Nonlocal Metapixels: Engineered Collective Response

• Nonlocal metapixels are engineered meta-units that use collective, spatially extended modes and momentum-space transfer functions for precise wavefront control.
• They enable multifunctional optical operations, such as edge detection and bright-field imaging, by leveraging guided-mode resonances and quasi-BICs.
• Their design utilizes phase-change materials and rigorous coupled-wave analysis for dynamic reconfigurability in flat optics applications.

Nonlocal metapixels are metasurface functional units whose response is governed by collective, spatially extended modes and by a designed transfer function in momentum space, rather than by a purely local phase–amplitude mask assigned independently at each point. In Fourier-optical terms, they are associated with a transfer law $$E_{\text{out}}(k_x,k_y)=H(k_x,k_y)E_{\text{in}}(k_x,k_y)$$, and in real space they act as convolution operators $E_{\text{out}}(x,y)=(h*E_{\text{in}})(x,y)$, with $H(k_x,k_y)=\mathcal{F}\{h(x,y)\}$ (Shastri et al., 2022). In the recent metasurface literature, the term is often implicit rather than formally standardized: a “metapixel” may denote a supercell supporting quasi-bound states in the continuum, a guided-mode-resonant region, or an entire spatially uniform aperture that behaves as one large analog filter. A representative implementation is the Sb$_2$Se$_3$-based phase-change nonlocal metasurface that switches between edge detection and bright-field imaging while retaining a fixed geometry (Yang et al., 2024).

1. Conceptual framework

In conventional local metasurfaces, subwavelength meta-atoms are treated as independent scatterers, neighbor–neighbor coupling is neglected, and the outgoing field is described by a multiplicative real-space mask $E_{\text{out}}(x,y)=t(x,y)E_{\text{in}}(x,y)$. In nonlocal metasurfaces, by contrast, the response depends nontrivially on the transverse wavevector $\mathbf{k}_\parallel=(k_x,k_y)$, so the operative object is a wavevector-dependent transfer function $T(\mathbf{k}_\parallel,\omega)$ rather than a position-only mask (Shastri et al., 2022). This distinction is the core of the nonlocal-metapixel concept.

The literature describes two closely related interpretations. One treats a nonlocal metapixel as a supercell or superperiod whose optical function is mediated by collective modes extending over many unit cells. The other treats the whole patterned region as a single large-area “pixel” in spatial-frequency space, particularly when the metasurface is laterally uniform and implements one global operator on the incident field. Both interpretations appear in current work. In “Multifunctional Nonlocal Metasurfaces,” nonlocal functionality is carried by quasi-BIC supermodes extending across superperiods such as $P=16a$ or $P_x=6.4\ \mu\mathrm{m}$, $E_{\text{out}}(x,y)=(h*E_{\text{in}})(x,y)$0, and each designed region acts as a spectrally addressable nonlocal metapixel (Overvig et al., 2020). In the phase-change image processor, by contrast, the entire uniform array behaves globally as one programmable, nonlocal analog filter (Yang et al., 2024).

A recurring misconception is that “metapixel” must imply an independently addressed digital pixel. The phase-change image-processing device explicitly does not define the metapixel as a separately addressed sub-region with independent functionality; each period of the periodic array is a meta-unit cell, but the optical function emerges from the collective response of the full lattice (Yang et al., 2024). This suggests that, in nonlocal flat optics, “pixelation” is often modal rather than geometric.

2. Physical mechanisms of nonlocal response

Nonlocal metapixels are realized by deliberately strengthening spatial dispersion. The dominant mechanisms identified across the literature are guided or leaky modes, Fano resonances, photonic-crystal-slab modes, guided-mode resonances, strong lattice coupling, and quasi-bound states in the continuum (Shastri et al., 2022). In each case, the field profile extends over many unit cells, so the scattering of one nominal “pixel” cannot be decoupled from its neighbors.

A central design route uses symmetry-protected BICs converted into quasi-BICs by a small symmetry-breaking perturbation. In that regime, the quality factor obeys $E_{\text{out}}(x,y)=(h*E_{\text{in}})(x,y)$1, where $E_{\text{out}}(x,y)=(h*E_{\text{in}})(x,y)$2 quantifies the perturbation strength, and the resulting high-$E_{\text{out}}(x,y)=(h*E_{\text{in}})(x,y)$3 resonance supplies a narrowband, strongly angle-dependent, and polarization-selective response (Overvig et al., 2020). Spatially varying perturbation orientation can then encode a geometric phase into the nonlocal mode, so a metapixel may store several channels $E_{\text{out}}(x,y)=(h*E_{\text{in}})(x,y)$4 within one supercell (Overvig et al., 2020).

Guided-mode-resonant platforms provide a complementary picture. In the atomically thin Si/Si$E_{\text{out}}(x,y)=(h*E_{\text{in}})(x,y)$5N$E_{\text{out}}(x,y)=(h*E_{\text{in}})(x,y)$6 guided-mode-resonance metasurface for spectrally decoupled wavefront manipulation, the relevant optical mode is a quasi-guided slab mode with long photon lifetime, propagation length $E_{\text{out}}(x,y)=(h*E_{\text{in}})(x,y)$7, and strong field enhancement. The response of a given region is therefore set by a collective slab mode rather than by independent nanoantennas, and the device behaves as a high-$E_{\text{out}}(x,y)=(h*E_{\text{in}})(x,y)$8, spectrally selective nonlocal metapixel (Song et al., 2021).

These mechanisms also explain why nonlocality has been viewed both as nuisance and opportunity. In traditional local metasurface design, strong coupling and angle sensitivity degrade the fidelity of a prescribed local phase mask. In nonlocal flat optics, the same coupling is intentionally engineered to realize wavevector-selective filtering, wavefront control restricted to a resonant band, energy channeling, and analog optical computing (Shastri et al., 2022).

3. Phase-change nonlocal metapixels

The most explicit recent realization of a reconfigurable nonlocal metapixel is the phase-change image processor based on Sb$E_{\text{out}}(x,y)=(h*E_{\text{in}})(x,y)$9Se$H(k_x,k_y)=\mathcal{F}\{h(x,y)\}$0 nanopillars on sapphire (Yang et al., 2024). The metasurface is a periodic array with pillar height $H(k_x,k_y)=\mathcal{F}\{h(x,y)\}$1, pillar diameter $H(k_x,k_y)=\mathcal{F}\{h(x,y)\}$2, and lattice period $H(k_x,k_y)=\mathcal{F}\{h(x,y)\}$3 in a square lattice. The patterned area is approximately $H(k_x,k_y)=\mathcal{F}\{h(x,y)\}$4, and the design wavelength is $H(k_x,k_y)=\mathcal{F}\{h(x,y)\}$5. Because the periodicity is subwavelength at the working wavelength, only the zeroth diffraction order propagates.

The paper’s metapixel interpretation is explicitly collective. Each nanopillar, together with its neighbors, supports resonant dipole modes, but the relevant transfer function $H(k_x,k_y)=\mathcal{F}\{h(x,y)\}$6 emerges only from the uniform periodic array. The device is therefore not a mosaic of different filters. It is “more akin to one large analog filter,” with programming performed not by lateral multiplexing in $H(k_x,k_y)=\mathcal{F}\{h(x,y)\}$7 but by switching the material phase between amorphous and crystalline states (Yang et al., 2024).

The nonlocality arises from angle-dependent excitation of in-plane Mie resonances, an out-of-plane vertically oriented dipole mode, and guided/leaky resonances. For p-polarized light in the amorphous phase, in-plane dipole resonances produce a broad low-transmission band between approximately $H(k_x,k_y)=\mathcal{F}\{h(x,y)\}$8 and $H(k_x,k_y)=\mathcal{F}\{h(x,y)\}$9, while the out-of-plane resonance compresses this band as incident angle increases. At $_2$0, the resulting response is a high-pass filter in spatial frequency. In the crystalline phase, the refractive-index change shifts the resonances away from the operating wavelength, leaving the transmission almost flat versus angle over the same numerical aperture range (Yang et al., 2024).

Sb$_2$1Se$_2$2 was chosen because it is optically low-loss in the near-IR, with imaginary part of $_2$3 for $_2$4, and because the refractive-index contrast between amorphous and crystalline phases is $_2$5 in the near-IR. The material is nonvolatile, so once switched it remains in the selected phase without holding power. In the reported experiment, switching is one-way, from amorphous to crystalline, by heating the sample on a hot plate at $_2$6 for 5 minutes; integrated microheaters are proposed for in-situ cycling (Yang et al., 2024).

4. Transfer functions and optical operations

The canonical operation implemented by a nonlocal metapixel is a prescribed spatial-frequency filter. In the Sb$_2$7Se$_2$8 device, the amorphous phase is designed to approximate a Laplacian-like transfer function,

$_2$9

while the crystalline phase approximates an all-pass transfer function,

$_3$0

In real space, the former corresponds to edge enhancement and the latter to bright-field imaging (Yang et al., 2024).

The device is polarization-sensitive. In the full tensor description, the metasurface is characterized by a $_3$1 transmission matrix in the $_3$2 basis. For the operating bandwidth and $_3$3, the cross-polarization terms $_3$4 and $_3$5 are small, with amplitude approximately $_3$6, corresponding to power approximately $_3$7. The authors therefore approximate the response in the amorphous phase by

$_3$8

and in the crystalline phase by

$_3$9

This yields a p-polarized Laplacian/high-pass channel in the amorphous phase and an approximately angle-independent bright-field channel in the crystalline phase (Yang et al., 2024).

Because of symmetry, input polarization selects directional derivatives. For x-polarized input, the transfer function is approximately $E_{\text{out}}(x,y)=t(x,y)E_{\text{in}}(x,y)$0, implementing $E_{\text{out}}(x,y)=t(x,y)E_{\text{in}}(x,y)$1 and highlighting vertical edges. For y-polarized input, $E_{\text{out}}(x,y)=t(x,y)E_{\text{in}}(x,y)$2, implementing $E_{\text{out}}(x,y)=t(x,y)E_{\text{in}}(x,y)$3 and highlighting horizontal edges. For circularly polarized input, $E_{\text{out}}(x,y)=t(x,y)E_{\text{in}}(x,y)$4, yielding full two-dimensional edge detection (Yang et al., 2024).

This transfer-function viewpoint generalizes beyond spatial filtering. “Temporal Signal Processing with Nonlocal Optical Metasurfaces” engineers a dispersive response $E_{\text{out}}(x,y)=t(x,y)E_{\text{in}}(x,y)$5 so that the transmitted envelope satisfies $E_{\text{out}}(x,y)=t(x,y)E_{\text{in}}(x,y)$6, i.e. first-order temporal differentiation (Cotrufo et al., 2024). “Space-Time Nonlocal Metasurfaces for Event-Based Image Processing” further realizes a mixed spatio-temporal operator with ideal transfer function $E_{\text{out}}(x,y)=t(x,y)E_{\text{in}}(x,y)$7, implementing $E_{\text{out}}(x,y)=t(x,y)E_{\text{in}}(x,y)$8 and transmitting only moving edges (Esfahani et al., 2024). This suggests that “nonlocal metapixel” is best understood as an operator-bearing unit whose designed response may live in space, in time, or jointly in both domains.

5. Fabrication, characterization, and demonstrations

The Sb$E_{\text{out}}(x,y)=t(x,y)E_{\text{in}}(x,y)$9Se$\mathbf{k}_\parallel=(k_x,k_y)$0 nonlocal metasurface was designed numerically with rigorous coupled-wave analysis using Reticolo. RCWA was used to compute zeroth-order transmittance as a function of wavelength and incident angle, derive p- and s-polarized transmittance coefficients, and calculate the two-dimensional transfer functions $\mathbf{k}_\parallel=(k_x,k_y)$1, $\mathbf{k}_\parallel=(k_x,k_y)$2, and the cross-polarized terms (Yang et al., 2024).

Fabrication proceeded by sputtering $\mathbf{k}_\parallel=(k_x,k_y)$3 Sb$\mathbf{k}_\parallel=(k_x,k_y)$4Se$\mathbf{k}_\parallel=(k_x,k_y)$5 on sapphire, spin-coating approximately $\mathbf{k}_\parallel=(k_x,k_y)$6 ma-N 2403 resist, patterning by e-beam lithography, transferring the pattern by fluorine-based RIE with CHF$\mathbf{k}_\parallel=(k_x,k_y)$7 at $\mathbf{k}_\parallel=(k_x,k_y)$8 and $\mathbf{k}_\parallel=(k_x,k_y)$9, and stripping the resist to leave the nanopillar array. SEM confirmed high-quality pillars. Crystallization was then achieved by heating at $T(\mathbf{k}_\parallel,\omega)$0 for 5 minutes (Yang et al., 2024).

Angle-resolved measurements at $T(\mathbf{k}_\parallel,\omega)$1 used a diode laser, a polarizer, sample rotation up to approximately $T(\mathbf{k}_\parallel,\omega)$2 corresponding to $T(\mathbf{k}_\parallel,\omega)$3, and collection of the transmitted zeroth order by a low-NA $T(\mathbf{k}_\parallel,\omega)$4 objective. In the amorphous phase for p polarization, the measured transmittance rose from approximately $T(\mathbf{k}_\parallel,\omega)$5 on axis to amplitude approximately $T(\mathbf{k}_\parallel,\omega)$6 at $T(\mathbf{k}_\parallel,\omega)$7, consistent with a high-pass edge-detection filter. In the crystalline phase for p polarization, the amplitude was flat at approximately $T(\mathbf{k}_\parallel,\omega)$8–$T(\mathbf{k}_\parallel,\omega)$9 across $P=16a$0, consistent with bright-field behavior. The s-polarized response remained low and angle-independent in both phases (Yang et al., 2024).

Imaging demonstrations used a coherent $P=16a$1 microscope with a 1951 USAF negative resolution test target, the metasurface placed about $P=16a$2 before an $P=16a$3 objective, polarization control for horizontal, vertical, and circular illumination, and CCD detection. In the amorphous phase, horizontal polarization produced $P=16a$4 edge enhancement, vertical polarization produced $P=16a$5, and circular polarization approximated $P=16a$6. Features larger than approximately $P=16a$7 were clearly processed, and edge lines had width approximately $P=16a$8, corresponding to the diffraction limit. In the crystalline phase, images with the metasurface were almost identical to those without it except for overall intensity reduction (Yang et al., 2024).

The reported metrics are notable. The effective numerical aperture is $P=16a$9. The theoretical diffraction limit is approximately $P_x=6.4\ \mu\mathrm{m}$0 at $P_x=6.4\ \mu\mathrm{m}$1. Bright-field imaging was demonstrated for features down to approximately $P_x=6.4\ \mu\mathrm{m}$2, while edge detection yielded distinguishable edges for features $P_x=6.4\ \mu\mathrm{m}$3. The amorphous p-polarized amplitude of approximately $P_x=6.4\ \mu\mathrm{m}$4 at high $P_x=6.4\ \mu\mathrm{m}$5 corresponds to approximately $P_x=6.4\ \mu\mathrm{m}$6 power transmission at edges. The crystalline p-polarized amplitude of approximately $P_x=6.4\ \mu\mathrm{m}$7–$P_x=6.4\ \mu\mathrm{m}$8 corresponds to approximately $P_x=6.4\ \mu\mathrm{m}$9–$E_{\text{out}}(x,y)=(h*E_{\text{in}})(x,y)$00 power transmission. The edge-detection signal-to-noise ratio was measured above $E_{\text{out}}(x,y)=(h*E_{\text{in}})(x,y)$01 (Yang et al., 2024).

The same platform was fabricated as a meta-coverslip on approximately $E_{\text{out}}(x,y)=(h*E_{\text{in}})(x,y)$02 sapphire. Demonstrations on onion epidermal cells and thin sodium carbonate crystals showed that the amorphous state clearly delineated cellular boundaries and crystal morphologies, whereas the crystalline state reverted to bright-field imaging with less-pronounced edges. The device therefore functioned as a dual-functionality meta-coverslip for microscopy (Yang et al., 2024).

6. Research landscape, trade-offs, and emerging directions

The broader nonlocal-flat-optics literature situates nonlocal metapixels between local metasurfaces and conventional resonant filters. Local metasurfaces offer rich wavefront control but poor spectral selectivity, while traditional nonlocal devices such as grating filters offer sharp resonances but limited spatial control. Quasi-BIC metasurfaces bridge these regimes by combining narrowband resonant behavior with spatially tailored wavefronts, including phase gradients, lenses, and multi-wavelength beam steering (Overvig et al., 2020). Active implementations add thermo-optic or mechanical tuning so that a nonlocal metapixel can switch between distinct wavefront functions without per-meta-atom addressing (Malek et al., 2020).

This broader landscape also clarifies the main trade-offs. Strong nonlocality is often narrowband because the response is resonance-based. High $E_{\text{out}}(x,y)=(h*E_{\text{in}})(x,y)$03 improves spectral discrimination but tightens angular tolerance and, in wavefront-shaping devices, constrains numerical aperture through band curvature (Overvig et al., 2020). In multi-mode supercells, crosstalk becomes non-negligible when several quasi-BIC channels are packed into one region. Reconfigurability is intrinsically more complicated than in local metasurfaces because changing one element perturbs the global modal structure (Shastri et al., 2022).

A second misconception is that nonlocality is synonymous with arbitrary inter-cell coupling or with fabrication imperfection. The recent roadmap instead treats nonlocality as a major design frontier, encompassing naturally occurring nonlocalities in plasmonic, polaritonic, and quantum materials, and stronger engineered nonlocal effects in metamaterials and metasurfaces. In that framing, nonlocality engineering is a route to enhanced wavefront control, spatial compression, multifunctional devices, and wave-based computing (Monticone et al., 1 Mar 2025). The same roadmap emphasizes that nonlocality also bears on topological, nonreciprocal, and time-varying systems (Monticone et al., 1 Mar 2025).

A plausible extension is the convergence of effective nonlocal metasurfaces with materials whose intrinsic response is itself strongly nonlocal. “Primordial Metamaterials” shows that structuring materials on the scale of their inherent nonlocality can reveal additional nonlocality-induced waves and strong overall nonlocality that remain observable at room temperature and in lossy materials (Ware et al., 26 Jun 2025). This suggests a future class of nonlocal metapixels in which the modal degrees of freedom are set not only by guided-mode engineering but also by the inherent spatial dispersion of the constituent medium.

In current usage, then, nonlocal metapixels are not simply subwavelength pixels with stronger coupling. They are operator-bearing photonic units whose defining variable is a designed $E_{\text{out}}(x,y)=(h*E_{\text{in}})(x,y)$04, whether implemented by a quasi-BIC supercell, a guided-mode-resonant slab, a space–time differentiator, or a phase-change aperture functioning as one large analog filter. Their significance lies in shifting flat-optics design from local wavefront assignment toward the direct engineering of collective modes and spatially dispersive transfer functions (Yang et al., 2024).