## 2 Materials and Methods  
Femtosecond pulses (50 fs, 806 nm, 2 mJ) from Ti:sapphire laser were focused inside the plasma plume to generate harmonics (Figure 1). The intensity of the driving pulses in the plasma area was maintained to be -2 \(\times\) 10\({}^{14}\) W cm\({}^{-2}\). The beamwaist diameter of the femtosecond driving pulses in the focal plane of the 400 mm focal length spherical lens was measured to be 90 \(\upmu\)m and the calculated Rayleigh length was 6 mm, which is notably larger than the sizes of target ablation.  
For the laser plasma formation, the 5 ns heating laser pulses (\(\lambda\) = 1064 nm, 5 mJ, 10 Hz) from Nd:YAG laser were used. The HP were focused by a 300 mm focal length spherical lens on the target surface at the spot with diameter 0.3 mm providing intensity of a few units of 10\({}^{9}\) W cm\({}^{-2}\) and fluence up to -7 J cm\({}^{-2}\). The lens used for focusing of heating pulses was approached to the target in such a way that the spot size on the target surface (0.3 mm) was notably larger that the beamwaist diameter in the focal plane (60 \(\upmu\)m). The variable defocusing was adjusted to keep the \"optimal\" fluence of heating radiation for each of used samples, which allowed us to achieve the maximal harmonic yield.  
The femtosecond pulses were delayed with regard to those from Nd:YAG laser using the digital delay generator to propagate through the formed plasma at maximal density of the ejected particles. The delay between pulses was tuned using the delay generator. The variable electronic delay range was equal to 0-10\({}^{6}\) ns. We analyzed plasma and harmonic dynamics from several nanoseconds to several hundred microsecond delays.  
Figure 1: Experimental setup of delayed excitation of nanosecond laser produced plasma for generation of high-order harmonics in carbon nanostructures contained plumes. DP: driving pulse; HP: heating pulse; LIP: laser-induced plasma; T: target; H: harmonic emission; XUVS: extreme ultraviolet spectrometer; CCD: charge-coupled device camera.  
The harmonics and plasma emission were detected using a flat field grazing-incidence XUV spectrometer.  
The pressed carbon-contained nanostructured targets were glued on microscopic glass slides fixed on a digitally controlled XYZ stage inside the target chamber. The powdered nanoparticles were placed in the press machine and pressed to form the 5 mm thick tablets of 10 mm diameter. We used the laser ablation fluence varied between 1 and 7 J cm\({}^{-2}\) depending on target properties. In each case, the conditions of ablation were chosen to achieve the maximal harmonic yield. The optimization of ablation conditions for each sample assumed a determination of the fluence. The plasma plume allowed the generation of harmonics at a specific delay between heating and driving pulses. The targets were moved around during HHG experiments to provide fresh surface for ablation. Using motorized rotating targets notably improved the stability of HHG from LIPs, particularly, in the case of powdered targets, and significantly minimized the modification of the target surface that could cause degradation of harmonic yield. The periodic change of the ablation zone allowed cooling down the heated area and maintained stable plasma formation. Six materials were used as the targets: bulk graphite (bulk C), C\({}_{60}\) powder, multi-walled carbon nanotube (MW CNT) powder (15 nm diameter; the length of tubes was varied in the range of 500 nm-20 \(\upmu\)m), carbon nanofibers (CNF, 100-200 nm diameter and a few tens micrometers length), diamond nanoparticles (DN, size 3-8 nm) and graphene (GR) wrapped sheets (sizes of sheets ~20 nm). All samples were purchased from Sigma-Aldrich (St. Louis, MO, USA).