  Beam self-imaging in nonlinear graded-index multimode optical fibers is of
interest for many applications, such as implementing a fast saturable absorber
mechanism in fiber lasers via multimode interference. We obtain an exact
solution for the nonlinear evolution of first and second order moments of a
laser beam carried by a graded-index multimode fiber, predicting that the
spatial self-imaging period does not vary with power. Whereas the amplitude of
the oscillation of the beam width is power-dependent. We have experimentally
studied the longitudinal evolution of beam self-imaging by means of femtosecond
laser pulse propagation in both the anomalous and the normal dispersion regime
of a standard telecom graded-index multimode optical fiber. Light scattering
out of the fiber core via visible fluorescence emission and harmonic wave
generation permits us to directly confirm that the self-imaging period is
invariant with power. Spatial shift and splitting of the self-imaging process
under the action of self-focusing are also emphasized.
