X-Ray Free-Electron Lasers with Variable Deflection Parameter of Undulators

The paper deals with X-ray free-electron lasers (FEL) with the variable deflection parameter of undulators with the electron buncher and harmonic amplifier. The analysis is given to the power evolution in the harmonic lasing self-seeded free electron laser (HLSS FEL), which is compared with the self-amplified spontaneous emission (SASE). Exact analytic expressions are used for Bessel coefficients in real beams and undulators as well as the analytic description of the FEL power evolution. The latest FELs with the variable deflection parameter are considered herein, namely: FLASH-II, European XFEL, SACLA and SwissFEL. A comparative analysis is presented for the FEL operation in the HLSS and SASE modes. The results of the analytical simulation of the FEL spectral parameters are in good agreement with the experimental data. It is shown that the HLSS FEL power grows more rapidly than during the self-amplified spontaneous emission at the same wavelength. The obtained results are compared with the experimental data on the FLASH-II FEL. It is found that the electron beam in the SACLA X-Ray FEL, which has a large energy spread, does not allow to use harmonic cascade amplification. As for the SwissFEL, the small spread in energy and the beam emittance provide an effective electron bunching on the second-harmonic wavelength. This allows reaching the HLSS FEL saturation power 10 m earlier than in the SASE, the radiation power and wavelength being constant. The calculations show that the effective harmonic amplification is also possible in the European XFEL.