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Perspective

Flatland Optics with Hyperbolic MetasurfacesJ.S. Gomez-Diaz, and Andrea Alu

ACS Photonics, Just Accepted Manuscript • DOI: 10.1021/acsphotonics.6b00645 • Publication Date (Web): 16 Nov 2016

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Flatland Optics with Hyperbolic Metasurfaces

J. S. Gomez-Diaz(1)

and Andrea Alù(2)*

(1)Department of Electrical and Computer Engineering, University of California, Davis,

Davis, CA 95616, USA

(2)Department of Electrical and Computer Engineering, The University of Texas at Austin,

Austin, TX 78712, USA

Keywords: Plasmonics, metasurfaces, uniaxial media, hyperbolic materials, graphene, black

phosphorus.

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 MTS 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 super-Planckian 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.

Ultrathin metasurfaces (MTSs) have recently gained significant attention, thanks to their

capability to locally modify the phase, amplitude and polarization of light in reflection and

transmission1

2,3,4,5. 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 splitters6,7,8,9

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 media10,11,12,13

. Undoubtedly, a large part of the success of planar MTSs is

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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-field8,14

. 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-plane propagation and near-field functionalities, enabled by confined surface

plasmon polaritons (SPPs)15,16,17,18

, with exciting applications in nanophotonics19

, planar

nanoantennas and transceivers20

, communication systems21

, extremely sensitive sensors22

, on-

chip networks and in-plane imaging23

.

This perspective focuses on recent developments in the theory and applications of in-plane SPP

optics using uniaxial metasurfaces, with the purpose of confining light into ultrathin structures

and then manipulate its in-plane propagation and near-field 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 dual of each other due to the additional constraints that the

reduced dimensionality of ultrathin metasurfaces imposes to electromagnetic wave propagation,

resulting in new, exciting propagation phenomena in two dimensions. We start by introducing

the concept of uniaxial metasurfaces and illustrating the different type of topologies and surface

plasmons that they can support26

. We pay particular attention to hyperbolic metasurfaces, due to

the fascinating properties that they can offer26,27,28

– such as extremely large wave confinement

and local density of states, as happens in bulk HMTMs13,29

. Several possibilities to realize

uniaxial metasurface 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). 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

super-Planckian thermal emission beyond the black-body limit. Finally, we outline our vision

for the future of uniaxial metasurfaces and their role in the coming generation of nanophotonic

devices.

Physics of uniaxial metasurfaces

The electromagnetic response of an infinitesimally-thin homogeneous anisotropic metasurface

can be modelled by its optical conductivity tensor

xx xy

yx yy

σ σσ

σ σ =

, (1)

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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[���] ≥0, Re[�] ≥ 0, and Re[��� + �] ≥ |�� + ��

∗ |, where ‘*’ denotes complex conjugate, under

a ���� time convention. In addition, the fact that we consider uniaxial metasurfaces imposes

that the conductivity tensor �� must be diagonal in a suitable reference coordinate system. Non-

diagonal conductivity terms may arise due to different factors, including i) the presence of

subwavelength inclusions with non-symmetrical shape with respect to the reference coordinate

system, associated with an in-plane bending of the propagating SPPs28

, ii) magneto-optical

effects19,32

associated to hybrid transverse magnetic-transverse electric (TM-TE) SPPs and in-

plane gyrotropic response33,34

, and iii) non-local effects35,36

, 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[���] and

Im[�], which determines the shape of the supported SPP isofrequency contours.

The dispersion relation of the surface modes supported by a free-standing anisotropic

metasurface is given by31,37,38,39

( )( ) ( ) ( )0

2 2 2 2

0 0 0 04 2 2 0

z xx yy xy yx xx yy x xx yy y x y xy yxk k k k k k kη σ σ σ σ η σ σ η σ σ σ σ+ − − + + + + = (2)

where 0k is the free-space wavenumber, decaying evanescent modes [ ]Im 0z

k > are enforced

away from the surface, and the reference coordinate system follows the one shown in Fig. 1.

Efficient techniques to solve this dispersion relation have been recently presented39

, thus

allowing to easily obtain the propagation properties of the supported SPPs over the frequency

range of interest. Fig. 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 identifying their properties. For instance, Fig. 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[���]� = sgn�Im[�]�, and it

can be associated to either quasi-TM (inductive, Im[���] > 0, Im[�] > 0 ) or quasi-TE

(capacitive, Im[���] < 0, Im[�] < 0 ) surface plasmons. The polarization of the supported

SPPs will be purely TM or TE only for isotropic metasurfaces, i.e., when ��� = �. 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., 0k kρ ≈ , 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 graphene18

, an inductive 2D material where

Im[���]=Im[�]>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 interactions30

, features that have been exploited to put forward a myriad of exciting

applications20,40,41,42

. An even more interesting scenario arises when one of the imaginary

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components of the metasurface conductivity tensor dominates over the other one, thus leading to

structures that supports SPPs with an anisotropic elliptical topology able to favor propagation

towards a specific direction. Fig. 1b illustrates an extreme case of this behavior, the �-near-zero

regime23,26

, able to canalize most of the energy towards the y-axis thanks to a Im�� ≈ 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 to a dramatic

enhancement of the local density of states. Lastly, Figs. 1c-d consider metasurfaces that support

quasi-TM SPPs with hyperbolic topology, which arises when the surface behaves as a dielectric

(capacitive, with Im[�] < 0) along one direction and as a metal (inductive, with Im[�]>0) along

the orthogonal one, i.e. sgn�Im[���]� ≠ sgn�Im[�]�. Even though such structures also support

weakly-confined quasi-TE plasmons26,28,39,

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 crystals44,45

. 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

(1,2) (1,2,3)y xk m k b≈ ± , (3)

where

2

(1,2)

1( ) ( ) 4

2xy yx xy yx xx yy

yy

m σ σ σ σ σ σσ = − + ± + − , (4)

being (1)m and

(2 )m associated to the positive and negative sign of the square root, respectively,

and

2

(1) 0 12 yy

Ab k

σ

= −

,

2

0(2,3)

(1,2)

12 xx

k Ab

m σ

= −

, (5)

with ( ) ( )2

0 0 0 0

2 2 2 24xx yy xy yx xx yy xy yx xx yyA σ σ σ σ σ σ σ σ σ σ

η η η η = + − ± + − −

. 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 0z

k > . In the common case of hyperbolic metasurfaces

defined by a diagonal conductivity tensor (i.e., with 0yx xyσ σ= = ) these equations reduce to

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(1,2)xx

yy

mσσ

= ± − ,

2

(1) 0

0

21

yy

b kη σ

= −

and

2

(2,3) (1,2) 0

0

21

xx

b m kη σ

= −

. (6)

Practical Implementation

Uniaxial ultrathin surfaces with exotic electromagnetic responses can indeed be found in natural

crystals46,47,48,

providing non-resonant 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 materials49

. 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 nm46

, 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 layers50,51,52

. 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 thanks

to its excellent compatibility with graphene optoelectronics54,55,56

. This material keeps its

exciting properties as it is thinned down to a thickness of about 1 nm49

and it has allowed the

experimental demonstration of low-loss hyperbolic phonon polaritons in the infrared57

. Another

interesting approach to realize uniaxial metasurfaces is to take advantage of emerging 2D

materials58,

59

, such as 2D chalcogenides and oxides. Among them, there has been an increasing

interest in black phosphorus (BP) for plasmonic and optoelectronic applications60,61,62,63,64

. BP is

an extremely anisotropic ultrathin crystalline structure, as illustrated in Fig. 2a, and it has

recently been isolated in a mono- and few- layers forms. BP possess exciting properties, such as

an intrinsic direct bandgap which 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 formalism62,64

. Figs. 2b-c shows the real and imaginary parts of the BP conductivity

components versus frequency for various values of chemical potential #$ . This potential is

defined here as the energy from the edge of the first conduction band to the Fermi level. Results

confirms 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[���] >0, Im�� > 0 ) and quasi-TE (Im[���] < 0, Im�� < 0 ) responses at low and very high

frequencies, respectively, as well as an intrinsic hyperbolic frequency band ( Im[���] >0, Im�� > 0 ) and two clearly-defined topological transitions that implement � -near-zero

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topologies. The hyperbolic nature of BP arises as the switch from plasmonic to dielectric

response, associated to interband transitions, happens at different 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 strategies65,66,11

, at the cost of relatively

larger losses and the need of 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, as shown in Fig. 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

graphene field effect67

. Indeed, simply applying a modest bias allows to manipulate the

metasurface band topology in real time, route the propagating surface plasmons to desired

directions within the plane, and to control light-matter interactions and associated sensing

capabilities. In optics, uniaxial metasurfaces may also be realized using metal gratings68

, as

recently reported experimentally at visible frequencies27

(see Fig. 3b) using single-crystalline

silver nanostructures, demonstrating exciting functionalities such as canalization, negative

refraction and polarization-dependent 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 ( ), ; ,xx xy yx yyσ σ σ σ σ= . Assuming a subwavelength

separation distance G between strips, their near-field coupling may be taken into account through

the effective grid conductivity ( ) ( )02 / ln csc / 2C effi L G Lσ ωε ε π π ≈ − , where % is the radial

frequency and &' and &()) 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 approach69

, yet it provides a powerful way to model the in-plane propagation

properties of complex metasurfaces. The effective conductivity tensor effσ of the metasurface

reads

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,, ,xy yxeff eff eff eff effxx c

xx xy xx yx xx

c xx xx xx

W W W

L W L L

σ σσ σσ σ σ σ σ

σ σ σ σ≈ ≈ ≈

+ (7)

eff eff

yx xy yx xyeff

yy yy eff

x cx xx

W W

L L

σ σ σ σσ σ

σ σ σ− +≈ , (8)

where we recall that a subwavelength periodicity has been assumed, i.e., SPPL λ<< , being

SPPλ

the plasmon wavelength. This approach is able to: i) easily homogenize ultrathin metasurfaces

such as those shown in Figs. 3a-b, leading to similar results as much more sophisticated

techniques such as the Kubo formalism70

, 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 Fig. 3a, this effective conductivity tensor effσ simplifies to

, 0,eff ef eff eff

xy yx x

fCx yy

C

W W

L W L

σσσ σ σ

σσσ

σ≈ ≈ = =

+, (9)

where σ is graphene scalar conductivity in the absence of magnetic bias. Figs. 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 0eff

yyσ > 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 0CL Wσ σ+ = . At low frequencies, the strong near-

field coupling between adjacent strips determines the capacitive response of this conductivity

component ( )Im 0eff

xxσ < , and it provides the typical hyperbolic response

( )Im 0, Im 0eff eff

xx yyσ σ < > of graphene-based metasurfaces26,39

. 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 evolve to capacitive at higher frequencies due to the intrinsic

contribution of interband transitions32

. Fig. 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.

Fig. 4 shows the isofrequency contours of the supported quasi-TM surface plasmons, and

highlights their evolution versus frequency. Specifically, Fig. 4a confirms that the SPPs presents

a σ -near-zero response that canalizes most of the energy towards 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 Fig. 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 infinite local density of states, wave confinement and

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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 x-direction23

(see

Fig. 4c). Finally, Fig. 4d confirms that at frequencies larger than the resonance the supported

quasi-TM 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 interactions36

. Fig. 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 Fig. 5a. In an ideal lossless case, considering a homogeneous graphene sheet

with infinite relaxation time, the isofrequency contour is open and completely unbounded. As

losses increase, the metasurface does no longer support plasmons with very large wavenumbers.

However, even in the case of very low graphene quality (* ≈ 0.05ps), 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 man-made HMTSs. As expected, the lattice

periodicity imposes a cutoff wavenumber at around -//, being L the unit-cell period of the

metasurface. Fig. 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 down71

. Our results

confirm that even for very small unit-cell periods, of just a few dozens of 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 (BGK) approach derived in72

. This model takes into account intraband

transitions in graphene, which is valid up to a few dozens of THz when the spatial variations of

the fields are smaller than the de Broglie wavelength of the particles (i.e., 01 < 20), where 0) is

the Fermi wavenumber), and it has been successfully applied to investigate the influence of

nonlocality in various graphene-based devices at THz73

. In the case of HMTSs made of other

materials rather than graphene, for instance noble metals27

, 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 01 ≈ �3 4)�⁄ 0' , where 4) 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 Fig. 5c. We do note that the nonlocal SPP

isofrequency contour suddenly disappears for high wavenumbers instead of closing down

towards the 0� axis. This behavior arises because as the wavenumber increases, the transverse

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component of graphene nonlocal conductivity dominates and pushes electrons towards the edge

of the strips thus effectively routing the plasmons towards non-supported 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 man-made HTMSs with a unit cell period / <678

$9:. This expression provides

a simple but powerful rule for the design of artificial HMTSs across the entire frequency region.

For instance, at infrared and visible frequencies, nonlocal effects are usually negligible35

because

the minimum cell period / is barely subwavelength, due to restrictions imposed by the

fabrication process resolution, thus becoming dominant. On the contrary, in the THz band

intrinsic nonlocal effects will dominate, and the above design rule becomes useful to easily

realize HMTSs with the largest unit-cell period / able to provide the maximum possible wave

confinement.

Applications

One of the most interesting possibilities offered by ultrathin metasurfaces is the routing of SPPs

towards desired directions within the surface, including functionalities such as in-plane beam

steering and negative refraction27

. As an example, we focus here on the simplest functionality,

i.e. the reflection / transmission of surface plasmons at the interface between two metasurfaces.

More specifically, we consider a SPP propagating along a planar layer defined by a conductivity

tensor (1)σ and impinging into a second metasurface characterized by (2)σ with an angle inθ

with respect to the direction normal to the interface, as illustrated in the inset of Fig. 5b. Even

though the rigorous solution of this problem requires the use of purely numerical techniques due

to the momentum mismatch of the evanescent waves carried out by plasmons traveling along

different surfaces, approximate closed-form expressions may be derived if we assume lossless

and quasi-TM SPPs that are extremely confined to the surface (i.e., 0zk → ). Under these

circumstances, the refracted angle of transmission reads

(2)

arcsiny

out

k

kρθ

≈

, (10)

where (1) (2)

y y yk k k= = is the transverse component of the wavenumber, which must be continuous

across the interface, and (2)kρ is the in-plane wavenumber in the second metasurface. The presence

of large dissipation losses may lead to complications that are not captured by this simple

formula, such as the presence of two refracted beams75

. As expected, and similarly to standard

bulk optics19

, the reflection angle is ref inθ θ= − . The transmission and reflection coefficients can

be then approximated as

(1) (2)

(1) (2)

x x

x x

k k

k k

−Γ ≈

+ and

(2) (1)

(1) (1) (2)

2 x

x x

k kt

k k k

ρ

ρ

≈+

, (11)

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leading to a reflectivity and transmissivity

2

R ≈ Γ , and

2(2) (1)

2

(1) (2)

x z

x z

k kT t

k k

≈

. (12)

This formulation allows to design uniaxial metasurfaces able to route surface plasmons with high

transmission efficiency towards a desired direction in the plane. For instance, Fig. 5a-b illustrate

the transmissivity and refraction angle of SPPs – propagating along an isotropic surface defined

by (1) 0.5i mSσ = and with an angle 56.4inθ ≈o with respect to the x-axis – versus the properties

of the metasurface on which they impinge. The results clearly show how an appropriate tailoring

of the metasurface conductivity tensor provides almost full control over the features of the

transmitted plasmons. We remark that similar design curves can be obtained as a function of the

features of incoming plasmons. In order to further validate this approach, we have selected four

different scenarios – namely points A, B, C, D in Figs. 5a-b– to illustrate highly efficient

transmission of SPPs across metasurfaces while we control the refraction angle and the

confinement of the transmitted waves. Figs. 5c-f show the electric field on the metasurfaces in

these cases, computed using full-wave numerical simulations (COMSOL Multiphysics76

). The

calculations confirm i) negative refraction from HMTSs, ii) large degree of control over the

transmitted plasmons, including features such as confinement and refracted angle, and iii) good

agreement with the approximate model presented above. It is important to highlight that,

contrary to negative refraction in double-negative metamaterials11

and similarly to the case of

bulk HMTMs29

, negative refraction in hyperbolic metasurfaces does not rely on a resonant

mechanism, and therefore it can be broadband and low-loss. The combination of Eqs. (10)-(13)

with the EMA employed to straightforwardly synthesize uniaxial metasurfaces provides a

powerful framework for the fast and accurate design of graded uniaxial metasurfaces able to

fully manipulate the direction and nature of confined SPPs. The experimental demonstration of

wavelength-dependent routing of SPPs using artificially made silver/air metasurfaces – as

described in Fig. 3b – was recently reported at visible frequencies27

. As illustrated in Fig. 7, these

structures are able to refract incoming surface plasmons propagating along patterned silver

towards positive and negative angles within the hyperbolic metasurface.

In a related context, the unusual �-near-zero topology supported by uniaxial metasurfaces can be

exploited to put forward planar hyperlenses able to canalize subwavelength images from a source

to an image plane without diffraction. This exciting functionality was investigated in Ref.23

by

using uniform graphene sheets modulated by closely-located corrugated ground planes, thus

achieving the desired extreme anisotropy. Similar responses have also been obtained in optics

using periodic metallic gratings68

, as experimentally reported in Ref.27

. In the ideal lossless case,

canalization along the y-axis requires that the metasurface conductivity tensor components

fulfill23

i

yσ → ∞ and 0i

xσ → , (13)

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where the superscript ‘i’ denotes the imaginary part of the conductivity. Under such conditions,

the wavenumber along the canalization direction becomes 0yk k= , completely independent of

xk . Consequently, all spatial harmonics of SPPs will propagate towards the y-direction with

exactly the same wavenumber, implementing the sought-after diffraction-free canalization. In

addition to the aforementioned realizations, planar hyperlenses can be synthesized at different

operation frequencies following the previously introduced EMA. More advanced control may be

achieved taking advantage of the gyrotropic properties of graphene under a magnetic bias77

,

leading to magnetically-induced non-resonant topological transitions that may enable, for

instance, tunable responses, non-reciprocal in-plane propagation and low-loss, in-plane

canalization.

In order to further investigate this latter response, let us consider two closely located dipoles

placed over a graphene sheet at 2 nm away from the interface with a planar hyperlens. Fig. 8a

illustrates this scenario, confirming that, despite the subwavelength separation between the

dipoles ( 0/ 5 / 500SPPd λ λ≈ ≈ , being SPPλ and ;' the graphene SPPs and free-space

wavelengths, respectively) and the presence of losses, the proposed planar hyperlens provides

dispersion-free propagation and resolves the sources with subwavelength details preserved. As

expected, this situation is different on the graphene layer, where the diffraction limit prohibits to

elucidate if the SPPs were generated by one or multiple emitters. Figs. 8b-d depict the

normalized electric field of the SPPs at 150 nm from both sides of the interface for various

separation distances between the dipoles. They show that resolutions larger than

0/10 /1000SPPλ λ≈ can be achieved, fully confirming the potential for subwavelength imaging.

It is important to stress that, even though it is not possible to exactly satisfy the conditions of Eq.

(13) and losses are unavoidable in practice, canalization-like propagation can still occur provided

that a large contrast between the diagonal components of the metasurface conductivity tensor

exists. Imperfect canalization will result in a deterioration of the imaging process, because the

different spatial components of the SPPs will travel with different phase velocities. Based on this

technique, an alternative planar approach to realize hyperlenses can be obtained by enforcing a

very large real part of the conductivity – instead of its imaginary component – along the

canalization direction, i.e., r

yσ → ∞ . This approach does not lead to large dissipative losses,

since materials with high conductivity actually provide low-loss responses because of the limited

field penetration78

. This unusual behavior can be found for instance at the resonance of

nanostructured graphene (see Fig. 3c-d and Fig. 4c), leading to strong canalization across the

graphene strips.

Another intriguing property of uniaxial metasurfaces is the dramatic enhancement of light-matter

interactions they may exhibit thanks to the ideally unbounded nature of their supported plasmon

spectrum. This feature can be exploited to boost the spontaneous emission rate (SER) of

arbitrarily-oriented emitters located nearby (see inset of Fig. 9c) by adequately tailoring the

components of the metasurface conductivity tensor. Specifically, the SER or Purcell factor of a

dipolar emitter can be computed in the framework of semi-classic electrodynamics as19

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( )0 0

0 0

61 Im , ,p S p

p

PSER G r r

P k

πµ ω µ

µ = = + ⋅ ⋅

r r r rr , (14)

where P and 0P are the powers emitted by the dipole in the inhomogeneous environment and in

free space, respectively; 0k is the free space wavenumber, 0rr

is the emitter position, ρµr

denotes

the unit vector along the dipole orientation eρ , and ( )0 0, ,SG r r ωr r

is the scattered component of

the dyadic Green’s function of the structure. Even though the purely numerical evaluation of this

Green’s function can be challenging39

, advanced techniques to efficiently treat the associated

Sommerfeld integrals have recently been reported79

.

At this point, it is important to note that near-field functionalities realized by bulk HMTMs can

also be achieved with common thin metal films, due to severe restrictions that appear in the

coupling of external evanescent waves to bulk hyperbolic modes80

. These restrictions can be

overcome using uniaxial metasurfaces, thus effectively providing a large enhancement of light-

matter interactions. This is illustrated in Fig. 9a, which shows the SER of a z-oriented dipolar

emitter located 5 nm above a lossless homogeneous surface versus the values of its conductivity

tensor components26

. The figure illustrates four well-defined quadrants, corresponding to the

different nature of SPPs supported by the metasurface. The first quadrant corresponds to the

elliptic region (Im[���]>0 and Im[�]>0), for which the metasurface supports quasi-TM SPPs

able to significantly interact with incoming waves thanks 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[���]<0 and Im[�]<0) is associated to quasi-TE SPPs barely-confined to the

surface that lead to negligible light-matter interactions and SER enhancement. Lastly, the second

and forth quadrants (Im[���]>0, Im[�]<0 and Im[���]<0, Im[�]>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 surface26

. The inset of Fig. 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

HMTSs43

, appears due to the flattering of the SPP isofrequency contour, which in turns permits

an even larger coupling of incoming waves. Fig. 9b illustrates how the SER response of uniaxial

metasurfaces strongly depends on the separation distance between the emitter and the surface.

When this separation increases the SER may become higher in elliptical metasurfaces than in

hyperbolic ones. This response arises because free-space behaves as a low-pass filter for

incoming waves, providing stronger attenuation to waves with larger wavenumbers, which are

usually coupled efficiently to HMTSs. In the opposite limit, the SER should tend to infinity as

the dipole get closer and closer to ideal hyperbolic surfaces, implying that an infinite amount of

energy may be coupled to the supported set of surface plasmons when the emitter is located

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exactly on the metasurface. However, as previously discussed, the presence of losses and the

influence of nonlocality will ultimately limit this singular response. This behavior is further

highlighted in Fig. 9c, which confirms that the intrinsic material nonlocality removes the SER

singularity, and it establishes an upper bound to the maximum possible light-matter interactions

available on the surface. This semi-classical analysis, focused on ideal dipolar emitters, can be

straightforwardly extended high-order multipolar transitions, spin-flip process, two-plasmon

phenomena, or multiquanta emission processes81

. Also these interactions are expected to be

significantly enhanced in HMTSs, thanks to the large wave-confinement and transverse wave

numbers that imply a fast variations of the fields on the surface, amenable to excited higher-order

multipolar transitions.

Engineering the isofrequency contour of uniaxial metasurfaces can also be applied to manipulate

the black body thermal emission from ultrathin surfaces, with exciting applications for energy

harvesting, noncontact measurements of temperature, thermophotovoltaics, and thermal

management82,83,84

. As it occurs in bulk HMTMs85,86,87

, the excitation of hyperbolic modes able

to support very large wavenumbers allows to overcome the black-body limit in a broadband

frequency region thanks to the near-field transport of energy carried out by the evanescent

spectrum. However, ultrathin metasurfaces present an even improved performance because the

supported modes are surface plasmons able to strongly interact with the surrounding media26,80

,

rather than bulk hyperbolic modes confined within a volumetric structure. In order to investigate

super-Planckian thermal emission and near-field radiative heat flux, let us consider two closely

located metasurfaces implemented using nanostructured graphene, as in Ref.88

(see Fig. 10a).

The top and bottom metasurfaces are assumed to be at temperatures 310 and 290 K, respectively.

Fig. 10b shows the spectral radiative heat flux of a single metasurface versus frequency

compared it to the one of pristine graphene. The results confirm a significant enhancement of the

heat flux in the band where the structure presents a hyperbolic response. Fig. 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 black body limit due to the excitation of confined surface plasmons88

.

However, the patterning of graphene dramatically increases the heat transfer by more than 1000

times compared to the black body 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 response88

.

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.

Conclusion and Outlook

The emerging field of uniaxial and hyperbolic metasurfaces holds a great promise to

significantly impact nanoscale optics and technology, thanks 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,

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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 to obtain 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 paper, we have first reviewed the unusual electromagnetic properties of ultrathin uniaxial

metasurfaces implemented either by natural or artificial materials, studying the different

topologies that they support –ranging from closed isotropic to ideally open hyperbolic, and going

through the �-near-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 light on the different mechanisms –namely losses and nonlocality– that close the otherwise

open hyperbolic isofrequency contour by imposing a cut-off on the supported SPPs

wavenumbers. In this regard, we have shown that the influence of the intrinsic material

nonlocality on the HMTSs 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 quasi-optimal 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 sub-diffractive 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 thanks to their

appealing properties and easy implementation on well-established platforms such as graphene at

THz and mid-IR or noble metal at optics. From a practical viewpoint, such metasurfaces face

important challenges that must be still 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, non-planar, or

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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 bends89

, 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 antennas91

. Inspired by these concepts and some recent developments in the context of

bulk, cylindrical indefinite media92,93

, we anticipate that interesting applications such as cloaking

and multi-modal 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 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 in-plane transformation optics using

MTSs96

, allowing, for example, to cloak planar defects or grain boundaries that may arise during

fabrication. Furthermore, magnetic-free nonreciprocal plasmon propagation based on spatio-

temporal 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 HMTMs98

, it is expected that the large field

enhancement in HMTSs may be exploited in nonlinear optics99

, 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 cloaks102,103

, double negative

refraction104

, and reflection/transmission coefficients which can be simultaneously equal or

greater than unity105

. 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 the space and providing efficient thermal

management at the nano-scale. 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/intra chip networks

and communications systems.

Corresponding author

*To whom correspondence should be addressed: alu@mail.utexas.edu

Acknowledgements

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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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Figures

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Figure 1. Topologies of uniaxial metasurfaces. Colormaps 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,

��� = � = 0.05 + <23.5μS ; (b) � -near-zero metasurface, ��� = 0.05 + <23.5μS, � =

0.05μS ; (c) hyperbolic metasurface, ��� = 0.05 @ <23.5μS, � = 0.05 + <23.5#A ; (d)

hyperbolic metasurface, ��� = 0.05 + <23.5#A, � = 0.05 @ <23.5#A.

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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 eV, 0.05 eV, and 0.1 eV, respectively. Black phosphorus thickness

is 10 nm, direct bandgap is 0.485eV, damping is 5 meV, and temperature is 300 K.

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Figure 3. Artificial hyperbolic metasurfaces implemented using an array of densely-packed

graphene strips26

(a) and nanostructured silver (from Ref27

) (b). Panels (c) and (d) show the

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 #$ = 0.3eV, its relaxation time *=1.0 ps. Panel (e) shows the

imaginary component of the effective conductivity tensor of the structure described in panels (c)-

(d) at 25 THz versus the graphene strip width W for various values of chemical potential.

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Figure 4. Isofrequency contours of quasi-TM surface plasmons supported by the artificial

hyperbolic metasurface illustrated and described in Fig. 3a. (a) Operation frequency: 2 THz, �-

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.

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Figure 5. Different mechanisms that limit the hyperbolic isofrequency contours. Quasi-TM

surface plasmons supported by the artificial hyperbolic metasurface illustrated in Fig. 3a. (a)

Influence of Ohmic losses. The metasurface is modelled using an effective medium approach

with periodicity L = 50nm, and graphene is characterized using the local Kubo formalism32

. (b)

Influence of periodicity. The metasurface is analyzed using a mode-matching full-wave

technique71

. Graphene is considered lossless and its imaginary part is characterized using the

local Kubo formalism. (c) Influence of nonlocality. The metasurface is modelled using an

effective medium approach with periodicity L = 50nm, graphene is considered lossless and its

imaginary part is characterized using a nonlocal conductivity model72

. Operation frequency is 15

TH, graphene chemical potential is 0.3 eV, and strip width is W = 0.5L.

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Figure 6. Transmission and reflection of surface plasmons. Transmissivity (a) and refraction

angle (b) of a SPP propagating along an isotropic surface (��G = <0.5HA) with an angle I�G ≈

56.4° and impinging onto a lossless uniaxial metasurface defined by the conductivity tensor ��

(see inset of panel b). The results are computed using Eqs. (10)-(12) versus the components of ��.

(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)-(b). Results are computed with

COMSOL Multiphysics76

. (c) Case A: ��� = � = 10�M + <0.25HA . (d) Case B: ��� =

10�M + <0.6HA , � = 10�M = <HA . (e) Case C: ��� = 10�M + <0.3HA , � = 10�M @

<0.3HA. (f) Case D: ��� = 10�M @ <0.1HA, � = 10�M + <0.62HA.

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Figure 7. Experimental demonstration of negative refraction on hyperbolic metasurfaces 27.

The color plots show measurements of surface plasmon polaritons propagating along silver and

being refracted at the interface with a hyperbolic metasurface (dashed box) implemented by

nanostructured silver (see Fig. 3b). I�G, INO� , and ; corresponds to the incidence angle of the SPP

with respect to the normal to the interface, the refracted angle, and the operation wavelength,

respectively. (a) ; = 640nm. (b) ; = 540nm. (c) ; = 490nm.

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Figure 8. Planar hyperlensing. (a) Interface between an isotropic metasurface (sheet 1) with

� = 5 ∙ 10�M + <5 ∙ 10�RmS and an anisotropic �-near-zero metasurface (sheet 2) with ��� = 5 ∙

10�R + <4mS and � = 5 ∙ 10�R + <4 ∙ 10�SmS. Sheet 1 is excited by two z-oriented dipoles

(depicted by magenta arrows) separated by a distance T = 60nm ≈ 0.21 ∙ λV ≈ 0.0021 ∙ λ' –

where λV and λ' are the plasmon wavelength on Sheet 1 and in free space, respectively– and

located on the layer at 2 nm from the interface. The color map illustrates the z-component of the

electric field along the sheets. Insets show the isofrequency contour of each layer. (b)

Normalized normal component of the electric field along the observation lines shown in (a). The

dipoles are separated by a distance T = 200nm ≈ 0.7 ∙ λV ≈ 0.007 ∙ λ'.(c) Same as (b) but with

a separation distance between the dipoles of T = 60nm ≈ 0.21 ∙ λV ≈ 0.0021 ∙ λ'. (d) Same as

(b) but with a separation distance between the dipoles of T = 30nm ≈ 0.1 ∙ λV ≈ 0.001 ∙ λ' .

The operation frequency is 10 THz.

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Figure 9. SER enhancement of a z-oriented dipole above a lossless uniaxial metasurface 26,

36. (a) Results are computed versus the conductivity components of the metasurface considering

an emitter located at d=5 nm above the structure. The inset details the SER enhancement at the

topological transition. (b) Results computed versus the distance d of the dipole above the sheet

and the yy-component of the metasurface conductivity. (c) Results are computed versus the

position d of the emitter with (solid blue) and without (solid red) considering the intrinsic

nonlocal response of the materials composing the metasurface. The structure is implemented

using graphene strips, assuming a chemical potential of #$ = 0.2eV , a relaxation time * =

0.3ps,strips with a width of W = 15 nm and a periodicity of L=50nm.

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Figure 10. Giant thermal emission from hyperbolic metasurfaces 88. (a) Schematic of the

configuration employed to study near-field radiative heat transfer: two hyperbolic metasurfaces

implemented by nanostructured graphene, located in free space, and separated by a distance d.

The temperature of the top (T1) and bottom (T2) sheets are 310 K and 290 K, respectively. (b)

Spectral radiative heat flux of an isolated metasurface and pristine graphene. The shaded region

corresponds to the spectral region where the metasurface presents a hyperbolic response. (c)

Ratio of the near-field radiative heat flux between two hyperbolic metasurfaces to that of

isotropic graphene sheets. Graphene’s chemical potential and relaxation are 0.5 eV and 0.1 ps,

respectively. Other parameters are W=30 nm and g=10 nm.

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Table of Content (ToC) Graphic

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