Measurement of Higher Order X-ray Optical Mixing*

X-ray optical wave mixing is a nonlinear diffraction method that gives direct information about the Ångstrom and femtosecond-scale structure of the local optically-induced charge density in bulk solids, information unavailable to purely optical methods. The first measurements of wave mixing between x rays and optical photons were reported for single crystal diamond [Glover et al., Nature 488, 603 (2012)]. Here we report x-ray optical wave mixing experiments using the Swiss-FEL and LCLS hard x-ray free-electron lasers. To measure the wave-mixing signal we use silicon crystal optics to monochromate the free-electron laser output and analyze the energy-angle dependent wave-mixing signal while rejecting the elastic background. The results include the first measurements from silicon and the first measurement of the higher-order wave-mixing process generating the sum frequency of two optical and one x-ray photon. The latter gives access to the lowest-order local nonlinear electronic response to strong optical fields. Here we observe a nontrivial laser-polarization dependence which we compare to theoretical predictions from first-principles electronic structure calculations using a Bloch-Floquet formalism. Preliminary analysis of these results is presented here.