factor $M_t^2$ departing from unity of no more than about $10^{-6}$.

One can think that the above-discussed constraints on seed laser may be true only for the particular case of EEHG. However, we can show that these constraints are actually of more general validity. For example, HGHG schemes can produce nearly transform-limited radiation spanning down to wavelengths of $1$ nm only under the same restrictions on temporal quality of the seed Ti:Sa laser. The key advantage of the EEHG scheme is that the amplitude of the achieved microbunching factor slowly decays with increasing harmonic number and that, consequently, generation of coherent soft X-ray emission within a single upshift stage becomes possible [@EEHG1; @EEHG2]. However, considering constraints on the seed laser $M_t^2$ factor, all harmonic generation schemes are similar, and must obey the universal result

$$\begin{aligned}
M_t^2-1 \lesssim \frac{1}{N^2}~. \label{unires}\end{aligned}$$

The requirement in the inequality (\[unires\]) can be somehow relaxed if the requirement of near-Fourier limit is relaxed as well. For example, the operation of a EEHG FEL is characterized by two microbunch bandwidth scales of interest. One is associated with inverse electron bunch duration $\Delta \omega_b = 1/\tau_b$, $\tau_b$ being the electron bunch duration. The other is the FEL amplification 