High harmonic generation in the undulators for free electron lasers

We present the analysis of the undulator radiation (UR) with account for major sources of the spectral line broadening. For relativistic electrons we obtain the analytical expressions for the UR spectrum, the intensity and the emission line shape with account for the finite size of the beam, the emittance and the energy spread. Partial compensation of the divergency by properly imposed weak constant magnetic component is demonstrated in the analytical form. Considering the examples of radiation from single and double frequency undulators, we study high harmonic generation with account for all major sources of homogeneous and inhomogeneous broadening with account for the characteristics of the electrons beam. We apply our analysis to free electron laser (FEL) calculations and we compare the obtained results with the radiation of a FEL on the supposition of the ideal undulator.