Self-focusing and nonlinear periodic beams in parabolic index optical fibres

The existence of exact nonlinear and periodic propagation modes in graded index nonlinear optical fibres that reproduce the periodic self-imaging property of the linear propagation regime is analytically proved. The stability of these periodically focused modes, which are constructed as self-similar solutions of the fibre nonlinear eigenmodes, is studied by means of standard linear stability analysis and numerical beam propagation. Self-similar solutions constructed from the fundamental nonlinear fibre mode are found to be linearly stable, whereas symmetry-breaking instabilities are found for higher-order nonlinear modes.