Characterization of Soft X-ray Echo-Enabled Harmonic Generation Free-Electron Laser Pulses in the Presence of Incoherent Electron Beam Energy Modulations

Echo-enabled harmonic generation free-electron lasers (EEHG FELs)are promising candidates to produce fully coherent soft x-ray pulses by virtue of efficient high harmonic frequency up-conversion from UV lasers.The ultimate spectral limit of EEHG, however, remains unclear, because of the broadening and distortions induced in the output spectrum by residual broadband energy modulations in the electron beam. We present a mathematical description of the impact of incoherent(broadband) energy modulations on the bunching spectrum produced by the microbunching instability through both the accelerator and the EEHG line. The model is in agreementwith a systematic experimental characterization of the FERMI EEHGFEL in the photon energy range $130-210$ eV. We find that amplification of electron beam energy distortions primarily in the EEHG dispersive sections explains an observed reductionof the FEL spectral brightness that is proportional to the EEHGharmonic number. Local maxima of the FEL spectral brightness and of the spectralstability are found for a suitable balance of the dispersive sections'strength and the first seed laser pulse energy. Such characterizationprovides a benchmark for user experiments and future EEHG implementations designed to reach shorter wavelengths