## 2 Experiment
### Scheme for the Formation of QPM Conditions in Plasma
CL, cylindrical lens; PHB, picosecond heating beam; MSM, multi-slit mask; MJP, multi-jet plasma; T, target; SL, spherical lens; FDB, femtosecond driving beam; HB, harmonic beam; XUVS, extreme ultraviolet spectrometer. Inset: The extended, six-, four-, and single-jet plasma images are shown from the top to bottom. PS, plasma shapes. relation for QPM (\(N_{\rm e}\approx 1\times 10^{18}\)/\(L_{\rm coh}\times H_{\rm qpm}\)), where (\(H_{\rm qpm}\)) is the maximally enhanced harmonic of the NIR radiation, and \(L_{\rm coh}\) is the coherence length, which is actually the length of single plasma jet [27]. This approach of determining the electron density provided almost similar value (\(\sim\!1\times 10^{17}\) cm\({}^{-3}\)), considering the uncertainties of some parameters. The plasma plume had a cone shape with angle of divergence of 20\({}^{0}\). The delay between driving and heating pulses was chosen from the point of view of maximal harmonic yield, which means that the driving femtosecond pulse propagated through the most \"optimal\" part of the plasma plume. The meaning of \"optimal plasma\" refers to the complex conditions when different impeding processes play a minimal role in the formation of the maximally favorable conditions for HHG. The fundamental radiation (810 nm or tunable NIR radiation) and its second harmonic, which was produced by propagating the fundamental beam through 0.3-mm thick (BBO) crystal installed inside the vacuum chamber, were focused onto the MJP. The same BBO was used to double both 810 nm and 1320 nm radiation at appropriate phase matching angle (29\({}^{0}\) and 21\({}^{0}\), respectively). The duration of the driving pulses was sufficiently long to provide the phase matching bandwidth in this crystal for both wavelengths. The position of the focused femtosecond beam was approximately 0.15 mm away from the targets surface. The Rayleigh range of femtosecond beam was 4 mm, i.e., close to the whole length of plasma, thus assuming that the driving beam propagated through the 5 mm long plasma at the condition of almost plane wave-plasma interaction, which was exactly simulated during this study. Thus, the role of Gouy phase was neglected in this study. The peak intensities of 810 nm and 1320 nm pumps in the plasma area were \(5\times 10^{14}\) and \(1.4\times 10^{14}\) W cm\({}^{-2}\), respectively. In the case of the TCP comprising of 1320 and 660 nm waves, due to the group velocity dispersion in type-I BBO crystal, the 1320-nm pulse was delayed by 26 fs with respect to the second harmonic pulse when using 0.3-mm-long BBO crystal. This delay is obtained from (\(\Delta_{\rm cryst}=d[(n^{\rm o}\omega)_{\rm group}/\rm c-(n^{\rm e}2\omega)_{ \rm group}/\rm c]\), where \(d\) is the crystal length, \(c/(n^{\rm o}\omega)_{\rm group}\) and \(c/(n^{\rm e}2\omega)_{\rm group}\) are the group velocities of the \(\omega\) and \(2\omega\) waves in the BBO crystal, \(\rm c\) is the light velocity, and \(n^{\rm o}\omega\) and \(n^{\rm e}2\omega\) are the refractive indices of the \(\omega\) (1320 nm) and \(2\omega\) (660 nm) waves. The duration of second harmonic pulse at the output of the 0.3-mm-long BBO crystal has a value of \(\sim\!90\) fs assuming the application of 1320 nm, 70-fs driving pulses. This value was estimated using \(t_{2\omega}\approx[(\Delta_{\rm cryst})^{2}+0.5(t_{\omega})^{2}]^{1/2}\)[28]. Hence, the 660 nm beam has longer pulse duration, corresponding to the induced delay and a certain percentage (\(\sim\!\!50\%\)) of the fundamental pulse duration. The latter results from the fact that the energy of the fundamental radiation is in general not high enough in the pulse wings to effectively generate the second-order harmonic. In the case of 810 nm driving pulses, the influence of group velocity dispersion is larger on the temporal overlap of this radiation and its second harmonic in the plasma area. As it is mentioned above, the group velocity dispersion in the BBO crystal leads to some temporal separation of two pulses.