Fig. 3

Theoretical model of the photon-acceleration-induced blue-shifted harmonics generation. a Time-dependent resonant frequency of the metasurface due to four-photon absorption free carrier generation plotted for three normalized peak intensities of the mid-infrared (MIR) pulse: \(\tilde I = 0.6\tilde I_{\mathrm{c}}\) (perturbative regime, blue curve), \(\tilde I = \tilde I_{\mathrm{c}}\) (critical regime, green curve), and \(\tilde I = 1.3\tilde I_{\mathrm{c}}\) (nonperturbative photon acceleration regime, red curve). Shaded areas: the frequency-domain (on the left) and time-domain (on the bottom) profiles of the incident pulse. b Spectra of the MIR electric field inside the metasurface. c, d Predicted third harmonic generation (THG) spectra for different input MIR intensities. In (c), a comparison between the predictions of the models of time-dependent metasurface model (thick lines; coloring as in (a)) and of the time-independent (thin lines) metasurfaces. The family of time-independent metasurfaces satisfies \(0 < \Delta \omega _{\mathrm{R}}/\omega _{\mathrm{R}}^0 < 4\%\) and \(\tilde I = 1.3\tilde I_{\mathrm{c}}\). Filled red area: the portion of the spectrum inaccessible within the time-independent framework. e Spectrum-integrated THG intensity versus input intensity for the unpatterned film (open circles) and resonant metasurface (closed circles). THG from the film is simulated as from a significantly nonresonant structure with γ_R = 10¹⁵ s⁻¹ with an intensity calibration coefficient that is adjusted for good agreement with the experiment at low intensities. Lines: guide-for-the-eye cubic dependences. f Intensity-dependent THG peak position (circles) and THG bandwidth (squares) generated by the metasurface. Shaded areas in (e) and (f): the nonperturbative photon acceleration regime