## 2 Numerical Section  
We have first used ARWEN 2D hydrodynamic code [13] developed by [PERSON], Spain, to clarify and then optimize the plasma amplifier conditions [14]. This code is a 2D hydrodynamic code with radiation-transport that solves radiation hydrodynamics equations enhanced with the treatment of electron thermal conduction. Navier-Stokes equations are solved with Eulerian scheme using a high-order Godunov type method with a real gas approximated Riemann solver. Electron thermal conduction and radiation transport are solved with matrix-free solvers and Sn multigroup methods respectively. Equations of State and opacities are obtained from the QEOS and BigBART programs. This code is based on the Adaptive Mesh Refinement technique, saving computational time by refining the mesh only in a small part of the simulation domain. Since [PERSON] _et al._[11] was the sole successful seeding experiment with a solid amplifier, we benchmarked our numerical codes against it. They used the so-called Transient Collisional Excitation Scheme (TCE), combining a long infrared pulse that creates the plasma at the right ionization stage followed after a variable delay by a short (ps) pulse that produces the population inversion. All modeling showed in this article is done using this scheme: the plasma is created by a 100 ps infrared pulse (\(I\) = 1.2 \(\times\) 10\({}^{12}\) W/cm\({}^{2}\)) preceded by a 1 ns pre-pulse (\(I\) = 1.2 \(\times\) 10\({}^{11}\) W/cm\({}^{2}\)); free-electrons are heated up directly by the short infrared pulse (0.5 ps, \(I\) = 1.1 \(\times\) 10\({}^{15}\) W/cm\({}^{2}\)) and later by thermal conductivity. Then, they collide with the lasing ionspopulating both the lower (1) and upper (2) lasing levels. On a simplified but realistic description, population inversion is created because the upper level is metastable (the radiative transition between the fundamental and the upper level is forbidden) and the lower level is coupled to the ground-state (0) by a very fast E2 transition. Lasing ions considered in this article are either neonlike (10 bound electrons) or nickellike (28 bound electrons). [PERSON]'s coefficients for the different transitions involved on this scheme are A\({}_{21}\) = 1.3 \(\times\) 10\({}^{10}\) s\({}^{-1}\) and A\({}_{10}\) = 8.3 \(\times\) 10\({}^{11}\) s\({}^{-1}\) for the 2_p_\({}^{5}\) 3_s_J_ = 1 \(\to\) 2_p_\({}^{5}\) 3\(p\), \(J\) = 0 neonlike Fe transition occurring at \(\lambda\) = 25.5 nm; A\({}_{21}\) = 1.8 \(\times\) 10\({}^{10}\) s\({}^{-1}\) and A\({}_{10}\) = 1.0 \(\times\) 10\({}^{12}\) s\({}^{-1}\) for the 2_p_\({}^{5}\) 3_s_J_ = 1 \(\to\) 2_p_\({}^{5}\) 3\(p\), \(J\) = 0 neonlike Zn transition at \(\lambda\) = 21.2 nm and A\({}_{21}\) = 4.4 \(\times\) 10\({}^{11}\) s\({}^{-1}\) and A\({}_{10}\) = 4.2 \(\times\) 10\({}^{11}\) s\({}^{-1}\) for the 4_p_\({}^{1}\)_J_ = 1 \(\to\) 4d\({}^{1}\)_J_ = 0 nickellike Mo transition occurring at \(\lambda\) = 18.9 nm. On Figure 1, a 3D map of the plasma and of the gain region is displayed. This plasma is assumed to be homogeneous along the longer (Z) dimension. In [PERSON] _et al._'s experiment [11], the pump laser was focused over a 30 \(\upmu\)m \(\times\) 5 mm line. At first glance, we may observe that the plasma is almost three times the size of the focal width along the target surface (Y). Such a strong plasma flow was not expected, leading to a larger gain region, about 50 \(\upmu\)m wide. It is interesting to observe that the gain zone extends only over 10 \(\upmu\)m along the pump laser axis (X). The gain peaks at about 62 cm\({}^{-1}\) agreeing very well with [PERSON] _et al._'s data. It decreases very fast when the distance from the target surface increases. The saturation fluence calculated over the gain region is roughly stable at 2.3 mJ/cm\({}^{2}\) agreeing well with [PERSON]'s article (fluence of 1.8 mJ/cm\({}^{2}\)). These two results are striking.