Harmonics generation in experiments with free-electron lasers in the X-ray wavelength range: a theoretical analysis

The generation of undulator X-ray radiation harmonics in free-electron lasers (FELs), which has been observed in different experiments, has been given an analytical explanation. Expressions for spectrum lines and radiation intensity are written in explicit form in terms of generalized Bessel and Airy functions with regard to the spreads of electron energy, electron beam size, emittance, spectrum line splitting, and magnetic field constant components. The theory proposed here adequately explains the spectrum and intensity of radiation harmonics in all experiments with a SACLA FEL, a Linac coherent light source (LCLS) based on an FEL, PAL-XFEL, SwissFEL, and other equipment conducted under different conditions in a wide wavelength range, including hard X-ray radiation.