I. INTRODUCTION

Section:

ChooseTop of pageABSTRACTI. INTRODUCTION <<II. METASURFACES—BASIC PR...III. NONLINEAR AND NONLOC...IV. OPPORTUNITIES AND CHA...V. CONCLUSIONIn the past few decades, controlled use of light has revolutionized science and technology, having a broad impact on our daily life, especially in connection with telecommunications and information technologies. However, continuous technological advancements have created increasing demands for light manipulation at ever smaller spatial and temporal scales.11. A. F. Koenderink, A. Alù, and A. Polman, “ Nanophotonics: Shrinking light-based technology,” Science 348(6234), 516–521 (2015). https://doi.org/10.1126/science.1261243 Nonlinear optical effects, such as frequency conversion, all-optical modulation, and generation of non-classical light, are highly desirable in many applications of optical science, but they are very challenging to observe and use in miniaturized optical devices. Since traditional optical materials have already reached their limits in conventional applications, the need for innovation in materials design, such as metamaterials, has emerged.22. M. Lapine, I. V. Shadrivov, and Y. S. Kivshar, “ Colloquium: Nonlinear metamaterials,” Rev. Mod. Phys. 86(3), 1093 (2014). https://doi.org/10.1103/RevModPhys.86.1093 Metamaterials can exhibit optical properties not found in natural materials, such as optical magnetism3,43. A. Alù and N. Engheta, “ Dynamical theory of artificial optical magnetism produced by rings of plasmonic nanoparticles,” Phys. Rev. B 78(8), 085112 (2008). https://doi.org/10.1103/PhysRevB.78.0851124. J. C. Ginn, I. Brener, D. W. Peters, J. R. Wendt, J. O. Stevens, P. F. Hines, L. I. Basilio, L. K. Warne, J. F. Ihlefeld, P. G. Clem et al., “ Realizing optical magnetism from dielectric metamaterials,” Phys. Rev. Lett. 108(9), 097402 (2012). https://doi.org/10.1103/PhysRevLett.108.097402 and negative refraction.5,65. J. B. Pendry, “ Negative refraction,” Contemp. Phys. 45(3), 191–202 (2004). https://doi.org/10.1080/001075104100016674346. J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “ Three-dimensional optical metamaterial with a negative refractive index,” Nature 455(7211), 376–379 (2008). https://doi.org/10.1038/nature07247 They are typically composed of subwavelength building blocks—the so-called meta-atoms. For the visible and near-infrared spectral ranges, the dimensions of these structures are in the nanoscale, making it difficult to fabricate 3D optical metamaterials. Therefore, the research has moved from 3D toward more feasible 2D structures—metasurfaces.77. C. Redit and C. Ommo, “Metasurfaces go mainstream,” Nat. Photonics 17(1), 1 (2023). https://doi.org/10.1038/s41566-022-01137-1Currently, after nearly two decades of developments, optical metasurfaces can provide essentially arbitrary control over the amplitude, phase, and polarization of light, serving as ultracompact elements for holographic devices, optical sources and detectors, sensors, imaging systems, and components for optical communication and information processing.8–148. A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “ Planar photonics with metasurfaces,” Science 339(6125), 1232009 (2013). https://doi.org/10.1126/science.12320099. N. Yu and F. Capasso, “ Flat optics with designer metasurfaces,” Nat. Mater. 13(2), 139–150 (2014). https://doi.org/10.1038/nmat383910. H.-T. Chen, A. J. Taylor, and N. Yu, “ A review of metasurfaces: Physics and applications,” Rep. Prog. Phys. 79(7), 076401 (2016). https://doi.org/10.1088/0034-4885/79/7/07640111. H.-H. Hsiao, C. H. Chu, and D. P. Tsai, “ Fundamentals and applications of metasurfaces,” Small Methods 1(4), 1600064 (2017). https://doi.org/10.1002/smtd.20160006412. O. Quevedo-Teruel, H. Chen, A. Díaz-Rubio, G. Gok, A. Grbic, G. Minatti, E. Martini, S. Maci, G. V. Eleftheriades, M. Chen et al., “ Roadmap on metasurfaces,” J. Opt. 21(7), 073002 (2019). https://doi.org/10.1088/2040-8986/ab161d13. A. Vaskin, R. Kolkowski, A. F. Koenderink, and I. Staude, “ Light-emitting metasurfaces,” Nanophotonics 8(7), 1151–1198 (2019). https://doi.org/10.1515/nanoph-2019-011014. C.-W. Qiu, T. Zhang, G. Hu, and Y. Kivshar, “ Quo vadis, metasurfaces?,” Nano Lett. 21(13), 5461–5474 (2021). https://doi.org/10.1021/acs.nanolett.1c00828 In nonlinear optics, metasurfaces provide a platform for frequency conversion, production of entangled photon pairs, all-optical switching and modulation, phase conjugation, optical limiting, saturable absorption, etc.15–2015. A. E. Minovich, A. E. Miroshnichenko, A. Y. Bykov, T. V. Murzina, D. N. Neshev, and Y. S. Kivshar, “ Functional and nonlinear optical metasurfaces,” Laser Photonics Rev. 9(2), 195–213 (2015). https://doi.org/10.1002/lpor.20140040216. G. Li, S. Zhang, and T. Zentgraf, “ Nonlinear photonic metasurfaces,” Nat. Rev. Mater. 2(5), 17010 (2017). https://doi.org/10.1038/natrevmats.2017.1017. A. Krasnok, M. Tymchenko, and A. Alù, “ Nonlinear metasurfaces: A paradigm shift in nonlinear optics,” Mater. Today 21(1), 8–21 (2018). https://doi.org/10.1016/j.mattod.2017.06.00718. S. Keren-Zur, L. Michaeli, H. Suchowski, and T. Ellenbogen, “ Shaping light with nonlinear metasurfaces,” Adv. Opt. Photonics 10(1), 309–353 (2018). https://doi.org/10.1364/AOP.10.00030919. G. Grinblat, “ Nonlinear dielectric nanoantennas and metasurfaces: Frequency conversion and wavefront control,” ACS Photonics 8(12), 3406–3432 (2021). https://doi.org/10.1021/acsphotonics.1c0135620. P. R. Sharapova, S. S. Kruk, and A. S. Solntsev, “ Nonlinear dielectric nanoresonators and metasurfaces: Toward efficient generation of entangled photons,” Laser Photonics Rev. 2023, 2200408. https://doi.org/10.1002/lpor.202200408 However, despite numerous experimental demonstrations, nonlinear metasurfaces are still far from wide commercial use, and many of their intriguing fundamental properties and potential functions have not yet been fully explored.Recent developments in the area of nonlocal metasurfaces are particularly interesting. It has been shown that collective nonlinear optical response of metasurfaces can be tailored by spatially extended resonances. The nonlocal optical modes associated with these resonances enter an advanced level of sophistication and make use of the concepts of parity–time (PT) symmetry,21–2321. R. Fleury, D. L. Sounas, and A. Alù, “ Negative refraction and planar focusing based on parity-time symmetric metasurfaces,” Phys. Rev. Lett. 113(2), 023903 (2014). https://doi.org/10.1103/PhysRevLett.113.02390322. F. Monticone, C. A. Valagiannopoulos, and A. Alù, “ Parity-time symmetric nonlocal metasurfaces: All-angle negative refraction and volumetric imaging,” Phys. Rev. X 6(4), 041018 (2016). https://doi.org/10.1103/PhysRevX.6.04101823. A. Cerjan, A. Raman, and S. Fan, “ Exceptional contours and band structure design in parity-time symmetric photonic crystals,” Phys. Rev. Lett. 116(20), 203902 (2016). https://doi.org/10.1103/PhysRevLett.116.203902 bound states in the continuum (BICs),24,2524. C. W. Hsu, B. Zhen, J. Lee, S.-L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “ Observation of trapped light within the radiation continuum,” Nature 499(7457), 188–191 (2013). https://doi.org/10.1038/nature1228925. C. W. Hsu, B. Zhen, A. D. Stone, J. D. Joannopoulos, and M. Soljačić, “ Bound states in the continuum,” Nat. Rev. Mater. 1(9), 16048 (2016). https://doi.org/10.1038/natrevmats.2016.48 and topological robustness.26,2726. J. Jin, X. Yin, L. Ni, M. Soljačić, B. Zhen, and C. Peng, “ Topologically enabled ultrahigh-Q guided resonances robust to out-of-plane scattering,” Nature 574(7779), 501–504 (2019). https://doi.org/10.1038/s41586-019-1664-727. X. Yin, J. Jin, M. Soljačić, C. Peng, and B. Zhen, “ Observation of topologically enabled unidirectional guided resonances,” Nature 580(7804), 467–471 (2020). https://doi.org/10.1038/s41586-020-2181-4 Nontrivial engineering of collective resonances can also involve effects that are unique to nonlinear optical interactions, such as nonlinear geometric phase,2828. A. Karnieli, S. Trajtenberg-Mills, G. Di Domenico, and A. Arie, “ Experimental observation of the geometric phase in nonlinear frequency conversion,” Optica 6(11), 1401–1405 (2019). https://doi.org/10.1364/OPTICA.6.001401 nonreciprocal parametric gain and loss,2929. M.-A. Miri and A. Alù, “ Nonlinearity-induced PT-symmetry without material gain,” New J. Phys. 18(6), 065001 (2016). https://doi.org/10.1088/1367-2630/18/6/065001 exotic symmetries of nonlinear susceptibility tensors,3030. R. Kolkowski, J. Szeszko, B. Dwir, E. Kapon, and J. Zyss, “ Non-centrosymmetric plasmonic crystals for second-harmonic generation with controlled anisotropy and enhancement,” Laser Photonics Rev. 10(2), 287–298 (2016). https://doi.org/10.1002/lpor.201500212 bistability,3131. Z. Geng, K. J. H. Peters, A. A. P. Trichet, K. Malmir, R. Kolkowski, J. M. Smith, and S. R. K. Rodriguez, “ Universal scaling in the dynamic hysteresis, and non-Markovian dynamics, of a tunable optical cavity,” Phys. Rev. Lett. 124(15), 153603 (2020). https://doi.org/10.1103/PhysRevLett.124.153603 and time-varying optical properties.3232. A. M. Shaltout, V. M. Shalaev, and M. L. Brongersma, “ Spatiotemporal light control with active metasurfaces,” Science 364(6441), eaat3100 (2019). https://doi.org/10.1126/science.aat3100 Combining them with novel material platforms, such as multi-quantum well nanostructures,3333. J. Lee, M. Tymchenko, C. Argyropoulos, P.-Y. Chen, F. Lu, F. Demmerle, G. Boehm, M.-C. Amann, A. Alù, and M. A. Belkin, “ Giant nonlinear response from plasmonic metasurfaces coupled to intersubband transitions,” Nature 511(7507), 65–69 (2014). https://doi.org/10.1038/nature13455 Dirac/Weyl semimetals,3434. L. Wu, S. Patankar, T. Morimoto, N. L. Nair, E. Thewalt, A. Little, J. G. Analytis, J. E. Moore, and J. Orenstein, “ Giant anisotropic nonlinear optical response in transition metal monopnictide Weyl semimetals,” Nat. Phys. 13(4), 350–355 (2017). https://doi.org/10.1038/nphys3969 quantum dot-2D material hybrid systems,3535. H. Hong, C. Wu, Z. Zhao, Y. Zuo, J. Wang, C. Liu, J. Zhang, F. Wang, J. Feng, H. Shen et al., “ Giant enhancement of optical nonlinearity in two-dimensional materials by multiphoton-excitation resonance energy transfer from quantum dots,” Nat. Photonics 15(7), 510–515 (2021). https://doi.org/10.1038/s41566-021-00801-2 and epsilon-near-zero materials,3636. O. Reshef, I. De Leon, M. Z. Alam, and R. W. Boyd, “ Nonlinear optical effects in epsilon-near-zero media,” Nat. Rev. Mater. 4(8), 535–551 (2019). https://doi.org/10.1038/s41578-019-0120-5 could ultimately yield nonlinear metasurfaces of unprecedented properties and superior performance. In this Perspective, we highlight selected hot topics in the field of nonlinear metasurfaces related to their possibly nonlocal optical responses. We provide a brief overview of the basic concepts underlying the optical properties of such metasurfaces and introduce their most important nonlinear functionalities. The discussions contain our personal viewpoints on the state of the art and the most interesting current and future research directions in the realm of nonlinear nonlocal metasurfaces.

II. METASURFACES—BASIC PRINCIPLES

Section:

ChooseTop of pageABSTRACTI. INTRODUCTIONII. METASURFACES—BASIC PR... <<III. NONLINEAR AND NONLOC...IV. OPPORTUNITIES AND CHA...V. CONCLUSIONThe nonlinear optical functionalities of photonic structures are tailored by the choice of the constituent materials determined by their nonlinear susceptibilities. However, micro- and nanostructuring affects both the linear and nonlinear optical properties such that the nonlinear response can become largely governed by the linear response.37,3837. S. Roke, M. Bonn, and A. V. Petukhov, “ Nonlinear optical scattering: The concept of effective susceptibility,” Phys. Rev. B 70(11), 115106 (2004). https://doi.org/10.1103/PhysRevB.70.11510638. K. O'Brien, H. Suchowski, J. Rho, A. Salandrino, B. Kante, X. Yin, and X. Zhang, “ Predicting nonlinear properties of metamaterials from the linear response,” Nat. Mater. 14(4), 379–383 (2015). https://doi.org/10.1038/nmat4214 In periodic metasurfaces, the main factors that determine their optical responses are the polarizabilities and spatial arrangement of the meta-atoms. In this section, we provide a brief overview of the main concepts underlying the nonlocal optical response of periodic metasurfaces.

A. General classification

In simple terms, metasurfaces can be classified as either diffractive or subdiffractive, depending on the distance between the meta-atoms relative to the wavelength. On the other hand, their local or nonlocal character is determined by the strength of multiple scattering between nearby meta-atoms, which depends on the polarizability of the meta-atoms and surface confinement of the scattered light. This simplified classification is graphically summarized in Fig. 1. Similar classifications of metasurfaces have been proposed also previously, e.g., in the recent review by Overvig and Alú,3939. A. Overvig and A. Alù, “ Diffractive nonlocal metasurfaces,” Laser Photonics Rev. 16(8), 2100633 (2022). https://doi.org/10.1002/lpor.202100633 where, in contrast to our classification, diffractive metasurfaces that are local are not considered. Here, we prefer to consider the nonlocality and the ability of the structure to diffract light as two independent degrees of freedom. For example, nonlocal/collective excitations, such as Fourier lattice resonances, can be realized in multipartite or even aperiodic structures,4040. T. L. Lim, Y. Vaddi, M. S. Bin-Alam, L. Cheng, R. Alaee, J. Upham, M. J. Huttunen, K. Dolgaleva, O. Reshef, and R. W. Boyd, “ Fourier-engineered plasmonic lattice resonances,” ACS Nano 16(4), 5696–5703 (2022). https://doi.org/10.1021/acsnano.1c10710 which at least in the conventional sense are not diffractive. Moreover, some diffractive structures can respond to the incident light either locally or nonlocally depending on the excitation conditions, as we explain further in the upcoming sections.

B. Polarizability of meta-atoms

In general, the optical polarizability of a particle, α, can be a complex tensorial quantity, with possible contributions of magnetoelectric coupling,4141. I. Sersic, C. Tuambilangana, T. Kampfrath, and A. F. Koenderink, “ Magnetoelectric point scattering theory for metamaterial scatterers,” Phys. Rev. B 83(24), 245102 (2011). https://doi.org/10.1103/PhysRevB.83.245102 higher-order multipolar terms,4242. P. Grahn, A. Shevchenko, and M. Kaivola, “ Electromagnetic multipole theory for optical nanomaterials,” New J. Phys. 14(9), 093033 (2012). https://doi.org/10.1088/1367-2630/14/9/093033 optical gain and loss,4343. R. Kolkowski and A. F. Koenderink, “ Lattice resonances in optical metasurfaces with gain and loss,” Proc. IEEE 108(5), 795–818 (2020). https://doi.org/10.1109/JPROC.2019.2939396 or even temporal modulation.4444. M. S. Mirmoosa, T. Koutserimpas, G. Ptitcyn, S. A. Tretyakov, and R. Fleury, “ Dipole polarizability of time-varying particles,” New J. Phys. 24, 063004 (2022). https://doi.org/10.1088/1367-2630/ac6b4c Among these contributions, the higher-order multipoles have been a subject of extensive research related to tailoring and enhancing of the nonlinear optical properties of nanostructured materials through localized plasmonic45–4745. S. Kujala, B. K. Canfield, M. Kauranen, Y. Svirko, and J. Turunen, “ Multipole interference in the second-harmonic optical radiation from gold nanoparticles,” Phys. Rev. Lett. 98(16), 167403 (2007). https://doi.org/10.1103/PhysRevLett.98.16740346. J. Butet, I. Russier-Antoine, C. Jonin, N. Lascoux, E. Benichou, and P.-F. Brevet, “ Sensing with multipolar second harmonic generation from spherical metallic nanoparticles,” Nano Lett. 12(3), 1697–1701 (2012). https://doi.org/10.1021/nl300203u47. J. Butet, P.-F. Brevet, and O. J. F. Martin, “ Optical second harmonic generation in plasmonic nanostructures: From fundamental principles to advanced applications,” ACS Nano 9(11), 10545–10562 (2015). https://doi.org/10.1021/acsnano.5b04373 and Mie resonances.48–5248. D. Smirnova and Y. S. Kivshar, “ Multipolar nonlinear nanophotonics,” Optica 3(11), 1241–1255 (2016). https://doi.org/10.1364/OPTICA.3.00124149. S. S. Kruk, R. Camacho-Morales, L. Xu, M. Rahmani, D. A. Smirnova, L. Wang, H. H. Tan, C. Jagadish, D. N. Neshev, and Y. S. Kivshar, “ Nonlinear optical magnetism revealed by second-harmonic generation in nanoantennas,” Nano Lett. 17(6), 3914–3918 (2017). https://doi.org/10.1021/acs.nanolett.7b0148850. L. Wang, S. Kruk, L. Xu, M. Rahmani, D. Smirnova, A. Solntsev, I. Kravchenko, D. Neshev, and Y. Kivshar, “ Shaping the third-harmonic radiation from silicon nanodimers,” Nanoscale 9(6), 2201–2206 (2017). https://doi.org/10.1039/C6NR09702B51. E. V. Melik-Gaykazyan, S. S. Kruk, R. Camacho-Morales, L. Xu, M. Rahmani, K. Z. Kamali, A. Lamprianidis, A. E. Miroshnichenko, A. A. Fedyanin, D. N. Neshev et al., “ Selective third-harmonic generation by structured light in Mie-resonant nanoparticles,” ACS Photonics 5(3), 728–733 (2018). https://doi.org/10.1021/acsphotonics.7b0127752. J. D. Sautter, L. Xu, A. E. Miroshnichenko, M. Lysevych, I. Volkovskaya, D. A. Smirnova, R. Camacho-Morales, K. Zangeneh Kamali, F. Karouta, K. Vora et al., “ Tailoring second-harmonic emission from (111)-GaAs nanoantennas,” Nano Lett. 19(6), 3905–3911 (2019). https://doi.org/10.1021/acs.nanolett.9b01112In the simplest case, a meta-atom can be considered as a point electric dipole with a moment p=α·Ein induced by an incoming field (where we used a convention in which ε0—required for the compliance with the SI units—is already included in α). The dipole radiates a secondary field Esc(r′)=G(r,r′)·p(r) shaped by the dyadic Green's function G(r,r′). Since the energy is radiated away, α is always complex, even if meta-atoms are made of a lossless dielectric (unless the radiative loss is exactly compensated for by gain). This imposes fundamental constraints on light manipulation by nanoscatterers, e.g., limiting their magnetoelectric response.5353. P. A. Belov, S. I. Maslovski, K. R. Simovski, and S. A. Tretyakov, “ A condition imposed on the electromagnetic polarizability of a bianisotropic lossless scatterer,” Tech. Phys. Lett. 29(9), 718–720 (2003). https://doi.org/10.1134/1.1615545In fact, higher-order multipoles are always excited in meta-atoms either by near-fields of neighboring meta-atoms or due to phase delays of the excitation field when it propagates through spatially extended meta-atoms. Usually, however, higher-order multipoles only weakly contribute to light scattering. The most significant of them are electric quadrupoles and magnetic dipoles, of which the latter are excited by the electric rather than magnetic field at optical frequencies. The magnetic response of meta-atoms can formally be described by introducing their magnetic, magnetoelectric, and electromagnetic polarizabilities (αmm, αme, and αem, respectively), as for bianisotropic metamaterials and metasurfaces.5454. V. S. Asadchy, A. Díaz-Rubio, and S. A. Tretyakov, “ Bianisotropic metasurfaces: Physics and applications,” Nanophotonics 7(6), 1069–1094 (2018). https://doi.org/10.1515/nanoph-2017-0132 The electric and magnetic dipole moments p and m are in this case given by p=αee·E+αem·H,m=αme·E+αmm·H,(1)where αee is the electric polarizability denoted above by α. Bianisotropy can result from a nonlocal response within each unit cell of the structure or from the meta-atomic excitations by the fields of neighboring meta-atoms, manifesting the phenomenon of spatial dispersion. Since usually, structural units of metamaterials and metasurfaces have non-trivial geometries, and for example, are fabricated on a substrate, spatial dispersion leads to the dependence of the response on the light propagation direction.55–5855. B. Gompf, B. Krausz, B. Frank, and M. Dressel, “ k-dependent optics of nanostructures: Spatial dispersion of metallic nanorings and split-ring resonators,” Phys. Rev. B 86(7), 075462 (2012). https://doi.org/10.1103/PhysRevB.86.07546256. P. Grahn, A. Shevchenko, and M. Kaivola, “ Interferometric description of optical metamaterials,” New J. Phys. 15(11), 113044 (2013). https://doi.org/10.1088/1367-2630/15/11/11304457. V. Kivijärvi, M. Nyman, A. Karrila, P. Grahn, A. Shevchenko, and M. Kaivola, “ Interaction of metamaterials with optical beams,” New J. Phys. 17(6), 063019 (2015). https://doi.org/10.1088/1367-2630/17/6/06301958. V. Kivijärvi, M. Nyman, A. Shevchenko, and M. Kaivola, “ An optical metamaterial with simultaneously suppressed optical diffraction and surface reflection,” J. Opt. 18(3), 035103 (2016). https://doi.org/10.1088/2040-8978/18/3/035103 This type of nonlocality, however, is effectively localized, since contribution of neighboring meta-atoms to the excitation weakens with the distance to them. In contrast, metasurfaces that we call nonlocal in this work have an effectively much larger range of nonlocality, because, in them, coupling between meta-atoms is provided by optical modes that are guided or diffracted along the metasurface. Higher-order multipoles can also be excited in such a metasurface, but they are not required for nonlocality. Furthermore, since higher-order multipoles are rather dark compared to electric dipoles, we can ignore them for simplicity of the description. In the next sections, multiple scattering of meta-atoms is described using the electric-dipole approximation.

C. Local response

If multiple scattering between meta-atoms is negligible (e.g., due to small scattering cross sections), then each meta-atom can be regarded as an independent source of a secondary field Esc. In this case, the optical response of the metasurface is governed by the far-field interference of Esc (Huygens principle). This approximation is commonly used in designing subdiffractive metasurfaces that behave as homogeneous phase-shifting interfaces obeying a modified Snell's law.5959. N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “ Light propagation with phase discontinuities: Generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011). https://doi.org/10.1126/science.1210713

D. Diffractive and subdiffractive metasurfaces

In the diffractive regime, the periodicity of the structures enables constructive interference of Esc in certain directions, which produces diffraction orders. In the parameter space spanned by frequency ω and in-plane momentum k||, the diffraction orders emerge at Rayleigh anomalies governed by condition |k||+K|=nω/c, where K represents the reciprocal lattice vectors and n is the refractive index of the embedding medium. This can be illustrated using the repeated zone scheme (Fig. 2), in which the dispersion cones |k|||=nω/c are repeated at the reciprocal lattice nodes K, forming the backbone of photonic band structure, known as the empty-lattice approximation.60,6160. N. W. Ashcroft and N. D. Mermin, Solid State Physics ( Cengage Learning, Boston, MA, 2011).61. C. Cherqui, M. R. Bourgeois, D. Wang, and G. C. Schatz, “ Plasmonic surface lattice resonances: Theory and computation,” Acc. Chem. Res. 52(9), 2548–2558 (2019). https://doi.org/10.1021/acs.accounts.9b00312Rayleigh anomalies are associated with a resonant response due to propagation of diffracted waves along the metasurface plane. Under such conditions, it is impossible to neglect multiple scattering between meta-atoms, which makes the optical response inherently nonlocal.3939. A. Overvig and A. Alù, “ Diffractive nonlocal metasurfaces,” Laser Photonics Rev. 16(8), 2100633 (2022). https://doi.org/10.1002/lpor.202100633 However, diffractive optical elements (such as diffraction gratings) may, in general, operate away from Rayleigh anomalies. In such a case, local response and paraxial approximation are sufficient to describe their optical properties.In addition to Rayleigh anomalies, resonant response can result from the propagation and multiple scattering of surface waves. A well-known example is associated with excitation of surface plasmon polaritons (SPPs) in periodic metal structures, which gives rise to the so-called Wood anomalies.6262. A. Hessel and A. A. Oliner, “ A new theory of Wood's anomalies on optical gratings,” Appl. Opt. 4(10), 1275–1297 (1965). https://doi.org/10.1364/AO.4.001275 Resonant excitation of SPP standing waves is responsible for the famous extraordinary optical transmission through periodic arrays of holes in metal films.6363. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “ Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391(6668), 667–669 (1998). https://doi.org/10.1038/35570 Due to the phase contribution from the complex-valued SPP propagation constants, Wood anomalies are often characterized by asymmetric spectra. They can also be distinguished from Rayleigh anomalies by their distinct polarization response: as opposed to waves scattered by nanoparticles in a dielectric medium, SPPs are launched along the metal surface in the direction parallel to the incident polarization.30,6430. R. Kolkowski, J. Szeszko, B. Dwir, E. Kapon, and J. Zyss, “ Non-centrosymmetric plasmonic crystals for second-harmonic generation with controlled anisotropy and enhancement,” Laser Photonics Rev. 10(2), 287–298 (2016). https://doi.org/10.1002/lpor.20150021264. J.-S. Bouillard, P. Segovia, W. Dickson, G. A. Wurtz, and A. V. Zayats, “ Shaping plasmon beams via the controlled illumination of finite-size plasmonic crystals,” Sci. Rep. 4(1), 7234 (2014). https://doi.org/10.1038/srep07234

E. Multiple scattering and collective resonances

Multiple scattering may become significant if |α| is large. For example, the scattering cross sections of resonant plasmonic nanoparticles may extend beyond the unit cell even in diffractive lattices. The coupling between meta-atoms can also be enhanced at Rayleigh anomalies, Wood anomalies, or through guided modes in waveguide structures.65,6665. A. Christ, S. G. Tikhodeev, N. A. Gippius, J. Kuhl, and H. Giessen, “ Waveguide-plasmon polaritons: Strong coupling of photonic and electronic resonances in a metallic photonic crystal slab,” Phys. Rev. Lett. 91(18), 183901 (2003). https://doi.org/10.1103/PhysRevLett.91.18390166. S. R. K. Rodriguez, S. Murai, M. A. Verschuuren, and J. G. Rivas, “ Light-emitting waveguide-plasmon polaritons,” Phys. Rev. Lett. 109(16), 166803 (2012). https://doi.org/10.1103/PhysRevLett.109.166803 A nonlocal interaction between meta-atoms can be regarded as an effective modification of their local electric-dipole response,6767. F. J. Garcia De Abajo, “ Colloquium: Light scattering by particle and hole arrays,” Rev. Mod. Phys. 79(4), 1267 (2007). https://doi.org/10.1103/RevModPhys.79.1267 pi=αi·Ein+αi·∑j≠iGij·pj=αeff,i·Ein.(2)In the above expression, the dipole moment of meta-atom i is induced by Ein and by the fields scattered by all other particles j. These scattered fields modify the polarizability of meta-atom i with respect to Ein from αi to αeff,i. In periodic metasurfaces, effective polarizabilities of meta-atoms are the same (in accordance with Bloch's theorem), but they depend on the in-plane momentum k|| and the corresponding phase difference between the neighboring unit cells, αeff(k||)=[α−1−Glattice(k||)]−1.(3)In general, lattices may contain many (identical or different) meta-atoms per unit cell, which turns Eq. (3) into a matrix equation. In such a case, α and αeff are block-diagonal matrices, and Glattice—the so-called lattice Green's function—describes the interactions between meta-atoms within the same sublattice (diagonal blocks) and between different sublattices (off-diagonal blocks).43,68,6943. R. Kolkowski and A. F. Koenderink, “ Lattice resonances in optical metasurfaces with gain and loss,” Proc. IEEE 108(5), 795–818 (2020). https://doi.org/10.1109/JPROC.2019.293939668. A. Kwadrin and A. Femius Koenderink, “ Diffractive stacks of metamaterial lattices with a complex unit cell: Self-consistent long-range bianisotropic interactions in experiment and theory,” Phys. Rev. B 89(4), 045120 (2014). https://doi.org/10.1103/PhysRevB.89.04512069. R. Kolkowski, S. Kovaios, and A. F. Koenderink, “ Pseudochirality at exceptional rings of optical metasurfaces,” Phys. Rev. Res. 3(2), 023185 (2021). https://doi.org/10.1103/PhysRevResearch.3.023185 Such complex metasurfaces can be designed to operate in either diffractive or subdiffractive regime, especially when the distance between meta-atoms d is smaller than λ, while the lattice period Λ is larger than λ.An important deviation from the description of nonlocal metasurfaces as perfectly periodic systems comes from the finite size effects, which should be carefully considered in any practical implementation. With any (finite) number of meta-atoms, the periodic boundary conditions of the scattered fields break down. Thus, the overall amplitude and phase of the field scattered by each meta-atom will become dependent on its position in the lattice.70,7170. L. Zundel and A. Manjavacas, “ Finite-size effects on periodic arrays of nanostructures,” J. Phys. Photonics 1(1), 015004 (2018). https://doi.org/10.1088/2515-7647/aae8a271. B. O. Asamoah, M. Nečada, W. Liu, J. Heikkinen, S. Mohamed, A. Halder, H. Rekola, M. Koivurova, A. I. Väkeväinen, P. Törmä et al., “ Finite size mediated radiative coupling of lasing plasmonic bound state in continuum,” arXiv:2206.05011 (2022).In subdiffractive metasurfaces, multiple scattering between periodically arranged meta-atoms may give rise to collective modes that are confined within the metasurface due to a momentum mismatch with freely propagating waves. The properties of such modes are strongly dependent on the properties of α,7272. P. Lunnemann and A. F. Koenderink, “ Dispersion of guided modes in two-dimensional split ring lattices,” Phys. Rev. B 90(24), 245416 (2014). https://doi.org/10.1103/PhysRevB.90.245416 which distinguishes them from the conventional Bloch modes of photonic crystals resulting from spatial modulation of the refractive index.In the diffractive regime, the metasurface modes and the Rayleigh anomalies appear in the light cone. As a result, the “empty lattice” backbone (Fig. 2) is imprinted on the