  Since its first demonstration in graded-index multimode fibers, spatial beam
self-cleaning has attracted a growing research interest. It allows for the
propagation of beams with a bell-shaped spatial profile, thus enabling the use
of multimode fibers for several applications, from biomedical imaging to
high-power beam delivery. So far, beam self-cleaning has been experimentally
studied under several different experimental conditions. Whereas it has been
theoretically described as the irreversible energy transfer from high-order
modes towards the fundamental mode, in analogy with a beam condensation
mechanism. Here, we provide a definitive theoretical description of beam
self-cleaning, by means of a semi-classical statistical mechanics model of wave
thermalization. This approach is confirmed by an extensive experimental
characterization, based on a holographic mode decomposition technique,
employing laser pulses with temporal durations ranging from femtoseconds up to
nanoseconds. An excellent agreement between theory and experiments is found,
which demonstrates that beam self-cleaning can be fully described in terms of
the basic conservation laws of statistical mechanics.
