Stimulated emission is a fundamental process in nature that deserves to be investigated and understood in the extreme ultra-violet (EUV) and x-ray regimes. Today, this is definitely possible through high energy density free electron laser (FEL) beams. In this context, we give evidence for soft-x-ray stimulated emission from a magnesium oxide solid target pumped by EUV FEL pulses formed in the regime of travelling-wave amplified spontaneous emission in backward geometry. Our results combine two effects separately reported in previous works: emission in a privileged direction and existence of a material-dependent threshold for the stimulated emission. We develop a novel theoretical framework, based on coupled rate and transport equations taking into account the solid-density plasma state of the target. Our model accounts for both observed mechanisms that are the privileged direction for the stimulated emission of the Mg L2,3 characteristic emission and the pumping threshold.
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