INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF OPTICS A: PURE AND APPLIED OPTICS J. Opt. A: Pure Appl. Opt. 4 (2002) 615–623 PII: S1464-4258(02)40194-8 Stable two-dimensional spinning solitons in a bimodal cubic–quintic model with four-wave mixing D Mihalache 1,2 , D Mazilu 1,2 , I Towers 3 , B A Malomed 3,4 and F Lederer 2 1 Department of Theoretical Physics, Institute of Atomic Physics, PO Box MG-6, Bucharest, Romania 2 Institute of Solid State Theory and Theoretical Optics, Friedrich-Schiller Universit¨ at Jena, Max-Wien-Platz 1, D-07743, Jena, Germany 3 Department of InterdisciplinaryStudies, Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel 4 Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, NM 87545, USA Received 30 July 2002, in final form 1 September 2002 Published 25 September 2002 Online at stacks.iop.org/JOptA/4/615 Abstract We show the formation of stable two-dimensional spinning solitons in a bimodal system described by coupled cubic–quintic nonlinear Schr¨ odinger equations. The cubic part of the model includes the self-phase modulation, cross-phase modulation, and four-wave mixing. Thresholds for the formation of both spinning and non-spinning solitons are found. Instability growth rates of perturbation eigenmodes with different azimuthal indices are calculated as functions of the solitons’ propagation constant. As a result, existence and stability domains are identified for the solitons with vorticity s = 0, 1, and 2 in the model’s parameter plane. The vortex solitons are found to be stable if their energy flux exceeds a certain critical value, so that, in typical cases, the stability domain of the s = 1 solitons occupies about 18% of their existence region, whereas that of the s = 2 solitons occupies 10% of the corresponding existence region. Direct simulations of the full nonlinear system are in perfect agreement with the linear-stability analysis: stable solitons easily self-trap from arbitrary initial pulses with embedded vorticity, while unstable vortex solitons split into a set of separating zero-spin fragments whose number is exactly equal to the azimuthal index of the strongest unstable perturbation eigenmode. Keywords: Ring solitons, competing nonlinearities, vortices 1. Introduction Spatial optical solitons in the form of cylindrical beams in a bulk medium, with an internal ‘hole’ induced by the vorticity (embedded phase dislocation), are objects of considerable interest as a class of fundamental (2 + 1)D ((2 + 1)- dimensional) solitons. In addition, they are of potential use in photonics applications as reconfigurable conduits for weak signal beams; unlike the usual optical vortices [1] (dark (2 + 1)D solitons), bright solitons with an embedded vorticity make it possible to design a multi-channel guiding system of this type [2]. The main problem in the study of the vortex spatial solitons is their stability. In simple models with a single type of nonlinearity, they are always subject to strong azimuthal instability [3]. However, they can be stabilized (as well as their (3 + 1)D counterparts spinning light bullets [4]) in models with competing nonlinearities, such as the cubic–quintic (CQ) nonlinear Schr¨ odinger (NLS) equation [5–7], or a χ (2) :χ (3) system, i.e. combining quadratic and self-defocusing cubic 1464-4258/02/060615+09$30.00 © 2002 IOP Publishing Ltd Printed in the UK 615