Diffractive beam splitter into a 7 × 7 spot array

Design an etched silica splitter that divides a 632.8 nm beam into a 7 × 7 array of equal beams, maximizing efficiency times uniformity.

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	1	+# Diffractive beam splitter into a 7 × 7 spot array
	2	+
	3	+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.
	4	+
	5	+## Task
	6	+
	7	+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
	8	+
	9	+$$U=\frac{\max_{i,j}\eta_{(i,j)}-\min_{i,j}\eta_{(i,j)}}{\max_{i,j}\eta_{(i,j)}+\min_{i,j}\eta_{(i,j)}},$$
	10	+
	11	+the score is
	12	+
	13	+$$\mathcal{M}=T\,(1-U),$$
	14	+
	15	+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.
	16	+
	17	+## Background
	18	+
	19	+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.
	20	+
	21	+## Data and evaluation
	22	+
	23	+- Instance: one fixed device, `invrs_gym.challenges.diffractive_splitter()` with default arguments.
	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 about 1000 Fourier terms. It also reports the minimum width and spacing in pixels. Submissions are `totypes` JSON parameter files.
	25	+- Hardware: none assumed; the score does not depend on run time.
	26	+
	27	+## What is known
	28	+
	29	+- 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.
	30	+- 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.
	31	+- All 24 leaderboard designs meet the 400 nm target and all come from the gym's author.
	32	+
	33	+## What would count as progress
	34	+
	35	+- A binary design meeting the 400 nm target with $\mathcal{M}>0.5964$, re-scored with the leaderboard code.
	36	+- A design with total efficiency above 0.70 and uniformity error below 0.10, which no leaderboard design reaches.
	37	+- A justified upper bound on $T$ for a single binary etched layer, to show how much room is left.
	38	+
	39	+## References
	40	+
	41	+1. M. F. Schubert. invrs-gym: a toolkit for nanophotonic inverse design research. 2024. [arXiv:2410.24132](https://arxiv.org/abs/2410.24132)
	42	+2. LightTrans. Design and rigorous analysis of non-paraxial diffractive beam splitter (use case, version 3.1). [LightTrans](https://www.lighttrans.com/use-cases/application/design-and-rigorous-analysis-of-non-paraxial-diffractive-beam-splitter.html)
	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. Diffractive splitter leaderboard (final state, archived October 2025). [Leaderboard page](https://invrs-io.github.io/leaderboard/notebooks/leaderboards/diffractive_splitter.html)
	45	+
	46	+## Source
	47	+
	48	+Imported from [invrs-gym diffractive_splitter](https://invrs-io.github.io/gym/notebooks/challenges/diffractive_splitter.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.