Continuous polarization–wavelength mapping with nonlocal metasurfaces

To overcome the inherent limitations in controlling photonic degrees of freedom—particularly the degeneracy between polarization and wavelength imposed by local dispersion—we propose a dimension-interlaced continuous design framework as illustrated in Fig. 1. As shown in Fig. 1a, we construct a nonlocal metasurface capable of generating continuous gradient polarization holography across the mid-infrared spectrum (2.7–4 μm), with each polarization state smoothly evolving with wavelength. Unlike conventional multiplexing strategies that discretely combine polarization–wavelength pairs, this design realizes continuous-domain modulation by reconstructing the far-field response from a nonlocal equivalent Jones matrix.

In Fig. 1b, we illustrate how conventional metasurface designs are constrained by the nearly constant group delay of structural eigenmodes (depicted by yellow and purple dashed lines on the left insert), which results in quasi-linear phase dispersion and limits the curvature (\({\partial }^{2}\varphi /\partial {\lambda }^{2}\)) needed for broadband, arbitrary modulation. The dispersion Jones matrix of such local structures can be expressed as:

Here, θ is the rotation angle of the meta-atom, and t_α,β, φ_α,β represent the amplitude and phase responses of the two eigen-polarization channels. Under the traditional C2-symmetrical structure with linear phase dispersion, θ becomes wavelength-invariant, precluding complex polarization evolution across spectrum.

To overcome this, we define a nonlocal equivalent Jones matrix by incorporating the spatial diffraction kernel f(x′, y′, z, λ), yielding:

This formalism introduces spatial-frequency dispersion and enables phase modulation beyond the local approximation, effectively unlocking an additional DoF associated with eigenvector evolution across the polarization–wavelength manifold. To link these responses to designable structural parameters, we construct a forward analytical model that relates the birefringent phase retardation Δφ = φ_β − φ_α to the target polarization path defined by ellipticity P₁(λ) and azimuth P₂(λ):

This expression allows us to extract the optimal rotation angle θ(λ) needed to reproduce any desired polarization trajectory on the Poincaré sphere (see Supplementary Note 3 and 4 for details). Using this perturbative mapping, we then define the far-field projection intensity along the conjugate polarization state as:

These fields form the basis of the nonlocal Jones response in the focal plane. To represent the dispersion behavior of the metaatoms, we further introduce a third-order polynomial model for both amplitude and phase:

Each meta-atom is characterized by a discrete set of coefficients [a, b, c, d, ap, bp, cp, dp]. By combining this model with the polarization path projection, we construct two hierarchical optimization layers—one for polarization channel matching and one for phase-dispersion perturbation fitting. This reduces the data complexity from 4n³ to 8n, enabling fast and globally optimal inverse design across a continuous polarization–wavelength parameter space (see Supplementary Note 5 for details). Together, this framework enables the first metasurface design strategy capable of simultaneously and densely sampled continuous controlling of polarization and wavelength, marking a transition from discrete channel manipulation to continuous-domain optical field engineering. Notably, the continuity here does not refer to an infinite number of experimentally resolvable states, but rather to a mathematically continuous mapping between input and output polarization–wavelength states defined by the nonlocal Jones-matrix formalism (Eq. 4). The Supplementary Note 2 provides more details.

The complete design and implementation flow of our dimension-interlaced continuous design framework is illustrated in Fig. 2. As shown in Fig. 2a, we integrate a continuous polarization–wavelength analytical model with a dimension-interlaced vectorial diffraction neural network, enabling inverse design of nonlocal metasurfaces over high-dimensional parameter spaces. Compared to conventional diffraction neural networks that rely on discrete pixel-wise phase targets, our approach introduces continuous variables in both wavelength and polarization domains, which allows the network to learn the projection relationship between conjugate polarization channels and nonlocal structural responses.

A key methodological innovation lies in the pre-compression of the meta-atom database and the transformation of the design problem into the conjugate polarization basis. Specifically, the effective “feature angle” of each meta-atom is extracted based on its birefringent response, and the projection intensity onto the conjugate polarization channel is computed. This compressed and polarization-aligned representation is then used as the input for the neural network, while the target Jones matrix in the focal plane—also expressed in the conjugate basis—serves as the excitation function. Additionally, a rotation angle perturbation layer is introduced to reflect the polarization-path-dependent phase response, enabling joint optimization over continuous θ and λ dimensions. This formulation not only reduces the optimization parameter space from cubic to quadratic complexity but also facilitates global convergence across the metasurface. The compressed dual-layer neural network architecture captures both the local phase-dispersion matching and global polarization trajectory customization. As a result, the design process becomes highly efficient, scalable, and physically interpretable.

To validate the platform’s physical feasibility, we construct a silicon-based meta-atom library using numerical simulations. As shown in Fig. 2b, the library is built over five representative wavelengths between 2.7–4 μm, covering structures with a fixed period of 1.5 μm and a height of 7 μm. Silicon (n ≈ 3.43) is chosen as the structural material due to its low loss in the mid-infrared and strong birefringent potential. With a mature deep reactive ion etching process, we achieve aspect ratios exceeding 20, which is essential for broadband dispersion engineering. Figure 2c presents the optimized phase profiles over the five spectral channels. These phase targets are weighted according to the laser energy attenuation at each wavelength to ensure uniform holographic fidelity across the band. Compared with the intrinsic quasi-linear behavior of raw meta-atom dispersion, the optimized phase profiles exhibit enhanced wavelength nonlinearity after projection into the conjugate polarization space, validating the efficiency of our compression strategy. The fabricated metadevice is shown in Fig. 2d. The 600 μm-sized sample contains over 160,000 anisotropic meta-units patterned with high fidelity. Tolerance analysis shows that deviations in sidewall angle ( < 0.5°), etch depth ( ± 200 nm), and linewidth ( ± 20 nm) introduce minimal influence on birefringent dispersion and overall holographic performance. These findings confirm that the proposed metasurface design is well within the capability of current nanofabrication technologies. The detailed tolerance analysis of fabrication errors can be found in Supplementary Note 6. Figure 2e depicts the optical measurement setup used for verifying holographic and polarization performance. A broadband mid-IR supercontinuum source is collimated, filtered via bandpass elements, and polarization-modulated using a liquid crystal variable retarder and wire-grid polarizers. The holographic output is then captured by an MWIR camera, enabling spatial and polarization-resolved imaging.

To verify the feasibility of our proposed framework, we designed a non-degenerate metasurface hologram that maps a densely sampled continuous set of polarization states onto distinct wavelengths. As shown in Fig. 3a, the five channels are represented on the Poincaré sphere, where each colored ellipse denotes a specific elliptical polarization state assigned to a particular wavelength. The input (blue) and output (red) states are designed to be conjugate pairs, symmetrically mirrored about the equator to maximize birefringent contrast and satisfy polarization-path coupling conditions. These polarization–wavelength mappings are not arbitrarily selected, but rather aligned along a physically realizable, continuously evolving dispersion trajectory enabled by our dimension-interlaced design model.

The broadband achromatic performance is summarized in Fig. 3b, where focal length deviations across channels are quantitatively evaluated. All five channels converge near the 700 μm focal plane, with acceptable variation, even under practical fabrication and material dispersion conditions. The slight deviation at 3.055 μm is attributed to atmospheric absorption near water vapor resonance and its underweighted contribution in the optimization process.

The designed holographic patterns are shown in Fig. 3c for each channel, including the target, simulation, and measured results. All outputs exhibit high fidelity, with minimal distortion, defocus, or image degradation. These results confirm that our metasurface achieves robust achromatic holography across both spectral and polarization channels, even under non-degenerate conditions. Importantly, this result cannot be achieved using conventional spatial interleaving or focal-plane-based holography, which fundamentally rely on discrete channel separation and local DoF control. Here, we demonstrate for the first time that a single-layer metasurface with nonlocal design and continuous-domain reconstruction can simultaneously realize (i) multicolor channel isolation, (ii) polarization selectivity, and (iii) broadband achromatic focusing—all in a co-encoded fashion. The effectiveness of this approach stems from our precise co-optimization of intrinsic meta-atom dispersion and nonlocal field propagation, alongside our data-compressed neural modeling strategy, which allows high-dimensional polarization–wavelength control to be mapped into a realizable structural design.

To further validate the versatility of our theoretical framework, we extend the metasurface design to support arbitrary polarization dispersion paths beyond the linear trajectories. As illustrated in Fig. 4a, each wavelength is mapped to an elliptical polarization state selected freely on the Poincaré sphere, unconstrained by equatorial symmetry or linear interpolation. This represents a non-degenerate polarization–wavelength mapping, where input (blue) and output (red) polarization states form conjugate elliptical pairs, distributed arbitrarily across the polarization manifold.

The target holographic image in Fig. 4b consists of five spatially distinct lobes, each encoding a different polarization–wavelength channel. Achieving this arbitrary dispersion mapping poses significant challenges due to the need for precise, channel-specific birefringence control under a shared metasurface structure. Moreover, the orthogonality among channels must be maintained despite their non-monotonic dispersion behavior and abrupt variations in polarization state. Such requirements exceed the capability of conventional interleaved metasurfaces or focal-plane multiplexing methods, which rely on spatial segregation or linear dispersion approximations.

In Fig. 4c, both simulation and experimental results confirm the successful reconstruction of each vectorial channel with minimal distortion or crosstalk. The high isolation among channels is attributed to our dual-DoF engineering approach, which simultaneously exploits wavelength-selective and polarization-selective responses within the nonlocal metasurface design. To quantitatively assess the inter-channel uniformity, Fig. 4d presents the relative efficiency for each channel, normalized to the maximum measured value. The achieved balance across channels highlights the effectiveness of our design in equalizing energy distribution even under asymmetric dispersion conditions. And the nonlocal optimization ensures that all functional modes lie within the subwavelength scale, thereby eliminating higher-order leakage and confining optical energy to the 0th order, which defines the reconstructed holographic field. Slight deviations at 3.055 μm and 3.35 μm are primarily attributed to atmospheric absorption and etching-induced structural variations, respectively.

Furthermore, to evaluate the polarization conversion fidelity, we compute the polarization contrast (PC) matrix shown in Fig. 4e. This metric compares the measured polarization vector \(|{\kappa }^{-} > ={\left[{A}_{1}{e}^{i{\delta }_{1}},{{A}}_{2}{e}^{i{\delta }_{2}}\right]}^{T}\) to the ideal target state ∣κ + 〉, using the contrast formula \({PC}={\left|{\left({\kappa }^{-}\right)}^{T}\cdot {\kappa }^{+}\right|}^{2}/\,{\left|{\kappa }^{+}\right|}^{2}\). The PC between target polarization channels exceeds 0.95 and low off-diagonal leakage confirm that the metasurface accurately distinguishes and reconstructs the intended elliptical states, even across varying wavelengths. The correlation coefficients and crosstalk metrics are provided in Supplementary Note 7. These quantitative results verify that the proposed nonlocal metasurface enables spectrally resolved and polarization-distinct reconstruction with minimal mutual interference. Additionally, although our experimental demonstration is limited to five discrete channels with ~200–300 nm channel spacing for visualization, the same nonlocal framework is scalable to tens or more polarization–wavelength states. Supplementary simulations confirm that by modestly relaxing isolation requirements, distinct holographic channels can be designed across sub-100 nm intervals, illustrating the pathway toward practically continuous modulation. Supporting simulations demonstrating these scalability pathways are provided in Supplementary Fig. S7.

It is worth noting that the current number of demonstrated channels is limited not by the physical framework as proved in previous content but by the finite sampling density of the meta-atom database. In principle, as the meta-atom library becomes denser, the neural network learns a higher-order interpolation between eigen-polarization and spectral response, enabling the design of densely spaced channels approaching a continuous mapping. We also provide a detailed discussion in the Supplementary Note 8 on the three key physical parameters (group-delay, eigen-polarizations, and spatial-frequency bandwidth) that fundamentally limit the infinite interpolation of holographic channels, along with an analysis of the challenges encountered in practical fabrication and optimization strategies.

As shown in Fig. 5a, we design a densely sampled continuous polarization trajectory from −30° to 60° linear polarization, conjugated across the Poincaré sphere. This path is projected onto nine representative wavelengths spanning the mid-infrared regime (2700–3920 nm), under a numerical aperture of 0.43. The resulting vectorial hologram shows consistent high-fidelity reconstructions across both dimensions without image deformation or spatial scaling artifacts. To quantitatively assess the broadband and polarization robustness, Fig. 5b presents the correlation coefficients between experimental and target images across all sampled channels. The high correlation values confirm the device’s achromatic and vectorially adaptive behavior, validating the effectiveness of our compressive forward model under continuous dual-DoF variation. As previously mentioned, the analytical form (Eq. 4) allows smooth spectral–polarization evolution that can be densely sampled in both simulation and experiment to verify continuity. In Supplementary Note 9, we further performed dense interpolation simulations and supplementary experiments for the continuous-gradient achromatic hologram. The system was sampled at 1° polarization and 15 nm wavelength intervals. The simulated Stokes parameters exhibit monotonic evolution on the Poincaré sphere, confirming smooth polarization rotation with wavelength. Experimentally interpolated channels using additional band-pass filters show consistent behavior, validating that intermediate states are physically reconstructable rather than extrapolated. These results substantiate the concept of continuity as a smooth, differentiable mapping in the polarization–wavelength space.

In contrast to previous metasurface holography works—where broadband achromaticity was typically limited to circular polarization channels, or polarization multiplexing was achieved only via pixel-level interleaving—our platform offers fully continuous polarization tuning across a broad infrared spectral band within a single-layer, non-pixelated metastructure. The progress from discrete to continuous control is summarized in Fig. 5c, which benchmarks the polarization and wavelength channel control of our metadevice against leading reported designs. Colored regions highlight whether independent polarization–wavelength decoupling was achieved. Detailed quantitative comparisons of fidelity, crosstalk, and efficiency among different design methods are provided in the Supplementary Note 10. Our method uniquely supports both continuous-domain variation and decoupled encoding, distinguishing it from state-of-the-art alternatives. These results establish a metasurface platform capable of continuous gradient polarization control with broadband achromatic fidelity, setting the stage for high-dimensional information encoding, secure optical communication, and wavelength-adaptive imaging systems across diverse photonic applications. Additionally, by replacing Si with low-loss, high-index dielectrics and leveraging advanced high-aspect-ratio nanofabrication, our nonlocal metasurface design can be directly extended to the visible range. The key challenges lie in maintaining fabrication precision and suppressing scattering losses, both of which are being progressively addressed by recent developments in visible-band dielectric metasurface technology^16,50.