do not affect the spatial quality of the FEL output radiation, which is the same for both perturbed and unperturbed wavefront cases. They only affect the input signal value at the target harmonic. However, because of the exponential dependence of the signal suppression factor on the wavefront errors, one obtains an appreciable FEL output only when phase errors are sufficiently small to give appreciable input signal. As a result, the seed UV laser beam must exhibit a nearly diffraction-limited wavefront in the waist plane, with very little phase variation. In particular for a target harmonic with wavelengths of about $1$ nm, the wavefront of the UV beam must be controlled to within a fraction of a degree across the electron beam area. These relatively small phase variations cause the signal at the entrance of the FEL amplifier to drop of a quantity of order of the ideal (diffraction-limited) performance. In contrast with phase variation in time, the spatial quality of UV seed laser beam can be improved by means of active optics and spatial filtering. However, these manipulations with laser beam usually cause significant losses in beam power.

To the best of our knowledge, the crucially important problem of seeding 