Microwave instability as a coherent light source

We suggest that the coherent radiation observed recently at SURF II and the National Snychrotron Light Source vacuum ultraviolet ring is due to coherent microwave instability or, equivalently, to {open_quotes}microbunching{close_quotes} of the electron beams in the storage rings. We formulate in this paper the problem of microwave instability in the time domain. A linear homogeneous integro-differential equation for the perturbed current distribution is derived to describe the microwave coherent motion inside the electron bunch. For a specific band-limited high-frequency impedance, the equation can be diagonalized analytically and the eigensolution manifests explicitly the characteristics of microbunching. Coherent radiation power is also calculated for this solvable model, assuming the instability to be initiated by the shot noise inherent in the electron beam. {copyright} {ital 1998} {ital The American Physical Society}