Superradiative instability of an electron cluster moving in an undulator field

Superradiative instability developing in a cluster of electrons moving in an undulator field or in an electromagnetic pump wave field is considered. The instability is shown to have no threshold and the instability increment is estimated. The development of instability results in the bunching of particles in the cluster and their coherent radiation at frequencies nearly equal to the frequencies of individual radiation (scattering) of particles. When the translational velocity of the cluster is close to the velocity of light, the frequency of the high-frequency (HF) component of radiation (along the translational motion) may be much higher than the oscillation frequency of particles.