A one-dimensional linear analysis is presented of the startup of a free-electron-laser oscillator. The model treats electron-beam pulses of arbitrary shape and many significant three-dimensional effects are included heuristically. A closed equation is derived for the ensemble-averaged electromagnetic energy density matrix which contains the spontaneous emission as a source, and which represents the small gain per pass and losses via coefficient matrices. Numerical solutions are compared with the Stanford experimental results.

• Received 10 January 1983

DOI:https://doi.org/10.1103/PhysRevLett.50.1775