Introduction

While early research in metamaterials predominantly focused on the design and fabrication of three-dimensional structures, it was later recognized that many unique light–matter interactions and associated applications could be effectively realized using their two-dimensional counterparts, known as metasurfaces. Optical metasurfaces consist of periodic arrays of subwavelength unit cells, known as meta-atoms, that enable precise control over all degrees of freedom of light at the nanoscale. This level of control has enabled a broad range of applications, including polarization conversion, holography, quantum state generation, and beam steering1,2,3,4,5,6. Among the various functionalities explored, increasing attention has been directed toward the realization of nonlinear optical effects, since the subwavelength thickness of metasurfaces significantly relaxes the stringent phase-matching conditions typically required in conventional bulk nonlinear materials7,8,9. In this context, single-layer nonlinear metasurfaces have shown a wide range of functionalities, such as nonlinear wavefront shaping, generation of entangled photon pairs, and structured light formation10,11,12,13,14,15,16,17,18,19,20,21,22. Although early research in nonlinear metasurfaces focused primarily on plasmonic metasurfaces, their intrinsic optical losses and low damage thresholds have shifted attention toward dielectric platforms23,24. Silicon-based dielectric metasurfaces are especially promising due to their large refractive index, low optical losses in the near-infrared spectral range, relatively high nonlinear susceptibility, compatibility with well-established complementary metal–oxide–semiconductor (CMOS) platforms, and the ability to support various resonant modes that enhance local electromagnetic fields and strengthen nonlinear light–matter interactions25,26,27,28. In this context, dielectric metasurfaces have been shown to support a wide range of resonant phenomena, including symmetry-protected modes, Mie-like resonances, and guided mode resonances (GMRs), which have been extensively investigated for their ability to significantly enhance the performance of applications such as sensing, imaging, electro-optic modulation, and nonlinear frequency conversion29,30,31,32,33,34,35.

However, despite their many advantages, single-layer metasurfaces face fundamental limitations, particularly in enabling efficient nonlinear optical functionalities and multitasking performance. While recent advances in meta-optics research have aimed to address some of these challenges through increasingly sophisticated design strategies, conventional single-layer metasurfaces remain inherently constrained by a limited number of available degrees of freedom and a restricted nonlinear interaction length36,37. One approach to achieve multifunctionality involves spatially interleaving distinct sets of meta-atoms, each designed for a specific optical task, enabling concurrent control over multiple optical properties. Although effective, the lattice periodicity in interleaved metasurfaces is constrained by the need to minimize cross-talk between adjacent elements, which limits the achievable resolution37,38. More importantly, interleaving does not overcome the fundamental limitation of nonlinear conversion efficiency imposed by the inherently short interaction length. An alternative approach involves multilayer metasurfaces, in which multiple metasurface layers are vertically stacked37,39,40,41,42,43. This configuration allows for the integration of distinct functionalities from each layer, enabling the development of multifunctional and highly efficient devices. Recent advancements in nanofabrication have greatly improved the practicality of multilayer designs, with numerous studies suggesting their potential in realizing achromatic metalenses, broadband responses, and wavelength-selective multifunctional devices36,44,45,46,47,48.

Bridging the gap between two-dimensional metasurfaces and three-dimensional metamaterials, vertically stacked structures introduce new degrees of freedom that give rise to rich physical phenomena, including the formation of a transmission gap, interlayer coupling, and symmetry breaking49,50,51. For example, Tanaka et al. demonstrated the potential of multilayer metasurfaces to enhance optical chirality52. Vincenti et al. theoretically predicted third-harmonic (TH) generation enhancement in a three-dimensional array of nanowires suspended in air, achieved by overlapping Mie resonances from individual metasurfaces, resulting in a transmission gap due to vertical stacking53. In such stacked architectures, the vertical spacing plays a critical role by controlling Fabry–Pérot (FP)-like cavity modes and the strength of interlayer coupling. Furthermore, the in-plane meta-atom fill fraction and horizontal displacement can break both transverse and vertical symmetries, thereby enriching the modal landscape. For instance, Alagappan et al. applied temporal coupled-mode theory to analyze FP resonances in bilayer metasurfaces, revealing how cavity length affects the resonant response and associated quality factor54. Similarly, Suh et al. theoretically investigated the coupling behavior between two photonic crystal slabs, showing that the reflection spectrum is highly sensitive to interlayer separation. Notably, when the gap between slabs becomes smaller than the periodicity of the structure, near-field coupling dominates the light–matter interaction, substantially altering the optical response55. Although less extensively explored, horizontal displacement between stacked layers has been shown to lift the degeneracy of singly degenerate states and to enable the emergence of novel optical modes55,56,57. In this context, Nguyen et al. engineered Moiré structures by tuning the relative periodicity of bilayer photonic crystal slabs58. In addition, Gromyko et al. demonstrated that horizontal displacement combined with in-plane meta-atom asymmetry can effectively control unidirectional optical chirality59. Tal et al. also theoretically showed that combining the bilayer structure with the geometric phase introduces an additional degree of freedom for controlling second-harmonic generation60.

While vertically stacked metasurfaces have emerged as a promising strategy to extend the functionality of planar photonic platforms, most studies have primarily focused on their linear optical properties. As a result, the nonlinear regime, particularly the experimental demonstration of harmonic generation in these structures, remains largely unexplored. In this work, we design and experimentally demonstrate nonlinear bilayer all-dielectric metasurfaces that achieve enhanced third-harmonic generation in the ultraviolet regime. This enhancement arises from the combined effects of transmission gap engineering and coupled guided mode resonances, as illustrated in Fig. 1a. The design process begins with a single-layer amorphous silicon metasurface that supports a Mie-type resonance, which serves as the fundamental building block for nonlinear enhancement, as shown in Case 1 of Fig. 1a. Building on this foundation, vertical stacking of Mie-resonant metasurfaces separated by a spin-on-glass spacer enables the formation and spectral tuning of a transmission gap. This configuration provides a higher nonlinear enhancement than the single-layer counterpart, as shown in Case 2. Further improvement is achieved by introducing a horizontal displacement between the stacked layers. This shift creates a new high-quality factor resonant mode that emerges from the simultaneous breaking of in-plane and out-of-plane symmetries, as illustrated in Case 3. The resulting interplay between vertical integration and horizontal asymmetry leads to strong field confinement and nonlinear enhancement that significantly exceeds those produced by transmission gap formation alone. These results are supported by both numerical simulations and experimental measurements, enabled by optimized fabrication procedures that ensure nanoscale precision in planarization and alignment. In addition to establishing a robust framework for high-efficiency nonlinear flat optics, it is demonstrated that multilayer metasurfaces offer a versatile platform for multifunctional nonlinear devices with precisely tailored spectral and spatial responses. This work opens new possibilities for compact ultraviolet light sources, advanced quantum photonics, and fully integrated nonlinear photonic systems.

Fig. 1: Schematic illustration of the proposed multilayer metasurface and its underlying physical mechanisms.
Fig. 1: Schematic illustration of the proposed multilayer metasurface and its underlying physical mechanisms.
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a The structure consists of two layers of a-Si gratings with \(w=250\) nm, \(h=268\) nm, and \(p=600\) nm, separated vertically by a 300 nm SOG spacer. Each single-layer metasurface (case 1) supports a Mie-type resonance at 1230 nm, providing the basic building block for nonlinear enhancement. When two identical layers are vertically stacked without horizontal displacement (case 2), the interaction between their resonances produces a transmission gap, shifting the spectral response and enabling greater control over light–matter interaction. Introducing a horizontal displacement between the stacked layers (case 3) simultaneously breaks in-plane and out-of-plane symmetries, giving rise to a newly emerged high-Q resonant mode near 1050 nm, resulting in strong field confinement and nonlinear enhancement exceeding that provided by transmission-gap effects alone. b Schematic illustration of the coupled vertically and horizontally displaced metasurfaces, each supporting forward- and backward-propagating modes ai±, i = 1,2, and vertical phase shift \(\psi\) (left) with different coupling mechanisms between four forward- and backward-propagating modes of two metasurfaces (right).

Results

Design of bilayer metasurfaces

Light-matter interactions in a bilayer metasurface can be described through the coupling of the forward- and backward-propagating modes of the two layers. These interactions may occur through different physical channels, such as forward- and backward-propagating (intra-)mode coupling within each metasurface, inter-metasurface mode coupling, and mode leaking into radiation58,61,62,63,64. The optical response of the stacked metasurface system can be described by the coupled-mode theory that can be written in an eigen equation of a 4 × 4 non-Hermitian Hamiltonian61, given by \(H=\left(\begin{array}{cc}{H}_{11} & {H}_{12}\\ {H}_{21} & {H}_{22}\end{array}\right),\) where \({H}_{11}\) and \({H}_{22}\) are 2 × 2 matrices that describe mode coupling inside each metasurface, and \({H}_{12}\,\) and \({H}_{21}\) describe light coupling between different metasurfaces. The schematic illustration of these coupling mechanisms is shown in Fig. 1b. The intralayer terms \({\kappa }_{\mathrm{1,2}}\), \({\gamma }_{\mathrm{1,2}}\), and \({\beta }_{\mathrm{1,2}}\) capture the coupling between counter-propagating modes within a single metasurface, the leakage of guided modes into radiation, and the evanescent influence of one metasurface on the other. The interlayer contributions, expressed through \({\kappa }_{d}\) and \({\kappa }_{c}\), account for diffractive/evanescent transfer between co-directional modes and evanescent coupling between counter-propagating modes in different metasurfaces61 (see Supplementary Material for additional details). When the metasurfaces are sufficiently separated, the coupling terms \({\beta }_{\mathrm{1,2}}\),\(\,{\kappa }_{d},\) and \({\kappa }_{c}\) tend to zero, and the only interaction that remains between the two metasurfaces is through radiation. This radiative pathway is governed by the vertical phase \(\psi\), which represents the phase accumulated within the metasurfaces and along the effective transverse optical distance between them. Additional phase factors further contribute to the mode interaction. In particular, the displacement phase \(\phi\) reflects the horizontal offset between the two metasurfaces, while the radiation-coupling phases \({\varphi }_{i\pm }\) describe how forward and backward propagating modes couple to the continuum. Together, these coupling coefficients and phases form the basis of the non-Hermitian Hamiltonian, enabling a unified description of the interplay between guided, evanescent, and radiative channels (see Supplementary Material 5 and Supplementary Fig. S6 for rigorous theoretical treatment).

We start with a basic design and progressively add complexity to reveal how each element contributes to the enhanced nonlinear response. In particular, we begin with a single metasurface layer that supports a Mie resonance, which provides the fundamental building block for our approach (case 1). The metasurface consists of amorphous silicon gratings with a width of 250 nm, a height of 268 nm, and a periodicity of 600 nm. Figure 2a illustrates the transmission spectrum of the single-layer metasurface, where an isolated dip appears near 1230 nm, and the corresponding field distribution in the inset confirms its magnetic dipole Mie-type resonance origin. When two such layers are vertically stacked without horizontal displacement (case 2), their interaction results in a transmission gap, characterized by a high-reflectivity region, as shown in Fig. 2a. In particular, numerical simulations of the electric field distribution for a fixed interlayer spacing of 300 nm show strong field enhancement at wavelengths near the band edge, as shown in the inset of Fig. 2a. We focus on a two-layer metasurface structure as a proof of concept due to its simpler fabrication procedure. In this two-layer configuration, the suppressed transmission region arises predominantly from coupling between the two resonant metasurfaces. Nevertheless, this coupling-induced suppression exhibits an optical behavior reminiscent of a bandgap, as illustrated in Supplementary Fig. S10 of the Supplementary File. In particular, the bilayer system supports a broad transmission gap of approximately \(\Delta {\rm{\lambda }}\approx 100\) nm that persists over incident angles spanning \(-0.35\) rad to \(0.35\) rad.

Fig. 2: Linear and nonlinear optical responses of vertically stacked metasurfaces.
Fig. 2: Linear and nonlinear optical responses of vertically stacked metasurfaces.
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a Comparison of transmission spectra for single-layer (blue curve) and double-layer (red curve) structures with a 300 nm interlayer spacing. The Mie resonance in the single-layer design evolves into a transmission gap upon stacking, resulting in stronger field confinement. b Nonlinear simulation results showing the THG conversion efficiency for four configurations of a single-layer unpatterned film (1 layer UF, black curve), a two-layer unpatterned (2 layer UF, green curve), a single-layer metasurface (1 layer MS, blue curve), and a two-layer metasurface (2 layer MS, red curve), evaluated at fundamental wavelengths of 1278 and 1300 nm. The curves are normalized by dividing each value by the maximum efficiency observed across all cases, enabling direct comparison. The single-layer metasurface shows an enhancement of 10 compared to the unpatterned single-layer film, while the two-layer metasurface achieves an enhancement of 140 relative to the two-layer unpatterned configuration, and further demonstrates a ~10-fold increase compared to the single-layer metasurface at the band-edge wavelength. c Simulated transmission spectra of the two-layer structure for varying interlayer spacings of 200 nm, 300 nm, and 400 nm as a function of operating wavelength.

To better understand the optical response of the multilayer metasurface, we examine the various mechanisms of symmetry breaking that govern its behavior. The first originates from in-plane asymmetry within each metasurface, due to an asymmetric filling fraction between the high- and low-index regions of the gratings, rather than an equal 50% split. The second source of asymmetry arises from the horizontal displacement between stacked layers, which simultaneously breaks in-plane and out-of-plane symmetries and gives rise to hybridized resonant modes. The third one stems from vertical asymmetry due to a refractive-index mismatch between the substrate and the superstrate. Together, these distinct mechanisms enable precise spectral control and enhance the nonlinear response. In addition to symmetry breaking, the vertical spacing between the metasurface layers provides another key parameter for tuning the spectral properties. The interlayer distance governs the strength of coupling between the two metasurfaces, thereby influencing both the position of the transmission gap and the degree of local field enhancement. Figure 2b demonstrates the nonlinear simulation results for third-harmonic generation (THG) from four different configurations of a single-layer unpatterned a-Si thin film (black curve), a two-layer a-Si unpatterned thin film (green line), a single-layer metasurface (blue curve), and a two-layer metasurface (red line). We note that for a direct comparison across configurations, all curves are normalized by dividing the THG conversion efficiency at each wavelength by the maximum observed across all structures, thereby highlighting relative enhancements and revealing how patterning and vertical stacking collectively strengthen the nonlinear response. To establish a clear baseline, we first simulated unpatterned thin films consisting of one and two layers with the same asymmetric material configuration and then compared these with their patterned metasurface counterparts. In particular, the single-layer metasurface shows a 10-fold enhancement compared to the unpatterned single-layer film, while the two-layer metasurface achieves a 140-fold enhancement compared to the two-layer unpatterned film. Beyond these improvements, introducing the second patterned layer produces strong interlayer coupling and resonance hybridization, which together lead to ≈10-fold higher THG efficiency at the band-edge wavelength compared to the single-layer metasurface [Fig. 2b] (see Methods for more details). The apparent spectral offset between the resonant dips in Fig. 2a and the wavelengths yielding the strongest nonlinear signal in Fig. 2b arises from the fact that efficient third-harmonic generation requires more than simply exciting a resonance at either the fundamental or harmonic wavelength. In general, the nonlinear response of a nanophotonic structure depends on both the magnitude of the local field enhancement and the spatial overlap of the interacting fields within the nonlinear material65,66,67,68. Thus, the wavelength associated with the strongest THG results from a balance between resonant enhancement at the fundamental frequency and the spatial overlap of the fundamental and third-harmonic modes inside the resonator. To quantify this effect, we computed the wavelength-dependent spatial mode overlap, which captures the combined influence of field enhancement and spatial matching between the interacting modes (see Supplementary Section 8 of the Supplementary File). The resulting overlap spectra show that the strongest nonlinear interaction occurs at wavelengths slightly detuned from the linear resonance dip, where the third-harmonic field attains its highest spatial overlap with the fundamental mode. Next, we investigate the effect of vertical spacing between the metasurfaces on the overall transmission spectrum. Figure 2c shows transmission spectra for the interlayer distances ranging from 200 nm to 400 nm, over the wavelength range of 1000 nm to 1500 nm (see Supplementary Material 3 and Supplementary Fig. S4 for additional results). As shown in this figure, varying the distance between the constituent metasurfaces not only broadens the transmission gap but also shifts it toward shorter wavelengths. This behavior is consistent with changes in interlayer coupling induced by the modified separation, confirming that both symmetry-breaking effects and vertical spacing collectively define the nonlinear optical response of the platform. We note that while vertical stacking functions as a critical parameter for tailoring the optical response of the bilayer metasurfaces, horizontal displacement between the layers can also introduce an additional degree of freedom, enabling enhanced control over the spectral characteristics. In this perspective, by extending this configuration, we introduce a deliberate horizontal displacement between the stacked layers, which simultaneously breaks in-plane and out-of-plane symmetries and gives rise to sharp guided-mode resonances (case 3).

To investigate this effect, we fixed the interlayer spacing at 300 nm and studied the transmittance spectra as a function of horizontal displacements ranging from \(0 < \Delta < 300\) nm. Figure. 3a illustrates the resulting optical response, where at \(\Delta =0\) nm and \(\Delta =300\) nm, no sharp resonance is observed. In contrast, at intermediate displacements, a distinct resonant mode emerges in the vicinity of 1050 nm, with its Q-factor strongly dependent on the displacement. In particular, as the horizontal shift increases from 0 nm to 150 nm (corresponding to half of the grating period), the emerged resonant mode broadens, reaching its maximum width at \(\Delta =\) 150 nm, and then narrows again as the displacement approaches \(\Delta =300\) nm. The disappearance of resonances at both zero and half-period displacements can be understood directly from Hamiltonian formalism (see Supplementary Note 5). In these two special cases, the phase parameter, which arises from the horizontal displacement \(\Delta\) between the gratings and governs higher-order diffraction processes, takes values of \(\phi =0\) (\(\Delta =0\)) or \(\phi =\pi\) (\(\Delta =P/2\)), reducing the coupling terms to real numbers. As a result, the Hamiltonian of the misaligned system becomes identical to that of the aligned structure (\({H}_{\Delta =0}={H}_{\Delta =P/2}\)), leading to eigenvalues that remain purely real and resonant modes that are dark under normal incidence. In particular, the Hamiltonian of the misaligned system and its corresponding eigenvalues are reduced exactly to those of the aligned case, meaning that the misaligned structure inherits the same behavior as its aligned counterpart. Moreover, because the resonances vanish with zero linewidth at both of these limits, the linewidth must evolve continuously between them. As the misalignment increases from zero toward a quarter period, the modes begin to leak, and the linewidth broadens, as shown in Fig. 3a. Beyond the quarter period, the linewidth gradually decreases again, ensuring that it reaches zero at the half-period displacement. This evolution explains the periodic behavior observed in the system, with the resonance linewidth expanding and then contracting to satisfy the boundary conditions imposed by the two symmetry-protected dark states. To further clarify the mechanism underlying the emerged resonant mode, we studied the electric field distributions for three representative cases of zero displacement (Δ = 0) and a horizontal displacement of \(\Delta =100\) nm, evaluated both on and off resonance as shown in Fig. 3b. As shown in these figures, in the absence of displacement, the electric field inside the meta-atom is primarily aligned with the incident polarization (perpendicular to the gratings), and no notable field enhancement is observed. In contrast, introducing a 100 nm displacement induces strong asymmetry in the field distribution, generating a nonzero z-component of the electric field within the meta-atom and resulting in pronounced field confinement at resonance. We note that the simulation results presented in Fig. 3a, b were obtained for an asymmetric configuration in which the first-layer metasurface was embedded in glass. At the same time, the second layer was exposed to air. This arrangement inherently introduces asymmetry between the substrate and superstrate, which can modify the resonance conditions and induce GMRs. To evaluate the influence of the surrounding dielectric environment, we performed additional simulations with the entire bilayer metasurface fully embedded in glass, thereby creating a symmetric configuration. Under this condition, we repeated the same set of horizontal displacement sweeps and corresponding electric field analyses used in the asymmetric glass–air case, with the results summarized in Fig. 3c, d. As shown in these figures, in the symmetric configuration, the resonant mode observed in the asymmetric case no longer appears as a single feature but instead splits into two distinct modes. In particular, as the horizontal displacement increases, these modes approach one another and merge at \(\Delta =150\) nm, which corresponds to one quarter of the grating period, before separating again as the displacement continues to grow. The difference between the spectral responses observed in Fig. 3a, c arises from the intrinsic sensitivity of the coupled bilayer system to its surrounding environment. In particular, all the coupling terms of the system, including radiation from each layer, interactions between the two layers, and coupling to external channels, are dictated by the modal field distributions, which depend strongly on the refractive indices of the media above and below the metasurfaces. When the environment is symmetric, the two layers radiate into identical optical channels, resulting in hybrid modes with balanced leakage rates. In contrast, an asymmetric environment leads each grating to radiate into a different optical channel, thereby introducing unequal loss pathways. This asymmetry thus alters the Hamiltonian of the system and yields the modification of the resonance frequencies, linewidths, and phase responses of the hybrid modes, leading to the distinct spectral behavior seen in the asymmetric configuration.

Fig. 3: Horizontal displacement-induced resonances and nonlinear enhancement in bilayer metasurfaces.
Fig. 3: Horizontal displacement-induced resonances and nonlinear enhancement in bilayer metasurfaces.
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a Simulated transmission spectra of the bilayer structure with the air on top as a function of horizontal displacement, showing a resonance that broadens and then sharpens again with increasing displacement. The dashed line highlights the corresponding misalignment cases that were experimentally measured. b Simulated electric field distribution and vector distribution for three cases of 0 nm displacement at 1050.6 nm, 100 nm displacement at 1050.6 nm (on-resonance), and 100 nm displacement at an off-resonance wavelength. c For comparison, simulated transmission spectra of the bilayer structure with a glass superstrate reveal two resonances that intersect and subsequently split as the horizontal displacement varies. d Corresponding electric field intensity and direction distributions for the glass-superstrate case, evaluated at 0 nm displacement at 1050.6 nm, 100 nm displacement at 1050.6 nm (on-resonance), and 100 nm displacement at an off-resonance wavelength. e THG conversion efficiency for the misaligned two-layer metasurface (\(\Delta \,=\,250\) nm, red), aligned two-layer metasurface (blue), and two-layer unpatterned film (green). The misaligned configuration shows an enhancement of 4 \(\times\)108 over the aligned case at the same spectral position, 2.5 \(\times\)108 over the unpatterned film, and 1.5 \(\times\)105 when compared at their respective resonances. f Total THG conversion efficiency decomposed into forward and backward components.

To examine the nonlinear implications of these resonances, we studied the THG conversion efficiency under a controlled horizontal misalignment between the two patterned layers, fixing the displacement at \(\varDelta \,=\,250\) nm and evaluating the response at the resonant wavelength of 1050.65 nm, as shown in Fig. 3e (red curve). This misaligned case was then directly compared with the perfectly aligned two-layer metasurface (blue) and the two-layer unpatterned film (green), with all results normalized by dividing the THG efficiency at each wavelength by the maximum value observed among the three configurations, thereby revealing how misalignment alters the strength of interlayer coupling and, in turn, reshapes the nonlinear response. As shown in this figure, the misaligned case yields an enhancement of 4 ×108 relative to the two-layer aligned configuration, since at the spectral position corresponding to the misaligned resonance, the aligned structure lacks a resonance and therefore lacks the local field enhancement necessary to boost the nonlinear response. Compared with the unpatterned two-layer film, the misaligned metasurface shows an additional enhancement of 2.5 \(\times\)108, clearly illustrating the effect of horizontal displacement. We note that although the resonant wavelengths of the aligned and misaligned structures do not coincide, for completeness, we also compared the misaligned case at its corresponding spectral resonance position with the aligned structure at its own resonance, revealing an additional enhancement of 1.5 \(\times\)105. This enhancement of the nonlinear response is attributed to the substantially stronger local-field buildup in the misaligned configuration compared to the aligned case. We also performed a linear simulation over the THG wavelength range to confirm that there is no resonant response in the structure at the third-harmonic wavelength in the linear transmission spectrum (see Supplementary Fig. S5 in the Supplementary Material).

Moreover, Fig. 3f shows that the extent of THG enhancement depends sensitively on the horizontal displacement, which we attribute to variations in the relative amplitudes of the coupled modes supported by the bilayer structure. To further assess this effect, we systematically varied the horizontal offset between the two layers and calculated the corresponding nonlinear response, confirming that different displacements lead to distinct resonance conditions and field distributions, thereby demonstrating that vertical stacking combined with controlled horizontal displacement offers multiple degrees of freedom for tailoring the spectral response and significantly boosting third-harmonic generation in the ultraviolet regime. We note that while recent advances have shown that single-layer metasurfaces supporting quasi–bound states in the continuum can exhibit strong nonlinear responses due to their high-Q resonances69,70,71, the approach introduced here differs in both concept and function. By vertically stacking two dielectric nanostructures and introducing a controlled in-plane displacement, the bilayer system accesses an interaction regime that cannot be realized in a single layer or in a perfectly aligned multilayer arrangement. The coupling between the two layers, mediated through radiative, nonradiative, and near-field interaction channels, provides a broad and flexible design space in which the overall optical response is not constrained to any particular resonance type. In this perspective, the same symmetry-breaking configuration can be applied to layers supporting identical or distinct resonant modes, including QBIC resonances, while maintaining independent control over their coupling strength.

Fabrication and characterization

To verify the theoretical predictions, we fabricated four multilayer metasurface samples with lateral displacements of Δ≈[0, 100, 150, and 250] nm, while maintaining a constant spacer thickness of 300 nm between the two layers in all cases. Due to fabrication constraints, we focus here on the asymmetric substrate–superstrate configuration (glass–air), noting that the underlying physical mechanisms described earlier remain valid for both symmetric and asymmetric configurations. Figure 4a demonstrates the fabrication flowchart where we implemented a multilayer process involving PECVD deposition of amorphous silicon, electron-beam lithography with a chromium mask, reactive ion etching, and the use of a SOG spacer to embed and separate the two metasurface layers, with alignment marks ensuring accurate registration (see Methods section). Representative scanning electron microscope (SEM) images, including tilted and cross-sectional views, are shown in Fig. 4b. The measured grating dimensions are in close agreement with the design specifications, and the cross-sectional SEM images confirm the intended horizontal misalignments, consistent with the target values (see Supplementary Fig. S3 for additional SEM images with varying displacements). We note that SEM measurements typically allow for an accuracy of ~20 nm, which implies that even in the nominally aligned case (\(\Delta =0\) nm), a slight horizontal shift on the order of 10 nm may still be present, and the effective degree of misalignment is therefore more reliably characterized through the optical response, as will be discussed later. To quantify structural characterization–induced deviations, we used focused ion beam milling followed by SEM imaging to obtain cross-sectional profiles at five locations: bottom left, bottom right, top left, top right, and center for the following nominal misalignment values \({\varDelta }_{{\rm{Nominal}}}\,=\,[\mathrm{0,100,150,250}]\) nm. The average measured values across these regions were found to be \({\varDelta }_{{\rm{Measured}}}\) = [11,102,151,245] nm with standard deviations of ~±10 nm. However, we note that while these measurements provide a reliable statistical estimate of alignment accuracy, it is important to emphasize that they are derived from images captured by the FIB/SEM system and are thus subject to the intrinsic resolution limits of the instrument. Even a marginal uncertainty in identifying the interface boundaries (equivalent to only a few pixels) can result in deviations of several tens of nanometers. These apparent variations arise primarily from limitations in image resolution and contrast, rather than from the measurement or analysis procedure itself. Although higher magnification or extended beam exposure can, in principle, yield finer spatial resolution, they commonly produce charging artifacts and image distortions, which, in turn, limit the reliability of the resulting measurements. As a result, the reported values should be viewed as our best achievable estimates, with the understanding that the actual misalignment may still vary slightly within this range. This consideration is particularly important for our system, which is highly sensitive to nanometer-scale variations in geometry, unlike Mie-resonant or non-resonant metasurfaces, where such small deviations have a negligible effect. Nevertheless, the primary objective of this work is to demonstrate the concept of multiple symmetry breakings in bilayer systems and to validate their effects in the nonlinear regime, and we expect that future advances in fabrication and alignment control will further reduce such deviations.

Fig. 4: Fabrication process and SEM images of the fabricated multilayer metasurfaces.
Fig. 4: Fabrication process and SEM images of the fabricated multilayer metasurfaces.
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a Flowchart of the developed nanofabrication process. The structures were fabricated through a multilayer process comprising PECVD deposition of amorphous silicon, electron-beam lithography with a chromium mask, reactive-ion etching, and a spin-on-glass spacer to embed and separate the two metasurface layers. b Tilted-view image showing the grating dimensions of a single layer. c Cross-sectional view of the multilayer structure with the in-plane displacement of 100 nm.

The fabricated samples were characterized using white-light transmission spectroscopy in an in-house-built setup to verify their optical response (see Method). As shown in Fig. 5a, b, the single-layer metasurface shows a pronounced dip in the transmittance spectrum near the designed resonance wavelength, and the excellent correspondence with the simulated response confirms the excitation of the targeted Mie-type mode. When transitioning from a single to a bilayer configuration, the transmission spectra exhibit the emergence of a transmission gap that aligns well with theoretical predictions. However, slight spectral shifts are evident, which can be attributed to fabrication-induced deviations in parameters such as grating dimensions, spacer thickness, and alignment accuracy. The role of horizontal displacement is further highlighted in Fig. 5c, which presents the measured spectra for four representative misalignment values. Clear resonance peaks appear near 1050 nm for the 100 nm and 150 nm displacement cases, with the latter showing a small redshift to 1070 nm, a behavior we ascribe to minor variations in the fabricated geometry. By contrast, for the nominally aligned (0 nm) and 250 nm displacement samples, the sharp resonances predicted by simulations are not distinctly resolved, a limitation we attribute to the finite spectral resolution of the measurement system, which may obscure narrow-linewidth modes. Taken together, these measurements substantiate the central design principle of the bilayer metasurface platform and demonstrate that horizontal displacement serves as a powerful degree of freedom for inducing and tuning guided-mode resonances.

Fig. 5: Experimental validation of the linear and nonlinear optical response of multilayer metasurfaces.
Fig. 5: Experimental validation of the linear and nonlinear optical response of multilayer metasurfaces.
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Simulated (blue) and measured transmission spectra (red) obtained from a white-light spectroscopy setup for a the single-layer and b aligned bilayer structures. c Measured transmission spectra for four samples with different horizontal displacements of \(\Delta \approx \left[0,\,100,150,\,250\right]\). d Measured THG signal of single-layer (black dots), aligned bilayer metasurface (red dots), and bare silicon film (blue dots) with pumping wavelength ranging from 1200 nm to 1400 nm. e Measured THG signal at the GMR resonance wavelength for the multilayer structure with different horizontal displacements, with pumping wavelength varying from 1050 nm to 1125 nm.

Next, we characterized the nonlinear response of the fabricated samples using a femtosecond pulse laser and an experimental setup described in the Methods section (see the Supplementary Material for more details). In these measurements, the pump wavelength of the fundamental beam was swept from 1200 nm to 1400 nm, covering the spectral range relevant for third-harmonic generation. Figure 5d compares the measured TH signals from the aligned bilayer metasurface, the single-layer metasurface, and a bare silicon thin film. The single-layer metasurface exhibits a peak efficiency at a pump wavelength of ≈1300 nm, consistent with a Gaussian fit associated with excitation of the Mie resonance, whereas the bilayer metasurface shows a peak response near 1330 nm. At their respective peak wavelengths, the bilayer metasurface produces a THG signal that is \(\approx 3.5\) times stronger than that of the single-layer device and nearly \(30\) times higher than that of the bare silicon film. We note that both resonances redshift relative to their designed spectral position, which can be attributed to two main factors: (i) the Kerr effect, where the pump increases the refractive index according to \(n={n}_{0}+{n}_{2}I\), thereby raising the effective index and shifting the wavelength to a longer regime, and (ii) fabrication deviations from the nominal design that further contribute to the observed displacement. It should also be noted that although the observed measurements represent a substantial experimental enhancement, the measured values fall below the simulated predictions, a discrepancy we attribute to fabrication imperfections and the finite size of the sample, which introduces additional optical losses and reduces the achievable conversion efficiency. The simulated spectrum for various key geometric parameters resulting from fabrication defects is shown in Supplementary File S8. In a separate set of measurements conducted at an average pump power of 50 µW, we investigated the wavelength-dependent THG signal by sweeping the pump wavelength from 1050 nm to 1125 nm in the vicinity of the predicted high-Q guided-mode resonance, as shown in Fig. 5e. As shown in this figure, strong enhancements in THG intensity were observed for samples with horizontal displacements of 100 nm, 150 nm, and 250 nm. In contrast, no enhancement was detected for the aligned (0 nm) case, thereby confirming that the nonlinear peak arises exclusively from nonzero horizontal displacement. The maximum THG intensities were measured at pump wavelengths of 1080 nm, 1085 nm, and 1075 nm for displacements of 100 nm, 150 nm, and 250 nm, respectively. Compared with the simulations, which show a resonance peak near 1050 nm, the experimental results consistently show a redshift. This trend is consistent with the behavior observed in Fig. 5d and can largely be attributed to the Kerr effect, where strong field confinement at resonance increases the effective refractive index (\(n={n}_{0}+{n}_{2}I)\) and shifts the mode to longer wavelengths.

As shown in Fig. 5e, the sample with \(\varDelta \,=\,250\) nm exhibits the strongest nonlinear response among the measured structures. We note that the efficiency of third-harmonic generation under ultrafast excitation depends not only on the strength of the field confinement provided by the resonance but also on the degree of spectral overlap between the resonance linewidth and the broadband spectrum of the pulse at fundamental frequency. In contrast to continuous-wave or narrowband excitation, where a higher Q-factor monotonically increases the local field enhancement, a femtosecond pulse interacts effectively only through the spectral components that fall within the resonant bandwidth. If the resonance is too narrow, only a small fraction of the pulse spectrum is enhanced; if it is too broad, the peak enhancement is reduced. The optimal nonlinear response is therefore obtained when the effective resonance linewidth matches the spectral width of the pulse. In our measurements, the resonances associated with the Δ = 150 nm and Δ = 250 nm structures are better spectrally overlapped with the 100 fs pulse compared to those of the Δ = 50 nm and Δ = 100 nm samples. This observation is fully consistent with the experimentally measured third-harmonic signals, which reach their maximum for the Δ = 250 nm sample, corresponding to the balance between strong resonant confinement and spectral overlap. Overall, the measured THG response reflects a competition between local field confinement, which dictates the strength of the nonlinear interaction within the meta-atoms, and spectral overlap with the broadband excitation, highlighting a fundamental design trade-off in resonant nonlinear metasurfaces.

Discussion

We designed and experimentally demonstrated double-layer all-dielectric metasurfaces that achieve efficient third-harmonic generation in the ultraviolet regime by combining transmission gap engineering with coupled guided-mode resonances. Vertical integration of Mie-resonant amorphous silicon metasurfaces, separated by a spin-on-glass spacer, allowed precise control over the photonic transmission gap. At the same time, horizontal displacement between the layers introduced simultaneous in-plane and out-of-plane symmetry breaking, giving rise to hybridized resonant modes with strong field confinement. These coupled resonances produced a nonlinear response far exceeding that enabled by transmission gap effects alone. Through optimized fabrication with nanoscale precision, we validated the predicted performance and demonstrated the feasibility of this multilayer approach. Taken together, our results establish multilayer metasurfaces with engineered asymmetries as a versatile platform for efficient nonlinear optical manipulation, offering extended geometric degrees of freedom to design complex, multifunctional responses and paving the way for compact quantum light sources. In particular, by implementing the design in second-order nonlinear materials, the same bilayer system can operate as a chip-scale spontaneous parametric down-conversion (SPDC) source with enhanced pair-generation efficiency. At the same time, appropriately tuned interlayer coupling enables tunable photon generation rates, which is critical for miniaturized quantum technologies.

Methods

Fabrication

A thin film of 300 nm a-Si was deposited onto a glass substrate using plasma-enhanced chemical vapor deposition (PECVD, Oxford Instruments Plasma Lab 100). Then, a 300 nm thick e-beam resist (ZEP 520A, Zeon) was spin-coated onto the Si thin film, followed by pre-baking at 180 °C to solidify the resist. To mitigate charging issues during electron-beam lithography (EBL), a 40-nm-thick discharge layer (DisCharge H2Ox2, DisChem) was spin-coated on top. The EBL process (Elionix ELS-7500 EX E-Beam Lithography System) was performed, followed by development to create the mask. A 30-nm-thick chromium (Cr) layer was then deposited using an electron-beam evaporator (CHA Industries Solution E-Beam), and the lift-off process was performed in 1165 solvent. The Cr mask was then used to etch the silicon layer down to a thickness of 268 nm, after which the Cr was removed via wet etching. For the spacer, SOG (Desert Silicon) was spin-coated on top of the first layer. The structure was then baked and cured in an N₂ environment. The liquid-phase SOG ensured precise filling of the structure without air voids, embedding the first layer metasurface entirely within the SiO₂ background. The fabrication process for the second layer followed the same steps, including Si deposition, EBL, Cr deposition, lift-off, Si etching, and Cr wet etching. Four alignment marks were used to ensure precise alignment of the two layers.

Numerical simulations

The numerical simulations were performed using the finite element method (FEM) implemented in the commercial software COMSOL Multiphysics. Specifically, the Wave Optics Module was employed to solve Maxwell’s equations in the frequency domain under appropriate boundary conditions. The metasurface unit cells were illuminated by a plane wave propagating along the z-axis with the electric field polarized along the x-axis. Periodic boundary conditions were applied, while perfectly matched layers (PMLs) were implemented along the z-axis to suppress spurious reflections. Periodic ports were also defined along the z-axis to launch the incident plane wave and collect the transmitted power. For the nonlinear simulations, the undepleted pump approximation was adopted through a two-step procedure. First, the linear scattering at the pump wavelength was calculated to determine the induced nonlinear polarization inside the meta-atoms. This nonlinear polarization was then introduced as a source in a subsequent electromagnetic simulation at the harmonic wavelength to compute the TH field. The THG conversion efficiency is then calculated as the ratio of the total output power at the THG wavelength to the total incident power of the fundamental pump beam.

Linear characterization

The linear transmittance of the designed metasurfaces was characterized using a supercontinuum white-light laser source (SuperK Fianium, NKT Photonics). The incident beam was first polarized along the desired direction by a Glan–Thompson polarizer (Thorlabs GTH10) and then weakly focused onto the sample. The transmitted light was collected using a 10× infinity-corrected objective with a numerical aperture of 0.25 (Olympus Plan Achromat Objective) and coupled into a multimode fiber connected to a wide-range optical spectrum analyzer (AQ6374 OSA). The schematic demonstration of the setup is shown in Supplementary Fig. S1. The transmittance spectrum was determined as \(T(\lambda )\,=\,I(\lambda )/{I}_{0}(\lambda )\), where \(I(\lambda )\) denotes the transmitted intensity spectrum measured with the sample inserted into the beam path and \({I}_{0}(\lambda )\) represents the reference intensity spectrum measured with only the bare substrate present.

Nonlinear measurements

An 800 nm fundamental beam was generated by a Ti: sapphire laser system (Libra, Coherent) operating at a 1 kHz repetition rate with a pulse duration of 100 fs. This output was directed into an ultrafast optical parametric amplifier (TOPAS-C, Light Conversion), providing tunable radiation across the spectral range of 260–2600 nm. A 4 f imaging system consisting of two lenses was employed for precise sample alignment and imaging of the metasurface arrays. The incident beam was weakly focused onto the sample using a 75 mm CaF₂ plano-convex lens, while the generated TH signal was collected with a high NA lens. Residual fundamental light was suppressed by a set of short-pass filters before detection. The TH photons were then coupled into a multimode fiber (NA = 0.50, Ø400 µm core, FP400URT, Thorlabs) connected to a UV–VIS–NIR spectrometer (Super Gamut, BaySpec Inc.). A manual filter wheel equipped with neutral density filters was used to control the incident pump power. The schematic demonstration of the setup is shown in Supplementary Fig. S2.