Crucial aspect of beam breakup in a steady-state free-electron laser in the microwave regime

The beam breakup instability caused by the interaction between beam and induction gaps in a steady-state free-electron laser in the microwave regime is considered. The large energy spread induced by free-electron laser performance is theoretically proved not to lead to Landau damping of the beam breakup instability when the synchrotron frequency is of order or larger than the betatron frequency.