Nonlinear X-Ray Spectroscopy
Ordinary nonlinear optical processes in x-ray regime are known to be very weak, while conventional x-ray sources suffer from insufficient brightness. Nevertheless, recent and expected improvements in brightness and beam quality of x-ray sources, together with new facilities such as the x-ray free-electron laser, offer the possibility of extending the concepts of nonlinear and quantum optics into x-ray energies. The new facilities with their increased power allow the observation of new x-ray nonlinear and quantum effects. Indeed, recently, the number of demonstrations of nonlinear and quantum effects in the x-ray regime, is growing rapidly. I will describe recently performed and proposed nonlinear experiments at x-ray wavelengths including x-ray and visible wave mixing , x-ray second harmonic generation  and x-ray parametric down-conversion into the ultraviolet and optical regimes [3,4]. I will discuss future directions of implementing nonlinear x-ray techniques as tools for spectroscopy and studies of ultrafast effects. For example, x-ray and visible mixing may lead to atomic scale resolution techniques to study chemical bonds. X-ray parametric down-conversion can be developed into techniques to study properties such as Fermi energies, plasmons, and the density of states in crystals. Nonlinear techniques are expected to be useful in the inspection of subfemtosecond temporal pulses.
 T. E. Glover et al. Nature 488, 603 (2012).
 S. Shwartz et al. Phys Rev. Lett. 112, 163901 (2014)
 D. Borodin , S. Levy , and S. Shwartz, App. Phys. Lett. 110, 131101 (2017).
 A. Schori et al. Phys. Rev. Lett. 119, 253902 (2017)