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Laser Photonics Rev. 00, No. 00, 1–9 (2015) / DOI 10.1002/lpor.201400457
dictated by the hyperbolic anisotropy of the metamaterial, may
be tuned by altering the geometrical parameters of the nanorod
composite.
Bulk plasmon-polaritons in hyperbolic nanorod metamaterial
waveguides
Nikolaos Vasilantonakis∗ , Mazhar E. Nasir, Wayne Dickson, Gregory A. Wurtz,
and Anatoly V. Zayats
1. Introduction
Light interacts with a resonant medium by forming
polaritonic waves. These are mixed excitations of the
electromagnetic field (photons) with quasiparticles related
to material resonances. These quasiparticles include
phonons, excitons in semiconductors and plasmons in
conductors [1] or atomic ensembles [2]. Exciton-polaritons
are the most intensely studied having numerous applications in semiconductor lasers. When electromagnetic
fields propagate in resonant media, polaritonic waves are
formed and their behavior is governed by the dispersion
determined by the material resonances around which
negative permittivity can be observed [1]. Both phononpolariton and exciton-polaritons can exist in the bulk of
the material and at the interface with the adjacent medium
as surface electromagnetic waves (surface polaritons).
On the other hand, at frequencies lower than the electron
plasma frequency, usually only surface plasmon-polaritons
(SPPs) have been considered, since electric fields do not
significantly penetrate metals over more than the skin depth
[3].
Plasmonic nanorod metamaterials possess unique optical properties [2] making them unrivalled for applications
in imaging [4], sensing [5, 6], ultrasound detection [7], designing nonlinear optical properties [8–10] and controlling quantum optical processes [11, 12]. These metamate-
rials exhibit hyperbolic isofrequency surfaces in a spectral
range where the real part of the diagonal components of the
permittivity tensor, corresponding to ordinary and extraordinary axis, have opposite signs. The hyperbolic dispersion allows various unusual modes to exist with very high
wavevectors and negative group velocity, opening up new
degrees of freedom for the development of both integrated
and free-space optical functionalities.
In this article, the behavior of bulk plasmon-polaritons
in a planar slab of an anisotropic metamaterial is studied
both experimentally and theoretically. It is found that such
extraordinary modes exhibit low or negative group velocity and low group-velocity dispersion (GVD). Furthermore,
a peculiar high-frequency cut-off for transverse-magnetic
(TM) polarized modes allows subwavelength single-mode
guiding with λ0 /6–λ0 /8 waveguide thickness, in contrast
to ordinary, transverse-electric (TE) modes which behave
as in conventional transparent dielectric waveguides. Bulk
plasmon-polaritons of an anisotropic plasmonic metamaterial slab can be considered in analogy to exciton-polaritons
in a semiconductor cavity. Such bulk plasmon-polariton
modes can be tailored by controlling the geometry of the
metamaterial design, and their properties can be utilized
in applications requiring sensitive control over both groupand phase-velocity dispersion, such as waveguides, sensors
or nonlinear optical devices.
The article is structured as follows. Section 2 describes
the theory and numerical simulations of the mode structure
Department of Physics, King’s College London, Strand, London, WC2R 2LS, UK
∗
Corresponding author: e-mail: nikolaos.vasilantonakis@kcl.ac.uk
This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction
in any medium, provided the original work is properly cited.
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Abstract Hyperbolic metamaterials comprised of an array of
plasmonic nanorods provide a unique platform for designing
optical sensors and integrating nonlinear and active nanophotonic functionalities. In this work, the waveguiding properties
and mode structure of planar anisotropic metamaterial waveguides are characterized experimentally and theoretically. While
ordinary modes are the typical guided modes of the highly
anisotropic waveguides, extraordinary modes, below the effective plasma frequency, exist in a hyperbolic metamaterial slab
in the form of bulk plasmon-polaritons, in analogy to planarcavity exciton-polaritons in semiconductors. They may have
very low or negative group velocity with high effective refractive indices (up to 10) and have an unusual cut-off from the
high-frequency side, providing deep-subwavelength (λ0 /6–λ0 /8
waveguide thickness) single-mode guiding. These properties,
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N. Vasilantonakis et al.: Bulk plasmon-polaritons in hyperbolic nanorod metamaterial waveguides
Figure 1 (a) Schematics of the planar metamaterial waveguide geometry consisting of an array of Au nanorods. Left: Schematic of the
metamaterial’s internal structure and top-view atomic force microscopy image of the metamaterial used in the experiment (topography
variations are less than 10 nm). Center: effective medium representation of the metamaterial slab placed between a semi-infinite
substrate (nsub ) and a superstrate (nsup ). Right: geometry and cross section of the unit cell of the metamaterial used in the numerical
simulations. (b) and (c) Spectral dependences of (b) real and (c) imaginary parts of the effective permittivity of the metamaterial
along ordinary and extraordinary directions for p = 0.32. Elliptic and hyperbolic dispersion ranges are shown with green and white
background colors, respectively.
of a metamaterial slab: in Section 2.1, the effective
permittivity of the anisotropic metamaterial is described
and the notion of effective plasma frequency is introduced;
in Section 2.2, the mode structure of the metamaterial
slab is studied both numerically and with an approximate
analytical formulation. Section 3 describes experimental
studies of the metamaterial’s modes.
2. Theoretical formulation
2.1. Optical properties of plasmonic nanorod
composites
We consider a planar metamaterial waveguide formed by a
finite thickness slab of aligned gold (Au) nanorods (Fig. 1a).
The anisotropic optical properties of the metamaterial can
be described in the effective medium theory (EMT) by
a diagonal permittivity tensor with nonzero components
eff
εxeff = εeff
y = εz , which can be expressed in the Maxwell–
Garnet approximation [13] as
eff
εx,y
=
pεAu εh + εh (1 − p)ε̃
,
pεh + (1 − p)ε̃
εzeff = pεAu + (1 − p)εh ,
(1)
(2)
where p = π (r/d)2 is the nanorod concentration, with r
being the nanorod radius and d being the period of a square
lattice, εAu and εh are the permittivities of the gold nanorods
[14] and the embedding porous alumina (AAO) medium
(εh = 2.56), respectively, and ε̃ = (εh + εAu )/2. Depending on the geometrical parameters of the metamaterial and
eff
the wavelength range, either elliptical with Re(εx,y
)>0
eff
eff
and Re(εz ) > 0 or hyperbolic with Re(εx,y ) > 0 and
Re(εzeff ) < 0 dispersion can be observed (Fig. 1b). In fact,
by selecting specific geometrical parameters, hyperbolic
dispersion can be achieved throughout the visible and nearinfrared spectral ranges. The spectral dependence of the
imaginary part of the permittivities follows a typical behavior for a resonant dielectric and an electron plasma along
the ordinary (x,y) and the extraordinary (z) directions, respectively (Fig. 1c).
We have introduced the effective plasma frequency of
the metamaterial to characterize their metal-like behavior for TM-polarized fields via the free-electron Drude
model [15]: Re(εzeff (ωpeff )) = 0. Note that this consideration
is not equivalent to the homogenization of the electron density in metallic components over the metamaterial volume,
which would not reflect the physical processes by eliminating anisotropy. Using the effective medium parameters
(Eqs. (1) and (2), the effective plasma frequency can be
eff
derived as Re(εAu
(ωpeff )) = (1 − p −1 )εh . An approximate
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Figure 2 Plasmon-polariton dispersions in an infinite Au nanorod metamaterial in ordinary and extraordinary directions: (a) the
metamaterial parameters are as in Fig. 1, (b) the same as in (a) but with losses artificially reduced 100 times to show the asymptotic
behavior, (c) the same as in (b) but for p = 0.13. The light line in AAO (magenta) is also shown. The dashed line shows the effective
plasma frequency for extraordinary waves. The EVL regime takes place between the dotted lines.
expression for the effective plasma frequency can then be
obtained assuming a lossless Drude-like permittivity for
Au given by εAu = ε∞ − (ωp /ωpeff )2 , where ε∞ is the highfrequency background and ωp is the free-electron plasma
frequency of Au:
ωpeff =
ωp
ε∞ + ( p −1 − 1)εh
.
(3)
Equation (3) shows that the metamaterial’s effective plasma
frequency can be tuned by varying the permittivity of the
host medium and/or concentration of nanorods. Note that
for a concentration p = 1, corresponding to a homogeneous
and isotropic metal layer, the plasma frequency of the bulk
metal is recovered.
In the case of a spatially infinite anisotropic material,
invariant in the z-direction, the electromagnetic wave dispersion can be plotted for both ordinary and extraordinary
waves (Fig. 2a). As expected, the ordinary wave has a typical dispersion for a transparent dielectric, lying to the right
of the light line in the AAO matrix since the effective refractive index is increased due to the presence of metal,
with the resonance–the so-called epsilon-very-large (EVL)
regime [16,17]–determined by cylindrical surface plasmon
(CSP) excitations on the rods [18]. At the same time, the
dispersion of extraordinary waves is that of a typical Drudelike metal with an effective plasma frequency ωpeff ≈ 2.2 eV
corresponding to the metamaterial’s transition from the elliptic to the hyperbolic regime. It is instructive to consider
a “gedanken” situation with reduced losses in Au that allows the nature of the dispersion properties of ordinary and
extraordinary waves to be clearly revealed (Fig. 2b). In the
eff
low-loss case, there is a range of frequencies where εx,y
<0
in the previously discussed EVL regime, where a bandgap
opens in the TE-mode dispersion. This TE-mode bandgap
and the effective plasma frequency for the TM mode can
be tuned with the geometry of the metamaterial realization.
While for low nanorod concentration, at least one of the
mode is always present and the metamaterial behaves either as an elliptic or as an hyperbolic medium, for higher
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concentrations (p > 0.28) and low loss, metallic behavior
can be observed in the frequency range where all diagonal
components of the effective permittivity tensor are negative
eff
< 0 and εzeff < 0, resulting in the dissimultaneously, εx,y
appearance of bulk modes (Fig. 2c). This regime, however,
takes place in the wavelength range and for loss parameters
where nonlocal, spatial dispersion effects occur and that
requires a different theoretical treatment [18, 19].
For a homogenized metamaterial described by EMT and
neglecting nonlocal corrections, the propagation of extraordinary waves, in which the electric field is solely polarized
along the long axis of the rods, is prohibited below the effective plasma frequency (hyperbolic regime), as expected for
conductors. However, as a result of the anisotropy of the
metamaterial, electron plasma oscillations may still give
rise to bulk plasmon-polaritons propagating in the metamaterial below the plasma frequency in directions determined
by the dispersion relation, k x2 /εzeff + k 2y /εzeff + k z2 /εxeff =
(ω/c0 )2 , where kx,y and kz are the wavevector components
along the x, y and z directions, respectively, ω is the electromagnetic field frequency, and c0 is the speed of light in
vacuum. In a microscopic consideration of the plasmonic
nanorod metamaterial studied here, these bulk plasmonpolaritons arise from interacting CSPs supported by individual nanorods forming the metamaterial [18], but can
also be observed in multilayered metal–dielectric–metal
metamaterials, where they arise from interacting smoothfilm SPP modes [20–23]. The isofrequency contours of
the metamaterial dispersion for TE (elliptic) and TM (hyperbolic) modes show striking differences in the allowed
wavevector ranges, which determines the dissimilar behavior of these modes in both the infinite metamaterial
as well as in a metamaterial slab (Fig. 3). In particular, for
any given frequency, an elliptic TE mode can only be either unbound, leaky or waveguided (the example shown in
Fig. 3a), depending on the respective position of the metamaterial permittivity isofrequency contour with respect to
that of the substrate and superstrate, while a hyperbolic
TM mode may be of all three types at the same frequency
(Fig. 3b).
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Figure 3 (a) and (b) Isofrequency contours in the first Brillouin zone calculated for a frequency corresponding to a free-space
wavelength λ0 = 700 nm for an infinite Au nanorod metamaterial with p = 0.5 for (a) ordinary, TE, and (b) extraordinary, TM, modes. In
the elliptic regime (a), the dispersion is bounded and corresponds to that of a typical anisotropic dielectric. The isofrequencies contours
in the superstrate (air) and the substrate (glass) are also shown. (c) and (d) The mode position of a metamaterial slab (400 nm in
thickness) shown as dots corresponding to intersection of the isofrequency contours of the infinite metamaterial with the quantized
values of kz = q(π/l), where q = ±1, ±2, ±3, . . . resulting from the finite size of the slab in the z-direction. (e) and (f) Angular spectra
of reflectance of the metamaterial slab as in (c) and (d) for λ0 = 700 nm calculated using TMM. The position of the modes obtained
analytically is shifted to higher wavevectors due to the analytic model assumptions, influencing the confinement of the modes.
2.2. Mode structure of a metamaterial slab
The modes supported by a metamaterial slab, when one (z-)
dimension is taken to be finite, were studied for a metamaterial slab placed on a silica substrate (n = 1.5) and with air as
a superstrate. The slab geometry quantizes the z-component
of the wavevector of the infinite metamaterial to values determined by the mode order and slab thickness, such as kz,
= q(π /l). The modal behavior at a given frequency is then
determined by the wavevectors satisfying both the quantization condition and bulk metamaterial dispersion (Figs. 3c
and d). These solutions have an x-component kx (kz ,q) of the
wavevector associated to each solution kz corresponding to
the propagation constant β = kx (kz ,q) of mode q in the slab.
In this instance, the elliptic dispersion allows for a finite
number of solutions with one unbound mode corresponding to a solution for the propagation constant within the
isofrequency contour of both superstrate and substrate, as
well as 3 waveguided modes. In contrast, the hyperbolic
dispersion allows modes for any q-value, with practical
limits eventually imposed by both losses and the geometry of the nanorod composite as the EMT breaks down for
wavevectors near the boundary of the Brillouin zone. For
the considered example (Fig. 3d), the modes are present
both within the light lines (Fabry–Perot modes) and confined to the metamaterial (waveguided modes).
The mode structure of the metamaterial slab can be
numerically simulated using the transfer matrix method
(TMM) approach in the complex incidence angle formulation [24]. In this approach, both reflected and transmitted intensities from the 3-layer substrate/metamaterial/
superstrate system are calculated using the effective
medium description for the metamaterial. The angle of incidence of a plane wave in the substrate varies from 0 to
π /2 (the angles corresponding to unbound and leaky modes
in the substrate) and then to the complex incidence angles
θ for which sin(θ ) > 1 in order to cover the waveguided
mode-dispersion region.
Using TMM, the mode dispersion of a typical planar
waveguide with a thickness of 400 nm and made of metamaterial comprised of 100 nm period and 40 nm radius
nanorods was calculated via angle resolved reflection spectra (Figs. 4a and b). A rich family of modes is observed for
both TE and TM polarizations with distinctively different
behavior. Modes in the dispersion correspond to reflectance
minima where the incident light is coupled to the modes.
The reflectance corresponding to the isofrequency contours
is in a good agreement with the solutions from the mode
quantization in the finite size slab, giving the same number of modes with a small discrepancy in their positions
due to the overestimated mode confinement (approximated
boundary conditions) in the analytic simulations (Figs. 3e
and f).
The complex mode structure of the slab emerges as
a consequence of the spatial confinement of plasmonpolaritons in the slab and is associated with both cavity
resonances and waveguided modes, above and below the
light line in air, respectively. In the cavity regime, the
dispersion of TM modes reveals discrete modes with very
low or negative group velocity. These unbound modes are
not confined to the metamaterial slab, being accessible to
plane waves in both the substrate and the superstrate. In this
regime, the metamaterial slab simply acts as a Fabry–Perot
(FP) cavity for bulk plasmon-polaritons resulting in
effects similar to those observed for cavity-polaritons in
semiconductors. Between the light lines in the substrate
and the superstrate, the modes are coupled to radiation in
the substrate (leaky modes), while being evanescent at the
metamaterial/superstrate interface. Due to this coupling,
the modes are ‘‘strongly bent’’ near the light lines and may
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Figure 4 Reflection dispersion of the metamaterial slab calculated using the complex-angle TMM method (a) and (b) and analytic
model (c) and (d) for a waveguide of 400 nm thickness with p = 0.5: (a) and (c) TE and (b) and (d) TM modes. The analytic model
shows the 5 lowest (1 q 5) modes. The dashed line indicates the effective plasma frequency. The vertical line corresponds to the
boundary of the Brillouin zone of the metamaterial realization when considering a square lattice of nanorods. The modes in (a) and
(b) correspond to the minima of reflectance. Light lines in the superstrate (magenta) and the substrate (orange) are also shown. (e)
and (f) The spatial distributions of the norm of the electric field for the q = 1–5 modes as obtained from the eigenmode simulations for
(e) TE and (f) TM modes.
have positive, negative, or vanishing values of the group
velocity (Fig. 4b). The modes below the substrate-line
are truly guided modes decaying exponentially in both
substrate and superstrate. A similar analysis holds for
TE-polarized modes, but in this regime the metamaterial
slab acts as a typical anisotropic dielectric waveguide
due to the orientation of the electric field normal to the
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nanorod axes (ordinary direction) and does not support
bulk plasmon-polaritons. The marked difference between
TE and TM modes is in the opposite sign of the group
velocity. Neglecting a strongly dispersive behavior in the
vicinity of the light lines, the TM modes always show
either negative or vanishing group velocity (Fig. 4b), while
the group velocity of TE modes is always positive (Fig. 4a).
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N. Vasilantonakis et al.: Bulk plasmon-polaritons in hyperbolic nanorod metamaterial waveguides
In order to understand the observed mode structure, we
adapted an analytic description of the dispersion of ordinary and extraordinary waves in a conventional anisotropic
waveguide [25] to a hyperbolic metamaterial. Within this
framework, a planar hyperbolic waveguide is considered
in the x–y plane with phase-insensitive reflections at the
metamaterial’s boundaries. Given the 2D geometry of the
waveguide, modes can be separated by TM and TE polarizations. The dispersions of TE and TM guided modes can
then be expressed as
2
= Re(εxeff )k02 −
βTE
2
βTM
=
Re(εzeff )k02
−
qπ 2
qπ 2
l
l
εeff
Re zeff
εx
(4)
,
,
(5)
where k0 = ω/c0 , q is a positive integer referring to the
mode number, and l is the thickness of the planar waveguide. The mode dispersions were calculated analytically for
the first five modes q = 1–5 for TE and TM polarizations
(Figs. 4c and d). Although the analytical model does not
take into account either substrate or superstrate, the analytical dispersions reproduce the TMM simulations with
very good agreement in both the cavity and true-waveguide
mode regimes, while not in the leaky mode regime. The
analytical model formulation does not allow the fields to
escape the waveguide and thus does not contain leaky
modes.
For TM modes, the group velocity is negative
below the light line in the substrate provided the condition εxeff ∂εzeff /∂ω > εzeff ∂εxeff /∂ω is satisfied. For bulk
plasmon-polaritons in a hyperbolic metamaterial slab,
this requirement is always fulfilled away from the resoeff
nances in εx,y
since εzeff < 0, ∂εxeff /∂ω > 0, εxeff > 0 and
eff
∂εz /∂ω > 0. In the vicinity of the εxeff resonance, the term
eff
/∂ω becomes negative and is to be taken into account
∂εx,y
to establish the hyperbolic behavior of the waveguide when
the ENZ frequency is close to this resonance. However,
in the epsilon near-zero (ENZ) regime, near the effective
bulk-plasma frequency ωeff
p , a different treatment is needed
to take into account nonlocal effects [18, 19], a scenario
not considered here. It is important to emphasize that for
typical metamaterial geometries, the condition for negative
group velocity is satisfied for all TM-mode orders and any
waveguide thickness.
The TE-mode dispersions are similar to those of
dielectric waveguides and cavities with a group velocity
eff
eff −1
υg = Re((2βc0 /ω)(ω∂εx,y
/∂ω + 2εx,y
) ). The sign of
the group velocity is always positive except when the meta
eff
material exhibits anomalous dispersion ∂εx,y
/∂ω < 0 ;
in this case, however, the losses near the εxeff resonance (Fig. 1c) prevent the existence of guided modes
[8].
Both TE- and TM-mode dispersions are bounded by a
high-frequency limit. TM modes converge, with increasing mode number q, to the effective plasma frequency of
the metamaterials (dashed line in Figs. 4a and b), above
which bulk plasmon-polaritons do not exist. At the same
time, TE modes are bounded by the EVL condition given
by εAu = −εh (1 + p)/(1 − p) (Fig. 2b). However, for each
mode number, while TE modes demonstrate conventional
behavior with a low-frequency cut-off determined by the
mode number at small β TE , the behavior of TM modes is
distinctly different and unusual. Each TM mode has a highfrequency cut-off for intermediate values of the propagation constant β TM where the turning point of the dispersion
curve is observed. No low-frequency cut-off is observed for
increasing β TM which is limited only by the Brillouin zone
of the metamaterial realization. Thus, hyperbolic metamaterial slabs of deep-subwavelength thickness can act as multimode TM waveguides, a behavior that finds its origin in
the inverse scaling law for hyperbolic metamaterial cavities
[26].
The high-frequency bound for TM-polarized modes
can be obtained by solving Eq. (5): ωcut−off =
2
eff ) , so that for frequenβTM
/εzeff + (q 2 π 2 )/(l 2 εx,y
c0 Re
cies higher than ωcut−off , propagation along the hyperbolic
2
slab waveguide for mode order q is prohibited as βTM
< 0.
Figure 5a shows the cut-off frequency dependence for the
q = 1–3 TM modes for various nanorod concentrations
obtained both analytically and with the TMM formalism.
As the nanorod concentration falls, the mode cut-off frequency monotonously increases for p > 0.2 and then decreases for p < 0.2. The nonmonotonous behavior and, in
particular, the decrease observed for smaller nanorod concentrations (smaller anisotropy of the metamaterial) is due
to the decrease of the effective plasma frequency ωpeff for
decreasing nanorod concentrations. Since the existence of
bulk plasmon-polaritons is determined by ωpeff , modes with
ωcut−off > ωpeff all adopt ωpeff as the cut-off frequency. This
occurs for p < 0.2 in Fig. 5b. As a result, the mode density
diverges as the frequency approaches the effective plasma
frequency.
An eigenmode analysis of the mode structure of the 2D
waveguide cavity allows the electromagnetic-field distributions associated with the modes to be mapped and confirms
the analytical identification of the modes (Figs. 4e and f).
As the mode number increases, the field distributions show
an increasing number of maxima/minima that are typical
of standing-wave distributions inside the waveguide, as expected from the analytical calculations. Depending on the
polarization, the waveguide acts either as a closed cavity
with a field maximum close to its center (TE case) or as
an open cavity with a field minimum at the center (TM
case) of the slab waveguide. This is a direct consequence
of boundary conditions due to the confinement along the
z-direction [27].
3. Experimental results
The modes supported by a metamaterial slab were measured
for the metamaterial placed on a silica substrate (n = 1.5)
and with air as a superstrate. The waveguides were fabricated as described in detail in Ref. [28] and are formed by
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Figure 5 (a) The dependence of the highfrequency TM mode cut-off ωcut-off on nanorod
concentration, p, for q = 1–3 modes: (lines) analytic model, (squares) TMM simulations. (b)
Effective plasma frequency, ωpeff , for the same
nanorod concentration range. The frequency
threshold below which the effective plasma
frequency determines the mode cut-off is for
q ≥ 3 at p = 0.2, for q ≥ 2 at p = 0.13, while
for p < 0.04, the cut-off for all modes converges
to the effective plasma frequency.
Au nanorods (100 nm period, 32 nm radius, giving an
average concentration p 0.32) in an AAO matrix (inset in
Fig. 1a). The mode dispersion of the slab was determined
experimentally via angle-resolved reflectance spectroscopy
under plane-wave illumination through a semicylinder silica substrate, so that incident angles both smaller and larger
than the angle of total internal reflection in silica were
probed and the coupling to both cavity and leaky modes
can be realized [29]. Geometrical constraints in the experimental set-up determine the low-wavevector limit in the
measured dispersions, while the high-wavevector limit is
determined by the light line in the substrate.
The measured and simulated reflectance dispersions reveal both cavity resonances and leaky modes, above and
below the light line in air, respectively (Fig. 6). More precisely, above the air-line, the dispersion of the TM modes reveals discrete modes with negative group velocity (Fig. 6a).
Between the light lines in the superstrate and substrate, the
modes are coupled to the modes radiating in the substrate
(leaky modes) while being evanescent at the metamaterial/air interface. Due to this coupling the modes (indicated
with curved dashed lines in Figs. 6a and b) can have positive, negative, or vanishing small group velocity. The truly
waveguided modes, present below the substrate-line, are
not accessible experimentally in the configuration used in
the experiment but had been examined analytically and
numerically in Section 2. A similar analysis for the dispersion of TE modes (Figs. 6c and d) shows that the guided
modes exhibit positive group velocity in all regimes. For
these modes, as was discussed above, due to the orientation of the electric field along the y-axis only (normal to
the nanorod axis), bulk plasmon-polaritons are not excited
and the metamaterial slab acts as an anisotropic dielectric
waveguide. It is thus possible to flip the sign of the group velocity of guided signals simply by altering the polarization
of the coupled light.
Based on the analytical model (Eqs. (4) and (5)), we
can now identify the modes in the experimental reflectance
as modes q = 1 to q = 3 (Fig. 6). For the TM modes,
as the mode order increases, both the group velocity and
group-velocity dispersion decrease, with the q = 3 mode
having negative group velocity even between the substrate
and superstrate light lines. In general, the experimental dispersions for the modes with q = 2 and q = 3 (Figs. 6a and
c) are in satisfactory agreement with the TMM modelled
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dispersions (Figs. 6b and d), further indicating that EMT is
still applicable for the metamaterial parameters and propagation constants considered, as long as the frequency is
away from the effective plasma frequency. It is important,
however, to explain the discrepancies that occur for higher
mode orders. It is evident from comparison of Figs. 6e and
f that the q = 4 mode is not observed experimentally. The
reason for this is a combination of higher loss and mode
density for increasing q’s at higher frequencies.
For practical purposes one can analyze the behavior
of the modal effective index for TM waveguided modes
(β>ω/c0 ) in a planar hyperbolic metamaterial, which can
eff , where A =
be expressed as n eff = εzeff − A2 εzeff /εx,y
qπ c0 /lω. From this expression it is clear that the effective index increases with q and decreases with l for a given
frequency, in contrast to TE modes’ effective index that
exhibits the opposite behavior. As an example, for a telecom wavelength of 1.5 µm with p = 0.5 and l = 500 nm,
the metamaterial slab supports multiple TM waveguided
modes, with the real part of the effective index taking values of about 2, 9 and 13 for q = 2, 3, and 4, respectively (the
TM mode with q = 1 is guided only at longer wavelength
for these metamaterial parameters due to the unusual highfrequency cut-off discussed above). At the same frequency,
for the same nanorod concentration p and l = 300, 400 and
500 nm, neff takes values of about 10, 6, and 2 for q = 2,
respectively. This strong inter-relationship of metamaterial
parameters and effective index provides a flexible way for
modal design throughout the visible and infrared spectral
ranges. Taking into account the wavevector cut-off due to
the metamaterial realization (the Brillouin zone) and the
high-frequency cut-off of the TM modes, the wavelength
range of around 2400–3100 nm represents the single-mode
guiding regime for p = 0.5, corresponding to a waveguide
thickness range of only λ0 /6–λ0 /8. The energy confinement
for the fundamental mode (q = 1) depends on the operating frequency, but, when measured as the ratio between
the power flow inside the waveguide to the total power
flow of the mode, is of the order of 40% at a wavelength
of 2480 nm. This ratio increases for higher-order modes,
as TM modes become increasingly confined to the waveguide. The dissipative nature of the hyperbolic waveguide
results in increasing losses for higher-order modes, as their
confinement in the guide increases.
C 2015 The Authors. Laser & Photonics Reviews published by Wiley-VCH Verlag GmbH & Co. KGaA Weinheim
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& PHOTONICS
REVIEWS
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N. Vasilantonakis et al.: Bulk plasmon-polaritons in hyperbolic nanorod metamaterial waveguides
Figure 6 Experimentally measured (a) and (c) and simulated (b) and (d) reflectance dispersions for a 340-nm thick metamaterial slab
with p 0.32 (the nanorod period is 100 nm and radius is 32 nm): (a) and (b) TM, (c) and (d) TE modes. The light lines in the
substrate (silica, green) and the superstrate (air, magenta) are shown. The Au nanorods are embedded in an AAO matrix. The modes
with q = 2 and 3 are tracked with dashed lines as a guide to the eye. The angular range measured in the experiment (20–75° in the
substrate) is indicated with white boxes in (b) and (d). (e) and (f) Experimental and simulated reflectance spectra at the angle of 60o
for TM polarization as extracted from (a) and (b). The first four modes (q = 1–4) are indicated with dashed lines.
4. Conclusions and outlook
We have investigated, experimentally and theoretically, the
mode structure of finite-thickness hyperbolic metamaterial
slabs. Similar to planar-cavity exciton-polaritons in semiconductors, planar hyperbolic metamaterial slabs support
bulk plasmon-polaritons for TM polarization, which in different regimes result in planar-cavity modes or waveguided
modes. TE-polarized modes are similar to anisotropic dielectric waveguide modes. It was shown that the spectral
range of hyperbolic waveguided modes is bounded from
the high-frequency side by the effective plasma frequency
of the metamaterial. TM waveguided modes have negative group velocity and an unusual high frequency cut-off
with no cut-off for high propagation constants, limited only
by the Brillouin zone of the metamaterial realization. The
negative group velocity as well as its dispersion can be
controlled by varying the anisotropy of the metamaterial,
i.e., the nanorod concentration. For the nanorod metamaterial studied, TM modes are slow modes with υ g down
to –0.03c0 and have a dispersion as low as 0.02 ps2 /mm.
Single-mode guiding can be achieved in planar waveguides
of λ0 /6–λ0 /8 thickness, depending on the anisotropy of the
metamaterial. These properties of hyperbolic metamaterial
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ORIGINAL
PAPER
Laser Photonics Rev. 00, No. 00 (2015)
9
waveguides are interesting for designing integrated deepsubwavelength sensors, nonlinear and active nanophotonic
devices, quantum information processing and optical datastorage components.
Planar hyperbolic metamaterial waveguides may be
important for designing integrated deep-subwavelength
chemical and biosensors, as well as nonlinear and active
nanophotonic devices, and for quantum information processing. Careful control of both group- and phase-velocity
dispersion is an important step towards the engineering
of spontaneous emission properties and may lead to new
nanoscale laser sources, including those based on slowlight properties. The fact that a slow-light regime can
be achieved in a tunable and controllable environment
opens up new possibilities in optical communications such
as network buffering, data synchronization and pattern
correlation.
[8]
[9]
[10]
[11]
[12]
[13]
Acknowledgements. This work has been supported by EPSRC
(UK) and the ERC iPLASMM project (321268). G. W. is grateful for
support from the People Programme (Marie Curie Actions) of the
EC FP7 project 304179. A. Z. acknowledges support from The
Royal Society and the Wolfson Foundation. The authors would
like to thank F. J. Rodrı́guez Fortuño for the helpful discussions
of the complex angle TMM simulations.
[14]
Received: 10 December 2014, Revised: 24 February 2015,
Accepted: 12 March 2015
[18]
Published online: 13 April 2015
[19]
Key words: Plasmonics, waveguides, Metamaterials, Hyperbolic.
[15]
[16]
[17]
[20]
[21]
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