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.

Nonlocal flat optics

Abstract

In electromagnetics and photonics, ‘nonlocality’ refers to the phenomenon by which the response/output of a material or system at a certain point in space depends on the input field across an extended region of space. Although nonlocal effects and the associated wavevector/momentum dependence have often been neglected or seen as a nuisance in the context of metasurfaces, the emerging field of nonlocal flat optics seeks to exploit strong effective nonlocality to enrich and enhance their response. Here we summarize the latest advances in this field, focusing on its fundamental principles and various applications, from optical computing to space compression. The convergence of local and nonlocal flat optics may open exciting opportunities in the quest to control light, in real and momentum space, using ultra-thin platforms.

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

Access options

Rent or buy this article

Get just this article for as long as you need it

$39.95

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

Fig. 1: Local versus nonlocal flat optics.
Fig. 2: Photonic platforms to realize strong artificial nonlocality.
Fig. 3: Applications and opportunities enabled by nonlocal flat optics.
Fig. 4: Space compression with nonlocal spaceplates.

Data availability

All relevant data are available from the corresponding author upon reasonable request.

References

  1. Seneca, L. A. Natural Questions (Univ. of Chicago Press, 2010).

  2. Davidson, N., Friesem, A. A. & Hasman, E. Computer-generated relief gratings as space-variant polarization elements. Opt. Lett. 17, 1541–1543 (1992).

    ADS  Google Scholar 

  3. Lalanne, P., Astilean, S., Chavel, P., Cambril, E. & Launois, H. Design and fabrication of blazed binary diffractive elements with sampling periods smaller than the structural cutoff. J. Opt. Soc. Am. A 16, 1143–1156 (1999).

    ADS  Google Scholar 

  4. Yu, N. et al. Light propagation with phase discontinuities: generalized laws of reflection and refraction. Science 334, 333–337 (2011).

    ADS  Google Scholar 

  5. Monticone, F., Estakhri, N. M. & Alu, A. Full control of nanoscale optical transmission with a composite metascreen. Phys. Rev. Lett. 110, 203903 (2013).

    ADS  Google Scholar 

  6. Pfeiffer, C. & Grbic, A. Metamaterial Huygens’ surfaces: tailoring wave fronts with reflectionless sheets. Phys. Rev. Lett. 110, 197401 (2013).

    ADS  Google Scholar 

  7. Zhan, A. et al. Low-contrast dielectric metasurface optics. ACS Photonics 3, 209–214 (2016).

    Google Scholar 

  8. Khorasaninejad, M. & Capasso, F. Metalenses: versatile multifunctional photonic components. Science 358, eaam8100 (2017).

    Google Scholar 

  9. Chen, M., Kim, M., Wong, A. M. H. & Eleftheriades, G. V. Huygens’ metasurfaces from microwaves to optics: a review. Nanophotonics 7, 1207–1231 (2018).

    Google Scholar 

  10. Banerji, S. et al. Imaging with flat optics: metalenses or diffractive lenses?’. Optica 6, 805–810 (2019).

    ADS  Google Scholar 

  11. Engelberg, J. & Levi, U. The advantages of metalenses over diffractive lenses. Nat. Commun. 11, 1991 (2020).

    ADS  Google Scholar 

  12. Miller, D. A. B. Why optics needs thickness. Preprint at https://arxiv.org/abs/2209.03552 (2022).

  13. Landau, L. D. et al. Electrodynamics of Continuous Media Vol. 8, 2nd edn (Elsevier, 2013).

  14. Belov, P. A. et al. Strong spatial dispersion in wire media in the very large wavelength limit. Phys. Rev. B 67, 113103 (2003).

    ADS  Google Scholar 

  15. Silveirinha, M. Generalized Lorentz-Lorenz formulas for microstructured materials. Phys. Rev. B 76, 245117 (2007).

    ADS  Google Scholar 

  16. Agranovich, V. M. & Vitaly, G. Crystal Optics with Spatial Dispersion, and Excitons Vol. 42 (Springer, 2013).

  17. Pozar, D. M. Microwave Engineering (Wiley, 2011).

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

    ADS  Google Scholar 

  19. Moses, C. A. & Engheta, N. An idea for electromagnetic feedforward-feedbackward media. IEEE Trans. Antennas Propag. 47, 918–928 (1999).

    ADS  Google Scholar 

  20. Silveirinha, M. G. Anomalous refraction of light colors by a metamaterial prism. Phys. Rev. Lett. 102, 193903 (2009).

    ADS  Google Scholar 

  21. Silveirinha, M. G. Additional boundary conditions for nonconnected wire media. New J. Phys. 11, 113016 (2009).

    ADS  Google Scholar 

  22. Ishimaru, A. Electromagnetic Wave Propagation, Radiation and Scattering: from Fundamentals to Applications (Wiley, 2017).

  23. Gerken, M. & Miller, D. A. B. Multilayer thin-film structures with high spatial dispersion. Appl. Opt. 42, 1330–1345 (2003).

    ADS  Google Scholar 

  24. Shastri, K., Reshef, O., Boyd, R. W., Lundeen, J. S. & Monticone, F. To what extent can space be compressed? Bandwidth limits of spaceplates. Optica 9, 738–745 (2022).

    ADS  Google Scholar 

  25. Gerken, M. & Miller, D. A. B. Limits on the performance of dispersive thin-film stacks. Appl. Opt. 44, 3349–3357 (2005).

    ADS  Google Scholar 

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

    ADS  MathSciNet  MATH  Google Scholar 

  27. Reshef, O. et al. An optic to replace space and its application towards ultra-thin imaging systems. Nat. Commun. 12, 3512 (2021).

    ADS  Google Scholar 

  28. Chen, A. & Monticone, F. Dielectric nonlocal metasurfaces for fully solid-state ultrathin optical systems. ACS Photonics 8, 1439–1447 (2021).

    Google Scholar 

  29. Monticone, F. & Alu, A. Leaky-wave theory, techniques, and applications: from microwaves to visible frequencies. Proc. IEEE 103, 793–821 (2015).

    Google Scholar 

  30. Limonov, M. F., Mikhail, V. R., Poddubny, A. N. & Kivshar, Y. S. Fano resonances in photonics. Nat. Photon. 11, 543–554 (2017).

    Google Scholar 

  31. Fan, S. & Joannopoulos, J. D. Analysis of guided resonances in photonic crystal slabs. Phys. Rev. B 65, 235112 (2002).

    ADS  Google Scholar 

  32. Snyder, W. C., Wan, Z. & Li, X. Thermodynamic constraints on reflectance reciprocity and Kirchhoff’s law. Appl. Opt. 37, 3464–3470 (1998).

    ADS  Google Scholar 

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

    ADS  Google Scholar 

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

    ADS  Google Scholar 

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

    ADS  Google Scholar 

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

    ADS  Google Scholar 

  37. Xue, W. & Owen, D. M. High-NA optical edge detection via optimized multilayer films. J. Opt. 23, 125004 (2021).

    ADS  Google Scholar 

  38. Guo, C., Wang, H. & Fan, S. Squeeze free space with nonlocal flat optics. Optica 7, 1133–1138 (2020).

    Google Scholar 

  39. Pagé, J. T. R. et al. Designing high-performance propagation-compressing spaceplates using thin-film multilayer stacks. Opt. Express 30, 2197–2205 (2022).

    ADS  Google Scholar 

  40. Mrnka, M. et al. Space squeezing optics: performance limits and implementation at microwave frequencies. APL Photonics 7, 076105 (2022).

    ADS  Google Scholar 

  41. Zhou, J. et al. Metasurface enabled quantum edge detection. Sci. Adv. 6, eabc4385 (2020).

    ADS  Google Scholar 

  42. Sihvola, A. Enabling optical analog computing with metamaterials. Science 343, 144–145 (2014).

    ADS  Google Scholar 

  43. Zangeneh-Nejad, F., Dimitrios, L. S., Alù, A. & Fleury, R. Analogue computing with metamaterials. Nat. Rev. Mater. 6, 207–225 (2021).

    ADS  Google Scholar 

  44. Goodman, J. W. Introduction to Fourier Optics (Roberts, 2005).

  45. Estakhri, N., Mohammadi & Alu, A. Wave-front transformation with gradient metasurfaces. Phys. Rev. 6, 041008 (2016).

    Google Scholar 

  46. Asadchy, V. S. et al. Perfect control of reflection and refraction using spatially dispersive metasurfaces. Phys. Rev. B 94, 075142 (2016).

    ADS  Google Scholar 

  47. Epstein, A. & Eleftheriades, G. V. Synthesis of passive lossless metasurfaces using auxiliary fields for reflectionless beam splitting and perfect reflection. Phys. Rev. Lett. 117, 256103 (2016).

    ADS  Google Scholar 

  48. Díaz-Rubio, A., Viktar, S. A., Elsakka, A. & Tretyakov, S. A. From the generalized reflection law to the realization of perfect anomalous reflectors. Sci. Adv. 3, e1602714 (2017).

    ADS  Google Scholar 

  49. Quan, L. & Alu, A. Passive acoustic metasurface with unitary reflection based on nonlocality. Phys. Rev. Appl. 11, 054077 (2019).

    ADS  Google Scholar 

  50. Zhu, H., Patnaik, S., Walsh, T. F., Jared, B. H. & Semperlotti, F. Nonlocal elastic metasurfaces: enabling broadband wave control via intentional nonlocality. Proc. Natl Acad. Sci. USA 117, 26099–26108 (2020).

    ADS  Google Scholar 

  51. Im, K., Kang, J.-H. & Park, Q.-H. Universal impedance matching and the perfect transmission of white light. Nat. Photon. 12, 143–149 (2018).

    ADS  Google Scholar 

  52. Horsley, S. Non-locality prevents reflection. Nat. Photon. 12, 127–128 (2018).

    ADS  Google Scholar 

  53. Zhu, Y. et al. Nonlocal acoustic metasurface for ultrabroadband sound absorption. Phys. Rev. B 103, 064102 (2021).

    ADS  Google Scholar 

  54. Lin, Z., Roques-Carmes, C., Christiansen, R. E., Soljačić, M. & Johnson, S. G. Computational inverse design for ultra-compact single-piece metalenses free of chromatic and angular aberration. Appl. Phys. Lett. 118, 041104 (2021).

    ADS  Google Scholar 

  55. Monticone, F., Constantinos, A. V. & Alu, A. Parity-time symmetric nonlocal metasurfaces: all-angle negative refraction and volumetric imaging. Phys. Rev. 6, 041018 (2016).

    Google Scholar 

  56. Valagiannopoulos, C. A., Monticone, F. & Alu, A. PT-symmetric planar devices for field transformation and imaging. J. Opt. 18, 044028 (2016).

    ADS  Google Scholar 

  57. Savoia, S. et al. Magnified imaging based on non-Hermitian nonlocal cylindrical metasurfaces. Phys. Rev. B 95, 115114 (2017).

    ADS  Google Scholar 

  58. Kamali, S. M. et al. Angle-multiplexed metasurfaces: encoding independent wavefronts in a single metasurface under different illumination angles. Phys. Rev. X 7, 041056 (2017).

    Google Scholar 

  59. Song, J.-H., van de Groep, J., Kim, S. J. & Brongersma, M. L. Non-local metasurfaces for spectrally decoupled wavefront manipulation and eye tracking. Nat. Nanotechnol. 16, 1224–1230 (2021).

    ADS  Google Scholar 

  60. Zhang, X. et al. Controlling angular dispersions in optical metasurfaces. Light Sci. Appl. 9, 76 (2020).

    ADS  Google Scholar 

  61. Overvig, A. C., Malek, S. C. & Yu, N. Multifunctional nonlocal metasurfaces. Phys. Rev. Lett. 125, 017402 (2020).

    ADS  Google Scholar 

  62. Overvig, A. C., Malek, S. C., Carter, M. J., Shrestha, S. & Yu, N. Selection rules for quasibound states in the continuum. Phys. Rev. B 102, 035434 (2020).

    ADS  Google Scholar 

  63. Lawrence, M. et al. High quality factor phase gradient metasurfaces. Nat. Nanotechnol. 15, 956–961 (2020).

    ADS  Google Scholar 

  64. Spägele, C. et al. Multifunctional wide-angle optics and lasing based on supercell metasurfaces. Nat. Commun. 12, 3787 (2021).

    ADS  Google Scholar 

  65. Overvig, A. & Alu, A. Wavefront-selective Fano resonant metasurfaces. Adv. Photonics 3, 026002 (2021).

    ADS  Google Scholar 

  66. Malek, S. C., Adam, C. O., Andrea, A. & Nanfang, Y. Multifunctional resonant wavefront-shaping meta-optics based on multilayer and multi-perturbation nonlocal metasurfaces. Light Sci. Appl. 11, 246 (2022).

    Google Scholar 

  67. Hsu, C. W. et al. Observation of trapped light within the radiation continuum. Nature 499, 188–191 (2013).

    ADS  Google Scholar 

  68. Monticone, F. & Alù, A. Bound states within the radiation continuum in diffraction gratings and the role of leaky modes. New J. Phys. 19, 093011 (2017).

    ADS  Google Scholar 

  69. Koshelev, K., Lepeshov, S., Liu, M., Bogdanov, A. & Kivshar, Y. Asymmetric metasurfaces with high-Q resonances governed by bound states in the continuum. Phys. Rev. Lett. 121, 193903 (2018).

    ADS  Google Scholar 

  70. Doeleman, H. M., Monticone, F., den Hollander, W., Alù, A. & Koenderink, A. F. Experimental observation of a polarization vortex at an optical bound state in the continuum. Nat. Photon. 12, 397–401 (2018).

    ADS  Google Scholar 

  71. Baranov, D. G. et al. Nanophotonic engineering of far-field thermal emitters. Nat. Mater. 18, 920–930 (2019).

    ADS  Google Scholar 

  72. Greffet, J.-J. et al. Coherent emission of light by thermal sources. Nature 416, 61–64 (2002).

    ADS  Google Scholar 

  73. Dahan, N. et al. Enhanced coherency of thermal emission: beyond the limitation imposed by delocalized surface waves. Phys. Rev. B 76, 045427 (2007).

    ADS  Google Scholar 

  74. Battula, A. & Chen, S. C. Monochromatic polarized coherent emitter enhanced by surface plasmons and a cavity resonance. Phys. Rev. B 74, 245407 (2006).

    ADS  Google Scholar 

  75. Drevillon, J., Joulain, K., Ben-Abdallah, P. & Nefzaoui, E. Far field coherent thermal emission from a bilayer structure. J. Appl. Phys. 109, 034315 (2011).

    ADS  Google Scholar 

  76. Lee, B. J., Fu, C. J. & Zhang, Z. M. Coherent thermal emission from one-dimensional photonic crystals. Appl. Phys. Lett. 87, 071904 (2005).

    ADS  Google Scholar 

  77. Biener, G., Dahan, N., Niv, A., Kleiner, V. & Hasman, E. Highly coherent thermal emission obtained by plasmonic bandgap structures. Appl. Phys. Lett. 92, 081913 (2008).

    ADS  Google Scholar 

  78. Overvig, A. C., Mann, S. A. & Alù, A. Thermal metasurfaces: complete emission control by combining local and nonlocal light-matter interactions. Phys. Rev. X 11, 021050 (2021).

    Google Scholar 

  79. Kondakci, H. E. & Abouraddy, A. F. Diffraction-free space–time light sheets. Nat. Photon. 11, 733–740 (2017).

    ADS  Google Scholar 

  80. Kondakci, H. E. & Abouraddy, A. F. Optical space-time wave packets having arbitrary group velocities in free space. Nat. Commun. 10, 929 (2019).

    ADS  Google Scholar 

  81. Guo, C., Xiao, M., Orenstein, M. & Fan, S. Structured 3D linear space–time light bullets by nonlocal nanophotonics. Light Sci. Appl. 10, 160 (2021).

    ADS  Google Scholar 

  82. Morizur, J.-F. et al. Programmable unitary spatial mode manipulation. J. Opt. Soc. Am. A 27, 2524–2531 (2010).

    ADS  Google Scholar 

  83. Tseng, E. et al. Neural nano-optics for high-quality thin lens imaging. Nat. Commun. 12, 6493 (2021).

    ADS  Google Scholar 

  84. Jiang, J. & Fan, J. A. Global optimization of dielectric metasurfaces using a physics-driven neural network. Nano Lett. 19, 5366–5372 (2019).

    ADS  Google Scholar 

  85. Lin, Z. et al. End-to-end metasurface inverse design for single-shot multi-channel imaging. Opt. Express 30, 28358–28370 (2022).

    ADS  Google Scholar 

  86. Li, Z., Pestourie, R., Lin, Z., Johnson, S. G. & Capasso, F. Empowering metasurfaces with inverse design: principles and applications. ACS Photonics 9, 2178–2192 (2022).

    Google Scholar 

  87. Bliokh, K. Y., Rodríguez-Fortuño, F. J., Bekshaev, A. Y., Kivshar, Y. S. & Nori, F. Electric-current-induced unidirectional propagation of surface plasmon-polaritons. Opt. Lett. 43, 963–966 (2018).

    ADS  Google Scholar 

  88. Morgado, T. A. & Silveirinha, M. G. Drift-induced unidirectional graphene plasmons. ACS Photonics 5, 4253–4258 (2018).

    Google Scholar 

  89. Hassani, G., Ali, S. & Monticone, F. Drifting electrons: nonreciprocal plasmonics and thermal photonics. ACS Photonics 9, 806–819 (2022).

    Google Scholar 

  90. Mortensen, N. A. Nonlocal formalism for nanoplasmonics: phenomenological and semi-classical considerations. Photon. Nanostruct. Fundamentals Appl. 11, 303–309 (2013).

    ADS  Google Scholar 

  91. Raza, S., Sergey, I. B., Wubs, M. & Mortensen, N. A. Nonlocal optical response in metallic nanostructures. J. Phys. Condens. Matter 27, 183204 (2015).

    ADS  Google Scholar 

  92. Khurgin, J., Tsai, W.-Y., Tsai, D. P. & Sun, G. Landau damping and limit to field confinement and enhancement in plasmonic dimers. ACS Photonics 4, 2871–2880 (2017).

    Google Scholar 

  93. Khurgin, J. B. Ultimate limit of field confinement by surface plasmon polaritons. Faraday Discuss. 178, 109–122 (2015).

    ADS  Google Scholar 

  94. Monticone, F. A truly one-way lane for surface plasmon polaritons. Nat. Photon. 14, 461–465 (2020).

    Google Scholar 

  95. Gangaraj, S. A. H. & Monticone, F. Physical violations of the bulk-edge correspondence in topological electromagnetics. Phys. Rev. Lett. 124, 153901 (2020).

    ADS  Google Scholar 

  96. Pollard, R. J. et al. Optical nonlocalities and additional waves in epsilon-near-zero metamaterials. Phys. Rev. Lett. 102, 127405 (2009).

    ADS  Google Scholar 

  97. Ginzburg, P. et al. Spontaneous emission in non-local materials. Light Sci. Appl. 6, e16273 (2017).

    Google Scholar 

  98. Agranovich, V. M., Shen, Y. R., Baughman, R. H. & Zakhidov, A. A. Linear and nonlinear wave propagation in negative refraction metamaterials. Phys. Rev. B 69, 165112 (2004).

    ADS  Google Scholar 

  99. Silveirinha, M. G. Metamaterial homogenization approach with application to the characterization of microstructured composites with negative parameters. Phys. Rev. B 75, 115104 (2007).

    ADS  Google Scholar 

  100. Shim, H., Monticone, F. & Miller, O. D. Fundamental limits to the refractive index of transparent optical materials. Adv. Mater. 33, 2103946 (2021).

    Google Scholar 

Download references

Acknowledgements

We acknowledge support from the Air Force Office of Scientific Research (grant no. FA9550-22-1-0204) through A. Nachman.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Francesco Monticone.

Ethics declarations

Competing interests

The authors declare no competing interests.

Peer review

Peer review information

Nature Photonics thanks Owen Miller 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.

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

Verify currency and authenticity via CrossMark

Cite this article

Shastri, K., Monticone, F. Nonlocal flat optics. Nat. Photon. 17, 36–47 (2023). https://doi.org/10.1038/s41566-022-01098-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/s41566-022-01098-5

This article is cited by

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