Femtosecond coherent pulses in the keV range from inner-shell transitions pumped by a betatron source

We present a new method to generate ultra-short X-ray laser pulses by using the recently demonstrated laser-driven betatron source to photo-pump inner-shell transitions. The proposed compact set-up will then open the route to a wide range of applications. The betatron spectrum and ion-population kinetics are modeled and the temporal evolution of the gain coefficient for the K-α transitions is assessed. Using measured values of divergence, duration, and number of photons per pulse of the betatron source as input parameters, local gain values close to 60 cm^-1 are calculated for nitrogen at 3.2 nm. Significant gain values are also numerically obtained at shorter wavelengths (for neon at 1.5 nm) when the betatron energy distribution is optimized as suggested by recent laser wakefield electron acceleration experiments.