of 9
ARTICLE
Optical magnetism in planar metamaterial
heterostructures
Georgia T. Papadakis
1
, Dagny Fleischman
1,2
, Artur Davoyan
1,2,3
, Pochi Yeh
4
& Harry A. Atwater
1
Harnessing arti
fi
cial optical magnetism has previously required complex two- and three-
dimensional structures, such as nanoparticle arrays and split-ring metamaterials. By contrast,
planar structures, and in particular dielectric/metal multilayer metamaterials, have been
generally considered non-magnetic. Although the hyperbolic and plasmonic properties of
these systems have been extensively investigated, their assumed non-magnetic response
limits their performance to transverse magnetic (TM) polarization. We propose and
experimentally validate a mechanism for arti
fi
cial magnetism in planar multilayer metama-
terials. We also demonstrate that the magnetic properties of high-index dielectric/metal
hyperbolic metamaterials can be anisotropic, leading to magnetic hyperbolic dispersion in
certain frequency regimes. We show that such systems can support transverse electric
polarized interface-bound waves, analogous to their TM counterparts, surface plasmon
polaritons. Our results open a route for tailoring optical arti
fi
cial magnetism in lithography-
free layered systems and enable us to generalize the plasmonic and hyperbolic properties to
encompass both linear polarizations.
DOI: 10.1038/s41467-017-02589-8
OPEN
1
Thomas J. Watson Laboratories of Applied Physics, California Institute of Technology, Pasadena, CA 91125, USA.
2
Kavli Nanoscience Institute, California
Institute of Technology, Pasadena, CA 91125, USA.
3
Resnick Sustainability Institute, California Institute of Technology, Pasadena, CA 91125, USA.
4
Department of Electrical and Computer Engineering, University of Santa Barbara, Santa Barbara, CA 93106, USA. Correspondence and requests for
materials should be addressed to G.T.P. (email:
gpapadak@caltech.edu
)
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DOI: 10.1038/s41467-017-02589-8
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1
1234567890():,;
I
n the optical spectral range, the magnetic response of most
materials, given by the magnetic permeability
μ
, is generally
weak. This is famously expressed by Landau et al.
1
:
there is no
meaning in using the magnetic susceptibility from the optical
frequencies onward, and in discussing such phenomena, we must
put
μ
=
1
. By contrast, in the optical regime, materials possess a
diverse range of dielectric properties, expressed through the
dielectric permittivity
ε
, which can be positive, negative, or zero.
The weak magnetic response in natural materials has moti-
vated a search for structures and systems that may exhibit mag-
netic properties arising from metamaterial design. Speci
fi
cally,
engineered displacement currents and conduction currents can
act as sources of arti
fi
cial magnetism when metamaterials are
illuminated with electromagnetic
fi
elds
2
. Nonetheless, until now,
the realization of such magnetic metamaterials has required
rather complex resonant geometries
2
4
, including arrays of paired
thin metallic strips
5
,
6
, split-ring resonators
7
9
and
fi
shnet meta-
materials
10
structures that require sophisticated fabrication
techniques at optical frequencies.
In contrast, the dielectric properties of metamaterials may be
engineered even in simple planar con
fi
gurations of layered media.
Hence, heterostructures of alternating metallic and dielectric layers,
termed hyperbolic metamaterials (HMMs), have been explored
intensively the last decade
11
13
due to their anisotropic dielectric
response that is described by the dielectric permittivity tensor
ε
eff
=
diag{
ε
o
,
ε
o
,
ε
e
}, where
ε
o
and
ε
e
are the ordinary and extraordinary
components of the tensor, with
ε
o
ε
e
<
0. Such a peculiar dielectric
response manifests itself in the hyperbolic dispersion for transverse
magnetic (TM) waves (i.e.,
k
H
=
0 whereas
k
E
0). Interesting
phenomena such as negative refraction
11
,
14
18
without the need of
a negative refractive index, hyper-lensing
19
, extreme enhancement
in the density of optical states
13
, and interface-bound plasmonic
modes
20
25
have been reported.
Nevertheless, all of the intriguing physics and applications for
such layered HMMs have been limited to TM polarization,
whereas phenomena related to transverse electric (TE) polarized
waves (i.e.,
k
E
=
0, whereas
k
H
0) have remained largely
unexplored. Utilizing the effective magnetic response (i.e.,
μ
eff
¼
diag
μ
o
;
μ
o
;
μ
e
fg
6
¼
I
) is necessary to harness and control
arbitrary light polarization (TE and TM). Namely, a multilayer
system with
ε
o
ε
e
<
0 and
μ
o
μ
e
<
0 could allow polarization inde-
pendent negative refraction (Fig.
1
a) and excitation of TE surface
waves (Fig.
1
b), the magnetic counterpart of surface plasmon
polaritons (SPPs). Furthermore, gaining control over the magnetic
permeability in planar systems can yield impedance-matched
epsilon-and-mu-near-zero (EMNZ) optical responses (Fig.
1
c)
26
.
Although it is straightforward to tailor the permittivity to cross
zero in planar metamaterials
27
, a simultaneously EMNZ meta-
material at optical frequencies has not yet been demonstrated.
In previous reports, the effective magnetic permeability in
planar layered media has been widely assumed to be unity
11
13
,
28
.
We note that Xu et al.
18
attributed their results of TM negative
refraction and negative index to a negative magnetic parameter.
This approach is valid in the case of isotropic media, however
planar HMMs are extremely anisotropic, namely uniaxial. The
index introduced in ref.
18
is the effective index of the mode
excited in their experiment and is not directly associated with
arti
fi
cial magnetism. Furthermore, refraction switches to positive
for TE polarization, similar to others reports
14
17
.
Here we propose a concept for tailoring the effective magnetic
response within planar, unpatterned, one-dimensional (1D)
multilayer structures. In contrast to previous generations of
magnetic metamaterials with complex three-dimensional struc-
tures such as split-ring resonators
7
9
,
fi
shnet structures
10
, and
nanoparticles
29
,
30
, pattern-free multilayers are readily realizable
with lithography-free thin-
fi
lm deposition, greatly simplifying
fabrication. We show theoretically and experimentally that the
magnitude and sign of the permeability tensor may be engineered
at will, enabling observation and use of TE polarization related
phenomena in simple layered structures. We further identify
implications that are associated with the observed arti
fi
cial
magnetism.
Results
Induced magnetic dipoles in planar systems
. A circulating
electric current can create a magnetic dipole and is the key to
inducing magnetism in magnet-free systems. Based on this
principle, induction coils generate and induce magnetic
fl
ux,
allowing to manipulate magnetic
fi
elds at radio frequencies (RFs).
The same concept is widespread in metamaterials design
31
,
32
;
similar to the RF regime, by properly shaping metamaterial ele-
ments to produce a circulating current
fl
ow, magnetic dipoles are
induced. Dielectric nanoparticles
29
,
30
,
33
36
and nanorods
37
,
38
have been the building blocks for three (3D)- and two (2D)-
dimensional magnetic metamaterial structures, respectively
(Figs.
2
a, b). We note that the magnetic response of these
arrangements is sometimes incorporated into an equivalent,
alternative, spatially dispersive permittivity. Although this is, in
principle, always possible
1
,
39
,
40
, we stress that, similar to naturally
occurring substances, described with a permittivity
ε
and a per-
meability
μ
, a metamaterial description based on (
ε
,
μ
) allows for
physical intuition and reduces complexity, especially when it is
straightforward to relate the dielectric (magnetic) response with
physical macroscopic electric (magnetic) moments. This can be
particularly useful for uniaxial planar and unpatterned multi-
layers, studied in this paper.
We start by considering a single subwavelength dielectric slab
of refractive index
n
diel
and thickness
d
. When illuminated at
a
bc
TE-Hyperbolic
k
y

o

o
~0
Magnetic plasmon

and

near zero

o
<0
k
k
x
k
z
Re(
E
)
a.u.
0
1
–1
Fig. 1
Implications of magnetic response in a planar geometry.
a
Transverse electric (TE) negative refraction of phase in a hyperbolic metamaterial with
μ
o
<
0 and
μ
e
>
0. The arrow indicates the direction of wavevector and the black line indicates the interface between air and the hyperbolic metamaterial.
Inset: 3D isofrequency diagram for
μ
o
<
0,
μ
e
>
0.
b
TE magnetic plasmon at the interface between air and magnetic material (
μ
<
0), analogous to
transverse magnetic polarized surface plasmon polaritons (
ε
<
0).
c
ε
and
μ
near zero (EMNZ): a
fi
eld propagating inside an EMNZ slab with vanishing
phase advance
ARTICLE
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normal incidence (
z
direction in Fig.
2
c), the displacement
current
J
d
¼
i
ωε
o
n
2
diel

1

E
induces a macroscopic effective
magnetization
M
eff
¼
1
=
2
μ
o
R
r
́
J
d
ðÞ
d
S
1
,
37
,
41
. By averaging the
magnetic
fi
eld,
H
avg
¼
R
d
=
2

d
=
2
H
ð
z
Þ
d
z
, we use
μ
eff
1
þ
M
eff
=
μ
o
H
avg

to obtain an empirical closed-form expression
for the magnetic permeability:
μ
eff
1

n
2
diel

1
2
n
2
diel

1
þ
n
diel
π
d
=
λ
tan
n
diel
π
d
=
λ
ðÞ

ð
1
Þ
By setting
n
diel
=
1, we recover the unity magnetic permeability
of free space. From Eq. (
1
), we see that
μ
eff
diverges when tan
(
n
diel
π
d
/
λ
)
=
0. This yields a magnetic resonant behaviour at free-
space wavelengths
λ
=
n
diel
d
/
ρ
, with
ρ
=
1, 2, ... At these
wavelengths, the displacement current distribution is anti-
symmetric, as shown in Fig.
2
d for
ρ
=
1, 2. This anti-
symmetric current
fl
ow closes a loop in
y
=
±
and induces a
magnetization
M
eff
that is opposite to the magnetic
fi
eld of the
incident wave (Fig.
2
c), leading to a magnetic resonance. Eq. (
1
)
enables estimating the design parameters for enhanced magnetic
response; in the long-wavelength limit, only the fundamental and
second order resonances,
λ
=
n
diel
d
,
n
diel
d
/2, play signi
fi
cant roles.
In the visible and near-infrared regime, with layer thicknesses on
the order of 10
100 nm, dielectric indices higher than
n
diel
~ 2 are
required for strong magnetic effects
42
. The same principle applies
for grazing incidence, with the displacement current inducing a
magnetic response in the extraordinary or, out-of-plane (
z
)
direction. So far, we have shown that the circular shape designed
to support a closed current loop is not a requirement for magnetic
metamaterials. A planar structure suf
fi
ces, for which the current
loop closes in
±
in
fi
nity.
In order to make this magnetic response signi
fi
cant, we extend
this principle to multilayer con
fi
gurations. We
fi
rst examine the
case of two in
fi
nite parallel wires in air, carrying opposite currents
(Fig.
2
e). Their net current distribution induces a magnetic
moment that scales with their distance, as dictated by
M
eff
r
×
J
.
This is directly equivalent to a layered con
fi
guration composed of
two high-index dielectrics separated by air. Their displacement
current distribution can be anti-symmetric on resonance, as
shown in Fig.
2
f. By calculating their magnetic permeability
μ
eff
,
we con
fi
rm the magnetic character of this arrangement. As shown
with the black curve in Fig.
2
g,
μ
eff
strongly deviates from unity.
The planar geometry does not require that the two high-index
layers be separated by air; any sequence of alternating high-low-
high refractive index materials will induce the same effect. For
example, replacing the air region with a layer of metal, with
n
metal
<
1 at visible wavelengths, does not drastically change the
magnetic response. This is shown in Fig.
2
g with the red curve for
a separation layer of silver. Therefore, at optical frequencies,
metals do not contribute signi
fi
cantly to the magnetic response in
this planar con
fi
guration. This is in contrast to the gigahertz
regime, where the conduction current in the metallic components
of resonant structures has been necessary for strong magnetic
effects
6
9
. From the magnetic
fi
eld distribution shown in the inset
of Fig.
2
g, one can see that the average magnetic
fi
eld faces in the
direction opposite to the magnetization, expressing a negative
magnetic response for the dielectric/silver unit cell (Supplemen-
tary Note
1
).
Combining hyperbolic dielectric and magnetic properties
.
Apart from the magnetic response described in the previous
section, multilayer systems composed of metals and dielectrics
have also been widely explored due to their distinct hyperbolic
dielectric response for TM polarization. These systems are
uniaxially anisotropic and, at wavelengths that are large com-
pared with the unit cell, they exhibit an in-plane metallic response
(
ε
o
<
0) due to the metallic layers, whereas
ε
e
>
0
11
. We show that
it is possible to induce a signi
fi
cant additional magnetic response
in planar dielectric/metal HMMs, if the dielectric layers are
composed of high-index materials that are capable of supporting
H
avg
d
z
x
y
c
z
(in
d
)
–1/2
–2
–1
2
1
0
1/2
0
Re (
J
d
) a.u.
Re (

eff
)
b
d
–2
–1
2
1
0
–1/2
1/2
0
f
g
e
z
(in
d
)
JJ
M
eff
a

=1

=2
M
eff
J
d
J
d
J
d
M
eff
M
eff
J
d
0
–5
5
With air
With Ag
Wavelength (in
d
)
0
2
4
6
789 1011
Re(
H
x
) a.u.
H
avg
M
eff
diel.
diel.
Ag
z
(in
d
)
–1/2
1/2
0
Re (
J
d
) a.u.
Fig. 2
Concept of arti
fi
cial magnetism in 3D, 2D, and 1D structures. A
circulating current
fl
ow
J
d
induces a magnetization
M
eff
in all three cases:
a
dielectric nanoparticles (three-dimensional metamaterials),
b
dielectric
nanorods (two-dimensional metamaterials), and
c
one-dimensional
dielectric slabs.
H
avg
is the average magnetic
fi
eld, which faces in the
direction opposite to
M
eff
.
d
Displacement current distribution at resonance
for
ρ
=
1,
ρ
=
2, for a 90 nm slab of refractive index
n
diel
=
4.5.
e
Two in
fi
nite
wires carrying opposite currents are equivalent to
f
two dielectric layers
(blue shaded regions) separated by air (pink shaded region) in terms of
their current distribution. Arrows in
f
indicate the direction of
J
d
, which is
anti-symmetric at resonance.
g
Effective permeability for two dielectric
layers separated by air and silver. Inset: tangential magnetic
fi
eld
distribution at resonance: average magnetic
fi
eld is opposite to
M
eff
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3
strong displacement currents at optical frequencies. Previously
reported dielectric/metal HMMs have primarily featured dielec-
tric layers with lower-refractive indices, such as LiF
43
,
Al2O3
28
,
44
,
45
, and TiO2
13
. Figure
3
shows that, for layer thick-
nesses below ~ 50 nm, these lower-index dielectric/metal systems
exhibit magnetic resonances in the ultraviolet (UV)-short wave-
length visible regime.
Previous approaches used effective medium approximations
(EMAs) for describing the dielectric response of dielectric/metal
HMMs, such as the Maxwell Garnett theory
46
. However, such
EMAs a priori assume a unity magnetic permeability along all
coordinate directions. By contrast, a uniaxial system is most
generally described in terms of both an effective permittivity
tensor
ε
eff
=
diag{
ε
o
,
ε
o
,
ε
e
} and an effective permeability tensor
μ
eff
=
diag{
μ
o
,
μ
o
,
μ
e
}. In order to capture the magnetic dipole
moments in multilayer structures, we use an exact parameter
retrieval, which relaxes the
μ
eff
¼
I
assumption
47
. We discuss and
compare these different approaches in the Methods section.
Experimental results
. We fabricate multilayer structures by
electron-beam evaporation and
fi
rst measure the optical constants
of the individual constituent layers with spectroscopic ellipso-
metry. We also determine their thicknesses with transmission
electron microscopy (TEM). Hence, we are able to homogenize
the layered metamaterials by assigning them effective parameters
ε
eff
and
μ
eff
47
, while taking into account fabrication imperfec-
tions. We then perform ellipsometric measurements of the full
metamaterials and
fi
t the experimental data with the effective
parameters
ε
o
,
ε
e
,
μ
o
, and
μ
e
in a uniaxial and Kramers
Kronig
consistent model, whereas the total metamaterial thickness is held
to the value determined through TEM. The
fi
tting is over-
determined as the number of incident angles exceeds the total
number of
fi
tted parameters (Supplementary Note
2
).
The fabricated metamaterials are composed of SiO
2
/Ag, TiO
2
/
Ag, and Ge/Ag alternating layers (TEM images and schematics in
insets of Figs.
3
e, f, j respectively). The indices of the selected
dielectric materials at optical frequencies are
n
SiO
2
1
:
5,
n
TiO
2
2, and
n
Ge
4
4.5. Figure
3
a shows that increasing the
dielectric index redshifts the magnetic resonance in the ordinary
direction
μ
o
; the SiO
2
/Ag metamaterial supports a magnetic
resonance in the long-wavelength UV regime (~ 300 nm),
whereas the TiO
2
/Ag and Ge/Ag metamaterials exhibit reso-
nances in the blue (450 nm) and red (800 nm) part of the
spectrum, respectively. The enhanced absorption in Ge at optical
frequencies leads to considerable broadening of the Ge/Ag
metamaterial magnetic resonance, yielding a broadband negative
magnetic permeability for wavelengths above 800 nm. As
expected, the losses in
μ
o
are increased at the magnetic resonance
frequency for all investigated heterostructures, similar to previous
reports on arti
fi
cial magnetism with split-ring resonators and
other magnetic metamaterials
48
50
.
The presence of Ag induces a negative ordinary permittivity
ε
o
(Fig.
3
b), which, for the Ge/Ag metamaterial, becomes positive
above 800 nm due to the high-index of Ge. Notably,
ε
o
crosses
zero at 800 nm, similar to
μ
o
, as emphasized with the asterisks in
Figs.
3
a, b. Thus, the Ge/Ag metamaterial exhibits an EMNZ
response at optical frequencies. The EMNZ condition is
con
fi
rmed by transfer-matrix analytical calculations of the
physical multilayer structure. As shown in the inset of Fig.
3
a,
the phase of the transmission coef
fi
cient vanishes at the EMNZ
wavelength, demonstrating that electromagnetic
fi
elds propagate
inside the metamaterial without phase advance
26
.
By comparing
μ
o
and
μ
e
in Figs.
3
a, c, respectively, one can
infer that increasing the dielectric index leads to enhanced
magnetic anisotropy. The parameter
μ
e
only slightly deviates from
μ
o
for the SiO
2
/Ag metamaterial, while the deviation is larger for
the TiO
2/
Ag one. For the Ge/Ag metamaterial,
μ
e
remains positive
beyond 800 nm, while
μ
o
<
0, indicating magnetic hyperbolic
response for TE polarization. Furthermore, all three heterostruc-
tures exhibit hyperbolic response for TM polarization, with
ε
o
<
0
and
ε
e
>
0 (Figs.
3
b, d). Consequently, the Ge/Ag metamaterial
possesses double hyperbolic dispersion.
Figures
3
e
j demonstrate the excellent agreement between
fi
tting and raw experimental data, where the parameters
Ψ
and
Δ
correspond to the conventional ellipsometric angles (Methods).
In Figs.
3
e, h, we also provide a Maxwell Garnett EMA-based
fi
t
for the SiO
2
/Ag metamaterial. The EMA fails to reproduce the
experimentally measured features, in both
Ψ
and
Δ
(gray-shaded
regions in Figs.
3
e, h), which correspond to magnetic permeability
resonances. Similar EMA-based
fi
ts for the TiO2/Ag and Ge/Ag
metamaterials lead to large disagreement with the experimental
–2
0
2
4
Re (

o
)
0
2
4
6
400
600
800
1,000
–10
–5
0
Re (

o
)
0
5
10
Wavelength (nm)
a
b
700
800
900
–50
0
50
100
Arg(T
TE
) (°)
wavelength (nm
)
0
1
2
0
0.5
1
1.5
–10
0
10
20
0
10
20
c
d
Im (

e
)
Re (

e
)
Re (

e
)
20
40
60
40
60
80
24
30
36
–50
100
250
–50
100
250
125
175
225
ψ
(°)
ψ
(°)
ψ
(°)
e
f
g
h
i
j
Δ
(°)
Δ
(°)
Δ
(°)
30 nm Ag
50 nm TiO
2
30 nm Ag
50 nm TiO
2
50 nm TiO
2
50 nm SiO
2
30 nm Ag
50 nm SiO
2
30 nm Ag
50 nm SiO
2
Im (

o
)
Im (

e
)
Im (

o
)
400
600
800
1,000
Wavelength (nm)
400
600
800
1,000
Wavelength (nm)
400
600
800
1,000
Wavelength (nm)
Wavelength (nm)
EMA
60° exp.
*
60° fit
65° exp.
*
65° fit
70° exp.
*
70° fit
50° exp.
*
50° fit
55° exp.
*
55° fit
50 nm Ge
30 nm Ag
30 nm Ag
50 nmGe
50 nm Ge
Fig. 3
Experimental veri
fi
cation of non-unity magnetic permeability in dielectric/metal metamaterials. Experimentally determined
a
μ
o
,
b
ε
o
,
c
μ
e
,
d
ε
e
for
SiO
2
/Ag-green, TiO
2
/Ag-blue, Ge/Ag-red metamaterial. Shaded regions in
a
indicate the regime of magnetic resonances in
μ
o
for the studied
metamaterials. Solid lines represent real parts while dashed lines represent imaginary parts. Asterisks in
a
and
b
indicate the
ε
o
and
μ
o
near zero (EMNZ)
wavelength for the Ge/Ag metamaterial. The EMNZ condition is con
fi
rmed by a vanishing phase of the transmission coef
fi
cient at the EMNZ wavelength,
shown in the inset of
a
.
e
g
and
h
j
show the agreement between raw experimental data,
Ψ
and
Δ
respectively (which are the conventional ellipsometric
angles), and the ellipsometric
fi
tting, for the SiO
2
/Ag metamaterial in
e
and
h
, for the TiO
2
/Ag metamaterial in
f
,
i
, and for the Ge/Ag metamaterial in
g
,
j
.
Shaded regions in
e
,
h
emphasize the disagreement between experimental data and the effective medium approximation (EMA). Insets in
e
,
f
,
j
show TEM
images of the fabricated samples. The scale bar is 50 nm for
e
,
f
and 100 nm for
j
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data across the whole visible-near-infrared spectrum and are,
thus, omitted. This disagreement is expected, as the EMA
approach is based on the assumption that the electric
fi
eld
exhibits negligible or no variation within the lattice period
46
,
which does not apply to high-index dielectric layers.
It should be noted that the dielectric hyperbolic response
ε
o
ε
e
<
0 is broadband in planar systems, as seen in Figs.
3
b, d. In
contrast, the magnetic permeabilities deviate from unity in a
resonant manner along both coordinate directions
μ
o
and
μ
e
,
thereby making TE polarization-based phenomena more narrow
band in nature.
Beyond
μ
eff
1 and TE polarization effects
. In the previous
sections we established, theoretically and experimentally, that
dielectric/metal layered systems may be described with an effec-
tive magnetic permeability that deviates from unity across all
coordinate directions. The purpose of introducing this parameter
is to build a simple and intuitive description for understanding
and predicting new phenomena, such as TE polarization response
in planar systems. In what follows we discuss how the non-unity
and, in particular the negative and anisotropic magnetic response
that we demonstrated (Fig.
3
) manifests itself in the character-
istics of TE-polarized propagating modes (Fig.
4
) and surface
waves (Fig.
5
).
We use an example system of dielectric/silver alternating
layers, similar to the one we investigate experimentally. To
emphasize that enhanced magnetic response at optical frequen-
cies requires high-index dielectrics, we let the refractive index of
the dielectric material
n
diel
vary. The calculations and full-wave
simulations presented here are performed in the actual, physical,
multilayer geometry (Figs.
4
a, d, e and
5
) and compared with the
homogeneous effective slab picture (
ε
eff
,
μ
eff
Figs.
4
b, c). This
helps assess the validity of our model and emphasize the
physicality of the magnetic resonances.
First, we perform transfer-matrix calculations for the example
multilayer metamaterial and we show in Fig.
4
a the angle
dependence for TE and TM re
fl
ectance. The strong angle
dependence for TM polarization is well understood in the
context of an equivalent homogeneous material with anisotropic
effective dielectric response
ε
o
ε
e
<
0. Bulk TM modes experience
dispersion
k
2
x
þ
k
2
y
ε
e
ð
ω
;
k
Þ
μ
o
ð
ω
;
k
Þ
þ
k
2
z
ε
o
ð
ω
;
k
Þ
μ
o
ð
ω
;
k
Þ
¼
k
2
o
ð
2
Þ
where
k
o
=
ω
/
c
. This dispersion is hyperbolic, as shown with
isofrequency diagrams in Fig.
4
b. Losses and spatial dispersion
perturb the perfect hyperbolic shape
12
. In contrast to the TM
modes, TE bulk modes interact with the magnetic anisotropy
through the dispersion equation
k
2
x
þ
k
2
y
ε
o
ð
ω
;
k
Þ
μ
e
ð
ω
;
k
Þ
þ
k
2
z
ε
o
ð
ω
;
k
Þ
μ
o
ð
ω
;
k
Þ
¼
k
2
o
ð
3
Þ
which is plotted in Fig.
4
c. For small wavenumbers (
k
//
/
k
o
<
1)
and small dielectric indices
n
diel
, the isofrequency diagrams are
circular, in other words, isotropic. This agrees well with our
experimental results; as shown in Figs.
3
a, c, for the SiO
2
/Ag
metamaterial, ordinary and extraordinary permeabilities do not
drastically deviate from each other. Increasing the dielectric index
opens the isofrequency contours, due to enhanced magnetic
response in the ordinary direction (
μ
o
), which leads to magnetic
anisotropy. We note that the displayed wavelengths are selected at
resonances of
μ
o
. Open TE polarization isofrequency contours for
n
diel
2 are also consistent with experimental results; as shown in
Fig.
3
for TiO
2
and Ge-based metamaterials, increasing
n
diel
enhances the anisotropy. This also agrees well with the picture of
the physical multilayer structure, as shown in Fig.
4
a; the TE
re
fl
ectance indeed exhibits extreme angle dependence for
increasing dielectric index. Strikingly, we observe a Brewster
angle effect for TE polarization, which is unattainable in natural
materials due to unity magnetic permeability at optical
frequencies
51
.
An open isofrequency surface can yield an enhancement in the
density of optical states relative to free space. Physically, this may
lead to strong interaction between incident light and a hyperbolic
structure, and enhanced absorption when it is possible to couple
to large wavenumbers from the surrounding medium
52
,
53
. So far,
only TM polarization has been considered to experience this
exotic hyperbolic response in planar dielectric/metal metamater-
ials, due to
ε
o
ε
e
<
0
12
,
13
,
28
. Based on the open isofrequency
surfaces for both TE and TM polarizations in Figs.
4
b, c, a high-
index dielectric/metal multilayer metamaterial may exhibit
–2
02
–2
–4
0
2
4
–2
–4
0
2
4
k
z
/
k
o
k
z
/
k
o
bc
–1
1
–2
02
–1
1
TM
TE
z
(
μ
m)
z
(
μ
m)
2
0
–4
–2
4
2
0
–4
–2
4
05
–5
10
x
(
μ
m)
05
–5
10
x
(
μ
m)

o

e
<0
E
TM
|
E
|
a.u.

o

e
<0
E
TE
Re(
E
y
)
a.u.
de
R
n
diel
=1.5,

=485 nm
n
diel
=2.5,

=500 nm
n
diel
=3.5,

=655 nm
n
diel
=4.5,

=825 nm
0
0.2
0.4
0.6
0.8
1
0
0.5
1
k
//
/
k
o
k
//
/
k
o
k
//
/
k
o
a
Fig. 4
Bulk propagating modes in magnetic hyperbolic layered
metamaterials. Analytical calculations of
a
re
fl
ectance and
b
,
c
isofrequency
diagrams for a metamaterial consisting of
fi
ve alternating layers of
dielectric
n
diel
: 55 nm/Ag: 25 nm. Solid lines in
a
correspond to TE
polarization whereas dashed lines correspond to TM polarization. Solid
lines in
b
,
c
correspond to real parts, whereas dashed lines correspond to
imaginary parts. Vertical black lines in
b
,
c
indicate the maximum free space
in-plane wavenumber
k
//
=
k
o
. Color code is the same for
a
c
.
d
,
e
Numerical simulation of a
fi
fty-
fi
ve layers dielectric (
n
diel
=
4)/Ag
multilayer metamaterial. The surrounding medium has index
n
sur
=
1.55,
allowing coupling of high-
k
modes. We increased the number of layers for
clear visibility of
fi
eld localization inside the structure. Strong
fi
eld
localization is the consequence of
d
dielectric hyperbolic dispersion for TM
polarization (
ε
o
ε
e
<
0) and
e
magnetic hyperbolic dispersion for TE
polarization (
μ
o
μ
e
<
0)
NATURE COMMUNICATIONS | DOI: 10.1038/s41467-017-02589-8
ARTICLE
NATURE COMMUNICATIONS
|
(2018) 9:296
|
DOI: 10.1038/s41467-017-02589-8
|
www.nature.com/naturecommunications
5