flatland optics with hyperbolic metasurfaces
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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: [email protected]
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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