An exact solution is given for the canonical model equations describing the electromagnetic fields on a free-electron-laser amplifier comprised of a high-quality relativistic electron beam propagating along the axis of a helical pump magnetic field. The case of a cold beam is analyzed in detail and it is shown that, depending upon the parameters, one can have gain when the system is stable, owing to constructive spatial interference of modes, or exponential gain when it is unstable. Gain curves, line shapes, and linewidths are presented for conditions of reported and planned experiments. Reasonable agreement is found with the reported experiments.

• Received 12 April 1979

DOI:https://doi.org/10.1103/PhysRevA.20.1661