Theoretical spectral analysis of FEL radiation from multi-harmonic undulators

A theoretical study of the spontaneous and stimulated undulator radiation (UR) from electrons in undulators with multiple periods in both transversal directions is presented. Exact expressions are derived for the UR intensities in terms of the generalized Bessel and Airy functions, accounting for undulator field harmonics of arbitrary strength and for real parameters of the beams and installations. Theoretical results are verified with numerical and experimental data for SWISS-XFEL, PAL-XFEL, LEUTL, LCLS etc. The spectrum, UR line shape and width, and the harmonic evolution along the undulators are analyzed and compared with the available data for these experiments. Moreover, the effect of the field harmonics is elucidated. It is demonstrated that the third field harmonic can cause distinct odd UR harmonics. The asymmetric undulator field configuration is identified, which allows intense radiation of these harmonics. The power evolution in a free-electron laser (FEL) with such an undulator is studied by means of an analytical FEL model. The latter is enhanced by a true description of the gradual power saturation of harmonics. A FEL with elliptic undulator and electron-photon phase-shifting is proposed and modeled. It is shown that the resulting harmonic power from the phase-shifted elliptic undulator can be significantly higher than from a planar undulator with the same phase-shifting.