## 2 Numerical Section
Although most plasma-based soft X-ray lasers are running at about 10 times the saturation fluence, in order to break the mJ threshold, one needs to produce a gain region of 0.1 cm\({}^{2}\) (to be compared to 5 \(\times\) 10\({}^{-6}\) cm\({}^{-2}\) for [PERSON] _et al_. [11]). We therefore explored the possibility of enlarging the gain region. The most accessible solution consists in enlarging the focal width while keeping the intensity constant. We modeled plasmas created with focal widths from 30 \(\upmu\)m up to 2 mm (Figure 2). It is striking to observe that peak gain increases very quickly from 62 to 126 cm\({}^{-1}\) when enlarging the focal width from 30 \(\upmu\)m  
Figure 1: Image in false colors of plasma created by two laser pulses and the gain region. This modeling has been used to benchmark ARWEN code against [PERSON] _et al._’s experiment. [11]. to 100 \(\upmu\)m. Then, peak gain remains constant for widths larger than 100 \(\upmu\)m [15]. Note that the gain region increases nearly 8 times along the X direction from 10 \(\upmu\)m to 76 \(\upmu\)m while the vertical flow becomes negligible maintaining the gain region near the focal width size. When the focal width is larger than 100 \(\upmu\)m, the gain zone is rather homogeneous. These changes are due to a reduction of the thermal losses by lateral conduction and radiation cooling as well as the inhibition of the lateral expansion when increasing the focal width [16]. Considering a 1 mm large plasma that may be created with a sub-PW laser, about 0.4 mJ is stored on the population inversion. This value, although too low for the most demanding experiments like biological imaging, represents a huge step forward. However, using only a hydrodynamic code is not sufficient to give a precise estimate of the output energy when seeding this amplifier with high harmonics. Several issues still remain to be studied numerically to define the best seeding conditions and thus the highest output energy and shortest duration achievable. In the second step, we used a time-dependent Maxwell-Bloch model as a post-processor for the plasma conditions, enabling us to answer the remaining questions by describing accurately the temporal structure of the soft X-ray pulse all along its amplification. In brief, the model describes the evolution of the electromagnetic field of the soft X-ray radiation (seed and spontaneous emission) during its amplification (Maxwell equations) coherently coupled to the time-dependent evolution of the population of the energy levels of the lasing ion (Bloch equations). The model includes a stochastic source term to mimic the spontaneous emission as well as a depolarization term due to electron-ion collisions. A thorough description of the model can be found in [17, 18, 19]. Our model is coupled to ARWEN output while previous similar models used _ab initio_ plasma conditions. It is worth noting that since HHG are linearly polarized, our Maxwell-Bloch model does not take into account the degeneracy of the lower level. If circularly polarized HHG were seeded (created using polarizing mirrors [20] or  
Figure 2: Image in false colors of plasmas created by 100 \(\upmu\)m (**left**) and 1 mm (**right**) focal width. For 1 mm large width, the plasma flow along the target surface is very weak. mixing two-color fields [21, 22]), the magnetic sub-levels must be taken into account since left polarized HHG induces transitions to the m = \(-\)1 level whereas right polarized HHG induces transitions to the m = 1 level. When modeling [PERSON] _et al._'s experiment, we observed the well-known weak amplification of the HHG seed followed by a wake field that is amplified [23, 24] and contains most of the energy in a 2 ps pulse (Figure 3). As explained by [PERSON] and collaborators, the incapacity of such plasma in amplifying the seed comes from the mismatch between the polarization characteristic time (\(\sim\)0.1-0.2 ps) and the HHG duration (30 fs).