Generation of soft X-ray radiation in a compact free-electron laser with harmonic multiplication

Single-pass free-electron lasers (FELs) with harmonic multiplication in the X-ray range are studied theoretically with the objective of producing intense radiation with the minimum possible FEL length, beam energy, and seed laser frequency. A cascade FEL with a two-frequency undulator for electron bunching at wavelengths of higher harmonics is proposed. The dynamics of radiation power in a FEL with harmonic multiplication cascades is investigated based on a well-proven phenomenological FEL model that covers all major losses for each harmonic in each FEL cascade. The effect of the beam quality on X-ray generation in a FEL is examined. It is demonstrated that electron beams with energy spread $\sigma_e\le$ 0.0002 are required for efficient use of harmonics in a cascade X-ray FEL. The evolution of power of harmonics in several cascade FELs at wavelengths of $\sim$ 2 – 3 nm with multiplication of seed harmonics at a wavelength of 13.51 nm (Mo/Si reflection peak) and 11.43 nm (MoRu/Be reflection peak) is studied. It is demonstrated that the power increases to $\sim$ 1 GW at a FEL length up to 40 m. In addition, the operation of multicascade FELs with a seed F$_2$ excimer 157-nm UV laser and harmonics of lasers with CO$_2$ and C$_2$H$_2$ at 30 nm with a beam of electrons with an energy of $\sim$ 0.6 GeV and a current of 1 kА is studied. The power of the modeled X-ray radiation of these FELs at $\lambda$ = 2.5 and 3.3 nm increases to $\sim$ 0.5 GW at $\sim$ 30 m.