Nonlinear Nonlocal Metasurfaces

• Nonlinear nonlocal metasurfaces are resonant planar nanostructures that use collective lattice modes rather than isolated resonances to facilitate enhanced frequency conversion processes.
• They integrate guided-mode resonances, quasi-BICs, and geometric phase engineering to achieve controlled imaging, electro-optic modulation, and harmonic generation across multiple nonlinear regimes.
• Design strategies balance trade-offs between high quality factors and usable bandwidth while enabling multifunctional control over amplitude, phase, momentum, and polarization.

Nonlinear nonlocal metasurfaces are resonant planar nanostructures in which nonlinear optical processes are mediated by collective, lattice-enabled modes rather than only by isolated local resonances. In the recent literature, this category includes guided-mode-resonant lithium niobate gratings for sum-frequency up-conversion imaging, silicon nonlocal phase-gradient metasurfaces based on quasi–bound states in the continuum for third-harmonic generation with wavefront control, lithium-niobate-on-insulator quasi-BIC arrays for electro-optical modulation in both the linear and nonlinear regime, plasmonic–lithium niobate lattices for continuous tuning of second-harmonic chirality, and topologically asymmetric all-dielectric metasurfaces supporting quasi-trapped modes for harmonic generation and Pancharatnam–Berry phase manipulation (Molina et al., 2024, Tian et al., 2 Feb 2026, Francescantonio et al., 2024, Liu et al., 16 Sep 2025, Sedeh et al., 8 Jul 2025).

1. Platform definitions and representative implementations

A recurring feature of these systems is that the metasurface periodicity, symmetry breaking, or lattice anisotropy couples otherwise dark or weakly radiative slab-guided modes to free space. The resulting resonances are described as guided-mode resonances, quasi-BICs, q-BICs, or quasi-trapped modes, depending on the structure and symmetry class. In nonlinear operation, these resonances enhance the local field overlap with $\chi^{(2)}$ or $\chi^{(3)}$ nonlinearities and thereby increase frequency-conversion efficiency or enable nonlinear wavefront engineering (Molina et al., 2024, Tian et al., 2 Feb 2026, Francescantonio et al., 2024, Liu et al., 16 Sep 2025, Sedeh et al., 8 Jul 2025).

Platform	Resonant structure	Demonstrated function
Etch-free LiNbO$_3$ metasurface	x-cut LiNbO$_3$ film, $t=307$ nm, with 1D SiO$_2$ ridges of period $a=879$ nm, width $d=500$ nm, height $h=200$ nm	SWIR-to-visible SFG imaging and edge detection
Silicon NPGM	260 nm silicon slab on glass with elliptical holes, period $P=750$ nm, q-BIC at $\chi^{(3)}$0 nm	THG, polarization-dependent deflection, dual-beam modulation
LiNbO$\chi^{(3)}$1 quasi-BIC nanowire array	x-cut LNOI, period $\chi^{(3)}$2 nm, asymmetric 1D array of nanowires	GHz electro-optic modulation and SHG modulation
Plasmonic–LiNbO$\chi^{(3)}$3 anisotropic lattice	Gold nanodisks on x-cut LN, $\chi^{(3)}$4 nm and $\chi^{(3)}$5 nm	Continuous SH chirality tuning
a-Si QTM metasurface	a-Si cylinders, $\chi^{(3)}$6 nm, $\chi^{(3)}$7 nm, off-center hole, square lattice	THG enhancement and resonant PB-phase control

The material set is correspondingly diverse. Lithium niobate appears in both purely dielectric and hybrid plasmonic–dielectric realizations because it combines Pockels tunability, a wide transparency window, and strong second-order nonlinearity. Silicon and amorphous silicon appear in third-order implementations because the q-BIC or QTM field enhancement boosts the cubic nonlinear polarization. A plausible implication is that the phrase “nonlinear nonlocal metasurface” now denotes a design class rather than a single geometry.

2. Resonant physics, nonlocality, and quality factor

In guided-mode-resonant implementations, the resonance is set by momentum matching between free-space illumination and a slab mode. For the LiNbO$\chi^{(3)}$8 up-conversion metasurface, the resonance condition is

$\chi^{(3)}$9

with $_3$0 and $_3$1 the guided-mode propagation constant. Near normal incidence, the resonance disperses as

$_3$2

so even a small in-plane wavevector shifts the resonance frequency. Because the guided mode propagates in-plane, an oblique incidence of even $_3$3 splits the resonance into a bright and a dark branch, which is the specific nonlocality emphasized in that system (Molina et al., 2024).

Temporal coupled-mode theory is used repeatedly to describe linewidth and loading. In the LiNbO$_3$4 imaging platform, the transmission is written as

$_3$5

with

$_3$6

Measured values were $_3$7–$_3$8 depending on fabrication, and the highest-Q samples reached approximately $_3$9 (Molina et al., 2024). In the lithium-niobate electro-optic modulator, the corresponding quasi-BIC linewidth was below $_3$0 nm, with $_3$1 and $_3$2 around $_3$3 nm (Francescantonio et al., 2024).

The symmetry mechanism differs across platforms but serves the same role. In the LNOI modulator, a folded guided mode is symmetry-protected at perfect mirror symmetry and becomes a leaky quasi-BIC once the unit cell is made asymmetric. In the a-Si harmonic-generation platform, an off-center cylindrical hole breaks the native $_3$4 symmetry to $_3$5, enabling a dark trapped mode to couple weakly to free space and become a high-Q quasi-trapped mode. There, temporal coupled-mode theory gives

$_3$6

and experiment reported $_3$7 as high as $_3$8 for $_3$9 nm and $t=307$0 nm (Sedeh et al., 8 Jul 2025).

The chirality-control platform realizes nonlocality differently, through two orthogonally propagating guided-mode resonances enabled by lattice anisotropy and LN birefringence. Under normal incidence, momentum matching excites a TE mode along $t=307$1 at $t=307$2 nm and a TE mode along $t=307$3 at $t=307$4 nm, with simulated quality factors $t=307$5 and $t=307$6 (Liu et al., 16 Sep 2025). This two-resonance structure is central to continuous polarization control at the harmonic frequency.

3. Nonlinear frequency conversion mechanisms

The nonlinear processes demonstrated in this class span sum-frequency generation, second-harmonic generation, and third-harmonic generation. In the LiNbO$t=307$7 infrared-imaging platform, the mechanism is SFG with

$t=307$8

where the pump is near $t=307$9 nm, the SWIR signal is resonant at $_2$0 nm, and the visible output is near $_2$1 nm. The nonlinear polarization is written as

$_2$2

and, in the undepleted-pump thin-film limit,

$_2$3

Because the film is sub-wavelength in $_2$4, longitudinal phase mismatch is negligible, while the grating supplies the in-plane momentum required for coupling (Molina et al., 2024).

The same work gives a standard approximate scaling for resonant SFG in a triply resonant cavity,

$_2$5

with $_2$6. Experimentally, the metasurface with $_2$7 reached a normalized efficiency of $_2$8 cm$_2$9/GW and produced a $a=879$0 stronger SFG signal than a bare LiNbO$a=879$1 film (Molina et al., 2024).

In the LNOI quasi-BIC modulator, the second-harmonic source is

$a=879$2

with radiated intensity scaling as

$a=879$3

By electrically shifting the resonance and therefore the local fundamental field, the device produced up to a factor-5 change in SHG on resonance and over one order of magnitude on the resonance slopes, corresponding to $a=879$4 V$a=879$5 for $a=879$6 V (Francescantonio et al., 2024).

Third-order realizations use analogous field-enhancement logic. In the silicon NPGM, the resonantly enhanced cubic polarization is

$a=879$7

and the measured THG conversion efficiency reached $a=879$8 at $a=879$9 nm under a pump intensity of $d=500$0 GW/cm$d=500$1; a log–log slope of $d=500$2 confirmed third-order scaling (Tian et al., 2 Feb 2026). In the all-dielectric a-Si QTM metasurface, the third-order polarization is written

$d=500$3

with simulated field enhancement $d=500$4 and an experimental THG yield enhancement of approximately $d=500$5 relative to an unstructured $d=500$6 nm a-Si film (Sedeh et al., 8 Jul 2025).

4. Wavefront engineering, geometric phase, and nonlinear chirality

One of the central technical themes in this area is the attempt to combine the efficiency of nonlocal resonances with the phase control usually associated with local metasurfaces. The silicon NPGM implements this through a nonlocal nonlinear geometric phase. Rotating each elliptical hole by an angle $d=500$7 gives, for right-circularly polarized pumping, nonlinear polarizations with phase factors $d=500$8 and $d=500$9 for the co-polarized and cross-polarized THG components, respectively. In the notation of that work, the nonlinear phase is therefore $h=200$0 for the co-polarized component and $h=200$1 for the cross-polarized component. Using 12 supercells with $h=200$2 varying in $h=200$3 steps over a full $h=200$4 cycle, the TH light is directed into discrete diffraction orders: under RCP pumping, the co-polarized THG appears in the $h=200$5nd order and the cross-polarized THG in the $h=200$6th order, while LCP reverses the sign to the negative orders (Tian et al., 2 Feb 2026).

The all-dielectric a-Si QTM platform uses a related but distinct PB-phase framework. For an $h=200$7th-order nonlinear process under circularly polarized pump of helicity $h=200$8, a rotated meta-atom imparts

$h=200$9

For THG this becomes

$P=750$0

By rotating $P=750$1 from $P=750$2 to $P=750$3, the co-polarized TH acquires full $P=750$4 phase coverage and the cross-polarized TH full $P=750$5 coverage. A specific result of that work is that a slight boundary perturbation switches the PB response on only at resonance: off resonance the phase is approximately zero, whereas at $P=750$6 the local hole region dominates and the nonlinear geometric phase follows the rotation angle (Sedeh et al., 8 Jul 2025).

Continuous control of nonlinear polarization is demonstrated most explicitly in the plasmonic–LiNbO$P=750$7 chirality platform. There, the orthogonally propagating guided-mode resonances are described by steady-state amplitudes

$P=750$8

which feed the tensorial $P=750$9 response of x-cut LN. The emitted SH chirality is quantified by

$\chi^{(3)}$00

Under the analytical model,

$\chi^{(3)}$01

At $\chi^{(3)}$02 nm in simulation, $\chi^{(3)}$03 varies from $\chi^{(3)}$04 at $\chi^{(3)}$05 to $\chi^{(3)}$06 at $\chi^{(3)}$07 while the SH intensity remains within $\chi^{(3)}$08 of its mean. Experimentally, at $\chi^{(3)}$09 nm, $\chi^{(3)}$10 was tuned from $\chi^{(3)}$11 at $\chi^{(3)}$12 to $\chi^{(3)}$13 at $\chi^{(3)}$14, with degree of polarization approximately $\chi^{(3)}$15 and intensity variation below $\chi^{(3)}$16 (Liu et al., 16 Sep 2025).

5. Imaging, electro-optical control, and analog processing

Nonlinear nonlocal metasurfaces are not limited to isolated frequency-conversion experiments; they also operate as image transducers and electrically tunable optical elements. In the SWIR up-conversion imaging system, the object is illuminated by the SWIR beam, lens $\chi^{(3)}$17 Fourier-images the object onto the metasurface so that the transfer function is uniform in momentum space, and $\chi^{(3)}$18 decodes the SFG back to real space on a standard silicon CMOS camera. This architecture was chosen because strong angular dispersion would otherwise suppress or distort spatial frequencies. Experimentally, images with high conversion efficiency and resolution quality were obtained despite strong nonlocality; the spatial resolution was approximately $\chi^{(3)}$19m on a Siemens star, and the limit was attributed to the 4-lens imaging chain rather than to the metasurface itself. The signal-to-noise ratio was reported to be on par with direct InGaAs detection and significantly better than a thin-film up-converter (Molina et al., 2024).

The same platform also demonstrates simultaneous imaging and edge detection. A spatially variant nonlocal 1D metagrating with a topological phase dislocation produces direct wavefront-preserving up-conversion in the zeroth SFG order,

$\chi^{(3)}$20

while the first diffraction orders implement a first-derivative filter,

$\chi^{(3)}$21

up to convolution with the pump. Numerically, the direct image is carried by $\chi^{(3)}$22 and high-contrast edges by $\chi^{(3)}$23 (Molina et al., 2024).

Electrical control is realized in the LNOI nonlocal metasurface through the Pockels effect. The first-order change of the inverse-index tensor is

$\chi^{(3)}$24

or, in scalar form for a principal direction,

$\chi^{(3)}$25

For x-cut LiNbO$\chi^{(3)}$26, the largest coefficient is $\chi^{(3)}$27 pm/V when the field $\chi^{(3)}$28 is aligned to the extraordinary axis. The resonance shift changes the reflectance according to

$\chi^{(3)}$29

The device yielded $\chi^{(3)}$30, with peak-to-peak relative modulation approximately $\chi^{(3)}$31 at $\chi^{(3)}$32 V$\chi^{(3)}$33, a linear modulation efficiency $\chi^{(3)}$34 V$\chi^{(3)}$35 for $\chi^{(3)}$36 V, a 3 dB bandwidth around $\chi^{(3)}$37 MHz, and detectable modulation up to $\chi^{(3)}$38 GHz (Francescantonio et al., 2024).

The silicon NPGM extends the control dimension from beam steering to active nonlinear intensity modulation by dual-beam interference. With two fundamental beams,

$\chi^{(3)}$39

the cubic nonlinearity gives

$\chi^{(3)}$40

By varying relative phase, polarization, and intensity ratio, the THG efficiency was tuned from $\chi^{(3)}$41 to $\chi^{(3)}$42, with near-unity modulation depth (Tian et al., 2 Feb 2026).

6. Trade-offs, misconceptions, and design directions

A persistent design trade-off in the field is the relation between quality factor and usable bandwidth. In the infrared up-conversion metasurface, higher-$\chi^{(3)}$43 samples near $\chi^{(3)}$44 produced only a modest additional gain of approximately $\chi^{(3)}$45–$\chi^{(3)}$46 over lower-$\chi^{(3)}$47 samples because narrow resonances filtered the femtosecond pulse spectrum. For pulse durations around $\chi^{(3)}$48 fs, that work identifies an optimal range of approximately $\chi^{(3)}$49–$\chi^{(3)}$50 (Molina et al., 2024). This directly qualifies the common assumption that larger $\chi^{(3)}$51 is always better for nonlinear conversion.

A second recurrent issue is the perceived incompatibility between nonlocal efficiency and meta-atom-level phase control. Both the silicon NPGM and the a-Si QTM platform explicitly target this point. One states that nonlocal metasurfaces enable high nonlinear conversion efficiency while local ones offer versatile wavefront control, yet achieving both within a single metasurface remains challenging; the other states that existing designs suffer from a trade off between the high efficiency of nonlocal metasurfaces and the precise wavefront control enabled by local ones (Tian et al., 2 Feb 2026, Sedeh et al., 8 Jul 2025). Their demonstrations of nonlinear geometric phase, order-selective deflection, and resonant-only PB control show that the two capabilities need not remain decoupled.

Strong nonlocality is also sometimes taken to imply poor imaging fidelity. The LiNbO$\chi^{(3)}$52 up-conversion system provides a counterexample under a precise condition: Fourier-plane up-conversion is required to avoid loss of spatial frequencies. Within that architecture, direct imaging and derivative filtering coexist on the same device (Molina et al., 2024). A related practical constraint is diffraction at the generated wavelength; in the same work, higher orders at $\chi^{(3)}$53 had to be filtered by the collection objective.

The present design directions are material, algorithmic, and system-level. Proposed materials include higher-$\chi^{(3)}$54, transparent platforms such as GaP and InGaP for stronger conversion. Proposed methodologies include inverse design and adjoint optimization for tailored dispersion and polarization independence, as well as 2D nonlocal metasurfaces for full-Stokes conversion and multi-channel analog computing (Molina et al., 2024). The chirality-control platform identifies compatibility with etched rib or ridge waveguides and large-area fabrication by nanoimprint or nano-transfer (Liu et al., 16 Sep 2025). The electro-optic LNOI device points toward reconfigurable free-space optical interconnects, LiDAR, spatial light modulators, compact scanners, multiplexers, and photonic neural nets (Francescantonio et al., 2024). The THG phase-gradient platforms identify on-chip frequency converters, switches, modulators, logic gates, nonlinear holography, vortex-beam generation, and possible extension to SPDC or four-wave mixing for quantum nonlinear optics (Tian et al., 2 Feb 2026, Sedeh et al., 8 Jul 2025).

Taken together, these results define nonlinear nonlocal metasurfaces as a research area centered on resonantly enhanced nonlinear response, strong lattice-mediated mode engineering, and increasingly sophisticated control over amplitude, phase, momentum, and polarization at the generated frequency. The current literature suggests that the field is moving from single-function enhancement devices toward multifunctional platforms that combine frequency conversion, analog processing, chirality control, and electro-optic tunability on the same ultrathin photonic interface.