slippage to the electrons. The radiator is composed by a sequence of undulators tuned to the desired output wavelength. This final section is similar to that used for SASE FELs. However it is shorter, and produces coherent radiation only because the beam has been coherently prebunched. The seed laser is assumed to be tuned at $200$ nm (or $270$ nm) corresponding to the fourth harmonic (or the third) of a Ti:Sapphire laser.

Limitations on the performance of EEHG schemes related with electron beam dynamics issues, as the beam goes through the various undulators and chicanes, has been extensively discussed in literature [@EETOL1; @EETOL2; @EETOL3] and goes beyond the scope of this paper. Here we will focus our attention on the first part of the harmonic generation process, discussing the constraints on the seed laser performance needed for reaching wavelengths of about $1$ nm. In fact, in chirped-pulse amplification systems (CPA) systems, both temporal and spatial quality of the beam can be degraded due to the propagation through the optical components, non-linear effects or inhomogeneous doping concentration in the amplifying media, and thermal effects linked to the pumping process. In particular, the aim of this work is to evaluate the impact 