Nonlinear geometric phase control via high-Q quasi-BIC resonance in all-dielectric metasurfaces

Abstract

Nonlinear geometric metasurfaces are an important platform for information encoding and light-field manipulation with wide applications in display encryption and beyond. Enhancing nonlinear conversion efficiency and controlling the nonlinear geometric phase are equally crucial for practical use. Recently, high-Q resonances in nonlocal metasurfaces have shown strong light–matter interactions, providing a foundation for efficient nonlinear conversion. However, simultaneously achieving high efficiency and effective phase control remains a key challenge. Here, based on the principle of geometric phase, we efficiently realize third-harmonic phase control, where the high-Q resonance corresponds to a magnetic dipole–related quasi-BIC mode. At the ultra-narrow resonance bandwidth, the electric field enhancement exceeds 50 times, with a quality factor over 3000 and high stability under rotation. A third-harmonic conversion efficiency of the order of 10⁻³ is achieved at a peak pump density of 100 MW cm⁻², outperforming previously reported nonlinear geometric metasurfaces. This work holds significant potential for on-chip nonlinear information processing and wavefront control requiring strong light–matter interaction.

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New concepts

Conventional nonlinear geometric metasurfaces typically face a trade-off that unifies high-efficiency THG with precise phase control because the resonance that boosts harmonic generation is tightly coupled to the meta-atom geometry used for wavefront encoding. Here, we introduce a nonlocal nonlinear PB platform that decouples resonance linewidth engineering from phase modulation, enabling high-efficiency THG while retaining flexible phase control. This is achieved with symmetry-broken amorphous-silicon cylindrical resonators with off-centered air holes that excite a magnetic-dipole QBIC, providing ultranarrow resonances and strong field enhancement while keeping the resonance wavelength nearly insensitive to in-plane rotation—thereby preserving the PB design freedom. This insight further enables multi-channel wavefront manipulation such as focusing and vortex-beam generation derived from a single rotational degree of freedom, eliminating the need for iterative optimization. By establishing a coherent approach that couples long-lived resonances with nonlinear phase engineering, this work opens a new direction for high-capacity metasurface encoding, scalable nonlinear nanophotonics, and compact on-chip information processing.

Introduction

Metasurfaces, by tailoring the properties of electromagnetic waves such as intensity, polarization, and wavelength, can precisely control the propagation of light or modulate its behavior to meet specific application requirements including light sources,^1,2 displays,^3–5 imaging,^6–9 sensing,^10,11 and detection.^12,13 Among these, nonlinear metasurfaces have attracted extensive attention owing to their intrinsic frequency conversion capabilities and the possibility of additional modulation of the generated light. They have exhibited unique advantages in areas such as multiplexed holographic encryption¹⁴ and optical logic operations.¹⁵ Consequently, nonlinear metasurfaces are regarded as an ideal platform for realizing multifunctional integration and multidimensional information encoding in photonic devices.

A commonly employed approach for encoding in nonlinear metasurfaces is the geometric phase, also named the Pancharatnam–Berry (P–B) phase. The geometric phase principle provides a dispersion-independent phase shift by employing a rotational arrangement of meta-atoms, making it particularly effective for phase control under circularly polarized illumination. This mechanism allows a continuous phase modulation, providing a straightforward means for full-phase control. In the past, the realization of the spatial phase control of generated light using nonlinear geometric metasurfaces primarily focused on the application of the spin-rotation coupling concept¹⁶ within nonlinear systems. Using metallic or dielectric structures with specific rotational symmetries, effective control of geometric phases has been realized in processes such as second-harmonic generation (SHG),¹⁷ third-harmonic generation (THG),^18,19 and four-wave mixing (FWM),^20,21 facilitating significant advances in multi-channel information encoding,^14,22 holographic imaging and color displays,²³ and vortex beam generation.²⁴ However, both the efficiency and control of harmonic generation are equally critical, as they directly determine the practical applicability of such systems.

In plasmonic metasurfaces, strong non-radiative absorption from free electrons at resonance leads to severe pump energy losses. Furthermore, the nonlinear response of metals primarily originates from their surfaces or metal–dielectric interfaces, resulting in a limited effective nonlinear volume. Even though hybrid plasmonic metasurfaces employing epsilon-near-zero effects²² or optimized antenna geometries²⁵ can enhance nonlinear processes, their conversion efficiencies remain low. In contrast, all-dielectric nanostructures with high refractive index and large nonlinear coefficients can substantially improve conversion efficiency while supporting multifunctional operation. For instance, nonlinear geometric-phase metasurfaces based on silicon have demonstrated THG conversion efficiencies on the order of 10⁻⁶–10⁻⁸,^18,19,26 though further improvement is still desirable.

Despite their success, conventional nonlinear geometric-phase metasurfaces rely on localized Mie or plasmonic resonances that inherently possess low quality factors (Q-factors), leading to insufficient nonlinear conversion efficiency. This limitation severely hinders their application in integrated photonics.²⁷ To overcome this bottleneck, bound states in the continuum (BICs) and their quasi-BIC (QBIC) counterparts have emerged as powerful mechanisms to achieve high-Q resonances,²⁸ strong field enhancement,^29,30 reduced scattering losses and great robustness.³¹ Through mechanisms such as symmetry breaking,^29,32 Friedrich-Wintgen interference,^31,33 and Fabry–Pérot type resonances,³⁴ QBICs have been experimentally realized in dielectric metasurfaces and photonic crystals with remarkably high Q-factors.^35,36 Their exceptional field confinement provides a robust physical foundation for enhancing various nonlinear optical processes. In THG, for example, asymmetric silicon metasurfaces have exhibited THG enhancements exceeding five orders of magnitude,³⁵ while AlGaAs nanoantennas have achieved superior conversion efficiencies through optimal coupling between the pump beam and QBIC modes.^37,38 Additional optimization strategies have further improved performance, such as coupling QBIC resonators with epsilon-near-zero substrates to improve phase matching,³⁹ or merging multiple BICs in momentum space to achieve ultra-high Q modes resilient to fabrication imperfections.³⁶ Nevertheless, a major limitation remains: most existing studies focus primarily on enhancing nonlinear conversion efficiency, while active phase control of the generated harmonics is largely overlooked.^30,37 A key remaining challenge is to integrate the high-efficiency characteristics of QBIC with the versatile wavefront manipulation offered by geometric-phase metasurfaces into a unified framework capable of both efficient nonlinear conversion and precise phase control.

Inspired by linear high-Q nonlocal metasurfaces^40–43 that sustain ultranarrow resonances while enabling independent wavefront shaping, we identify an underexplored opportunity for nonlinear light-field control: decoupling resonance-linewidth engineering from phase modulation.^40,41 Here we target a long-standing bottleneck in nonlinear metasurfaces to simultaneously achieve high nonlinear conversion efficiency and flexible, high-precision nonlinear wavefront engineering. We extend high-Q geometric-phase metasurface concepts into the nonlinear regime by using high-index amorphous silicon and symmetry-broken cylindrical resonators with off-centered air holes to excite QBIC modes. The resulting ultranarrow resonances provide strong field enhancement, enabling substantially improved THG. Under circularly polarized pumping, we further realize linear geometric-phase control in both co- and cross-polarized nonlinear channels via antenna rotation. Based on this nonlinear geometric-phase mechanism, we demonstrate multifunctional wavefront control, including triple-channel focusing and vortex-beam generation, as illustrated in Fig. 1(a and b). This single-platform integration overcomes the conventional efficiency–phase-control trade-off, enabling high-capacity nonlinear optical information encoding.

Fig. 1 Schematic illustration of multifunctional metasurfaces capable of simultaneously manipulating linear and nonlinear optical generation. (a) Under left-handed circularly polarized (LCP) excitation, the metasurface produces focused beams of right-handed circularly polarized fundamental light (RCP-FF) and both RCP-TH and LCP-TH components, each with distinct focal lengths. (b) For LCP incident light, the metasurface simultaneously generates focused vortex beams of RCP-FF, RCP-TH, and LCP-TH light, exhibiting different focal lengths and topological charges. (c) The unit cell consists of amorphous silicon (a-Si) on a fused-silica substrate, with optimized geometric parameters: period P = 740 nm, silicon pillar radius R = 220 nm, height H = 210 nm, air-hole offset l = 90 nm, and hole radius r = 30 nm.

Results and discussion

Engineering symmetry-broken meta-atoms for tailored high-Q linear resonances

The meta-atom consists of cylindrical nanoresonators with off-axis air holes, fabricated from amorphous silicon on a quartz substrate. Numerical simulations were performed using the finite element method (FEM) (more details can be found in the SI). By fixing all other parameters, we calculated the influence of the air-hole radius (r) and offset distance (l) on the transmission spectra. As shown in Fig. 2(a and b), increasing either r or l causes a blue shift of the resonance dip accompanied by a gradual broadening of the linewidth. Within the wavelength range of 1.3–1.55 µm, only a single resonance mode appears as r and l vary, indicating that no additional resonances are introduced. Consequently, the optimal parameters are determined by considering the resonance position, radiative characteristics, and fabrication feasibility, as shown in Fig. 1(c). This configuration introduces an in-plane asymmetry within an otherwise symmetric cylindrical resonator, forming a model of BICs. Under circularly polarized excitation, the simulated linear transmission spectrum of the periodic structure exhibits a sharp resonance peak near 1.45 µm with a linewidth of approximately 0.4 nm, corresponding to QBIC with magnetic dipole (MD) characteristics, as shown in Fig. 2(c). The electric field distribution in the x–y plane as depicted in Fig. 2(d) reveals an asymmetric vortex pattern, with a strong field enhancement near the air hole. The maximum total electric field enhancement is 56 times. The corresponding magnetic field in the y–z plane is oriented parallel to the cylinder axis.

Fig. 2 Simulated transmission characteristics of the metasurface under circularly polarized illumination. (a) The transmission spectra as a function of the air-hole offset l. (b) The transmission spectra for different air-hole radii r. (c) Simulated transmission spectrum of the optimized structure with all parameters fixed. (d) Electromagnetic field distribution of the QBIC mode within a single unit cell, where red arrows indicate the in-plane electric field direction.

To further investigate the nature of the QBIC, we analyzed a symmetric nanocylinder without asymmetry breaking (for details of the design, see the SI). As shown in the inset of Fig. S1b, the electric field in the x–y plane forms a symmetric vortex, representing a vertical magnetic dipole mode aligned with the cylinder axis. The calculated band structure and Q-factors demonstrate that as the wavevector approaches the Γ point, the Q-factor increases exponentially, exceeding 10⁸, confirming the existence of a symmetry-protected BIC that cannot couple to free space. This occurs because the mode's in-plane symmetry is incompatible with that of a normally incident plane wave, and the subwavelength lattice constant restricts the non-normal incident wavevector. These characteristics eliminate radiative decay channels, resulting in ultra-high Q-factors and long lifetime resonances. Such long lifetime BIC modes are crucial for efficient nonlinear frequency conversion, as they enable prolonged interaction between the localized field and the induced nonlinear polarization. By introducing an eccentric air hole, the in-plane symmetry of the nanocylinder is broken, opening radiative channels and transforming the ideal BIC into a QBIC that can couple to free-space radiation. The corresponding electric and magnetic field distributions are shown in Fig. 2(d). Compared with the field of the symmetric cylinder, the field vortex in the air-hole structure remains largely intact but exhibits enhanced localization near the hole, giving rise to a nonzero in-plane electric dipole moment consistent with the symmetry of the incident plane wave.

Fig. 3(a) illustrates the variation of the resonance wavelength and Q-factor as a function of air-hole radius while keeping other parameters fixed. As the radius increases, the resonance wavelength undergoes a blue shift, and the Q-factor decreases significantly. Fig. 3(b) shows that the resonance wavelength remains nearly independent of the rotation angle, which is advantageous for subsequent geometric-phase control. Although the Q-factor exhibits slight periodic modulation with rotation, it remains within the same order of magnitude, implying negligible influence on the resonance quality. To further characterize the resonance behavior of the asymmetric unit cell, the offset distance l of the air hole was introduced as an asymmetry parameter. As shown in Fig. 3(c), the Q-factor decreases rapidly with increasing l, approaching infinity as l tends toward zero. Theoretically, the Q-factor follows an inverse-square relationship (Q ∝ 1/l⁻²), a hallmark of symmetry-protected BICs. Linear fitting yields a slope of 1.97, in excellent agreement with theoretical expectations. Furthermore, the multipole decomposition analysis of the periodic structure in Fig. 3(d) confirms that the magnetic dipole dominates the resonance at the QBIC position, verifying that the sharp MD resonance originates from symmetry-breaking-induced QBIC behavior.

Fig. 3 Analysis of resonance characteristics and multipole responses of the metasurface. (a) Evolution of the resonance wavelength and the Q-factor of the BIC mode as a function of the air-hole radius r. (b) Dependence of the resonance wavelength and Q-factor on the rotation angle θ of the structure. (c) Variation of the Q-factor with the air-hole offset l, serving as the asymmetry parameter; the red dashed line indicates the fitted trend (log–log scale). (d) Normalized scattering cross-sections obtained from multipole decomposition under identical conditions.

Nonlinear geometric phase control and three-channel nonlinear metasurface for focusing and OAM beam generation

Based on the above discussion, the symmetry-broken cylindrical structure enables strong electromagnetic field enhancement under circularly polarized excitation, which is favorable for improving the THG conversion efficiency. Meanwhile, due to its C₁ rotational symmetry, the structure naturally supports the introduction of a nonlinear geometric phase. In plasmonic metasurfaces, the effective nonlinear surface current can generate THG carrying a spin-dependent nonlinear geometric phase. The meta-atom can be approximately regarded as a planar medium, where its in-plane symmetry determines the form of the nonlinear susceptibility tensor, and only the in-plane components of the electric field need to be considered. However, in contrast to plasmonic geometric-phase metasurfaces, the nonlinear harmonics in dielectric metasurfaces originate from the induced volume nonlinear polarization, resulting in collective radiation. Owing to their large aspect ratio and high refractive index, strong light confinement and wave coupling occur within the dielectric medium, leading to localized field interactions. The coupling among different components of the fundamental wave influences the phase evolution of the generated harmonics, thereby introducing an additional nonlinear geometric phase in the emitted radiation.¹⁹ If the dielectric geometric-phase metasurface is theoretically treated as an equivalent two-dimensional interface, the z-directional coupling can be neglected. Under normal incidence of left-handed circularly polarized fundamental light (LCP-FF), the third-order nonlinear polarization can be expressed as:¹⁹Here, eqn (1) indicates that the generated LCP-TH component carries a geometric phase of exp(2iθ), while the RCP-TH component carries exp(4iθ). P^(3)eff_L and P^(3)eff_R represent the LCP and RCP components of the volume effective nonlinear polarization, respectively. ε₀ is the permittivity of the medium. In the circular coordinate system, the effective third-order susceptibility tensor can be obtained via coordinate transformation, where α, β, γ and δ denote the circular coordinates (L, R, z), and i, j, k, and l represent the Cartesian coordinates (x, y, z). Λ^α_i denotes the transformation matrix between the two coordinate systems.

The simulated nonlinear phase distributions and THG intensities are shown in Fig. 4(a and b). As the rotation angle of the meta-atom varies from 0° to 360°, the co-polarized THG component exhibits a phase variation from 0 to 4π, while the cross-polarized component varies from 0 to 8π, in agreement with theoretical predictions. The cross-polarized THG intensity is notably stronger than that of the co-polarized component, with the peak intensity reaching nearly an order of magnitude. Due to the linear dependence of the geometric phase on the rotation angle, it is essential to select an operating wavelength within the resonance range that simultaneously ensures great phase linearity and high nonlinear conversion efficiency. The theoretical conversion efficiency is defined as η = P_TH/P_in. As shown in Fig. 4(c), at an input intensity of 100 MW cm⁻², the conversion efficiency is 2.09 × 10⁻³ at a working wavelength of 1.453 µm, outperforming previously reported nonlinear geometric-phase metasurfaces. The inset of Fig. 4(c) shows that although the THG efficiency slightly fluctuates with the rotation angle due to near-field coupling between adjacent meta-atoms, it remains within the same order of magnitude. The mean conversion efficiency is 2.5 × 10⁻³, with a standard deviation of 0.039%. Furthermore, the THG signal exhibits high spectral sensitivity to the excitation wavelength near the QBIC resonance. As the excitation wavelength deviates from resonance, the THG intensity decreases sharply, confirming that the QBIC mode is the dominant contributor to enhance conversion efficiency. The electric field distributions of the two THG polarization components are plotted in Fig. 4(d).

Fig. 4 Simulation results for THG in the silicon resonator. (a) The simulated nonlinear geometric phase is plotted for both co-polarized and cross-polarized components as the silicon resonator unit is rotated from 0° to 360° in 5° increments. This demonstrates the phase modulation capability of the unit. (b) Following the phase analysis, this panel shows the transmitted intensity of the co- and cross-polarized components as a function of wavelength, characterizing the spectral response of the device. (c) The simulated THG conversion efficiency is presented as a function of wavelength. The inset further details the THG conversion efficiency at varying rotation angles of the resonator unit, again from 0° to 360° in 5° steps. (d) Finally, the electric field distributions for the co- and cross-polarized TH components are visualized, providing insight into the spatial characteristics of the generated nonlinear signal.

To further verify the above phase relationship under various polarization and wavelength conditions, we designed a three-channel (RCP–FF, RCP–TH, LCP–TH) encoding metasurface capable of generating independent optical responses carrying different information. For multi-focal metalens, when illuminated by LCP fundamental light at 1.453 µm, three focusing channels with different focal lengths can be designed. The optical path difference between an arbitrary point on the metalen surface (x,y) and the focal point corresponding to focal length f can be expressed as , leading to a corresponding phase difference φ = k·Δ. For the RCP–FF channel with focal length f_RCP–FF = 100 µm, the phase distribution ϕ_RCP–FF is shown in Fig. 1(a). The focusing behavior along the propagation direction is obtained using the angular spectrum propagation method, as illustrated in Fig. 5(a and c). The theoretical focal lengths for other channels are f_RCP–TH = 150 µm and f_LCP–TH = 300 µm, respectively. Although minor deviations exist between the simulated and theoretical results due to phase discretization during design, the results clearly demonstrate the feasibility of multi-focal, three-channel nonlinear focusing.

Fig. 5 Simulated performance of multifunctional metasurfaces under left-handed circularly polarized (LCP) illumination. (a)–(c) Field profiles of a spin-selective metalen that focuses different spin and frequency components to distinct points. The outputs shown are for the right-handed circularly polarized fundamental frequency (RCP-FF), RCP third-harmonic (RCP-TH), and LCP third-harmonic (LCP-TH). (d)–(f) Intensity profiles of a spin-selective metasurface that generates focused vortex beams. The outputs correspond to the RCP-FF beam with topological charge l = 6, the RCP-TH beam with l = 3, and the LCP-TH beam with l = 6. For all panels, the insets show the intensity distribution in the XY focal plane.

Similarly, we designed a metasurface capable of generating three-channel focused vortex beams. The combined phase profiles of the lens and vortex plate are shown in Fig. 1(b), expressed as:

where l is the topological charge of the emitted optical vortex, and k is the wavevector of the fundamental light. For the three channels, the focal lengths and topological charges are designed as f_RCP–FF = 200 µm, l_RCP–FF = 6; f_RCP–TH = 300 µm, l_RCP–TH = 3; f_LCP–TH = 600 µm, l_LCP–FF = 6, respectively. The simulated results in Fig. 5(d–f) confirm that each channel generates a vortex beam with the designed orbital angular momentum (OAM) state. This demonstrates that, by exploiting the phase relationship between polarization channels, only a single-phase distribution needs to be designed—phase maps for the other channels can be directly derived without additional optimization or iterative retrieval algorithms. This approach simplifies the design process while enabling multifunctional nonlinear optical control within a compact metasurface platform.

As shown in Table 1, the performance comparison summarizes the essential characteristics of different types of nonlinear metasurfaces, including conversion efficiency, phase modulation, and design degrees of freedom. Finally, our approach simultaneously combines high nonlinear conversion efficiency and continuous phase tunability, unifying efficient nonlinear generation and flexible wavefront control within a single metasurface platform, which is significantly higher than that of most comparable designs.

Table 1 Comparison of the current state of the art of THG nonlinear metasurfaces

Material	Method	Structure	Wavelength (µm)	Function of PB phase modulation	Efficiency	Ref.
Si	SP-QBIC	Dimer-hole	1.5	No	3.6 × 10^−6@1 GW cm^−2	44
Si	High-order Mie resonance	Block	1.2	No	3.25 × 10^−5@23.8 GW cm^−2	45
Si	SP-QBIC	Block	1.55	No	1.26 × 10^−5@9.2 GW cm^−2	46
Si	Multi-mode Fano	Elliptical resonators	1.5	No	2.8 × 10^−7@1.2 GW cm^−2	47
Si	Doubly degenerate QBIC	Nanodisks	1.539	No	1.03 × 10^−5@5.85 GW cm^−2	48
Si	SP-QBIC	Kite-shaped	1.36	No	1.75 × 10^−6@8.2 GW cm^−2	49
Si	SP-QBIC	Disk with air hole	1.345	No	∼10^−5@1 GW cm^−2	50
Si	QBIC&Magnetic Mie mode	Disk with air hole	1.3	Yes	1.4 × 10^−6@0.32 GW cm^−2	26
Si	Localized modes	block	1.3	Yes	∼10^−10@50 mW	18,19
Si	Localized modes	C3/C4 meta-atom	1.25	Yes	∼10^−8@1.33 GW cm^−2	21
Au	Plasmonic resonance	Block	1.55	Yes	2.63 × 10^−10@55 MW cm^−2	25
Si	SP-QBIC	Disk with air hole	1.45	Yes	2.09 × 10^−3@100 MW cm^−2	This work

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Conclusions

In this work, we have successfully designed and theoretically demonstrated a nonlocal nonlinear metasurface based on high-Q QBICs. This platform enables independent phase control under an ultranarrow linewidth by eliminating the dependence of the resonance wavelength on resonator rotation, thereby achieving both efficient nonlinear frequency conversion and flexible geometric-phase modulation. The metasurface employs symmetry-broken cylindrical nanoresonators to excite magnetic-dipole-associated QBIC modes, yielding electric field enhancement exceeding 50-fold and Q factors above 3000 in the near-infrared regime. Under a pump intensity of 100 MW cm⁻², a THG conversion efficiency on the order of 10⁻³ is achieved, which is significantly higher than that reported for previously demonstrated nonlinear metasurfaces. Through theoretical analysis and numerical simulations, we further verify the continuous tunability of the nonlinear geometric phase supported by this structure and demonstrate multifunctional wavefront engineering capabilities, including multichannel focusing and vortex beam generation. This work successfully integrates high-Q resonant enhancement with geometric-phase control, overcoming the long-standing trade-off between nonlinear efficiency and phase tunability in conventional nonlinear metasurfaces. Moreover, it provides a promising design paradigm for integrated nonlinear photonic devices, high-capacity optical information encoding, and dynamic wavefront engineering. Future efforts may focus on optimizing the radiative properties and diffraction states of the structure to further enhance nonlinear conversion efficiency and phase-control performance, thereby advancing the practical application of this platform in nonlinear optical processing and on-chip photonic information systems.

Author contributions

He-Rui Yang: investigation, conceptualization, data curation, formal analysis, methodology, software, validation, visualization and writing the original draft; Wen-Juan Shi: methodology, validation and editing; Cong-Fu Zhang: methodology, validation and editing; Wang-Hao Zhu: methodology, validation and editing; Di Ma: methodology, validation and editing; Shao-Jie Li: methodology, validation and editing; Zhao-Lu Wang: validation and editing; Hong-Jun Liu: funding acquisition, supervision, project administration, validation and editing. All authors reviewed the manuscript.

Conflicts of interest

There are no conflicts to declare.

Data availability

The data supporting the findings of this study, including numerical simulation data, and processed datasets, are presented in the main text and supplementary information (SI). The custom simulation procedures were implemented using commercially available electromagnetic simulation software, and all essential modeling parameters and configurations required for reproducibility are provided in the Methods section and SI. See DOI: https://doi.org/10.1039/d6nh00054a.

Additional raw data related to this work are available from the corresponding author upon reasonable request.

Acknowledgements

This work is supported by the National Natural Science Foundation of China (NFSC) (Project ID 61975232).

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