Diffractive beam splitter into a 7 × 7 spot array

Design an etched glass surface that splits one laser beam into 49 beams of equal power, fanning out over ±15°. Such splitters are used in structured-light projection and multifocal microscopy, and the hard part is to keep both the total efficiency high and the 49 beams equal at the same time.

Task

The device is a biperiodic pattern etched in silica (SiO₂, refractive index 1.46) with a 7.2 µm × 7.2 µm unit cell. A design consists of a binary etch pattern on a 40 nm grid (180 × 180 pixels), the thickness of the patterned layer in $[0.5,1.5]$ µm, and the thicknesses of two thin auxiliary layers, each in $[0,0.1]$ µm. A TE-polarized 632.8 nm plane wave arrives from air at normal incidence. Let $\eta_{(i,j)}$ be the efficiency of the transmitted diffraction order $(i,j)$, for $i,j\in\{-3,\dots,3\}$, summed over the two output polarizations (TE→TE plus TE→TM). With total efficiency $T=\sum_{i,j}\eta_{(i,j)}$ and uniformity error

$$U=\frac{\max_{i,j}\eta_{(i,j)}-\min_{i,j}\eta_{(i,j)}}{\max_{i,j}\eta_{(i,j)}+\min_{i,j}\eta_{(i,j)}},$$

the score is

$$\mathcal{M}=T\,(1-U),$$

higher is better (a perfect splitter scores 1). A valid design is fully binary, with minimum feature width and minimum gap of at least 10 pixels (400 nm), measured with the imageruler package, the challenge's default target.

Background

The problem follows a LightTrans case study of a non-paraxial diffractive beam splitter, whose designs used a grid of square pillars and a hybrid of the iterative Fourier transform algorithm, the thin-element approximation and a final rigorous coupled-wave analysis. Schubert (2024) implemented it in the invrs-gym with the FMMAX Fourier modal method, which matches the LightTrans results closely and is fast enough to optimize with the accurate model throughout. The published designs suggest a trade-off between total efficiency and uniformity, so the problem is a natural test for multi-objective optimization and for how the two goals are combined into one score.

Data and evaluation

• Instance: one fixed device, invrs_gym.challenges.diffractive_splitter() with default arguments.
• Evaluation code: the invrs leaderboard scorer (invrs_leaderboard.utils.evaluate_solutions_to_challenge), on CPU in 64-bit precision with about 1000 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

• The LightTrans designs, re-simulated by Schubert (2024), have total efficiencies of about 0.68 to 0.74 but uniformity errors between 0.40 and 0.80, so their scores stay below 0.45.
• Best on the final leaderboard: 0.5964, a design by M. F. Schubert (April 2024) with 400 nm features, total efficiency 0.641 and uniformity error 0.069.
• All 24 leaderboard designs meet the 400 nm target and all come from the gym's author.

What would count as progress

• A binary design meeting the 400 nm target with $\mathcal{M}>0.5964$, re-scored with the leaderboard code.
• A design with total efficiency above 0.70 and uniformity error below 0.10, which no leaderboard design reaches.
• A justified upper bound on $T$ for a single binary etched layer, to show how much room is left.

References

1. M. F. Schubert. invrs-gym: a toolkit for nanophotonic inverse design research. 2024. arXiv:2410.24132
2. LightTrans. Design and rigorous analysis of non-paraxial diffractive beam splitter (use case, version 3.1). LightTrans
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
4. invrs.io. Diffractive splitter leaderboard (final state, archived October 2025). Leaderboard page

Source

Imported from invrs-gym diffractive_splitter, 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.