Self-Trapping of “Necklace” Beams in Self-Focusing Kerr Media

Solitons in Kerr media are the most well-studied solitons in nature. The reason for that is twofold. First, the Kerr nonlinearity can be found in many systems: It represents a weak symmetric anharmonicity, which is equivalent to a weak saturation in a simple harmonic oscillator. For electromagnetic waves propagating in a weakly nonlinear centrosymmetric dielectric media, the Kerr nonlinearity manifests itself in the cubic nonlinear Schrödinger equation (NLSE) [1], which in many cases describes the envelope of waves in plasmas, shallow water, deep water, gravity, etc. [2]. The second reason is that Kerr solitons are mathematically elegant: The cubic NLSE is integrable in s1 1 1d dimensions. Its solitons can be found analytically and form a closed set; in their collisions, the total power and momentum in the solitons, and the number of solitons are always conserved [3]. The s2 1 1dD NLSE, although not integrable, has many conserved quantities, but, in the context of self-focusing, is haunted by stability problems [3]; s2 1 1dD bright Kerr solitons are unstable and undergo catastrophic collapse or expansion [4], and s1 1 1dD bright Kerr solitons in a 3D medium suffer from transverse instability [5]. These instabilities occur for solitons of all orders, including, e.g., the higher order self-trapped s2 1 1dD solutions [6]. In optics, bright Kerr solitons are observed only as temporal solitons [7], which are inherently s1 1 1dD, or as s1 1 1dD spatial solitons in single mode waveguides [8], for which transverse instability is eliminated by stringent boundary conditions. Thus interactions between bright solitons are restricted to planar systems. Consequently, much of the beautiful similarity between solitons and particles is lost; e.g., angular momentum has no equivalent in the strictly planar system of bright solitons represented by the s1 1 1dD NLSE. Here, we present self-trapped bright “necklace”-ring beams that exhibit stable propagation for very large distances (.50 diffraction lengths) in Kerr media. The intensities of the necklace beams are azimuthally periodically modulated (in the form of “pearls”), and the widths of the beams are very narrow compared to their radia. A necklace beam is actually a ring array of s2 1 1dD quasisolitons (pearls), which we find to be stable whenever the azimuthal period length of the ring is smaller than or equal to the width of the ring. Computer simulations indicate that this necklace ring is stable in the absolute sense, although we cannot prove this analytically. The necklace ring slowly expands, with a rate of expansion dependent on the number of pearls in the ring, the width of the ring, the initial peak intensity, and ring’s diameter. When the number of pearls is large, holding the parameters of each pearl fixed, the beams are almost fully stationary, and in some cases allow approximate analytic solutions. The normalized cubic s2 1 1dD NLSE is