Abstract

We analyze and model the nonlocal response of ultrathin hyperbolic metasurfaces (HMTSs) by applying an effective medium approach. We show that the intrinsic spatial dispersion in the materials employed to realize the metasurfaces imposes a wavenumber cutoff on the hyperbolic isofrequency contour, inversely proportional to the Fermi velocity, and we compare it with the cutoff arising from the structure granularity. In the particular case of HTMSs implemented by an array of graphene nanostrips, we find that graphene nonlocality can become the dominant mechanism that closes the hyperbolic contour – imposing a wavenumber cutoff at around 300k0 – in realistic configurations with periodicity L<π/(300k0), thus providing a practical design rule to implement HMTSs at THz and infrared frequencies. In contrast, more common plasmonic materials, such as noble metals, operate at much higher frequencies, and therefore their intrinsic nonlocal response is mainly relevant in hyperbolic metasurfaces and metamaterials with periodicity below a few nm, being very weak in practical scenarios. In addition, we investigate how spatial dispersion affects the spontaneous emission rate of emitters located close to HMTSs. Our results establish an upper bound set by nonlocality to the maximum field confinement and light-matter interactions achievable in practical HMTSs, and may find application in the practical development of hyperlenses, sensors and on-chip networks.

© 2015 Optical Society of America

1. Introduction

Hyperbolic metasurfaces (HMTSs) are extremely anisotropic 2D structures whose transverse surface conductivity changes the sign of its imaginary part as a function of the polarization of the in-plane electric field [1], thus behaving as a dielectric along one direction and as a metal along the orthogonal one within the sheet. The ultrathin planar nature of HTMSs solves some of the major practical challenges of bulk hyperbolic metamaterials (HMTMSs) [2–4], thanks to a large resilience to volumetric losses, a relatively simple fabrication using standard lithographic and etching techniques [5], and easy external access to the propagating energy, while being fully compatible with integrated circuits and optoelectronic components. In contrast to HMTMs, HMTSs support the propagation of low-loss and extremely confined surface plasmon polaritons (SPPs) with evanescent fields in the direction perpendicular to the surface, providing access to a drastic enhancement of light-matter interactions on a surface [1]. Furthermore, HMTSs allow a large degree of freedom to manipulate the supported SPPs, including routing them towards specific directions within the sheet, dispersion-free propagation (canalization), and negative refraction [5]. The intriguing electromagnetic response of HMTSs arises from their hyperbolic isofrequency contour, associated with an ideally infinite local density of states (LDOS). In classical terms, LDOS and the maximum guided wave number are limited by two mechanisms, namely the presence of losses [1] and the granularity of the metasurface [1, 6], that impose a wavenumber cutoff and close the otherwise open hyperbolic contour. In the case of HMTMs, a third fundamental mechanism has recently been revealed by considering the actual nonlocal response of the metals composing the structure [7], a phenomenon that gives rise to a wavenumber cutoff inversely proportional to the Fermi velocity of electrons in the material [7].

There are several possibilities to implement HMTSs in practice. At visible frequencies, HMTSs based on single-crystalline silver nanostructures have been experimental demonstrated in [5]. Another approach relies on using anisotropic subwavelength metallic scatterers, as envisaged in [8]. An interesting alternative operating at THz and near infrared frequencies uses a densely packed array of graphene nanostrips [1]. This configuration – analyzed using an effective medium approach (EMA) in [1,6] and the Kubo formalism in [9]– allows tuning in real time the wavenumber contour topology, from closed elliptical to open hyperbolic going through the extremely anisotropic σ-near zero case [1], by simply applying a gate bias. This structure takes advantage of the inherent reconfigurable capabilities of graphene plasmonics [10,11], a field that has recently emerged as an excellent platform for strong light-matter interactions and has triggered a myriad of exciting applications [12–16]. In this context, the intrinsic spatial dispersion of graphene [17–19] is known to significantly affect the response of guided THz waveguides and components [20–22] and to blueshift the plasmonic resonances appearing in graphene nanostructures [23].

In this contribution, we investigate the electromagnetic nonlocal response of hyperbolic metasurfaces, focusing on the case of a densely-packed array of graphene strips at THz and near infrared frequencies. We have chosen this configuration because the intrinsic nonlocal effects of graphene may be stronger than those usually found in noble metals, and they can be strong enough to become the fundamental mechanism that closes the hyperbolic dispersion relation of the structure. However, we also consider implementations based on other 2D materials, assuming that their nonlocal conductivity is available. Our main goal is to isolate and independently investigate nonlocalities arising due to (i) granularity and (ii) the intrinsic spatially-dispersive response of the constituent materials. The former case is analyzed using a full-wave mode-matching approach [24], where graphene is characterized using a local model. As expected, the lattice periodicity L closes the hyperbolic isofrequency contour of the structure by introducing a cutoff wavenumber at π/L. In the latter scenario, we isolate the influence of graphene nonlocality in the response of the entire HMTSs by applying an effective medium approach that homogenizes the 2D structure in the L0 limit, thus removing the spatial dispersion introduced by the lattice. This allows us to demonstrate that the inherent nonlocal response of graphene imposes a cutoff to the HMTSs wavenumbers inversely proportional to the Fermi velocity of the material, as occurs in HMTMs [7]. We do stress that graphene possesses a relatively high Fermi velocity (vF106 m/s) able to significantly modify the propagation of surface waves with kρ>300k0. Consequently, graphene nonlocality becomes the dominant mechanism that controls SPPs propagation in hyperbolic metasurfaces with periodicity L<π/300k0, thus setting a simple rule for the design of practical HMTSs. Taking into account the operation frequency of these structures, nonlocal effects play a key role in the response of realistic HMTSs with periodicity ranging between 500 to 20 nm. This result is in clear contrast to the case of conventional optical HMTMs, where intrinsic spatial dispersion effects of the materials may be relevant only for subwavelength composites – with periodicity below a few nm –very challenging to produce even with advanced nanofabrication techniques. Therefore, they are usually of limited practical impact. We conclude our study showing that nonlocality affects the spontaneous emission rate (SER) of emitters located very close to HMTSs, imposing a fundamental limit associated to the large wavevector cutoff. Throughout the paper, full-wave simulations based on the commercial software COMSOL Multiphysics are employed to validate our analytical study.

2. Homogenization of nonlocal HMTSs

In this section, we first describe the hyperbolic metasurface under study – a densely packed array of 2D strips – and derive an EMA able to homogenize the structure in the general case of ribbons characterized by a fully-populated conductivity tensor. Then, we introduce and discuss spatially dispersive models employed to characterize graphene and 2D materials. Next, we study the effective conductivity of homogenized nonlocal HTMSs. Finally, we derive the dispersion relation of theses structures and provide brief guidelines to its numerical solution.

2.1 EMA of periodic 2D strips defined by a fully-populated conductivity tensor

Let us consider an array of densely packed 2D strips, as illustrated in Fig. 1, where each strip is described by a fully populated conductivity tensor

σm¯¯=(σxxmσxymσyxmσyym)
that depends on the specific features of the 2D material employed. Assuming a subwavelength separation distance between the strips, their near-field coupling can be approximated through the effective conductivity [1, 25]
σc¯¯=(σxxc000)    with  σxxc=iωε0εeffLπln[csc(πG2L)]
where ω is the angular frequency, ε0 and εeff are the permittivity of free-space and the relative one of the surrounding medium, respectively, L is the periodicity of the unit-cell, and G is the separation distance between two adjacent strips. This description allows transforming the original structure into an equivalent one composed of strips with different conductivities, as shown in Fig. 1.

 

Fig. 1 Schematic of a hyperbolic metasurface composed of 2D strips with widths W and unit cell period L. The inset shows an equivalent representation of one unit-cell modelled by two strips described by effective conductivity tensors σ¯¯ and σC¯¯.

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The main goal of this subsection is to derive an EMA to characterize the effective conductivity tensor of the metasurface in the limit of infinitesimally small periodicity, i.e., when L0. To this purpose, we extend the EMA introduced in [1,26]. Specifically, applying boundary conditions to the equivalent unit-cell (see inset of Fig. 1) we note that (i) the electric field Ey must be continuous at the strip edges, and (ii) there is no specific restriction that prevents the continuity of the surface current across the ribbons. Taking advantage of the subwavelength nature of the structure, and given the boundary conditions on the parallel strips (continuous normal current distribution and continuous tangential electric field at the strip edges), we enforce a constant current Jx and electric field Ey across theentire unit cell. Then, applying Ohm’s law, one finds

Ex(x)=Jx1σxx(x)Eyσxyσxx(x),Jy(x)=Jxσyx(x)σxx(x)+Ey[σyy(x)σyx(x)σxy(x)σxx(x)],
where σ¯¯(x)=σm¯¯ in the strip (i.e., where x[0,W]) and σ¯¯(x)=σC¯¯ in the air gap (i.e., where x[W,L]. Averaging the electric field, the effective homogenized conductivity tensor σeff¯¯ of the metasurface reads
1σxxeff=1LL1σxx(x)dx,σxyeff=σxxeffLLσxy(x)σxx(x)dx,σxyeff=σxxeffLLσxy(x)σxx(x)dx,
σyyeff=1LLσyy(x)dx1LLσxy(x)σyx(x)σxx(x)dx+σxyeffσyxeffσxxeff,
which, considering the specific conductivity profile of our metasurface, simplifies to
σxxeff=LσxxmσxxcWσxxc+(LW)σxxg,σxy,eff=σxy,eff=σxx,effWLσxymσxxm.
σyyeff=WLσyymWLσyxmσxymσxxmσxx0+σyxeffσxyeffσxxeff.
The relevance of this simple homogenization approach is threefold. First, it provides an EMA able to homogenize ultrathin metasurfaces –composed of any combination of 2D strips– in the deep subwavelength region (L0), thus providing useful physical insight about their electromagnetic response. Second, it allows modeling magnetically active 2D materials described by a fully populated conductivity tensor such as graphene [27], and therefore it may find applications to model complex phenomena, such as non-reciprocal responses, Faraday rotation [15], and magnetoplasmons [28,29]. Third, it also permits to accurately take into account nonlocal effects of 2D materials. We do note here that the numerical modelling of spatially-dispersive materials usually requires the use of additional boundary conditions [7, 30, 31]. However, our approach based on imposing a uniform current Jx and electric field Ey within the unit-cell is valid here, since the possible spatial variations of these quantities – i.e., Ey(x) and Jx(x) – arising from the intrinsic nonlocal effects of 2D materials, are negligible compared to the periodicity of the cell. Consequently, spatially-dispersive materials can be safely taken into account in EMAs.

2.2 Intrinsic nonlocal response of 2D materials

We analyze here the influence of nonlocal effects in the electromagnetic response of 2D bare materials. We focus on graphene at THz and near infrared frequencies, a material that has recently led to the development of exciting HTMSs. Then, we briefly discuss the possible nonlocal response of usual metals assuming ultrathin (2D) configurations. Throughout this work, we model spatially-dispersive graphene using the Bhatnagar-Gross-Krook (BGK) approach derived in [19]. This model is based on the Boltzmann transport equation, taking into account intraband contributions in graphene, and it is accurate up to tens of THz when the spatial variations of the fields are lower than the de Broglie wavelength of the particles [19] (i.e. kρ<2kF, where kF is the Fermi wavenumber). According to this model, graphene is modelled by a fully-populated conductivity tensor σg¯¯ defined by

σxxg(kx,ky)=γIϕxx+γDΔky(IϕxxkyIϕyxkx)Dσ,
σxyg(kx,ky)=γIϕxy+γDΔky(IϕxykyIϕyykx)Dσ,
σyxg(kx,ky)=γIϕyx+γDΔkx(IϕyxkxIϕxxky)Dσ,
σyyg(kx,ky)=γIϕyy+γDΔkx(IϕyykxIϕxyky)Dσ,
with
Iϕxx(kx,ky)=2πvF2ky2kρ2RαvFkxkq2α2kq2(1R)vF2(α+vFkx)kρ4,
Iϕxy(kx,ky)=Iϕyx(kx,ky)=2πkxkyvF2kρ2R+2αvFkx+α2kq2(1R)vF2(α+vFkx)kρ4,
Iϕyy(kx,ky)=2πvF2kx2kρ2R+αvFkxkq2+α2kq2(1R)vF2(α+vFkx)kρ4,
and
γ=iqe2kBTπ22log{2[1+cosh(μckBT)]},γD=ivF2πωτ,Dσ=1+γDΔkρ2,
Δ=2πvFkρ2(1αα2vF2kρ2),R(kx,ky)=α+vFkxα2vF2kρ2,α=ω+i/τ,
kρ2=kx2+ky2,kq2=kx2ky2
where qe is the charge of an electron, is the reduced Planck’s constant, kB is the Boltzmann’s constant, vF is the Fermi velocity, τ is the electron relaxation time and μc is the chemical potential. Figure 2 briefly illustrates the behavior of nonlocal graphene conductivity (normalized versus the local response of the material) as a function of real kx/k0 and ky/k0 wavenumbers. Note that these wavenumbers do not correspond to any resonant mode of the structure, but they are considered here just to show the response of nonlocal graphene in a basic scenario. As expected, spatial dispersion barely affects the graphene conductivity for very small wavenumbers, converging to the local response as kx and ky 0. As the wavenumbers increase, however, the influence of nonlocality becomes more apparent, slightly boosting graphene conductivity. As we further increase the wavenumbers and get closer to kρ (c/vF)k0, the response drastically changes. Specifically, the diagonal components of the conductivity change their usual metallic response (Im[σ]>0) into a dielectric one (Im[σ]<0) for large wavevectors parallel to them, a known phenomenon [19] associated with a significant increase of losses along that direction. We stress here that these losses are not dissipative, but are associated to the coupling with the cross-terms of the conductivity tensor. This behavior can be better understood by changing from rectangular (x,y) to polar coordinates (ρ, ϕ) through the transformation σg¯¯=M¯¯Tσg¯¯M¯¯, where M¯¯=1/(kρ)[kx, ky;ky,kx], leading to a longitudinal (σρ) and a transverse (σϕ) conductivity, i.e., σ¯¯g=(σρ, 0;0, σϕ), as shown in Fig. 3 (see [19] for further details). In the case of low wavenumbers, both components present a similar metallic nature (Im[σ]>0) that supports the propagation of confined TM plasmons. However, as the wavenumber increases, the longitudinal conductivity becomes dielectric-like – starting at kρ (c/vF)k0 –, and the transverse component remains the only metallic contribution to the conductivity. As a consequence, the propagating energy undergoes an in-plane rotation, in agreement with the cross conductivity terms that appear in Cartesian coordinates. We do remark that, within the limits of validity of the BGK model, nonlocal graphene is isotropic in the xy plane [19].

 

Fig. 2 Nonlocal conductivity tensor elements of graphene in Cartesian coordinates, normalized versus the material local response. Results are given as a function of real kx/k0 and ky/k0 in-plane wavenumbers. Other parameters are f=2.5 THz,  μc=0.02 eV, and τ=0.3 ps.

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Fig. 3 Imaginary parts of the (a) longitudinal and (b) transverse nonlocal conductivity components of graphene normalized versus the local response. Results are given as a function of real kx/k0 and ky/k0 in-plane wavenumbers. Other parameters are f=2.5 THz, μc=0.02 eV, and τ=0.3 ps

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Although we are mainly interested in graphene-based HMTSs in this work, it might be instructive to analyze the possible nonlocal conductivity of other 2D metals. For instance, assuming for a moment that the hydrodynamic Drude model within the Thomas-Fermi approximation [7, 32] holds as we thin down a metal and that we operate above the plasma frequency, its effective in-plane conductivity can be expressed in polar coordinates as

σd¯¯(kρ)=(σρ(kρ)00σϕ(kρ)),
where σρ(kρ)=iωε0t[εmρ(kρ)1] and σϕ(kρ)=iωε0t[εmϕ(kρ)1]are the longitudinal and transverse components of the conductivity, respectively, ‘t’ is the metal thickness, and
εmρ(kρ)=1ωp2ω2+iωγβ2kρ2,εmϕ(kρ)=1ωp2ω2+iωγ,
are the longitudinal and transverse dielectric functions, being ωp the plasma frequency and γ the Drude damping. The term β=3/5vF takes nonlocal effects of longitudinal waves into account. A simple visual inspection of the equations above reveals that for large wavenumbers (with kρ k0c/vF), the longitudinal component of the conductivity will change their response from metallic to dielectric, and therefore the transverse contribution will dominate the metallic response of the 2D material. This analysis is fully consistent with previous predictions using the BGK model of nonlocal graphene and with usual nonlocal bulk media [7].

2.3 Nonlocal effective parameters of homogenized HMTSs

The effective conductivity tensor σeff¯¯(kx,ky) of nonlocal HMTSs composed of densely packed 2D strips can easily be derived by combining the EMA derived above with the spatially-dispersive model of the materials that compose the structure. In the general case, it can be expressed as

σeff¯¯(kx,ky)=(σxxeff(kx,ky)σxyeff(kx,ky)σyxeff(kx,ky)σyyeff(kx,ky)).
In order to investigate the behavior of homogenized nonlocal HMTSs, Fig. 4 illustrates the effective conductivity tensor of a metasurface with periodicity L=100 nm and graphene strips with widths W=50 nm at the operation frequency f=2.5 THz. Even though we show the conductivity components versus real kx/k0 and ky/k0 wavenumbers, we remark again that these wavenumbers do not correspond to any supported mode, and they are just employed here to illustrate the overall trends of σeff¯¯. It is clear that the xx component of the effective conductivity tensor is dominated by the near-field coupling between the strips, thus leading to an almost constant dielectric (capacitive) behavior for all wavenumbers. However, spatial dispersion may significantly affect the response of the other conductivity components. In the case of low wavenumbers, σyyeff keeps its local metallic nature and the cross-terms are negligible, thus leading to the usual description of hyperbolic metasurfaces [1].

 

Fig. 4 Effective conductivity tensor of a nonlocal graphene-based HMTS. The structure comprises an array of spatially-dispersive graphene strips with width W=50 nm and period L=100 nm. Other parameters are f=2.5 THz, μc=0.2 eV, and τ=0.3 ps.

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This scenario is different as the wavenumbers approach |kρ/k0|c/vF300, where the transverse part of nonlocal graphene conductivity dominates the response of the material. As a consequence, σyyeff changes its nature from metallic to dielectric for large ky wavenumbers, while the cross-terms also contributes to the response. Intuitively, this behavior can be better understood in polar coordinates, since the nonlocal longitudinal conductivity is strongly reduced –and changes the nature of its response from metallic to dielectric–preventing energy propagation along the strips, whereas the dominant transverse nonlocal component rotates the energy flow towards the ribbons edges. We emphasize again that the behavior described above is not by any means restricted to the case of graphene, but it also applies to other HMTSs composed of nonlocal 2D materials, such as those described by the hydrodynamic Drude model within the Thomas-Fermi approximation.

2.4 Dispersion relation of homogenized nonlocal HMTSs

The dispersion relation of homogeneous ultrathin metasurfaces defined by an effective conductivity tensor σeff¯¯. can be expressed as [33]

k0kz[4+η02{σxxeff(kx,ky)σyyeff(kx,ky)σxyeff(kx,ky)σyxeff(kx,ky)}]+2k02η0[σxxeff(kx,ky)+σyyeff(kx,ky)]2η0[σxxeff(kx,ky)kx2+σyyeff(kx,ky)ky2+kxky(σxyeff(kx,ky)+σxyeff(kx,ky))]=0,
where the identity kz2=k02kx2ky2 holds. We stress that the metasurface supports SPPs only when the wavenumbers perpendicular to the structure are evanescent, i.e., Im[kz]>0 [33]. In the expression, we have explicitly highlighted the possible nonlocal response of the effective conductivity components.

In the local case, Eq. (21) can be solved analytically by fixing the direction of propagation of the supported SPPs and then physically rotating the metasurface, as described in [6]. However, in the nonlocal case this approach cannot be applied due to the intrinsic spatial dispersion of σeff¯¯, and one has to resort to numerical methods. In this work, we solve Eq. (21) by applying the Newton–Raphson approach [34]. In order to avoid finding local minima in the complex plane, and to speed up the convergence of the algorithm, it is in general most efficient to first find the solution in the y-direction, where the hyperbolic plasmons are less confined and the local dispersion relation provides a good starting point, and then progressively track the solution in other directions until cutoff.

3. Influence of nonlocalities on the electromagnetic response of HTMSs

In this Section, we discuss the influence of different nonlocalities on the isofrequency contours of HTMSs and on the SER of emitters located in their vicinity, and we determine under which conditions they may dominate the electromagnetic response of the entire structure. To this purpose, we apply the local and nonlocal versions of the EMA derived in Section II, comparing the predicted dispersion relations with results computed by two independent full-wave methods. First, to isolate the influence of spatial dispersion associated with the macroscopic periodicity, we utilize a mode matching approach that considers a local, scalar σg, and is able to accurately predict the closing of the hyperbolic contour due to lattice granularity [1, 6, 24]. Then, in order to include the intrinsic nonlocal response of graphene, we use the commercial FEM solver COMSOL Multiphysics to find the complex eigenfrequencies of the unit cell for arbitrary macroscopic wavevectors. Here, graphene is modelled as a surface current with the BGK model described above, thus taking into account both types of spatial dispersion. However, we stress that our COMSOL simulations are only strictly valid in the L0 limit, when the spatial variation of the fields within the unit cell is only associated to the macroscopic propagation along the metasurface, and provide a very good approximation otherwise. A completely rigorous numerical solution would require the development of an additional 2D boundary condition, as similarly done for nonlocal bulk media [7, 30, 31]. We do remark that this approach is justified here, since we are mainly interested in finding the inherent limitations of nonlocal HMTSs even when structure granularity can be neglected. Moreover, in the case of larger unit cells, the hyperbolic contour will be closed before the nonlocal conductivity becomes relevant, as it will be shown below.Figure 5 shows the isofrequency contour of an HMTS with a fixed graphene filling factor of 0.5, operating at f=2.5 THz, with μc=0.02 eV and τ=0.3 ps, for several values of the lattice period L. This approach allows us to easily identify the different physical mechanisms responsible for closing the otherwise open hyperbolic isofrequency contour. In case of extremely subwavelength unit cells [Fig. 5(a), L=30 nm], the lattice periodicity imposes a very large wavenumber cutoff, and therefore the approaches based on local material predict a very similar response. In this scenario, the intrinsic nonlocal response of graphene enforces a wavenumber cutoff at around |kρ|(c/vF)k0, fully consistent with the arguments given in Section II. Importantly, the supported SPPs abruptly disappear instead of following the expected closing towards the kx axis. This behavior arises because at large wavenumbers the transverse component of nonlocal graphene conductivity dominates, thus forcing energy towards the ribbons edges and effectively rotating the SPPs angle of propagation within the sheet. In addition, we do note that HMTSs only supports propagation through a well-defined set of directions, as described in [8]. The combination of these two factors fully explains the mode disappearance in the nonlocal isofrequency contour of Fig. 5(a). We stress the excellent agreement found between our nonlocal EMA approach and full-wave simulations using COMSOL. If we consider now L=100 nm [Fig. 5(b)], the graphene spatial dispersion remains the dominant mechanism, but the structure granularity becomes non-negligible near cutoff, as shown by the slight deviation of the numerical results from the predicted EMA dispersion. In this case, full-wave simulations using local conductivity are still very inaccurate compared to the simple EMA developed here. A complementary scenario is shown in Fig. 5(c) (L=300 nm), where granularity is the main mechanism responsible of the closing of the contour, but graphene spatial dispersion further deforms it towards the x direction. In this case, both nonlocal phenomena should be taken into account. Finally, Fig. 5(d) considers L=700 nm. Now, graphene nonlocality is negligible, since the hyperbola is closed at low wavenumbers due to the large lattice period.

 

Fig. 5 Isofrequency contours of the hyperbolic plasmons supported by an array of graphene strips with W=0.5 L for (a) L=30 nm, (b) L=100 nm, (c) L=300 nm, and (d) L=700 nm. Solid blue (green) lines computed with EMA considering local (nonlocal) graphene conductivity σg¯¯. Dashed red lines computed using a full-wave mode matching approach with local σg¯¯. Markers computed with FEM eigenfrequency solver COMSOL Multyphysics using nonlocal σg¯¯.

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Our analysis of nonlocal graphene-based HMTSs provides important design guidelines for practical devices. Specifically, we have demonstrated that reducing the size of the unit cell in order to increase the LDOS is only beneficial until LvFπ/(ck0), since the inherent nonlocal response of electrons becomes dominant at this point. Therefore, nonlocality imposes a practical limit to the HMTSs periodicity that leads to realizable structures at THz and near infrared frequencies - fully within the capabilities of current nanofabrication technology [36] - able to reach the maximum possible wave confinement and LDOS.

Once we have validated the accuracy of our proposed nonlocal EMA, we apply it to illustrate the response of graphene-based HMTSs versus some physical parameters of the structure. First, we investigate in Fig. 6(a) the influence of the strip width W, for a fixed period L. Interestingly, the slightly open branches associated to a hyperbolic dispersion with reduced σyy (small W/L, red lines) obtained using a local EMA are compensated by graphene nonlocality, thus leading to an almost perfect canalization. For other strip widths, nonlocal effects shows a relatively minor influence on the hyperbolic contour until cutoff. Figure 6(b) illustrates the robustness of the hyperbolic response of spatially dispersive HMTSs versus graphene loss, similarly to previous studies that considered local conductivity [1]. We note, however, that the effective propagation length of the supported hyperbolic plasmons is strongly dependent on graphene therelaxation time τ. In this sense, we have observed that the figure of merit ratio Re[kρ]/Im[kρ] is slightly lower than the case of bare graphene [1, 6]. Finally, Figs. 6(c)-6(d) showcase the reconfiguration capabilities of HMTS through the control of graphene chemical potential μc, easily achieved through electrostatic biasing of the strips [11]. We consider two scenarios: Fig. 6(c) corresponds to an HMTS designed to operate far from potential topological transitions, with a hyperbolic dispersion that is widely tunable. Spatial dispersion limits the maximum confinement in all cases, most noticeably for μc=0.01 eV due to the larger minimum wavenumber supported. On the other hand, Fig. 6(d) depicts the isofrequency contour of a HMTS operating in the σxx near-zero regime, where small modifications of the chemical potential may result in topological transitions from elliptical to hyperbolic. Interestingly, this study demonstrates nonlocality-induced topological transitions [35]. For μc=0.01  eV both local and nonlocal EMA models predict a closed contour. However, spatial dispersion causes the transition to hyperbolic topology to occur for different values of μc, as shown by the plots for μc=0.075 eV and μc=0.2 eV. Consequently, while the local model predicts a closed contour, nonlocality forces a topological transition that leads to open hyperbolic dispersion. This behavior further demonstrates the importance of correctly modelling nonlocality in the design of HTMSs, since it shows that the local description of the materials may predict an incorrect topology.

 

Fig. 6 Isofrequency contours of plasmons supported by an array of graphene strips versus (a) strip width W (L=100 nm, f=4 THz,  μc=0.02 eV, τ = 0.3 ps) and (b) electron relaxation time τ (L=100 nm, W=30 nm, f=2.5 THz, μc=0.3 eV). (c)-(d) shows similar results versus chemical potential μc with (c) L=100 nm, W=25 nm, and f=2.5 THz and (d) L=120 nm, W=110 nm, and f=4.0 THz. Solid (dashed) lines computed with EMA considering local (nonlocal) graphene conductivity σg¯¯.

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Enhancement of light-matter interactions is one of the most attractive features of HMTSs [1, 6], so it is indeed of interest to study how spatial dispersion affects the SER of emitters located in their vicinity. To this end, we calculate the SER of a z-oriented dipole placed above a graphene-based HMTSs with L= 50 nm, W=25 nm, μc = 0.02 eV, τ=0.3 ps, andf=2.5 THz. Results are obtained following the approach recently introduced in [1, 6], modified here to include graphene nonlocal response. Figures 7(a)-7(b) show the emitter SER versus the ribbon width and the distance d between the source and the surface, applying the local and nonlocal EMAs. In order to reveal the influence of nonlocality to the SER, we focus here on emitters closely located to HTMSs, thus preventing free space from filtering waves with high spatial frequency. For local graphene, very high SER is predicted close to the canalization regime – implemented by very narrow ribbons – due to the extremely high light confinement enabled by this topology [1,6], up to thousands of times larger than k0. Essentially, this design maximizes the range of kx components radiated by the emitter that can be guided by the metasurface. Since we are considering small distances between the emitter and the HMTS, these components are not fully filtered by free space and a large portion of their power is coupled to the HMTS. In the case of nonlocal graphene, however, components with kρ>300k0 are forbidden, leading to a reduced SER. Contrary to the local case, Fig. 7(b) shows that for small distances, spatial dispersion causes the SER to increase with W, reaching an optimum at 41 nm. We have verified that this W corresponds precisely to the design that supports the widest range of wavenumbers, consistent with the conclusions in the nondispersive scenario. Note that for larger separations between emitter and HMTS both approaches leads to very similar results, due to the filtering of evanescent waves by free-space. Interestingly, these results also show that spatial dispersion does not necessarily cause a reduction of the SER for every HMTS design, although it does impose upper limits associated to the high wavevector cutoff.

 

Fig. 7 SER (in logarithm scale) of a z-oriented dipole as a function of its distance from an array of graphene strips, computed with EMA derived in Section II. (a)-(b) Results versus strip width W for local and nonlocal graphene responses, respectively. Parameters are L=50 nm, f=2.5 THz, μc=0.2 eV, and τ = 0.3 ps.

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In order to further investigate this limit, we investigate in Fig. 8(a) the SER of a z-oriented dipole versus its distance to a lossless homogenized metasurface. We have removed in this example the presence of losses, since the standard definition of SER [39] does not hold when the emitter in embedded in lossy media [40, 41]. As expected [7], this leads to a significant decrease of the emitter SER compared to the examples shown in Figs. 7(a)-7(b). In the local case, neglecting the influence of the granularity, the emitter SER diverges as the emitter gets closer to the HMTSs, ideally providing an infinite SER when the dipole is placed exactly on the sheet. The picture is drastically different when the nonlocal response of graphene is taken into account, since it removes the singularity and provides a finite and realistic response. This example clearly demonstrates how nonlocality imposes a fundamental limit to the response of hyperbolic metasurfaces. The features of this limit mainly depend on the Fermi velocity of theelectrons within the nonlocal 2D material, and increases as the strip width or chemical potential is reduced [see Fig. 8(b)]. As expected, emitters located on structures that support surface waves with elliptic dispersion provide SER orders of magnitude lower than in the hyperbolic case.

 

Fig. 8 Fundamental limits imposed by nonlocality to the SER of a z-oriented emitter. (a) SER (in logarithm scale) versus the dipole distance to the metasurface. The width of the graphene strips is set to W=15 nm and μc=0.2 eV. (b) SER (in logarithm scale) of an emitter located exactly on the HMTSs versus the structure physical dimensions and chemical potential. Parameters are L=100 nm, f=2.5 THz, and τ = 0.3 ps.

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Finally, we would like to emphasize that the qualitative behavior of nonlocal HMTSs discussed in this section and the fundamental limits that they impose on the LDOS and wave confinement arise by the finite Fermi velocity of electrons, and therefore they are not restricted to graphene-based nanostructures, but they hold for any HMTS.

4. Conclusions

In conclusion, we have investigated the nonlocal response of HMTSs in detail. To this purpose, we have derived an accurate EMA to homogenize nonlocal 2D metasurfaces assuming an infinitesimally small periodicity (L0). This approach has allowed us to remove the spatial dispersion effects introduced by the lattice in order to focus on the influence of the material nonlocality in the response of HMTSs. We do note that our approach constitutes a first approximation to the rigorous analysis of nonlocal HTMSs. More advanced models should take into account: (i) the influence of the lattice in the homogenization process [37], and (ii) an additional boundary condition able to impose the continuity of the tangential surface current in 2D media with intrinsic spatial dispersion [7, 30, 31]. Applying our EMA, we have demonstrated that material nonlocality imposes a wavenumber cutoff to the open isofrequency contour of HMTSs that is inversely proportional to the electron Fermi velocity, a phenomenon also found in bulk HMTMs [7]. We have then compared this wavenumber cutoff to the usual one arising from the metasurface granularity, delimiting the conditions under which each type of nonlocality dominates the HMTSs response. Finally, we have studied the influence of spatial dispersion in the SER of emitters located very close to HMTSs.

Our results are particularly relevant in case of graphene-based HMTSs operating at THz and near infrared frequencies. There, the intrinsic spatial dispersion of graphene becomes the dominant mechanism to close the hyperbolic contour – at around kρ300k0 – in practical implementations with periodicities L<π/(300k0), i.e., in the range of 500-20 nm. This result clearly illustrates the tremendous importance of nonlocal effects in the correct analysis and design of realistic HMTSs. We stress that similar graphene-based nanostructures have already been designed and fabricated for other purposes [36, 38]. These configurations may therefore provide an excellent scenario to indirectly measure graphene nonlocality. In the case of conventional HMTSs and HMTMs at optical wavelengths, we have found that the intrinsic nonlocal effects of metals are only relevant in configurations with periodicity below a few nanometers. Since these devices are usually very challenging to produce, even with modern nanofabrication techniques, this type of spatial dispersion is usually expected to be very weak in practice. Even though the intrinsic nonlocal effects of materials impose upper bounds to the field confinement and LDOS achievable in HMTSs, there is still plenty of room for exciting applications in realistic structures operating from THz to optics, including planar hyperlenses, imaging systems, sensors exploiting the extremely large (but finite) light-matter interactions, and on-chip networks.

Acknowledgments

The authors wish to thank Dr. D. Sounas (The University of Texas of Austin, US) for fruitful discussions. This work was supported by the Air Force Office of Scientific Research (AFOSR) with grant No. FA9550-13-1-0204, the Welch foundation with grant No. F-1802, and the National Science Foundation with grant No. ECCS-1406235.

References and links

1. J. S. Gomez-Diaz, M. Tymchenko, and A. Alù, “Hyperbolic Plasmons and Topological Transitions Over Uniaxial Metasurfaces,” Phys. Rev. Lett. 114(23), 233901 (2015). [CrossRef]   [PubMed]  

2. A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, “Hyperbolic metamaterials,” Nat. Photonics 7(12), 948–957 (2013). [CrossRef]  

3. V. P. Drachev, V. A. Podolskiy, and A. V. Kildishev, “Hyperbolic metamaterials: new physics behind a classical problem,” Opt. Express 21(12), 15048–15064 (2013). [CrossRef]   [PubMed]  

4. H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological transitions in metamaterials,” Science 336(6078), 205–209 (2012). [CrossRef]   [PubMed]  

5. A. A. High, R. C. Devlin, A. Dibos, M. Polking, D. S. Wild, J. Perczel, N. P. de Leon, M. D. Lukin, and H. Park, “Visible-frequency hyperbolic metasurface,” Nature 522(7555), 192–196 (2015). [CrossRef]   [PubMed]  

6. J. S. Gomez-Diaz, M. Tymchenko, and A. Alù, “Hyperbolic metasurfaces: surface plasmons, light-matter interactions, and physical implementation using graphene strips,” Opt. Mater. Express 5(10), 2313–2329 (2015). [CrossRef]  

7. W. Yan, M. Wubs, and N. A. Mortensen, “Hyperbolic metamaterials: Nonlocal response regularizes broadband supersingularity,” Phys. Rev. B 86(20), 205429 (2012). [CrossRef]  

8. O. Y. Yermakov, A. I. Ovcharenko, M. Song, A. A. Bogdanov, I. V. Iorsh, and Yu. S. Kivshar, “Hybrid waves localized at hyperbolic metasurfaces,” Phys. Rev. B 91(23), 235423 (2015). [CrossRef]  

9. I. Trushkov and I. Iorsh, “Two-dimensional hyperbolic medium for electrons and photons based on the array of tunnel-coupled graphene nanoribbons,” Phys. Rev. B 92(4), 045305 (2015). [CrossRef]  

10. F. H. L. Koppens, D. E. Chang, and F. J. García de Abajo, “Graphene Plasmonics: A Platform for Strong Light-Matter Interactions,” Nano Lett. 11(8), 3370–3377 (2011). [CrossRef]   [PubMed]  

11. F. J. García de Abajo, “Graphene Plasmonics: Challenges and Opportunities,” ACS Photonics 1(3), 135–152 (2014). [CrossRef]  

12. J. S. Gómez-Díaz and J. Perruisseau-Carrier, “Graphene-based plasmonic switches at near infrared frequencies,” Opt. Express 21(13), 15490–15504 (2013). [CrossRef]   [PubMed]  

13. J. S. Gómez-Díaz, M. Esquius-Morote, and J. Perruisseau-Carrier, “Plane wave excitation-detection of non-resonant plasmons along finite-width graphene strips,” Opt. Express 21(21), 24856–24872 (2013). [CrossRef]   [PubMed]  

14. P. Chen, H. Huang, D. Akinwande, and A. Alù, “Graphene-Based Plasmonic Platform for Reconfigurable Terahertz Nanodevices,” ACS Photonics 1(8), 647–654 (2014). [CrossRef]  

15. D. L. Sounas, H. S. Skulason, H. V. Nguyen, A. Guermoune, M. Siaj, T. Szkopek, and C. Caloz, “Faraday rotation in magnetically biased graphene at microwave frequencies,” Appl. Phys. Lett. 102(19), 191901 (2013). [CrossRef]  

16. S. Thongrattanasiri, F. H. L. Koppens, and F. J. García de Abajo, “Complete Optical Absorption in Periodically Patterned Graphene,” Phys. Rev. Lett. 108(4), 047401 (2012). [CrossRef]   [PubMed]  

17. L. A. Falkovsky and A. A. Varlamov, “Space-time dispersion of graphene conductivity,” Eur. Phys. J. B 56(4), 281–284 (2007). [CrossRef]  

18. M. Jablan, H. Buljan, and M. Soljacic, “Plasmonics in graphene at infrared frequencies,” Phys. Rev. B Condens. Matter 80(24), 245435 (2009). [CrossRef]  

19. G. Lovat, G. W. Hanson, R. Araneo, and P. Burghignoli, “Semiclassical spatially dispersive intraband conductivity tensor and quantum capacitance of graphene,” Phys. Rev. B Condens. Matter 87(11), 115429 (2013). [CrossRef]  

20. D. Correas-Serrano, J. S. Gomez-Diaz, and A. Alvarez-Melcon, “On the Influence of Spatial Dispersion on the Performance of Graphene-Based Plasmonic Devices,” IEEE Anten. Wirel. Propag. Lett. 13, 345–348 (2014).

21. D. Correas-Serrano, J. S. Gomez-Diaz, J. Perruisseau-Carrier, and A. Alvarez-Melcon, “Spatially Dispersive Graphene Single and Parallel Plate Waveguides: Analysis and Circuit Models,” IEEE Trans. Microw. Theory Tech. 61(12), 4333–4344 (2013). [CrossRef]  

22. J. S. Gomez-Diaz, J. Mosig, and A. Alvarez-Melcon, “Effect of Spatial Dispersion on Surface Waves Propagating Along Graphene Sheets,” IEEE Trans. Antenn. Propag. 61, 3589–3596 (2013).

23. A. Fallahi, T. Low, M. Tamagnone, and J. Perruisseau-Carrier, “Nonlocal electromagnetic response of graphene nanostructures,” Phys. Rev. B 91(12), 121405 (2015). [CrossRef]  

24. M. Tymchenko, A. Y. Nikitin, and L. Martín-Moreno, “Faraday Rotation Due to Excitation of Magnetoplasmons in Graphene Microribbons,” ACS Nano 7(11), 9780–9787 (2013). [CrossRef]   [PubMed]  

25. O. Luukkonen, C. Simovski, G. Grant, G. Goussetis, D. Lioubtchenko, A. Raisanen, and S. Tretyakov, “Simple and Accurate Analytical Model of Planar Grids and High-Impedance Surfaces Comprising Metal Strips or Patches,” IEEE Trans. Antenn. Propag. 56, 1624 (2008).

26. E. Forati, G. W. Hanson, A. B. Yakovlev, and A. Alù, “Planar hyperlens based on a modulated graphene monolayer,” Phys. Rev. B 89(8), 081410 (2014). [CrossRef]  

27. V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, “Magneto-optical conductivity in graphene,” J. Phys. Condens. Matter 19(2), 026222 (2007). [CrossRef]  

28. I. Crassee, M. Orlita, M. Potemski, A. L. Walter, M. Ostler, T. Seyller, I. Gaponenko, J. Chen, and A. B. Kuzmenko, “Intrinsic Terahertz Plasmons and Magnetoplasmons in Large Scale Monolayer Graphene,” Nano Lett. 12(5), 2470–2474 (2012). [CrossRef]   [PubMed]  

29. J. S. Gómez-Díaz and J. Perruisseau-Carrier, “Propagation of hybrid transverse magnetic-transverse electric plasmons on magnetically biased graphene sheets,” J. Appl. Phys. 112(12), 124906 (2012). [CrossRef]  

30. W. L. Mochan, M. del CastilloMussot, and R. G. Barrera, “Effect of plasma waves on the optical properties of metal-insulator superlattices,” Phys. Rev. B 35(3), 1088–1098 (1987). [CrossRef]   [PubMed]  

31. M. G. Silveirinha, “Additional boundary condition for the wire medium,” IEEE Trans. Antenn. Propag. 54, 1766–1780 (2006).

32. A. D. Boardman, Electromagnetic Surface Modes, John Wiley and Sons, Chichester (1982).

33. H. J. Bilow, “Guided Waves on a Planar Tensor Impedance Surface,” IEEE Trans. Antenn. Propag. 51(10), 2788–2792 (2003). [CrossRef]  

34. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C + + : the art of scientific computing, 3rd edition (Cambridge University Press, 2007).

35. L. Chen, C. Zhang, J. Zhou, and L. J. Guo, “Probing the Ultrathin Limit of Hyperbolic Meta-material: Nonlocality Induced Topological Transitions,” in CLEO:2015, OSA Technical Digest (online) (Optical Society of America, 2015), paper FTu4C.6.

36. D. Rodrigo, O. Limaj, D. Janner, D. Etezadi, F. J. García de Abajo, V. Pruneri, and H. Altug, “Mid-infrared plasmonic biosensing with graphene,” Science 349(6244), 165–168 (2015). [CrossRef]   [PubMed]  

37. A. Alù, “First-principles homogenization theory for periodic metamaterials,” Phys. Rev. B 84(7), 075153 (2011). [CrossRef]  

38. J. G. Son, M. Son, K. J. Moon, B. H. Lee, J. M. Myoung, M. S. Strano, M. H. Ham, and C. A. Ross, “Sub-10 nm Graphene Nanoribbon Array Field-Effect Transistors Fabricated by Block Copolymer Lithography,” Adv. Mater. 25(34), 4723–4728 (2013). [CrossRef]   [PubMed]  

39. L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University Press, 2006).

40. N. A. P. Nicorovici, R. C. McPhedran, and L. C. Botten, “Relative local density of states for homogeneous lossy materials,” Phys. B Condens. Matter 405, 2915 (2010).

41. O. Di Stefano, N. Fina, S. Savasta, R. Girlanda, and M. Pieruccini, “Calculation of the local optical density of states in absorbing and gain media,” J. Phys. Condens. Matter 22(31), 315302 (2010). [CrossRef]   [PubMed]  

References

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  1. J. S. Gomez-Diaz, M. Tymchenko, and A. Alù, “Hyperbolic Plasmons and Topological Transitions Over Uniaxial Metasurfaces,” Phys. Rev. Lett. 114(23), 233901 (2015).
    [Crossref] [PubMed]
  2. A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, “Hyperbolic metamaterials,” Nat. Photonics 7(12), 948–957 (2013).
    [Crossref]
  3. V. P. Drachev, V. A. Podolskiy, and A. V. Kildishev, “Hyperbolic metamaterials: new physics behind a classical problem,” Opt. Express 21(12), 15048–15064 (2013).
    [Crossref] [PubMed]
  4. H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological transitions in metamaterials,” Science 336(6078), 205–209 (2012).
    [Crossref] [PubMed]
  5. A. A. High, R. C. Devlin, A. Dibos, M. Polking, D. S. Wild, J. Perczel, N. P. de Leon, M. D. Lukin, and H. Park, “Visible-frequency hyperbolic metasurface,” Nature 522(7555), 192–196 (2015).
    [Crossref] [PubMed]
  6. J. S. Gomez-Diaz, M. Tymchenko, and A. Alù, “Hyperbolic metasurfaces: surface plasmons, light-matter interactions, and physical implementation using graphene strips,” Opt. Mater. Express 5(10), 2313–2329 (2015).
    [Crossref]
  7. W. Yan, M. Wubs, and N. A. Mortensen, “Hyperbolic metamaterials: Nonlocal response regularizes broadband supersingularity,” Phys. Rev. B 86(20), 205429 (2012).
    [Crossref]
  8. O. Y. Yermakov, A. I. Ovcharenko, M. Song, A. A. Bogdanov, I. V. Iorsh, and Yu. S. Kivshar, “Hybrid waves localized at hyperbolic metasurfaces,” Phys. Rev. B 91(23), 235423 (2015).
    [Crossref]
  9. I. Trushkov and I. Iorsh, “Two-dimensional hyperbolic medium for electrons and photons based on the array of tunnel-coupled graphene nanoribbons,” Phys. Rev. B 92(4), 045305 (2015).
    [Crossref]
  10. F. H. L. Koppens, D. E. Chang, and F. J. García de Abajo, “Graphene Plasmonics: A Platform for Strong Light-Matter Interactions,” Nano Lett. 11(8), 3370–3377 (2011).
    [Crossref] [PubMed]
  11. F. J. García de Abajo, “Graphene Plasmonics: Challenges and Opportunities,” ACS Photonics 1(3), 135–152 (2014).
    [Crossref]
  12. J. S. Gómez-Díaz and J. Perruisseau-Carrier, “Graphene-based plasmonic switches at near infrared frequencies,” Opt. Express 21(13), 15490–15504 (2013).
    [Crossref] [PubMed]
  13. J. S. Gómez-Díaz, M. Esquius-Morote, and J. Perruisseau-Carrier, “Plane wave excitation-detection of non-resonant plasmons along finite-width graphene strips,” Opt. Express 21(21), 24856–24872 (2013).
    [Crossref] [PubMed]
  14. P. Chen, H. Huang, D. Akinwande, and A. Alù, “Graphene-Based Plasmonic Platform for Reconfigurable Terahertz Nanodevices,” ACS Photonics 1(8), 647–654 (2014).
    [Crossref]
  15. D. L. Sounas, H. S. Skulason, H. V. Nguyen, A. Guermoune, M. Siaj, T. Szkopek, and C. Caloz, “Faraday rotation in magnetically biased graphene at microwave frequencies,” Appl. Phys. Lett. 102(19), 191901 (2013).
    [Crossref]
  16. S. Thongrattanasiri, F. H. L. Koppens, and F. J. García de Abajo, “Complete Optical Absorption in Periodically Patterned Graphene,” Phys. Rev. Lett. 108(4), 047401 (2012).
    [Crossref] [PubMed]
  17. L. A. Falkovsky and A. A. Varlamov, “Space-time dispersion of graphene conductivity,” Eur. Phys. J. B 56(4), 281–284 (2007).
    [Crossref]
  18. M. Jablan, H. Buljan, and M. Soljacic, “Plasmonics in graphene at infrared frequencies,” Phys. Rev. B Condens. Matter 80(24), 245435 (2009).
    [Crossref]
  19. G. Lovat, G. W. Hanson, R. Araneo, and P. Burghignoli, “Semiclassical spatially dispersive intraband conductivity tensor and quantum capacitance of graphene,” Phys. Rev. B Condens. Matter 87(11), 115429 (2013).
    [Crossref]
  20. D. Correas-Serrano, J. S. Gomez-Diaz, and A. Alvarez-Melcon, “On the Influence of Spatial Dispersion on the Performance of Graphene-Based Plasmonic Devices,” IEEE Anten. Wirel. Propag. Lett. 13, 345–348 (2014).
  21. D. Correas-Serrano, J. S. Gomez-Diaz, J. Perruisseau-Carrier, and A. Alvarez-Melcon, “Spatially Dispersive Graphene Single and Parallel Plate Waveguides: Analysis and Circuit Models,” IEEE Trans. Microw. Theory Tech. 61(12), 4333–4344 (2013).
    [Crossref]
  22. J. S. Gomez-Diaz, J. Mosig, and A. Alvarez-Melcon, “Effect of Spatial Dispersion on Surface Waves Propagating Along Graphene Sheets,” IEEE Trans. Antenn. Propag. 61, 3589–3596 (2013).
  23. A. Fallahi, T. Low, M. Tamagnone, and J. Perruisseau-Carrier, “Nonlocal electromagnetic response of graphene nanostructures,” Phys. Rev. B 91(12), 121405 (2015).
    [Crossref]
  24. M. Tymchenko, A. Y. Nikitin, and L. Martín-Moreno, “Faraday Rotation Due to Excitation of Magnetoplasmons in Graphene Microribbons,” ACS Nano 7(11), 9780–9787 (2013).
    [Crossref] [PubMed]
  25. O. Luukkonen, C. Simovski, G. Grant, G. Goussetis, D. Lioubtchenko, A. Raisanen, and S. Tretyakov, “Simple and Accurate Analytical Model of Planar Grids and High-Impedance Surfaces Comprising Metal Strips or Patches,” IEEE Trans. Antenn. Propag. 56, 1624 (2008).
  26. E. Forati, G. W. Hanson, A. B. Yakovlev, and A. Alù, “Planar hyperlens based on a modulated graphene monolayer,” Phys. Rev. B 89(8), 081410 (2014).
    [Crossref]
  27. V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, “Magneto-optical conductivity in graphene,” J. Phys. Condens. Matter 19(2), 026222 (2007).
    [Crossref]
  28. I. Crassee, M. Orlita, M. Potemski, A. L. Walter, M. Ostler, T. Seyller, I. Gaponenko, J. Chen, and A. B. Kuzmenko, “Intrinsic Terahertz Plasmons and Magnetoplasmons in Large Scale Monolayer Graphene,” Nano Lett. 12(5), 2470–2474 (2012).
    [Crossref] [PubMed]
  29. J. S. Gómez-Díaz and J. Perruisseau-Carrier, “Propagation of hybrid transverse magnetic-transverse electric plasmons on magnetically biased graphene sheets,” J. Appl. Phys. 112(12), 124906 (2012).
    [Crossref]
  30. W. L. Mochan, M. del CastilloMussot, and R. G. Barrera, “Effect of plasma waves on the optical properties of metal-insulator superlattices,” Phys. Rev. B 35(3), 1088–1098 (1987).
    [Crossref] [PubMed]
  31. M. G. Silveirinha, “Additional boundary condition for the wire medium,” IEEE Trans. Antenn. Propag. 54, 1766–1780 (2006).
  32. A. D. Boardman, Electromagnetic Surface Modes, John Wiley and Sons, Chichester (1982).
  33. H. J. Bilow, “Guided Waves on a Planar Tensor Impedance Surface,” IEEE Trans. Antenn. Propag. 51(10), 2788–2792 (2003).
    [Crossref]
  34. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C + + : the art of scientific computing, 3rd edition (Cambridge University Press, 2007).
  35. L. Chen, C. Zhang, J. Zhou, and L. J. Guo, “Probing the Ultrathin Limit of Hyperbolic Meta-material: Nonlocality Induced Topological Transitions,” in CLEO:2015, OSA Technical Digest (online) (Optical Society of America, 2015), paper FTu4C.6.
  36. D. Rodrigo, O. Limaj, D. Janner, D. Etezadi, F. J. García de Abajo, V. Pruneri, and H. Altug, “Mid-infrared plasmonic biosensing with graphene,” Science 349(6244), 165–168 (2015).
    [Crossref] [PubMed]
  37. A. Alù, “First-principles homogenization theory for periodic metamaterials,” Phys. Rev. B 84(7), 075153 (2011).
    [Crossref]
  38. J. G. Son, M. Son, K. J. Moon, B. H. Lee, J. M. Myoung, M. S. Strano, M. H. Ham, and C. A. Ross, “Sub-10 nm Graphene Nanoribbon Array Field-Effect Transistors Fabricated by Block Copolymer Lithography,” Adv. Mater. 25(34), 4723–4728 (2013).
    [Crossref] [PubMed]
  39. L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University Press, 2006).
  40. N. A. P. Nicorovici, R. C. McPhedran, and L. C. Botten, “Relative local density of states for homogeneous lossy materials,” Phys. B Condens. Matter 405, 2915 (2010).
  41. O. Di Stefano, N. Fina, S. Savasta, R. Girlanda, and M. Pieruccini, “Calculation of the local optical density of states in absorbing and gain media,” J. Phys. Condens. Matter 22(31), 315302 (2010).
    [Crossref] [PubMed]

2015 (7)

J. S. Gomez-Diaz, M. Tymchenko, and A. Alù, “Hyperbolic Plasmons and Topological Transitions Over Uniaxial Metasurfaces,” Phys. Rev. Lett. 114(23), 233901 (2015).
[Crossref] [PubMed]

A. A. High, R. C. Devlin, A. Dibos, M. Polking, D. S. Wild, J. Perczel, N. P. de Leon, M. D. Lukin, and H. Park, “Visible-frequency hyperbolic metasurface,” Nature 522(7555), 192–196 (2015).
[Crossref] [PubMed]

J. S. Gomez-Diaz, M. Tymchenko, and A. Alù, “Hyperbolic metasurfaces: surface plasmons, light-matter interactions, and physical implementation using graphene strips,” Opt. Mater. Express 5(10), 2313–2329 (2015).
[Crossref]

O. Y. Yermakov, A. I. Ovcharenko, M. Song, A. A. Bogdanov, I. V. Iorsh, and Yu. S. Kivshar, “Hybrid waves localized at hyperbolic metasurfaces,” Phys. Rev. B 91(23), 235423 (2015).
[Crossref]

I. Trushkov and I. Iorsh, “Two-dimensional hyperbolic medium for electrons and photons based on the array of tunnel-coupled graphene nanoribbons,” Phys. Rev. B 92(4), 045305 (2015).
[Crossref]

A. Fallahi, T. Low, M. Tamagnone, and J. Perruisseau-Carrier, “Nonlocal electromagnetic response of graphene nanostructures,” Phys. Rev. B 91(12), 121405 (2015).
[Crossref]

D. Rodrigo, O. Limaj, D. Janner, D. Etezadi, F. J. García de Abajo, V. Pruneri, and H. Altug, “Mid-infrared plasmonic biosensing with graphene,” Science 349(6244), 165–168 (2015).
[Crossref] [PubMed]

2014 (4)

D. Correas-Serrano, J. S. Gomez-Diaz, and A. Alvarez-Melcon, “On the Influence of Spatial Dispersion on the Performance of Graphene-Based Plasmonic Devices,” IEEE Anten. Wirel. Propag. Lett. 13, 345–348 (2014).

E. Forati, G. W. Hanson, A. B. Yakovlev, and A. Alù, “Planar hyperlens based on a modulated graphene monolayer,” Phys. Rev. B 89(8), 081410 (2014).
[Crossref]

P. Chen, H. Huang, D. Akinwande, and A. Alù, “Graphene-Based Plasmonic Platform for Reconfigurable Terahertz Nanodevices,” ACS Photonics 1(8), 647–654 (2014).
[Crossref]

F. J. García de Abajo, “Graphene Plasmonics: Challenges and Opportunities,” ACS Photonics 1(3), 135–152 (2014).
[Crossref]

2013 (10)

J. S. Gómez-Díaz and J. Perruisseau-Carrier, “Graphene-based plasmonic switches at near infrared frequencies,” Opt. Express 21(13), 15490–15504 (2013).
[Crossref] [PubMed]

J. S. Gómez-Díaz, M. Esquius-Morote, and J. Perruisseau-Carrier, “Plane wave excitation-detection of non-resonant plasmons along finite-width graphene strips,” Opt. Express 21(21), 24856–24872 (2013).
[Crossref] [PubMed]

G. Lovat, G. W. Hanson, R. Araneo, and P. Burghignoli, “Semiclassical spatially dispersive intraband conductivity tensor and quantum capacitance of graphene,” Phys. Rev. B Condens. Matter 87(11), 115429 (2013).
[Crossref]

D. L. Sounas, H. S. Skulason, H. V. Nguyen, A. Guermoune, M. Siaj, T. Szkopek, and C. Caloz, “Faraday rotation in magnetically biased graphene at microwave frequencies,” Appl. Phys. Lett. 102(19), 191901 (2013).
[Crossref]

A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, “Hyperbolic metamaterials,” Nat. Photonics 7(12), 948–957 (2013).
[Crossref]

V. P. Drachev, V. A. Podolskiy, and A. V. Kildishev, “Hyperbolic metamaterials: new physics behind a classical problem,” Opt. Express 21(12), 15048–15064 (2013).
[Crossref] [PubMed]

D. Correas-Serrano, J. S. Gomez-Diaz, J. Perruisseau-Carrier, and A. Alvarez-Melcon, “Spatially Dispersive Graphene Single and Parallel Plate Waveguides: Analysis and Circuit Models,” IEEE Trans. Microw. Theory Tech. 61(12), 4333–4344 (2013).
[Crossref]

J. S. Gomez-Diaz, J. Mosig, and A. Alvarez-Melcon, “Effect of Spatial Dispersion on Surface Waves Propagating Along Graphene Sheets,” IEEE Trans. Antenn. Propag. 61, 3589–3596 (2013).

M. Tymchenko, A. Y. Nikitin, and L. Martín-Moreno, “Faraday Rotation Due to Excitation of Magnetoplasmons in Graphene Microribbons,” ACS Nano 7(11), 9780–9787 (2013).
[Crossref] [PubMed]

J. G. Son, M. Son, K. J. Moon, B. H. Lee, J. M. Myoung, M. S. Strano, M. H. Ham, and C. A. Ross, “Sub-10 nm Graphene Nanoribbon Array Field-Effect Transistors Fabricated by Block Copolymer Lithography,” Adv. Mater. 25(34), 4723–4728 (2013).
[Crossref] [PubMed]

2012 (5)

I. Crassee, M. Orlita, M. Potemski, A. L. Walter, M. Ostler, T. Seyller, I. Gaponenko, J. Chen, and A. B. Kuzmenko, “Intrinsic Terahertz Plasmons and Magnetoplasmons in Large Scale Monolayer Graphene,” Nano Lett. 12(5), 2470–2474 (2012).
[Crossref] [PubMed]

J. S. Gómez-Díaz and J. Perruisseau-Carrier, “Propagation of hybrid transverse magnetic-transverse electric plasmons on magnetically biased graphene sheets,” J. Appl. Phys. 112(12), 124906 (2012).
[Crossref]

H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological transitions in metamaterials,” Science 336(6078), 205–209 (2012).
[Crossref] [PubMed]

W. Yan, M. Wubs, and N. A. Mortensen, “Hyperbolic metamaterials: Nonlocal response regularizes broadband supersingularity,” Phys. Rev. B 86(20), 205429 (2012).
[Crossref]

S. Thongrattanasiri, F. H. L. Koppens, and F. J. García de Abajo, “Complete Optical Absorption in Periodically Patterned Graphene,” Phys. Rev. Lett. 108(4), 047401 (2012).
[Crossref] [PubMed]

2011 (2)

F. H. L. Koppens, D. E. Chang, and F. J. García de Abajo, “Graphene Plasmonics: A Platform for Strong Light-Matter Interactions,” Nano Lett. 11(8), 3370–3377 (2011).
[Crossref] [PubMed]

A. Alù, “First-principles homogenization theory for periodic metamaterials,” Phys. Rev. B 84(7), 075153 (2011).
[Crossref]

2010 (2)

N. A. P. Nicorovici, R. C. McPhedran, and L. C. Botten, “Relative local density of states for homogeneous lossy materials,” Phys. B Condens. Matter 405, 2915 (2010).

O. Di Stefano, N. Fina, S. Savasta, R. Girlanda, and M. Pieruccini, “Calculation of the local optical density of states in absorbing and gain media,” J. Phys. Condens. Matter 22(31), 315302 (2010).
[Crossref] [PubMed]

2009 (1)

M. Jablan, H. Buljan, and M. Soljacic, “Plasmonics in graphene at infrared frequencies,” Phys. Rev. B Condens. Matter 80(24), 245435 (2009).
[Crossref]

2008 (1)

O. Luukkonen, C. Simovski, G. Grant, G. Goussetis, D. Lioubtchenko, A. Raisanen, and S. Tretyakov, “Simple and Accurate Analytical Model of Planar Grids and High-Impedance Surfaces Comprising Metal Strips or Patches,” IEEE Trans. Antenn. Propag. 56, 1624 (2008).

2007 (2)

V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, “Magneto-optical conductivity in graphene,” J. Phys. Condens. Matter 19(2), 026222 (2007).
[Crossref]

L. A. Falkovsky and A. A. Varlamov, “Space-time dispersion of graphene conductivity,” Eur. Phys. J. B 56(4), 281–284 (2007).
[Crossref]

2006 (1)

M. G. Silveirinha, “Additional boundary condition for the wire medium,” IEEE Trans. Antenn. Propag. 54, 1766–1780 (2006).

2003 (1)

H. J. Bilow, “Guided Waves on a Planar Tensor Impedance Surface,” IEEE Trans. Antenn. Propag. 51(10), 2788–2792 (2003).
[Crossref]

1987 (1)

W. L. Mochan, M. del CastilloMussot, and R. G. Barrera, “Effect of plasma waves on the optical properties of metal-insulator superlattices,” Phys. Rev. B 35(3), 1088–1098 (1987).
[Crossref] [PubMed]

Akinwande, D.

P. Chen, H. Huang, D. Akinwande, and A. Alù, “Graphene-Based Plasmonic Platform for Reconfigurable Terahertz Nanodevices,” ACS Photonics 1(8), 647–654 (2014).
[Crossref]

Altug, H.

D. Rodrigo, O. Limaj, D. Janner, D. Etezadi, F. J. García de Abajo, V. Pruneri, and H. Altug, “Mid-infrared plasmonic biosensing with graphene,” Science 349(6244), 165–168 (2015).
[Crossref] [PubMed]

Alù, A.

J. S. Gomez-Diaz, M. Tymchenko, and A. Alù, “Hyperbolic Plasmons and Topological Transitions Over Uniaxial Metasurfaces,” Phys. Rev. Lett. 114(23), 233901 (2015).
[Crossref] [PubMed]

J. S. Gomez-Diaz, M. Tymchenko, and A. Alù, “Hyperbolic metasurfaces: surface plasmons, light-matter interactions, and physical implementation using graphene strips,” Opt. Mater. Express 5(10), 2313–2329 (2015).
[Crossref]

P. Chen, H. Huang, D. Akinwande, and A. Alù, “Graphene-Based Plasmonic Platform for Reconfigurable Terahertz Nanodevices,” ACS Photonics 1(8), 647–654 (2014).
[Crossref]

E. Forati, G. W. Hanson, A. B. Yakovlev, and A. Alù, “Planar hyperlens based on a modulated graphene monolayer,” Phys. Rev. B 89(8), 081410 (2014).
[Crossref]

A. Alù, “First-principles homogenization theory for periodic metamaterials,” Phys. Rev. B 84(7), 075153 (2011).
[Crossref]

Alvarez-Melcon, A.

D. Correas-Serrano, J. S. Gomez-Diaz, and A. Alvarez-Melcon, “On the Influence of Spatial Dispersion on the Performance of Graphene-Based Plasmonic Devices,” IEEE Anten. Wirel. Propag. Lett. 13, 345–348 (2014).

J. S. Gomez-Diaz, J. Mosig, and A. Alvarez-Melcon, “Effect of Spatial Dispersion on Surface Waves Propagating Along Graphene Sheets,” IEEE Trans. Antenn. Propag. 61, 3589–3596 (2013).

D. Correas-Serrano, J. S. Gomez-Diaz, J. Perruisseau-Carrier, and A. Alvarez-Melcon, “Spatially Dispersive Graphene Single and Parallel Plate Waveguides: Analysis and Circuit Models,” IEEE Trans. Microw. Theory Tech. 61(12), 4333–4344 (2013).
[Crossref]

Araneo, R.

G. Lovat, G. W. Hanson, R. Araneo, and P. Burghignoli, “Semiclassical spatially dispersive intraband conductivity tensor and quantum capacitance of graphene,” Phys. Rev. B Condens. Matter 87(11), 115429 (2013).
[Crossref]

Barrera, R. G.

W. L. Mochan, M. del CastilloMussot, and R. G. Barrera, “Effect of plasma waves on the optical properties of metal-insulator superlattices,” Phys. Rev. B 35(3), 1088–1098 (1987).
[Crossref] [PubMed]

Belov, P.

A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, “Hyperbolic metamaterials,” Nat. Photonics 7(12), 948–957 (2013).
[Crossref]

Bilow, H. J.

H. J. Bilow, “Guided Waves on a Planar Tensor Impedance Surface,” IEEE Trans. Antenn. Propag. 51(10), 2788–2792 (2003).
[Crossref]

Bogdanov, A. A.

O. Y. Yermakov, A. I. Ovcharenko, M. Song, A. A. Bogdanov, I. V. Iorsh, and Yu. S. Kivshar, “Hybrid waves localized at hyperbolic metasurfaces,” Phys. Rev. B 91(23), 235423 (2015).
[Crossref]

Botten, L. C.

N. A. P. Nicorovici, R. C. McPhedran, and L. C. Botten, “Relative local density of states for homogeneous lossy materials,” Phys. B Condens. Matter 405, 2915 (2010).

Buljan, H.

M. Jablan, H. Buljan, and M. Soljacic, “Plasmonics in graphene at infrared frequencies,” Phys. Rev. B Condens. Matter 80(24), 245435 (2009).
[Crossref]

Burghignoli, P.

G. Lovat, G. W. Hanson, R. Araneo, and P. Burghignoli, “Semiclassical spatially dispersive intraband conductivity tensor and quantum capacitance of graphene,” Phys. Rev. B Condens. Matter 87(11), 115429 (2013).
[Crossref]

Caloz, C.

D. L. Sounas, H. S. Skulason, H. V. Nguyen, A. Guermoune, M. Siaj, T. Szkopek, and C. Caloz, “Faraday rotation in magnetically biased graphene at microwave frequencies,” Appl. Phys. Lett. 102(19), 191901 (2013).
[Crossref]

Carbotte, J. P.

V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, “Magneto-optical conductivity in graphene,” J. Phys. Condens. Matter 19(2), 026222 (2007).
[Crossref]

Chang, D. E.

F. H. L. Koppens, D. E. Chang, and F. J. García de Abajo, “Graphene Plasmonics: A Platform for Strong Light-Matter Interactions,” Nano Lett. 11(8), 3370–3377 (2011).
[Crossref] [PubMed]

Chen, J.

I. Crassee, M. Orlita, M. Potemski, A. L. Walter, M. Ostler, T. Seyller, I. Gaponenko, J. Chen, and A. B. Kuzmenko, “Intrinsic Terahertz Plasmons and Magnetoplasmons in Large Scale Monolayer Graphene,” Nano Lett. 12(5), 2470–2474 (2012).
[Crossref] [PubMed]

Chen, P.

P. Chen, H. Huang, D. Akinwande, and A. Alù, “Graphene-Based Plasmonic Platform for Reconfigurable Terahertz Nanodevices,” ACS Photonics 1(8), 647–654 (2014).
[Crossref]

Correas-Serrano, D.

D. Correas-Serrano, J. S. Gomez-Diaz, and A. Alvarez-Melcon, “On the Influence of Spatial Dispersion on the Performance of Graphene-Based Plasmonic Devices,” IEEE Anten. Wirel. Propag. Lett. 13, 345–348 (2014).

D. Correas-Serrano, J. S. Gomez-Diaz, J. Perruisseau-Carrier, and A. Alvarez-Melcon, “Spatially Dispersive Graphene Single and Parallel Plate Waveguides: Analysis and Circuit Models,” IEEE Trans. Microw. Theory Tech. 61(12), 4333–4344 (2013).
[Crossref]

Crassee, I.

I. Crassee, M. Orlita, M. Potemski, A. L. Walter, M. Ostler, T. Seyller, I. Gaponenko, J. Chen, and A. B. Kuzmenko, “Intrinsic Terahertz Plasmons and Magnetoplasmons in Large Scale Monolayer Graphene,” Nano Lett. 12(5), 2470–2474 (2012).
[Crossref] [PubMed]

de Leon, N. P.

A. A. High, R. C. Devlin, A. Dibos, M. Polking, D. S. Wild, J. Perczel, N. P. de Leon, M. D. Lukin, and H. Park, “Visible-frequency hyperbolic metasurface,” Nature 522(7555), 192–196 (2015).
[Crossref] [PubMed]

del CastilloMussot, M.

W. L. Mochan, M. del CastilloMussot, and R. G. Barrera, “Effect of plasma waves on the optical properties of metal-insulator superlattices,” Phys. Rev. B 35(3), 1088–1098 (1987).
[Crossref] [PubMed]

Devlin, R. C.

A. A. High, R. C. Devlin, A. Dibos, M. Polking, D. S. Wild, J. Perczel, N. P. de Leon, M. D. Lukin, and H. Park, “Visible-frequency hyperbolic metasurface,” Nature 522(7555), 192–196 (2015).
[Crossref] [PubMed]

Di Stefano, O.

O. Di Stefano, N. Fina, S. Savasta, R. Girlanda, and M. Pieruccini, “Calculation of the local optical density of states in absorbing and gain media,” J. Phys. Condens. Matter 22(31), 315302 (2010).
[Crossref] [PubMed]

Dibos, A.

A. A. High, R. C. Devlin, A. Dibos, M. Polking, D. S. Wild, J. Perczel, N. P. de Leon, M. D. Lukin, and H. Park, “Visible-frequency hyperbolic metasurface,” Nature 522(7555), 192–196 (2015).
[Crossref] [PubMed]

Drachev, V. P.

Esquius-Morote, M.

Etezadi, D.

D. Rodrigo, O. Limaj, D. Janner, D. Etezadi, F. J. García de Abajo, V. Pruneri, and H. Altug, “Mid-infrared plasmonic biosensing with graphene,” Science 349(6244), 165–168 (2015).
[Crossref] [PubMed]

Falkovsky, L. A.

L. A. Falkovsky and A. A. Varlamov, “Space-time dispersion of graphene conductivity,” Eur. Phys. J. B 56(4), 281–284 (2007).
[Crossref]

Fallahi, A.

A. Fallahi, T. Low, M. Tamagnone, and J. Perruisseau-Carrier, “Nonlocal electromagnetic response of graphene nanostructures,” Phys. Rev. B 91(12), 121405 (2015).
[Crossref]

Fina, N.

O. Di Stefano, N. Fina, S. Savasta, R. Girlanda, and M. Pieruccini, “Calculation of the local optical density of states in absorbing and gain media,” J. Phys. Condens. Matter 22(31), 315302 (2010).
[Crossref] [PubMed]

Forati, E.

E. Forati, G. W. Hanson, A. B. Yakovlev, and A. Alù, “Planar hyperlens based on a modulated graphene monolayer,” Phys. Rev. B 89(8), 081410 (2014).
[Crossref]

Gaponenko, I.

I. Crassee, M. Orlita, M. Potemski, A. L. Walter, M. Ostler, T. Seyller, I. Gaponenko, J. Chen, and A. B. Kuzmenko, “Intrinsic Terahertz Plasmons and Magnetoplasmons in Large Scale Monolayer Graphene,” Nano Lett. 12(5), 2470–2474 (2012).
[Crossref] [PubMed]

García de Abajo, F. J.

D. Rodrigo, O. Limaj, D. Janner, D. Etezadi, F. J. García de Abajo, V. Pruneri, and H. Altug, “Mid-infrared plasmonic biosensing with graphene,” Science 349(6244), 165–168 (2015).
[Crossref] [PubMed]

F. J. García de Abajo, “Graphene Plasmonics: Challenges and Opportunities,” ACS Photonics 1(3), 135–152 (2014).
[Crossref]

S. Thongrattanasiri, F. H. L. Koppens, and F. J. García de Abajo, “Complete Optical Absorption in Periodically Patterned Graphene,” Phys. Rev. Lett. 108(4), 047401 (2012).
[Crossref] [PubMed]

F. H. L. Koppens, D. E. Chang, and F. J. García de Abajo, “Graphene Plasmonics: A Platform for Strong Light-Matter Interactions,” Nano Lett. 11(8), 3370–3377 (2011).
[Crossref] [PubMed]

Girlanda, R.

O. Di Stefano, N. Fina, S. Savasta, R. Girlanda, and M. Pieruccini, “Calculation of the local optical density of states in absorbing and gain media,” J. Phys. Condens. Matter 22(31), 315302 (2010).
[Crossref] [PubMed]

Gomez-Diaz, J. S.

J. S. Gomez-Diaz, M. Tymchenko, and A. Alù, “Hyperbolic Plasmons and Topological Transitions Over Uniaxial Metasurfaces,” Phys. Rev. Lett. 114(23), 233901 (2015).
[Crossref] [PubMed]

J. S. Gomez-Diaz, M. Tymchenko, and A. Alù, “Hyperbolic metasurfaces: surface plasmons, light-matter interactions, and physical implementation using graphene strips,” Opt. Mater. Express 5(10), 2313–2329 (2015).
[Crossref]

D. Correas-Serrano, J. S. Gomez-Diaz, and A. Alvarez-Melcon, “On the Influence of Spatial Dispersion on the Performance of Graphene-Based Plasmonic Devices,” IEEE Anten. Wirel. Propag. Lett. 13, 345–348 (2014).

D. Correas-Serrano, J. S. Gomez-Diaz, J. Perruisseau-Carrier, and A. Alvarez-Melcon, “Spatially Dispersive Graphene Single and Parallel Plate Waveguides: Analysis and Circuit Models,” IEEE Trans. Microw. Theory Tech. 61(12), 4333–4344 (2013).
[Crossref]

J. S. Gomez-Diaz, J. Mosig, and A. Alvarez-Melcon, “Effect of Spatial Dispersion on Surface Waves Propagating Along Graphene Sheets,” IEEE Trans. Antenn. Propag. 61, 3589–3596 (2013).

Gómez-Díaz, J. S.

Goussetis, G.

O. Luukkonen, C. Simovski, G. Grant, G. Goussetis, D. Lioubtchenko, A. Raisanen, and S. Tretyakov, “Simple and Accurate Analytical Model of Planar Grids and High-Impedance Surfaces Comprising Metal Strips or Patches,” IEEE Trans. Antenn. Propag. 56, 1624 (2008).

Grant, G.

O. Luukkonen, C. Simovski, G. Grant, G. Goussetis, D. Lioubtchenko, A. Raisanen, and S. Tretyakov, “Simple and Accurate Analytical Model of Planar Grids and High-Impedance Surfaces Comprising Metal Strips or Patches,” IEEE Trans. Antenn. Propag. 56, 1624 (2008).

Guermoune, A.

D. L. Sounas, H. S. Skulason, H. V. Nguyen, A. Guermoune, M. Siaj, T. Szkopek, and C. Caloz, “Faraday rotation in magnetically biased graphene at microwave frequencies,” Appl. Phys. Lett. 102(19), 191901 (2013).
[Crossref]

Gusynin, V. P.

V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, “Magneto-optical conductivity in graphene,” J. Phys. Condens. Matter 19(2), 026222 (2007).
[Crossref]

Ham, M. H.

J. G. Son, M. Son, K. J. Moon, B. H. Lee, J. M. Myoung, M. S. Strano, M. H. Ham, and C. A. Ross, “Sub-10 nm Graphene Nanoribbon Array Field-Effect Transistors Fabricated by Block Copolymer Lithography,” Adv. Mater. 25(34), 4723–4728 (2013).
[Crossref] [PubMed]

Hanson, G. W.

E. Forati, G. W. Hanson, A. B. Yakovlev, and A. Alù, “Planar hyperlens based on a modulated graphene monolayer,” Phys. Rev. B 89(8), 081410 (2014).
[Crossref]

G. Lovat, G. W. Hanson, R. Araneo, and P. Burghignoli, “Semiclassical spatially dispersive intraband conductivity tensor and quantum capacitance of graphene,” Phys. Rev. B Condens. Matter 87(11), 115429 (2013).
[Crossref]

High, A. A.

A. A. High, R. C. Devlin, A. Dibos, M. Polking, D. S. Wild, J. Perczel, N. P. de Leon, M. D. Lukin, and H. Park, “Visible-frequency hyperbolic metasurface,” Nature 522(7555), 192–196 (2015).
[Crossref] [PubMed]

Huang, H.

P. Chen, H. Huang, D. Akinwande, and A. Alù, “Graphene-Based Plasmonic Platform for Reconfigurable Terahertz Nanodevices,” ACS Photonics 1(8), 647–654 (2014).
[Crossref]

Iorsh, I.

I. Trushkov and I. Iorsh, “Two-dimensional hyperbolic medium for electrons and photons based on the array of tunnel-coupled graphene nanoribbons,” Phys. Rev. B 92(4), 045305 (2015).
[Crossref]

A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, “Hyperbolic metamaterials,” Nat. Photonics 7(12), 948–957 (2013).
[Crossref]

Iorsh, I. V.

O. Y. Yermakov, A. I. Ovcharenko, M. Song, A. A. Bogdanov, I. V. Iorsh, and Yu. S. Kivshar, “Hybrid waves localized at hyperbolic metasurfaces,” Phys. Rev. B 91(23), 235423 (2015).
[Crossref]

Jablan, M.

M. Jablan, H. Buljan, and M. Soljacic, “Plasmonics in graphene at infrared frequencies,” Phys. Rev. B Condens. Matter 80(24), 245435 (2009).
[Crossref]

Jacob, Z.

H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological transitions in metamaterials,” Science 336(6078), 205–209 (2012).
[Crossref] [PubMed]

Janner, D.

D. Rodrigo, O. Limaj, D. Janner, D. Etezadi, F. J. García de Abajo, V. Pruneri, and H. Altug, “Mid-infrared plasmonic biosensing with graphene,” Science 349(6244), 165–168 (2015).
[Crossref] [PubMed]

Kildishev, A. V.

Kivshar, Y.

A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, “Hyperbolic metamaterials,” Nat. Photonics 7(12), 948–957 (2013).
[Crossref]

Kivshar, Yu. S.

O. Y. Yermakov, A. I. Ovcharenko, M. Song, A. A. Bogdanov, I. V. Iorsh, and Yu. S. Kivshar, “Hybrid waves localized at hyperbolic metasurfaces,” Phys. Rev. B 91(23), 235423 (2015).
[Crossref]

Koppens, F. H. L.

S. Thongrattanasiri, F. H. L. Koppens, and F. J. García de Abajo, “Complete Optical Absorption in Periodically Patterned Graphene,” Phys. Rev. Lett. 108(4), 047401 (2012).
[Crossref] [PubMed]

F. H. L. Koppens, D. E. Chang, and F. J. García de Abajo, “Graphene Plasmonics: A Platform for Strong Light-Matter Interactions,” Nano Lett. 11(8), 3370–3377 (2011).
[Crossref] [PubMed]

Kretzschmar, I.

H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological transitions in metamaterials,” Science 336(6078), 205–209 (2012).
[Crossref] [PubMed]

Krishnamoorthy, H. N. S.

H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological transitions in metamaterials,” Science 336(6078), 205–209 (2012).
[Crossref] [PubMed]

Kuzmenko, A. B.

I. Crassee, M. Orlita, M. Potemski, A. L. Walter, M. Ostler, T. Seyller, I. Gaponenko, J. Chen, and A. B. Kuzmenko, “Intrinsic Terahertz Plasmons and Magnetoplasmons in Large Scale Monolayer Graphene,” Nano Lett. 12(5), 2470–2474 (2012).
[Crossref] [PubMed]

Lee, B. H.

J. G. Son, M. Son, K. J. Moon, B. H. Lee, J. M. Myoung, M. S. Strano, M. H. Ham, and C. A. Ross, “Sub-10 nm Graphene Nanoribbon Array Field-Effect Transistors Fabricated by Block Copolymer Lithography,” Adv. Mater. 25(34), 4723–4728 (2013).
[Crossref] [PubMed]

Limaj, O.

D. Rodrigo, O. Limaj, D. Janner, D. Etezadi, F. J. García de Abajo, V. Pruneri, and H. Altug, “Mid-infrared plasmonic biosensing with graphene,” Science 349(6244), 165–168 (2015).
[Crossref] [PubMed]

Lioubtchenko, D.

O. Luukkonen, C. Simovski, G. Grant, G. Goussetis, D. Lioubtchenko, A. Raisanen, and S. Tretyakov, “Simple and Accurate Analytical Model of Planar Grids and High-Impedance Surfaces Comprising Metal Strips or Patches,” IEEE Trans. Antenn. Propag. 56, 1624 (2008).

Lovat, G.

G. Lovat, G. W. Hanson, R. Araneo, and P. Burghignoli, “Semiclassical spatially dispersive intraband conductivity tensor and quantum capacitance of graphene,” Phys. Rev. B Condens. Matter 87(11), 115429 (2013).
[Crossref]

Low, T.

A. Fallahi, T. Low, M. Tamagnone, and J. Perruisseau-Carrier, “Nonlocal electromagnetic response of graphene nanostructures,” Phys. Rev. B 91(12), 121405 (2015).
[Crossref]

Lukin, M. D.

A. A. High, R. C. Devlin, A. Dibos, M. Polking, D. S. Wild, J. Perczel, N. P. de Leon, M. D. Lukin, and H. Park, “Visible-frequency hyperbolic metasurface,” Nature 522(7555), 192–196 (2015).
[Crossref] [PubMed]

Luukkonen, O.

O. Luukkonen, C. Simovski, G. Grant, G. Goussetis, D. Lioubtchenko, A. Raisanen, and S. Tretyakov, “Simple and Accurate Analytical Model of Planar Grids and High-Impedance Surfaces Comprising Metal Strips or Patches,” IEEE Trans. Antenn. Propag. 56, 1624 (2008).

Martín-Moreno, L.

M. Tymchenko, A. Y. Nikitin, and L. Martín-Moreno, “Faraday Rotation Due to Excitation of Magnetoplasmons in Graphene Microribbons,” ACS Nano 7(11), 9780–9787 (2013).
[Crossref] [PubMed]

McPhedran, R. C.

N. A. P. Nicorovici, R. C. McPhedran, and L. C. Botten, “Relative local density of states for homogeneous lossy materials,” Phys. B Condens. Matter 405, 2915 (2010).

Menon, V. M.

H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological transitions in metamaterials,” Science 336(6078), 205–209 (2012).
[Crossref] [PubMed]

Mochan, W. L.

W. L. Mochan, M. del CastilloMussot, and R. G. Barrera, “Effect of plasma waves on the optical properties of metal-insulator superlattices,” Phys. Rev. B 35(3), 1088–1098 (1987).
[Crossref] [PubMed]

Moon, K. J.

J. G. Son, M. Son, K. J. Moon, B. H. Lee, J. M. Myoung, M. S. Strano, M. H. Ham, and C. A. Ross, “Sub-10 nm Graphene Nanoribbon Array Field-Effect Transistors Fabricated by Block Copolymer Lithography,” Adv. Mater. 25(34), 4723–4728 (2013).
[Crossref] [PubMed]

Mortensen, N. A.

W. Yan, M. Wubs, and N. A. Mortensen, “Hyperbolic metamaterials: Nonlocal response regularizes broadband supersingularity,” Phys. Rev. B 86(20), 205429 (2012).
[Crossref]

Mosig, J.

J. S. Gomez-Diaz, J. Mosig, and A. Alvarez-Melcon, “Effect of Spatial Dispersion on Surface Waves Propagating Along Graphene Sheets,” IEEE Trans. Antenn. Propag. 61, 3589–3596 (2013).

Myoung, J. M.

J. G. Son, M. Son, K. J. Moon, B. H. Lee, J. M. Myoung, M. S. Strano, M. H. Ham, and C. A. Ross, “Sub-10 nm Graphene Nanoribbon Array Field-Effect Transistors Fabricated by Block Copolymer Lithography,” Adv. Mater. 25(34), 4723–4728 (2013).
[Crossref] [PubMed]

Narimanov, E.

H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological transitions in metamaterials,” Science 336(6078), 205–209 (2012).
[Crossref] [PubMed]

Nguyen, H. V.

D. L. Sounas, H. S. Skulason, H. V. Nguyen, A. Guermoune, M. Siaj, T. Szkopek, and C. Caloz, “Faraday rotation in magnetically biased graphene at microwave frequencies,” Appl. Phys. Lett. 102(19), 191901 (2013).
[Crossref]

Nicorovici, N. A. P.

N. A. P. Nicorovici, R. C. McPhedran, and L. C. Botten, “Relative local density of states for homogeneous lossy materials,” Phys. B Condens. Matter 405, 2915 (2010).

Nikitin, A. Y.

M. Tymchenko, A. Y. Nikitin, and L. Martín-Moreno, “Faraday Rotation Due to Excitation of Magnetoplasmons in Graphene Microribbons,” ACS Nano 7(11), 9780–9787 (2013).
[Crossref] [PubMed]

Orlita, M.

I. Crassee, M. Orlita, M. Potemski, A. L. Walter, M. Ostler, T. Seyller, I. Gaponenko, J. Chen, and A. B. Kuzmenko, “Intrinsic Terahertz Plasmons and Magnetoplasmons in Large Scale Monolayer Graphene,” Nano Lett. 12(5), 2470–2474 (2012).
[Crossref] [PubMed]

Ostler, M.

I. Crassee, M. Orlita, M. Potemski, A. L. Walter, M. Ostler, T. Seyller, I. Gaponenko, J. Chen, and A. B. Kuzmenko, “Intrinsic Terahertz Plasmons and Magnetoplasmons in Large Scale Monolayer Graphene,” Nano Lett. 12(5), 2470–2474 (2012).
[Crossref] [PubMed]

Ovcharenko, A. I.

O. Y. Yermakov, A. I. Ovcharenko, M. Song, A. A. Bogdanov, I. V. Iorsh, and Yu. S. Kivshar, “Hybrid waves localized at hyperbolic metasurfaces,” Phys. Rev. B 91(23), 235423 (2015).
[Crossref]

Park, H.

A. A. High, R. C. Devlin, A. Dibos, M. Polking, D. S. Wild, J. Perczel, N. P. de Leon, M. D. Lukin, and H. Park, “Visible-frequency hyperbolic metasurface,” Nature 522(7555), 192–196 (2015).
[Crossref] [PubMed]

Perczel, J.

A. A. High, R. C. Devlin, A. Dibos, M. Polking, D. S. Wild, J. Perczel, N. P. de Leon, M. D. Lukin, and H. Park, “Visible-frequency hyperbolic metasurface,” Nature 522(7555), 192–196 (2015).
[Crossref] [PubMed]

Perruisseau-Carrier, J.

A. Fallahi, T. Low, M. Tamagnone, and J. Perruisseau-Carrier, “Nonlocal electromagnetic response of graphene nanostructures,” Phys. Rev. B 91(12), 121405 (2015).
[Crossref]

D. Correas-Serrano, J. S. Gomez-Diaz, J. Perruisseau-Carrier, and A. Alvarez-Melcon, “Spatially Dispersive Graphene Single and Parallel Plate Waveguides: Analysis and Circuit Models,” IEEE Trans. Microw. Theory Tech. 61(12), 4333–4344 (2013).
[Crossref]

J. S. Gómez-Díaz and J. Perruisseau-Carrier, “Graphene-based plasmonic switches at near infrared frequencies,” Opt. Express 21(13), 15490–15504 (2013).
[Crossref] [PubMed]

J. S. Gómez-Díaz, M. Esquius-Morote, and J. Perruisseau-Carrier, “Plane wave excitation-detection of non-resonant plasmons along finite-width graphene strips,” Opt. Express 21(21), 24856–24872 (2013).
[Crossref] [PubMed]

J. S. Gómez-Díaz and J. Perruisseau-Carrier, “Propagation of hybrid transverse magnetic-transverse electric plasmons on magnetically biased graphene sheets,” J. Appl. Phys. 112(12), 124906 (2012).
[Crossref]

Pieruccini, M.

O. Di Stefano, N. Fina, S. Savasta, R. Girlanda, and M. Pieruccini, “Calculation of the local optical density of states in absorbing and gain media,” J. Phys. Condens. Matter 22(31), 315302 (2010).
[Crossref] [PubMed]

Poddubny, A.

A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, “Hyperbolic metamaterials,” Nat. Photonics 7(12), 948–957 (2013).
[Crossref]

Podolskiy, V. A.

Polking, M.

A. A. High, R. C. Devlin, A. Dibos, M. Polking, D. S. Wild, J. Perczel, N. P. de Leon, M. D. Lukin, and H. Park, “Visible-frequency hyperbolic metasurface,” Nature 522(7555), 192–196 (2015).
[Crossref] [PubMed]

Potemski, M.

I. Crassee, M. Orlita, M. Potemski, A. L. Walter, M. Ostler, T. Seyller, I. Gaponenko, J. Chen, and A. B. Kuzmenko, “Intrinsic Terahertz Plasmons and Magnetoplasmons in Large Scale Monolayer Graphene,” Nano Lett. 12(5), 2470–2474 (2012).
[Crossref] [PubMed]

Pruneri, V.

D. Rodrigo, O. Limaj, D. Janner, D. Etezadi, F. J. García de Abajo, V. Pruneri, and H. Altug, “Mid-infrared plasmonic biosensing with graphene,” Science 349(6244), 165–168 (2015).
[Crossref] [PubMed]

Raisanen, A.

O. Luukkonen, C. Simovski, G. Grant, G. Goussetis, D. Lioubtchenko, A. Raisanen, and S. Tretyakov, “Simple and Accurate Analytical Model of Planar Grids and High-Impedance Surfaces Comprising Metal Strips or Patches,” IEEE Trans. Antenn. Propag. 56, 1624 (2008).

Rodrigo, D.

D. Rodrigo, O. Limaj, D. Janner, D. Etezadi, F. J. García de Abajo, V. Pruneri, and H. Altug, “Mid-infrared plasmonic biosensing with graphene,” Science 349(6244), 165–168 (2015).
[Crossref] [PubMed]

Ross, C. A.

J. G. Son, M. Son, K. J. Moon, B. H. Lee, J. M. Myoung, M. S. Strano, M. H. Ham, and C. A. Ross, “Sub-10 nm Graphene Nanoribbon Array Field-Effect Transistors Fabricated by Block Copolymer Lithography,” Adv. Mater. 25(34), 4723–4728 (2013).
[Crossref] [PubMed]

Savasta, S.

O. Di Stefano, N. Fina, S. Savasta, R. Girlanda, and M. Pieruccini, “Calculation of the local optical density of states in absorbing and gain media,” J. Phys. Condens. Matter 22(31), 315302 (2010).
[Crossref] [PubMed]

Seyller, T.

I. Crassee, M. Orlita, M. Potemski, A. L. Walter, M. Ostler, T. Seyller, I. Gaponenko, J. Chen, and A. B. Kuzmenko, “Intrinsic Terahertz Plasmons and Magnetoplasmons in Large Scale Monolayer Graphene,” Nano Lett. 12(5), 2470–2474 (2012).
[Crossref] [PubMed]

Sharapov, S. G.

V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, “Magneto-optical conductivity in graphene,” J. Phys. Condens. Matter 19(2), 026222 (2007).
[Crossref]

Siaj, M.

D. L. Sounas, H. S. Skulason, H. V. Nguyen, A. Guermoune, M. Siaj, T. Szkopek, and C. Caloz, “Faraday rotation in magnetically biased graphene at microwave frequencies,” Appl. Phys. Lett. 102(19), 191901 (2013).
[Crossref]

Silveirinha, M. G.

M. G. Silveirinha, “Additional boundary condition for the wire medium,” IEEE Trans. Antenn. Propag. 54, 1766–1780 (2006).

Simovski, C.

O. Luukkonen, C. Simovski, G. Grant, G. Goussetis, D. Lioubtchenko, A. Raisanen, and S. Tretyakov, “Simple and Accurate Analytical Model of Planar Grids and High-Impedance Surfaces Comprising Metal Strips or Patches,” IEEE Trans. Antenn. Propag. 56, 1624 (2008).

Skulason, H. S.

D. L. Sounas, H. S. Skulason, H. V. Nguyen, A. Guermoune, M. Siaj, T. Szkopek, and C. Caloz, “Faraday rotation in magnetically biased graphene at microwave frequencies,” Appl. Phys. Lett. 102(19), 191901 (2013).
[Crossref]

Soljacic, M.

M. Jablan, H. Buljan, and M. Soljacic, “Plasmonics in graphene at infrared frequencies,” Phys. Rev. B Condens. Matter 80(24), 245435 (2009).
[Crossref]

Son, J. G.

J. G. Son, M. Son, K. J. Moon, B. H. Lee, J. M. Myoung, M. S. Strano, M. H. Ham, and C. A. Ross, “Sub-10 nm Graphene Nanoribbon Array Field-Effect Transistors Fabricated by Block Copolymer Lithography,” Adv. Mater. 25(34), 4723–4728 (2013).
[Crossref] [PubMed]

Son, M.

J. G. Son, M. Son, K. J. Moon, B. H. Lee, J. M. Myoung, M. S. Strano, M. H. Ham, and C. A. Ross, “Sub-10 nm Graphene Nanoribbon Array Field-Effect Transistors Fabricated by Block Copolymer Lithography,” Adv. Mater. 25(34), 4723–4728 (2013).
[Crossref] [PubMed]

Song, M.

O. Y. Yermakov, A. I. Ovcharenko, M. Song, A. A. Bogdanov, I. V. Iorsh, and Yu. S. Kivshar, “Hybrid waves localized at hyperbolic metasurfaces,” Phys. Rev. B 91(23), 235423 (2015).
[Crossref]

Sounas, D. L.

D. L. Sounas, H. S. Skulason, H. V. Nguyen, A. Guermoune, M. Siaj, T. Szkopek, and C. Caloz, “Faraday rotation in magnetically biased graphene at microwave frequencies,” Appl. Phys. Lett. 102(19), 191901 (2013).
[Crossref]

Strano, M. S.

J. G. Son, M. Son, K. J. Moon, B. H. Lee, J. M. Myoung, M. S. Strano, M. H. Ham, and C. A. Ross, “Sub-10 nm Graphene Nanoribbon Array Field-Effect Transistors Fabricated by Block Copolymer Lithography,” Adv. Mater. 25(34), 4723–4728 (2013).
[Crossref] [PubMed]

Szkopek, T.

D. L. Sounas, H. S. Skulason, H. V. Nguyen, A. Guermoune, M. Siaj, T. Szkopek, and C. Caloz, “Faraday rotation in magnetically biased graphene at microwave frequencies,” Appl. Phys. Lett. 102(19), 191901 (2013).
[Crossref]

Tamagnone, M.

A. Fallahi, T. Low, M. Tamagnone, and J. Perruisseau-Carrier, “Nonlocal electromagnetic response of graphene nanostructures,” Phys. Rev. B 91(12), 121405 (2015).
[Crossref]

Thongrattanasiri, S.

S. Thongrattanasiri, F. H. L. Koppens, and F. J. García de Abajo, “Complete Optical Absorption in Periodically Patterned Graphene,” Phys. Rev. Lett. 108(4), 047401 (2012).
[Crossref] [PubMed]

Tretyakov, S.

O. Luukkonen, C. Simovski, G. Grant, G. Goussetis, D. Lioubtchenko, A. Raisanen, and S. Tretyakov, “Simple and Accurate Analytical Model of Planar Grids and High-Impedance Surfaces Comprising Metal Strips or Patches,” IEEE Trans. Antenn. Propag. 56, 1624 (2008).

Trushkov, I.

I. Trushkov and I. Iorsh, “Two-dimensional hyperbolic medium for electrons and photons based on the array of tunnel-coupled graphene nanoribbons,” Phys. Rev. B 92(4), 045305 (2015).
[Crossref]

Tymchenko, M.

J. S. Gomez-Diaz, M. Tymchenko, and A. Alù, “Hyperbolic metasurfaces: surface plasmons, light-matter interactions, and physical implementation using graphene strips,” Opt. Mater. Express 5(10), 2313–2329 (2015).
[Crossref]

J. S. Gomez-Diaz, M. Tymchenko, and A. Alù, “Hyperbolic Plasmons and Topological Transitions Over Uniaxial Metasurfaces,” Phys. Rev. Lett. 114(23), 233901 (2015).
[Crossref] [PubMed]

M. Tymchenko, A. Y. Nikitin, and L. Martín-Moreno, “Faraday Rotation Due to Excitation of Magnetoplasmons in Graphene Microribbons,” ACS Nano 7(11), 9780–9787 (2013).
[Crossref] [PubMed]

Varlamov, A. A.

L. A. Falkovsky and A. A. Varlamov, “Space-time dispersion of graphene conductivity,” Eur. Phys. J. B 56(4), 281–284 (2007).
[Crossref]

Walter, A. L.

I. Crassee, M. Orlita, M. Potemski, A. L. Walter, M. Ostler, T. Seyller, I. Gaponenko, J. Chen, and A. B. Kuzmenko, “Intrinsic Terahertz Plasmons and Magnetoplasmons in Large Scale Monolayer Graphene,” Nano Lett. 12(5), 2470–2474 (2012).
[Crossref] [PubMed]

Wild, D. S.

A. A. High, R. C. Devlin, A. Dibos, M. Polking, D. S. Wild, J. Perczel, N. P. de Leon, M. D. Lukin, and H. Park, “Visible-frequency hyperbolic metasurface,” Nature 522(7555), 192–196 (2015).
[Crossref] [PubMed]

Wubs, M.

W. Yan, M. Wubs, and N. A. Mortensen, “Hyperbolic metamaterials: Nonlocal response regularizes broadband supersingularity,” Phys. Rev. B 86(20), 205429 (2012).
[Crossref]

Yakovlev, A. B.

E. Forati, G. W. Hanson, A. B. Yakovlev, and A. Alù, “Planar hyperlens based on a modulated graphene monolayer,” Phys. Rev. B 89(8), 081410 (2014).
[Crossref]

Yan, W.

W. Yan, M. Wubs, and N. A. Mortensen, “Hyperbolic metamaterials: Nonlocal response regularizes broadband supersingularity,” Phys. Rev. B 86(20), 205429 (2012).
[Crossref]

Yermakov, O. Y.

O. Y. Yermakov, A. I. Ovcharenko, M. Song, A. A. Bogdanov, I. V. Iorsh, and Yu. S. Kivshar, “Hybrid waves localized at hyperbolic metasurfaces,” Phys. Rev. B 91(23), 235423 (2015).
[Crossref]

ACS Nano (1)

M. Tymchenko, A. Y. Nikitin, and L. Martín-Moreno, “Faraday Rotation Due to Excitation of Magnetoplasmons in Graphene Microribbons,” ACS Nano 7(11), 9780–9787 (2013).
[Crossref] [PubMed]

ACS Photonics (2)

F. J. García de Abajo, “Graphene Plasmonics: Challenges and Opportunities,” ACS Photonics 1(3), 135–152 (2014).
[Crossref]

P. Chen, H. Huang, D. Akinwande, and A. Alù, “Graphene-Based Plasmonic Platform for Reconfigurable Terahertz Nanodevices,” ACS Photonics 1(8), 647–654 (2014).
[Crossref]

Adv. Mater. (1)

J. G. Son, M. Son, K. J. Moon, B. H. Lee, J. M. Myoung, M. S. Strano, M. H. Ham, and C. A. Ross, “Sub-10 nm Graphene Nanoribbon Array Field-Effect Transistors Fabricated by Block Copolymer Lithography,” Adv. Mater. 25(34), 4723–4728 (2013).
[Crossref] [PubMed]

Appl. Phys. Lett. (1)

D. L. Sounas, H. S. Skulason, H. V. Nguyen, A. Guermoune, M. Siaj, T. Szkopek, and C. Caloz, “Faraday rotation in magnetically biased graphene at microwave frequencies,” Appl. Phys. Lett. 102(19), 191901 (2013).
[Crossref]

Eur. Phys. J. B (1)

L. A. Falkovsky and A. A. Varlamov, “Space-time dispersion of graphene conductivity,” Eur. Phys. J. B 56(4), 281–284 (2007).
[Crossref]

IEEE Anten. Wirel. Propag. Lett. (1)

D. Correas-Serrano, J. S. Gomez-Diaz, and A. Alvarez-Melcon, “On the Influence of Spatial Dispersion on the Performance of Graphene-Based Plasmonic Devices,” IEEE Anten. Wirel. Propag. Lett. 13, 345–348 (2014).

IEEE Trans. Antenn. Propag. (4)

M. G. Silveirinha, “Additional boundary condition for the wire medium,” IEEE Trans. Antenn. Propag. 54, 1766–1780 (2006).

H. J. Bilow, “Guided Waves on a Planar Tensor Impedance Surface,” IEEE Trans. Antenn. Propag. 51(10), 2788–2792 (2003).
[Crossref]

O. Luukkonen, C. Simovski, G. Grant, G. Goussetis, D. Lioubtchenko, A. Raisanen, and S. Tretyakov, “Simple and Accurate Analytical Model of Planar Grids and High-Impedance Surfaces Comprising Metal Strips or Patches,” IEEE Trans. Antenn. Propag. 56, 1624 (2008).

J. S. Gomez-Diaz, J. Mosig, and A. Alvarez-Melcon, “Effect of Spatial Dispersion on Surface Waves Propagating Along Graphene Sheets,” IEEE Trans. Antenn. Propag. 61, 3589–3596 (2013).

IEEE Trans. Microw. Theory Tech. (1)

D. Correas-Serrano, J. S. Gomez-Diaz, J. Perruisseau-Carrier, and A. Alvarez-Melcon, “Spatially Dispersive Graphene Single and Parallel Plate Waveguides: Analysis and Circuit Models,” IEEE Trans. Microw. Theory Tech. 61(12), 4333–4344 (2013).
[Crossref]

J. Appl. Phys. (1)

J. S. Gómez-Díaz and J. Perruisseau-Carrier, “Propagation of hybrid transverse magnetic-transverse electric plasmons on magnetically biased graphene sheets,” J. Appl. Phys. 112(12), 124906 (2012).
[Crossref]

J. Phys. Condens. Matter (2)

V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, “Magneto-optical conductivity in graphene,” J. Phys. Condens. Matter 19(2), 026222 (2007).
[Crossref]

O. Di Stefano, N. Fina, S. Savasta, R. Girlanda, and M. Pieruccini, “Calculation of the local optical density of states in absorbing and gain media,” J. Phys. Condens. Matter 22(31), 315302 (2010).
[Crossref] [PubMed]

Nano Lett. (2)

I. Crassee, M. Orlita, M. Potemski, A. L. Walter, M. Ostler, T. Seyller, I. Gaponenko, J. Chen, and A. B. Kuzmenko, “Intrinsic Terahertz Plasmons and Magnetoplasmons in Large Scale Monolayer Graphene,” Nano Lett. 12(5), 2470–2474 (2012).
[Crossref] [PubMed]

F. H. L. Koppens, D. E. Chang, and F. J. García de Abajo, “Graphene Plasmonics: A Platform for Strong Light-Matter Interactions,” Nano Lett. 11(8), 3370–3377 (2011).
[Crossref] [PubMed]

Nat. Photonics (1)

A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, “Hyperbolic metamaterials,” Nat. Photonics 7(12), 948–957 (2013).
[Crossref]

Nature (1)

A. A. High, R. C. Devlin, A. Dibos, M. Polking, D. S. Wild, J. Perczel, N. P. de Leon, M. D. Lukin, and H. Park, “Visible-frequency hyperbolic metasurface,” Nature 522(7555), 192–196 (2015).
[Crossref] [PubMed]

Opt. Express (3)

Opt. Mater. Express (1)

Phys. B Condens. Matter (1)

N. A. P. Nicorovici, R. C. McPhedran, and L. C. Botten, “Relative local density of states for homogeneous lossy materials,” Phys. B Condens. Matter 405, 2915 (2010).

Phys. Rev. B (7)

W. Yan, M. Wubs, and N. A. Mortensen, “Hyperbolic metamaterials: Nonlocal response regularizes broadband supersingularity,” Phys. Rev. B 86(20), 205429 (2012).
[Crossref]

O. Y. Yermakov, A. I. Ovcharenko, M. Song, A. A. Bogdanov, I. V. Iorsh, and Yu. S. Kivshar, “Hybrid waves localized at hyperbolic metasurfaces,” Phys. Rev. B 91(23), 235423 (2015).
[Crossref]

I. Trushkov and I. Iorsh, “Two-dimensional hyperbolic medium for electrons and photons based on the array of tunnel-coupled graphene nanoribbons,” Phys. Rev. B 92(4), 045305 (2015).
[Crossref]

W. L. Mochan, M. del CastilloMussot, and R. G. Barrera, “Effect of plasma waves on the optical properties of metal-insulator superlattices,” Phys. Rev. B 35(3), 1088–1098 (1987).
[Crossref] [PubMed]

A. Fallahi, T. Low, M. Tamagnone, and J. Perruisseau-Carrier, “Nonlocal electromagnetic response of graphene nanostructures,” Phys. Rev. B 91(12), 121405 (2015).
[Crossref]

E. Forati, G. W. Hanson, A. B. Yakovlev, and A. Alù, “Planar hyperlens based on a modulated graphene monolayer,” Phys. Rev. B 89(8), 081410 (2014).
[Crossref]

A. Alù, “First-principles homogenization theory for periodic metamaterials,” Phys. Rev. B 84(7), 075153 (2011).
[Crossref]

Phys. Rev. B Condens. Matter (2)

M. Jablan, H. Buljan, and M. Soljacic, “Plasmonics in graphene at infrared frequencies,” Phys. Rev. B Condens. Matter 80(24), 245435 (2009).
[Crossref]

G. Lovat, G. W. Hanson, R. Araneo, and P. Burghignoli, “Semiclassical spatially dispersive intraband conductivity tensor and quantum capacitance of graphene,” Phys. Rev. B Condens. Matter 87(11), 115429 (2013).
[Crossref]

Phys. Rev. Lett. (2)

S. Thongrattanasiri, F. H. L. Koppens, and F. J. García de Abajo, “Complete Optical Absorption in Periodically Patterned Graphene,” Phys. Rev. Lett. 108(4), 047401 (2012).
[Crossref] [PubMed]

J. S. Gomez-Diaz, M. Tymchenko, and A. Alù, “Hyperbolic Plasmons and Topological Transitions Over Uniaxial Metasurfaces,” Phys. Rev. Lett. 114(23), 233901 (2015).
[Crossref] [PubMed]

Science (2)

H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological transitions in metamaterials,” Science 336(6078), 205–209 (2012).
[Crossref] [PubMed]

D. Rodrigo, O. Limaj, D. Janner, D. Etezadi, F. J. García de Abajo, V. Pruneri, and H. Altug, “Mid-infrared plasmonic biosensing with graphene,” Science 349(6244), 165–168 (2015).
[Crossref] [PubMed]

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W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C + + : the art of scientific computing, 3rd edition (Cambridge University Press, 2007).

L. Chen, C. Zhang, J. Zhou, and L. J. Guo, “Probing the Ultrathin Limit of Hyperbolic Meta-material: Nonlocality Induced Topological Transitions,” in CLEO:2015, OSA Technical Digest (online) (Optical Society of America, 2015), paper FTu4C.6.

A. D. Boardman, Electromagnetic Surface Modes, John Wiley and Sons, Chichester (1982).

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Figures (8)

Fig. 1
Fig. 1 Schematic of a hyperbolic metasurface composed of 2D strips with widths W and unit cell period L. The inset shows an equivalent representation of one unit-cell modelled by two strips described by effective conductivity tensors σ ¯ ¯ and σ C ¯ ¯ .
Fig. 2
Fig. 2 Nonlocal conductivity tensor elements of graphene in Cartesian coordinates, normalized versus the material local response. Results are given as a function of real k x / k 0 and k y / k 0 in-plane wavenumbers. Other parameters are f=2.5 THz,   μ c =0.02 eV , and τ=0.3 ps.
Fig. 3
Fig. 3 Imaginary parts of the (a) longitudinal and (b) transverse nonlocal conductivity components of graphene normalized versus the local response. Results are given as a function of real k x / k 0 and k y / k 0 in-plane wavenumbers. Other parameters are f=2.5 THz,  μ c =0.02 eV , and τ=0.3 ps
Fig. 4
Fig. 4 Effective conductivity tensor of a nonlocal graphene-based HMTS. The structure comprises an array of spatially-dispersive graphene strips with width W=50 nm and period L=100 nm. Other parameters are f=2.5 THz, μ c =0.2 eV, and τ=0.3 ps.
Fig. 5
Fig. 5 Isofrequency contours of the hyperbolic plasmons supported by an array of graphene strips with W=0.5 L for (a) L=30 nm, (b) L=100 nm, (c) L=300 nm, and (d) L=700 nm. Solid blue (green) lines computed with EMA considering local (nonlocal) graphene conductivity σ g ¯ ¯ . Dashed red lines computed using a full-wave mode matching approach with local σ g ¯ ¯ . Markers computed with FEM eigenfrequency solver COMSOL Multyphysics using nonlocal σ g ¯ ¯ .
Fig. 6
Fig. 6 Isofrequency contours of plasmons supported by an array of graphene strips versus (a) strip width W ( L=100 nm, f=4 THz,   μ c =0.02 eV, τ = 0.3 ps) and (b) electron relaxation time τ ( L=100 nm, W=30 nm, f=2.5 THz, μ c =0.3 eV). (c)-(d) shows similar results versus chemical potential μ c with (c) L=100 nm, W=25 nm, and f=2.5 THz and (d) L=120 nm, W=110 nm, and f=4.0 THz. Solid (dashed) lines computed with EMA considering local (nonlocal) graphene conductivity σ g ¯ ¯ .
Fig. 7
Fig. 7 SER (in logarithm scale) of a z-oriented dipole as a function of its distance from an array of graphene strips, computed with EMA derived in Section II. (a)-(b) Results versus strip width W for local and nonlocal graphene responses, respectively. Parameters are L=50 nm, f=2.5 THz, μ c =0.2 eV, and τ = 0.3 ps.
Fig. 8
Fig. 8 Fundamental limits imposed by nonlocality to the SER of a z-oriented emitter. (a) SER (in logarithm scale) versus the dipole distance to the metasurface. The width of the graphene strips is set to W=15 nm and μ c =0.2 eV. (b) SER (in logarithm scale) of an emitter located exactly on the HMTSs versus the structure physical dimensions and chemical potential. Parameters are L=100 nm, f=2.5 THz, and τ = 0.3 ps.

Equations (21)

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σ m ¯ ¯ =( σ xx m σ xy m σ yx m σ yy m )
σ c ¯ ¯ =( σ xx c 0 0 0 )    with   σ xx c =i ω ε 0 ε eff L π ln[ csc( πG 2L ) ]
E x ( x )= J x 1 σ xx ( x ) E y σ xy σ xx ( x ) , J y ( x )= J x σ yx ( x ) σ xx ( x ) + E y [ σ yy ( x ) σ yx ( x ) σ xy ( x ) σ xx ( x ) ],
1 σ xx eff = 1 L L 1 σ xx (x) dx , σ xy eff = σ xx eff L L σ xy (x) σ xx (x) dx , σ xy eff = σ xx eff L L σ xy (x) σ xx (x) dx ,
σ yy eff = 1 L L σ yy (x)dx 1 L L σ xy (x) σ yx (x) σ xx (x) dx + σ xy eff σ yx eff σ xx eff ,
σ xx eff = L σ xx m σ xx c W σ xx c +(LW) σ xx g , σ xy,eff = σ xy,eff = σ xx,eff W L σ xy m σ xx m .
σ yy eff = W L σ yy m W L σ yx m σ xy m σ xx m σ xx0 + σ yx eff σ xy eff σ xx eff .
σ xx g ( k x , k y )=γ I ϕ xx + γ D Δ k y ( I ϕ xx k y I ϕ yx k x ) D σ ,
σ xy g ( k x , k y )=γ I ϕ xy + γ D Δ k y ( I ϕ xy k y I ϕ yy k x ) D σ ,
σ yx g ( k x , k y )=γ I ϕ yx + γ D Δ k x ( I ϕ yx k x I ϕ xx k y ) D σ ,
σ yy g ( k x , k y )=γ I ϕ yy + γ D Δ k x ( I ϕ yy k x I ϕ xy k y ) D σ ,
I ϕ xx ( k x , k y )=2π v F 2 k y 2 k ρ 2 Rα v F k x k q 2 α 2 k q 2 (1R) v F 2 (α+ v F k x ) k ρ 4 ,
I ϕ xy ( k x , k y )= I ϕ yx ( k x , k y )=2π k x k y v F 2 k ρ 2 R+2α v F k x + α 2 k q 2 (1R) v F 2 (α+ v F k x ) k ρ 4 ,
I ϕ yy ( k x , k y )=2π v F 2 k x 2 k ρ 2 R+α v F k x k q 2 + α 2 k q 2 (1R) v F 2 (α+ v F k x ) k ρ 4 ,
γ=i q e 2 k B T π 2 2 log{ 2[ 1+cosh( μ c k B T ) ] }, γ D =i v F 2πωτ , D σ =1+ γ D Δ k ρ 2 ,
Δ= 2π v F k ρ 2 ( 1 α α 2 v F 2 k ρ 2 ),R( k x , k y )= α+ v F k x α 2 v F 2 k ρ 2 ,α=ω+i/τ,
k ρ 2 = k x 2 + k y 2 , k q 2 = k x 2 k y 2
σ d ¯ ¯ ( k ρ )=( σ ρ ( k ρ ) 0 0 σ ϕ ( k ρ ) ),
ε m ρ ( k ρ )=1 ω p 2 ω 2 +iωγ β 2 k ρ 2 , ε m ϕ ( k ρ )=1 ω p 2 ω 2 +iωγ ,
σ eff ¯ ¯ ( k x , k y )=( σ xx eff ( k x , k y ) σ xy eff ( k x , k y ) σ yx eff ( k x , k y ) σ yy eff ( k x , k y ) ).
k 0 k z [ 4+ η 0 2 { σ xx eff ( k x , k y ) σ yy eff ( k x , k y ) σ xy eff ( k x , k y ) σ yx eff ( k x , k y ) } ]+2 k 0 2 η 0 [ σ xx eff ( k x , k y )+ σ yy eff ( k x , k y ) ] 2 η 0 [ σ xx eff ( k x , k y ) k x 2 + σ yy eff ( k x , k y ) k y 2 + k x k y ( σ xy eff ( k x , k y )+ σ xy eff ( k x , k y ) ) ]=0,

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