Four-wave mixing (FWM) processes, based on third-order nonlinear light–matter interactions, can combine ultrafast time resolution with energy and wavevector selectivity, and enable the exploration of dynamics inaccessible by linear methods1,2,3,4,5,6,7. The coherent and multi-wave nature of the FWM approach has been crucial in the development of advanced technologies, such as silicon photonics8, subwavelength imaging9 and quantum communications10. All these technologies operate at optical wavelengths, which limits the spatial resolution and does not allow the probing of excitations with energy in the electronvolt range. Extension to shorter wavelengths—that is, the extreme ultraviolet and soft-X-ray ranges—would allow the spatial resolution to be improved and the excitation energy range to be expanded, as well as enabling elemental selectivity to be achieved by exploiting core resonances5,6,7,11,12,13,14. So far, FWM applications at such wavelengths have been prevented by the absence of coherent sources of sufficient brightness and of suitable experimental set-ups. Here we show how transient gratings, generated by the interference of coherent extreme-ultraviolet pulses delivered by the FERMI free-electron laser15, can be used to stimulate FWM processes at suboptical wavelengths. Furthermore, we have demonstrated the possibility of observing the time evolution of the FWM signal, which shows the dynamics of coherent excitations as molecular vibrations. This result opens the way to FWM with nanometre spatial resolution and elemental selectivity, which, for example, would enable the investigation of charge-transfer dynamics5,6,7. The theoretical possibility of realizing these applications has already stimulated ongoing developments of free-electron lasers16,17,18,19,20: our results show that FWM at suboptical wavelengths is feasible, and we hope that they will enable advances in present and future photon sources.
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We acknowledge support from the Italian Ministry of University and Research through grants FIRB-RBAP045JF2 and FIRB-RBAP06AWK3, and from the European Research Council through grant 202804-TIMER. We thank M. Svandrlik and the FERMI-FEL team for support.
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Extended data figures and tables
Extended Data Figure 1 Experimental set-up for FEL-based FWM measurements.
a, Top-view layout of the experimental set-up used to split and recombine the FEL beams. b, Top-view layout of the experimental set-up used to control the optical beam. c, Top-view picture of the set-up: the two FEL paths (FP1 and FP2) downstream of M1 and the trajectory of the optical pulse are indicated. d, Sketch of the movements needed to change 2θ keeping ΔtEUV-EUV fixed. e, Sketch of the movements needed to change ΔtEUV-EUV keeping 2θ fixed. See Methods for details of a–e. f, g, Optical reflectivity changes in Si3N4 induced by the FEL beam propagating through FP1 (green dots) and FP2 (magenta dots). In f, the mirrors were displaced with respect to the nominal position; a poor time coincidence and a different fluence level in the interaction region can be seen. g, Same measurements as in f after optimization of the geometry; the superposition of the two traces indicates a large improvement in the time coincidence and a similar FEL fluence in the interaction region.
Extended Data Figure 2 AFM topographies.
AFM topographies of 8 × 8 μm2 areas of the sample surface as follows: in a region that was not irradiated (a), in an area irradiated by ∼300 FEL shots at a fluence larger than 50 mJ cm−2 (b), and in an area continuously irradiated by FEL pulses at low fluence (c). d–f, Representative depth profiles of the sample surface along the green lines shown in a–c, respectively. The power spectral densities (PSD) corresponding to data reported in d–f are shown in g–i, respectively.
Extended Data Figure 3 Time sequence of acquired data.
Black open and crossed circles connected by lines are data shown in Fig. 3; crossed circles correspond to a scan made several hours after the one corresponding to data shown as open circles; in both scans the time delay was continuously increased. Green dots are data collected before these two scans; here we had not yet optimized the FWM signal at Δt = 0 (these data are scaled by a factor to fit the peak intensity of the data shown as black circles). Blue and red lines are the same as shown in Fig. 3.
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Bencivenga, F., Cucini, R., Capotondi, F. et al. Four-wave mixing experiments with extreme ultraviolet transient gratings. Nature 520, 205–208 (2015). https://doi.org/10.1038/nature14341
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