Broadband meta-atom library for visible metasurfaces

Design eight tiny titanium-dioxide structures, the building blocks of a large-area metasurface, so that a grating assembled from them sends visible light of every color into one diffraction order. Libraries like this let lenses and other flat optics be built by tiling a few well-designed cells, and broadband operation is what makes them hard.

Task

Each of the eight meta-atoms occupies a 400 nm square cell on an SiO₂ substrate, in air. A design consists of

eight binary patterns of TiO₂ and air on a 5 nm grid (80 × 80 pixels), each free of TiO₂ in a 30 nm border so that cells can be tiled, and each mirror-symmetric so that the response does not depend on polarization;
one TiO₂ thickness in $[0.5,0.7]$ µm.

For each meta-atom $n$ and wavelength $\lambda_j$ the simulation gives zeroth-order complex transmission coefficients for circularly polarized light, both polarization-conserving and converting. Placing the atoms $n=0,\dots,7$ side by side approximates a grating whose diffraction amplitudes are

S^{j}_{k}=\frac{1}{8}\sqrt{\frac{n_{\mathrm{sub}}}{n_{\mathrm{amb}}}}\sum_{n=0}^{7}t^{j}_{n}\,e^{-i\pi kn/4},

computed separately for each output polarization; $\eta^{j}_{k}$ sums their squared moduli. The relative efficiency at $\lambda_j$ is $\eta^{j}_{+1}/\sum_k\eta^{j}_{k}$ . Each atom may be rotated in its cell, and the rotations with the highest wavelength-averaged relative efficiency are used. The score $\mathcal{M}$ is the minimum relative efficiency over the wavelengths 450, 470, …, 650 nm and both circular input polarizations; higher is better. A valid design is fully binary, with minimum feature width and gap of at least 12 pixels (60 nm), measured with the imageruler package, the challenge's default target.

Background

The challenge follows Chen et al. (2023), who designed dispersion-engineered TiO₂ meta-atoms with particle swarm optimization and a commercial FDTD solver and reported broadband relative diffraction efficiency around 90%. Schubert (2024) made it an invrs-gym challenge backed by the FMMAX Fourier modal method, which reproduces the published phase responses. Because the grating response is approximated by stitching single-cell transmissions, a higher score does not guarantee a better fabricated grating; the broadband objective and eight coupled design regions make this one of the gym's more advanced problems.

Data and evaluation

Instance: one fixed device, invrs_gym.challenges.meta_atom_library() with default arguments.
Evaluation code: the invrs leaderboard scorer (invrs_leaderboard.utils.evaluate_solutions_to_challenge), on CPU in 64-bit precision, with eleven wavelengths from 450 to 650 nm and about 300 Fourier terms. It also reports the minimum width and spacing in pixels. Submissions are totypes JSON parameter files.
Hardware: none assumed; the score does not depend on run time.

What is known

Best on the final leaderboard that meets the 60 nm target: 0.9237, a library by M. F. Schubert (October 2024).
The highest score overall, 0.9514, comes from the same batch but has 50 nm features, below the target.
Libraries obtained by fine-tuning the designs of Chen et al. (2023) in the gym reach up to 0.918, with 45 nm features (August 2024).

What would count as progress

A library meeting the 60 nm target with $\mathcal{M}>0.9237$ , re-scored with the leaderboard code.
A score above 0.9514 with features of at least 50 nm, beating every library on the leaderboard.
A full-wave simulation of a grating built from a top library, showing how well the stitched estimate predicts the real efficiency.

References

W. T. Chen, J.-S. Park, J. Marchioni, et al. Dispersion-engineered metasurfaces reaching broadband 90% relative diffraction efficiency. Nature Communications 14, 2544 (2023). doi:10.1038/s41467-023-38185-2
M. F. Schubert. invrs-gym: a toolkit for nanophotonic inverse design research. 2024. arXiv:2410.24132
M. F. Schubert, A. M. Hammond. Fourier modal method for inverse design of metasurface-enhanced micro-LEDs. Optics Express 31, 42945 (2023). doi:10.1364/OE.503481
invrs.io. Meta-atom library leaderboard (final state, archived October 2025). Leaderboard page

Source

Imported from invrs-gym meta_atom_library, curated by Martin F. Schubert (invrs.io). The gym and its leaderboard were archived in October 2025 and no longer take submissions; the scores above are the leaderboard's final state. The statement above is written in our own words; see the source for the original wording, history and discussion.