## 2 Numerical Section
The same features shown in [PERSON] _et al._'s [9] article have been observed. The HHG is barely amplified and is preceded and followed by strong ASE. Most of the output energy is contained on the post-HHG ASE. The duration of each spike (about 150 fs) is physical while for [PERSON]'s experiment it was convolved by the streak camera resolution (40 ps). Such behavior (very strong post-HHG ASE) was not observed on previous seeding experiments ([PERSON] _et al._[10] and [PERSON] _et al._[11] experiments). In order to clarify this issue, we concentrated our attention on the interaction of the HHG with the amplifier. As expected, the HHG saturates the amplification (gain drops to near zero). However, the gain is recovered extremely fast, in about 2 ps, allowing spontaneous emission to be amplified. To maintain the coherent amplification inhibiting spontaneous emission, the simplest solution would be to seed a new HHG pulse every 2 ps (or less). In this case, modeling shows a very different picture from the original experiment (Figure 6b). The amplified HHG pulses contain 5 mJ while the ASE drops to 0.1 mJ. Although it is possible to generate a HHG pulse train [27], it seems barely realistic to consider synchronizing over 100s of pulses. We then shifted from a discrete multiple-pulse seed to a single-pulse but continuous seed by seeding a 100s of ps HHG pulses. In this case, we consider as amplifier the plasma reported in a recent experiment on zinc soft X-ray laser (21.2 nm) since it exhibited the highest energy beam (12 mJ) reached to-date. In this case the gain lasts about 200 ps,  
Figure 6: Intensity of the amplified HHG (red) and ASE (blue) at different plasma lengths in **(a)** [PERSON]’s experiment; **(b)** when seeding a train of 100 HHG (only ten HHG are shown); and **(c)** when a stretched HHG matching the duration of the gain is seeded into the plasma. meaning that the seed must have a duration of 200 ps (and an energy of 1 uJ in-band). The result is very positive: the coherent amplified pulse has 12 mJ after 10 mm of plasma while the ASE remains negligible at 0.1 uJ (Figure 6c). Still, the problem with this scheme is the realization of a seed with a duration of hundreds of picosecond, as it is well-known that HHG pulses are produced by using ultrashort infrared pulses from few fs to 1 ps, leading to sub-ps duration at maximum. We thus propose to transpose the Chirped Pulse Amplification technique [28] to the soft X-rays. The full architecture is displayed in Figure 7 showing the seed HHG, a pre-amplifier followed by a soft X-ray stretcher (grating pair + telescope) the main plasma and the soft X-ray compressor (grating pair). Compressor efficiency as high as 50% is realistic if using a single-pass compressor and off-axis gratings [12,29]. After recompression, a 6 mJ, 200 fs, fully coherent in space and in time soft X-ray laser may be produced. The pulse duration has been estimated considering the spectral narrowing as well as the chirp acquired during the amplification. Compensation of quadratic phase terms by the compressor is not included. The required IR pump laser energy reaches 100 J but in sub-ns pulses, that is commercially available with repetition-rate of 1 shot per minute. Figure 7: Schematic description of a high energy X-ray Chirped Pulse Amplification (CPA) chain. The same laser is split into three beams to generate the HHG, to pump the pre-amplifier using the Transient Collisional Excitation Scheme (TCE) scheme and to pump the main amplifier with ns duration. Note that the stretcher is placed after the pre-amplifier to keep the intensity high enough for efficiently seeding the different plasmas. This scheme is fully scalable to higher or lower energies. In that last case, pre-amplifier is not necessary.