We show that a non-Hermitian modulation of the potential along the nonlinear multimode fibers controls dynamics of propagating radiation. Specifically we consider simultaneous modulation of the refraction index and gain/loss profile. We predict and observe that the non-Hermitian modulation introduces a unidirectional and controllable coupling towards the lower/higher order transverse modes, depending on the potential parameters. Such effect may enhance the beam self-cleaning phenomena, i.e. improve the spatial structure of light in propagation. On the contrary, coupling towards higher order modes may enhance pulsing, turbulence and, eventually help in super-continuum generation.
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