Hyperbolic metasurfaces: surface plasmons, light-matter interactions, and physical implementation using graphene strips [Invited]

J. S. Gomez-Diaz; M. Tymchenko; A. Alù

doi:10.1364/OME.5.002313

1. Introduction

The unique electromagnetic properties of hyperbolic metamaterials (HMTMs) [1–5] have enabled a wide variety of applications, including lensing with subwavelength resolution [6–8], negative refraction [4, 9, 10], large absorption [4], heat transfer [11], and strong and broadband enhancement of the density of states, able to boost the efficiency of nearby emitters [4, 12, 13]. HMTMs respond as a dielectric for electric fields polarized along one direction, and as a metal in the perpendicular one, and they can be usually characterized using an anisotropic uniaxial permittivity tensor with longitudinal and transversal components of opposite sign. Several practical HMTM implementations have been proposed and analyzed, including wire-media [4], stacks of multilayers alternating dielectrics and metals at optics [3,4,8], or dielectrics and graphene sheets in the terahertz band [14,15], and transmission lines loaded with lumped components at RF and microwave frequencies [16]. Even though these metamaterials have been shown to provide unconventional ways to manipulate and control light, they require complex fabrication techniques and suffer from volumetric dissipative losses – especially at high frequencies – that have significantly hindered their performance. Ultrathin metasurfaces (MTSs) [17–20], the two dimensional counterpart of metamaterials (MTMs) [21], have recently been proposed to alleviate akin problems arising in bulk MTMs, while providing similar or superior functionalities and being fully compatible with integrated circuits and optoelectronic components. Graphene plasmonics [22,23] has further provided unusual and exciting possibilities to MTSs, enabling a platform for guided and radiative reconfigurable devices and applications [24–28] based on surface plasmons polaritons (SPPs) [29].

Merging the concept of bulk hyperbolic metamaterials and ultrathin metasurfaces has recently led to hyperbolic and extremely anisotropic $σ$ -near zero uniaxial metasurfaces [30–32]. In particular, Ref [30]. analyzed the properties of such structures, including extreme confinement of the supported SPPs and their channeling towards specific directions within the sheet, topological transitions, and the associated dramatic enhancement of light-matter interactions. It also proposed a simple and effective implementation at terahertz and near infrared frequencies using a densely-packed array of graphene strips. Other implementations, based on anisotropic subwavelength scatters, have also been envisaged [31]. Simultaneously, Ref [32]. reported the experimental realization of hyperbolic metasurfaces at visible frequencies using single-crystalline silver nanostructures, demonstrating characteristics such as negative refraction and diffraction-free propagation over the surface.

In this contribution, we first present an in-depth study of hyperbolic and extremely anisotropic $σ$ -near-zero ultrathin metasurfaces defined by an arbitrary fully-populated conductivity tensor. To this purpose, we formulate the Green’s function of a dipole placed above an anisotropic 2D MTS, an approach that provides useful physical insights, allows an easy derivation of the SPP dispersion relation and spontaneous emission rate (SER) or Purcell factor. The dispersion relation is solved using a new formulation that avoids searching for roots in the complex plane, and is then asymptotically approximated to obtain an analytical closed-form expression that establishes a clear connection between the conductivity components and the HMTS optical response. Then, we apply this general approach to investigate i) isotropic elliptic MTSs, implemented by electrically and magnetically-biased graphene sheets, and ii) strongly anisotropic elliptic and hyperbolic metasurfaces, implemented by an array of densely-packed graphene strips and characterized using an effective medium approach. Specifically, we analyze the features of the extremely confined plasmons supported by these structures, and we demonstrate spontaneous emission enhancement of quantum emitters using realistic configurations. Furthermore, we exploit the inherent reconfiguration capabilities of graphene to propose practical metasurfaces able to electrically control their topology, i.e. from elliptic to hyperbolic, going through the $σ$ -near-zero scenario, associated with a large control of the guided plasmons. We finish our study by delimiting the accuracy of our analytical homogenized model, analyzing the influence of the strip periodicity in the actual HMTS response. Our results open the door to novel and exciting phenomena for reconfigurable devices in the field of nanophotonics, including extreme miniaturization of optical components, planar transceivers and nanoantennas for communication and tagging systems, enhanced resolution in sensors and detectors, near-field imaging, hyperlenses, and on-chip networks.

2. Wave propagation in anisotropic metasurfaces

In this section, we first describe and analyze different topologies supported by anisotropic MTSs. Then, we develop a Green’s function approach to rigorously investigate their electromagnetic features, including the associated enhancement of light-matter interactions and the dispersion relation of the supported SPPs.

2.1 Metasurface topologies

The electromagnetic response of an infinitesimally-thin homogeneous anisotropic metasurface can be described by the general conductivity tensor

\bar{\bar{σ}} = (\begin{matrix} σ_{x x} & σ_{x y} \\ σ_{y x} & σ_{y y} \end{matrix}),

in which the tensor elements are generally complex. In what follows, we restrict ourselves to passive metasurfaces, requiring

Re [σ_{x x}] \geq 0

,

Re [σ_{y y}] \geq 0

, and

Re [σ_{x x} + σ_{y y}] \geq | σ_{x y} + σ_{y x}^{*} |

[33], and we also assume an

e^{- i ω t}

time convention. Strictly speaking, in order to determine the metasurface topology, the tensor

\bar{\bar{σ}}

must be diagonal in our reference coordinate system. The presence of non-diagonal terms can arise due to i) magneto-optic effects [34, 35], leading to an in-plane gyrotropic response that usually entails a change in SPP polarization, ii) shape and possible rotation of the subwavelength inclusions with respect to the reference coordinate system, associated with a bending of the SPP direction of propagation within the sheet. We focus here on lossless metasurfaces, whose topology depends on the signs of

Im [σ_{x x}]

and

Im [σ_{y y}]

. We do note, however, that the presence of loss may relax the topological conditions, especially in case of extreme anisotropy.

Figure 1 illustrates the $E_{z}$ field distribution of SPPs propagating along metasurfaces with different topologies and excited at the origin of the reference system. Specifically, Fig. 1(a) refers to an isotropic elliptic topology, which allows wave propagation along all directions within the sheet with similar properties. This topology arises when $sgn (Im [σ_{y y}]) = sgn (Im [σ_{x x}])$ , and it can implement either inductive ( $Im [σ_{x x}], Im [σ_{y y}] > 0$ ) or capacitive ( $Im [σ_{x x}], Im [σ_{y y}] < 0$ ) metasurfaces. In case that the signs of the imaginary parts are still equal but $σ_{x x} \neq σ_{y y}$ , the structure presents an anisotropic elliptical topology, which may favor SPP propagation towards a specific direction. Figure 1(b) considers a hyperbolic topology, achieved when $sgn (Im [σ_{x x}]) \neq sgn (Im [σ_{y y}])$ . Contrary to the previous case, this topology allows wave propagation –with ideally infinite field confinement – towards the directions defined by theratio between $σ_{x x}$ and $σ_{y y}$ [30]. These directions can be manipulated by adding cross-terms to the conductivity tensor, as shown in Fig. 1(c), while keeping the hyperbolic topology. Finally, Fig. 1(d) illustrates the $σ$ -near zero regime, able to canalize most of the energy towards a specific direction thanks to the large contrast between conductivity components. The insets in Fig. 1 shows possible practical realizations of the discussed topologies at terahertz and near-infrared frequencies using nanostructured graphene layers.

Fig. 1 $E_{z}$ field component of surface plasmons excited by a z-oriented electric dipole located 10 nm above homogeneous metasurfaces defined by various conductivity tensors. (a) Isotropic elliptical metasurface with $σ_{x x} = σ_{y y} = 0.05 + 0.3 i mS$ . (b) Hyperbolic metasurface with $σ_{x x} = 0.05 - 0.3 i mS$ and $σ_{y y} = 0.05 + 0.3 i mS$ . (c) Hyperbolic metasurface with $σ_{x x} = 0.05 + 0.15 i mS$ , $σ_{y y} = 0.05 - 0.15 i mS$ , $σ_{x y} = σ_{y x} = 0.26 i mS$ . (d) $σ$ -near zero metasurface with $σ_{x x} = 0.05 + 0.1 i mS$ and $σ_{x x} = 0.05 - 5 i mS$ . The insets show possible realizations of the different metasurface topologies using pristine or nanostructured graphene layers.

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2.2 SER of an emitter located nearby a metasurface

The spontaneous emission rate, or Purcell factor, of an arbitrarily-oriented emitter located nearby a metasurface can be computed by [34]

S E R = \frac{P}{P_{0}} = 1 + \frac{6 π}{| {\vec{μ}}_{p} | k_{0}} {\vec{μ}}_{p} \cdot Im [{\bar{\bar{G}}}_{S} ({\vec{r}}_{0}, {\vec{r}}_{0}, ω)] \cdot {\vec{μ}}_{p},

where

{\bar{\bar{G}}}_{S} ({\vec{r}}_{0}, {\vec{r}}_{0}, ω)

is the scattered component of the dyadic Green’s function taking into account the presence of the metasurface,

{\vec{μ}}_{p}

is the unit vector along the emitter orientation

{\vec{e}}_{ρ}

,

k_{0}

is the free-space wavenumber,

{\vec{r}}_{0}

is the source position, and

P

,

P_{0}

are the powers emitted by the dipole in the inhomogeneous environment and in free-space, respectively. Following the procedure outlined in [36], the scattered tensor Green’s function at the source position can be obtained as

{\bar{\bar{G}}}_{S} ({\vec{r}}_{0}, {\vec{r}}_{0}, ω) = \frac{i}{8 π^{2}} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} (Γ_{s s} {\bar{\bar{M}}}_{s s} + Γ_{s p} {\bar{\bar{M}}}_{s p} + Γ_{p s} {\bar{\bar{M}}}_{p s} + Γ_{p p} {\bar{\bar{M}}}_{p p}) e^{i 2 k_{z} z_{0}} d k_{x} d k_{y},

where

{\vec{r}}_{0} = x_{0} {\vec{e}}_{x} + y_{0} {\vec{e}}_{y} + z {\vec{e}}_{z}

,

\begin{array}{l} {\bar{\bar{M}}}_{s s} = \frac{1}{k_{z} k_{ρ}^{2}} (\begin{matrix} k_{y}^{2} & - k_{x} k_{y} & 0 \\ - k_{x} k_{y} & k_{x}^{2} & 0 \\ 0 & 0 & 0 \end{matrix}), {\bar{\bar{M}}}_{p p} = \frac{k_{z}}{k_{0}^{2} k_{ρ}^{2}} (\begin{matrix} - k_{x}^{2} & - k_{x} k_{y} & - k_{x} k_{ρ}^{2} / k_{z} \\ - k_{x} k_{y} & - k_{y}^{2} & - k_{y} k_{ρ}^{2} / k_{z} \\ k_{x} k_{ρ}^{2} / k_{z} & k_{y} k_{ρ}^{2} / k_{z} & k_{ρ}^{4} / k_{z}^{2} \end{matrix}), \\ {\bar{\bar{M}}}_{s p} = \frac{1}{k_{0} k_{ρ}^{2}} (\begin{matrix} - k_{x} k_{y} & - k_{y}^{2} & - k_{y} k_{ρ}^{2} / k_{z} \\ k_{x}^{2} & k_{x} k_{y} & k_{x} k_{ρ}^{2} / k_{z} \\ 0 & 0 & 0 \end{matrix}), {\bar{\bar{M}}}_{p s} = \frac{1}{k_{0} k_{ρ}^{2}} (\begin{matrix} k_{x} k_{y} & - k_{x}^{2} & 0 \\ k_{y}^{2} & - k_{x} k_{y} & 0 \\ - k_{y} k_{ρ}^{2} / k_{z} & k_{x} k_{ρ}^{2} / k_{z} & 0 \end{matrix}), \end{array}

and the tensor reflection coefficient

\bar{\bar{Γ}}

related to incident ‘s’ and ‘p’ polarized waves is

\bar{\bar{Γ}} = (\begin{matrix} Γ_{s s} & Γ_{s p} \\ Γ_{p s} & Γ_{p p} \end{matrix}) = (\begin{matrix} \frac{- η_{0} {σ^{'}}_{y y} (2 Z^{p} + η_{0} σ'_{x x}) + η_{0}^{2} {σ^{'}}_{x y} {σ^{'}}_{y x}}{(2 Z^{s} + η_{0} {σ^{'}}_{y y}) (2 Z^{p} + η_{0} {σ^{'}}_{x x}) - η_{0}^{2} {σ^{'}}_{x y} {σ^{'}}_{y x}} & \frac{- 2 c^{p} Z^{p} η_{0} {σ^{'}}_{x y}}{[(2 Z^{s} + η_{0} {σ^{'}}_{y y}) (2 Z^{p} + η_{0} {σ^{'}}_{x x}) - η_{0}^{2} {σ^{'}}_{x y} {σ^{'}}_{y x}]} \\ \frac{- 2 Z^{s} η_{0} {σ^{'}}_{y x}}{c^{p} [(2 Z^{s} + η_{0} {σ^{'}}_{y y}) (2 Z^{p} + η_{0} {σ^{'}}_{x x}) - η_{0}^{2} {σ^{'}}_{x y} {σ^{'}}_{y x}]} & \frac{- η_{0} {σ^{'}}_{x x} (2 Z^{s} + η_{0} {σ^{'}}_{y y}) + η_{0}^{2} {σ^{'}}_{x y} {σ^{'}}_{y x}}{(2 Z^{s} + η_{0} {σ^{'}}_{y y}) (2 Z^{p} + η_{0} {σ^{'}}_{x x}) - η_{0}^{2} {σ^{'}}_{x y} {σ^{'}}_{y x}} \end{matrix}),

being

η_{0}

the free-space impedance,

Z^{s} = k_{z} / k_{0}

,

Z^{p} = k_{0} / k_{z}

,

c^{p} = k_{z} / k_{0}

, and the identity

k_{0}^{2} = k_{x}^{2} + k_{y}^{2} + k_{z}^{2}

holds. In addition,

{\bar{\bar{σ}}}^{'}

denotes the rotated conductivity tensor

{\bar{\bar{σ}}}^{'} = (\begin{matrix} {σ^{'}}_{x x} & {σ^{'}}_{x y} \\ {σ^{'}}_{y x} & {σ^{'}}_{y y} \end{matrix}) = {\bar{\bar{R}}}^{T} ​ \bar{\bar{σ}} \bar{\bar{R}} = \frac{1}{k_{ρ}^{2}} (\begin{matrix} k_{x}^{2} σ_{x x} + k_{y}^{2} σ_{y y} + k_{x} k_{y} (σ_{x y} + σ_{y x}) & k_{x}^{2} σ_{x y} - k_{y}^{2} σ_{y x} + k_{x} k_{y} (σ_{y y} - σ_{x x}) \\ k_{x}^{2} σ_{y x} - k_{y}^{2} σ_{x y} + k_{x} k_{y} (σ_{y y} - σ_{x x}) & k_{x}^{2} σ_{y y} + k_{y}^{2} σ_{x x} - k_{x} k_{y} (σ_{x y} + σ_{y x}) \end{matrix}),

where

\bar{\bar{σ}}

is the conductivity tensor defined in Eq. (1) and

\bar{\bar{R}} = (\begin{matrix} \cos (φ) & - \sin (φ) \\ \sin (φ) & \cos (φ) \end{matrix}) = \frac{1}{k_{ρ}} (\begin{matrix} k_{x} & - k_{y} \\ k_{y} & k_{x} \end{matrix}), {\bar{\bar{R}}}^{T} = (\begin{matrix} \cos (φ) & \sin (φ) \\ - \sin (φ) & \cos (φ) \end{matrix}) = \frac{1}{k_{ρ}} (\begin{matrix} k_{x} & k_{y} \\ - k_{y} & k_{x} \end{matrix}),

are standard rotation matrixes. They are expressed in the transformed domain by using the identities

k_{x} = k_{ρ} \cos (φ)

and

k_{y} = k_{ρ} \sin (φ)

, where

φ

is the rotation angle within the sheet.

In order to compute the SER of an emitter located nearby an anisotropic metasurface, we numerically solve Eq. (3) by applying standard complex integration techniques, and then we use the result in Eq. (2).

2.3 Dispersion relations

The modes supported by an anisotropic metasurface are provided by the poles of the Green’s function [see Eqs. (3)-(5)], i.e.,

(2 Z^{s} + η_{0} σ'_{y y}) (2 Z^{p} + η_{0} σ'_{x x}) - η_{0}^{2} σ'_{x y} σ'_{y x} = 0.

After some straightforward manipulations, this equation can be simplified to become the usual dispersion relation of SPPs propagating along infinitesimally-thin metasurfaces [33, 37, 38]. Unfortunately, the solution of Eq. (8) is quite challenging, since it involves searching for roots in the complex plane. This equation is usually solved using purely numerical algorithms, such as the Newton–Raphson method [39], that depend on the starting point during the search and do not guarantee that all physical solutions are found. Here, we propose an alternative approach that avoids searching in the complex plane, and at the same time it provides physical insights into the nature of the supported plasmons. This procedure is based on fixing the SPPs direction of propagation along one specific angle within a reference coordinate system, for instance the ‘x’ axis (i.e.,

k_{y} = 0

), and then physically rotate the metasurface an angle

φ

through Eq. (6) in order to evaluate the features of SPPs propagating along that direction (see inset of Fig. 2). Following this approach, the two possible solutions for the transverse wavenumber

k_{z}

at any direction

φ

are analytically obtained as

k_{z}^{\pm} (φ) = \frac{k_{0}}{2 {σ^{'}}_{x x}} [- (\frac{2}{η_{0}} + \frac{η_{0}}{2} ({σ^{'}}_{x x} {σ^{'}}_{y y} - {σ^{'}}_{x y} {σ^{'}}_{y x})) \pm \sqrt{{(\frac{2}{η_{0}} + \frac{η_{0}}{2} ({σ^{'}}_{x x} {σ^{'}}_{y y} - {σ^{'}}_{x y} {σ^{'}}_{y x}))}^{2} - 4 {σ^{'}}_{x x} {σ^{'}}_{y y}}] .

An infinitesimally-thin surface supports SPPs only when the transverse wavenumbers are evanescent, i.e., Im

[k_{z} (φ)] > 0

[33]. Therefore, each particular direction

φ

may support SPP propagation, depending on the metasurface conductivity tensor in Eq. (1). The complete solution of the dispersion relation is then retrieved by

k_{ρ} (φ) = \sqrt{k_{0}^{2} - k_{z}^{2} (φ)}

, sweeping the rotation angle

φ

and using Eq. (9), thus avoiding the use of computationally-intense numerical routines. Lossless metasurfaces with sgn

(Im [σ'_{x x} (φ)])

= sgn

(Im [σ'_{y y} (φ)])

support – along the direction

φ

– either quasi transverse-magnetic (quasi-TM) or quasi transverse-electric (quasi-TE) modes, as a function of the positive/negative sign of the imaginary part of the diagonal conductivity components. Lossless metasurfaces with sgn

(Im [σ'_{x x} (φ)]) \neq

sgn

(Im [σ'_{y y} (φ)])

may simultaneously support quasi-TE and quasi-TM modes along that direction. As in the case of isotropic surfaces [29], TE surface waves are weakly guided, presenting propagation features similar to those found in plane waves traveling in free-space, and therefore do not provide a significant increase in light-matter interactions. On the other hand, TM plasmons are usually confined to the structure, and may provide exciting phenomena and applications in nanoplasmonics [23].

Fig. 2 Isofrequency contours of several lossless anisotropic metasurfaces. (a) Elliptic isotropic (red line) and anisotropic (blue line) configurations, with $σ_{x x} = σ_{y y} = 0.5 i mS$ and $σ_{x x} = 5 i m S, σ_{y y} = 0.37 i mS$ , respectively. (b) Hyperbolic metasurface, with $σ_{x x} = - 5 i m S, σ_{y y} = 5 i mS$ . (c) Hyperbolic metasurface, with $σ_{x x} = - 5 i m S, σ_{y y} = 5 i mS, σ_{x y} = σ_{y x} = - 15 i m S$ . The green line is computed using the asymptotic expression of Eq. (10). The inset (top right) illustrates the procedure employed to accurately solve the dispersion relation of Eq. (8). Operation frequency is 10 THz.

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Our approach effectively implies that surface wave propagation along a direction $φ$ within an anisotropic metasurface experiences an effective conductivity depending on the type of mode and the angle of propagation. However, we do note that not all angles of propagation within the surface are allowed [31], since in each direction the structure must support evanescent fields normal to the metasurface Im $[k_{z} (φ)] > 0$ . It is important to stress that each term of the rotated tensor ${\bar{\bar{σ}}}^{'}$ depends on all components of the metasurface conductivity tensor $\bar{\bar{σ}}$ [see Eq. (6)], which indeed contribute to define the plasmon characteristics. We remark that an anisotropic metasurface may support SPPs with significantly different propagation features as a function of their direction. Consequently, an adequate tailoring of the conductivity tensor $\bar{\bar{σ}}$ can allow propagation – and full control – of SPPs towards specific angles, and at the same time forbid wave propagation in other directions. The latter can be realized by forcing Im $[k_{z} (φ)] < 0$ at desired angles within the sheet, thus preventing SPPs from propagating in these directions. Therefore, the energy coupled to HMTSs is not cylindrically spread through the entire surface, as it usually occurs in isotropic surfaces, but it may be directed towards desired direction [see Figs. 1(b) and 1(d)].

As a special case of interest, we focus on HMTSs, for which sgn $(Im [σ_{x x}]) \neq$ sgn $(Im [σ_{y y}])$ and the supported quasi-TM plasmons present a dispersion relation that follows a hyperbolic surface [30]. Taking into account the presence of the non-diagonal terms in the tensor $\bar{\bar{σ}}$ , the branches of the hyperbola can be asymptotically approximated by

k_{y} = m k_{x} \pm b,

where

m_{(1, 2)} = \frac{1}{2 σ_{y y}} [- (σ_{x y} + σ_{y x}) \pm \sqrt{{(σ_{x y} + σ_{y x})}^{2} - 4 σ_{x x} σ_{y y}}],

being

m_{(1)}

and

m_{(2)}

related to the positive and negative sign of the square root, and

b_{(1, 2)} = - m_{(1, 2)} k_{0} \sqrt{1 - {[\frac{A}{2 σ_{x x}}]}^{2}} and b_{(3)} = k_{0} \sqrt{1 - {[\frac{A}{2 σ_{y y}}]}^{2}},

with

A = [\frac{2}{η_{0}} + \frac{η_{0}}{2} (σ_{x x} σ_{y y} - σ_{x y} σ_{y x})] \pm \sqrt{{[\frac{2}{η_{0}} + \frac{η_{0}}{2} (σ_{x x} σ_{y y} - σ_{x y} σ_{y x})]}^{2} - 4 σ_{x x} σ_{y y}}

. Note that Eq. (10) presents multiple solutions, but only those with Im

[k_{z}] > 0

are physical. In the case of HMTSs defined by a diagonal conductivity tensor (i.e., with

σ_{x y} = σ_{y x} = 0

), Eqs. (11)-(12) reduces to

m_{(1, 2)} = \pm \sqrt{- \frac{σ_{x x}}{σ_{y y}}}, b_{(1, 2)} = m_{(1, 2)} k_{0} \sqrt{1 + {(\frac{2}{η_{0} σ_{x x}})}^{2}} and b_{(3)} = k_{0} \sqrt{1 + {(\frac{2}{η_{0} σ_{y y}})}^{2}} .

Figure 2 illustrates the isofrequency contour of several anisotropic lossless metasurfaces, including elliptic and hyperbolic topologies. In particular, Fig. 2(b) shows the electromagnetic response of a hyperbolic metasurface, highlighting its unclosed contour that translates into ideally infinitely confined SPPs – i.e., infinite local density of states – propagating towards specific directions within the sheet. In practice, hyperbolic isofrequency contours are closed due to the presence of realistic losses and the granularity of the composing subwavelength inclusions, as will be discussed later, but the closing wavenumber may be significantly larger than the free-space wavenumber. Finally, Fig. 2(c) confirms that the proper tailoring of the conductivity cross-terms allows to further control the direction of energy flow [see Fig. 1(c)].

3. Physical Implementation

This section analyses elliptic and hyperbolic reconfigurable metasurfaces implemented by arrays of densely packed graphene strips, operating in the terahertz and near infrared frequency regions. Real time tunability is achieved – neglecting the possible presence of hysteresis in graphene [43] – by applying an electrostatic field perpendicular to the patterned sheet, thus controlling graphene’s chemical potential through the field effect [20–22]. Our numerical results consider realistic graphene [40,41], modelled through the Kubo formalism [42] using a relaxation time $τ = 0.1$ ps at 300 K.

3.1 Graphene sheets: isotropic elliptic metasurfaces

A natural example of elliptic isotropic metasurfaces is graphene [22,23], illustrated in the inset of Fig. 1(a), a 2D material able to intrinsically support the propagation of SPPs [29]. Neglecting spatial dispersion effects [44], graphene can be electromagnetically modelled using the conductivity tensor

\bar{\bar{σ}} = (\begin{matrix} σ_{d} & σ_{h} \\ - σ_{h} & σ_{d} \end{matrix}),

where

σ_{d}

and

σ_{h}

denote the direct and Hall conductivities [42]. These terms depend on graphene intrinsic features, operation frequency, environmental temperature and the applied electric and magnetic field. The SPP properties in graphene have been extensively studied in the literature [22, 23], both in the presence of electric [24, 29] and magnetic bias [45,46]. Besides, graphene provides a platform for strong light-matter interactions [23], significantly enhancing the local density of states. Here, we briefly review the features of graphene as a 2D elliptic metasurface and extend previous studies to investigate the SER enhancement that electrically and magnetically-biased graphene provides to a nearby emitter.

Figure 3(a) shows the isofrequency contour of a graphene sheet as a function of its chemical potential $μ_{c}$ at 10 THz. As expected, graphene behaves as an isotropic elliptic metasurface, providing similar features to the supported TM SPPs independently of their direction of propagation [see Fig. 1(a)]. Importantly, plasmon features can be tuned by modifying $μ_{c}$ , thus enabling reconfigurable isotropic metasurfaces that have recently led to various potential applications [24–28]. Figure 3(b) presents a similar study, but considering now that a magnetostatic field is applied to the direction perpendicular to the graphene sheet. The presence of the magnetic bias does not modify the isotropic elliptic nature of graphene [45,46], but it modifies the polarization of the plasmons – which are now hybrid TE-TM – and slightly decreases their absolute wavenumber at this frequency. Not shown here for the sake of brevity, the influence of the magnetic bias on the SPP features increases for low values of $μ_{c}$ [45,46].

Fig. 3 Electrically and magnetically-biased graphene as a natural isotropic elliptic metasurface. Isofrequency contour of the supported plasmons at 10 THz versus (a) chemical potential, with $B = 0 T$ , (b) magnetic field, with $μ_{c} = 0.2 eV$ . SER in logarithm scale of a z-oriented emitter versus its position d above graphene [see inset of Fig. 1(b)] as a function of: (c) chemical potential, with $B = 0 T$ at 10 THz, (d) magnetic field, with $μ_{c} = 0.2 eV$ at 10 THz, (e) frequency, with $B = 0 T$ and $μ_{c} = 0.05 eV$ , and (f) frequency, with $B = 5 T$ and $μ_{c} = 0.05 eV$ . Other parameters are the graphene relaxation time $τ = 0.1 ps$ , and temperature $T = 300 K$ .

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The properties of SPPs supported by graphene are closely related to the enhanced light-matter interaction that this 2D material offers [23]. To further investigate these interactions, Fig. 3(c) analyses the SER of a z-oriented emitter located at a distance d above graphene versus its chemical potential $μ_{c}$ at 10 THz. As expected, the SER increases for lower values of $μ_{c}$ , because they allow propagation of highly confined SPPs [see Fig. 3(a)], able to interact with a broader part of the spectrum radiated by the emitter. Moreover, the SER decreases as the emitter is moved away from the sheet, which is attributed to the natural filtering of evanescent waves associated with free-space decay. Figure 3(d) illustrates a similar scenario, but versus the value of the applied magnetic bias. Results reveal that the SER remains almost constant in all cases, and that it is even slightly reduced as the magnetic bias increases. This is because the magnetic bias mainly affects the polarization of the supported SPPs [45,46], but it has little influence over the SPPs absolute wavenumber in this frequency range, as clearly illustrated in Fig. 3(b). Figures 3(e)-3(f) investigate the SER of the dipole versus frequency, considering graphene with low chemical potential ( $μ_{c} = 0.05 eV$ ). Figure 3(e) analyzes this scenario in the absence of magnetic bias. Results indicate that the confined SPPs supported by this configuration are able to interact with waves with large wavenumbers, thus boosting the SER. Importantly, the SER diminishes at higher frequencies, due to the intrinsic dispersion of graphene conductivity [42]. Figure 3(f) presents a similar study but considering an applied magnetostatic bias $B = 5 T$ . As previously pointed out, graphene with low chemical potential, as the one employed in this example, maximizes the influence of the applied magnetic bias on the SPP wavenumber [45,46]. Results confirm that the emitter SER moderately increases in this case, especially at low frequencies – due to higher dissipation losses – and when the emitter is very close to the surface, enhancement that can be attributed to the hybrid TE-TM nature of the supported plasmons that allow them to better interact with arbitrarily-polarized incoming waves.

3.2 Array of densely packed graphene strips: hyperbolic and $σ$ -near zero metasurfaces

An array of densely packed graphene strips (see insets of Fig. 1) is an ideal platform to implement any metasurface topology, ranging from isotropic to hyperbolic propagation, and going through the extremely anisotropic $σ$ -near zero case [30]. Here, we present a detailed study of this structure using an effective medium approach (EMA). Specifically, we investigate the type of band topology, the properties of the supported SPPs, and the dramatic SER enhancement for nearby emitters versus the geometrical parameters of the strips and frequency. Finally, we delimit the influence of the ribbon granularity in our EMA approach by comparing our results with full-wave simulations based on mode-matching [47].

Assuming that the condition $L ≪ λ_{S P P}$ is satisfied, where $L$ is the strip periodicity and $λ_{S P P}$ is the plasmon wavelength, the in-plane effective conductivity tensor ${\bar{\bar{σ}}}^{e f f}$ of an array of densely-packed graphene strips can be analytically derived using effective medium theory [30] as

σ_{x x}^{e f f} = \frac{L σ σ_{C}}{W σ_{C} + G σ} and σ_{y y}^{e f f} = σ \frac{W}{L},

where

W

is the width of the ribbons,

G

is the separation distance between consecutive strips,

σ

is the graphene conductivity and

σ_{C} = - i ω ε_{0} ε_{e f f} (\frac{L}{π}) ln [csc (\frac{π G}{2 L})]

is an effective conductivity related to the near-field coupling between adjacent strips obtained through an electrostatic approach. In this last expression,

ε_{e f f}

is the effective permittivity of the media that embed the ribbons. A simple inspection of Eq. (15) reveals that the propagation along the strips [y-direction, see inset of Fig. 1(b)] is purely inductive, i.e.

Im [σ_{y y}^{e f f}] > 0

, whereas the near-field coupling across the strips (x-direction) can be either inductive

(Im [σ_{x x}^{e f f}] > 0)

or capacitive

(Im [σ_{x x}^{e f f}] < 0)

, as a function of the geometrical parameters.

The overall behavior of the effective conductivity tensor versus frequency and the strip width $G$ is illustrated in Fig. 4, assuming an array of ribbons with periodicity $L = 50 n m$ and $μ_{c} = 0.2 e V$ . Figures 4(a)-4(b) confirm that $σ_{y y}^{e f f}$ provides an inductive response in all cases. More interestingly, Fig. 4(d) shows the behavior of $Im [σ_{x x}^{e f f}]$ , clearly determining the two main topologies implemented by this structure, namely hyperbolic – blue/green region on the left, providing negative $Im [σ_{x x}^{e f f}]$ – and elliptic, red/yellow region on the right, leading to positive $Im [σ_{x x}^{e f f}]$ . The transition point between these two topologies is given by the expression $W σ_{C} + G σ = 0$ , i.e., the pole of $σ_{x x}^{e f f}$ . This point provides a resonant response similar to the one found in bulk MTMs [21], associated with a large amount of dissipative losses [see Fig. 4(c)]. It is important to point out that the dispersive behavior of graphene conductivity modifies this transition point, which becomes frequency dependent. In addition, this configuration provides an extremely anisotropic $σ$ -near zero response for very narrow graphene strips, thanks to the almost negligible $σ_{x x}^{e f f}$ found there. This response is similar to the canalization regime found in HMTMs, and usually applied for hyperlensing [6, 7].

Fig. 4 Effective conductivity tensor ${\bar{\bar{σ}}}^{e f f}$ (in mS) of an array of densely packed graphene strips versus frequency and ribbon width $W$ , computed using the effective medium approach (15). (a) $Re [σ_{y y}^{e f f}]$ , (b) $Im [σ_{y y}^{e f f}]$ , (c) $Re [σ_{x x}^{e f f}]$ , (d) $Im [σ_{x x}^{e f f}]$ . The periodicity is set to $L = 50 n m$ , the temperature is $T = 300 K$ , and the graphene relaxation time and chemical potential are $τ = 0.1 p s$ and $μ_{c} = 0.2 e V$ , respectively.

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The properties of the SPPs supported by this structure at 10 THz are investigated in Fig. 5. Specifically, Figs. 5(a)-5(b) show the mode confinement and figure of merit – related to the number of wavelengths that SPPs can propagate before being strongly attenuated – versus the angle of propagation within the sheet for various strip widths $W$ . The results clearly demonstrate the ability of this configuration to implement any topology: i) isotropic elliptic, found when $W = L$ , i.e., a usual graphene sheet (black line); ii) anisotropic elliptic, found when $W > G Im [\frac{- σ}{σ_{c}}]$ (red line); and iii) hyperbolic, found when $W < G Im [\frac{- σ}{σ_{c}}]$ (green and blue lines). In the latter case, the SPP wavenumber gets more and more confined as the strip width $W$ decreases, situation in which the metasurface response tends to the canalization regime. Moreover, and contrary to the case of the elliptic topology, wave propagation is only allowed towards specific directions within the sheet, as detailed in Section II. It is important to remark that losses remain very similar to those found in pristine graphene, and that in all cases they are mainly related to the intrinsic features of the employed material [30]. These results demonstrate that HMTSs support extremely confined low-loss and dispersion-free SPPs that can be focused towards any desired direction. Figures 5(c)-5(d) present a similar study, but keeping the strip width fixed to $W = 44 n m$ – close to the transition point between elliptic and hyperbolic topologies – and modifying the graphene chemical potential. Our results reveal that an array of graphene strips can dynamically modify their topology, going from elliptic to hyperbolic by simply applying a modest DC bias. More interestingly, as the chemical potential increases the proposed structure further confines the supported SPPs towards a specific direction, clearly converging towards an extremely anisotropic $σ$ -near zero case. This unusual response provides interesting opportunities for guiding and confining light. Figures 5(e)-5(f) complete our study by illustrating the isofrequency contours of graphene strips in the aforementioned scenarios. Specifically, Fig. 5(e) shows the evolution of the dispersion relation versus strip width $W$ , highlighting how the metasurface topology can be controlled by adjusting this physical dimension. Figure 5(f) shows a similar study, but versus graphene’s chemical potential, further highlighting the tunability of the metasurface topology. Finally, it is worth pointing out that the otherwise open isofrequency contours associated with hyperbolic topologies are closed here – thus leading to a finite field confinement and local density of states – due to the presence of realistic losses in graphene.

Fig. 5 Properties of the surface plasmons supported by an array of densely packed graphene strips. Normalized phase constant (a) and figure of merit (b) of the SPPs versus their angle of propagation $φ$ within the metasurface, for different strip widths W. The graphene chemical potential is $μ_{c} = 0.2 eV$ . Panels (c) and (d) show similar data versus graphene chemical potential, keeping the strip width fix to $W = 44 n m$ . Isofrequency contours (dispersion relation) of the metasurface versus (e) strip width W, fixing graphene chemical potential to $μ_{c} = 0.2 eV$ and (f) chemical potential, keeping the strip width fix to $W = 44 nm$ . Other parameters are strip periodicity $L = 50 nm$ , graphene relaxation time $τ = 0.1 ps$ , operation frequency 10 THz and temperature $T = 300 K$ .

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One of the main advantages of HMTSs consists in the dramatic enhancement of light matter interactions that they provide compared to more conventional elliptic topologies [4, 30]. This is further investigated in Fig. 6, where the SER of a z-oriented emitter located at a distance d above a densely packed array of graphene strips with periodicity $L = 50 nm$ is illustrated as a function of the strip width $W$ . Figures 6(a)-6(b) study this scenario versus the operation frequency, considering graphene with $μ_{c} = 0.2 eV$ and $μ_{c} = 1.0 eV$ , respectively. In the first case, the lowchemical potential allows propagation of highly confined plasmons [see Fig. 3(a)] along a pristine graphene sheet, thus providing a large SER [22]. The patterning of graphene strips leads a topological transition in the metasurface, associated to a dramatic enhancement – several orders of magnitude – of SER. This enhancement is further increased as the strip width is reduced, leading to metasurfaces able to support extremely confined hyperbolic SPPs. Interestingly, very narrow strips at high frequencies may strongly decrease the SER, due to the flattening of the metasurface dispersion relation that occurs in the $σ$ -near zero regime around very large wavenumbers [see Fig. 5(e)], thus hindering the coupling of incident waves with low/moderate wavenumbers. Importantly, the SER boost is more significant at low frequencies,due to the intrinsic dispersive response of graphene. The picture is similar in the second case, illustrated in Fig. 6(b), however the larger chemical potential of graphene reduces the field confinement of plasmons in the elliptic metasurface, thus providing a highest contrast between the SER enhancement of patterned and pristine graphene. Figures 6(c)-6(d) present a similar study but analyzing the influence of the distance d between the dipole and the metasurface at 10 and 30 THz, respectively. In the former case, hyperbolic metasurfaces are able to couple most of the evanescent waves radiated by the dipole, thus leading to a strong SER enhancement that slowly decreases as the dipole is moved away from the structure. In the latter case, the SER enhancement is most significant for emitters closely located to the metasurface, but is attenuated for large values of d. This response is due to the graphene dispersion, which leads to effectively low-loss hyperbolic metasurfaces operating in the canalization regime that are unable to couple incoming waves with reduced wavenumbers. On the contrary, conventional isotropic graphene sheets may provide larger SER here, thanks to their ability to interact with these waves.

Fig. 6 SER in logarithmic scale for a z-oriented emitter located above an array of densely packed graphene strips versus the ribbon width $W$ . The SER is computed using Eq. (2), modelling the structure using the effective conductivity tensor of Eq. (15). In the upper row, the SER of a dipole placed at d = 10 nm above the metasurface is shown versus frequency. Graphene’s chemical potential is set to (a) $μ_{c} = 0.2 eV$ and (b) $μ_{c} = 1.0 eV$ . The lower row depicts the SER of an emitter versus its distance d towards the metasurface. Graphene’s chemical potential is $μ_{c} = 0.2 eV$ , and frequency is set to (c) 10 THz and (d) 30 THz. Other parameters are strips periodicity $L = 50 n m$ , graphene relaxation time $τ = 0.1 ps$ and temperature $T = 300 K$ .

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Finally, we investigate the influence of the strip granularity, and how it provides an upper cutoff for the SPPs wavenumber in HMTSs. To this purpose, Fig. 7(a) studies the SER of emitters located nearby a homogeneous hyperbolic metasurface versus the periodicity of the composing strips, concluding that it barely depends on this physical parameter. However, as previously pointed out, it is important to keep in mind that our numerical approach is only valid when the nanocomposite structure can be regarded as a homogeneous metasurface, i.e., when the condition $L ≪ λ_{S P P}$ is satisfied. In case that the supported SPPs are extremely confined, their wavelength may be comparable to the features of the structure ( $L \approx λ_{S P P}$ ) thus invalidating our homogeneous model. In order to fully analyze this scenario, Figs. 7(b)-7(d) show theisofrequency contour in the first Brillouin zone of an array of densely packed graphene strips for different values of $L$ . These results have been computed with a full-wave mode-matching approach [47] (color maps) and are compared to those obtained through the homogeneous EMA technique proposed here (white dashed lines). It can be observed that both approaches agree well for low and moderate wavenumbers, since the EMA technique is more and more accurate as the period decreases. These results confirm the accuracy of the numerical study presented here, since the presence of losses (see Fig. 4) imposes a similar limitation to the propagation of extremely confined SPPs as the granularity of the strips employed there (see Fig. 5). Importantly, our full-wave analysis brings additional insights into the physical properties of the proposed hyperbolic metasurface. First, it clearly illustrates how granularity closes the isofrequency contour of the hyperbolic structure, imposing a cutoff at $π / L$ that limits the maximum SPPs wavenumber. Second, it confirms that the propagation along the strips (y-direction) is mainly determined by graphene features [see Eq. (15)] and that is the coupling across the strips (x-direction) which provides the hyperbolic nature to the metasurface. Finally, it highlights how decreasing the period $L$ significantly up-shifts the cutoff wavenumber, thus leading to homogeneous configurations.

Fig. 7 Influence of periodicity in the electromagnetic response of graphene-based hyperbolic metasurfaces. (a) SER in logarithm scale of x, y and z-oriented emitters located above an array of densely packed graphene strips versus their periodicity $L$ . The SER is computed using Eq. (2), modelling the structure using the effective conductivity tensor of Eq. (15). Panels (b), (c) and (d) shows the dispersion relation of the structure, computed using a full-wave mode-matching approach [47] (colormaps) and the homogenous metasurface model of Eq. (15) (white dashed line). (b) $L = 15 nm$ , (c) $L = 50 nm$ , and (d) $L = 85 nm$ , corresponding to points B, C and D of panel (a). The strip width is $W = L / 2$ , operation frequency is 10 THz, and graphene relaxation time, chemical potential and temperature are $τ = 0.1 ps$ , $μ_{c} = 0.2 eV$ , and $T = 300 K$ , respectively.

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4. Conclusions

In conclusion, we have analyzed surface wave propagation and spontaneous emission rate in ultrathin metasurfaces defined by arbitrary anisotropic conductivity tensors using a Green function approach. We have performed an extensive study of the unusual electromagnetic properties of hyperbolic metasurfaces, including the presence of extremely confined surface plasmons directed towards specific directions within the sheet and their associated dramatic enhancement of light-matter interactions. The dispersion relation of anisotropic MTSs has been derived and accurately solved avoiding searching roots in the complex plane. Furthermore, we have studied its asymptotic behavior establishing a clear connection between the different components of the conductivity tensor and the HMTSs topology. We have applied this approach to investigate isotropic elliptic metasurfaces implemented by electrically and magnetically-biased graphene sheets at terahertz and near infrared frequencies, and to study the behavior of an array of densely packed graphene ribbons described by an in-plane effective medium approach. This latter configuration provides exciting and unique electromagnetic characteristics, including the possibility to control in real-time its topology, enabling an unprecedented dynamic degree of tunability of the extremely confined low-loss and dispersion-free SPPs supported, while simultaneously boosting the local density of states. We have also investigated the influence of the ribbons granularity on the metasurface response, delimiting the cutoff that they impose on the SPPs wavenumber. Hyperbolic and $σ$ -near-zero metasurfaces can also be implemented using different technologies, such as patterned graphene-based parallel-plate waveguides [43] or properly tailored subwavelength inclusions made of metals [32] or highly-doped semiconductors printed on a host sheet. We envision that these metasurfaces may pave the way towards realistic ultrathin plasmonic devices able to strongly interact with the incoming light, allowing the dynamic manipulation and processing of extremely confined dispersion-free SPPs, with direct application in a wide variety of areas such as imaging, sensing, hyperlenses, communications, and on chip-networks.

Acknowledgments

This work was supported by the Air Force Office of Scientific Research (AFOSR) grant No. FA9550-13-1-0204, the Welch foundation with grant No. F-1802, and the National Science Foundation with grant No. ECCS-1406235.

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Hyperbolic metasurfaces: surface plasmons, light-matter interactions, and physical implementation using graphene strips [Invited]

Abstract

Corrections

1. Introduction

2. Wave propagation in anisotropic metasurfaces

2.1 Metasurface topologies

2.2 SER of an emitter located nearby a metasurface

2.3 Dispersion relations

3. Physical Implementation

3.1 Graphene sheets: isotropic elliptic metasurfaces

3.2 Array of densely packed graphene strips: hyperbolic and $σ$ -near zero metasurfaces

4. Conclusions

Acknowledgments

References and links

Cited By

Figures (7)

Equations (15)

Optical Materials Express

Abstract

Corrections

1. Introduction

2. Wave propagation in anisotropic metasurfaces

2.1 Metasurface topologies

2.2 SER of an emitter located nearby a metasurface

2.3 Dispersion relations

3. Physical Implementation

3.1 Graphene sheets: isotropic elliptic metasurfaces

3.2 Array of densely packed graphene strips: hyperbolic and σ-near zero metasurfaces

4. Conclusions

Acknowledgments

References and links

Cited By

Figures (7)

Equations (15)

Optical Materials Express

3.2 Array of densely packed graphene strips: hyperbolic and $σ$ -near zero metasurfaces