Harmonic radiation in a double-frequency undulator with account for broadening

Undulator radiation (UR) was studied taking the broadening of spectral lines via homogeneous and nonhomogeneous contributions into account. Exact analytical expressions for the UR spectrum and intensity in the case of relativistic highenergy electrons were obtained. Analytical expressions that characterize the interrelation between the constant components of the magnetic field in undulators and the electron beam divergence within them were derived. These expressions suggest that this divergence may be partially compensated. The radiation of a doublefrequency undulator and the higher harmonic radiation in an ordinary planar undulator were investigated taking the broadening of spectral lines into account. We discuss examples of UR in actual devices and demonstrate that the developed method and the results may be used in the calculation of induced UR.