bandwidth $\Delta \omega_a$. One can relax the requirement of near-Fourier limit substituting it by the requirement to achieve an output radiation bandwidth narrower than the SASE bandwidth $\Delta \omega_a$. On the one hand, the product of bunch duration by amplification bandwidth can be estimated in the order of $\tau_b \Delta \omega_a \sim 10^2$ in the soft X-ray wavelength range. On the other hand, the FEL radiation bandwidth broadening due to the effect of linear frequency chirp is about $\Delta \omega \sim
|\alpha_m|/\tau_b$. Therefore, in the case when

$$\begin{aligned}
|\alpha_m| >   (\tau_b \Delta \omega_a) \sim 10^2 ~,\label{alpham}
\label{alpham}\end{aligned}$$

the output signal has a bandwidth larger than the SASE bandwidth, and harmonic generation techniques have no practical applications. However, if, for example, we have a microbunching chirp parameter $|\alpha_m| \sim 10$, the effective radiation bandwidth becomes ten times narrower than the SASE bandwidth, although is ten times wider compared to the ideal transform-limited bandwidth. Following this discussion, a weaker constraint on the temporal quality factor of seed laser is $M_t^2 - 1 < 10^2/N^2$. For a Ti:Sa laser seed and a radiation wavelength of $1$ nm it is possible to discuss about harmonic generation techniques applications only when $M_t^2 - 1 <
