Breathing and beating dynamics of Gaussian beam in planar graded-index absorbing nonlinear waveguides

The spatial dynamics of laser beams in absorbing planar waveguides with a parabolic index profile in a saturable or cubic-quintic medium are calculated using the "collective variable approach" technique. In the absence of losses, we construct diagrams which define regions of self-focusing and self-diffractive beam propagation for both types of media. It is found that propagating pulses exhibit an oscillatory pattern, similar to breathing behavior in homogeneous media. If the incident pulse spatial profile and the center of the index profile are not aligned, the pulse oscillates around the index origin with a beat frequency that depends on the graded index. Both the breathing and the beat frequencies are also calculated for other graded-index profiles, such as those with additional higher-power terms, and are found to be extremely sensitive to the index profile. In media with linear and nonlinear absorption, we demonstrate the difference between the breathing behavior in graded-index and homogeneous waveguides.