Inverse design of metasurfaces with non-local interactions

Optical metasurfaces enable wavefront manipulation by spatially arranging sub-wavelength phase-shifter elements on a flat surface. Almost all existing designs rely on libraries of pre-determined meta-atoms that introduce localized phase shifts to the incident wavefront, requiring electromagnetic independence between neighboring elements. Thus most wavefront-control metasurfaces use high-index-contrast, high-aspect-ratio structures to localize light and minimize inter-element coupling, which complicates fabrication and large-scale integration. There is growing demand for thinner, simpler metasurfaces that retain high efficiency to facilitate integration with electronics, MEMS, and deformable substrates. Attempts using thinner waveguides have led to incomplete phase modulation and reduced performance. Ultrathin dielectric nanoresonators offering electric and magnetic dipolar resonances can provide high transmission and full 2π phase control, but their long evanescent fields couple strongly to neighbors, making conventional library-based phase-gradient design ineffective. Consequently, thin resonator metasurfaces have been limited to slow phase gradients or longer wavelengths with higher-index materials. Limited high-index materials in the visible further constrain ultrathin designs, and conventional local metasurface approaches suffer efficiency limits for extreme wave manipulation. Strong non-local metasurfaces offer a promising alternative. Inspired by computational inverse design, the authors develop an evolutionary optimization (EO) strategy to design ultrathin (t ≈ λ/5) visible metalenses, explicitly leveraging inter-resonator interactions. A genetic algorithm (GA) interfaces with finite-difference time-domain (FDTD) simulations to optimize the spatial arrangement of nanoresonators to maximize focusing efficiency. The optimal designs are experimentally fabricated and validated.

Design and simulation: The metasurfaces use TiO2 circular nanoresonators on fused silica, embedded in HSQ to provide environmental passivation and mechanical protection. Initial FDTD simulations (Lumerical) characterized homogeneous periodic arrays (square and hexagonal) of identical resonators, sweeping disc radius r, edge-to-edge gap g, and thickness t (t = 115 nm optimized). Transmission and phase versus geometry were computed under normal incidence to reveal strong dependence on local arrangement and shifts in Huygens conditions (electric and magnetic dipole spectral matching), evidencing non-local coupling in ultrathin resonators. Inverse design via evolutionary optimization (EO): The target device is a cylindrical metalens at λ = 532 nm with specified focal length f; objective is to maximize focusing efficiency (fraction of incident power in the focal spot). Workflow: (1) Randomly generate an initial population N0 (~40) of 1D layouts (x positions and radii of discs along x; periodic in y) obeying constraints: 0 ≤ x ≤ xmax − r, rmin ≤ r ≤ rmax, and nearest-neighbor gap ≥ gmin. (2) For each member, perform full-device FDTD to obtain axial intensity |E|^2, fit to Gaussian to extract focal spot power and FWHM; define objective function (OF) as focusing efficiency (focal spot power normalized by a glass region of same size). (3) Rank members by OF; select m parents (~16) with probability proportional to fitness fi = (OFi − OFmin)/(OFmax − OFmin). (4) Apply genetic operations to generate offspring: mutation (perturb positions/radii with normally distributed probability), crossover (1-, 2-, or 3-point swaps between two parents), inversion (mirror positions about center), and replacement by random structure. Assigned operation probabilities: mutation 0.6, crossover 0.2, inversion 0.1, random 0.1. Overlaps are resolved by deleting overlapping discs and re-inserting if constraints allow. (5) Combine parents and offspring; retain best N0 for next generation. (6) Iterate until convergence when the best Nb = 8 members differ by ≤2% in OF. Populations are clustered into performance-based classes (“tribal competition”) to enhance diversity and exploration near global optima. Multiple GA runs (tens) from different random seeds are performed; ~10 best configurations from each run are retained. Scalability strategy: To manage computational cost as device size grows (FDTD scales with number of mesh cells), optimization proceeds incrementally by increasing lens radius in steps (e.g., 6, 12, 18, 24 µm) at constant f = 10 µm. The previously optimized central region is fixed, and GA optimizes only the new peripheral ring, using a 1 µm inward offset as a buffer zone to account for non-local interactions; the buffer re-optimizes with each step. Fabrication: 115 nm TiO2 films were deposited by ALD on glass. A PMMA bilayer resist stack was spun, and a 10 nm Au conductive layer was sputtered. Single-step e-beam lithography defined patterns; Au was removed by wet etch; development in MIBK/IPA (3:1) at 4 °C with ultrasonication (1 min). A 10 nm Cr hard mask was deposited (e-beam evaporation) and lifted off; TiO2 was dry-etched, Cr was wet-etched to yield TiO2 nanoresonator arrays, subsequently embedded in HSQ. Optical characterization: An inverted microscope (Olympus IX73) imaged transmission. A 532 nm laser illuminated the metasurface. Focal plane images quantified focusing efficiency and spot size by Gaussian fitting of measured intensity. Efficiency was defined as focal spot intensity divided by the intensity through a glass area of identical size. Z-stacks were acquired with a motorized focus controller to reconstruct x–z intensity distributions. Simulated and measured focusing efficiencies were compared for both EO and conventional library-based designs across multiple NAs.

- Ultrathin resonant metasurfaces (t ≈ λ/5; TiO2 thickness 115 nm at λ = 532 nm) exhibit strong non-local interactions; transmission and phase depend strongly on local arrangement (square vs hexagonal arrays yield different Huygens condition loci for identical r and g). - Inverse design via evolutionary optimization (EO) that explicitly leverages inter-resonator coupling produces aperiodic metalens layouts with significantly higher focusing efficiencies than conventional library-based designs across all tested numerical apertures (NAs). - Quantitative efficiency gains: For NA ≈ 0.17, conventional ≈ 70% vs EO ≈ 77%; for NA ≈ 0.51, conventional ≈ 30% vs EO ≈ 60%. Efficiency decreases with increasing NA for both methods, but EO maintains a large advantage. - EO-designed lenses exhibit diffraction-limited focal spots and measured axial intensity distributions that agree with simulations. - EO layouts feature clustered discs with broader size variation and systematically lower filling factors than conventional constant-pitch layouts; despite lower filling factor, EO devices show higher efficiency, indicating that collective modes of resonator clusters plus surrounding voids act as effective elements to direct light to the focus and suppress satellite lobes. - The study reports, to date, the highest efficiencies for the thinnest transmissive visible metalenses and validates the designs experimentally. - The stepwise optimization strategy (expanding lens radius with a buffer zone) efficiently manages computational cost while accounting for non-local interactions at region boundaries.

The results demonstrate that embracing, rather than suppressing, non-local electromagnetic interactions among ultrathin resonators enables global control of the metasurface response and higher focusing efficiencies, especially at higher NAs where phase varies rapidly and conventional local designs break down. EO discovers aperiodic, lower-filling-factor layouts that form collective optical modes to funnel energy into the desired diffraction-limited focus and reduce power lost to sidelobes and off-axis scattering. Agreement between simulation and experiment across multiple NAs corroborates the approach. The work suggests that performance is not yet at the fundamental limit for ultrathin visible metalenses; as NA increases, non-local effects become more pronounced, potentially allowing further gains. Future optimization avenues include topology optimization exploiting multimode interference, metagratings with tailored bianisotropy for unitary efficiencies without deeply subwavelength discretization, and AI-guided searches (e.g., deep learning, Monte Carlo Tree Search). However, practical considerations such as training data requirements for deep learning and convergence times for evolutionary algorithms must be balanced against full-wave simulation costs.

This work introduces and validates an inverse design framework for ultrathin visible metalenses that explicitly leverages non-local interactions among TiO2 nanoresonators via an evolutionary optimization coupled to FDTD. The EO approach yields aperiodic, low-filling-factor layouts that significantly outperform conventional library-based designs in focusing efficiency across a broad range of NAs, while maintaining subwavelength thickness (≈λ/5) and diffraction-limited performance. The methodology offers a scalable route to high-efficiency, fabrication-friendly metasurfaces compatible with integration platforms. Future research should explore hybrid and alternative optimization strategies (topology optimization, metagratings with bianisotropy, AI-guided search), extend to 2D spherical metalenses and broadband/achromatic targets, and further reduce computational overheads via surrogate modeling or multi-fidelity optimization.

- Computational cost: Full-device FDTD-driven EO requires thousands of simulations and scales poorly with device size; although a stepwise radius-expansion strategy with buffer zones mitigates this, optimization of large-area or fully 2D (spherical) lenses remains expensive. - Convergence speed: GA-based methods can converge slowly and may get trapped near local optima; multiple runs are needed to sample solution space. - Scope: Experimental validation focuses on cylindrical (1D) metalenses at a single wavelength (532 nm) and fixed thickness (115 nm); generalization to broadband, achromatic, or polarization-tailored devices is not demonstrated here. - Material/system constraints: Limited high-index, low-loss materials in the visible restrict achievable confinement; strong non-local coupling complicates predictive design without full-wave optimization. - Efficiency at very high NA: Although EO improves performance, efficiencies still decrease with increasing NA, indicating room for improvement and potential fundamental trade-offs.