Free-Electron Lasers in the X-Ray Region: The Quantum Regime

The quantum regime for a free-electron laser is defined as that in which the energy lost by an electron upon emission of a photon is larger than the homogeneous broadening due to the finite interaction time. We anticipate that soft x-ray free-electron lasers pumped by an optical wave (rather than a permanent-magnet wiggler), will operate in that regime. The small-signal gain in this regime is given by an expression which is explicitly quantum-mechanical.