We analyzed propagation of elliptical Gaussian beams in nonlinear non-graded waveguides and waveguides with a parabolic graded-index (GRIN) profile. To describe the nonlinearity, we considered both saturable and cubic-quintic models. Besides, we took into account both linear and nonlinear absorption.

 

For lossless media, we constructed diagrams which define regions of self-focusing and self-diffractive beam propagation for both models in non-graded and GRIN waveguides. The widths of the propagating elliptic beam exhibit "breathing" and "beating" behavior. Two types of beating oscillations are observed in both models.

 

We calculated the dependence of the ratio of the "beating" to "breathing" oscillation periods on the beam ellipticity and the GRIN. We have detected a remarkable difference in this dependence between saturable and cubic-quintic media: in the saturable model, the ratio is a monotonic function of ellipticity, whereas in the cubic-quintic model, it is characterized by singularities, which correspond to transitions between the types of beat oscillations.

 

This allows one to identify the type of nonlinearity of the medium and determine parameters of high order nonlinearity. For lossy media, we discuss the difference between the breathing behavior in non-graded and GRIN waveguides.