Intensity-asymmetric wavefront shaping in nonlocal meta-lens

Domain: physics
Published: 2026-01-26
Editorial outcome: minor_revision

Abstract
Asymmetric wavefront shaping can unlock essential functionalities for next-generation photonic systems, such as unidirectional transmission, nonlinear signal routing, and nonreciprocal light control. However, achieving these functionalities in passive nonlinear metasurfaces remains challenging due to the required trade-offs among nonlinear efficiency, phase modulation, and directional asymmetry, particularly on low-index substrates with weak intrinsic asymmetry. Here, we propose a meta-lens that effectively integrates local Mie-type resonances with nonlocal quasi-bound states in the continuum, tailoring their interaction. Without any additional layers, this hybrid design produces a strong directional response originating from asymmetries induced by the silica substrate, while preserving high nonlinear efficiency and robust phase modulation. Experimentally, directional focusing is achieved across the fundamental wavelength and its second and third harmonics, with a maximum forward-to-backward power ratio exceeding 10 dB. This work opens new opportunities for nonreciprocal LIDAR and optical computing applications.

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 modulation4–6, 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 asymmetry9–12. 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 modulation16–20. They have been widely explored in applications, such as narrowband wavefront shaping21–23, 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 D = 620 nm and height 330 nm with another cylinder of diameter D = 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.

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 |E_F/E_B|^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 lineshape 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.

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_{ee} E_x \pm \alpha_{em} H_y (1)

m_y = \pm \alpha_{me} E_x + \alpha_{mm} H_y (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 \alpha_{ee}, \alpha_{mm}, and \alpha_{me} correspond to the electric, magnetic, and magneto-electric polarizability terms. Due to the presence of the magneto-electric coupling term \alpha_{me}, the in-plane magnetic field H_y plays a key role in enhancing the asymmetry. For a single resonance, E_x and H_y generally have limited spatial overlap, making it difficult to simultaneously maximize nonlinear efficiency and asymmetry. In our design, the q-BIC resonance primarily enhances E_x and H_z, leading to high nonlinear efficiency while weak asymmetry. By contrast, the Mie-type MD resonance enhances H_y, 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 results13,28,38–42. A detailed performance comparison is provided in Supplementary Note 12.

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:

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

where r is the radial coordinate, \lambda is the fundamental operating wavelength, and f = 220 \mu m is the designed focal length. This corresponds to a numerical aperture (NA) of 0.2, given a meta-lens radius of 45 \mu m. 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 \theta as 1\theta/3\theta and 2\theta/4\theta, 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 \mu m, respectively, which are close to their corresponding diffraction limits (0.61\lambda/NA) of 2.81, 4.71, 2.01, and 2.93 \mu m. 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 T_{RL(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 T_{RL(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 |E_F/E_B|^2 primarily determines the nonreciprocal intensity range (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.

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 \alpha-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 \mu m) 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.

Data availability
Data underlying the results are available from the corresponding authors upon request.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "The paper by J. Yao et. al. describes a series of nonlinear focusing measurements involving the illumination of nonlocal metalenses with femtosecond laser pulses. The nonlocal resonance in the metalens leads to an amplification of 2nd and 3rd harmonic generation as well as a reduction on illumination power required to induce thermo-optic nonlinear limiting. Moreover, the authors reveal that weak structural symmetry breaking caused by the elevated index of the substrate produces directional nearfield build up within the nonlocal resonance, causing the efficiencies of the various nonlinear focusing phenomena to vary dramatically depending on which side is illuminated. The underlying platform has been introduced by the authors before; all be it without directional nonlinearity. That said, the use of nonlocal metasurfaces to directionally boost wavefront shaped nonlinear optical effects is timely. While the resonant enhancement is modest and many of the mechanisms discussed, including geometric phase based nonlocal meta focusing and asymmetry from magneto-electric coupling, are not new, I do not believe asymmetric nonlinear focusing has been boosted by a nonlocal resonance before. Further improvements to the platform to bring the nonlinear efficiency gains over other platforms from small to large, would be exciting to see. I do, therefore, recommend the manuscript for publication after minor corrections have been made.\nDetailed comments are provided below.\n1. On line 70, the statement “Moreover, asymmetric wavefront shaping based on optical nonlinearity remains unexplored” is not true. Explorations may be few and may have limitations, but they do exist. The authors should refine this statement.\n2. On lines 72-73, “These challenges highlight opportunities for new design strategies to effectively balance these trade-offs.” is a very clumsy sentence. It is not clear which challenges and trade-offs will be addressed.\n3. Panel c in figure 1 is confusing and misleading.\n4. It is not clear what the different curves in Fig. 3A-B represent, nor what the colors mean. This should be clearly stated in the caption and discussed in the text.\n5. The final section is labeled “Nonreciprocal wavefront shaping in the linear regime”, but the measurements performed do not compare reciprocal illumination. The reciprocal of plane wave illumination from one side is collimation from the other side.", "reviewer_2": "This manuscript demonstrates asymmetric nonlinear wavefront shaping using silicon metasurfaces that combine local Mie-type and nonlocal quasi-bound state in the continuum (q-BIC) resonances. They further show nonreciprocal focusing, forming a custom power diode, which is particularly enticing. The key is a vertical asymmetry of the near field when pumping from the substrate or superstrate, an effect enhanced by the high Q-factor of the q-BIC resonance (similar to Ref 31, but here with local phase control).\nThe work presents interesting physics and novel design principles, extending the authors’ previous results on similarly-looking devices from linear to nonlinear experiments. It is well explained, and the results are compelling. We find it likely to make a significant impact in the field, and find it appropriate for the journal. However, several issues must be addressed before accepting it to the journal.\n1. The authors mentioned that a 300 mW pump power was used, corresponding to a very high peak power density of 1.37 GW/cm². While this was necessary to observe the impressive nonlinear effects, the data also indicates that saturation effects begin to appear above 100 mW, which is well below the maximum power used. Could the authors perhaps comment on the metasurface's long-term stability and optical damage threshold under these high-power conditions? Furthermore, it would be insightful to hear their perspective on how the device might perform in lower-power regimes, which are more common in practical applications. A brief discussion on these points would be a valuable addition, helping to bridge this significant scientific demonstration with the considerations needed for real-world technological integration.\n2. The inset SEM image in Figure 1b is quite dark and has low contrast. If possible, enhancing the brightness and contrast would make the structure's details more visible.\n3. The authors claim 'effective sub-diffraction-limit focusing', presumably comparing the “expected” diffraction limited spot size of the pump to the actual harmonic component (which has a smaller wavelength). More discussion of this point is appropriate: what is exactly being compared? To fully substantiate this strong claim, a quantitative analysis of the focal spots shown in Figure 3 should be carried out. For example, a cross-sectional intensity profile with a measurement of the full width at half maximum (FWHM) could be directly compared to the calculated diffraction limit; Strehl ratio; comparison to ideal, etc.\n4. A small improvement in figure labeling would enhance clarity. The inconsistency varies across the figures: in Figure 3f and 4c, the x-axis label is included for the first bottom panel but absent from the other. In Figure 3g, the label is missing entirely from all panels (maybe at least x-axis label should be there). To resolve these issues, adding the x-axis label or a scale bar to all relevant xy-plane images would ensure consistency and make the figures much easier to interpret. In figure 2d, the overlapping of several pieces of information promotes conciseness, but is difficult to parse. It would be easier to read with a separate panel for the transmission colormap and the geometric phase profiles (line graphs/spheres/dots).", "reviewer_3": "Yao et al. present in their manuscript an asymmetric wave propagation while simultaneously controlling the spatial wavefront at the example of a meta-lens. Such asymmetry in propagation was already demonstrated earlier by metasurfaces for frequency conversion. Here, the only novelty seems to be the additional spatial wavefront control for second- and third-harmonic generation. Overall, the manuscript is well written and the explanation of the underlying concepts is convincingly discussed. However, I feel the topic is more suitable for a specialized optics journal like Light:Science&Applications rather than Nature Communications. One reason for that is the poor performance (contrast) of asymmetry for the fundamental wavelength.\nFor a revised version of the manuscript, I would recommend to address a few points:\n• The title seems somehow misleading to me. I am not sure if “asymmetric” is the best word to choose here because it can give the idea the wavefronts are laterally asymmetrically shaped which is a standard process. Although the authors explain their definition of asymmetry in the text later, I personally feel there are better ways of expressing the propagation properties of the light for the two directions.\n• What do the authors mean with magneto-electric coupling within the IRUs? An explanation would be helpful.\n• If the authors use the passive nonlinearity of the system, then why is there a saturation effect observable at these relative low powers? For polarization induced nonlinear processes these effects are normally not be able to observe so easily. Did the authors check that the resonance is not shifting spectrally with the higher powers due to Kerr or thermal effects? Such an effect has nothing to do with saturation. I feel the authors need to be here more precise in their wording.\n• I also do not agree with the term “nonreciprocal focusing in the linear regime.” In the linear optical regime (without losses or gain) the system is always reciprocal. Linear optical regime also means that there are no dependencies of the properties on the light intensity. I understand that the authors are unexperienced in the field of nonlinear optics but what they mean is that they observe a nonlinear effect at the fundamental wavelength. That has nothing do to with linear optics. I recommend to rephrase these statements.\n• How are the phase values in Fig. 2d calculated? For a single unit or a period pattern where all elements are oriented in the same way? What is the error in phase if the elements like in the metalens are all differently oriented?\n• The assumed nonlinear third-order permittivity of silicon to model the refractive index change seems extremely high. Typical values in the NIR range are 10E-18 to 10E-19 m2/V2. This is three to four orders of magnitude smaller than the value the authors used in their simulations.", "reviewer_4": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts."}, "author_response": "We thank the Reviewer for his/her positive comments and constructive suggestions on our manuscript. We addressed the comments point-to-point as follows.\nSpecific Comments:\n1. On line 70, the statement “Moreover, asymmetric wavefront shaping based on optical nonlinearity remains unexplored” is not true. Explorations may be few and may have limitations, but they do exist. The authors should refine this statement.\nResponse:\nWe thank the Reviewer for the constructive comment regarding our insufficiently rigorous description. Indeed, the expression “remains unexplored” may have been overstated. Our intention was to clarify one of the main motivations of this work, namely that asymmetric wavefront shaping still represents a significant challenge (rather than an unexplored topic) in nonlinear optical metasurfaces. Existing local and nonlocal designs generally struggle to simultaneously achieve high nonlinear efficiency, precise wavefront control, and strong directional asymmetry.\nWe have revised the corresponding statement on Page 4 of the Introduction section to improve its rigor and clarity:\n“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.”\n2. On lines 72-73, “These challenges highlight opportunities for new design strategies to effectively balance these trade-offs.” is a very clumsy sentence. It is not clear which challenges and trade-offs will be addressed.\nResponse: We thank the Reviewer for the helpful comment. The challenges arise from the trade-off among three key performance requirements, including high nonlinear efficiency, precise wavefront control, and strong directional asymmetry, together with the structural complexity of additional asymmetric layers mentioned above in the main text.\nWe have modified this sentence on Page 4 of the Introduction section to make it clearer:\n“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.”\n3. Panel c in figure 1 is confusing and misleading.\nResponse: We thank the Reviewer for the valuable comment on this point. Figure 1c illustrates the design principle of asymmetric wavefront shaping in nonlinear metasurfaces, which requires the simultaneous realization of high nonlinear efficiency, precise wavefront control, and strong directional asymmetry, which is one of the main motivations of our work. 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.\nWe have revised the relevant descriptions on Pages 5 and 6 of the Results section to make them more comprehensive:\n“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.”\nWe have also adjusted the axis length and added auxiliary lines indicating the coordinate values of the blocks in Fig. 1c, and revised the figure caption on Page 6 of the Results section to improve clarity:\nRevised Fig. 1c: Design principle illustrating the simultaneous realization of high nonlinear efficiency, precise wavefront shaping ability, and strong directional asymmetry. Blue and red blocks denote typical local and nonlocal resonances in dielectric metasurfaces.\n4. It is not clear what the different curves in Fig. 3A-B represent, nor what the colors mean. This should be clearly stated in the caption and discussed in the text.\nResponse: We thank the Reviewer for the comment. Different curves in Fig. 3a and 3b, from left to right, represent the normalized experimental nonlinear spectra under pump central wavelengths ranging from 1540 to 1600 nm with a step of 10 nm. The SHG spectrum under a 1600 nm pump is too weak to be detected and is therefore not shown.\nWe have added the legends in Fig. 3a and 3b and revised the caption on Page 12 of the Results section to make it clearer:\nRevised Fig. 3a and 3b: Normalized experimental SHG (a) and THG (b) spectra under forward and backward excitations with pump central wavelengths ranging from 1540 to 1600 nm in 10 nm increments.\n5. The final section is labeled “Nonreciprocal wavefront shaping in the linear regime”, but the measurements performed do not compare reciprocal illumination. The reciprocal of plane wave illumination from one side is collimation from the other side.\nResponse: We thank the Reviewer for the insightful comment. In our experiment, we demonstrated asymmetric wavefront shaping by changing the incident direction of the pump plane wave in the linear regime, rather than by employing a reversed point source for strict nonreciprocity testing. One key reason is that the reversed point source cannot effectively excite the nonlocal resonance due to its intrinsic strong incident-angle dispersion, which would introduce undesired additional asymmetric responses. We agree with the Reviewer that, although the overall system exhibits nonreciprocal behavior under nonlinear thermo-optic effects, the presented results correspond to asymmetric rather than nonreciprocal responses.\nTo ensure accuracy, we have revised the subtitle on Page 13 of the Results section and replaced all relevant mentions of “nonreciprocal” with “asymmetric”:\n“Asymmetric wavefront shaping at the fundamental wavelength”\nFor Reviewer 2:\nGeneral Comment:\nThis manuscript demonstrates asymmetric nonlinear wavefront shaping using silicon metasurfaces that combine local Mie-type and nonlocal quasi-bound state in the continuum (q-BIC) resonances. They further show nonreciprocal focusing, forming a custom power diode, which is particularly enticing. The key is a vertical asymmetry of the near field when pumping from the substrate or superstrate, an effect enhanced by the high Q-factor of the q-BIC resonance (similar to Ref 31, but here with local phase control).\nThe work presents interesting physics and novel design principles, extending the authors’ previous results on similarly-looking devices from linear to nonlinear experiments. It is well explained, and the results are compelling. We find it likely to make a significant impact in the field, and find it appropriate for the journal. However, several issues must be addressed before accepting it to the journal.\nResponse:\nWe thank the Reviewer for his/her positive comments and constructive suggestions on our manuscript. We addressed the comments point-to-point as follows.\nSpecific Comments:\n1. The authors mentioned that a 300 mW pump power was used, corresponding to a very high peak power density of 1.37 GW/cm². While this was necessary to observe the impressive nonlinear effects, the data also indicates that saturation effects begin to appear above 100 mW, which is well below the maximum power used. Could the authors perhaps comment on the metasurface's long-term stability and optical damage threshold under these high-power conditions? Furthermore, it would be insightful to hear their perspective on how the device might perform in lower-power regimes, which are more common in practical applications. A brief discussion on these points would be a valuable addition, helping to bridge this significant scientific demonstration with the considerations needed for real-world technological integration.\nResponse:\nWe thank the Reviewer for this insightful comment. 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 a 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 nonlinear harmonic powers tend to level off above a pump power of 100 mW, mainly due to thermo-optic refractive index variations that induce a resonance wavelength shift rather than structural deterioration. The simulated influence of thermo-optic effects on the meta-lens is provided in Supplementary Fig. 8. This resonance shift reduces the near-field enhancement and thus the conversion efficiency at the pump wavelength. In addition, silicon possesses an intrinsically high optical damage threshold. Reported silicon-based nonlinear metasurfaces can typically withstand peak power densities of several GW/cm2 or even higher [Nano Lett. 15, 7388-7393 (2015); ACS Photonics 6, 1639-1644 (2019); Nano Lett. 19, 6585-6591 (2019); Nano Lett. 18, 8054-8061 (2018); Nano Lett. 24, 2257-2263 (2024); Nat. Photonics 16, 561-565 (2022)], which exceeds those used in our experiments.\nSupplementary Fig. 8: a,b,c, Simulated transmission polarization conversion efficiency T (a), RL(LR) averaged refractive index increase Δn within the silicon IRU (b), and wavelength shift Δλ (c) as a function of pump power along forward and backward directions.\nWe have added some discussions on the stability and damage threshold on Pages 10 and 11 of the Results section:\n“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 results, despite the relatively low pump intensity. A detailed performance comparison is provided in Supplementary Note 12.”\nFor the perspective on the lower-power regimes, as shown in Fig. 3c, the nonlinear signal decreases nearly quadratically with pump power below 100 mW. 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. This demonstrates that the meta-lens can operate reliably under low-power conditions. Considering the fundamental wavelength, the enhanced nonlocal resonance effectively reduces the required average pump power to 15.3 kW/cm2. This value is limited by the spectral bandwidth of the femtosecond laser (~20 nm), which causes averaging over multiple wavelengths. Despite this limitation, the required power remains comparable to previously reported works [Nat. Photonics 18, 81-90 (2024); Nano Lett. 25, 3259-3264 (2025)]. Further reduction of the operating power could be achieved by increasing the Q-factor of the nonlocal resonances or employing materials with higher nonlinear susceptibility.\nWe have added some discussions on Page 10 of the Results section:\n“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.”\nand some perspectives on Page 14 of the Results section:\n“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.”\n2. The inset SEM image in Figure 1b is quite dark and has low contrast. If possible, enhancing the brightness and contrast would make the structure's details more visible.\nResponse: We thank the Reviewer for the constructive comment. We have enhanced the brightness and contrast of the inset SEM image in Fig. 1b on Page 6 of the Results section to improve its visibility.\nRevised Fig. 1b: Schematic diagram and tilted SEM image of the individual IRU.\n3. The authors claim 'effective sub-diffraction-limit focusing', presumably comparing the “expected” diffraction limited spot size of the pump to the actual harmonic component (which has a smaller wavelength). More discussion of this point is appropriate: what is exactly being compared? To fully substantiate this strong claim, a quantitative analysis of the focal spots shown in Figure 3 should be carried out. For example, a cross-sectional intensity profile with a measurement of the full width at half maximum (FWHM) could be directly compared to the calculated diffraction limit; Strehl ratio; comparison to ideal, etc.\nResponse: We thank the Reviewer for the constructive suggestion. The full-widths at half-maximum (FWHMs) of the forward SHG and THG focal spots for different polarizations in Figs. 3d and 3g 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. The THG focusing exhibits sub-diffraction-limit behavior, while the SHG focusing approaches the diffraction limit. The measured Strehl ratios are 0.81, 0.69, 0.72, and 0.73, respectively. These values are mainly influenced by the fact that the phase profile is designed only at the fundamental wavelength, while the SHG and THG follow proportional wavelengths and nonlinear geometric phases to achieve approximate focusing, which inevitably leads to slight degradation in focal quality. Despite this effect, the experimental Strehl ratios remain close to the diffraction limit.\nIn addition, we have simulated the phases extracted from the nonlocal meta-lens for the linear, SHG, and THG cases, shown in Supplementary Fig. 6. The obtained phase distributions closely match the designed ones, demonstrating the potential for effective focusing capability. The minor nonuniformities arise from the limited mesh resolution, as the full-lens simulation involves an extremely large data volume, but they do not affect our conclusions.\nWe have revised the inaccurate description of “sub-diffraction-limit” to “close to the diffraction limit” and added relevant discussions on the focusing performance on Page 11 of the Results section:\n“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 full-widths at half-maximum (FWHMs) of the forward SHG and THG focal spots for different polarizations in Fig. 3d and 3g 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.”\nand on Page 9 of Supplementary Note 6 section:\n“To demonstrate that the designed phases are correctly implemented at their corresponding locations, the simulated phase distributions extracted from the nonlocal meta-lens for the linear, SHG, and THG cases are presented in Supplementary Fig. 6. The obtained phase distributions closely match the designed ones, demonstrating the potential for effective focusing capability. The minor nonuniformities arise from the limited mesh resolution, as the full-lens simulation involves an extremely large data volume, but they do not affect our conclusions.”\nSupplementary Fig. 6: Designed and simulated phase profiles of the proposed nonlocal meta-lens for linear, SHG, and THG with different circular polarizations.\n4. A small improvement in figure labeling would enhance clarity. The inconsistency varies across the figures: in Figure 3f and 4c, the x-axis label is included for the first bottom panel but absent from the other. In Figure 3g, the label is missing entirely from all panels (maybe at least x-axis label should be there). To resolve these issues, adding the x-axis label or a scale bar to all relevant xy-plane images would ensure consistency and make the figures much easier to interpret. In figure 2d, the overlapping of several pieces of information promotes conciseness, but is difficult to parse. It would be easier to read with a separate panel for the transmission colormap and the geometric phase profiles (line graphs/spheres/dots).\nResponse: We thank the Reviewer for the comment. Figure 3f and 3g share the same axes. As suggested, we have corrected the previously inaccurate axis information and added detailed axis descriptions for all relevant subfigures in Fig. 3f and 3g on Page 12 of the Results section to enhance the clarity:\nRevised Fig. 3f and 3g: f, Experimental xz- and xy-plane SHG focusing profiles under forward and backward excitations for two output circular polarization states. Xy-plane intensity profiles are extracted at the respective focal planes. g, Corresponding THG focusing profiles.\nand Fig. 4c on Page 14 of the Results section:\nRevised Fig. 4c: Normalized xz- and xy-plane focusing profiles under varying pump power.\nFigure 2d is intended to demonstrate the rotation robustness of the proposed metalens, as indicated by the stable resonant wavelength, transmission, and phase modulation. While we carefully considered using separate panels, we retained the combined view to emphasize the correlation between these quantities.\nTo further improve clarity and readability, we have reduced the font size and the size of the different markers in Fig. 2d on Page 9 of the Results section:\nRevised Fig. 2d: Transmission (color map) and phase (markers) as functions of the IRU rotation angle. Phases are extracted regarding the resonant wavelength of 1570 nm in the periodic IRU. LCP, RCP: Left and right circular polarization.\nFor Reviewer 3:\nGeneral Comment:\nYao et al. present in their manuscript an asymmetric wave propagation while simultaneously controlling the spatial wavefront at the example of a meta-lens. Such asymmetry in propagation was already demonstrated earlier by metasurfaces for frequency conversion. Here, the only novelty seems to be the additional spatial wavefront control for second- and third-harmonic generation. Overall, the manuscript is well written and the explanation of the underlying concepts is convincingly discussed. However, I feel the topic is more suitable for a specialized optics journal like Light:Science&Applications rather than Nature Communications. One reason for that is the poor performance (contrast) of asymmetry for the fundamental wavelength.\nFor a revised version of the manuscript, I would recommend to address a few points:\nResponse:\nWe thank the Reviewer for the constructive comments and valuable guidance. We address the points raised as follows. Regarding the concern about the relatively low asymmetry contrast at the fundamental wavelength:\n(1) In our work, three key factors, high nonlinear efficiency, precise wavefront shaping, and strong directional asymmetry, require to be simultenously considered. The directional asymmetric is quantified by the asymmetric electric field enhancement ratio |E /E |2, which directly determines the intensity asymmetry for harmonic waves, but not for the fundamental wavelength. For the fundamental wavelength, this ratio instead determines the non-reciprocal intensity range (NRIR), defined as the pump intensity range over which asymmetric responses occur. In our design, the value of the NRIR is around three (3) according to Fig. 4b, which is comparable to or even higher than previous studies without phase modulation capability [Nat. Photonics 18, 81-90 (2024); Nat. Commun. 15, 5077 (2024)].\n(2) In addition to the three key factors considered in (1), high efficiency at the fundamental wavelength is also required to be additionally considered, which inevitably influences the design for high intensity asymmetry. There are two theoretical limits for circularly polarized asymmetric responses. The first is the transmission polarization conversion efficiency of approximately 25% in ultrathin metasurfaces, which has been overcome in the previous work [Nat. Commun. 15, 6543 (2024)]. The second is the asymmetric transmission limit of 4NRIR/(NRIR+1)2 [Nat. Photonics 18, 81-90 (2024)]. Taking both limits into account and given that our experimental value of NRIR is approximately 3, the theoretical efficiency limit for a single-mode ultrathin metasurface is about 18.75%. Benefiting from Fano-like coupling between local and nonlocal resonances, our design achieves a high experimental polarization conversion efficiency of approximately 60%, surpassing this theoretical bound. Supplementary Fig. 9 analyzes the correlation between polarization conversion efficiency and the forward-to-backward transmission ratio. As the offset L increases, the coupling strength between local and nonlocal resonances is tuned, resulting in a gradual reduction of efficiency while effectively enhancing the forward-to-backward transmission ratio. A high ratio of 15.6 (11.9 dB) is achieved at L = 340 nm, where the efficiency remains 21%, still exceeding the theoretical limit of 18.75%. We note that an intrinsic balance exists between efficiency and contrast ratio, representing an inevitable trade-off to be further optimized in future studies. Our present work primarily focuses on overcoming the trade-offs among nonlinear conversion efficiency, wavefront shaping capability, and directional asymmetry, while achieving their simultaneous operation at the fundamental, SHG, and THG wavelengths.\n(3) Although the observed contrast of 4.3 dB is not high compared with conventional devices, it is comparable to or higher than that of nonlinear metasurfaces without a high-Q factor or phase modulation [Nat. Commun. 15, 5077 (2024); Nano Lett. 25, 3259–3264 (2025)]. In addition to the moderate reduction in efficiency mentioned in (2), the contrast can be further improved by reducing the spectral bandwidth of the femtosecond pump laser, which is currently approximately 20 nm, thereby minimizing averaging effects over multiple wavelengths.\nWe have added the relevant discussion on how to improve the asymmetry contrast on Page 13 of the Results section:\n“Since the design of high asymmetric enhancement |E /E |2 primarily determines the non-reciprocal intensity range (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.”\nand on Pages 13 and 14 of the Supplementary Note 9 section:\n“In addition to the three key factors of high nonlinear efficiency, precise wavefront shaping, and strong directional asymmetry, high efficiency at the fundamental wavelength is also required to be additionally considered, which inevitably influences the design for high intensity asymmetry. There are two theoretical limits for circularly polarized asymmetric responses. The first is the transmission polarization conversion efficiency of approximately 25% in ultrathin metasurfaces, which has been overcome in the previous work. The second is the asymmetric transmission limit of 4NRIR/(NRIR+1)2. Taking both limits into account and given that our experimental value of NRIR is approximately 3, the theoretical efficiency limit for a single-mode ultrathin metasurface is about 18.75%. Benefiting from Fano-like coupling between local and nonlocal resonances, our design achieves a high experimental polarization conversion efficiency of approximately 60%, surpassing this theoretical bound. Supplementary Fig. 9 analyzes the correlation between polarization conversion efficiency and the forward-to-backward transmission ratio. As the offset L increases, the coupling strength between local and nonlocal resonances is tuned, resulting in a gradual reduction of efficiency while effectively enhancing the forward-to-backward transmission ratio. A high ratio of 15.6 (11.9 dB) is achieved at L = 340 nm, where the efficiency remains 21%, still exceeding the theoretical limit of 18.75%. We note that an intrinsic balance exists between efficiency and contrast ratio, representing an inevitable trade-off to be further optimized in future studies. Our present work primarily focuses on overcoming the trade-offs among nonlinear conversion efficiency, wavefront shaping capability, and directional asymmetry, while achieving their simultaneous operation at the fundamental, SHG, and THG wavelengths.”\nSupplementary Fig. 9: Simulated polarization conversion efficiency and forward-backward transmission ratio with varying offset L.\nSpecific Comments:\n1. The title seems somehow misleading to me. I am not sure if “asymmetric” is the best word to choose here because it can give the idea the wavefronts are laterally asymmetrically shaped which is a standard process. Although the authors explain their definition of asymmetry in the text later, I personally feel there are better ways of expressing the propagation properties of the light for the two directions.\nResponse: We thank the Reviewer for the constructive comment. We agree that the term “asymmetric” may potentially imply that the wavefronts themselves are different. In our work, the intention is to emphasize that the proposed design maintains identical wavefront shaping capability for both propagation directions, while exhibiting distinct transmitted intensities. To better reflect this meaning and avoid possible misunderstanding, we have revised the Title on Page 1 as follows:\n“Intensity-asymmetric wavefront shaping in nonlocal metalens”\n2. What do the authors mean with magneto-electric coupling within the IRUs? An explanation would be helpful.\nResponse: We thank the Reviewer for this insightful comment. The magneto-electric coupling is primarily discussed in Fig. 2c and Eqs. (1) and (2) in the main text, which form the foundation of the asymmetric near-field responses. Figure 2c presents the electric dipole (ED) and magnetic dipole (MD) contributions for both propagation directions, obtained through multipole decomposition. 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 modes [Nat. Photonics 16, 561-565 (2022)]:\np x x x e e E x xm ye H y α α = ± (1)\nm y ym ym H y xm ye E x α α = ± − (2)\nwhere 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 α , α , and α correspond ee mm me to the electric, magnetic, and magneto-electric polarizability terms. Due to the presence of the magneto-electric coupling term α , the in-plane magnetic field H plays a key role in me y enhancing the asymmetry. For a single resonance, E and H generally have limited spatial x y overlap, making it difficult to simultaneously maximize nonlinear efficiency and asymmetry. In our design, the q-BIC resonance primarily enhances E and H , leading to high nonlinear x z efficiency while weak asymmetry. By contrast, the Mie-type MD resonance enhances H , y thereby enabling strong asymmetry for the high-Q q-BIC resonance through mode coupling. The insets in Fig. 2c display the normalized electric field distributions, which reveal similar field profiles but different enhancement levels. This asymmetry in field enhancement offers the potential for directional nonlinear responses.\nTo clarify this point, we have revised the discussions on Pages 7 and 8 of the Results section:\n“Figure 2c presents the electric dipole (ED) and magnetic dipole (MD) contributions for both propagation directions, obtained through multipole decomposition. 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 modes:\np x x x e e E x xm ye H y α α = ± (1)\nm y ym ym H y xm ye E x α α = ± − (2)\nwhere 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 α , α , and α correspond ee mm me to the electric, magnetic, and magneto-electric polarizability terms. Due to the presence of the magneto-electric coupling term α , the in-plane magnetic field H plays a key role in me y enhancing the asymmetry. For a single resonance, E and H generally have limited spatial x y overlap, making it difficult to simultaneously maximize nonlinear efficiency and asymmetry. In our design, the q-BIC resonance primarily enhances E and H , leading to high nonlinear x z efficiency while weak asymmetry. By contrast, the Mie-type MD resonance enhances H , y thereby enabling strong asymmetry for the high-Q q-BIC resonance through mode coupling. The insets in Fig. 2c display the normalized electric field distributions, which reveal similar field profiles but different enhancement levels. This asymmetry in field enhancement offers the potential for directional nonlinear responses.”\n3. If the authors use the passive nonlinearity of the system, then why is there a saturation effect observable at these relative low powers? For polarization induced nonlinear processes these effects are normally not be able to observe so easily. Did the authors check that the resonance is not shifting spectrally with the higher powers due to Kerr or thermal effects? Such an effect has nothing to do with saturation. I feel the authors need to be here more precise in their wording.\nResponse: We thank the Reviewer for this insightful comment. We agree with the Reviewer that the nonlinear harmonic powers tend to level off in Fig. 3c, which is primarily attributed to the resonance spectral shift induced by the thermo-optic effect. This process is a stable or quasi-stable process governed by the average pump intensity. The simulated influence of the thermo-optic effect on the meta-lens is provided in Supplementary Fig. 8. With increasing pump power, the forward direction experiences a larger refractive index change and a correspondingly greater resonance wavelength shift, resulting in a more pronounced decrease in transmission.\nSupplementary Fig. 8: a,b,c, Simulated transmission polarization conversion efficiency T (a), RL(LR) averaged refractive index increase Δn within the silicon IRU (b), and wavelength shift Δλ (c) as a function of pump power along forward and backward directions.\nTo improve clarity and accuracy, we have revised the term “saturation” to “the nonlinear harmonic powers tend to level off” throughout the text. Fig. 3c has been accordingly updated, and its caption has been modified on Page 12 of the Results section:\nRevised Fig. 3c: Dependence of nonlinear output power on the pump power. The gray-shaded region indicates the range where the harmonic powers level off.\nWe have revised relevant discussions on Pages 10 and 11 of the Results section:\n“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. 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 results, despite the relatively low pump intensity. A detailed performance comparison is provided in Supplementary Note 12.”\nand on Page 13 of the Results section:\n“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 onset of leveling-off 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.”\n4. I also do not agree with the term “nonreciprocal focusing in the linear regime.” In the linear optical regime (without losses or gain) the system is always reciprocal. Linear optical regime also means that there are no dependencies of the properties on the light intensity. I understand that the authors are unexperienced in the field of nonlinear optics but what they mean is that they observe a nonlinear effect at the fundamental wavelength. That has nothing do to with linear optics. I recommend to rephrase these statements.\nResponse: We would like to express our gratitude to the Reviewer for helping us improve the clarity. We agree that the correct description should refer to the nonlinear effects at the “fundamental wavelength”, rather than in the “linear regime”.\nWe have therefore revised all occurrences of “in the linear regime” to “at the fundamental wavelength” throughout the manuscript. For example, the subtitle on Page 13 of the Results section has been updated to:\n“Asymmetric wavefront shaping at the fundamental wavelength”\n5. How are the phase values in Fig. 2d calculated? For a single unit or a period pattern where all elements are oriented in the same way? What is the error in phase if the elements like in the metalens are all differently oriented?\nResponse: We thank the Reviewer for the helpful comment. The phase value is calculated in periodic units because the proposed meta-lens requires a unit array to fully excite the nonlocal q-BIC resonance. Our design requires the resonance to be nonlocal for a high Q factor while keeping the phase nearly local for stable wavefront shaping, i.e., the rotation of units provides a stable geometric phase while having little influence on the resonance. To demonstrate this point, we discuss the following four aspects:\n(1) In the unit design, the resonant wavelength, transmission, and Q factor all remain stable with varying rotation angles, and the generated geometric phases exhibit a nearly linear dependence on the rotation angle, as shown in Fig. 2d.\n(2) In the meta-lens design, the simulated phases extracted from the nonlocal meta-lens for the linear, SHG, and THG cases are shown in Supplementary Fig. 6. The obtained phase distributions are nearly consistent with the designed ones. The minor nonuniformities arise from the limited mesh resolution, as the full-lens simulation involves an extremely large data volume, but they do not affect our conclusions.\n(3) In the transmission spectra, experimental spectra of the meta-lens composed of units with different rotation angles agree well with the simulation spectra based on periodic units, as shown in Fig. 2b. A Fano-like line shape with a high-Q (~100) resonance peak is observed around 1570 nm.\n(4) In the light-field focusing profile, all the linear, SHG, and THG signals with different circular polarizations exhibit an effective focusing property. The experimental results show good agreement with the simulated results presented in Supplementary Figs. 7 and 8. The full-widths at half-maximum (FWHMs) of the forward SHG and THG focal spots for different polarizations in Fig. 3d and 3g 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.\nAccording to the above-mentioned four aspects, to elaborate on the phase design, we have added and revised some descriptions in the caption of Fig. 2d on Page 9 of the Results section:\nRevised Fig. 2d: Transmission (color map) and phase (markers) as functions of the IRU rotation angle. Phases are extracted regarding the resonant wavelength of 1570 nm in the periodic IRU. LCP, RCP: Left and right circular polarization.\nand on Page 11 of the Results section:\n“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 full-widths at half-maximum (FWHMs) of the forward SHG and THG focal spots for different polarizations in Fig. 3d and 3g 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.”\nand on Page 9 of Supplementary Note 6 section:\n“To demonstrate that the designed phases are correctly implemented at their corresponding locations, the simulated phase distributions extracted from the nonlocal meta-lens for the linear, SHG, and THG cases are presented in Supplementary Fig. 6. The obtained phase distributions closely match the designed ones, demonstrating the potential for effective focusing capability. The minor nonuniformities arise from the limited mesh resolution, as the full-lens simulation involves an extremely large data volume, but they do not affect our conclusions.”\nSupplementary Fig. 6: Designed and simulated phase profiles of the proposed nonlocal meta-lens for linear, SHG, and THG with different circular polarizations.\n6. The assumed nonlinear third-order permittivity of silicon to model the refractive index change seems extremely high. Typical values in the NIR range are 10E-18 to 10E-19 m2/V2. This is three to four orders of magnitude smaller than the value the authors used in their simulations.\nResponse: We thank the Reviewer for this insightful comment. The nonlinear refractive index change discussed in the main text originates from the thermo-optic effect, which is a stable or quasi-stable process governed by the average pump intensity. This steady-state effect can be described by an effective intensity-dependent nonlinear susceptibility (3 ) e ff χ that appears several orders of magnitude higher than the intrinsic transient nonlinearity, similar to the values reported in previous studies (~10-14 m2/V2) [Nat. Photonics 18, 81-90 (2024)].\nIn contrast, the nonlinear optical processes related to harmonic generation, as described in Supplementary Note 5, are governed by the intrinsic transient nonlinear susceptibility and are dependent on the peak pump intensity. The transient Kerr effect is not considered here, as our measurements do not involve ultrafast time-resolved characterization.\nWe have added the following clarification on Pages 11 and 12 of Supplementary Note 8 section:\n“This value is four orders of magnitude higher than the intrinsic transient nonlinearity used in Supplementary Note 5. This is because the nonlinear refractive index change discussed in our work originates from the thermo-optic effect, which is a stable or quasi-stable process governed by the average pump intensity. The transient Kerr effect, which depends on the peak pump intensity, is not considered here, as our measurements do not involve ultrafast time-resolved characterization.”\nFor Reviewer 4:\nGeneral Comment:\nI co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts.\nResponse:\nWe sincerely appreciate the time, effort, and thoughtful feedback contributed by both reviewers. Their detailed and constructive comments have greatly helped us improve the clarity, rigor, and overall quality of our manuscript. We are also grateful for this collaborative review process and for the opportunity to benefit from the expertise of both the primary and co-reviewer."}, {"round": 2, "reviewer_comments": {"reviewer_1": "The authors have satisfactorily addressed my comments.", "reviewer_2": "The authors have addressed my concerns to my satisfaction. I recommend publication.", "reviewer_3": "With their revision the authors addressed important aspects of the work and clarified several open questions. The explanations are now more detailed and better phrased. However, I still believe that the observed effect is more an incremental improvement of previous systems without any useful application. Therefore, I still believe that a more specialized optics journal would be more suited for publication.", "reviewer_4": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts."}, "author_response": "For Reviewer 1:\nGeneral Comment:\nThe authors have satisfactorily addressed my comments.\nResponse:\nWe thank the reviewer for the positive comments and the recommendation of our work.\nFor Reviewer 2:\nGeneral Comment:\nThe authors have addressed my concerns to my satisfaction. I recommend publication.\nResponse:\nWe thank the reviewer for the positive comments and the recommendation of our work.\nFor Reviewer 3:\nGeneral Comment:\nWith their revision the authors addressed important aspects of the work and clarified several open questions. The explanations are now more detailed and better phrased. However, I still believe that the observed effect is more an incremental improvement of previous systems without any useful application. Therefore, I still believe that a more specialized optics journal would be more suited for publication.\nResponse:\nWe thank the reviewer for the constructive comments to improve our manuscript. Our work focuses on overcoming the trade-off among nonlinear efficiency, phase modulation, and directional asymmetry in conventional local and nonlocal metasurface designs. Relying solely on silica-substrate-induced asymmetry without additional structural layers, the device enables efficient directional focusing across the fundamental, second-, and third-harmonic wavelengths. The combination of compactness, high efficiency, wavefront modulation, and directional asymmetry opens new opportunities for highly demanded asymmetric devices, such as nonreciprocal LIDAR and all-optical computing applications.\nFor Reviewer 4:\nGeneral Comment:\nI co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts.\nResponse:\nWe thank the reviewer for the positive comments and the recommendation of our work."}]

Decision letter