  The input power-induced transformation of the transverse intensity profile at
the output of graded-index multimode optical fibers from speckles into a
bell-shaped beam sitting on a low intensity background is known as spatial beam
self-cleaning. Its remarkable properties are the output beam brightness
improvement and robustness to fiber bending and squeezing. These properties
permit to overcome the limitations of multimode fibers in terms of low output
beam quality, which is very promising for a host of technological applications.
In this review, we outline recent progress in the understanding of spatial beam
self-cleaning, which can be seen as a state of thermal equilibrium in the
complex process of modal four-wave mixing. In other words, the associated
nonlinear redistribution of the mode powers which ultimately favors the
fundamental mode of the fiber can be described in the framework of statistical
mechanics applied to the gas of photons populating the fiber modes. On the one
hand, this description has been corroborated by a series of experiments by
different groups. On the other hand, some open issues still remain, and we
offer a perspective for future studies in this emerging and controversial field
of research.
