  We present a general theory of thermalization of light in multimode optical
fibers, including optical beams with nonzero orbital angular momentum or vortex
beams. A generalized Rayleigh-Jeans distribution of asymptotic mode composition
is obtained, based on the conservation of the angular momentum. We confirm our
predictions by numerical simulations and experiments based on holographic mode
decomposition of multimode beams. This establishes new constraints for the
achievement of spatial beam self-cleaning, giving previously unforeseen
insights into the underlying physical mechanisms.
