Introduction

High-harmonic generation (HHG) stands as a fundamental phenomenon at the intersection of strong-field physics, ultrafast science, and nonlinear optics. Initially observed in noble gases, where intense laser fields prompt electrons to re-collide with their parent ions emitting high-energy photons, HHG has revolutionised attosecond science by enabling the production of ultrashort pulses and the real-time probing of electron dynamics. This foundational mechanism has subsequently been extended to condensed matter physics, where HHG allows to uncover the intricate interplay between intense light fields and the electronic structure of solids. In this context, HHG in bulk crystals not only offers a pathway to more compact and scalable device platforms, but also serves as a sensitive probe of crystal symmetry, electronic topology, and ultrafast carrier dynamics. A diverse array of materials underpins these advances, including non-centrosymmetric compounds such as ZnO1 and LiNbO32, elemental semiconductors like Si3, and III-V compound semiconductors like InP4 and AlGaAs5. These material platforms enable HHG applications ranging from compact extreme ultraviolet sources6 to spectroscopy of electronic band structures7,8, with the orders of generated harmonics reaching record high values of several tens9,10.

With the advent of nanophotonics, metasurfaces have emerged as tools that offer unprecedented control over local electromagnetic fields and optical resonances which proved particularly useful for studying nonlinear phenomena and expanding their scope11. A key breakthrough has been the discovery of bound states in the continuum (BICs) and quasi-BICs (q-BICs), which enable ultrahigh-Q resonances by suppressing radiative losses through symmetry protection or mode superposition12. Recent experiments have demonstrated exceptionally high Q-factors by harnessing q-BICs13,14. These resonances have been exploited to enhance various nonlinear optical processes15,16,17,18,19, including HHG5,20,21, in ways unattainable in conventional bulk systems. For example, perovskite metasurfaces supporting q-BIC modes have demonstrated efficient fifth-harmonic generation under picosecond pulsed excitation22. In contrast, symmetry-broken silicon metasurfaces pumped by picosecond pulses exhibit even harmonics (second and fourth) that deviate from expected power laws, pointing to resonant symmetry breaking and field enhancement effects23. Further pushing the capabilities of BIC-enabled metasurfaces, Zograf et al. demonstrated odd harmonics up to the 11-th order generated from dielectric structures supporting BIC resonances21. While they probably observed some unconventional features of HHG, they have not been understood or recognized. The observation of HHG in other nanophotonic systems, including GST24, CdT25, ZnO26 and ENZ materials27, reflects a broader trend toward ultra-thin and strongly resonant platforms for high-field light-matter interaction. However, the interpretation of these phenomena is frequently complicated by multifaceted interactions with substrates, thermal effects, and material absorption.

Free-standing dielectric membranes provide a minimal, substrate-free platform for HHG, effectively mitigating the asymmetries introduced by underlying substrates. Recent studies have demonstrated that silicon membranes, which support Fabry-Pérot resonances, support HHG at least up to the seventh order, with power-law exponents that align well with theoretical predictions28. Furthermore, high-Q free-standing membranes have been proposed for sensing applications due to their narrow resonant line-widths and exceptional mechanical stability29,30.

Herein, we consider HHG in structured free-standing membrane metasurfaces supporting ultrahigh-Q resonances arising from a q-BIC mode. Such a system uniquely integrates the strong field enhancement with the simplicity inherent to a substrate-free design. Our experimental results reveal a striking enhancement in the HHG signals as compared to unpatterned membrane, exceeding three orders of magnitude for the seventh harmonic. It is also accompanied by the emergence of the ninth harmonics observable exclusively in the vicinity of resonance. Notably, our findings indicate that the harmonic power dependencies deviate substantially from the classical perturbative laws, manifesting non-integer power scaling. Theoretical simulations reveal that the observation of this phenomenon is enabled by the strong field enhancement at the q-BIC resonance, which induces a complex cross-talk between the nonlinear polarizations of different orders otherwise achievable only for significantly higher laser fluences. These results establish free-standing resonant membranes as a promising and novel platform for probing and controlling the onset of extreme regimes of high harmonic generation.

Results

Metasurface design and optical properties

Our metasurface design is based on a free-standing crystalline silicon membrane perforated with a hexagonal grid of elliptical apertures (Fig. 1). It is fabricated using wafer-scale compatible clean-room processes (see “Methods” section) from a h = 1 μm thick silicon-on-insulator base, while both silicon oxide buffer layer and thick silicon substrate are removed from underneath the structure. The resulting free-standing membrane meatsurface has lateral dimensions of 500 μm × 500 μm. SEM image of the structure is shown in Figure 1b, with an inset showing a zoomed-in area with a marked unit cell.

Fig. 1: Dielectric membrane metasurface for high-harmonic generation.
Fig. 1: Dielectric membrane metasurface for high-harmonic generation.
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a Concept of high-harmonic generation from resonant membrane metasurfaces. The inset shows the structure’s unit cell and the definition of its geometrical parameters. b SEM image of the metasurface and its unit cell. The metasurface structural parameters are h = 1 μm, d = 1.69 μm, a = 0.73 μm, and b = 0.4 μm.

Our design is based on a symmetry-protected BIC mode that is supported by a membrane with a regular hexagonal lattice of circular apertures31,32. This BIC mode can then be transformed into a q-BIC upon changing the ellipticity of the aperture, which leads to the manifestation of a narrow transmission resonance. We optimize the structure parameters to match the resonance frequency of this mode with the center of the tuning range of the laser used in nonlinear experiments. The resulting design features a regular hexagon unit cell with short diagonal d = 1.73 μm, an elliptical aperture with axes of a = 0.73 μm and b = 0.39 μm, oriented along the main diagonal of the unit cell (Figure 1a, b); the material permittivity was set at εSi = 12.3. The calculated mid-IR transmission spectrum (red curve) that reveals several pronounced resonant features, including a Fano-shaped q-BIC resonance at 3.96 μm is shown in Fig. 2a. The electric and magnetic field distributions within a unit cell are illustrated in Figure 2b, revealing significant electromagnetic field localization. The electric field enhancement factor, averaged over the membrane volume, reaches approximately 6 at resonance, while the local maximum of the electric field enhancement reaches about 20 (see Section 1 of the Supplementary Information). The experimental linear transmission spectrum, shown with a blue curve in Figure 2a, aligns exceptionally well with the theoretical predictions (see “Methods” section for further experimental details). This close agreement not only validates our numerical models but also attests to the high quality of the fabricated metasurfaces, as evidenced by a simulated Q-factor of 580 for the q-BIC versus an experimentally measured value of 480.

Fig. 2: Linear properties of resonant metasurfaces.
Fig. 2: Linear properties of resonant metasurfaces.
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a Theoretical and experimental linear transmission spectra of the metasurface, excited by linearly polarized light along the long semiaxis of the ellipses. b Magnetic- and electric-field distributions of the eigenmode at the resonance wavelength \({\lambda }_{{{{\rm{res}}}}}=3.96\,\mu \,{{{\rm{m}}}}\).

High-harmonic generation

In the nonlinear experiments, we pump the metasurface with a tunable femtosecond laser operating in the 3.85-4.05 μm window and capture the harmonic signals in transmission both in near-infrared and visible spectral ranges (see also “Methods” section). In Fig. 3a, the third harmonic generation (THG) spectrum is plotted as a function of the pump wavelength at a fixed pump power of 20 mW, which corresponds to a fluence of ≈2 mJ ⋅ cm−2 (we estimate the pump pulse duration at  ≈ 800 fs). When the pump wavelength is detuned from the q-BIC resonance, the THG spectrum shows a broad peak with a spectral width of 10 nm defined by the pump pulse linewidth, which experiences a respective linear spectral shift. In stark contrast, when the pump wavelength approaches the q-BIC resonance, a narrow peak with a spectral width of 5 nm emerges from the broad harmonic spectrum. This peak aligns precisely with the tripled frequency of the q-BIC mode and shows considerably stronger THG signal. The red curve in Figure 3d tracks the integrated THG efficiency for each pump wavelength, revealing that the off-resonance THG conversion efficiency remains constant at approximately 10−8, while the resonant excitation amplifies the efficiency up to fivefold. For comparison, the same measurements were performed on an unpatterned membrane (the raw THG spectra are provided in Section 2 of the Supplementary Information). The extracted THG efficiency for the unpatterned membrane, shown by the gray curve in Figure 3d, remains constant over the entire pump tuning range.

Fig. 3: Experimental high-harmonic generation from q-BIC resonant membrane metasurface.
Fig. 3: Experimental high-harmonic generation from q-BIC resonant membrane metasurface.
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Intensity maps of (a) third, (b) fifth, and (c) seventh harmonics pumped at different wavelengths. The third and fifth harmonics are pumped at 20 mW, whereas the seventh harmonic is pumped at 55 mW. The pump powers were selected to ensure optimal visibility of the spectral features of HHG around the resonance, while minimizing the interference from the photoluminescence background driven by multiphoton absorption. Conversion efficiency of (d) third, (e) fifth and (f) seventh harmonics for the metasurface (color lines) and unpatterned membrane (gray lines), as a function of the pump wavelength. Note that the higher harmonic efficiency for the seventh harmonic is due to the higher pump power used.

We further expand our study to higher odd harmonics. The dependence of the fifth-harmonic intensity on the pump wavelength at a constant pump power of 20 mW is presented in Figure 3b. Consistent with the observations for THG, the fifth harmonic remains relatively uniform away from resonance, with an enhancement in the vicinity of the q-BIC resonance. The dependences of peak fifth harmonic conversion efficiencies on the pump wavelength presented in Figure 3e show that the metasurface (green curve) provides over 7-fold enhancement over the unpatterned membrane (gray curve) near the q-BIC resonance. Notably, in addition to the resonant harmonic signal enhancement, the measured spectra reveal broad background emission when the pump wavelength matches the q-BIC resonance. We attribute this background to silicon photoluminescence33,34,35 driven by multiphoton absorption, which is enhanced by the resonant field localization similarly to the harmonics.

Finally, the map of the seventh harmonic spectra collected at a fixed pump power of 55 mW is shown in Figure 3c. It reveals that the seventh harmonic signal manifests exclusively in the vicinity of the resonance. The extracted spectrum of conversion efficiency, plotted in Figure 3f (blue curve), indicates that under the resonant conditions it reaches 4.7⋅10−11, which is even higher than for the fifth harmonic due to the higher pump power used. This brings the enhancement over the unpatterned membrane (gray curve in Figure 3f) to more than three orders of magnitude. Moreover, we also observe ninth-harmonic generation from the metasurface in the vicinity of the resonance, with a maximum conversion efficiency of approximately 2⋅10−15 (see Section 3 in Supplementary Information for more details). No ninth-harmonic signal was detectable with our setup from the unpatterned membrane.

However, the most interesting HHG phenomena are revealed in the power dependence of harmonic generation from the resonant metasurface. For comparison, we start with measurements from an unpatterned membrane, which are shown in Fig. 4a. It displays together the spectra of the third, fifth, and seventh harmonics generated at a pump wavelength of 4.96 μm with an average power of 55 mW. For clarity, the signals of the fifth and seventh harmonics are multiplied by scaling factors of 400 and 105, respectively, to facilitate direct comparison. The corresponding power dependencies for each harmonic are plotted in Figure 4b and show excellent agreement with the expected perturbative power-law scaling, with slopes perfectly matching the harmonic orders: 3 ± 0.08 for the third harmonic, 5 ± 0.28 for the fifth harmonic, and 7 ± 0.36 for the seventh harmonic.

Fig. 4: Experimental high-harmonic generation.
Fig. 4: Experimental high-harmonic generation.
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Comparison of HHG (a, c) spectra and (b, d) power dependencies, from (a, b) the unpatterned free-standing membrane and (c, d) the membrane metasurface. Spectra are obtained from a pump power of 55 mW and wavelength of 3.96 μm.

The harmonic spectra recorded from the q-BIC membrane metasurface at the maximum pump power of 55 mW are presented in Figure 4c. One can immediately notice that the metasurface-driven enhancement improves drastically for higher harmonics, bringing the efficiency of the seventh harmonic on par with the fifth (the intensity multipliers are added for visual clarity, similarly to Figure 4a). Furthermore, as noted previously, we detect the ninth-harmonic signal from the metasurface, while no corresponding signal is observed from the unpatterned membrane. The ninth-harmonic conversion efficiency is comparable to that of the seventh harmonic generated by the unpatterned membrane. It is also worth noting that the measured third harmonic spectrum deviates considerably from the gaussian shape, which might be due to the contribution of additional modes supported by the metasurface at the wavelength of THG.

The power dependencies of the harmonic signals obtained from the metasurface are illustrated Figure 4d. In stark contrast to the unpatterned membrane, for which the harmonics adhere to conventional scaling laws, the metasurface exhibits a modified, non-integer power scaling. Specifically, the third harmonic demonstrates a best-fit slope of 2.3 ± 0.1 at higher pump powers, accompanied by an inflection at lower powers, while the fifth harmonic follows an effective slope of 3 ± 0.08. In addition, the seventh and ninth harmonics are characterized by slopes of 4.6 ± 0.34 and 4.5 ± 0.22, respectively. As we show further, this unconventional behavior is directly connected to the excitation of the high-Q mode and the associated strong local field enhancement.

To elucidate the underlying physics driving the unconventional HHG trends, we performed rigorous numerical simulations by solving the nonlinear wave equation in the frequency domain. In the simulations, we accounted for the spatial field distribution within the metasurface and incorporated all pertinent nonlinear source terms derived from a perturbative expansion of the nonlinear polarization (further details on the numerical results are available in Section “Methods"). The calculated conversion efficiencies for the transmitted third, fifth, seventh, and ninth harmonic signals are shown in Fig. 5a. The simulations were carried out assuming a fixed pump power of 50 mW and a pump wavelength tuned across the q-BIC resonance (3.96 μm). The harmonic wavelengths were determined as the pump wavelength divided by the corresponding harmonic order. The peak conversion efficiencies were estimated to be 6.4⋅10−9 for the third harmonic, 2.1⋅10−12 for the fifth harmonic, 4.4⋅10−13 for the seventh harmonic and 1.7⋅10−16 for the ninth harmonic, in good agreement with the experimentally measured values. The simulations reveal that the spectral width of HHG enhancement becomes progressively narrower with increasing harmonic order, as a consequence of the higher-order nonlinear processes36. The experimental data do not exhibit such a pronounced narrowing of the resonance across different pump wavelengths. This discrepancy can be attributed primarily to the broad spectral bandwidth of the femtosecond excitation employed in the experiments, whereas the simulations assume an idealized tunable monocromatic continuous wave excitation. Additional deviations may arise from the finite numerical aperture of the focused laser beam, which introduces angular averaging, in contrast to the simulations that consider normal incidence illumination.

Fig. 5: Calculated high-harmonic generation from the resonant membrane metasurfaces.
Fig. 5: Calculated high-harmonic generation from the resonant membrane metasurfaces.
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a Spectra obtained with a pump power of 50 mW and (b) power dependencies of generated harmonics as a function of input power.

We then numerically estimated the output power scaling as a function of the input pump power. To reproduce the experimentally observed power-dependent trends, the simulations included nonlinear susceptibilities up to the ninth order, together with feedback from the generated harmonics at the pump frequency, such as self-phase modulation. In this framework, the linear polarization of material P(ω) acquires a weak nonlinear contribution due to the self-phase modulation effect, while the nonlinear polarizations contain contributions from higher-order nonlinear susceptibilities (see Section 4 of the Supplementary Information for details). As illustrated in Figure 5b, the simulations reveal that even under resonant conditions, a perturbative framework can accurately capture the main features of the experimental response.

The power dependencies of the individual harmonics exhibit slopes that qualitatively agree with the measurements: 2.4 for the third harmonic, 3.9 for the fifth harmonic, 5.5 for the seventh harmonic, and 7.4 for the ninth harmonic. However, once the pump power reaches 60 mW, the harmonic outputs begin to deviate from the ideal slopes. This deviation originates from the nonlinear modification to the linear polarization P(ω), which makes the refractive index effectively dependent on the pump intensity. The resulting resonance shift reduces the spectral overlap between the pump and the resonant mode, thereby diminishing the efficiency of the HHG process and giving rise to an apparent saturation effect. In contrast, when the same calculations are performed for the unpatterned membrane, the harmonic powers follow the expected scaling laws exactly, as shown in Section 5 of the Supplementary Information.

As the manifestation of non-integer scaling is directly connected to the local field amplitude, this regime could, in principle, appear even in an unpatterned membrane. However, its observation would require extremely high pump intensities. The fact that we are able to observe this phenomenon at moderate pump fluences of a few GW/cm2 is directly related to the temporal dynamics of strong field enhancement that are enabled by low non-radiative loss of the q-BIC resonance of the membrane as well as by optimal balance of its radiative loss with the pump pulse linewidth. In principle, increasing the Q-factor should enhance HHG efficiency, as has been demonstrated for third harmonic generation in recent works37,38. However, this expectation is based on simplified conditions, such as continuous-wave or narrowband excitation, where the incident light is spectrally well matched to the resonance. Under femtosecond excitation, the broad spectral bandwidth of the pulses strongly modifies this behavior. In such cases, the conversion efficiently does not necessarily scale with Q, and can even decrease for higher-Q resonances due to reduced spectral overlap. This counterintuitive effect has been recently observed for THG, where shorter pump pulses led to lower conversion efficiency from the same resonance39.

To determine the optimal conditions for HHG enhancement, we analyse the excitation of the metasurface via a Gaussian pulse in the vicinity of the resonance using temporal coupled mode theory40,41 (see Section 6 of the Supplementary Information for details). Within this approach, three parameters define the dynamics of the field in the mode: its radiative and non-radiative losses, as well as the pump pulse duration τpulse = 1/γpulse. Systematic variation of these parameters (see Supplementary Fig. S6) shows that the ratio between the pulse duration and the modal Q-factor plays a decisive role in the excitation dynamics and electric-field enhancement. Specifically, low non-radiative losses are critical for achieving the maximum field enhancement. Furthermore, in the limit when γnrad is zero, the maximum is achieved when the mode radiative decay rate is twice larger than the temporal pulse width, γpulse ≈ 2γrad.

Our metasurface provides an ideal platform for reaching maximum local field, as the lack of substrate in the membrane design minimizes the non-radiative loss, while the radiative part can be precisely tuned using the q-BIC mechanics. In particular, for our design the radiative Q-factor is 480, while the nonradiative Q-factor extracted from modeling reaches  ~ 3000, bringing it close to the optimal coupling condition for the pump laser pulse parameters, τpulse ≈ 800 fs. This is further highlighted by the fact that for metasurfaces with Q-factors exceeding  ~1000 we did not observe the non-integer scaling behavior in the available range of laser fluences. We attribute this to the worse spectral overlap between the femtosecond pump pulses and the ultranarrow resonance, which prevents efficient coupling. Consequently, the resulting harmonic signals become indistinguishable from those generated by the unpatterned membrane even under resonant excitation (see Section 7 in Supplementary Information for more details).

These results highlight that the observed non-integer-power scaling in the metasurface is a direct consequence of strong field enhancement associated with the q-BIC resonance. This enhancement amplifies the contribution of higher-order nonlinear susceptibilities and thereby modifies the effective nonlinearity of the system, establishing a clear link between resonant field confinement and the unconventional HHG scaling. For more accurate power scaling predictions, additional effects such as free-carrier generation and the magnetic component of the Lorentz force could be incorporated28,42, but these are beyond the scope of the present model.

Discussion

We have studied the high-harmonic generation from a free-standing crystalline silicon membrane metasurfaces supporting q-BIC resonances. We have demonstrated a strong resonant enhancement of the HHG signal when the pump matches the q-BIC wavelength. The enhancement factor grows with the order of the nonlinear process involved, and it allows the observation of the ninth-harmonic that is beyond the detection limit for an unpatterned membrane.

Importantly, we have revealed the deviation from the conventional power scaling of the generated harmonics near the q-BIC resonance. In a stark contrast to the unpatterned membranes, for which the power laws strictly adhere to the order of the nonlinear optics processes, for resonant q-BIC metasurfaces we observe non-integer power dependencies. Our numerical analysis further demonstrates that a complete, generalized perturbative framework, incorporating higher-order nonlinear susceptibilities as well as cross-phase modulation between the harmonics, is sufficient to capture all salient features of the highly resonant metaphotonic system. This highlights that strong field enhancement enabled by the physics of q-BIC invokes extreme regimes of high-harmonic generation, providing deep insights into the transformative role of high-Q metasurfaces in nonlinear optical processes.

Methods

Device Fabrication

Silicon (Si) membrane metasurfaces were fabricated starting from Silicon on insulator (SOI) wafers: 1 μm Device Si (crystalline, 10Ω cm) − 1 μm buried oxide (SiO2) − 250 μm Handle Si. First, the backside openings are defined lithographically in 3 μm SiO2 (Plasma-Therm Corial D250L PECVD) using direct laser writing (Heidelberg Instruments Maskless Aligner MLA 150, AZ ECI 3027, 3000 rpm) and fluorine based deep reactive ion etching (DRIE) (SPTS Advanced Plasma System). Second, the device Si is patterned by a single step electron beam lithography (Raith EBPG5000, PMMA 495k 4wt% in Anisole, 4000 rpm) followed by DRIE (Alcatel AMS 200 SE). For unpatterned membranes, the EBL and device-layer patterning steps were omitted. Third, the membranes are opened by etching the handle silicon through the previously defined SiO2 mask using a DRIE Bosch process (Alcatel AMS 200 SE). Last, the buried oxide layer is removed from under the membranes by Hydrofluoric acid (HF) vapor etching (SPTS μEtch). All metasurfaces and unpatterned free-standing membranes were fabricated on the same SOI wafer to ensure identical fabrication conditions.

Infrared spectroscopy

We obtained the infrared (IR) transmission spectra at normal incidence using a Bruker Vertex 80v FT-spectrometer with an IR Microscope attachment (HYPERION 3000) equipped with a liquid nitrogen cooled MCT detector. The metasurfaces are excited using a ZnSe lens with the focal length of 50 mm mildly focusing linearly polarized IR light on the sample surface. Transmitted light was collected with another 25 mm lens equipped with an additional iris placed at its back focal plane. Closing the iris allowed for limiting the numerical aperture of the system down to approximately 0.02 and thus suppressing the unwanted signal from oblique excitation angles. Signal collection area is restricted to an approximately 300 μm square central region of the membrane by a double-blade aperture placed in the conjugate image plane of the IR microscope. The sample chamber was constantly purged with dry air to provide a stable low level of humidity.

Nonlinear spectroscopy

The sample was pumped in the mid-IR range from 3.85 μm to 4.05 μm. The laser system consists of 1030 nm laser (Ekspla Femtolux 3) and an optical parametric amplifier (MIROPA from Hotlight Systems). The laser has a pulse duration of 250 fs and a repetition rate of 1.49 MHz, which results in the MIR pulse duration of  ~ 800 fs. The optical parameter amplifier produces mid-IR radiation as idler pulses from the amplification of continuous wave spectrally narrow seed lasers in the near-IR spectrum. The mid-IR radiation was focused with the CaF2 lens with an NA = 0.03 and a focus of 40 mm on the sample. The laser spot had a diameter of about 30 μm. The maximum laser system power is 55 mW, which corresponds to a fluence of 5.5 mJ/cm2. At this power level, no damage was observed on the exposed metasurface, indicating that the actual threshold exceeds this value. The harmonic signal was collected by Mitutoyo Plan Apo NIR Infinity Corrected Objective X20 NA = 0.4 microscope objective and detected with a Peltier-cooled spectrometer Ocean Optics QE Pro for the visible range and Ocean Optics NIR Quest for near-IR. The setup diagram can be found in Section 8 of the Supplementary Information. To measure the conversion efficiency of the generated harmonics, we used an Ophir PD300-IR power meter and spectrometers. The conversion efficiency is defined as the ratio between the average power of the harmonic and that of the pump. The power of the third harmonic was measured directly with the power meter, while its spectrum was recorded by the spectrometer. This allowed us to calibrate the spectrometer counts against absolute power, particularly in cases where the signal was below the detection threshold of the power meter. Using this calibration, along with the wavelength-dependent quantum efficiency of the spectrometer, we estimated the conversion efficiencies of the higher-order harmonics based on the spectrometer measurements.

Numerical modelling

We performed our numerical simulations using the finite element method implemented in COMSOL Multiphysics. In our approach, we solve the nonlinear wave equation in the frequency domain, rigorously accounting for the spatial field distribution within the metasurface and incorporating all pertinent nonlinear source terms derived from a perturbative expansion of the nonlinear polarization. The silicon bulk nonlinear susceptibilities for the third, fifth, seventh, and ninth orders are assumed to be isotropic and are integrated with a dispersion profile extracted via an experimental-numerical ellipsometry procedure43. This dispersion is subsequently extended to higher orders using an atomic field scaling approach28,42,43,44. In our implementation, the electric field is expanded as a vector sum over its harmonic components; however, we retain only those nonlinear terms that depend on powers of the fundamental pump field, deliberately neglecting contributions arising solely from higher harmonics. This assumption is well justified by the experimentally observed low power levels at harmonic frequencies. Conversely, owing to the strong field localization in the silicon metasurface, we preserve all nonlinear terms that contribute to pump depletion, self-phase modulation among the harmonics, and higher-order effects at each harmonic frequency driven by the pump. The simulation details are provided in Section 4 of the Supplementary Information.