Random excitation of intense periodic highly-localized single-cycle light pulses in a stochastic background by continuous-wave stimulated Brillouin scattering in long optical fibers with weak feedback is found experimentally. Events with low period numbers are dominant and the optical feedback is crucial for the phenomenon. A three-wave coupling model for the phenomenon is proposed. The results are in good qualitative agreement with the observed phenomenon. The latter should be relevant to the understanding of similar rogue wave events in other nonlinear dissipative systems.
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