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A numerical study of the effect of betatron oscillations on the second harmonic generation in free-electron lasers (FELs) is presented. Analytical expressions for the effective coupling strength factors are derived that clearly distinguish all contributions in subharmonics and each polarization of the radiation. A three-dimensional time-dependent numerical FEL code that takes into account the main FEL effects and the individual contribution of each electron to the second harmonic generation is presented. Also, the X- and Y-polarizations of the second harmonic are analyzed. The second harmonic was detected in experiments at the Advanced Photon Source (APS) Low Energy Undulator Test Line (LEUTL) and Linac Coherent Light Source (LCLS) in the soft X-ray regime. The approach presented in the article can be useful for a comprehensive study and diagnostics of XFELs. In the paper, the LCLS and Pohang Accelerator Laboratory X-ray Free-Electron Laser (PAL-XFEL) experiments are modeled. The simulation results are in a good agreement with the experimental data.

The harmonic power and bunching evolution in X-ray single-pass free-electron lasers (FELs) is modelled and the harmonic generation in a phase-shifted two-frequency FEL is explored. The advanced phenomenological FEL model, which is validated numerically and experimentally, is employed. The model accounts for major losses for each harmonic individually; it is compared with reported experimental data and with PERSEO numerical simulations, which are performed here for a variety of experiments. The latter cover the radiation wavelength range 0.15–300 nm. The phenomenological description is based on a few key FEL parameters: electron beam section, current, energy and its spread and divergence. The model is employed for modelling harmonic bunching and power evolution in a phase-shifted X-ray FEL with a two-frequency undulator, where lower harmonics with numbers less than nth are suppressed by the electron–photon phase shift of kπ/n, k = 2, 4, …, between the undulator sections. The benefits of the two-frequency phase-shifted FEL are highlighted. FEL-induced energy spread is shown to be three times lower than in a FEL without the phase-shift. The high-power harmonic and sub-harmonic radiation in such a FEL is demonstrated. In particular, powerful ∼14 GW X-ray radiation at λ5 = 0.15 nm from electrons with energy of 5.47 GeV and beam current ∼3.66 kA is possible in a two-frequency phase-shifted FEL at 30 m; this constitutes half of a FEL length where a common planar undulator radiates the same wavelength and power at the fundamental harmonic. Moreover, about a three times lower energy spread is induced by the dominant fifth harmonic, and the harmonic power can be thousands of times higher than in a common planar undulator FEL.