## 1 Introduction  
Over the last decade, ultra-intense X-ray sources have opened new avenues in physics, creating new states of matter, probing or imaging the most intimate components of living [1] or inert matter or realizing movies of samples excited by ultra-fast sources [2]. Within the panorama of available ultra-intense X-ray sources, free-electron lasers (FEL) have a strong leadership, as they deliver pulses combining femtosecond duration and 10s of microjoules to millijoule energy [3; 4]. Considering the applications developed so far or under consideration, pump (infrared laser)-probe (FEL) experiments attract a lot of interest. Although such technique has been widely used on other X-ray sources like synchrotrons or high harmonic generation (HHG, high order harmonics of infrared lasers), free-electron lasers are just opening new horizons by freezing during a few femtoseconds the non-reproducible evolution of samples excited by an external source. As an example, we may cite the work of [PERSON] and co-authors [5] that imaged using X-ray holography the local deformation of nano-domains of a magnetic multilayer after being heated up by a femtosecond laser. This experiment raised two key issues. First, simpler techniques like X-ray diffraction suffer from information loss, being incapable of catching the tiniest change of the magnetic structure observed by X-ray holography as it averages the size and shape over all the nano-domains. Second, X-ray holography, even using the most efficient techniques, requires extreme energies of few mJ per pulse. Indirectly, this energy range has been confirmed by [PERSON] _et al_[6]. who accumulated 0.1 mJ, corresponding to nearly a million high harmonic pulses for producing a high quality soft X-ray diffraction image (at 60 eV) of a magnetic sample similar to those used by [PERSON] _et al_. [5]. The list can be extended to many imaging experiments and beyond to other fields, of which we would like to draw the attention on the study of [PERSON] and collaborators on imaging of biological samples [7]. Here, 1500 FEL pulses needed to be accumulated for reaching a total of 15 mJ on target, to produce high-resolution image of algae by soft X-ray holography (at 8 nm). Nowadays, milli-Joule energy in a single pulse is only achieved at the LCLS free-electron laser that combines a high number of photons, several 10\({}^{12}\)/pulse, and photon energies up to 12 keV. For the rest of the article, we differentiate the soft x-ray domain that extends from typically 30 eV to 1 keV from hard X-ray (photon energy above 1 keV). In parallel, laser-pumped soft X-ray lasers (SXRL) show exciting promise having demonstrated the most energetic monochromatic soft X-ray pulse, up to 12 mJ but with poor coherence and long (200 ps) pulses [8]. Increasing the coherence of plasma-based SXRL has been a subject of active research during the last two decades. Many techniques have been tested, like the use of a half or full cavity, all being far from producing fully coherent pulses. The first ground-breaking step forward was realized by [PERSON] and collaborators who seeded a plasma with an external ultrafast (200 fs) and fully coherent high harmonic pulse [9]. Results were not at the level of expectations: the seed was amplified only 3 times reaching less than 1 \(\upmu\)J while the amplified spontaneous emission, ASE, _i.e._, stochastic incoherent noise, reached 5 mJ. Analysis of the experiment by [PERSON] _et al_. pointed out the problem of refraction in plasmas created from solid targets as well as several technical problems. Based on their conclusions, we realized the first successful demonstration of amplification of high harmonic pulses in plasma created from gas targets [10]. Gas amplifiers present negligible refraction effects and due to the pumping technique (longitudinal pumping) they eased the location of the gain region in space and time. This work was followed in 2008 by the strong amplification, up to 600 times, of a high harmonic pulse in plasma created from solid target [11].