Self-trapping of "necklace-ring" beams in self-focusing kerr media

Recently, we suggested a type of self-trapped optical beams that can propagate in a stable form in (2+1)D self-focusing Kerr media: Necklace-ring beams [M. Soljacic, S. Sears, and M. Segev, Phys. Rev. Lett. 81, 4851 (1998)]. These self-trapped necklaces slowly expand their ring diameter as they propagate as a result of a net radial force that adjacent "pearls" (azimuthal spots) exert on each other. Here, we revisit the self-trapped necklace beams and investigate their properties analytically and numerically. Specifically, we use two different approaches and find analytic expressions for the propagation dynamics of the necklace beams. We show that the expansion dynamics can be controlled and stopped for more than 40 diffraction lengths, making it possible to start thinking about interaction-collision phenomena between self-trapped necklaces and related soliton effects. Such self-trapped necklace-ring beams should also be observable in all other nonlinear systems described by the cubic (2+1)D nonlinear Shrodinger equation-in almost all nonlinear systems in nature that describe envelope waves.