Difference frequency generation of ultraviolet from x-ray pulses in opaque materials

We suggest a new approach for observing x-ray nonlinear wave mixing in opaque materials. We focus on difference frequency generation of ultraviolet radiation from two short x-ray pulses by measuring the depletion of the pumping pulses. Like other processes involving nonlinear interactions between x-rays and longer wavelengths, our method can lead to the development of a probe for spectroscopy of valence electrons at the atomic scale resolution. The two main advantages of the method we propose over the direct observation of the generated signal are the ability to probe the properties of materials at wavelengths where they are opaque and the higher predicted efficiency in the ultraviolet regime. We describe a possible experimental setup with realistic specifications optimized with respect to the characteristics of the input pulses. We expect that experimental observations of the effect will be feasible with the new emerging high-repetition-rate x-ray free-electron lasers.