Four-wave mixing experiments with extreme ultraviolet transient gratings

Abstract

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 methods. The coherent and multi-wave nature of the FWM approach has been crucial in the development of advanced technologies, such as silicon photonics, subwavelength imaging and quantum communications. 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 resonances. 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 laser, 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 dynamics. The theoretical possibility of realizing these applications has already stimulated ongoing developments of free-electron lasers: our results show that FWM at suboptical wavelengths is feasible, and we hope that they will enable advances in present and future photon sources.

Extended Data Figure 1. Experimental setup 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 M1 and the trajectory of the optical pulse are indicated. d, sketch of the movements needed to change 2θ keeping fix Δt_EUV-EUV. e, sketch of the movements needed to change Δt_EUV-EUV keeping fix 2θ. f and 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 fluece level in the interaction region can be appreciated. 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 in a not irradiated region of the sample (a), in an area irradiated by ≈300 FEL shots at fluence larger than 50 mJ/cm² (b) and in an area continuously irradiated by FEL pulses at low fluence (c). d-f are representative depth profiles of the sample surface along the green lines shown in panels a-c. The power spectral densities (PSD) corresponding to data reported in d-f are shown in g-i.

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 shown in Fig. 3.

Figure 1. FWM experiments with EUV transient gratings

a, sketch of the FEL-based FWM experiment: 2θ=6.16°, θ_B=49.9°, λ_EUV=27.6 nm and λ_opt=392.8 nm are the crossing angle between the two FEL beams that generate the EUV dynamic grating, the angle between their bisector (dashed black line) and the optical beam, the FEL and laser wavelength, respectively. A CCD sensor is placed along the expected propagation direction of the FWM signal beam (k_FWM), which is determined by the “phase matching” (shown in (b); here k_opt, k_EUV1 and k_EUV2 are the wavevectors of the optical and of the two FEL pulses, respectively).

Figure 2. FWM signal stimulated by EUV transient gratings

Images from the CCD placed along k_FMW (X and Y are the directions parallel and orthogonal to the scattering plane, respectively). The images were acquired at Δt=0 (a), Δt=−0.5 ps (b) and Δt=70 ps (c), respectively.

Figure 3. Time evolution of the FWM signal

a, black circles connected by lines are the time dependence of the FWM signal, scaled to the intensity of the input beams (error bars are estimated as one standard deviation of the set of CCD images corresponding to the same Δt value); the blue and red lines are R_cc and the expected signal modulation due to acoustic modes, respectively. b, black circles connected by lines is the FWM signal after R_cc is subtracted. The red line is the modulation due to oscillations at frequencies ν₁=1.15 THz and ν₂=4.1 THz.

Figure 4. Outlooks

a, energy-wavevector (ω_ex-k_ex) range of typical excitations in condensed matter and corresponding time-length scales. The green area sketches the range accessible by optical FWM while the double-ended blue vertical arrow is the range probed in the present work; the horizontal dashed line is the time duration of the excitation pulses. The area delimited by the thick blue lines represents the (ω_ex-k_ex) range accessible by EUV/SXR FWM, with the addition of a EUV/SXR probe and the development of EUV/SXR CRS. b, wavevector arrangement for a CRS process involving two-colours FEL excitation pulses (|k_EUV1|≠|k_EUV2|); k₃ and k_FWM are the wavevectors of the probe and signal beams, respectively. c, level scheme in case ω_EUV1 is tuned to the energy of a core state (CS) transition and ω_EUV1-ω_EUV2 to a lower energy excitation (ω_ex). GS and ES are the ground and excited states, respectively, while VS is a virtual state (in standard CRS the CS is replaced by a VS); τ is the eventual time delay between the two excitation pulses while ω₃ and ω_FWM=ω₃+ω_ex are the photon energies of the probe and signal beams, respectively. d, the independent control of three EUV/SXR input beams might allow to separately tune the energy of the excitation and probe beams to CS’s of two distinct atoms (atom-A and atom-B), hence allowing for monitoring the excitation dynamics among two different atomic sites^, (a valence band exciton in this sketch; here VB and CB are the valence and conduction bands, respectively).