Theory of a Free-Electron Laser

A classical, linear theory is given for the gain of a short-wavelength free-electron laser. Waves propagating along the relativistic beam are unamplified when the beam is cold. Two of the six geometric optics modes have phase velocities $<c$. When thermal effects are included, the theory predicts the possibility of substantial amplification of these two modes for asymmetric electron distribution functions, due to wave-particle resonance.