LAUSR

Abstract Analytical formalism for the description of free electron laser (FEL) spectrum is proposed with account for the electron beam size, divergences, beam energy spread, the off-axis deflection and the periodic harmonic and non-periodic constant magnetic field components in undulators. The exact analytical expressions for the spontaneous undulator radiation (UR) are obtained in terms of generalized Bessel and Airy functions, which describe the spectrum line shape and intensities. The obtained analytical formulae distinguish contributions of each installation and beam parameter in the FEL harmonic radiation. The enhanced phenomenological model is employed for the FEL modeling; it describes the harmonic powers evolution along the undulators and gradual saturation around the saturation length with account for the saturated powers oscillations. With its help we study the generation of FEL harmonics, including even, in several instances of FEL experiments LCLS, LEUTL and PAL-XFEL. We analytically demonstrate the even harmonics reasons in the LCLS and LEUTL FEL experiments. The spectrum line split due to the beam size is demonstrated analytically and compared with experimentally measured spectral characteristics. The PAL-XFEL experiment is analyzed and compared with LCLS experiment; the possibility of high harmonic radiation in it is explored. The modeling results fully agree with the measured values in the studied FELs. The analytical formalism shows physical background for each harmonic in every FEL. It allows the analysis of the radiation and beam alignment in projected, built and working FELs and it can help minimizing losses and correcting magnetic fields and projecting experiments with FELs.