10^{-4}$.

To complete the picture, we should note that an alternative method to harmonic generation setups, called self-seeding [@SELF; @TREU; @MARI], is available, and allows for the generation of temporally coherent radiation in XFELs. A self-seeded soft X-ray FEL consists of two undulators separated by a monochromator installed within a magnetic chicane. The remarkable temporal quality of the output radiation and the wavelength tunability of self-seeding schemes has stimulated interest in using this technique to generate nearly transform-limited soft- X-ray pulses. A project of self-seeding schemes with grating monochromator is now under development at LCLS II [@LCLS2]-[@FENG]. EEHG output will compete with self-seeding output only when the temporal quality of the seed laser beam obeys the mores stringent requirement (\[unires\]).

\[sub:self\] Self-phasing and constraints on field amplitude variation
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The seeding pulse from the Ti:Sa laser must necessarily propagate through vacuum window and BBO crystals without experiencing temporal phase distortions. Above a power density of $1 \mathrm{GW/cm^2}$, the refractive index $n$ becomes intensity-dependent according to the well-known expression

$$\begin{aligned}
n = n_0 + n_2 I ~, \label{refrin}\end{aligned}$$

where $n_0$ is the index of refraction at low intensity and $I$ is the laser intensity. Due to temporal variations of the laser 