Abstract
Metasurfaces offer great potential to control near and farfields through engineering optical properties of elementary cells or metaatoms. Such perspective opens a route to efficient manipulation of the optical signals both at nanoscale and in photonics applications. In this paper we show that a local surface conductivity tensor well describes optical properties of a resonant plasmonic hyperbolic metasurface both in the farfield and in the nearfield regimes, where spatial dispersion usually plays a crucial role. We retrieve the effective surface conductivity tensor from the comparative analysis of experimental and numerical reflectance spectra of a metasurface composed of elliptical gold nanoparticles. Afterwards, the restored conductivities are validated by semianalytic parameters obtained with the nonlocal discrete dipole model with and without interaction contribution between metaatoms. The effective parameters are further used for the dispersion analysis of surface plasmons localized at the metasurface. The obtained effective conductivity describes correctly the dispersion law of both quasiTE and quasiTM plasmons in a wide range of optical frequencies as well as the peculiarities of their propagation regimes, in particular, topological transition from the elliptical to hyperbolic regime with eligible accuracy. The analysis in question offers a simple practical way to describe properties of metasurfaces including ones in the nearfield zone with effective conductivity tensor extracting from the convenient farfield characterization.
Introduction
Miniaturization of integrated optical circuits requires an effective control of light on the subwavelength scale. Significant advances in this field have been achieved with the help of metamaterials^{1,2,3} – artificially created media, whose electromagnetic properties can drastically differ from the properties of the natural materials. However, a threedimensional structure of metamaterials, related fabrication challenges and high costs, especially for optical applications, form significant obstacles for their implementation in integrated optical circuits.
An alternative way is to use metasurfaces – twodimensional analogues of metamaterials. There are also natural twodimensional anisotropic materials such as hexagonal boron nitride^{4,5}, transition metal dichalcogenides^{6,7}, black phosphorus^{8}. In the visible and the nearIR range, metasurfaces can be implemented using subwavelength periodic arrays of plasmonic or highindex dielectric nanoparticles^{9,10,11,12}. A nanostructured graphene could also be considered as a metasurface for THz frequencies^{13,14}. In the microwave range, metasurfaces can be implemented by using LCcircuits, splitring resonators, arrays of capacitive and inductive elements (strips, grids, mushrooms), wire medium etc^{15,16}. Despite subwavelength or even monoatomic thicknesses, the metasurfaces offer unprecedented control over light propagation, reflection and refraction^{15,17}.
Metasurfaces exhibit a lot of intriguing properties for a wide area of applications such as nearfield microscopy, imaging, holography, biosensing, photovoltaics etc.^{10,15,16,17,18,19}. For instance, it was shown that metasurfaces based on Si nanoparticles can exhibit nearly 100% reflectance^{20} and transmittance^{21} in a broadband frequency range. Moreover, metasurfaces can serve as light control elements: frequency selectors, antennas, lenses, perfect absorbers^{16}. They offer an excellent functionality with polarization conversion, beam shaping and optical vortices generation^{22,23}. Besides, metasurfaces provide an efficient control over dispersion and polarization of surface waves^{24,25,26,27,28,29,30}. Surface plasmonpolaritons propagating along a metasurface assist pushing, pulling and lateral optical forces in its vicinity^{31,32}. Metasurfaces are prospective tools for spincontrolled optical phenomena^{33,34,35,36} and holographic applications^{37,38,39}. Particular attention is paid to hyperbolic metasurfaces, which are promising tools for many applications including the manipulation over surface waves^{26,40,41,42}. The main advantages of metasurfaces, such as relative manufacturing simplicity, rich functionality and planar geometry, fully compatible with modern fabrication technologies, create a promising platform for the photonic metadevices. It has been recently pointed out that alldielectric metasurfaces and metamaterials can serve as a prospective lowloss platform, which could replace plasmonic structures^{43}. However, one of the main advantages of plasmonic structures, unachievable with dielectric ones, is that the plasmonic structures can be resonant in the visible range keeping at the same time a deep subwavelength thickness and period. Thus, here we concentrate on plasmonic metasurfaces allowing light manipulation with a deep subwavelength structure.
The common feature of bulk metamaterials and metasurfaces is that due to the subwavelength structure they can be considered as homogenized media described by effective material parameters. For bulk metamaterials, such effective parameters are permittivity \({\hat{\varepsilon }}_{{\rm{eff}}}\) and/or permeability \({\hat{\mu }}_{{\rm{eff}}}\). Retrieving effective parameters is one of the most important problems in the study of metamaterials. Generally, the effective parameters are tensorial functions of frequency ω, wavevector k, and intensity I. Homogenization of micro and nanostructured metamaterials can become rather cumbersome, especially taking into account nonlocality^{44,45,46}, chirality^{47}, bianisotropy^{48,49} and nonlinearity^{50,51}.
Analogous homogenization procedures are relevant for metasurfaces. Apparently, homogenization procedures for 2D structures were firstly developed in radiophysics and microwaves (equivalent surface impedance) in applications to thin films, highimpedance surfaces and wire grids etc^{52,53,54}. It has been recently pointed out that twodimensional structures, like graphene, silicene and metasurfaces, can be described within an effective conductivity approach^{24,25,55,56,57,58}. In virtue of a subwavelength thickness, a metasurface could be considered as a twodimensional equivalent current and, therefore, characterized by effective electric \({\hat{\sigma }}_{e}(\omega ,{{\bf{k}}}_{\tau })\) and magnetic \({\hat{\sigma }}_{m}(\omega ,{{\bf{k}}}_{\tau })\) surface conductivity tensors, where k_{τ} is the component of the wavevector in the plane of the metasurface^{15,16}. Importantly, such effective surface conductivity describes the properties of the metasurface both in the farfield when \({{\bf{k}}}_{\tau } < \omega /c\) (reflection, absorption, refraction, polarization transformation etc.) and in the nearfield (surface waves, Purcell effect, optical forces), when \({{\bf{k}}}_{\tau } > \omega /c\).
In this paper, we focus our study on a resonant plasmonic anisotropic metasurface represented by a twodimensional periodic array of gold nanodisks with the elliptical base. We derive and analyze the electric surface conductivity tensor of the anisotropic metasurface in three ways: (i) numerically by combining the optical measurements of the fabricated metasurface, simulations of the experiment and analytical approach (zerothickness approximation); (ii) experimentally by characterization of the metasurface reflection spectra and a subsequent fitting with DrudeLorentz formula; and (iii) theoretically by using the nonlocal discrete dipole model. We reveal that the effective surface conductivity tensor extracted from the farfield measurements well describes nearfield properties of metasurface such as the spectrum of surface waves and their behaviour in all possible regimes  capacitive, inductive, and hyperbolic. By using the discrete dipole model we study the effects of spatial dispersion on the eigenmodes spectrum and define the limitations of the effective model applicability.
Sample Design and Fabrication
We consider a metasurface composed of gold anisotropic nanoparticles placed on a fused silica substrate. The design of the sample is shown in Fig. 1. The target structure consists of 20 nm thick gold nanodisks with the elliptical base packed in the square lattice with a period of 200 nm. The average long and short axes of the disks are a_{x} = 134 nm and a_{y} = 103 nm, respectively. The distribution of the nanodisks sizes is provided in Fig. S1 (See Supplementary Information S1).
The sample was fabricated via electron beam lithography on a fused silica substrate. Before the electron beam exposure process, the resist layer (PMMA) was covered with a thin gold layer to prevent local charge accumulation. After the exposure, a 20 nm thick gold layer was sputtered via thermal evaporation. During the last step of the fabrication process, the remains of the resist were removed via the liftoff procedure. Finally, the sample was immersed in a liquid with a refractive index nearly matching the glass substrate. Thus, we obtained the metasurface with a homogeneous ambient medium with permittivity ε = 2.1. The SEM image of the fabricated sample is shown in Fig. 1a.
Effective Conductivity Tensor
The plasmonic resonant metasurface shown in Fig. 1 is anisotropic and nonchiral. Asymmetry of each particle splits its inplane dipole plasmonic resonance with frequency Ω into two resonances with frequencies Ω_{⊥} and Ω_{}^{24,29}. Consideration of metasurfaces as an absolutely flat object might be restricted due to the emergence of the outofplane polarizability caused by the finite thickness of the plasmonic particles. In our case, the outofplane polarizability α_{z} could be neglected due to a small thickness of the particles as it is shown in Fig. S2 (See Supplementary Information S2). Therefore, this metasurface can be described by a twodimensional effective surface conductivity tensor, which is diagonal in the principal axes (when the axes of the coordinates system are parallel to the axes of the elliptical base of the nanodisks).
Numerical vs Experimental Characterization
To extract the effective surface conductivity of the fabricated sample, we apply a procedure based on the combination of the optical experiments, numerical simulations and theoretical calculations.
First, we measure the intensity of the reflectance for the light polarized along and across the principle axes of the metasurface under normal incidence (Fig. 2a). Both spectra demonstrate single peaks corresponding to the individual localized plasmon resonances of the nanodisks. The phase retrieved by the fitting of the experimental reflectance with the intensity calculated by the use of the Drude formula (See Supplementary Information S3) is shown in Fig. 2a by the red lines.
Then, we model the experiment with CST Microwave Studio (Fig. 2b). The difference in the intensity of the peaks in Fig. 2a,b can be attributed to roughness and inhomogeneity of the sample. We obtain similar values of reflectance spectra from the experiment and simulation by increasing the imaginary part of the gold permittivity in the simulation (See Supplementary Information S3). The minor shift of the resonance position can be associated with a small difference between the immersion liquid and fused silica substrate permittivities (See Supplementary Information S7). In order to retrieve a complex conductivity tensor we take the intensity and the phase of reflection coefficient from the simulation, wherein we obtain good matching between the simulated and experimental shapes of the reflectance spectra (Fig. 2a,b).
Basing on the calculated complex reflection coefficient we find an effective surface conductivity using the zerothickness approximation (ZTA). Within this approximation we replace the real structure of finite thickness H by the effective twodimensional plane disposed at distance H/2 from the substrate. This technique can be applied only for deeply subwavelength structures. The limitation can be formulated as H/λ ≪ 1 according to the NicolsonRossWeir method^{59,60}.
Considering a twodimensional layer with effective conductivity σ sandwiched between two media with refractive indices n_{1} (superstrate) and n_{2} (substrate) one can find Fresnel’s coefficients^{55,61,62} and express the effective surface conductivity as follows
where \({S}_{11}^{x,y}\) is the component of the Smatrix. Indices x, y correspond to different orientations of the electric field of the incident wave. Hereinafter we use the Gauss system of units and express surface conductivity in the dimensionless units \(\sigma =4\pi \tilde{\sigma }/c\). The proposed extraction technique may be applied even in the case of the arbitrary incident angle (See Supplementary Information S4). In this case, one can obtain the dependence of the retrieved conductivity on the inplane wavevector, which allows to define the impact of spatial dispersion.
In order to obtain the proper conductivity of a metasurface one should retain only the phase of the reflection coefficient related to the metasurface properties. In the simulation, the total phase of the Sparameters has two contributions \({\rm{\arg }}({S}_{11}^{x,y})={\rm{\Delta }}{\phi }_{1}+{\rm{\Delta }}{\phi }_{2}\). The first one arises directly when the wave reflects from the metasurface. The second phase arises because of the wave propagation from the port to the metasurface and back \({\rm{\Delta }}{\phi }_{2}=2{k}_{0}L\). Here, the time dependence is defined through the factor e^{iωt}, \({k}_{0}={n}_{1}\omega /c\), L is the distance between the excitation port and the metasurface. The problem is how to correctly determine distance L if the metasurface has a finite thickness? We found that the correct results not breaking the energy conservation law (See Supplementary Information S5) are obtained only if L is defined as the distance to the middle of the metasurface. Thus, the effective twodimensional layer has to be disposed exactly at distance H/2 from the substrate. The obvious analogue of ZTA is the transfer matrix method (TMM), which originates from Fresnel’s reflection and transmission coefficients. For the metasurface under consideration ZTA and TMM give the results with the average relative error of 1%. However, the advantage of ZTA over TMM is that it is necessary to know only one either reflection or transmission coefficient to extract the effective parameters. The effective conductivity retrieved from the farfield measurements correctly predicts the nearfield properties. In particular, it describes well the surface waves spectrum. It is possible due to the noncritical contribution of the spatial dispersion.
The extracted conductivities for both polarizations are presented in Fig. 2c. For the light wave polarized along the long axis (TMpolarization) the plasmon resonance is observed at 670 nm, while for light polarized along the short axis (TEpolarization) the resonance corresponds to 780 nm.
Discrete Dipole Model
In order to derive surface conductivity of a metasurface analytically we apply the discrete dipole model (DDM), in many works it is also called the pointdipole model. This technique has been implemented for 1D, 2D and 3D structures^{63,64,65,66,67}. Within this approach we consider a 2D periodic array of the identical scatterers as an array of point dipoles.
In the framework of the DDM it is more convenient to operate with an effective polarizability of the structure, which is straightforwardly connected to the effective conductivity tensor as follows:
In the case under consideration, the thickness of the scatterers is deeply subwavelength and, therefore, we can neglect the polarizability of the particles in the direction perpendicular to the plane of the metasurface. Thus, we can describe the metasurface by either twodimensional polarizability tensor \({\hat{\alpha }}_{{\rm{eff}}}\) or conductivity tensor \({\hat{\sigma }}_{{\rm{eff}}}\) with zero offdiagonal components (in the basis of the principal axes). Rigorous derivation of the effective polarizability of a twodimensional lattice of resonant scatterers is performed in Refs.^{49,65,68}. The effective polarizability of the metasurface can be written as
Here, \({\hat{\alpha }}_{0}(\omega )\) is the polarizability of the individual resonant scatterer, and \(\hat{C}(\omega ,{{\bf{k}}}_{{\boldsymbol{\tau }}})\) is the socalled dynamic interaction constant^{65}. The latter contains the lattice sum, which takes into account interaction of each dipole with all others. We approximate the polarizability of the disk with the elliptical base \({\hat{\alpha }}_{0}\) by the polarizability of an ellipsoid with the same volume and aspect ratio (See Supplementary Information S2). We calculate the interaction between the identical scatterers by using the Green’s function formalism:
Here \(\hat{G}(\omega ,{{\bf{r}}}_{ij})\) is the dyadic Green’s function and r_{ij} are the coordinates of the dipoles. This sum has slow convergence. So, we calculate the interaction term in Eq. (4) within the Ewald summation technique^{63,66,69,70} applied for a twodimensional periodic structure, which ensures fast convergence of the sum (See Supplementary Information S6).
The discrete dipole model can be successfully applied for many types of metasurfaces. It is applicable for twodimensional periodic structures under three main conditions:

1.
Quasistatic condition: na ≪ λ. Here n is the refractive index of the environment, a is the lattice constant, λ is the incident wavelength.

2.
Dipole approximation: f ≪ 1 (or d ≪ a, where d is the characteristic size of a scaterrer). Here f = A/a^{2} is the filling factor, A is the area occupied by the scatterer in the unit cell (in our case, A = πa_{x}a_{y}), a^{2} is the area of the square unit cell. When the scatterers are not sufficiently small one has to take into account higher order multipoles.

3.
Quasitwodimensionality: a_{zz} ≪ min(a_{xx}, a_{yy}) and H ≪ λ. This condition is achieved, when thickness of a metasurface is less than both characteristic inplane sizes of metaatoms (H < min{a_{x}, a_{y}}) and skin depth δ (H < δ).
For the metasurface sample under consideration H/a_{y} = 0.19, and f = 0.27. Although the applicability condition of the dipole approximation is poorly satisfied, the DDM gives eligible results. Parameters na/λ and H/λ lie in the interval from 0.25 to 0.75 and from 0.02 to 0.05, respectively, for wavelengths λ = 400–1200 nm. Skin depth δ for gold is around 20–40 nm in the wavelength range under consideration^{71}.
One can see in Fig. 3a,c that neglecting interaction term \(\hat{C}(\omega ,{{\bf{k}}}_{\tau })\) in Eq. (3) results in a blue shift of the conductivity spectra by several tens of nanometers for both polarizations. Accounting these interactions brings the DDM into almost perfect agreement with the ZTA (Fig. 3b,d). However, matching for σ_{y} is better than for σ_{x}. It could be explained by the fact that polarizability of an ellipsoid approximates polarizability of the elliptical disk in the y direction better that in the x direction.
The spectral dependences of the extracted surface conductivities along the principal axes are shown in Figs. 2c and 3. They clearly show that the fabricated metasurface is characterized by a highly anisotropic resonant conductivity tensor:
One can see from Fig. 2c that the metasurface supports three different regimes depending on wavelength λ of the incident light. These regimes can be classified by the signs of (i) \({\rm{\det }}[{\rm{Im}}(\hat{\sigma })]\) and (ii) \({\rm{tr}}[{\rm{Im}}(\hat{\sigma })]\). Specifically, when \({\rm{\det }}[{\rm{Im}}(\hat{\sigma })] > 0\) and \({\rm{tr}}[{\rm{Im}}(\hat{\sigma })] > 0\) (for λ < 670 nm) the inductive regime of the metasurface is observed. In this case, the metasurface corresponds to the conventional metal sheet and only a TMpolarized surface wave can propagate. For \(det[{\rm{I}}{\rm{m}}(\hat{\sigma })] > 0\) and \({\rm{tr}}[{\rm{Im}}(\hat{\sigma })] < 0\) (for λ > 780 nm) the capacitive regime of the metasurface is met, so only a TEpolarized surface wave can propagate. When \({\rm{\det }}[{\rm{Im}}(\hat{\sigma })] < 0\) (between the resonances, i.e. for wavelengths from 670 to 780 nm), a metasurface supports the socalled hyperbolic regime, in which simultaneous propagation of both TE and TMmodes is possible^{24}.
Surface Waves
The dispersion equation of the surface waves supported by an anisotropic metasurface, described by the effective conductivity tensor (5), can be straightforwardly derived from Maxwell’s equations and boundary conditions at the metasurface^{24}:
Here, σ_{ij} are the tensor components in the coordinate system rotated by angle φ (see Fig. 1b), ε_{1}, μ_{1}, κ_{1} and ε_{2}, μ_{2}, κ_{2} are the permittivity, permeability and inverse penetration depths of the wave in the superstrate and substrate, respectively. The latter is defined as \({\kappa }_{i}=\sqrt{{{{\bf{k}}}_{\tau }}^{2}{\varepsilon }_{i}{\mu }_{i}{\omega }^{2}/{c}^{2}}\), where k_{τ} is the wavevector in the plane of the metasurface. In our case Eq. (6) is simplified since we consider the metasurface in nonmagnetic (μ_{1} = μ_{2} = 1) and homogeneous environment with the permittivity corresponding to fused silica ε = ε_{1} = ε_{2} = 2.1.
The first and the second factors in the left side of Eq. (6) correspond to the dispersion of purely TEpolarized and TMpolarized surface waves, respectively. The right side of Eq. (6) is the coupling factor responsible for the mixing of TE and TM modes. If an electromagnetic wave propagates along a principal axis the coupling factor is zero, so either a conventional TMplasmon or TEplasmon exists. However, due to anisotropy (φ ≠ 0°) the coupling factor can become nonzero giving rise to hybrid surface waves of mixed TETM polarizations. Despite the hybridization, only one type of polarization is predominant for each mode. Therefore, it is logical to refer to such modes as quasiTM and quasiTE surface plasmons.
It is important to note that for a number of practical problems it is necessary to take into account nonlocal effects caused by spatial dispersion. For instance, it was shown that the intrinsic graphene nonlocality may have a significant impact on the properties of surface plasmons propagating along the hyperbolic metasurface based on the graphene strips^{72}. Unfortunately, it can not be accounted for in the framework of the effective surface conductivity extracted from the normal incidence measurements. However, it can be calculated by using lattice sums. In this case, the dispersion equation for the eigenmodes has the following form:
Equation (7) can be transformed into Eq. (6) under the assumption that d ≪ a ≪ λ.
Figure 4 shows the dispersion of the surface waves localized at the studied metasurface sample for different propagation angles φ = 0, 45, 90°. In Figs. 4a–c we compare the effective model and the discrete dipole model taking into account spatial dispersion \(({{\bf{k}}}_{\tau }\ne {\bf{0}})\). One can see that the difference in the dispersions obtained within the local and nonlocal models is significant. It can be explained by quite a large filling factor f, which sharply limits the accounting for nonlocal effects in the framework of the discrete dipole model. Nevertheless, both models are qualitatively similar. For instance, the resonant frequencies are close in both models for all propagation angles. Both models predict the frequency gap between TM and TEplasmons for φ = 0° which shrinks with increasing of φ. At φ = 90°, the gap disappears and both surface modes can propagate at the same frequency, that is in accordance with the results of fullwave numerical simulations (see Fig. 4f). Better matching between the results of DDM and fullwave simulations could be obtained if we account for anisotropy of the dynamic interaction constant, but this theoretical extension is the subject of our further research.
To check the applicability of the effective conductivities extracted from the farfield measurements in characterization of the nearfield phenomena, we compare dispersion of the surface waves from Figs. 4ac with the results from fullwave numerical simulations carried out in COMSOL Multiphysics (Figs. 4d–f). One can see good correspondence of bands at low frequencies (for the quasiTE mode). At high frequencies, i.e. small wavelengths, the effective model works worse but it is still eligible for qualitative results.
It is convenient to present dispersion of surface waves in terms of equal frequency contours, which can be visualized in reflection experiments with a high index ZnSe prism in Otto geometry. We calculate reflection of a light wave in such a configuration by using the transfer matrix method^{73}. When \({\rm{\det }}[{\rm{Im}}(\hat{\sigma })] > 0\), the equal frequency contours have an elliptic shape (Fig. 5a,c,d,f). For a hyperbolic regime, when \({\rm{\det }}[{\rm{Im}}(\hat{\sigma })] < 0\) (λ = 730 nm), the equal frequency contours represent a set of hyperbolas for the quasiTE mode (Fig. 5b) and arcs for the quasiTM mode (Fig. 5e). This drastic change of the shape is often called topological transition. One can see that in the hyperbolic regime both quasiTE and quasiTM modes are present, i.e. simultaneous propagation of two types of surface plasmons is observed (Fig. 5b,e), which is consistent with bands dispersion in Fig. 4c,f. For the capacitive and inductive regimes only a single mode propagates. However, each mode has hybrid TETM polarization, so it is observed in both polarizations as shown in Fig. 5. Although polarization of the surface mode at 660 nm is predominantly similar to polarization of a conventional TMplasmon (Fig. 5d), TEpolarization is also visible (Fig. 5a). The opposite situation takes place for a quasiTE plasmon at λ = 900 nm (Fig. 5c,f). The exceptions are the principal axes directions where polarization of surface modes is strictly either purely TE or purely TM due to the lack of anisotropy.
Conclusions
To conclude, we have suggested a practical concept to describe the full set of optical properties of a metasurface. Our approach is based on extraction of the effective surface conductivity. It allows to study various phenomena in the farfield as well as to calculate the spectrum of surface waves. We have developed two techniques to retrieve the effective conductivity and discussed their limitations. There are three different regimes of the local diagonal conductivity tensor of the anisotropic metasurface composed of elliptical gold nanodisks: inductive (metallike), capacitive (dielectriclike) and hyperbolic (like in an indefinite medium). In contrast to an isotropic metasurface such anisotropic metasurface supports two modes of hybrid polarizations. We have shown the influence of nonlocality on dispersion of the surface waves. Finally, we have demonstrated the topological transition of the equal frequency contours and the hybridization of two eigenmodes in the optical and midIR ranges. We believe these results will be highly useful for a plethora of metasurfaces applications in nanophotonics, plasmonics, sensing and optoelectronics.
Additional information
Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
References
 1.
Smith, D. R., Pendry, J. B. & Wiltshire, M. C. Metamaterials and negative refractive index. Science 305, 788–792 (2004).
 2.
Engheta, N. & Ziolkowski, R. W. Metamaterials: physics and engineering explorations (John Wiley & Sons, 2006).
 3.
Shalaev, V. M. Optical negativeindex metamaterials. Nat. Photonics 1, 41–48 (2007).
 4.
Dai, S. et al. Subdiffractional focusing and guiding of polaritonic rays in a natural hyperbolic material. Nat. Commun. 6, 6963 (2015).
 5.
Li, P. et al. Hyperbolic phononpolaritons in boron nitride for nearfield optical imaging. Nat. Commun. 6, 7507 (2015).
 6.
Hamm, J. M. & Hess, O. Two twodimensional materials are better than one. Science 340, 1298–1299 (2013).
 7.
Glazov, M. et al. Exciton fine structure and spin decoherence in monolayers of transition metal dichalcogenides. Phys. Rev. B 89, 201302 (2014).
 8.
CorreasSerrano, D., GomezDiaz, J., Melcon, A. A. & Alù, A. Black phosphorus plasmonics: anisotropic elliptical propagation and nonlocalityinduced canalization. J. Opt. 18, 104006 (2016).
 9.
Kildishev, A. V., Boltasseva, A. & Shalaev, V. M. Planar photonics with metasurfaces. Science 339, 1232009 (2013).
 10.
Yu, N. & Capasso, F. Flat optics with designer metasurfaces. Nat. Mater. 13, 139 (2014).
 11.
Meinzer, N., Barnes, W. L. & Hooper, I. R. Plasmonic metaatoms and metasurfaces. Nat. Photonics 8, 889–898 (2014).
 12.
Kuznetsov, A. I., Miroshnichenko, A. E., Brongersma, M. L., Kivshar, Y. S. & Luk’yanchuk, B. Optically resonant dielectric nanostructures. Science 354, aag2472 (2016).
 13.
Christensen, J., Manjavacas, A., Thongrattanasiri, S., Koppens, F. H. & Garcia de Abajo, F. J. Graphene plasmon waveguiding and hybridization in individual and paired nanoribbons. ACS Nano 6, 431–440 (2011).
 14.
Trushkov, I. & Iorsh, I. Twodimensional hyperbolic medium for electrons and photons based on the array of tunnelcoupled graphene nanoribbons. Phys. Rev. B 92, 045305 (2015).
 15.
Holloway, C. L. et al. An overview of the theory and applications of metasurfaces: The twodimensional equivalents of metamaterials. IEEE Antenn. Propag. M. 54, 10–35 (2012).
 16.
Glybovski, S. B., Tretyakov, S. A., Belov, P. A., Kivshar, Y. S. & Simovski, C. R. Metasurfaces: From microwaves to visible. Phys. Rep. 634, 1–72 (2016).
 17.
Yu, N. & Capasso, F. Optical metasurfaces and prospect of their applications including fiber optics. J. Lightwave Technol. 33, 2344–2358 (2015).
 18.
Chen, H.T., Taylor, A. J. & Yu, N. A review of metasurfaces: physics and applications. Rep. Prog. Phys. 79, 076401 (2016).
 19.
Ding, F., Pors, A. & Bozhevolnyi, S. I. Gradient metasurfaces: a review of fundamentals and applications. Rep. Prog. Phys. 81, 026401 (2018).
 20.
Moitra, P., Slovick, B. A., Yu, Z. G., Krishnamurthy, S. & Valentine, J. Experimental demonstration of a broadband alldielectric metamaterial perfect reflector. Appl. Phys. Lett. 104, 171102 (2014).
 21.
Decker, M. et al. Highefficiency dielectric Huygens’ surfaces. Adv. Opt. Mater. 3, 813–820 (2015).
 22.
Yang, Y. et al. Dielectric metareflectarray for broadband linear polarization conversion and optical vortex generation. Nano Lett. 14, 1394–1399 (2014).
 23.
Desiatov, B., Mazurski, N., Fainman, Y. & Levy, U. Polarization selective beam shaping using nanoscale dielectric metasurfaces. Opt. Express 23, 22611–22618 (2015).
 24.
Yermakov, O. Y. et al. Hybrid waves localized at hyperbolic metasurfaces. Phys. Rev. B 91, 235423 (2015).
 25.
GomezDiaz, J. S., Tymchenko, M. & Alù, A. Hyperbolic plasmons and topological transitions over uniaxial metasurfaces. Phys. Rev. Lett. 114, 233901 (2015).
 26.
GomezDiaz, J. & Alù, A. Flatland optics with hyperbolic metasurfaces. ACS Photonics 3, 2211–2224 (2016).
 27.
Takayama, O., Bogdanov, A. A. & Lavrinenko, A. V. Photonic surface waves on metamaterial interfaces. J. Phys. Condens. Matter. 29, 463001 (2017).
 28.
Low, T. et al. Polaritons in layered twodimensional materials. Nat. Mater. 16, 182–194 (2017).
 29.
Samusev, A. et al. Polarizationresolved characterization of plasmon waves supported by an anisotropic metasurface. Opt. Express 25, 32631–32639 (2017).
 30.
Ding, F., Deshpande, R. & Bozhevolnyi, S. I. Bifunctional gapplasmon metasurfaces for visible light: polarizationcontrolled unidirectional surface plasmon excitation and beam steering at normal incidence. Light Sci. Appl. 7, 17178 (2018).
 31.
RodríguezFortuño, F. J., Engheta, N., Martnez, A. & Zayats, A. V. Lateral forces on circularly polarizable particles near a surface. Nat. Commun. 6, 8799 (2015).
 32.
Petrov, M. I., Sukhov, S. V., Bogdanov, A. A., Shalin, A. S. & Dogariu, A. Surface plasmon polariton assisted optical pulling force. Laser Photonics Rev. 10, 116–122 (2016).
 33.
Shitrit, N. et al. Spinoptical metamaterial route to spincontrolled photonics. Science 340, 724–726 (2013).
 34.
Aiello, A., Banzer, P., Neugebauer, M. & Leuchs, G. From transverse angular momentum to photonic wheels. Nat. Photonics 9, 789–795 (2015).
 35.
Bliokh, K. Y. & Nori, F. Transverse and longitudinal angular momenta of light. Phys. Rep. 592, 1–38 (2015).
 36.
Yermakov, O. Y. et al. Spin control of light with hyperbolic metasurfaces. Phys. Rev. B 94, 075446 (2016).
 37.
Huang, L. et al. Threedimensional optical holography using a plasmonic metasurface. Nat. Commun. 4, 2808 (2013).
 38.
Ni, X., Kildishev, A. V. & Shalaev, V. M. Metasurface holograms for visible light. Nat. Commun. 4, 2807 (2013).
 39.
Zheng, G. et al. Metasurface holograms reaching 80% efficiency. Nat. Nanotechnol. 10, 308–312 (2015).
 40.
Sedighy, S. H., Guclu, C., Campione, S., Amirhosseini, M. K. & Capolino, F. Wideband planar transmission line hyperbolic metamaterial for subwavelength focusing and resolution. IEEE Transactions Antenn. Propag. 61, 4110–4117 (2013).
 41.
High, A. A. et al. Visiblefrequency hyperbolic metasurface. Nature 522, 192 (2015).
 42.
Li, P. et al. Infrared hyperbolic metasurface based on nanostructured van der Waals materials. Science 359, 892–896 (2018).
 43.
Jahani, S. & Jacob, Z. Alldielectric metamaterials. Nat. Nanotechnol. 11, 23–36 (2016).
 44.
Simovski, C. R. On electromagnetic characterization and homogenization of nanostructured metamaterials. J. Opt. 13, 013001 (2010).
 45.
Pors, A., Tsukerman, I. & Bozhevolnyi, S. I. Effective constitutive parameters of plasmonic metamaterials: homogenization by dual field interpolation. Phys. Rev. E 84, 016609 (2011).
 46.
Chebykin, A., Orlov, A., Simovski, C., Kivshar, Y. S. & Belov, P. A. Nonlocal effective parameters of multilayered metaldielectric metamaterials. Phys. Rev. B 86, 115420 (2012).
 47.
Andryieuski, A. et al. Homogenization of resonant chiral metamaterials. Phys. Rev. B 82, 235107 (2010).
 48.
Ouchetto, O., Qiu, C.W., Zouhdi, S., Li, L.W. & Razek, A. Homogenization of 3D periodic bianisotropic metamaterials. IEEE T. Microw. Theory 54, 3893–3898 (2006).
 49.
Alù, A. Firstprinciples homogenization theory for periodic metamaterials. Phys. Rev. B 84, 075153 (2011).
 50.
Mackay, T. G. Linear and nonlinear homogenized composite mediums as metamaterials. Electromagn. 25, 461–481 (2005).
 51.
Larouche, S. & Smith, D. R. A retrieval method for nonlinear metamaterials. Opt. Commun. 283, 1621–1627 (2010).
 52.
MacFarlane, G. Surface impedance of an infinite parallelwire grid at oblique angles of incidence. J. Inst. Electr. Eng. Part IIIA: Radiolocation 93, 1523–1527 (1946).
 53.
Klein, N. et al. The effective microwave surface impedance of high T thin films. J. Appl. Phys. 67, 6940–6945 (1990).
 54.
Tretyakov, S. & Maslovski, S. Thin absorbing structure for all incidence angles based on the use of a highimpedance surface. Microw. Opt. Techn. Lett. 38, 175–178 (2003).
 55.
Andryieuski, A. & Lavrinenko, A. V. Graphene metamaterials based tunable terahertz absorber: effective surface conductivity approach. Opt. Express 21, 9144–9155 (2013).
 56.
Tabert, C. J. & Nicol, E. J. Magnetooptical conductivity of silicene and other buckled honeycomb lattices. Phys. Rev. B 88, 085434 (2013).
 57.
Danaeifar, M., Granpayeh, N., Mortensen, N. A. & Xiao, S. Equivalent conductivity method: straightforward analytical solution for metasurfacebased structures. J. Phys. D Appl. Phys. 48, 385106 (2015).
 58.
Nemilentsau, A., Low, T. & Hanson, G. Anisotropic 2D materials for tunable hyperbolic plasmonics. Phys. Rev. Lett. 116, 066804 (2016).
 59.
BakerJarvis, J., Vanzura, E. J. & Kissick, W. A. Improved technique for determining complex permittivity with the transmission/reflection method. IEEE Transactions Microw. Theory 38, 1096–1103 (1990).
 60.
Luukkonen, O., Maslovski, S. I. & Tretyakov, S. A. A stepwise NicolsonRossWeirbased material parameter extraction method. IEEE Antennas Wirel. Propag. Lett. 10, 1295–1298 (2011).
 61.
Merano, M. Fresnel coefficients of a twodimensional atomic crystal. Phys. Rev. A 93, 013832 (2016).
 62.
Huidobro, P. A., Kraft, M., Maier, S. A. & Pendry, J. B. Graphene as a tunable anisotropic or isotropic plasmonic metasurface. ACS Nano 10, 5499–5506 (2016).
 63.
Moroz, A. Exponentially convergent lattice sums. Opt. Lett. 26, 1119–1121 (2001).
 64.
Lunnemann, P. & Koenderink, A. F. Dispersion of guided modes in twodimensional split ring lattices. Phys. Rev. B 90, 245416 (2014).
 65.
Belov, P. A. & Simovski, C. R. Homogenization of electromagnetic crystals formed by uniaxial resonant scatterers. Phys. Rev. E 72, 026615 (2005).
 66.
Poddubny, A. N., Belov, P. A., Ginzburg, P., Zayats, A. V. & Kivshar, Y. S. Microscopic model of Purcell enhancement in hyperbolic metamaterials. Phys. Rev. B 86, 035148 (2012).
 67.
Chebykin, A. V., Gorlach, M. A. & Belov, P. A. Spatialdispersioninduced birefringence in metamaterials with cubic symmetry. Phys. Rev. B 92, 045127 (2015).
 68.
Tretyakov, S. A., Viitanen, A. J., Maslovski, S. I. & Saarela, I. E. Impedance boundary conditions for regular dense arrays of dipole scatterers. IEEE T. Antenn. Propag. 51, 2073–2078 (2003).
 69.
Silveirinha, M. G. & Fernandes, C. A. A new acceleration technique with exponential convergence rate to evaluate periodic Green functions. IEEE Transactions Antenn. Propag. 53, 347–355 (2005).
 70.
Capolino, F., Wilton, D. R. & Johnson, W. A. Efficient computation of the 3D Green’s function for the Helmholtz operator for a linear array of point sources using the Ewald method. J. Comput. Phys. 223, 250–261 (2007).
 71.
Olmon, R. L. et al. Optical dielectric function of gold. Phys. Rev. B 86, 235147 (2012).
 72.
CorreasSerrano, D., GomezDiaz, J., Tymchenko, M. & Alù, A. Nonlocal response of hyperbolic metasurfaces. Opt. Express 23, 29434–29448 (2015).
 73.
Dmitriev, P. kitchenknif/pyatmm: V1.0.0a1, https://doi.org/10.5281/zenodo.1041040 (2017).
Acknowledgements
This work was partially supported by the Villum Fonden, Denmark through the DarkSILD project (No. 11116), Megagrant (No. 14.Y26.31.0015), the Ministry of Education and Science of the Russian Federation (3.1365.2017/4.6, 3.1668.2017/4.6, 3.8891.2017/8.9), RFBR (163260123, 163760064, 170201234, 183200739), the Grant of the President of the Russian Federation (MK403.2018.2) and the Foundation for the Advancement of Theoretical Physics and Mathematics “BASIS” . The authors are thankful to M. Gorlach and A. Poddubny for the fruitful discussions and critical comments and to K. Ladutenko for the additional simulations.
Author information
Affiliations
Department of Nanophotonics and Metamaterials, ITMO University, St. Petersburg, 197101, Russia
 Oleh Y. Yermakov
 , Dmitry V. Permyakov
 , Pavel A. Dmitriev
 , Anton K. Samusev
 , Ivan V. Iorsh
 , Andrei V. Lavrinenko
 & Andrey A. Bogdanov
Department of Biblical Studies, St. Petersburg Theological Academy, St. Petersburg, 191167, Russia
 Filipp V. Porubaev
DTU Fotonik, Technical University of Denmark, Oersteds pl. 343, DK2800, Kongens Lyngby, Denmark
 Radu Malureanu
 & Andrei V. Lavrinenko
Authors
Search for Oleh Y. Yermakov in:
Search for Dmitry V. Permyakov in:
Search for Filipp V. Porubaev in:
Search for Pavel A. Dmitriev in:
Search for Anton K. Samusev in:
Search for Ivan V. Iorsh in:
Search for Radu Malureanu in:
Search for Andrei V. Lavrinenko in:
Search for Andrey A. Bogdanov in:
Contributions
O.Y. and F.P. performed the theoretical calculations. O.Y., D.P. and P.D. performed the numerical calculations. R.M. fabricated the samples. D.P. and A.S. conducted the measurements. A.S., I.I., A.A. and A.L. supervised the project. The paper was written by O.Y., A.A. and A.L. All authors discussed the results and reviewed the manuscript.
Competing Interests
The authors declare no competing interests.
Corresponding author
Correspondence to Oleh Y. Yermakov.
Electronic supplementary material
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.