse modulation instability. Solitary wave (self-bound) solutions of nonlinear propagation equations are a fascinating topic of nonlinear dynamics. The analysis of their stability and nonlinear evolution properties is one of the most crucial parts of the problem of self-trapping of optical beams. Much research has been devoted to this topic in connection with the question of the existence of optical solitons in media with Kerr-type nonlinearity (see, e.g., ref. [1]-[8]). In this analysis it is important to understand the relationship between a particular solitary solution and possibly more general equations from which it has been derived. Despite an exact mathematical analogy between self-bound solutions their evolution may turn out to be completely different in physically different situations. Consider, e.g., a two-dimensional solitary wave solution following from the two-dimensional basic set of equations (a pulse propagating in a single-mode fiber). This solution is a soliton since it can experience perturbations of only a certain limited class and is stable with respect to these perturbations [2]. The same two-dimensional solitary solution viewed in the context of, say, a cw beam propagating in a bulk medium is unstable [3]. The reason for this is that the same solution now belongs to a subclass of all possible solutions and has lower dimension (1 + 1) than the dimension (2 + 1) of the basic set of equations. It contains a “hidden” homogeneous coordinate (is independent of this coordinate) and is now subject to a broader class of perturbations. Of particular importance are perturbations that affect this hidden coordinate. The corresponding instability is called a transverse modulation instability. It goes back to papers by Bespalov and Talanov [4] and Benjamin and Feir [5] who discussed its manifestations for a homogeneous (plane-wave) ground state. Analyses of stability of two-dimensional self-bound solutions in a Kerr medium have been carried out in [7]-[8]. In this paper we analyze the evolution of two-dimensional bright spatial solitons that propagate in a three-dimensional bulk nonlinear photorefractive medium (the theoretical analysis (∗) Permanent address: Institute for Problems in Mechanics, Russian Academy of Sciences, Prospekt Vernadskogo 101, Moscow, 117526 Russia. c © Les Editions de Physique 26 EUROPHYSICS LETTERS of this paper has been reported in ref. [9]). We show that they are destroyed because of the symmetry breaking and the appearance of a spatial structure along the hidden homogeneous coordinate. We present experimental demonstrations of the break-up and the dependence of the instability on the parameters of the problem. Observations indicative of this instability for the case of solitary waves behind a ship were reported in [10]. The present work is, to our knowledge, the first experimental demonstration of the transverse modulation instability of two-dimensional solitary waves in any nonlinear optical medium. We shall describe propagation of an optical beam B(r) in a photorefractive medium in the presence of an externally applied electric field and/or photogalvanic nonlinearity by the set of equations [9], [11] [ ∂ ∂x − i 2 ∇ ] B(r) = i ∂φ