Joint work with David Ianetz, Department of Mathematics, Bar-Ilan University and Holon Institute of Technology

In previous work, by one of us (DI) and other collaborators, a set of ordinary differential equations was derived to (approximately) describe the evolution of beam widths of an asymmetric Gaussian beam in nonlinear waveguides with cubic-quintic and saturable nonlinearities and a parabolic graded-index (GRIN) prole. Numerical solution of these equations revealed that the beam widths exhibit "beating" - the widths oscillate rapidly, but with a slowly varying amplitude. Two types of beating are observed. In type I breathing the amplitude of oscillation of the beam width in one direction remains greater than the amplitude of oscillation in the other direction. In type II, there is an interchange between the widths in the two transverse directions. As the parameters of the system and the initial beam eccentricity are changed there can be a change in the type of beating, which is characterized by the beating period becoming infinite.

We provide an analytic model to describe the beating phenomena, and, in particular, to study transitions between the types. The differential equations describing the beam widths are equations of motion of a 2 degree-of-freedom Hamiltonian system with a fixed point which is close to 1 : 1 resonance. We show how small oscillations near a fixed point close to 1 : 1 resonance can be approximated using an integrable Hamiltonian and, ultimately, by a single first order differential equation. In particular, the beating transitions can be located from coincidences of roots of a pair of quadratic equations, with coefficients determined (in a highly complex manner) by the internal parameters and initial conditions of the original system. The analytic approximation we develop (which is relevant for a wide class of Hamiltonians) is applied to the specific systems describing evolution of beam widths, and excellent agreement is found with numerics, over a wide range of parameter values and initial conditions.