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The input power-induced transformation of the transverse intensity profile atthe output of graded-index multimode optical fibers from speckles into abell-shaped beam sitting on a low intensity background is known as spatial beamself-cleaning. Its remarkable properties are the output beam brightnessimprovement and robustness to fiber bending and squeezing. These propertiespermit to overcome the limitations of multimode fibers in terms of low outputbeam quality, which is very promising for a host of technological applications.In this review, we outline recent progress in the understanding of spatial beamself-cleaning, which can be seen as a state of thermal equilibrium in thecomplex process of modal four-wave mixing. In other words, the associatednonlinear redistribution of the mode powers which ultimately favors thefundamental mode of the fiber can be described in the framework of statisticalmechanics applied to the gas of photons populating the fiber modes. On the onehand, this description has been corroborated by a series of experiments bydifferent groups. On the other hand, some open issues still remain, and weoffer a perspective for future studies in this emerging and controversial fieldof research.

On spatial beam self-cleaning from the perspective of optical wave thermalization in multimode graded-index fibers