We perform phenomenological modelling of nonlinear harmonic generation in free electron laser (FEL) experiments. We model gradual saturation of the harmonic power and its evolution around the saturated regime. The model includes all major losses, accounted for each harmonic individually. This reflects the fact that the undulator radiation (UR) spectral line and the electron-wave interaction at high harmonic wavelengths is more sensitive to the losses, such as the beam energy spread, divergence etc. The results of the phenomenological modelling are compared with the experimental data and with numerical simulations. The proposed description of nonlinear harmonic generation appears in very good agreement with the FEL experiments SALCA, LEUTL, LCLS and other in various conditions. The results also match numerical simulations; differently from the latter, our model allows fast, still accurate modelling of FEL harmonic power and bunching evolution with a PC or even an engineering calculator. The model allows studying FEL radiation from undulators with arbitrary magnetic field configuration. With the help of the analytical FEL model we demonstrate that, contrary to the helical undulator, in whose spectrum the fundamental tone and the 2nd harmonic prevail, the FEL radiation from the elliptic undulator with the 3rd field harmonic has distinct 5th UR harmonic besides the fundamental, while other harmonics are negligible even close to the saturation length.
41.60.Cr Free-electron lasers
41.60.Ap Synchrotron radiation
$^1$Department of Theoretical Physics, Faculty of Physics, M.V.Lomonosov Moscow State University,