## 2 Numerical Section
Also, the long lasting (several picosecond) wake originates from the long duration of the polarization induced by the seed. Indeed, depolarization mainly occurs through electron-ion collisions that have a characteristic time of several picoseconds. Therefore, [PERSON] and collaborators proposed and tested to seed a longer HHG beam (200 fs) in high-density plasma that presents a faster depolarization timescale and thus a shorter wake [18]. ARWEN modeling showed that plasmas larger than 100 \(\upmu\)m have the adequate conditions. Seeding with this parameters (and 50 nJ energy to maintain the seeding intensity) a 100 \(\upmu\)m wide plasma enhances the coupling HHG-plasma. The combination of the fast depolarization of the plasma (due to electron collisions) and the strong saturation of the population inversion by the HHG field inhibits the creation and amplification of the wake, conserving the femtosecond duration of the seed. Furthermore, as observed on classical lasers [25, 26], amplification in this deep-saturation regime is only efficient during the first part of the high harmonic pulse while the second half of the pulse encountering a reduced population inversion is weakly amplified. This phenomenon results in the pulse shortening observed for lengths above 2 mm (Figure 3 right). This scheme leads to the generation of a 1.4 \(\upmu\)J, 80 fs pulse after passing through a 2.5 mm length plasma (Figure 3). It is not worth to use longer plasmas as the ASE and the wake are strongly developed after this optimal length. Figure 3: Evolution of the intensity _versus_ time along the 30 \(\upmu\)m (**left**, high harmonic generation (HHG) = 30 fs) and 100 \(\upmu\)m width (**right**, HHG = 200 fs) plasma columns. The green, red and blue lines stand for the ASE, seed and wake fields. The vertical scales are not constant to account for amplification. When considering direct amplification in a 1 mm large plasma, we observe that optimal seeding conditions require using 1 \(\upmu\)J seed. However, even for this high-density plasma, the spectral width of the soft X-ray laser is about 30 times narrower than the spectral width of the HHG, meaning that we need a 30 \(\upmu\)J HHG beam. Such energy is currently unrealistic. However, 100 \(\upmu\)m plasma seeded with HHG delivers a beam with optimal properties (1.4 \(\upmu\)J, 80 fs and similar bandwidth) as the input of the 1 mm one (Figure 4). At the end of the main amplifier (1.5 mm long, 1 mm large), the pulse is totally dominated by the amplified coherent HHG seed, reaching at best 18 \(\upmu\)J in 80 fs (Figure 5). About 10 J is necessary to create the main plasma, which is achievable using commercial, 10 Hz lasers. Figure 4: Artistic view of an amplification chain starting, from left to right, with the HHG source, focusing optics, pre and main amplifiers. Figure 5: **Left**, temporal evolution of the HHG intensity at the first plasma entrance (blue) and at its exit (red) (_cf_ Figure 3). **Right**, same image but at the entrance and exit of the main amplifier. One can observe although ASE exists in both cases, the optimized seeding conditions allow it to be kept at a negligible level. Although a complete study of seeding soft x-ray lasers in a transient collisional scheme opened very interesting paths for delivering beams of interest to users, the highest energy achievable is still far from our goal of several millijoules. We thus studied the seminal experiment of [PERSON] _et al._'s [9]. This experiment was using kilojoule class laser with nanosecond duration to create a plasma emitting ASE soft X-ray laser during about 1.2 ns. Such modeling is particularly delicate due to the very different time-scales under consideration, from 20 fs for the HHG, to 100s of ps for the spontaneous emission and ns for the plasma hydrodynamic. We first benchmarked our Maxwell-Bloch model against [PERSON] _et al._'s experiment [9]. The plasma was seeded with 200 fs, 0.5 nJ in-band HHG. Figure 6a displays the temporal evolution of the emitted beam at different plasma lengths.