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Achromatic RGB metalens with 100 nm features

Design a 10 µm binary metalens that focuses 450, 550 and 650 nm light to one point, maximizing the weakest intensity enhancement.

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1	# Achromatic RGB metalens with 100 nm features
2
3	Design a flat, one-dimensional lens that focuses blue, green and red light to the same point, and make the weakest of the three foci as bright as possible. Bringing several wavelengths to one focus is a widely studied problem in flat optics, and this test case lets design methods be compared on equal terms.
4
5	## Task
6
7	The lens is a 10 µm wide, 1 µm tall region filled with a binary pattern of a dielectric with refractive index 2.4 and air (index 1), on a 20 nm grid. Plane waves at 450, 550 and 650 nm, with the electric field in the plane of the lens, arrive at normal incidence from a substrate of index 2.4. For each wavelength $\lambda_j$, the intensity enhancement is
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9	$$e_j=\frac{\lvert\mathbf{E}_j(\mathbf{r}_0)\rvert^2}{\lvert\mathbf{E}^0_j(\mathbf{r}_0)\rvert^2},$$
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11	where $\mathbf{r}_0$ is the target focus, 2.4 µm above the centre of the lens, $\mathbf{E}_j$ is the field with the lens and $\mathbf{E}^0_j$ the field without it. The score is
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13	$$\mathcal{M}=\min_j e_j,$$
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15	higher is better. A valid design is fully binary, with minimum feature width and minimum gap of at least 5 pixels (100 nm), measured with the imageruler package; this is the challenge's default target.
16
17	## Background
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19	The problem is the RGB metalens test case of the photonics inverse-design testbed of Chen et al. (2024), where designs with length scales from 70 to 250 nm were made by topology optimization backed by finite-element and FDTD solvers. Schubert (2024) implemented it in the invrs-gym with the FMMAX Fourier modal method, slicing the lens into thin layers, and found close agreement with the published finite-element and FDTD results. The challenge is a good yardstick because many existing solutions span a wide range of feature sizes, and the score falls as the minimum feature grows.
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21	## Data and evaluation
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23	- Instance: one fixed device, `invrs_gym.challenges.metalens()` with default arguments (x-polarized incidence).
24	- 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 more layers (30) and Fourier terms (340) than the optimization default. It also reports the minimum width and spacing in pixels. Designs can be submitted as a CSV density array or a `totypes` JSON file.
25	- Hardware: none assumed; the score does not depend on run time.
26
27	## What is known
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29	- Best on the final leaderboard that meets the 100 nm target: 18.80, a design by M. F. Schubert (July 2024) with exactly 5-pixel width and spacing.
30	- The highest score overall, 19.95, belongs to a 70 nm finite-element design from the testbed of Chen et al. (2024), whose measured features (60 to 80 nm) fall below the target.
31	- In the testbed designs re-simulated by Schubert (2024), the minimum enhancement drops from about 20 for the 70 nm design to about 8 for the 256 nm design.
32
33	## What would count as progress
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35	- A binary design meeting the 100 nm target with $\mathcal{M}>18.80$, re-scored with the leaderboard code.
36	- A design with 150 nm or larger features whose score beats the best published designs at that size.
37	- An upper bound on the achievable enhancement for this geometry, to show how far the designs are from optimal.
38
39	## References
40
41	1. M. Chen, R. E. Christiansen, J. A. Fan, et al. Validation and characterization of algorithms and software for photonics inverse design. Journal of the Optical Society of America B 41, A161–A176 (2024). [doi:10.1364/JOSAB.506412](https://doi.org/10.1364/JOSAB.506412)
42	2. M. F. Schubert. invrs-gym: a toolkit for nanophotonic inverse design research. 2024. [arXiv:2410.24132](https://arxiv.org/abs/2410.24132)
43	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)
44	4. invrs.io. Metalens leaderboard (final state, archived October 2025). [Leaderboard page](https://invrs-io.github.io/leaderboard/notebooks/leaderboards/metalens.html)
45
46	## Source
47
48	Imported from [invrs-gym metalens](https://invrs-io.github.io/gym/notebooks/challenges/metalens.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.