Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

  • Article
  • Published:

Solving integral equations in free space with inverse-designed ultrathin optical metagratings

Nature Nanotechnology volume 18, pages 365–372 (2023)Cite this article

Subjects

Abstract

As standard microelectronic technology approaches fundamental limitations in speed and power consumption, novel computing strategies are strongly needed. Analogue optical computing enables the processing of large amounts of data at a negligible energy cost and high speeds. Based on these principles, ultrathin optical metasurfaces have been recently explored to process large images in real time, in particular for edge detection. By incorporating feedback, it has also recently been shown that metamaterials can be tailored to solve complex mathematical problems in the analogue domain, although these efforts have so far been limited to guided-wave systems and bulky set-ups. Here, we present an ultrathin Si metasurface-based platform for analogue computing that is able to solve Fredholm integral equations of the second kind using free-space visible radiation. A Si-based metagrating was inverse-designed to implement the scattering matrix synthesizing a prescribed kernel corresponding to the mathematical problem of interest. Next, a semitransparent mirror was incorporated into the sample to provide adequate feedback and thus perform the required Neumann series, solving the corresponding equation in the analogue domain at the speed of light. Visible wavelength operation enables a highly compact, ultrathin device that can be interrogated from free space, implying high processing speeds and the possibility of on-chip integration.

This is a preview of subscription content, access via your institution

Access options

Buy this article

  • Purchase on Springer Link
  • Instant access to full article PDF
Buy now

Prices may be subject to local taxes which are calculated during checkout

Fig. 1: Si metagrating-based integral equation solver.
Fig. 2: Kernel design.
Fig. 3: Analogue matrix inversion.
Fig. 4: Fabrication and optical characterization of the optical analogue integral equation solver.
Fig. 5: Optical characterization of S2T.
Fig. 6: Estimated experimental solution.

Similar content being viewed by others

Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Code availability

All codes produced during this research are available from the corresponding author upon reasonable request.

References

  1. Athale, R. & Psaltis, D. Optical computing: past and future. Opt. Photon. News 27, 32–39 (2016).

    Article  Google Scholar 

  2. Solli, D. R. & Jalali, B. Analog optical computing. Nat. Photon. 9, 704–706 (2015).

    Article  CAS  Google Scholar 

  3. Zangeneh-Nejad, F., Sounas, D. L., Alù, A. & Fleury, R. Analogue computing with metamaterials. Nat. Rev. Mater. 6, 207–225, (2020).

    Article  Google Scholar 

  4. Silva, A. et al. Performing mathematical operations with metamaterials. Science 343, 160–163 (2014).

    Article  CAS  Google Scholar 

  5. Zhu, T. et al. Plasmonic computing of spatial differentiation. Nat. Commun. 8, 15391 (2017).

    Article  CAS  Google Scholar 

  6. Zhu, T. et al. Generalized spatial differentiation from the spin Hall effect of light and its application in image processing of edge detection. Phys. Rev. Appl. 11, 034043 (2019).

    Article  CAS  Google Scholar 

  7. Zhu, T. et al. Topological optical differentiator. Nat. Commun. 12, 680 (2021).

    Article  CAS  Google Scholar 

  8. Guo, C., Xiao, M., Minkov, M., Shi, Y. & Fan, S. Photonic crystal slab Laplace operator for image differentiation. Optica 5, 251–256 (2018).

    Article  Google Scholar 

  9. Guo, C., Xiao, M., Minkov, M., Fan, S. & Shi, Y. Isotropic wavevector domain image filters by a photonic crystal slab device. J. Opt. Soc. Am. A 35, 1685–1691 (2018).

    Article  Google Scholar 

  10. Wang, H., Guo, C., Zhao, Z. & Fan, S. Compact incoherent image differentiation with nanophotonic structures. ACS Photon. 7, 338–343 (2020).

    Article  CAS  Google Scholar 

  11. Kwon, H., Sounas, D., Cordaro, A., Polman, A. & Alù, A. Nonlocal metasurfaces for optical signal processing. Phys. Rev. Lett. 121, 173004 (2018).

    Article  CAS  Google Scholar 

  12. Cordaro, A. et al. High-index dielectric metasurfaces performing mathematical operations. Nano Lett. 19, 8418–8423 (2019).

    Article  CAS  Google Scholar 

  13. Youssefi, A., Zangeneh-Nejad, F., Abdollahramezani, S. & Khavasi, A. Analog computing by Brewster effect. Opt. Lett. 41, 3467–3470 (2016).

    Article  Google Scholar 

  14. Momeni, A., Rajabalipanah, H., Abdolali, A. & Achouri, K. Generalized optical signal processing based on multioperator metasurfaces synthesized by susceptibility tensors. Phys. Rev. Appl. 11, 064042 (2019).

    Article  CAS  Google Scholar 

  15. Abdollahramezani, S., Hemmatyar, O. & Adibi, A. Meta-optics for spatial optical analog computing. Nanophotonics 9, 4075–4095 (2020).

    Article  Google Scholar 

  16. Moeini, M. M. & Sounas, D. L. Discrete space optical signal processing. Optica 7, 1325–1331 (2020).

    Article  Google Scholar 

  17. Zhou, Y., Zheng, H., Kravchenko, I. I. & Valentine, J. Flat optics for image differentiation. Nat. Photon. 14, 316–323 (2020).

    Article  CAS  Google Scholar 

  18. Pors, A., Nielsen, M. G. & Bozhevolnyi, S. I. Analog computing using reflective plasmonic metasurfaces. Nano Lett. 15, 791–797 (2015).

    Article  CAS  Google Scholar 

  19. Bykov, D. A. et al. First-order optical spatial differentiator based on a guided-mode resonant grating. Opt. Express 26, 10997–11006 (2018).

    Article  CAS  Google Scholar 

  20. Kwon, H., Cordaro, A., Sounas, D., Polman, A. & Alù, A. Dual-polarization analog 2D image processing with nonlocal metasurfaces. ACS Photon. 7, 1799–1805 (2020).

    Article  CAS  Google Scholar 

  21. Bogaerts, W. et al. Programmable photonic circuits. Nature 586, 207–216 (2020).

    Article  CAS  Google Scholar 

  22. Shen, Y. et al. Deep learning with coherent nanophotonic circuits. Nat. Photon. 11, 441–446 (2017).

    Article  CAS  Google Scholar 

  23. Shastri, B. J. et al. Photonics for artificial intelligence and neuromorphic computing. Nat. Photon. 15, 102–114 (2021).

    Article  CAS  Google Scholar 

  24. van de Burgt, Y. et al. A non-volatile organic electrochemical device as a low-voltage artificial synapse for neuromorphic computing. Nat. Mater. 16, 414–418 (2017).

    Article  Google Scholar 

  25. Van De Burgt, Y., Melianas, A., Keene, S. T., Malliaras, G. & Salleo, A. Organic electronics for neuromorphic computing. Nat. Electron. 1, 386–397 (2018).

    Article  Google Scholar 

  26. Zangeneh-Nejad, F. & Fleury, R. Performing mathematical operations using high-index acoustic metamaterials. New J. Phys. 20, 073001 (2018).

  27. Zangeneh-Nejad, F. & Fleury, R. Topological analog signal processing. Nat. Commun. 10, 2058 (2019).

    Article  Google Scholar 

  28. Hughes, T. W., Williamson, I. A. D., Minkov, M. & Fan, S. Wave physics as an analog recurrent neural network. Sci. Adv. 5, eaay6946 (2019).

    Article  Google Scholar 

  29. Mohammadi Estakhri, N., Edwards, B. & Engheta, N. Inverse-designed metastructures that solve equations. Science 363, 1333–1338 (2019).

    Article  CAS  Google Scholar 

  30. Camacho, M., Edwards, B. & Engheta, N. A single inverse-designed photonic structure that performs parallel computing. Nat. Commun. 12, 1466 (2021).

    Article  CAS  Google Scholar 

  31. Arfken, G. B., Weber, H. J. & Harris, F. E. Mathematical Methods for Physicists (Elsevier, 2013).

  32. Oldenburger, R. Infinite powers of matrices and characteristic roots. Duke Math. J. 6, 357–361 (1940).

    Article  Google Scholar 

  33. Molesky, S. et al. Inverse design in nanophotonics. Nat. Photon. 12, 659–670 (2018).

    Article  CAS  Google Scholar 

  34. Piggott, A. Y. et al. Inverse design and demonstration of a compact and broadband on-chip wavelength demultiplexer. Nat. Photon. 9, 374–377 (2015).

    Article  CAS  Google Scholar 

  35. Sell, D., Yang, J., Doshay, S., Yang, R. & Fan, J. A. Large-angle, multifunctional metagratings based on freeform multimode geometries. Nano Lett. 17, 3752–3757 (2017).

    Article  CAS  Google Scholar 

  36. Lalau-Keraly, C. M., Bhargava, S., Miller, O. D. & Yablonovitch, E. Adjoint shape optimization applied to electromagnetic design. Opt. Express 21, 21693 (2013).

    Article  Google Scholar 

  37. Hughes, T. W., Minkov, M., Williamson, I. A. D. & Fan, S. Adjoint method and inverse design for nonlinear nanophotonic devices. ACS Photon. 5, 4781–4787 (2018).

    Article  CAS  Google Scholar 

  38. Green, M. A. Self-consistent optical parameters of intrinsic silicon at 300 K including temperature coefficients. Sol. Energy Mater. Sol. Cells 92, 1305–1310 (2008).

    Article  CAS  Google Scholar 

  39. Kelly, R. L. Program of the 1972 Annual Meeting of the Optical Society of America. J. Opt. Soc. Am. 62, 1336 (1972).

    Article  Google Scholar 

  40. Malitson, I. H. Interspecimen comparison of the refractive index of fused silica. J. Opt. Soc. Am. 55, 1205–1209 (1965).

    Article  CAS  Google Scholar 

  41. Jiang, J. et al. Free-form diffractive metagrating design based on generative adversarial networks. ACS Nano 13, 8872–8878 (2019).

    Article  CAS  Google Scholar 

  42. Goodman, J. W., Dias, A. R. & Woody, L. M. Fully parallel, high-speed incoherent optical method for performing discrete Fourier transforms. Opt. Lett. 2, 1–3 (1978).

    Article  CAS  Google Scholar 

  43. Athale, R. A. & Collins, W. C. Optical matrix–matrix multiplier based on outer product decomposition. Appl. Opt. 21, 2089–2090 (1982).

    Article  CAS  Google Scholar 

  44. Farhat, N. H., Psaltis, D., Prata, A. & Paek, E. Optical implementation of the Hopfield model. Appl. Opt. 24, 1469–1475 (1985).

    Article  CAS  Google Scholar 

  45. Zhu, W., Zhang, L., Lu, Y., Zhou, P. & Yang, L. Design and experimental verification for optical module of optical vector–matrix multiplier. Appl. Opt. 52, 4412–4418 (2013).

    Article  Google Scholar 

  46. Spall, J., Guo, X., Barrett, T. D. & Lvovsky, A. I. Fully reconfigurable coherent optical vector–matrix multiplication. Opt. Lett. 45, 5752–5755 (2020).

    Article  Google Scholar 

  47. Rosenblatt, G., Simkhovich, B., Bartal, G. & Orenstein, M. Nonmodal plasmonics: controlling the forced optical response of nanostructures. Phys. Rev. X 10, 011071 (2020).

    CAS  Google Scholar 

  48. Li, L. Bremmer series, R-matrix propagation algorithm, and numerical modeling of diffraction gratings. J. Opt. Soc. Am. A 11, 2829–2836 (1994).

    Article  Google Scholar 

  49. Sukham, J., Takayama, O., Lavrinenko, A. V. & Malureanu, R. High-quality ultrathin gold layers with an APTMS adhesion for optimal performance of surface plasmon polariton-based devices. ACS Appl. Mater. Interfaces 9, 25049–25056 (2017).

    Article  CAS  Google Scholar 

  50. Verschuuren, M. A., Knight, M. W., Megens, M. & Polman, A. Nanoscale spatial limitations of large-area substrate conformal imprint lithography. Nanotechnology 30, 345301 (2019).

    Article  CAS  Google Scholar 

  51. Lalanne, P., Hugonin, J. P. & Chavel, P. Optical properties of deep lamellar gratings: a coupled Bloch-mode insight. J. Light. Technol. 24, 2442–2449 (2006).

    Article  Google Scholar 

Download references

Acknowledgements

This work is part of the research programme of the Dutch Research Council (NWO) and is supported by the Air Force Office of Scientific Research’s Multidisciplinary Research Program of the University Research Initiative with grant no. FA9550-17-1-0002. V.N.’s effort is supported by the National Science Foundation Materials Research Science and Engineering Centers programme under award no. DMR-1720530.

Author information

Authors and Affiliations

  1. Institute of Physics, University of Amsterdam, Amsterdam, The Netherlands

    Andrea Cordaro

  2. Center for Nanophotonics, Institute AMOLF, NWO, Amsterdam, The Netherlands

    Andrea Cordaro & Albert Polman

  3. Department of Electrical and Systems Engineering, University of Pennsylvania, Philadelphia, PA, USA

    Brian Edwards, Vahid Nikkhah & Nader Engheta

  4. Photonics Initiative, Advanced Science Research Center, City University of New York, New York, NY, USA

    Andrea Alù

  5. Physics Program, Graduate Center, City University of New York, New York, NY, USA

    Andrea Alù

Authors
  1. Andrea Cordaro
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Brian Edwards
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Vahid Nikkhah
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Andrea Alù
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Nader Engheta
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Albert Polman
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

A.C. designed and fabricated the samples, performed the numerical simulations and performed the optical measurements. A.C., B.E. and V.N. performed the theoretical analyses. A.A., N.E. and A.P. supervised the project. All authors contributed to the analysis and writing of the paper.

Corresponding author

Correspondence to Andrea Cordaro.

Ethics declarations

Competing interests

The authors declare no competing interests.

Peer review

Peer review information

Nature Nanotechnology thanks Arkaprovo Das, Xiaowen Dong and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Additional information

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary information

Supplementary Information

Supplementary Figs. 1–8 and Discussion.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cordaro, A., Edwards, B., Nikkhah, V. et al. Solving integral equations in free space with inverse-designed ultrathin optical metagratings. Nat. Nanotechnol. 18, 365–372 (2023). https://doi.org/10.1038/s41565-022-01297-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/s41565-022-01297-9

This article is cited by

Search

Advanced search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing