Modelling of transient Ni-like silver X-ray lasers

We have modelled an experiment performed at the LULI facility (Ecole Polytechnique, Palaiseau, France). This experiment was devoted to the measurement of the temporal coherence of the transient Ni-like silver X-ray laser at the wavelength of 13.9 nm. In population kinetics studies of saturated lasers, it is necessary to account for the interaction between the X-ray laser electric field and the lasing ions. To this end, we have used the Maxwell–Bloch formalism in the paraxial approximation. The Zeeman sublevels (JM) associated with the lower lasing level (J = 1) are not identically affected by the X-ray laser field. As a result, their populations are different. However, elastic collisions between free electrons and lasing ions have the opposite effect: they tend to restore equilibrium between the sublevel populations. Therefore, elastic collision rates obtained in the distorted wave approximation have been included in the rate equations. Refraction of the X-ray beam, due to electron density gradients, is taken into account by using a ray-trace code which works as a post-processor of the hydro-code EHYBRID. We have checked that the Voigt profile is a good approximation for lasing lines in Ni-like ions. This allowed us to implement a subroutine calculating the Voigt profile in the Maxwell–Bloch code. Whilst the FWHM of the spontaneous emission profile is 12 mA, the amplified X-ray line shows a smaller width of ∼3 mA. This is known as the gain narrowing effect. We notice the saturation of the line-width for a propagation length of 2–3 mm. Comparison with experiment is discussed.