Undulator and free-electron laser radiation with field harmonics and off-axis effects taken into account analytically

We present an analytic description of the generation of undulator radiation (UR) harmonics in a generalized elliptic undulator with field harmonics. The obtained analytic expressions for generalized Bessel and Airy functions describe the UR spectral lines and radiation intensity in the general and particular cases of two-frequency planar and helical undulators and also for other elliptic and planar undulators. We analytically describe the effect of finite electron beam size, beam emittance, off-axis deflection of electrons, electron energy spread, and beam-bending permanent magnetic components. The obtained analytic expressions distinguish the contributions made by each field component and by the beam and undulator parameters to the generation of radiation harmonics. Using an analytic model of single-pass free-electron lasers (FELs), we analyze the power evolution of harmonics in the LCLS, LEUTL, and SACLA FEL experiments. We discuss the effect that the beam and undulator parameters have on the generation of harmonics. It is shown that the strong second harmonics of radiation observed in experiments are caused by the detected beam deviation from the axis by $15~\mu$m over one gain length of 1.5 m in the X-ray LCLS FEL and by the large beam cross section of $\sim 200~\mu$m in the LEUTL FEL. The results of modeling are fully consistent with experiments, which confirms the validity of the presented theoretical description of FEL power and radiation spectral density.