where ϵAl(λ) is the relative dielectric permittivity of Al, reported in Fig. 1d. Specifically, by means of the FEL of the Trieste FERMI facility, we observe at the carrier wavelength λ ≃ 44 nm a strong self-phase modulation (SPM) enhancement dependent on incidence angle θ. For the investigated Al thin foils, the NZI FB-resonance θ = θNZI(λ) arises over the entire 25 nm < λ < 85 nm broadband range as an effect of the mitigated Al absorption. Self-induced NL spectral modulation in Al mainly results from ultrafast heating that is modelled by an effective temporally nonlocal and nonlinearly saturated polarisation. Theoretical predictions based on such an effective model find experimental confirmation in the angle and power dependence of the transmitted spectrum.

The FERMI EIS-TIMEX experimental setup, realised to measure plasmon-enhanced SPM by FB mode excitation, is sketched in Fig. 1a, b. The experimental approach employs the interaction of TM polarised FEL pulses (by a single-shot raster scan reported in Materials and Methods) with a pristine Al free-standing foil of thickness d = 300 nm, at an incidence angle θ. The FEL pulse associated electric field can be expressed as , where the vectorial envelope E(r, t) lies in the x-z plane spanned by the unit vectors, is the carrier wavevector with unit vector , λ and ω are the carrier wavelength and angular frequency, respectively. Owing to the large FEL peak intensity TW/cm (where ϵ is the vacuum dielectric permittivity and c is the speed of light in vacuum), electrons in Al undergo ultrafast heating followed by relaxation via electron-phonon scattering. Such complex electron dynamics are characterised by a photo-induced transient electronic temperature T(r, t), leading, in turn, to a spatio-temporal refractive index modulation. We phenomenologically account for ultrafast electron heating by a NL polarisation , where is the effective refractive index, A(r, t) is the vectorial electric field envelope within the Al foil, and is the polarisation unit vector. Finally, Δϵ(r, t) accounts for the NL dielectric permittivity modulation, which originates from coherent electron dynamics (providing instantaneous Kerr effect), the delayed thermal nonlinearity produced by ultrafast heating, and the saturation due to electron-phonon collision quenching, see Materials and Methods for further details. The latter saturation effect, not observed in ref. , is unveiled by the enhancement induced by NZI resonance. The NL dielectric permittivity correction Δϵ(r, t) produces asymmetric redshifted spectral broadening due to the combination of the above mentioned effects, analogously to stimulated Raman-shifted SPM in dielectrics. In our measurements, the NL-induced spectral modulation of the transmitted pulses is quantified by the difference of normalised spectra upstream (I(λ)) and downstream (I(λ)) of the sample (ΔI(λ) = I(λ) - I(λ)), see Materials and Methods. Owing to the ENZ permittivity ϵ(λ) of Al around the plasma frequency (corresponding to λ ~85 nm), the FEL pulses excite FB modes enabling transient radiation trapping and NL enhancement. In turn, while in Indium tin oxide (ITO) FB mode excitation mainly occurs at the plasma frequency (arising at the ENZ condition reported in Fig. 1c and is angle insensitive), in Al it occurs over the entire ENZ region (where , i.e. 25 nm < λ < λ) for distinct excitation angles θ(λ) matching the NZI condition in Eq. (1), benefiting from the reduced absorption of Al, accounted for by , reported in Fig. 1d. Such a broadband resonant FB mode excitation in the Al-based foil produces a NL enhancement f ≃ 20, see Fig. 1e and Materials and Methods. Such a parameter accounts for the boosting of electric field intensity within the ENZ material and its reduced effective refractive index, see Materials and Methods, thus quantifying the enhancement of third-order NL effects. As illustrated in Fig. 1e, for every fixed wavelength λ < λ, f is maximised at a peculiar angle θ(λ) such that Eq. (1) is satisfied.

The experimental evidence of ENZ-based NL enhancement of self-driven spectral modulation is summarised in Fig. 2a, illustrating the measured normalised spectral difference, ΔI(λ), for diverse excitation angles and fixed FEL pulse fluence of 10.5 J/cm and time duration of 37 fs full width at half maximum (FWHM), corresponding to a peak intensity of 267 TW/cm. In agreement with the expected enhancement depicted in Fig. 1e, the spectral modulation signal at θ = 60 is amplified by the FB mode excitation. As an effect of the delayed thermal nonlinearity of out-of-equilibrium hot conduction electrons in Al, the spectral modulation is dominated by a strong redshift, also evident in the transmitted spectra reported by coloured lines in Fig. 2c. Moreover, the shoulder-like feature across 44.18 nm highlights a resonant spectral broadening around θ = 60, ascribable to instantaneous SPM (ISPM). Our theoretical model accounts for both of these mechanisms through the effective NL polarisation P(r, t), similarly to previously reported NL spectral broadening in Mg below the absorption edge of core electrons. Figure 2c also illustrates the comparison between theoretically assumed and experimentally observed upstream spectra in solid and dashed black lines, respectively, highlighting excellent agreement. The theoretically predicted spectral modulation is depicted in Fig. 2b for the same incidence angles adopted in the experiments reported in Fig. 2a. In particular, experimental data are reproduced by a thermal ratio f = 0.996, i.e., the relative weight of delayed thermal response and saturated instantaneous Kerr nonlinearity. However, for the considered FEL pulse peak intensity, the temperature saturation significantly attenuates the delayed nonlinearity contribution (see Fig. 4 for further details). Figure 2d further illustrates the λ, θ dependence of the cumulative spectral modulation (CSM), quantified by . Similarly to Fig. 1e, the CSM enhancement is maximised at θ(λ) such that Eq. (1) is satisfied. Notably, the NL polarisation saturation induces a blurring of θ(λ), especially for short wavelengths. Interestingly, the intensity of the signal is larger for shorter wavelengths, owing to reduced absorption accounted for by a lower in such frequency region (see Fig. 1d).

Besides spectral blurring, NL polarisation saturation strongly affects the FEL peak intensity dependence of SPM. As illustrated in Fig. 3a, the SPM signal increases with the FEL peak intensity in a saturated fashion even at the lowest considered FEL intensity, i.e., the one producing the minimum signal detectable by the downstream spectrometer (1.14 TW/cm). However, a drastic ISPM suppression is observed upon FEL peak intensity reduction, quantified by the peak around 44.18 nm (see Fig. 3a). This saturated dependence of the NL signal is reproduced by our phenomenological model obtained with the same parameters as in Fig. 2b. The excellent agreement with the experimental results demonstrates the accuracy and predictive power of spectral modulation models (see Fig. 3b). Moreover, the theoretical model demonstrates the possibility of observing SPM signal in the NZI condition even at 380 GW/cm. The dependence of the CSM intensity on the incidence angle and the peak intensity of the FEL pulse is illustrated in Fig. 3c. To rationalise this agreement, we should consider that the NL redshift is related to the material optical properties modulation over the FEL pulse timescale, producing instantaneous and delayed temperature-dependent saturation. In turn, the time-dependent NL saturation occurring during pulse propagation produces a NL signal mainly from the leading part of the pulse. The strong saturation of delayed thermal nonlinearity achieved at the peak intensity of 267 TW/cm implies a smaller NL modulation than that produced at 15 TW/cm. Differently, for ISPM, the presence of only one saturation mechanism implies an increase in the NL signal with the peak intensity. Such distinct saturation mechanisms, observed for delayed thermal and instantaneous Kerr nonlinearities, can also help to rationalise previously reported results concerning SPM below the absorption edge of metallic Mg.

The reported NZI nonlinearity enhancement at remarkably low FEL intensities opens the door to new investigations of NL phenomena with tabletop HHG-based systems, e.g., by means of experimental schemes similar to the one illustrated in Fig. 1a. Specifically, this could be achieved by focusing HHG radiation on a tilted thin film of Al, producing sub-nm spectral redshifts analogously to the results reported here. Since the FB resonant excitation angle θ(λ) strongly depends on λ (see Fig. 2d), each harmonic of the HHG system would then experience a different shift arising from the NL enhancement of the effect. The capability to obtain from a tabletop XUV system two beams with tuneable sub-nm frequency shift can overcome current limitations in frequency resolution dictated by the IR pump generating the HHG spectrum, thus enabling the realisation of innovative pump-probe schemes with unprecedented resolution at specific electronic resonances.

Moreover, HHG occurring in reflection via the plasma mirror effect can, in principle, be enhanced by FB excitation in the Al thin foils reported here, thus providing further potential applications in the advancement of XUV strong-field physics. However, spectral measurements in reflection from a rotating sample are currently beyond the capabilities of the FEL Fermi facility and will require further advancements in order to investigate HHG via plasma mirror.

Interestingly, although Al is a centrosymmetric material, second-order nonlinearities may still arise due to symmetry breaking induced by photon momentum at the surface under tilted excitation. In addition, engineered non-centrosymmetric Al samples -achieved, for instance, through strain engineering, surface patterning, or by coating with non-centrosymmetric materials such as LiNbO -could be employed to enable NZI-enhanced second-order NL processes. This possibility could offer qualitatively new insights into NZI-XUV NLO, particularly due to the ability of ENZ media to relax phase-matching constraints, and thus merits further investigation.