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

Recently, we suggested a type of self-trapped optical beams that can propag ate in a stable form in (2 +1)D self-focusing Ken media: Necklace-ring beam s [M. Soljacic, S. Sears, and M. Segev, Phys. Rev. Lett. 81, 4851 (1998)]. These self-trapped necklaces slowly expand their ring diameter as they prop agate as a result of a net radial force that adjacent "pearls" (azimuthal s pots) exert on each ether. Here, we revisit the self-trapped necklace beams and investigate their properties analytically and numerically. Specificall y, we use two different approaches and find analytic expressions for the pr opagation dynamics of the necklace beams. We show that the expansion dynami cs can be controlled and stopped for more than 40 diffraction lengths, maki ng it possible to start thinking about interaction-collision phenomena betw een self-trapped necklaces and related soliton effects. Such self-trapped n ecklace-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.