Beam optics of gain-guided soft-x-ray lasers in cylindrical plasmas

We analyze soft-x-ray beam propagation and amplification in gain-guided amplifiers of cylindrical geometry for arbitrary gain and density profiles. A general relation that must be fulfilled for refraction not to be an impediment to the exponential growth of the intensity is obtained. It is shown that for sufficiently long plasma columns the effective gain, reduced by refraction, is determined solely by the gain and the curvature of the density profile at the position of maximum density, even when the location of the gain and the density maxima do not coincide. We analyze the case of amplifiers with gain–length approaching saturation and show that refraction reduces the effective gain–length product at which gain saturation occurs. The theoretical results are used to analyze the output of a capillary discharge soft-x-ray laser.