Quantum kinetic theory of free-electron lasers in external fields of arbitrary amplitude

The quantum kinetic equation method is used outside the framework of the theory of perturbations of the amplitude of an external field to calculate the gain and saturation of a freeelectron laser with a collinear low-current relativistic beam and two circularly polarized electromagnetic waves in vacuum. An analysis is made of the mechanism of saturation of the useful signal in an undulator free-electron laser. The saturation amplitude of the signal is found by direct quantum-kinetic calculations and also on the basis of the Klein–Gordon equation. The two sets of results are shown to agree apart from a constant factor. An exact allowance for the external field cannot be reduced to a simple replacement of the real mass of an electron with its effective mass. This is manifested by a difference between the formula for the wavelength of the useful signal emitted by a free-electron laser and the wavelength formula known from the published literature, as well as by the presence of an additional contribution to the saturation coefficient.