We show that a high-order Bessel beam propagating in a medium with nonlinearabsorption is not completely absorbed, but survives in the form of a newpropagation invariant vortex beam in which the beam energy and orbital angularmomentum are permanently transferred to matter and at the same time refueled byspiral inward currents of energy and angular momentum. Unlike vortex solitonsand dissipative vortex solitons, these vortex beams are not supported byspecific dispersive nonlinearities (self-focusing or self-defocusing) and donot require gain. Propagation invariance in presence of multiple absorption ofphotons carrying (possibly high) orbital angular momentum makes these beamsattractive for optical pumping of angular momentum over long distances.
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