Bunching phase and constraints on echo enabled harmonic generation

A simple mathematical description is developed for the bunching spectrum in echo enabled harmonic generation (EEHG) that incorporates the effect of additional electron beam energy modulations. Under common assumptions, they are shown to contribute purely through the phase of the longitudinal bunching factor, which allows the spectral moments of the bunching to be calculated directly from the known energy modulations. In particular, the second moment (spectral bandwidth) serves as simple constraint on the amplitude of the energy modulations to maintain a transform-limited seed. We show that, in general, the impact on the spectrum of energy distortions that develop between the EEHG chicanes scales like the harmonic number compared to distortions that occur upstream. This may limit the parameters that will allow EEHG to reach short wavelengths in high brightness FELs.