consists in the third or in the fourth harmonic of the Ti:Sa laser beam. Usually, laser frequency multipliers are based on the use of Beta Barium Borate (BBO) crystals. The effect of frequency multiplication on phase variation amounts again to multiplication of the phase variations. Therefore we may say that when we study constraints on the performance of Ti:Sa seed laser for EEHG schemes, the total frequency multiplication chain consists of two stages. The first stage is the BBO crystals with a frequency multiplication factor $N_1 =3$ (or $N_1 =4$). The second stage is the EEHG setup itself, with frequency multiplication factor up to $N_2 \sim 270$ (or $N_2
\sim 200$). If the final required output radiation is around wavelengths of $1$ nm, the total frequency multiplication factor $N
= N_1 N_2$ is about $N \sim 800$.

From the previously discussed condition $\alpha \lesssim 1/N$ it can be seen that the Ti:Sa laser produces nearly transform-limited microbunching at wavelengths around $1$ nm only when the laser chirp parameter $\alpha \lesssim 10^{-3}$. Thus, for most purposes, if the total multiplication factor is around $800$ or exceeds it, we may formulate the constraint on the Ti:Sa laser quality by requiring a quality 