A 2021 preprint demonstrated that a multi-level diffractive lens (MDL) — a flat lens of concentric, height-quantized rings designed by inverse design — can focus achromatically across the 400–800 nm band at 100 mm aperture and f/2, apparently circumventing theoretical size-bandwidth limits on flat lenses. The demonstration left its central design question explicitly open: its rings (5 µm and 2.5 µm wide) are far wider than the ~833 nm diffraction-limited spot size, forcing reliance on higher diffraction orders that raise point-spread-function background and lower Strehl ratio, and wavelengths not sampled during design (25 nm spectral spacing) show elevated background. The problem posed here is the quantitative account of how ring width and spectral over-sampling determine the achievable bandwidth, efficiency, Strehl ratio, and background of large-aperture achromatic MDLs, and whether sub-diffraction-limited ring widths can be realized over ~100 mm apertures. The question remains open and actively disputed: the same group's 2025 successor device still uses 5 µm rings with low efficiency, a published comment contests its achievable focusing efficiency and broadband resolution, and Fresnel-number-based limit theorems remain unreconciled with the demonstrations.

Background

A flat lens focuses light using structure on or near a surface rather than bulk refraction, enabling optics that are thin and lightweight. Two families dominate current research: metalenses, composed of subwavelength scatterers ("meta-atoms") whose local, wavelength-dependent phase shifts are engineered through waveguiding or resonant effects, and diffractive lenses, in which the focusing phase profile is realized by concentric zones or rings. Both families suffer from chromatic aberration — the phase a diffractive structure imprints varies strongly with wavelength, so the focal length changes with wavelength — and making a single flat element achromatic (focal length fixed across a broad band) is a central difficulty of the field.

A multi-level diffractive lens (MDL) is a diffractive lens made of concentric rings of fixed width whose heights are quantized to discrete levels; the heights are found by inverse design — numerical optimization (for example, a gradient-assisted direct binary search) minimizing the difference between the simulated point-spread function (PSF; the image of a point source) and the ideal diffraction-limited Airy pattern at each of a set of design wavelengths. Quality is quantified by the Strehl ratio (peak PSF intensity relative to an ideal aberration-free lens of the same aperture), the focusing efficiency (fraction of incident power concentrated within a region comparable to the focal spot), and the level of background haze outside the focal spot. The numerical aperture (NA; the sine of the half-angle of the cone of converging light) sets the diffraction-limited spot size, roughly 0.5λ/NA0.5\lambda/\mathrm{NA}.

Several theoretical results bound what achromatic flat lenses can do. Presutti and Monticone (2020) showed that in any time-invariant focusing system the product of achievable time delay and bandwidth is bounded, limiting achromatic metalenses whose meta-atoms must supply group delay from finite index contrast. Engelberg and Levy (2021) derived corresponding limits for diffractive lenses: an achromatic diffractive lens can be near-diffraction-limited only at low Fresnel numbers, FN=R2/(λf)≲pΔn+1\mathrm{FN}=R^2/(\lambda f)\lesssim p\Delta n+1, where RR is the lens radius, ff the focal length, pp the maximum height normalized to the wavelength, and Δn\Delta n the available index contrast. For realistic thin devices this bound is of order a few — apparently forbidding large aperture, high NA, and broad bandwidth simultaneously.

Majumder, Meem, Brimhall, and Menon (arXiv:2112.15157, 2021) argued these bounds can be circumvented. Building on earlier work in which allowing the image-plane phase to be a free variable let a ~1 mm MDL hold a constant focal length from the visible to the longwave infrared (Meem et al., Optics Express 2021), they designed, fabricated (by grayscale lithography, which writes the quantized height profile directly in photoresist), and characterized a 100 mm diameter, f/2 (NA=0.24\mathrm{NA}=0.24), 2.4 µm thick MDL whose focal length is fixed from 400 nm to 800 nm, and combined it with a refractive lens into a hybrid telescope. They further argued, via the coherence length of the illumination, that the relative-bandwidth bound Δλ/λ0≤2\Delta\lambda/\lambda_0\le 2 is independent of lens diameter and NA, so that coherence itself imposes no fundamental limit and the true limiter is the image sensor's finite integration time and bandwidth rather than the lens.

The demonstration, however, came with acknowledged open problems. The design used discrete 25 nm spectral sampling, and wavelengths between samples show elevated PSF background. The fabricated ring widths (5 µm and 2.5 µm) far exceed the diffraction-limited full width at half maximum of about 0.5×400 nm/0.24≈8330.5\times 400\,\mathrm{nm}/0.24\approx 833 nm, so focusing relies on higher diffraction orders, which increases background, reduces the Strehl ratio, and extends the depth of focus. The device's 25 g mass is dominated by a glass wafer that serves only as mechanical support and has no optical function. The authors speculated that ring widths below the diffraction limit combined with spectral over-sampling might eliminate the spectral sensitivity, writing that elucidating these limits is a subject for future research — the question this problem states.

Problem Statement

For large-aperture (order 100 mm and scalable beyond) achromatic multi-level diffractive lenses operating over broad visible bands such as 400–800 nm, establish the quantitative limits of the design space, jointly addressing: (1) How the design ring width ww — in particular whether it lies below the diffraction-limited spot size ∼0.5λmin⁡/NA\sim 0.5\lambda_{\min}/\mathrm{NA} (about 833 nm at NA=0.24\mathrm{NA}=0.24 and λmin⁡=400\lambda_{\min}=400 nm) — and the spectral sampling or over-sampling used in the design optimization together determine the achievable operating bandwidth, focusing efficiency, Strehl ratio, and PSF background, including at wavelengths not sampled during design; (2) Whether sub-diffraction-limited ring widths can be fabricated over such apertures with acceptable fabrication complexity, and what the resulting trade-offs between ring width, background, Strehl ratio, and achievable bandwidth are. These address one overarching scientific objective: a validated quantitative account of the ring-width and spectral-sampling design space and its limits for large-aperture broadband achromatic flat lenses.

References

  1. Apratim Majumder, Monjurul Meem, Nicole Brimhall, Rajesh Menon (2021). Circumventing size-bandwidth limits in imaging with flat lenses. arXiv preprint. DOI: 10.48550/arXiv.2112.15157. https://arxiv.org/abs/2112.15157
  2. Apratim Majumder, Monjurul Meem, Alexander Ingold, Paul Ricketts, Tanner Obray, Nicole Brimhall, Rajesh Menon (2025). Color astrophotography with a 100 mm-diameter f/2 polymer flat lens. Applied Physics Letters 126, 051701. DOI: 10.1063/5.0242208. https://doi.org/10.1063/5.0242208
  3. G. K. Skinner, J. F. Krizmanic (2025). Comment on 'Color astrophotography with a 100 mm-diameter f/2 polymer flat lens' [Appl. Phys. Lett. 126, 051701 (2025)]. Applied Physics Letters 127, 226101. DOI: 10.1063/5.0295471. arXiv: 2512.00054. https://arxiv.org/abs/2512.00054
  4. G. K. Skinner (2024). Design of achromatic diffractive lenses. Optics Express 32(1), 230–247. DOI: 10.1364/oe.509946. https://doi.org/10.1364/oe.509946
  5. Federico Presutti, Francesco Monticone (2020). Focusing on bandwidth: achromatic metalens limits. Optica 7(6), 624–631. DOI: 10.1364/optica.389404. https://doi.org/10.1364/optica.389404
  6. Jacob Engelberg, Uriel Levy (2021). Achromatic flat lens performance limits. Optica 8(6), 834. DOI: 10.1364/optica.422843. https://doi.org/10.1364/optica.422843
  7. Jacob Engelberg, Uriel Levy (2022). Standardizing flat lens characterization. Nature Photonics 16, 171–173. DOI: 10.1038/s41566-022-00963-7. https://doi.org/10.1038/s41566-022-00963-7
  8. Monjurul Meem, Apratim Majumder, Sourangsu Banerji, Juan C. Garcia, Orrin B. Kigner, Philip W. C. Hon, Berardi Sensale-Rodriguez, Rajesh Menon (2021). Imaging from the visible to the longwave infrared wavelengths via an inverse-designed flat lens. Optics Express 29(13), 20715–20723. DOI: 10.1364/oe.423764. https://doi.org/10.1364/oe.423764
  9. Yanhao Chu, Xiaopeng Li, Xingjian Xiao, Shining Zhu, Tao Li (2025). Achromatic multi-level diffractive lenses and related applications. Advances in Physics: X 10, 2580626. DOI: 10.1080/23746149.2025.2580626. https://doi.org/10.1080/23746149.2025.2580626
  10. Xingjian Xiao, Yunwei Zhao, Xin Ye, Chen Chen, Xinmou Lu, Yansen Rong, Junhong Deng, Guixin Li, Shining Zhu, Tao Li (2022). Large-scale achromatic flat lens by light frequency-domain coherence optimization. Light: Science & Applications 11, 323. DOI: 10.1038/s41377-022-01024-y. https://doi.org/10.1038/s41377-022-01024-y
  11. Shanshan Hu, Xingjian Xiao, Xin Ye, Rongtao Yu, Yanhao Chu, Ji Chen, Shining Zhu, Tao Li (2023). Deep learning enhanced achromatic imaging with a singlet flat lens. Optics Express 31(21), 33873. DOI: 10.1364/oe.501872. https://doi.org/10.1364/oe.501872
  12. Yuning Ye, Mengdi Luo, Jisen Wen, Dazhao Zhu, Cuifang Kuang, Xu Liu (2025). High-efficiency, large-scale, high-numerical-aperture achromatic flat lens via focal phasor matching. Photonics Research 13(12), 3431. DOI: 10.1364/prj.571096. https://doi.org/10.1364/prj.571096
  13. Shuyi Wang, Shaobo Ge, Zehua Sun, Lujun Shen, Yibo Zhao, Jiying Luo, Shun Zhou, Jin Zhang, Rui Chen, Xingjian Xiao, Tao Li, Weiguo Liu (2024). Large-Area, Broadband Multi-Level Diffractive Lens of Achromatic Imaging from Visible to Near-Infrared. SSRN preprint. DOI: 10.2139/ssrn.4900600. https://doi.org/10.2139/ssrn.4900600
  14. Joon-Suh Park, Soon Wei Daniel Lim, Arman Amirzhan, Hyukmo Kang, Karlene Karrfalt, Daewook Kim, Joel Leger, Augustine Urbas, Marcus Ossiander, Zhaoyi Li, Federico Capasso (2024). All-Glass 100 mm Diameter Visible Metalens for Imaging the Cosmos. ACS Nano 18(4), 3187–3198. DOI: 10.1021/acsnano.3c09462. https://doi.org/10.1021/acsnano.3c09462
  15. Johannes E. Fröch, Praneeth Chakravarthula, Jipeng Sun, Ethan Tseng, Shane Colburn, Alan Zhan, Forrest Miller, Anna Wirth-Singh, Quentin A. A. Tanguy, Zheyi Han, Karl F. Böhringer, Felix Heide, Arka Majumdar (2025). Beating spectral bandwidth limits for large aperture broadband nano-optics. Nature Communications 16, 3025. DOI: 10.1038/s41467-025-58208-4. https://doi.org/10.1038/s41467-025-58208-4
  16. Nikolay Ivliev, Viktoria Evdokimova, Vladimir Podlipnov, Maxim Petrov, Sofiya Ganchevskaya, Ivan Tkachenko, Dmitry Abrameshin, Yuri Yuzifovich, Artem Nikonorov, Roman Skidanov, Nikolay Kazanskiy, Victor Soifer (2022). First Earth-Imaging CubeSat with Harmonic Diffractive Lens. Remote Sensing 14(9), 2230. DOI: 10.3390/rs14092230. https://doi.org/10.3390/rs14092230
  17. Yuanjian Huang, Qiangqiang Liu, Ying Guo, Xiaoyin Li, Guoquan Fu, Qiong He, Qi Zhang, Fei Zhang, Mingfeng Xu, Mingbo Pu, Junwen Zhang, Nan Chi, Xiangang Luo (2026). Ultralight 110-mm-Aperture Planar Liquid-Crystal Diffractive Lens for Photon-Counting LiDAR. Photonics Research. DOI: 10.1364/prj.599707. https://doi.org/10.1364/prj.599707