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# Effective surface conductivity of optical hyperbolic metasurfaces: from far-field characterization to surface wave analysis

## Abstract

Metasurfaces offer great potential to control near- and far-fields through engineering optical properties of elementary cells or meta-atoms. 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 far-field and in the near-field 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 semi-analytic parameters obtained with the nonlocal discrete dipole model with and without interaction contribution between meta-atoms. 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 quasi-TE and quasi-TM 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 near-field zone with effective conductivity tensor extracting from the convenient far-field 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 metamaterials1,2,3 – artificially created media, whose electromagnetic properties can drastically differ from the properties of the natural materials. However, a three-dimensional 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 – two-dimensional analogues of metamaterials. There are also natural two-dimensional anisotropic materials such as hexagonal boron nitride4,5, transition metal dichalcogenides6,7, black phosphorus8. In the visible and the near-IR range, metasurfaces can be implemented using subwavelength periodic arrays of plasmonic or high-index dielectric nanoparticles9,10,11,12. A nanostructured graphene could also be considered as a metasurface for THz frequencies13,14. In the microwave range, metasurfaces can be implemented by using LC-circuits, split-ring resonators, arrays of capacitive and inductive elements (strips, grids, mushrooms), wire medium etc15,16. Despite subwavelength or even monoatomic thicknesses, the metasurfaces offer unprecedented control over light propagation, reflection and refraction15,17.

Metasurfaces exhibit a lot of intriguing properties for a wide area of applications such as near-field 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% reflectance20 and transmittance21 in a broadband frequency range. Moreover, metasurfaces can serve as light control elements: frequency selectors, antennas, lenses, perfect absorbers16. They offer an excellent functionality with polarization conversion, beam shaping and optical vortices generation22,23. Besides, metasurfaces provide an efficient control over dispersion and polarization of surface waves24,25,26,27,28,29,30. Surface plasmon-polaritons propagating along a metasurface assist pushing, pulling and lateral optical forces in its vicinity31,32. Metasurfaces are prospective tools for spin-controlled optical phenomena33,34,35,36 and holographic applications37,38,39. Particular attention is paid to hyperbolic metasurfaces, which are promising tools for many applications including the manipulation over surface waves26,40,41,42. 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 all-dielectric metasurfaces and metamaterials can serve as a prospective low-loss platform, which could replace plasmonic structures43. 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 nonlocality44,45,46, chirality47, bi-anisotropy48,49 and nonlinearity50,51.

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, high-impedance surfaces and wire grids etc52,53,54. It has been recently pointed out that two-dimensional structures, like graphene, silicene and metasurfaces, can be described within an effective conductivity approach24,25,55,56,57,58. In virtue of a subwavelength thickness, a metasurface could be considered as a two-dimensional 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 kτ is the component of the wavevector in the plane of the metasurface15,16. Importantly, such effective surface conductivity describes the properties of the metasurface both in the far-field when $$|{{\bf{k}}}_{\tau }| < \omega /c$$ (reflection, absorption, refraction, polarization transformation etc.) and in the near-field (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 two-dimensional 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 (zero-thickness approximation); (ii) experimentally by characterization of the metasurface reflection spectra and a subsequent fitting with Drude-Lorentz formula; and (iii) theoretically by using the nonlocal discrete dipole model. We reveal that the effective surface conductivity tensor extracted from the far-field measurements well describes near-field 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 ax = 134 nm and ay = 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 lift-off 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 non-chiral. Asymmetry of each particle splits its in-plane 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 out-of-plane polarizability caused by the finite thickness of the plasmonic particles. In our case, the out-of-plane polarizability αz 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 two-dimensional 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 zero-thickness approximation (ZTA). Within this approximation we replace the real structure of finite thickness H by the effective two-dimensional 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 Nicolson-Ross-Weir method59,60.

Considering a two-dimensional layer with effective conductivity σ sandwiched between two media with refractive indices n1 (superstrate) and n2 (substrate) one can find Fresnel’s coefficients55,61,62 and express the effective surface conductivity as follows

$${\sigma }_{x,y}=\frac{{n}_{1}-{n}_{2}-{S}_{11}^{x,y}({n}_{1}+{n}_{2})}{1+{S}_{11}^{x,y}},$$
(1)

where $${S}_{11}^{x,y}$$ is the component of the S-matrix. 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 in-plane 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 S-parameters 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 eiωt, $${k}_{0}={n}_{1}\omega /c$$, 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 two-dimensional 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 far-field measurements correctly predicts the near-field properties. In particular, it describes well the surface waves spectrum. It is possible due to the non-critical 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 (TM-polarization) the plasmon resonance is observed at 670 nm, while for light polarized along the short axis (TE-polarization) 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 point-dipole model. This technique has been implemented for 1D, 2D and 3D structures63,64,65,66,67. 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:

$${\hat{\sigma }}_{{\rm{eff}}}=-\,i\frac{4\pi \varepsilon \omega }{c{a}^{2}}{\hat{\alpha }}_{{\rm{eff}}}\mathrm{.}$$
(2)

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 two-dimensional polarizability tensor $${\hat{\alpha }}_{{\rm{eff}}}$$ or conductivity tensor $${\hat{\sigma }}_{{\rm{eff}}}$$ with zero off-diagonal components (in the basis of the principal axes). Rigorous derivation of the effective polarizability of a two-dimensional lattice of resonant scatterers is performed in Refs.49,65,68. The effective polarizability of the metasurface can be written as

$${\hat{\alpha }}_{{\rm{eff}}}^{-1}(\omega ,{{\bf{k}}}_{{\boldsymbol{\tau }}})={\hat{\alpha }}_{0}^{-1}(\omega )+\hat{C}(\omega ,{{\bf{k}}}_{{\boldsymbol{\tau }}}\mathrm{).}$$
(3)

Here, $${\hat{\alpha }}_{0}(\omega )$$ is the polarizability of the individual resonant scatterer, and $$\hat{C}(\omega ,{{\bf{k}}}_{{\boldsymbol{\tau }}})$$ is the so-called dynamic interaction constant65. 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:

$$\hat{C}(\omega ,{{\bf{k}}}_{{\boldsymbol{\tau }}})=\sum _{i,j}\hat{G}(\omega ,{{\bf{r}}}_{ij}){e}^{i{{\bf{k}}}_{\tau }{{\bf{r}}}_{ij}}\mathrm{.}$$
(4)

Here $$\hat{G}(\omega ,{{\bf{r}}}_{ij})$$ is the dyadic Green’s function and rij are the coordinates of the dipoles. This sum has slow convergence. So, we calculate the interaction term in Eq. (4) within the Ewald summation technique63,66,69,70 applied for a two-dimensional 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 two-dimensional periodic structures under three main conditions:

1. 1.

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

2. 2.

Dipole approximation: f 1 (or da, where d is the characteristic size of a scaterrer). Here f = A/a2 is the filling factor, A is the area occupied by the scatterer in the unit cell (in our case, A = πaxay), a2 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. 3.

Quasi-two-dimensionality: azz min(axx, ayy) and Hλ. This condition is achieved, when thickness of a metasurface is less than both characteristic in-plane sizes of meta-atoms (H < min{ax, ay}) and skin depth δ (H < δ).

For the metasurface sample under consideration H/ay = 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 consideration71.

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 σy is better than for σx. 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:

$${\hat{\sigma }}_{{\rm{e}}{\rm{f}}{\rm{f}}}=(\begin{array}{cc}{\sigma }_{x} & 0\\ 0 & {\sigma }_{y}\end{array}).$$
(5)

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 TM-polarized 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 TE-polarized 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 so-called hyperbolic regime, in which simultaneous propagation of both TE- and TM-modes is possible24.

## 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 metasurface24:

$$(\frac{c{\kappa }_{1}}{{\mu }_{1}\omega }+\frac{c{\kappa }_{2}}{{\mu }_{2}\omega }-i{\sigma }_{xx})(\frac{{\varepsilon }_{1}\omega }{c{\kappa }_{1}}+\frac{{\varepsilon }_{2}\omega }{c{\kappa }_{2}}+i{\sigma }_{yy})={\sigma }_{xy}{\sigma }_{yx}.$$
(6)

Here, σij are the tensor components in the coordinate system rotated by angle φ (see Fig. 1b), ε1, μ1, κ1 and ε2, μ2, κ2 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 kτ is the wavevector in the plane of the metasurface. In our case Eq. (6) is simplified since we consider the metasurface in non-magnetic (μ1 = μ2 = 1) and homogeneous environment with the permittivity corresponding to fused silica ε = ε1 = ε2 = 2.1.

The first and the second factors in the left side of Eq. (6) correspond to the dispersion of purely TE-polarized and TM-polarized 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 TM-plasmon or TE-plasmon exists. However, due to anisotropy (φ ≠ 0°) the coupling factor can become non-zero giving rise to hybrid surface waves of mixed TE-TM polarizations. Despite the hybridization, only one type of polarization is predominant for each mode. Therefore, it is logical to refer to such modes as quasi-TM and quasi-TE 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 strips72. 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:

$${\rm{\det }}|{\hat{\alpha }}_{{\rm{eff}}}^{-1}(\omega ,{{\bf{k}}}_{\tau })|={\rm{\det }}|{\hat{\alpha }}_{0}^{-1}(\omega )+\hat{C}(\omega ,{{\bf{k}}}_{\tau })|=0.$$
(7)

Equation (7) can be transformed into Eq. (6) under the assumption that daλ.

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 TE-plasmons 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 full-wave numerical simulations (see Fig. 4f). Better matching between the results of DDM and full-wave 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 far-field measurements in characterization of the near-field phenomena, we compare dispersion of the surface waves from Figs. 4a-c with the results from full-wave numerical simulations carried out in COMSOL Multiphysics (Figs. 4d–f). One can see good correspondence of bands at low frequencies (for the quasi-TE 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 method73. 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 quasi-TE mode (Fig. 5b) and arcs for the quasi-TM mode (Fig. 5e). This drastic change of the shape is often called topological transition. One can see that in the hyperbolic regime both quasi-TE and quasi-TM 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 TE-TM 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 TM-plasmon (Fig. 5d), TE-polarization is also visible (Fig. 5a). The opposite situation takes place for a quasi-TE 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 far-field 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 (metal-like), capacitive (dielectric-like) 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 non-locality 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 mid-IR ranges. We believe these results will be highly useful for a plethora of metasurfaces applications in nanophotonics, plasmonics, sensing and opto-electronics.

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## 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 (16-32-60123, 16-37-60064, 17-02-01234, 18-32-00739), the Grant of the President of the Russian Federation (MK-403.2018.2) 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

1. #### 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
2. #### Department of Biblical Studies, St. Petersburg Theological Academy, St. Petersburg, 191167, Russia

• Filipp V. Porubaev
3. #### DTU Fotonik, Technical University of Denmark, Oersteds pl. 343, DK-2800, Kongens Lyngby, Denmark

•  & Andrei V. Lavrinenko

### 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.