We present a theoretical and experimental study of the propagation, decay, and interaction of optical vortices in media with an anisotropic nonlocal nonlinearity. The initial stage of decay of a circular vortex is characterized by charge dependent rotation, and stretching of the vortex. Our results suggest that a compact vortex of unit topological charge cannot exist in such media, but that a counterrotating vortex pair can form a bound state. Some analogies and differences between optical

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... ces and the classical dynamics of vortices in an ideal fluid are described. [S0031-9007(96)01742-5] PACS numbers: 42.65.Tg, 42.65.Hw, 42.65.Sf, 47.32.Cc Vortex solitons exist in media with a defocusing nonlinearity. Their stability depends on the structure of the nonlinearity and the dimensionality of the medium. Evolution in a medium with cubic nonlinearity in the limit of one transverse dimension is described by the ͑1 1 1͒D nonlinear Schrödinger equation which is integrable, and admits solitary wave solutions both for focusing and defocusing nonlinearities [1]. Planar ͑1 1 1͒D solutions are modulationally unstable in bulk media described by a ͑2 1 1͒D system of equations. This was known theoretically at an early stage [2-4], and has been seen experimentally in water waves [5] and in nonlinear optics both in atomic vapors with a Kerr type response [6] and in photorefractive media [7] . Three dimensional solitary vortex solutions of the nonlinear Schrödinger equation were first considered in the context of superfluidity [8] and have attracted a great deal of recent interest in nonlinear optics [9] . Kerr type optical media with a cubic, isotropic, and local nonlinearity support ͑1 1 1͒D vortex solitons [10] . A Kerr type nonlinearity is, however, a simplified idealized model of a nonlinear response. The evolution of vortices in media with a more complex nonlinear response has not been studied extensively. The bulk photorefractive nonlinear medium used in the experiments reported here exhibits a nonlinearity that is both anisotropic and nonlocal. This leads to some remarkable qualitative differences in the spatial dynamics of light beams. In particular an optical vortex with initial circular symmetry rotates and stretches, aligning itself so that its major axis coincides with the direction of greatest material nonlinearity. The stretching proceeds unchecked so that the vortex becomes more and more delocalized. Both theory and experiment indicate that, in contradiction to results reported earlier [11] , localized optical vortex solutions and, in particular, soliton vortex solutions of unit topological charge do not exist in these media. The nonexistence of localized vortex solutions is not a simple consequence of the anisotropy alone since, in related work done in the same material but with a self-focusing nonlinearity, we have demonstrated convergence to elliptical soliton solutions [12] . Despite the above unique features of spatial dynamics of single vortices the initial development of beams with several embedded vortices demonstrates spatial dynamics reminiscent of that known from the classical theory of point vortices in fluids [13] and superfluids [14] . This is not unexpected since there is a close mathematical analogy between the nonlinear Schrödinger equation and the mechanics of fluids in two dimensions [8, 15] . The spatial evolution of the optical field B͑ r͒ is described in dimensionless form by the equation The differential operator = ŷ≠͞≠y 1ẑ≠͞≠z acts on coordinates y and z perpendicular to the direction of propagation of the beam x, and the operatorf͑jBj 2 ͒ describes the nonlinearity [see Eq. (4) ]. Time evolution is replaced here by propagation along the coordinate x. Writing the optical field in the form B p r e ic , we obtain the continuity equation where r is the density and y =c is identified as the velocity, together with the Euler equation for the velocity field y x 1 y ? = y = Equations (2) and (3) are valid where the velocity field is irrotational away from the zeros of r. In Kerr mediaf r and the pressure is given by p r 2 ͞2, while the last term on the right hand side of (3) is a nonlinear correction to the pressure. In the photorefractive media considered here the material response is anisotropic and is given by [16] f a≠f͞≠z, where the electrostatic potential f satisfies = 2 f 1 = ln͑1 1 jBj 2 ͒ ? =f ≠ ≠z ln͑1 1 jBj 2 ͒ . (4) The nonlinearity coefficient a is proportional to the amplitude of an externally applied electric field: a~E ext (for details see [7, 16] ). The pressure is therefore anisotropic 4544 0031-9007͞96͞77(22)͞4544(4)$10.00