  We predict that two electron beams can develop an instability when passing
through a slab of left-handed media (LHM). This instability, which is inherent
only for LHM, originates from the backward Cherenkov radiation and results in a
self-modulation of the beams and radiation of electromagnetic waves. These
waves leave the sample via the rear surface of the slab (the beam injection
plane) and form two shifted bright circles centered at the beams. A simulated
spectrum of radiation has well-separated lines on top of a broad continuous
spectrum, which indicates dynamical chaos in the system. The radiation
intensity and its spectrum can be controlled either by the beams' current or by
the distance between the two beams.
