Optical Magnetism in Planar Metamaterial Heterostructures
Georgia T. Papadakis,
1,
∗
Dagny Fleischman,
1, 2
Artur Davoyan,
1, 2, 3
Pochi Yeh,
4
and Harry A. Atwater
1
1
Thomas J. Watson Laboratories of Applied Physics,
California Institute of Technology, California 91125, USA
2
Kavli Nanoscience Institute, California Institute of Technology, California 91125, USA
3
Resnick Sustainability Institute, California Institute of Technology, California 91125, USA
4
Department of Electrical and Computer Engineering,
University of Santa Barbara, California 93106, USA
(Dated: May 9, 2017)
Harnessing artificial optical magnetism has required rather complex two- and three-dimensional
structures, examples include split-ring and fishnet metamaterials and nanoparticles with non-trivial
magnetic properties. By contrast, dielectric properties can be tailored even in planar and pattern-
free, one-dimensional (1D) arrangements, for example metal/dielectric multilayer metamaterials.
These systems are extensively investigated due to their hyperbolic and plasmonic response, which,
however, has been previously considered to be limited to transverse magnetic (TM) polarization,
based on the general consensus that they do not possess interesting magnetic properties. In this
work, we tackle these two seemingly unrelated issues simultaneously, by proposing conceptually and
demonstrating experimentally a mechanism for artificial magnetism in planar, 1D metamaterials.
We show experimentally that the magnetic response of metal/high-index dielectric hyperbolic meta-
materials can be anisotropic, leading to frequency regimes of magnetic hyperbolic dispersion. We
investigate the implications of our results for transverse electric (TE) polarization and show that
such systems can support TE interface-bound states, analogous to their TM counterparts, surface
plasmon polaritons. Our results simplify the structural complexity for tailoring artificial magnetism
in lithography-free systems and generalize the concept of plasmonic and hyperbolic properties to
encompass both TE and TM polarizations at optical frequencies.
PACS numbers:
78.67.Pt, 73.20.Mf, 75.30.Gw, 78.20.Ls
I. INTRODUCTION
In the optical spectral range, the magnetic response
of most materials, given by the magnetic permeability
μ
, is generally weak. This is famously expressed in the
textbook by Landau and Lifshitz [1]: “
there is no mean-
ing in using the magnetic susceptibility from the optical
frequencies onward, and in discussing such phenomena,
we must put
μ
= 1”. The magnetic properties of natu-
ral materials arise from microscopic orbital currents and
intrinsic spins and typically vanish at frequencies above
the GHz range. This has motivated a search for struc-
tures and systems that may exhibit artificial optical mag-
netism by utilizing principles of metamaterial design. In
this regime, engineered displacement and conduction cur-
rents, induced when metamaterials are illuminated with
electromagnetic fields, act as sources of artificial mag-
netism [2].
Maxwell
'
s equations exhibit a duality with respect to
dielectric permittivity
and magnetic permeability
μ
for
the two linear polarizations of light; transverse magnetic
(TM) and transverse electric (TE), respectively. Despite
this symmetry, at frequencies beyond THz, far more dis-
cussion in the literature has been devoted to tailored
dielectric properties of metamaterials than to artificial
magnetic properties. This imbalance is understandable
∗
gpapadak@caltech.edu
because, until now, the realization of magnetic metama-
terials has required rather complex resonant geometries
[2–4], such as arrays of paired thin metallic strips [5, 6],
split ring resonators [7–9] or fishnet structures [10], which
are challenging to realize experimentally at optical fre-
quencies.
In contrast, engineered dielectric properties of meta-
materials are achievable even in simple planar multilayer
configurations. In fact, heterostructures of alternating
metallic and dielectric layers, termed hyperbolic metama-
terials (HMMs), have been explored intensively the last
decade [11–13]. They are often described with an effec-
tive permittivity tensor,
~
~
eff
=
diag
{
o
,
o
,
e
}
, where the
subscript-o (e) indicates the ordinary (extraordinary) di-
rection. Their in-plane response is metallic (
o
<
0) while
their out-of-plane response is dielectric (
e
>
0). HMMs
support interesting electromagnetic phenomena, includ-
ing negative refraction [11, 14] without the need of a neg-
ative refractive index, diverging density of optical states
for Purcell-factor enhancement [13], and hyper-lensing
[15]. Furthermore, the negative dielectric permittivity
o
leads to surface-propagating plasmonic modes [16, 17],
similar to surface plasmon polaritons (SPPs) supported
on noble metals’ surfaces [18] or metal/dielectric arrange-
ments and waveguides [19–22].
The plasmonic and hyperbolic properties of planar,
multilayer heterostructures have featured prominently
in photonics, however, their relevance has been limited
to TM polarization, based on their effective dielectric
response. TE polarization-related phenomena have re-
arXiv:1608.02909v2 [physics.optics] 6 May 2017
2
mained unexplored, as the effective magnetic permeabil-
ity of such systems has been widely considered to be
unity [11–13, 23]. By contrast, fishnet metamaterials
are known to exhibit a magnetic response and, recently,
a magnetic hyperbolic metamaterial was demonstrated
[10]. However, the fishnet structure is by definition biax-
ial, thus, TE and TM polarizations cannot be indepen-
dently manipulated.
Here, we focus on the magnetic properties of unpat-
terned, one-dimensional (1D) multilayer uniaxial sys-
tems, where TE and TM polarizations are uncoupled.
Inducing an artificial magnetic response in these systems
is of interest for generalizing their hyperbolic and plas-
monic properties to encompass both TE and TM polar-
izations. For example, a metal/dielectric planar system
with opposite magnetic permeabilities along different co-
ordinate directions (
μ
o
μ
e
<
0) is double hyperbolic with
unbound wavevectors for TE polarization (Fig. 1a and
inset). Furthermore, no TE counterpart of the surface
plasmon polariton, i.e. a magnetic surface plasmon (Fig.
1b), has been reported at optical frequencies, due to lack
of negative magnetic response. Artificial epsilon-and-mu-
near-zero (EMNZ) metamaterials at optical frequencies
are interesting building blocks for electrostatic-like sys-
tems, due to near-zero phase advance in the material [24]
(Fig. 1c). While it is straightforward to tailor the per-
mittivity to cross zero in planar metamaterials [25], a
simultaneously EMNZ metamaterial at optical frequen-
cies has not yet been demonstrated.
(a) TE-Hyperbolic
k
k
z
k
x
k
y
(c)
ΕΜΝΖ
ε
o
μ
o
~0
(b) Magnetic plasmon
μ
o
<0
k
k
FIG. 1.
(a) TE hyperbolic refraction in type II HMMs
(
μ
o
<
0,
μ
e
>
0)-inset: 3D isofrequency diagram (b) TE
magnetic plasmon: TE polarized surface state at the inter-
face between air and magnetic material (
μ <
0), analogous to
TM polarized surface plasmon polaritons (
<
0). (c)
and
μ
near zero (EMNZ) regime: phase diagram demonstrating
vanishing phase advance at EMNZ wavelengths.
In this paper, we perform a comprehensive study of
artificial magnetism in planar, 1D multilayer metama-
terials. The practical importance of our results lies in
the drastic simplification of the structural complexity of
previous generation magnetic metamaterials; the realiza-
tion of split-ring resonators [7–9], fishnet structures [10],
and nanoparticles [26, 27] at optical frequencies requires
multi-step lithographic processes and synthesis. By con-
trast,
pattern-free
multilayer metamaterials are readily
realizable with lithography-free thin-film deposition. We
start by introducing the physical concept for inducing
an artificial magnetic response in 1D systems (Sec.II).
In Sec.III A, we briefly discuss a simple approach based
on which this magnetic response can be taken into ac-
count in the design of multilayer metamaterials by relax-
ing previously made assumptions in widely used effective
medium theories [11–13, 28, 29]. In Sec. III B, we experi-
mentally confirm our findings and demonstrate magnetic
resonances at optical frequencies in multilayer HMMs.
Motivated by the non-trivial effective magnetic response
that we observe experimentally, in Sec. IV we theoreti-
cally investigate its implications using a simple transfer-
matrix approach. Contrary to the majority of work in
planar plasmonics and HMMs, we investigate TE polar-
ization phenomena. We find that concepts previously
discussed for TM polarization, based on engineering the
dielectric permittivity, are generalizable for both linear
polarizations. The proposed effective description of 1D
systems in terms of effective dielectric
and
magentic pa-
rameters provides a simple and intuitive understanding
of the underlying physics.
II. PHYSICAL MECHANISM: INDUCED
MAGNETIC DIPOLES IN 1D SYSTEMS
Magnetic fields at radio frequencies are usually ma-
nipulated with induction coils that generate and induce
magnetic flux. They operate based on circulating con-
duction currents in coil loops that can be approximated
as magnetic dipoles. This concept is widespread in meta-
materials design [30, 31], where the conduction current
is often replaced by displacement current in artificially
magnetic structures at higher frequencies. Similar to the
RF regime, by properly shaping metamaterial elements
to produce a circulating current flow, magnetic dipoles
are induced. Dielectric nanoparticles [26, 27, 32–35] and
nanorods [36, 37] have been the building blocks for three
(3D)- and two (2D)-dimensional magnetic metamaterial
structures, respectively (Fig. 2a, b). In both cases, the
circular geometry allows for loop-like current flow, gener-
ating a magnetic moment. We note that the magnetic re-
sponse of these arrangements is sometimes incorporated
into an equivalent, alternative, spatially dispersive per-
mittivity. Although this is, in principle, always possible
[1, 38, 39], we stress that, similar to naturally occuring
substances, described with a permittivity
and a per-
meability
μ
, a metamaterial description based on (
,
μ
)
allows for physical intuition and reduces complexity, es-
pecially 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 1D multilayers, as, in
this case, TE and TM linear polarizations are decoupled
and directly associated with
μ
and
, respectively.
Here, we demonstrate, a principle for strong mag-
netic response in 1D layered metamaterial heterostruc-
tures.
We start by considering a single subwave-
length dielectric slab of refractive index
n
diel
and
thickness
d
.
When illuminated at normal incidence
(
z
direction in Fig.
2c), its displacement current
~
J
d
=
iω
o
(
n
2
diel
−
1)
~
E
induces a macroscopic effective
3
Μ
eff
J
d
(a)
Μ
eff
J
d
Μ
eff
H
avg
J
d
J
d
d
z
x
y
(c)
ρ
=1
ρ
=2
z (in d)
-1/2
-2
-1
2
1
0
1/2
0
Re(J
d
), a.u.
wavelength (in d)
79
11
Re(
μ
eff
)
6
4
2
0
with air
with Ag
(f)
-5
0
5
Re(H
x
) (a. u)
z (in d)
H
avg
-1/2
1/2
0
M
eff
diel.
diel.
Ag
y(nm)
H
avg
M
eff
(b)
(d)
-2
-1
2
1
0
z (in d)
-1/2
1/2
0
Re(J
d
), a.u.
(e)
FIG. 2. Induced magnetization in (a) dielectric nanoparticles
(three-dimensional metamaterials) (b) in dielectric nanorods
(two-dimensional metamaterials) and (c) in a one-dimensional
dielectric slab. (d) Displacement current distribution at res-
onance, for
ρ
= 1,
ρ
= 2 for a 90nm slab of refractive index
n
diel
= 4
.
5. (e) Displacement current distribution for two
dielectric layers separated by air. (f) Effective permeability
for two dielectric layers separated by air-black and silver-red.
Inset: tangential magnetic field profile at resonance: average
magnetic field is opposite to
M
eff
.
magnetization
~
M
eff
= 1
/
2
μ
o
∫
(
~r
×
~
J
d
)
·
~
dS
[1, 36, 40].
By averaging the magnetic field,
〈
H
〉
=
∫
d/
2
−
d/
2
H
(
z
)
dz
,
we use
μ
eff
'
1 +
M
eff
/
(
μ
o
〈
H
〉
) to obtain an empirical
closed-form expression for the magnetic permeability:
μ
eff
'
1
−
n
2
diel
−
1
2
n
2
diel
{
−
1 +
n
diel
πd/λ
tan(
n
diel
πd/λ
)
}
(1)
By setting
n
diel
= 1, we recover the unity mag-
netic permeability of free space. From Eq.(1), we see
that the magnetic permeability
μ
eff
will diverge when
tan(
n
diel
πd/λ
) = 0. This yields a magnetic resonant
behavior at free-space wavelengths
λ
=
n
diel
d/ρ
, with
ρ
= 1
,
2
,..
. At these wavelengths, the displacement cur-
rent distribution is anti-symmetric, as shown in Fig. 2d
for
ρ
= 1
,
2. This anti-symmetric current flow closes a
loop in
y
=
±∞
and induces a magnetization
~
M
eff
which
is opposite to the incoming magnetic field (Fig. 2c), lead-
ing to a diamagnetic response. Eq.(1) serves to estimate
the design parameters for enhanced magnetic response;
in the long-wavelength limit, only the fundamental and
second resonances,
λ
=
n
diel
d,n
diel
d/
2, play significant
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. The same principle applies for grazing incidence,
with the displacement current inducing a magnetic re-
sponse in the out-of-plane (
z
) direction.
In order to make this magnetic response significant,
we consider the case of two parallel metallic wires in air,
carrying opposite currents; their magnetic moment scales
with their distance, as dictated by
~
M
∝
~r
×
~
J
. In the
planar geometry considered here, an equivalent scheme
is represented by two high-index layers separated by air,
as shown in Fig. 2e. Indeed, as demonstrated with the
black curve in Fig. 2f, the magnetic permeability
μ
eff
of this system strongly deviates from unity. In fact, the
separation layer is not required to be air; any high-low-
high refractive index sequence will induce the same ef-
fect. For example, replacing the air region with a layer
of metal, with
n
metal
1 at optical frequencies, does
not drastically change the magnetic response. This is
shown for a separation layer of silver in Fig. 2f with the
red curve. Therefore, at optical frequencies, where the
conduction current in metallic layers is small, metals do
not contribute significantly to the magnetic response, in
contrast to the GHz regime, where the metallic compo-
nent in resonant structures has been necessary for strong
magnetic effects [6–9]. From the inset of Fig. 2f, one
can see that the average magnetic field faces in the direc-
tion opposite to the magnetization, expressing a negative
magnetic response for the dielectric/silver unit cell.
III. COMBINING HYPERBOLIC DIELECTRIC
AND MAGNETIC PROPERTIES
Alternating layers of metals and dielectrics have a dis-
tinct
dielectric
response, which is hyperbolic for TM po-
larization; the metallic component allows for
o
<
0,
while the dielectric layers act as barriers of conduction in
the out-of-plane (
z
) direction, leading to
e
>
0. We com-
bine this concept with the principle for creating magnetic
resonances in planar systems, discussed in Sec.II. We
show that it is possible to induce an additional
magnetic
response in planar dielectric/metal hyperbolic metama-
terials, if the dielectric layers are composed of high-
index materials, capable of supporting strong displace-
ment currents at optical frequencies. Previous consider-
ations mostly pertained to lower-refractive index dielec-
tric layers, for example, LiF [41] or Al
2
O
3
[23, 42, 43] and
TiO
2
[13]. As can be inferred from Fig. 3 in what follows,
for layer thicknesses below
∼
50nm, these lower-index di-
electric/metal systems exhibit magnetic resonances in the
ultraviolet (UV)-short wavelength visible regime.
The effective magnetic response of planar multilayer
metamaterials is a uniaxial tensor
μ
~
~
eff
=
diag
{
μ
o
,μ
o
,μ
e
}
and, as we demonstrate below, it may also be extremely
anisotropic, which has interesting implications for TE po-
larization (See Sec.IV). We point out, however, that the
dielectric hyperbolic response
o
e
<
0 in planar systems
is broadband. In contrast, the magnetic permeabilities
4
along both coordinate directions
μ
o
and
μ
e
deviate from
unity in a resonant manner, thus, TE polarization-based
phenomena are more narrow-band in nature.
A. Relaxing the
μ
eff
= 1
constraint
Prior to delving into experimental results, we briefly
discuss our computational method, which allows relax-
ing the previously made
μ
eff
= 1 assumption.
The
most extensively used approach for describing the ef-
fective response of hyperbolic multilayer metamaterials
is the Maxwell Garnett effective medium approximation
(EMA) [11] (and references therein), [12, 13], based on
which the in-plane dielectric permittivity is given by
o
,
MG
=
f
m
+ (1
−
f
)
d
and the out-of-plane extraor-
dinary permittivity is
−
1
e
,
MG
=
f
−
1
m
+ (1
−
f
)
−
1
d
, where
f
is the metallic filling fraction [28], while
μ
eff
is
a pri-
ori
set to unity. Another commonly used approach is
the Bloch formalism, based on which, a periodic A-B-A-
B. . . superlattice is described with a Bloch wavenumber
[44], which is directly translated to an effective dielectric
permittivity [29]. These approaches are useful and simple
to use, however, they are both based on the assumption
of an infinite and purely periodic medium, without ac-
counting for the finite thickness of realistic stacks.
By contrast, metamaterials other than planar ones,
which are, in general, more complicated in structure, for
example split-ring resonators [7–9, 45], nanoparticles [27],
fishnet structures [46, 47] and many others, are modeled
with exact S-parameter retrieval approaches [48, 49]. S-
parameter retrievals solve the inverse problem of deter-
mining the effective dielectric permittivity and magnetic
permeability,
eff
and
μ
eff
respectively, of a homogeneous
slab with the same scattering properties, namely trans-
mission
T
and reflection
R
coefficients, as the arbitrary
inhomogeneous, composite metamaterial system of finite
thickness
d
.
By lifting the assumption of an infinite medium, one
is able to compute both transmission
T
and reflection
R
coefficients, and utilize them in S-parameter approaches.
This allows to obtain an effective wavenumber
k
eff
to-
gether with an effective impedance
Z
eff
for the system
under study [48, 49]. These parameters are then used
to decouple the effective permittivity from the perme-
ability through
k
eff
=
√
eff
μ
eff
ω
c
and
Z
eff
=
√
μ
eff
eff
. By
contrast, Bloch-based approaches [29, 44] only consider a
Bloch wavenumber
K
Bloch
(based on periodicity), with no
other information available for allowing decoupling
μ
eff
from
eff
. Both the Maxwell Garnett result [28] and its
Bloch-based generalizations (for example [29]) are based
on the assumption that
μ
eff
= 1. For a schematic com-
parison of the two approaches, see Figs.3a, b.
Contrary to the extensive use of EMAs, we utilize the
S-parameter approach to describe dielectric/metal mul-
tilayer metamaterials of finite thickness. By letting the
magnetic permeability
μ
eff
be a free parameter, instead
of
a priori
setting
μ
eff
= 1, we obtain magnetic reso-
nances at wavelengths where magnetic dipole moments
25 layers
T
I
n
diel
=4
Ag
α
(a)
α
Arbitrary
composite
medium
Z
eff
,
k
eff
IR
T
(b)
x
y
z
ordinary (o)
extraordinary (e)
d
n
diel
=4
A
B
A
B
A
B
A
A
K
Bloch
|T|
2
Z
eff
Z
μ
=1
Z
EMA
0
0.5
1
(c)
|T|
2
, Z
eff
20
40
60
80
wavelength (in
α
)
FIG. 3. Comparison between (a) traditional EMA and Bloch
approaches and (b) the general concept of S-parameter re-
trievals. (c) Impedance-matching sanity check at normal in-
cidence for a 25 layers dielectric/metal stack with
n
diel
= 4.
The transmittance
|
T
|
2
calculation was performed with the
transfer-matrix formalism [44] for the physical multilayer sys-
tem in the lossless limit. The dielectric and magnetic effective
model (
o
,
μ
o
) accurately captures the structures resonances
unlike the non-magnetic approach (
o
,μ
=1
) and the Maxwell
Garnett EMA.
occur, as demonstrated in Figs.2e, f. This confirms the
physicality of the non-unity
μ
; based on the arguments
discussed in Sec.II, magnetic resonances arise at wave-
lengths at which systems support circular or loop-like
current distributions.
By accounting for the uniaxial anisotropy in planar
heterostructures, we obtain both the ordinary and the
extraordinary permeabilities
μ
o
and
μ
e
, together with
their dielectric permittivity counterparts,
o
and
e
. As a
sanity check, we first consider homogeneous metallic and
dielectric slabs with known dielectric permittivity
o
=
e
and
μ
o
=
μ
e
= 1, which we recover upon application of
our retrieval [50].
Another way to establish the validity of the the
effective parameters is to perform an impedance-
matching sanity check. Based on electromagnetic the-
ory, the impedance of a structure at normal-incidence,
Z
eff
=
√
μ
o
o
, must be unity at transmittance
|
T
|
2
max-
ima. As seen in Fig. 3c, the retrieved parameters
o
and
μ
o
accurately describe the scattering properties of pla-
nar dielectric/metal arrangements of finite thickness. On
the contrary, not accounting for a magnetic permeabil-
ity leads to inaccurate prediction of transmittance max-
ima. This is seen both by utilizing our S-retrieval-based
approach while setting
a priori
the magnetic permeabil-
ity to unity (
Z
μ
=1
in Fig. 3c), and with the traditional