Low-frequency spontaneous radiation in a free-electron laser in the trapped regime.

Abstract

The trapped regime of a free-electron laser is considered in terms of the relativistic quantum-mechanical Klein-Gordon equation in the presence of the undulator and the laser field. The electron which is trapped in the wells of the ponderomotive potential occupies discrete equidistant levels. Transitions between these quantum-mechanical levels are shown to lead to the well-known classical sidebands. As a new feature in spontaneous emission we obtain a low frequency \ensuremath{\delta}\ensuremath{\omega} emitted slightly off axis, where \ensuremath{\delta}\ensuremath{\omega} denotes the shift between the first sideband and the small-signal laser frequency. The low frequency is again classical and constitutes a sideband of the zeroth harmonic \ensuremath{\omega}=0. Under current experimental conditions this low-frequency spontaneous emission is at the borderline of detectability.