with beam wavefront distortions was only recently reported in workshops [@WORK1; @WORK2], where the impact of wavefront errors on the EEHG performance was discussed, based on numerical simulations, in the case of the highest target harmonic at $13$ nm. Results of [@WORK1; @WORK2] are consistent with our analysis, which has been performed purely analytically.

The suppression of the output signal due to phase variations in space seems somehow in contrast with the effects of phase variations in time, where phase errors affect the temporal quality of the output radiation, but not the FEL output power. From this viewpoint, it should be noted that the radiation field is characterized by notions such as temporal and spatial coherence. The transverse coherence of FEL radiation develops automatically, without laser seeding. This happens due to transverse eigenmode selection: due to different gains of the FEL transverse eigenmodes, only one survives at the end of the FEL process. The coherence time is defined by the inverse FEL amplification bandwidth. For conventional soft X-ray FELs the typical amplification bandwidth is much wider than the Fourier transform limited value corresponding to the radiation pulse duration, meaning that the coherence time is much shorter than the pulse duration. 