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  • Review Article
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Nonlocal metamaterials and metasurfaces

Nature Reviews Physics volume 7, pages 299–312 (2025)Cite this article

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Abstract

The aim of rationally designed composites called metamaterials or metasurfaces is to achieve effective properties that go beyond those of their constituent parts. For periodic architectures, the design can draw on concepts from solid-state physics, such as crystal symmetries, reciprocal space, band structures and Floquet–Bloch eigenfunctions. Recently, nonlocality has emerged as a design paradigm, enabling both static and dynamic properties that are unattainable with a local design. In principle, all material properties described by linear response functions can be nonlocal, but for ordinary solids, local descriptions are mostly good approximations, leaving nonlocal effects as corrections. However, metamaterials and metasurfaces can be designed to go far beyond local behaviour. This Review covers these anomalous behaviours in elasticity, acoustics, electromagnetism, optics and diffusion. In the dynamic regime, nonlocal interactions enable versatile band structure and refraction engineering. In the static regime, they result in large decay lengths of ‘frozen’ evanescent Bloch modes, leading to strong size effects. For zero modes, the decay length diverges.

Key points

  • Metamaterials are rationally designed composites with effective properties that go beyond their constituents. Following this definition, metamaterials include photonic and phononic crystals.

  • (Meta)material response functions can be phenomenological or result from a theoretical homogenization procedure. For nonlocal materials, the response function at a given location depends not only on the field at that location but also on the field at other locations.

  • We review nonlocal metamaterials for diverse physical fields according to three unified physical mechanisms to incorporate nonlocality, that is, beyond-nearest-neighbour interactions, chirality and delocalized zero modes.

  • The nonlocal mechanisms mentioned earlier can lead to interesting wave properties, such as roton-like dispersion, topological insulators with large winding number, chiral eigenmodes and frequency splitting and anomalous dispersion cones.

  • Nonlocality also induces static properties with anomalously large characteristic lengths, connected to frozen evanescent modes (evanescent Bloch modes at zero frequency) emerging from local minima on dispersion relations.

  • Time dependence is introduced as a separate and emerging method to achieve nonlocal responses in metamaterials, that is, the time-reflection coefficient and time-refraction coefficient become spatially dispersive or wavenumber-dependent.

  • Nonlocal metasurfaces often use leaky-wave modes or physical coupling to build nonlocal interactions in metasurfaces, enabling high frequency selectivity or performing spatial derivatives on incident wave fields.

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Fig. 1: Illustration of nonlocal interactions in a tight-binding model.
Fig. 2: Gallery of nonlocal metamaterials based on different mechanisms.
Fig. 3: Beyond-nearest-neighbouring interactions for tailoring dispersion bands and topological properties.
Fig. 4: Delocalized zero modes for nonlocality.
Fig. 5: Frequency splitting resulting from chirality.
Fig. 6: Nonlocal metasurfaces.

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Acknowledgements

The authors acknowledge financial support by the Deutsche Forschungsgemeinschaft (German Research Foundation) under Germany’s Excellence Strategy via the Excellence Cluster ‘3D Matter Made to Order’, EXC-2082/1-390761711, by the Carl Zeiss Foundation through the ‘Carl-Zeiss-Foundation-Focus@HEiKA’, by the State of Baden-Württemberg and by the Helmholtz programme ‘Materials Systems Engineering’. R.F. acknowledges the support of the Swiss National Science Foundation under the Eccellenza award PCEGP2_181232 for the research project titled ‘Ultra-compact wave devices based on deep sub wavelength spatially dispersive effects’. G.H. thanks the support of National Natural Science Foundation of China (Grant No. 11991030).

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Authors and Affiliations

  1. Institute of Nanotechnology, Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany

    Yi Chen & Martin Wegener

  2. Institute of Applied Physics, Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany

    Yi Chen & Martin Wegener

  3. Laboratory of Wave Engineering, École Polytechnique Fédéral de Lausanne (EPFL), Lausanne, Switzerland

    Romain Fleury

  4. Institut de Mathematiques de Toulon, Universite de Toulon, Toulon, France

    Pierre Seppecher

  5. School of Aerospace Engineering, Beijing Institute of Technology (BIT), Beijing, China

    Gengkai Hu

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Y.C. and M.W. drafted the initial version. All authors contributed to the discussions and the revision of this paper.

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Glossary

Characteristic length scales

For sample sizes much larger than the characteristic length, size effects become negligible.

Chiral metamaterials

A metamaterial lacking inversion symmetry, mirror planes and rotation–reflection symmetries.

Interlaced wire media

An electromagnetic structure composed of interconnected metal wire meshes.

Non-Bloch solutions

Non-Bloch solutions do not obey Bloch’s theorem but are still solutions of the periodic problem.

Saint-Venant’s principle

The linear elastic response of a material in the far field becomes insensitive to the precise location and distribution of the loading.

Size effects

We refer to size effects as the dependence of material properties, for example, the Young’s modulus, on the size of the sample.

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Chen, Y., Fleury, R., Seppecher, P. et al. Nonlocal metamaterials and metasurfaces. Nat Rev Phys 7, 299–312 (2025). https://doi.org/10.1038/s42254-025-00829-1

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