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Nov 16, 2016 - Flatland Optics with Hyperbolic Metasurfaces. J. S. Gomez-Diaz† and Andrea Alù‡. † Department of Electrical and Computer Enginee...
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Perspective pubs.acs.org/journal/apchd5

Flatland Optics with Hyperbolic Metasurfaces J. S. Gomez-Diaz† and Andrea Alù*,‡ †

Department of Electrical and Computer Engineering, University of California, Davis, Davis, California 95616, United States Department of Electrical and Computer Engineering, The University of Texas at Austin, Austin, Texas 78712, United States

‡

ABSTRACT: In this Perspective, we discuss the physics and potential applications of planar hyperbolic metasurfaces (MTSs), with emphasis on their in-plane and near-field responses. After revisiting the governing dispersion relation and properties of the supported surface plasmon polaritons (SPPs), we discuss the different topologies that uniaxial MTSs can implement. Particular attention is devoted to the hyperbolic regime, which exhibits unusual features, such as an ideally infinite wave confinement and local density of states. In this context, we clarify the different physical mechanisms that limit the practical implementation of these ideal concepts using materials found in nature, and we describe several approaches to realize hyperbolic MTSs, ranging from the use of novel 2D materials such as black phosphorus to artificial nanostructured composites made of graphene or silver. Some exciting phenomena and applications are then presented and discussed, including negative refraction and the routing of SPPs within the surface, planar hyperlensing, dramatic enhancement and tailoring of the local density of states, and broadband superPlanckian thermal emission. We conclude by outlining our vision for the future of uniaxial MTSs and their potential impact for the development of nanophotonics, on-chip networks, sensing, imaging, and communication systems. KEYWORDS: plasmonics, metasurfaces, uniaxial media, hyperbolic materials, graphene, black phosphorus

U

plane propagation and near-field functionalities, enabled by confined surface plasmon polaritons (SPPs),15−18 with exciting applications in nanophotonics,19 planar nanoantennas and transceivers,20 communication systems,21 extremely sensitive sensors,22 on-chip networks, and in-plane imaging.23 This Perspective focuses on recent developments in the theory and applications of in-plane SPP optics using uniaxial metasurfaces, with the purpose of conf ining light into ultrathin structures and then manipulating its in-plane propagation and near-f ield features, including refraction/reflection, polarization, interaction with matter, and canalization at the nanoscale. To do so, we translate onto 2D surfaces the unusual optical interactions found in the bulk by uniaxial materials and hyperbolic metamaterials (HMTMs).24,25 We stress that, even though these scenarios are analogous, they are not equal to each other due to the additional constraints that the reduced dimensionality of ultrathin metasurfaces imposes on electro-

ltrathin metasurfaces (MTSs) have recently gained significant attention, thanks to their capability to locally modify the phase, amplitude, and polarization of light in reflection and transmission.1−5 MTSs are usually composed of subwavelength scatterers, suitably tailored to enable advanced functionalities, mimicking the response of common optical components such as lenses, polarizers, or beam splitters6−9 in planar, ultrathin configurations. Even more exotic scattering responses, such as negative refraction, hyperlensing, and the generation of vortex beams, have been engineered in ultrathin MTSs borrowing concepts from optical metamaterials (MTMs) and uniaxial media.10−13 Undoubtedly, a large part of the success of planar MTSs is due to their ability to significantly alleviate some of the challenges of bulk MTMs, including simplifying the fabrication process, removing volumetric losses, providing an easy access and process of the stored energy, and allowing compatibility with other planar devices. To date, most MTSs have been designed to operate as optical elements located in free space, aiming to fully control light coming from the far field.8,14 Recent works have also suggested that metasurfaces may provide the basis for a powerful ultrathin platform to realize guided and radiative devices based on in© 2016 American Chemical Society

Received: Revised: Accepted: Published: 2211

August 26, 2016 November 14, 2016 November 16, 2016 November 16, 2016 DOI: 10.1021/acsphotonics.6b00645 ACS Photonics 2016, 3, 2211−2224

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Figure 1. Topologies of uniaxial metasurfaces. Color maps show the z-component of the electric field excited by a z-directed dipole (black arrow) located 25 nm above the surface. The insets present the isofrequency contour of each metasurface topology: (a) elliptic metasurface, σxx = σyy = 0.05 + i23.5 μS; (b) σ-near-zero metasurface, σxx = 0.05 + i23.5 μS, σyy = 0.05 μS; (c) hyperbolic metasurface, σxx = 0.05 − i23.5 μS, σyy = 0.05 + i23.5 μS; (d) hyperbolic metasurface, σxx = 0.05 + i23.5 μS, σyy = 0.05 − i23.5 μS.

⎛ σxx σxy ⎞ ⎟ σ =⎜ ⎝ σyx σyy ⎠

magnetic wave propagation, resulting in new, exciting propagation phenomena in two dimensions. We start by introducing the concept of uniaxial metasurfaces and illustrating the different types of topologies and surface plasmons that they can support.26 We pay particular attention to hyperbolic metasurfaces, due to the fascinating properties that they can offer,26−28 such as extremely large wave confinement and local density of states, as happens in bulk HMTMs.13,29 Several possibilities to realize uniaxial metasurfaces are then considered, ranging from the use of novel 2D materials to man-made ultrathin structures with electromagnetic responses tailored at will. In this context, graphene18,30 has emerged as an excellent candidate to implement such artificial devices, thanks to its ultrathin nature, intrinsic tunability by simply applying a modest bias voltage, and the ability to support confined surface plasmons at terahertz (THz) frequencies. Next, we present and discuss some exciting applications enabled by uniaxial metasurfaces, including negative refraction and SPP routing through the interface between planar MTSs, in-plane hyperlenses with deeply subwavelength resolution, dramatic enhancement of light−matter interactions, and broadband superPlanckian thermal emission beyond the blackbody limit. Finally, we outline our vision for the future of uniaxial metasurfaces and their role in the coming generation of nanophotonic devices.

(1)

where the different components may be in general complex. Through this work, we will focus on passive uniaxial metasurfaces. In particular, passivity here31 enforces the conditions Re[σxx] ≥ 0, Re[σyy] ≥ 0, and Re[ σxx + σyy] ≥ |σxy + σyx * |, where * denotes a complex conjugate, under a e−iωt time convention. In addition, the fact that we consider uniaxial metasurfaces imposes that the conductivity tensor σ must be diagonal in a suitable reference coordinate system. Nondiagonal conductivity terms may arise due to different factors, including (i) the presence of subwavelength inclusions with nonsymmetrical shape with respect to the reference coordinate system, associated with an in-plane bending of the propagating SPPs,28 (ii) magneto-optical effects19,32 associated with hybrid transverse magnetic−transverse electric (TM−TE) SPPs and inplane gyrotropic response,33,34 and (iii) nonlocal effects,35,36 associated with the finite Fermi velocity of electrons in the metasurface-composing materials. In order to identify the band topology of a particular metasurface, it is important to consider the relative signs of Im[σxx] and Im[σyy], which determine the shape of the supported SPP isofrequency contours. The dispersion relation of the surface modes supported by a free-standing anisotropic metasurface is given by31,37−39

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PHYSICS OF UNIAXIAL METASURFACES The electromagnetic response of an infinitesimally thin homogeneous anisotropic metasurface can be modeled by its optical conductivity tensor

k 0kz(4 + η02(σxxσyy − σxyσyx)) − 2η0k 02(σxx + σyy) + 2η0(kx2σxx + σyyk y2 + kxk yσxyσyx) = 0 2212

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where k0 is the free-space wavenumber, decaying evanescent modes Im[kz] > 0 are enforced away from the surface, and the reference coordinate system follows the one shown in Figure 1. Efficient techniques to solve this dispersion relation have been recently presented,39 thus allowing easily obtaining the propagation properties of the supported SPPs over the frequency range of interest. Figure 1 illustrates the electric field distribution of the plasmons when excited by a z-oriented dipole located above uniaxial metasurfaces that support SPPs with various canonical topologies. These topologies allow classifying metasurfaces as a function of their conductivity tensor shape, and they will help in identifying their properties. For instance, Figure 1a shows an isotropic elliptic topology, for which the excited TM SPPs propagate in all directions within the sheet with similar features. This topology appears when sgn(Im[σxx]) = sgn(Im[σyy]), and it can be associated with either quasi-TM (inductive, Im[σxx] > 0, Im[σyy] > 0) or quasiTE (capacitive, Im[σxx] < 0, Im[σyy] < 0) surface plasmons. The polarization of the supported SPPs will be purely TM or TE only for isotropic metasurfaces, i.e., when σxx = σyy. Analyzing the supported plasmons, it is easy to realize that quasi-TE SPPs present a dispersion relation similar to the one of free space, i.e., kρ ≈ k0, thus leading to responses with almost negligible wave confinement and light−matter interactions that are of little practical interest. On the contrary, quasi-TM plasmons can provide fascinating properties. An example of isotropic elliptic metasurfaces able to support TM SPPs is graphene,18 an inductive 2D material where Im[σxx] = Im[σyy] > 0. Graphene has recently emerged as a platform able to support tunable and extremely confined plasmons at terahertz and infrared frequencies, while providing large light−matter interactions,30 features that have been exploited to put forward a myriad of exciting applications.20,40−42 An even more interesting scenario arises when one of the imaginary components of the metasurface conductivity tensor dominates over the other one, thus leading to structures that support SPPs with an anisotropic elliptical topology able to favor propagation toward a specific direction. Figure 1b illustrates an extreme case of this behavior, the σnear-zero regime,23,26 able to canalize most of the energy toward the y-axis due to an Im[σyy] ≈ 0 conductivity. This regime usually appears at the metasurface topological transition,26,43 where the topology evolves from elliptic to hyperbolic or vice versa, and it is associated with a dramatic enhancement of the local density of states. Lastly, Figure 1c,d consider metasurfaces that support quasi-TM SPPs with hyperbolic topology, which arises when the surface behaves as a dielectric (capacitive, with Im[σ] < 0) along one direction and as a metal (inductive, with Im[σ] > 0) along the orthogonal one, i.e., sgn(Im[σxx]) ≠ sgn(Im[σyy]). Even though such structures also support weakly confined quasi-TE plasmons,26,28,39 we focus in this work on quasi-TM hyperbolic plasmons due to their exciting in-plane response that translates into ideally confined waves, i.e., infinite local density of states, that can propagate in a limited range of directions within the sheet.26,28 These modes can be seen as the two-dimensional version of Dyakonov surface states that appear along the interface between anisotropic 3D crystals.44,45 As it happens in the bulk case, their dispersion relation can be largely simplified by asymptotically approximating the branches of the resulting hyperbola in eq 2, leading to39 k y ≈ m(1,2)kx ± b(1,2,3)

m(1,2) =

1 [−(σxy + σyx)] ± 2σyy

(σxy + σyx)2 − 4σxxσyy (4)

where m(1) and m(2) are associated with the positive and negative sign of the square root, respectively, and b(1) = k 0

⎡ A ⎤2 ⎡ A ⎤2 k0 ⎥ , b(2,3) = 1−⎢ 1−⎢ ⎥ m(1,2) ⎢⎣ 2σyy ⎥⎦ ⎣ 2σxx ⎦ (5)

with ⎛2 ⎞ 2 A = ⎜⎜ + (σxxσyy − σxyσyx)⎟⎟ η0 ⎝ η0 ⎠ ±

⎛2 ⎞2 2 ⎜⎜ + (σxxσyy − σxyσyx)⎟⎟ − 4σxxσyy η0 ⎝ η0 ⎠

The different branches and signs of the square roots can easily be selected by enforcing decaying evanescent modes away from the structure, i.e., Im[kz] > 0. In the common case of hyperbolic metasurfaces defined by a diagonal conductivity tensor (i.e., with σyx = σxy = 0) these equations reduce to σ

m(1,2) = ± − σxx , b(1) = k 0 1 − yy

b(2,3) = m(1,2)k 0 1 −

■

2

( ) 2 η0σxx

⎛ 2 ⎞2 ⎜ ⎟ , and ⎝ η0σyy ⎠ (6)

PRACTICAL IMPLEMENTATION Uniaxial ultrathin surfaces with exotic electromagnetic responses can indeed be found in natural crystals,46−48 providing nonresonant responses and avoiding the use of complex nanofabrication processes and their associated tolerances and increased losses. Possibly the most straightforward approach to realize them is to simply reduce the profile of well-known bulk uniaxial materials.49 Such materials include for instance graphite, which is composed of parallel graphene layers that provide a metallic in-plane response. These layers are held together through van der Waals forces, forming a natural hyperbolic media in the wavelength range around 240−280 nm,46 i.e., within the frequency band where this coupling is strongly capacitive. This natural response has inspired recent developments of artificial HMTMs in the THz band using arrangements of graphene layers.50−52 Another crystal with similar features is magnesium diboride (MgB2), for which graphene-like layers of boron are alternated by densely packed layers of Mg. Other materials, such as tetradymites, provide the sought-after extreme anisotropic and hyperbolic response in the visible part of the spectrum. High-quality hexagonal boron nitride (hBN) is undoubtedly one of the most promising candidates in this category49,53 partially due to its excellent compatibility with graphene optoelectronics.54−56 This material keeps its exciting properties as it is thinned down to a thickness of about 1 nm,49 and it has allowed the experimental demonstration of low-loss hyperbolic phonon polaritons in the infrared.57 Another interesting approach to realize uniaxial metasurfaces is to take advantage of emerging 2D materials,58,59 such as 2D chalcogenides and oxides. Among them, there has been an increasing interest in black phosphorus (BP) for

(3)

where 2213

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Figure 2. Naturally hyperbolic 2D materials: the case of black phosphorus. (a) Lattice structure of monolayer black phosphorus. Different colors are used for visual clarification. (b) Imaginary part of black phosphorus conductivity components versus frequency for several values of chemical potential. (c) Real part of black phosphorus conductivity components versus frequency for several values of chemical potential. Solid, dashed, and dotted lines correspond to chemical potentials of 0.005, 0.05, and 0.1 eV, respectively. Black phosphorus thickness is 10 nm, direct band gap is 0.485 eV, damping is 5 meV, and temperature is 300 K.

Figure 3. Artificial hyperbolic metasurfaces implemented using an array of densely packed graphene strips26 (a) and nanostructured silver (from ref 27, reprinted by permission from Macmillan Publishers Ltd: Nature, 522, 192−196, doi: 10.1038/nature14477, copyright 2015) (b). (c and d) Imaginary and real components of the effective conductivity tensor of an array of graphene strips versus frequency. Periodicity L is set to 150 nm and graphene strip width W is 130 nm. Graphene chemical potential is μc = 0.3 eV; its relaxation time τ = 1.0 ps. (e) Imaginary component of the effective conductivity tensor of the structure described in panels (c) and (d) at 25 THz versus the graphene strip width W for various values of chemical potential.

plasmonic and optoelectronic applications.60−64 BP is an extremely anisotropic ultrathin crystalline structure, as illustrated in Figure 2a, and it has recently been isolated in monoand few-layer forms. BP possesses exciting properties, such as an intrinsic direct band gap that may range from around 2 eV in monolayers (phosphorne) to ∼0.3 eV in its bulk configuration, tunable electric response versus thickness, externally applied electric/magnetic fields and mechanical strain, and the support of confined surface plasmons. Similarly to the case of graphene, the ultrathin nature of BP allows a simple electromagnetic characterization in terms of optical conductivity, which may be accurately derived applying the Kubo formalism.62,64 Figure 2b,c shows the real and imaginary parts of the BP conductivity

components versus frequency for various values of chemical potential μc. This potential is defined here as the energy from the edge of the first conduction band to the Fermi level. Results confirm the dispersive, tunable, and extremely anisotropic response of BP in the infrared. This response is indeed very rich, and it includes anisotropic elliptic quasi-TM (Im[σxx] > 0, Im[σyy] > 0) and quasi-TE (Im[σxx] < 0, Im[σyy] < 0) responses at low and very high frequencies, respectively, as well as an intrinsic hyperbolic frequency band (Im[σxx] > 0, Im[σyy] > 0) and two clearly defined topological transitions that implement σ-near-zero topologies. The hyperbolic nature of BP arises as the switch from plasmonic to dielectric response, associated with interband transitions, happens at different 2214

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techniques such as the Kubo formalism,70 (ii) give physical insights about the structure response, (iii) take complex phenomena such as magnetic bias and nonlocal effects into account, and (iv) provide useful and simple rules to engineering any response at a desired operation frequency. In the particular case of a uniaxial metasurface implemented by an array of graphene strips, as shown in Figure 3a, this effective conductivity tensor σ eff simplifies to

frequencies for each conductivity component. Despite their advantages, the electromagnetic response of BP and all natural ultrathin uniaxial materials is intrinsically associated with their lattice structure, therefore limiting the flexibility to be directly applied to the design of advanced plasmonic and optoelectronic components. An additional degree of flexibility in the design of such devices may be achieved by engineering uniaxial metasurfaces following metamaterial-inspired strategies,65,66,11 at the cost of relatively larger losses and the need for nanofabrication processes. This powerful approach is based on combining planar materials on the same surface in different geometries, to tailor their macroscopic electromagnetic response at will. For instance, ref 26 introduced an array of densely packed graphene strips to design hyperbolic and extremely anisotropic metasurfaces at THz frequencies, as shown in Figure 3a. This structure provides intriguing optical properties combined with an easy fabrication, large field confinement, and full compatibility with integrated circuits and optoelectronic components. Its major advantage resides in its intrinsic tunability, enabled by the graphene field effect.67 Indeed, simply applying a modest bias allows manipulating the metasurface band topology in real time, routing the propagating surface plasmons to desired directions within the plane, and controlling light−matter interactions and associated sensing capabilities. In optics, uniaxial metasurfaces may also be realized using metal gratings,68 as recently reported experimentally at visible frequencies27 (see Figure 3b) using single-crystalline silver nanostructures, demonstrating exciting functionalities such as canalization, negative refraction, and polarizationdependent routing. An insightful and practical technique to design these and other uniaxial metasurfaces is the effective medium approach (EMA).65,26,36 This technique is based on averaging the different constitutive materials of the structure in order to macroscopically model its electromagnetic response. Let us consider, for the sake of illustration, a uniaxial metasurface composed of unit cells with periodicity L made of infinitely long 2D strips with width W, characterized by the fully populated conductivity tensor σ = (σxx,σxy ; σyx, σyy). Assuming a subwavelength separation distance G between strips, their near-field coupling may be taken into account through the effective grid conductivity σC ≈ −i2ωε0εeff(L/π) ln[csc(πG/ 2L)], where ω is the radial frequency and ε0 and εeff are the permittivity of free space and the one relative to the surrounding medium. It should be emphasized that this is an approximate result derived using an electrostatic approach,69 yet it provides a powerful way to model the in-plane propagation properties of complex metasurfaces. The effective conductivity tensor σ eff of the metasurface reads Wσxxσc W σxy eff W σyx , σxyeff ≈ σxxeff , σyx ≈ σxxeff σxxeff ≈ Lσc + Wσxx L σxx L σxx

σxxeff ≈

WσσC W eff , σyyeff ≈ σ , σxyeff = σyx =0 LσC + Wσ L

(9)

where σ is graphene scalar conductivity in the absence of magnetic bias. Figure 3c,d show the imaginary and real part of the effective conductivity tensor of such uniaxial metasurfaces assuming unit cells with periodicity L = 150 nm and strip widths W = 130 nm. The figures confirm that propagation along the strips, i.e., y-direction, is low-loss and inductive (Im[σeff yy ] > 0) for the entire frequency band under analysis. The response across the strips, i.e., x-direction, is quite different, due to its resonant response at LσC + Wσ = 0. At low frequencies, the strong near-field coupling between adjacent strips determines the capacitive response of this conductivity eff component (Im[σxx ] < 0), and it provides the typical eff hyperbolic response (Im[σeff xx ] < 0, Im[σyy ] > 0) of graphene26,39 based metasurfaces. At frequencies larger than the resonance, the inductive response of graphene dominates, while it also slowly decreases as operation frequency further increases. Not shown in the graphs, the response of both conductivity components evolves to capacitive at higher frequencies due to the intrinsic contribution of interband transitions.32 Figure 3e illustrates the possibility to fully engineer the metasurface response at any desired frequency by using simple techniques such as adjusting the strip width or manipulating graphene’s chemical potential through the field effect.

(7)

σyyeff

eff eff σyx σxy W W σyxσxy ≈ + σyy − eff L L σxxσc σxx

(8) Figure 4. Isofrequency contours of quasi-TM surface plasmons supported by the artificial hyperbolic metasurface illustrated and described in Figure 3a. (a) Operation frequency: 2 THz, σ-near-zero topology. (b) Operation frequency: 15 THz, hyperbolic topology. (c) Operation frequency: 24.0 THz, canalization regime. (d) Operation frequency: 40.0 THz, elliptic anisotropic topology.

where we recall that a subwavelength periodicity has been assumed, i.e., L ≪ λSPP, with λSPP being the plasmon wavelength. This approach is able to (i) easily homogenize ultrathin metasurfaces such as those shown in Figures 3a,b, leading to similar results to those of much more sophisticated 2215

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Figure 5. Different mechanisms that limit the hyperbolic isofrequency isofrequency contours of quasi-TM surface plasmons supported by the artificial hyperbolic metasurface illustrated in Figure 3a. (a) Influence of ohmic losses. The metasurface is modeled using an effective medium approach with periodicity L = 50 nm, and graphene is characterized using the local Kubo formalism.32 (b) Influence of periodicity. The metasurface is analyzed using a mode-matching full-wave technique.71 Graphene is considered lossless, and its imaginary part is characterized using the local Kubo formalism. (c) Influence of nonlocality. The metasurface is modeled using an effective medium approach with periodicity L = 50 nm, graphene is considered lossless, and its imaginary part is characterized using a nonlocal conductivity model.72 Operation frequency is 15 THz, graphene chemical potential is 0.3 eV, and strip width is W = 0.5L.

man-made HMTSs. As expected, the lattice periodicity imposes a cutoff wavenumber at around π/L, with L being the unit-cell period of the metasurface. Figure 5b clearly shows the influence of the HMTSs’ periodicity on the isofrequency contours of the supported quasi-TM mode, analyzing such structures with a full-wave mode-matching technique in the regime for which the assumptions of EMA break down.71 Our results confirm that even for very small unit-cell periods, of just a few dozen nanometers, the periodicity strongly dominates over dissipation losses to shape the SPP isofrequency contours. The last mechanism to be considered to explain the closing of isofrequency contours is the intrinsic nonlocal response of the HMTS constitutive materials. In our particular implementation, made of a densely packed array of graphene strips, we model nonlocal graphene using the Bhatnagar−Gross−Krook approach derived in ref 72. This model takes into account intraband transitions in graphene, which is valid up to a few dozen THz when the spatial variations of the fields are smaller than the de Broglie wavelength of the particles (i.e., kρ < 2kf, where kf is the Fermi wavenumber), and it has been successfully applied to investigate the influence of nonlocality in various graphene-based devices at THz frequencies.73 In the case of HMTSs made of materials other than graphene, for instance noble metals,27 techniques such as the hydrodynamic Drude model within the Thomas−Fermi approximation35,74 can be applied to model their intrinsic nonlocality. Generally speaking, the intrinsic nonlocal response of materials enforces a wavenumber cutoff to the supported SPPs at around kρ ≈ (c/ νf)k0, where νf is the Fermi velocity of electrons in the material, even in the ideal case in which losses and periodicity are not the limiting factor. This response is illustrated in Figure 5c. We do note that the nonlocal SPP isofrequency contour suddenly disappears for high wavenumbers instead of closing down toward the kx axis. This behavior arises because as the wavenumber increases, the transverse component of graphene nonlocal conductivity dominates and pushes electrons toward the edge of the strips, thus effectively routing the plasmons toward nonsupported directions of propagation within the sheet.36 Therefore, the intrinsic nonlocal response of the involved materials becomes the dominant mechanism that closes the isofrequency contour in natural HMTSs or in those

Figure 4 shows the isofrequency contours of the supported quasi-TM surface plasmons and highlights their evolution versus frequency. Specifically, Figure 4a confirms that the SPPs present a σ-near-zero response that canalizes most of the energy toward the y-axis, i.e., along the strips, in the low-THz band. At higher frequencies, around 15 THz, the supported mode shows a typical hyperbolic response, as depicted in Figure 4b. These two examples correspond to isofrequency contours that would be open in the ideal case and are closed here because of the intrinsic dissipation losses of graphene. In the ideal lossless and homogeneous scenario, hyperbolic metasurfaces would possess an inf inite local density of states, wave confinement, and singular light−matter interactions. However, as discussed in some detail below, there exist different physical mechanisms that in reality contribute to close these isofrequency contours in practice and limit the response of hyperbolic metasurfaces. At the resonant frequency, around 24 THz, the real part of the conductivity across the strips is significantly larger than its orthogonal counterpart, leading to a relatively low-loss canalization phenomenon along the xdirection23 (see Figure 4c). Finally, Figure 4d confirms that at frequencies larger than the resonance the supported quasiTM modes acquire a more common anisotropic and elliptical response. In the specific case of metasurfaces with hyperbolic topology, it is instructive to understand the physical mechanisms that close the otherwise open isofrequency contours of the supported SPPs and that establish an upper bound to the maximum achievable level of wave confinement and light− matter interactions.36 Figure 5 isolates the influence of these mechanisms on the response of a graphene-based HMTS. The most basic mechanism is related to the presence of dissipative losses, as shown in Figure 5a. In an ideal lossless case, considering a patterned graphene sheet with infinite relaxation time, the isofrequency contour is open and completely unbounded. As losses increase, the metasurface no longer supports plasmons with very large wavenumbers. However, even in the case of very low graphene quality (τ ≈ 0.05 ps), losses just filter out SPPs with very large wavenumbers, which may be difficult to excite in practice. The second mechanism responsible for closing SPPs’ isofrequency contour in realistic metasurfaces is the nonlocality associated with the periodicity of 2216

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Figure 6. Transmission and reflection of surface plasmons. Transmissivity (a) and refraction angle (b) of an SPP propagating along an isotropic surface (σin = i0.5 mS) with an angle θin ≈ 56.4° and impinging onto a lossless uniaxial metasurface defined by the conductivity tensor σ (see inset of panel b). The results are computed using eqs 10−12 versus the components of σ . (c−f) Top view of the z-component of the electric field induced at the interface between two metasurfaces, following cases A−D detailed in panels (a) and (b). Results are computed with COMSOL Multiphysics.76 (c) Case A: σxx = σyy = 10−3 + i0.25 mS. (d) Case B: σxx = 10−3 + i0.06 mS, σyy = 10−3i ms. (e) Case C: σxx = 10−3 + i0.3 mS, σyy = 10−3 − i0.3 mS. (f) Case D: σxx = 10−3 − i0.1 mS, σyy = 10−3 + i0.62 mS.

man-made HTMSs with a unit-cell period L
0 and Im[σyy] > 0), for which the metasurface supports quasi-TM SPPs able to significantly interact with incoming waves due to their relatively largebut always finitewavenumbers. As expected, lowering the imaginary part of the conductivity components increases the emitter SER, due to the stretching of the metasurface isofrequency contour. The third quadrant (Im[σxx] < 0 and Im[σyy] < 0) is associated with quasi-TE SPPs barely confined to the surface that lead to negligible light− matter interactions and SER enhancement. Lastly, the second and fourth quadrants (Im[σxx] > 0, Im[σyy] < 0 and Im[σxx] < 0, Im[σyy] > 0, respectively) implement hyperbolic metasurfaces with ideally open isofrequency contours, able to couple and interact with incoming waves with arbitrarily large wavenumbers, thus leading to an impressive enhancement of the dipole SER. Importantly, this enhancement is finite even in this ideal case due to the filtering of evanescent waves carried out by the free-space region located between the dipole and the metasurface. The slight decrease of SER found when the conductivity increases is attributed to the progressive shift of the hyperbolic branches, which prevents the coupling of incoming waves with low wavenumbers to the surface.26 The inset of Figure 9a depicts the topological transition between hyperbolic and elliptic topologies, associated with a further enhancement of light−matter interactions. This behavior, similar to the one found in bulk HMTSs,43 appears due to 2220

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flux, let us consider two closely located metasurfaces implemented using nanostructured graphene, as in ref 88 (see Figure 10a). The top and bottom metasurfaces are assumed to be at temperatures 310 and 290 K, respectively. Figure 10b shows the spectral radiative heat flux of a single metasurface versus frequency compared to that of pristine graphene. The results confirm a significant enhancement of the heat flux in the band where the structure presents a hyperbolic response. Figure 10c shows the ratio of the near-field heat flux between the two metasurfaces compared to the case of pristine graphene sheets versus the separation distance between the layers. As expected, this ratio significantly increases as the metasurfaces are closer and closer to each other, which is attributed to the large influence of evanescent waves in the energy transfer. For instance, considering a separation distance of d = 50 nm, the heat flux between pristine graphene sheets is already about 120 times higher than the blackbody limit due to the excitation of confined surface plasmons.88 However, the patterning of graphene dramatically increases the heat transfer by more than 1000 times compared to the blackbody limit. It is important to stress that over 80% of the entire heat flux comes from the frequency region in which the metasurfaces present a hyperbolic response.88 As expected, this scenario becomes very different when the separation distance between the metasurfaces increases. Then, the overall heat transfer significantly decreases due to the filtering of the evanescent spectrum imposed by free space.

light on the different mechanisms, namely, losses and nonlocality, that close the otherwise open hyperbolic isofrequency contour by imposing a cutoff on the supported SPP wavenumbers. In this regard, we have shown that the influence of the intrinsic material nonlocality on the HMTS dispersion is inversely proportional to the electron Fermi velocity and that it may be dominant over the nonlocality arising from the metasurface granularity. This has allowed us to derive a practical rule to design artificial HMTSs with quasioptimal physical dimensions. Finally, we have exploited the large degree of flexibility provided by man-made uniaxial metasurfaces to explore some of the exciting applications that such ultrathin structures may offer. First, we have analyzed the propagation of surface plasmons across the boundary between two different metasurfaces, illustrating phenomena such as negative refraction and beam-steering. Then, we have designed metasurfaces operating in the canalization regime, demonstrating planar hyperlensing with deeply subdiffractive resolution even in the presence of losses. Next, we have illustrated the dramatic enhancement of light−matter interactions offered by uniaxial metasurfaces and its straightforward application to boost the SER of emitters located nearby. As expected, this enhancement cannot become singular, and it is limited in practice by the presence of losses and nonlocality. Lastly, we have discussed how hyperbolic metasurfaces may enable broadband super-Planckian thermal radiation far beyond the blackbody limit. Uniaxial metasurfaces have opened new perspectives in the field of plasmonics due to their appealing properties and easy implementation on well-established platforms such as graphene at THz and mid-IR or noble metal at optical wavelengths. From a practical viewpoint, such metasurfaces face important challenges that must still be overcome, such as the intrinsic losses of plasmonic materials, the fabrication of patterned structures with lower tolerances and better quality, and, more importantly, the in- and out-coupling of external electromagnetic waves. Advances in all these challenges will contribute to a bright future for uniaxial metasurfaces and will further broaden their impact in practical scenarios. In this Perspective we have focused on planar hyperbolic metasurfaces. Even though technologically more challenging, nonplanar, or conformal, hyperbolic metasurfaces may provide another degree of freedom to induce exciting phenomena, as well as interesting applications. For instance, hyperbolic plasmons can certainly be routed through curved surfaces and bends,89 which is indeed desirable in flexible circuits and biocompatible devices. In addition, surface plasmons over cylindrical tubes made of 2D materials have gained significant attention in the context of new nanophotonic components90 and terahertz antennas.91 Inspired by these concepts and some recent developments in the context of bulk, cylindrical indefinite media,92,93 we anticipate that interesting applications such as cloaking and multimodal optical fibers will significantly benefit from the thrilling possibilities offered by hyperbolic metasurfaces. We further envision that such responses will lead to extreme electrodynamical light trapping94 and enhanced sensing95 in deeply subwavelength objects mantled by HTMSs. The broad range of potential applications enabled by hyperbolic metasurfaces has yet to be fully explored. For instance, some functionalities such as the spin control of light and polarization-dependent routing of surface plasmons27 might find application in advanced chiral optical components and quantum information science. Moreover, the use of

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CONCLUSION AND OUTLOOK The emerging field of uniaxial and hyperbolic metasurfaces holds great promise to significantly impact nanoscale optics and technology, due to a combination of fascinating phenomena and unusual optical properties within a reduced dimensionality, opening the door to a wide variety of exciting applications. Compared to uniaxial and hyperbolic bulk materials, metasurfaces exhibit important advantages, including a simple fabrication, compatibility with integrated circuits and optoelectronic components, lack of volumetric losses, easy access and subsequent process of stored energy using near-field techniques, and strong light−matter interactions. Uniaxial metasurfaces have taken advantage of the large flexibility provided by the metamaterial approach, enabling the design of periodic planar configurations able to exhibit unusual band topologies, with thrilling functionalities at any desired operation frequency. Artificial metasurfaces also face some challenges, such as an increased level of losses near the structure resonance and the influence of nonlocality due to the inherent periodicity. As a promising alternative, it should be highlighted the availability of natural 2D materials with intrinsic hyperbolic dispersion. These materials allow obtaining samples of large size and may avoid complex nanofabrication processes, imperfections, and associated losses. Unfortunately, the frequency band of hyperbolic dispersion is determined solely by the intrinsic properties of such 2D materials, which may also suffer from intrinsic material losses. In this Perspective, we have first reviewed the unusual electromagnetic properties of ultrathin uniaxial metasurfaces implemented by either natural or artificial materials, studying the different topologies that they supportranging from closed isotropic to ideally open hyperbolic and going through the σnear-zero caseand their associated surface plasmon properties versus the features of the metasurface conductivity tensor components. In the particular case of HMTSs, we have shed 2221

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reconfigurable materials such as graphene enables the development of tunable surfaces able to manipulate their topology in real time. Related applications include the realization of inplane transformation optics using MTSs,96 allowing, for example, to cloak planar defects or grain boundaries that may arise during fabrication. Furthermore, magnetic-free nonreciprocal plasmon propagation based on spatiotemporal modulation, as recently reported in ref 97, can be extended to provide controlled in-plane wave-mixing and frequency conversion enhanced by hyperbolic responses. In a related context, and following recent development in bulk HMTMs,98 it is expected that the large field enhancement in HMTSs may be exploited in nonlinear optics,99 leading to very large effective nonlinear responses100 and enabling intriguing applications such as third-harmonic generation and self-focusing, among others. Another interesting direction still to be explored may be the development of parity-time symmetric101 HMTSs, giving rise to the foundation of unconventional gain−loss surfaces with application in unidirectional cloaks,102,103 double-negative refraction,104 and reflection/transmission coefficients that can be simultaneously equal to or greater than unity.105 Last but not least, graded uniaxial metasurfaces may allow to manipulate and route plasmons along the surface at will, while simultaneously directing super-Planckian thermal radiation to any desired direction in space and providing efficient thermal management at the nanoscale. These fascinating properties and functionalities open unprecedented venues for the realization of ultrathin plasmonic devices with exciting applications in sensing, imaging, energy harvesting, quantum optics, inter/ intrachip networks, and communications systems.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the Air Force Office of Scientific Research Grant No. FA9550-13-1-0204, the Welch Foundation with Grant No. F-1802, the Simons Foundation, and the National Science Foundation with Grant No. ECCS-1406235.

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REFERENCES

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DOI: 10.1021/acsphotonics.6b00645 ACS Photonics 2016, 3, 2211−2224