Broadband meta-atom library for visible metasurfaces

Design eight TiO₂ meta-atoms whose stitched grating keeps a high relative diffraction efficiency from 450 to 650 nm, with features of at least 60 nm.

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	1	+# Broadband meta-atom library for visible metasurfaces
	2	+
	3	+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.
	4	+
	5	+## Task
	6	+
	7	+Each of the eight meta-atoms occupies a 400 nm square cell on an SiO₂ substrate, in air. A design consists of
	8	+
	9	+- 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;
	10	+- one TiO₂ thickness in $[0.5,0.7]$ µm.
	11	+
	12	+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
	13	+
	14	+$$S^{j}_{k}=\frac{1}{8}\sqrt{\frac{n_{\mathrm{sub}}}{n_{\mathrm{amb}}}}\sum_{n=0}^{7}t^{j}_{n}\,e^{-i\pi kn/4},$$
	15	+
	16	+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.
	17	+
	18	+## Background
	19	+
	20	+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.
	21	+
	22	+## Data and evaluation
	23	+
	24	+- Instance: one fixed device, `invrs_gym.challenges.meta_atom_library()` with default arguments.
	25	+- Evaluation code: the [invrs leaderboard](https://github.com/invrs-io/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.
	26	+- Hardware: none assumed; the score does not depend on run time.
	27	+
	28	+## What is known
	29	+
	30	+- Best on the final leaderboard that meets the 60 nm target: 0.9237, a library by M. F. Schubert (October 2024).
	31	+- The highest score overall, 0.9514, comes from the same batch but has 50 nm features, below the target.
	32	+- 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).
	33	+
	34	+## What would count as progress
	35	+
	36	+- A library meeting the 60 nm target with $\mathcal{M}>0.9237$, re-scored with the leaderboard code.
	37	+- A score above 0.9514 with features of at least 50 nm, beating every library on the leaderboard.
	38	+- A full-wave simulation of a grating built from a top library, showing how well the stitched estimate predicts the real efficiency.
	39	+
	40	+## References
	41	+
	42	+1. 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](https://doi.org/10.1038/s41467-023-38185-2)
	43	+2. M. F. Schubert. invrs-gym: a toolkit for nanophotonic inverse design research. 2024. [arXiv:2410.24132](https://arxiv.org/abs/2410.24132)
	44	+3. 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](https://doi.org/10.1364/OE.503481)
	45	+4. invrs.io. Meta-atom library leaderboard (final state, archived October 2025). [Leaderboard page](https://invrs-io.github.io/leaderboard/notebooks/leaderboards/meta_atom_library.html)
	46	+
	47	+## Source
	48	+
	49	+Imported from [invrs-gym meta_atom_library](https://invrs-io.github.io/gym/notebooks/challenges/meta_atom_library.html), 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.