Introduction

Optical asymmetry refers to the unequal response of a system to forward and backward signals, enabling functionalities, such as unidirectional transmission, isolation, and asymmetric frequency conversion in photonic platforms1. While Lorentz reciprocity can be broken by active modulations, including magneto-optical effects2,3 and spatio-temporal modulation4,5,6, asymmetric responses can also arise from passive optical nonlinearities7. Among various approaches, nonlinear metasurfaces with magneto-electric coupling8 of resonant modes have attracted substantial interest for realizing bias-free and compact devices exhibiting directional asymmetry9,10,11,12. By employing patterned dielectric resonators with engineered nonlinearities to encode distinct emission profiles, flexible asymmetric frequency conversion and image generation have been demonstrated at nanoscales13. Using InGaP metasurfaces, continuous tuning of asymmetric biphoton polarization entanglement can be realized through wavelength-controlled nonlinear optical responses14. For asymmetric transmission, hybrid silicon-VO₂ metasurfaces exploit optically induced phase transitions to achieve broadband operation over 100 nm and low-threshold responses at intensities as low as 150 W/cm²15. Despite these advances, current mechanisms primarily rely on localized nonlinear or phase-transition responses, limiting further improvements in conversion efficiency and spectral modulation capabilities.

Nonlocal metasurfaces, which leverage inter-unit interactions as an additional degree of freedom, offer promising opportunities for spectral and dispersion modulation16,17,18,19,20. They have been widely explored in applications, such as narrowband wavefront shaping21,22,23, optical computing24,25, spatial radiation control26, and lasing27. Due to their intrinsically high quality (Q) factors, these systems are well suited for enhancing nonlinear optical processes, including harmonic generation28, spontaneous parametric down-conversion29, and intensity-dependent effects30. Nonlocal metasurfaces have also been demonstrated for nonlinear nonreciprocal control of free-space radiation31,32. Ultrathin silicon-based metasurfaces exploiting thermo-optic third-order nonlinearities, combined with both out-of-plane and in-plane symmetry breaking, enable large nonreciprocal transmission contrasts exceeding 10 dB at moderate intensities below 3 kW/cm², while maintaining low insertion loss31. However, the asymmetry induced by a low-refractive-index substrate typically results in a weak directional response, prompting the use of additional asymmetric layers to enhance performance33,34, particularly in high-Q nonlocal metasurfaces where the ultrathin structure causes weak out-of-plane asymmetry. This added structural complexity complicates device architecture and increases sensitivity to fabrication imperfections. Moreover, asymmetric wavefront shaping remains challenging in nonlinear optical metasurfaces, as existing local and nonlocal designs struggle to simultaneously support high nonlinear efficiency, precise wavefront control, and strong directional asymmetry. The challenges arising from the trade-off among multiple performance requirements, together with the structural complexity, highlight opportunities for new design strategies to effectively balance these factors.

In this work, we demonstrate a nonlocal meta-lens to overcome these limitations, enabling efficient asymmetric wavefront shaping through controlled nonlinear responses of silicon integrated-resonant units (IRUs)35. A nonlocal quasi-bound state in the continuum (q-BIC) resonance, induced by in-plane geometric asymmetry, is excited to achieve a high Q factor ( ~ 100) and thus a high nonlinear efficiency. Simultaneously, local Mie-type resonance is involved to effectively interact with the nonlocal resonance, enabling further tailoring of the near field within the IRUs. This interplay leads to strong directional enhancement even with only silica-substrate-induced asymmetry, accompanied by robust geometric phase modulation. Experimental results indicate that the proposed nonlocal meta-lens exhibits nonlinear power ratios of forward to backward focusing exceeding 7 dB for second harmonic generation (SHG) and 10 dB for third harmonic generation (THG). Asymmetric narrowband focusing at the fundamental near-infrared wavelength is also achieved by manipulating the interaction between two resonances and enhancing the asymmetric intensity-dependent effect.

Results

Design and simulation of the nonlocal meta-lens

The proposed nonlocal meta-lens for asymmetric nonlinear focusing is schematically illustrated in Fig. 1a. With identical fundamental pump illumination from the air (forward) and silica substrate (backward) sides of the meta-lens, nonlinear focusing of the generated harmonic wave exhibits distinct intensities. The schematic diagram and tilted scanning electron microscope (SEM) image of the individual IRU are shown in Fig. 1b. The designed silicon crescent-shaped IRU, arranged in a hexagonal lattice with a period of P = 1000 nm, is formed by trimming a cylinder of diameter D1 = 620 nm and height 330 nm with another cylinder of diameter D2 = 540 nm, offset by L = 300 nm. The design principle of simultaneous realization of high nonlinear efficiency, precise wavefront shaping, and strong directional asymmetry, which are the three key requirements for asymmetric nonlinear wavefront shaping, is illustrated in Fig. 1c. In typical dielectric metasurfaces, local Mie-type resonances (blue block) offer nearly independent phase modulation capability and enhanced magneto-electric coupling, enabling accurate wavefront control and moderate directional asymmetry. In contrast, nonlocal resonances (red block) rely on strong interactions between neighboring units and slight geometric asymmetry to achieve a high Q factor, which enhances nonlinear efficiency but restricts individual phase modulation and limits strong directional asymmetry. By effectively controlling the Fano-like interaction between local and nonlocal resonances, our design (yellow star) achieves stronger directional asymmetry than single-mechanism counterparts. High-Q nonlocal resonance ensures enhanced nonlinear efficiency, while local phase modulation remains decoupled. Such synergistic properties overcome the inherent trade-offs among nonlinear efficiency, wavefront control, and directional asymmetry that typically constrain conventional metasurface designs.

Fig. 1: Nonlocal meta-lens for asymmetric nonlinear wavefront shaping.
Fig. 1: Nonlocal meta-lens for asymmetric nonlinear wavefront shaping.
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a Schematic illustration of the nonlocal meta-lens for asymmetric nonlinear focusing. b Schematic diagram and tilted SEM image of the individual integrated resonant unit. c Design principle illustrating the simultaneous realization of high nonlinear efficiency, precise wavefront shaping ability, and strong directional asymmetry. Metasurfaces based on local (blue) and nonlocal (red) resonances are shown in the diagram at their typical location in this illustrative space.

The interaction process between local and nonlocal resonances by varying the offset L is shown in Fig. 2a. The proposed IRU can excite the nonlocal q-BIC resonance induced by in-plane geometric asymmetry with a high Q factor (red region). The local Mie-type resonance with relatively low Q factor (blue region) is introduced to effectively interact with the nonlocal resonance, enabling further tailoring of the near field and magneto-electric coupling within the IRUs. As a result, although only silica-substrate-induced asymmetry, the directional response can be strongly enhanced, achieving the maximum asymmetric electric field enhancement ratio |EF/EB | 2 = 3.8 with offset L = 300 nm. This value exceeds those with individual resonances. The local and nonlocal origins of the two excited resonances are detailed in Supplementary Note 4. The simulated and experimental transmission spectra under forward and backward incidences are presented in Fig. 2b. The spectra for both directions are nearly identical, and the simulated results agree well with the experimental data. A Fano-like line shape with a high-Q ( ~ 100) resonance peak is observed around 1570 nm. The rationale for selecting this Q factor level is explained in Supplementary Note 3. It is worth noting that, due to fabrication imperfections and the finite array size, the resonant wavelength is slightly blue-shifted compared to that in Fig. 2a. To account for this shift, the simulated spectra are obtained by reducing the offset L to 275 nm. This adjustment facilitates comparison and does not affect the underlying design principle.

Fig. 2: Design principle and numerical simulation of the integrated resonant unit.
Fig. 2: Design principle and numerical simulation of the integrated resonant unit.
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a Asymmetric electric field enhancement ratio |EF/EB | 2 (colored square and spheres) and resonant wavelengths as a function of offset L. Blue and red regions represent the bandwidths of local and nonlocal resonances, respectively. EF(B) represents the forward (backward) average electric field amplitude inside the integrated-resonant unit (IRU). b Simulated (top) and experimental (bottom) transmission spectra under forward (solid lines) and backward (dashed) incidences. c Electric dipole (ED) and magnetic dipole (MD) contributions for both propagation directions. Insets display the electric field distributions in the xy plane, taken at the center of IRU, normalized by the incident field amplitude ( | EF(B)/E0 | ). d Transmission (color map) and phase (markers) as functions of the IRU rotation angle. Phases are extracted at the resonant wavelength of 1570 nm in the periodic IRU, selecting right (RCP) or left (LCP) circular polarization in the fundamental (linear) and harmonic waves.

Although the far-field responses are identical, the near-field distributions exhibit directional asymmetry. The electric dipole (ED) and magnetic dipole (MD) contributions for both propagation directions are presented in Fig. 2c, obtained through multipole decomposition36. In the forward direction, the response is mainly governed by the ED contribution, as the asymmetric out-of-plane MD can induce an in-plane ED. However, in the backward direction, the response consists of a combination of both ED and MD moments. This process can be described by the magneto-electric coupling between these two modes13:

$${p}_{x}={\alpha }_{{{{\rm{ee}}}}}^{xx}{E}_{x}\pm {\alpha }_{{{{\rm{me}}}}}^{xy}{H}_{y}$$
(1)
$${m}_{y}=\pm {\alpha }_{{{{\rm{mm}}}}}^{yy}{H}_{y}-{\alpha }_{{{{\rm{me}}}}}^{xy}{E}_{x}$$
(2)

where p and m denote the ED and MD moments, respectively, while E and H represent the local electric and magnetic fields inside the IRU. The coefficients αee, αmm, and αme correspond to the electric, magnetic, and magneto-electric polarizability terms. Due to the presence of the magneto-electric coupling term αme, the in-plane magnetic field Hy plays a key role in enhancing the asymmetry. For a single resonance, Ex and Hy generally have limited spatial overlap, making it difficult to simultaneously maximize nonlinear efficiency and asymmetry. In our design, the q-BIC resonance primarily enhances Ex and Hz, leading to high nonlinear efficiency while weak asymmetry. By contrast, the Mie-type MD resonance enhances Hy, thereby enabling strong asymmetry for the high-Q q-BIC resonance through mode coupling. The normalized electric field distributions are displayed in the insets of Fig. 2c, which reveal similar field profiles but different enhancement levels. This asymmetry in field enhancement offers the potential for directional nonlinear responses. Furthermore, to achieve stable wavefront shaping, it is important that the resonance excitation remains robust under phase modulation. The transmission and phase as functions of the IRU rotation angle are plotted in Fig. 2d. The resonant wavelength, transmission, and Q factor all remain stable with varying rotation angles. The generated geometric phases exhibit a linear dependence on the rotation angle. Owing to its C1 symmetry, the proposed IRU is compatible with the linear regime, symmetry-breaking artificial SHG, and natural THG for different circular polarizations37. This robustness and versatility support the use of the IRU for efficient asymmetric wavefront shaping across diverse scenarios.

Asymmetric wavefront shaping in harmonic generations

Based on the analysis of the asymmetric near-field distribution, experimental SHG and THG spectra under forward and backward excitations are shown in Fig. 3a, b. Both spectra exhibit a maximum at the resonant wavelength of 1570 nm, with nonlinear power ratios exceeding 5 (7 dB) for SHG and 10 (10 dB) for THG between the forward and backward directions. The experimental results agree with the simulated results presented in Supplementary Fig. 5. The broader full width at half maximum (FWHM) of the experimental efficiency peak is primarily attributed to the spectral bandwidth of the femtosecond laser ( ~ 20 nm), which causes averaging over multiple wavelengths. The dependence of the nonlinear output power on the pump power is shown in Fig. 3c. The SHG and THG signals closely follow the theoretical predictions, with power-law slopes of 2 and 3, respectively. The strong field enhancement of nonlocal resonances enables the nonlinear signals to remain detectable even at pump powers as low as 10 mW or below. When the pump power exceeds 100 mW, the nonlinear harmonic powers tend to level off in the forward case due to stronger near-field enhancement, which induces thermo-optic refractive index variations and results in a wavelength shift of the resonances rather than structural deterioration. In our experiments, only a portion of the pump power is coupled into the nanoresonators because of their high Q factor. No observable degradation or damage was detected on the metasurface after repeated measurements under the maximum pump power of 300 mW (corresponding to a peak power density of 1.37 GW/cm2), indicating good stability and a relatively high optical damage threshold. The maximum conversion efficiencies for total transmitted SHG and THG reach 5.6 × 10-10 and 2.9 × 10-6, respectively, under the 300-mW pump. With the additional functionalities of wavefront shaping and asymmetric response, these values remain competitive with, or even surpass, previously reported results13,28,38,39,40,41,42. A detailed performance comparison is provided in Supplementary Note 12.

Fig. 3: Asymmetric focusing in harmonic generations.
Fig. 3: Asymmetric focusing in harmonic generations.
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a, b Experimental SHG (a) and THG (b) spectra under forward and backward excitations, as a function of the pump central wavelength, ranging from 1540 to 1600 nm, with 10 nm increments. c Dependence of the nonlinear output power on the input pump power. The shaded gray region indicates the range of pump power where the harmonic power levels off. d Schematic layout of the meta-lens units, whose rotation angle is encoded in the color map. e SEM image of the fabricated sample. The inset shows an enlarged view of a selected region of interest. f Experimental xz- and xy-plane intensity profiles of the focused second harmonics, under forward (top half) and backward (bottom half) excitations and for two output circular polarization states. Xy-plane intensity profiles are extracted at the respective focal planes. g Analogous data to (f), shown for the focused third harmonics.

The schematic layout of the meta-lens units with encoded rotation angles is shown in Fig. 3d. To ensure that the proposed nonlocal meta-lens operates effectively in both linear and nonlinear regimes, the arrangement follows a spherical phase profile at the fundamental wavelength, given by ref. 43:

$$\varphi=-\frac{2{{{\rm{\pi }}}}}{\lambda }(\sqrt{{r}^{2}+{f}^{2}}-f)$$
(3)

where r is the radial coordinate, λ is the fundamental operating wavelength, and f = 220 μm is the designed focal length. This corresponds to a numerical aperture (NA) of 0.2, given a meta-lens radius of 45 μm. Since the wavelengths of the SHG and THG are proportional to those of the fundamental wave, and their spin-dependent geometric phases scale with the meta-atom rotation angle θ as 1θ/3θ and 2θ/4θ, respectively, both harmonic waves are expected to focus accordingly, with corresponding focal lengths. The SEM image of the fabricated sample is presented in Fig. 3e, corresponding to the design in Fig. 3d. Experimental xz- and xy-plane SHG and THG focusing profiles under forward and backward excitations for two output circular polarization states are shown in Fig. 3f, g. Benefiting from the stable phase modulation shown in Fig. 2d, the SHG and THG signals with different circular polarizations exhibit an effective focusing property. The FWHMs of the forward SHG and THG focal spots for different polarizations in Fig. 3f, g are 2.96, 4.9, 1.71, and 2.72 μm, respectively, which are close to their corresponding diffraction limits (0.61λ/NA) of 2.81, 4.71, 2.01, and 2.93 μm. Consistent with the spectral responses in Fig. 3a, b, the forward excitation yields substantially higher intensity than the backward case, in good agreement with the simulations shown in Supplementary Fig. 7. The focusing performance can be further improved by enlarging the meta-lens and beam spot size and employing a more collimated incident beam to fully activate the nonlocal effects.

Asymmetric wavefront shaping at the fundamental wavelength

In addition to nonlinear frequency conversion, intensity-dependent effects (e.g., thermo-optic effect) can also be exploited to achieve directional responses, which operate at the fundamental wavelength. The schematic illustration of the nonlocal meta-lens for asymmetric narrowband focusing is given in Fig. 4a. Under broadband illumination, the backward direction exhibits high-efficiency focusing at narrowband resonant wavelengths, whereas the forward direction exhibits suppressed transmission due to strong intensity-dependent effects. The experimental transmission polarization conversion efficiency TRL(LR) as a function of pump power along forward and backward directions is shown in Fig. 4b. The Fano-like interaction between local and nonlocal resonances enables a high initial transmission TRL(LR) of 62%. As the pump power increases, the backward transmission remains nearly constant at around 60%, while the forward transmission drops sharply above 100 mW and stabilizes at around 25% beyond 200 mW. This behavior primarily arises from the wavelength shift induced by the intensity-dependent refractive index, as discussed in Supplementary Note 8. The unequal nonlinear responses of the two resonances also contribute, since their mutual interaction becomes weakened at high pump powers. The threshold behavior around 100 mW corresponds well to the saturation-like onset observed in Fig. 3c. The maximum forward-to-backward transmission ratio reaches approximately 2.7 (4.3 dB), in good agreement with the simulation results shown in Supplementary Fig. 8. Since the design of high asymmetric enhancement |EF/EB | 2 primarily determines the non-reciprocal intensity range31 (NRIR, around 3 in our case), the transmission ratio can be further improved by appropriately tuning the coupling between local and nonlocal resonances. Such optimization involves a moderate yet unavoidable reduction in polarization conversion efficiency, as discussed in Supplementary Note 9. To investigate the light-field characteristics, normalized xz- and xy-plane focusing profiles under varying pump power are shown in Fig. 4c. Consistent with the spectral response in Fig. 4b, the forward efficiency decreases with increasing pump power, while the backward case remains nearly unchanged, demonstrating asymmetric focusing behavior at high pump powers. Considering low-power regimes relevant to practical technological integrations, the operating power could be further reduced by increasing the Q factor of the nonlocal resonances or by employing materials with higher nonlinear susceptibility.

Fig. 4: Asymmetric focusing at the fundamental wavelength.
Fig. 4: Asymmetric focusing at the fundamental wavelength.
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a Schematic illustration of the nonlocal meta-lens for asymmetric narrowband focusing. b Experimental polarization conversion efficiency in transmission TRL(LR), as a function of the pump power and for forward and backward propagation. c Normalized intensity profile of the focused signal in the xz- and xy-planes, with respect to the varying pump powers (as indicated by the titles).

Discussion

In this work, we propose a nonlocal meta-lens for asymmetric wavefront shaping by tailoring the asymmetric nonlinear responses of silicon integrated-resonant units (IRUs). The carefully designed IRUs not only excite a nonlocal q-BIC resonance with a high Q factor ( ~ 100), but also generate a Fano-like interaction between the q-BIC mode and a local Mie-type resonance. Their interaction yields a large asymmetric electric field enhancement ratio ( | EF/EB | 2 = 3.8), solely relying on asymmetries induced by the silica substrate. The asymmetric response can be further enhanced by tuning the substrate refractive index and the incident angle, as detailed in Supplementary Notes 10 and 11. This design ensures robustness to geometric rotation of the IRUs, providing a stable phase modulation in both linear and nonlinear regimes. Leveraging these effects, the achieved nonlinear conversion efficiency is comparable to reported works lacking wavefront shaping and asymmetric control. The nonlinear power ratios of forward to backward focusing demonstrated here exceed 7 dB for SHG and 10 dB for THG, whereas asymmetric narrowband focusing is also achieved at the fundamental near-infrared wavelength by increasing the pump power. This ultimately modifies the interaction between the two resonances of the structure, enhancing the asymmetric intensity-dependent effect and achieving a maximum transmission ratio of approximately 4.3 dB. This nonlocal meta-lens could additionally induce asymmetric phase modulation at the IRU level to realize distinct functionalities for the two propagation directions. The proposed nonlocal meta-lens opens new avenues for efficient asymmetric wavefront shaping in both transmission and harmonic generation, with promising applications in LIDAR, optical computing, and communications.

Methods

Simulation

The electromagnetic responses of both IRUs and meta-lenses are simulated using the finite element method (FEM) in COMSOL Multiphysics. Perfectly matched layers (PMLs) are applied at the top and bottom to truncate the open boundaries. Periodic boundary conditions are used along the x and y directions to represent the periodic IRU array, while for near-field simulations of isolated IRUs or meta-lenses, the periodic conditions are replaced by PMLs. Far-field responses are obtained by propagating the near-field results using scalar diffraction theory.

Nonlinear simulations of SHG and THG are performed under the undepleted pump approximation via a two-step process44. First, the electromagnetic fields at the fundamental wavelength are computed, from which the induced nonlinear polarization in silicon is derived. This polarization serves as the source term for calculating the harmonic emission. The refractive index dispersion of silicon is shown in Supplementary Fig. 1, and the silica substrate is assumed to have a constant refractive index of 1.45.

Fabrication

A 330 nm-thick α-Si film was deposited onto a SiO₂ substrate at 0.5 Å/s, followed by the deposition of a 22 nm Cr hard mask at the same rate using electron beam evaporation. An 80 nm PMMA layer was then spin-coated and baked at 180 °C for one hour. The PMMA resist was patterned by electron beam lithography (Raith E-line, 30 kV) and developed in MIBK/IPA at 0 °C for 30 s. After development, inductively coupled plasma (ICP) etching (Oxford ICP180) was used to sequentially etch the Cr and Si layers. Finally, the residual Cr mask was removed by immersion in a chromium etchant for 10 min.

Characterization

The optical measurement setup is schematically illustrated in Supplementary Fig. 2. For linear spectral measurements, a supercontinuum laser (NKT, FIU-6) provides broadband coherent light. Circularly polarized light is generated by combining a linear polarizer and a quarter-wave plate. The incident beam is collimated using a lens and an objective lens (Mitutoyo, 20× magnification, NA = 0.4). The transmitted light is collected by an identical objective lens. For nonlinear spectral and light-field profile measurements, the pump source is changed to a femtosecond Ti: Sapphire laser (pulse duration 140 fs, repetition rate 80 MHz) combined with an optical parametric oscillator (OPO). A lens with a 10 cm focal length focuses the beam onto the sample, ensuring high power density while maintaining nearly collimated incidence. The laser spot diameter focused onto the meta-lens ( ~ 50 μm) is smaller than its aperture, ensuring efficient and uniform illumination across the structure. After transmission or harmonic generation, the output passes through a quarter-wave plate, linear polarizer, and appropriate spectral filters to isolate the desired circular polarization components (LCP or RCP). The filtered light is then analyzed by a spectrometer or imaged by a camera for spectral or spatial characterization, respectively.