Quasi-linear theory of multimode free-electron lasers with an inhomogeneous frequency broadening

assuming that the number of excited modes is large and their phases are random a quasi-linear theory of interaction of ultrarelativistic electron beam with the field of a high-quality cavity is developed. The saturation effects in FELs with an inhomogeneous frequency broadening are described. It is shown that in the case of very large excesses over threshold the efficiency of FELs tends to 50% but the emitted energy is distributed in a very wide spectral interval.