Infrared Dielectric Resonator Metamaterial
James C. Ginn∗and Igal Brener
Sandia National Laboratory, Albuquerque, NM 87185, USA. and
Center for Integrated Nanotechnologies, Sandia National Laboratory, Albuquerque, NM 87185, USA.
David W. Peters, Joel R. Wendt, Jeffrey O. Stevens, Paul F. Hines, Lorena I.
Basilio, Larry K. Warne, Jon F. Ihlefeld, Paul G. Clem, and Michael B. Sinclair
Sandia National Laboratory, Albuquerque, NM 87185, USA.
(Dated: August 25, 2011)
We demonstrate, for the first time, an all-dielectric metamaterial resonator in the mid-wave
infrared based on high-index tellurium cubic inclusions. Dielectric resonators are desirable compared
to conventional metallo-dielectric metamaterials at optical frequencies as they are largely angular
invariant, free of ohmic loss, and easily integrated into three-dimensional volumes. With these
low-loss, isotropic elements, disruptive optical metamaterial designs, such as wide-angle lenses and
cloaks, can be more easily realized.
PACS numbers: 81.05.Xj, 78.67.Pt, 85.50.-n
The unique properties of metamaterials have yielded
many exciting electromagnetic phenomena including sub-
diffraction-limited imaging , cloaking , and perfect
absorption . In spite of the rapid advances in this
field, passive metamaterials at optical frequencies have
often proven impractical due to significant conductor loss
from the metallic resonators comprising these volumes
. Additionally, the inherent geometrical asymmetry of
these resonators further restricts metamaterial behaviour
to a small range of incident angles even when assembled
into three-dimensional structures . Three-dimensional
dielectric resonators, unlike their metallic counterpart,
have significantly less material loss, support resonant
modes that are invariant to the excitation angle, and can
be easily integrated into thick volumes. In this letter, we
describe the development of a dielectric resonator based
metamaterial in the infrared with spectral regions of neg-
ative magnetic and electric effective properties. Addi-
tional insight is also provided in addressing material lim-
itations imposed on dielectric metamaterials at optical
Lewin  theoretically demonstrated that an array of
sub-wavelength dielectric resonators can exhibit spectral
regions of Lorentzian-like effective permittivity and per-
meability. Following Mie theory , the effective electric
and magnetic polarizabilities of a densely packed array
of sub-wavelength spheres can be altered by changing the
dimensions, composition, and packing fraction of the in-
clusions. Similar behavior can be achieved using cubic
dielectric resonators (CDRs)  which, unlike spheres,
are compatible with existing nano-scale lithography tech-
niques and can be integrated into multi-layer composites
through repeated steps of thin-film deposition, etching,
and planarization. Since there are no analytical expres-
sions for predicting the behavior of a cubic scatterer 
and approximations are problematic due to the high in-
∗Present Address: Plasmonics Inc, Orlando, FL 32826, USA.;
Electronic Mail: email@example.com
FIG. 1. (color online). (a) Excitation configuration of an
isolated sphere (top row) and a 1:1 periodic cube array (bot-
tom row). (b) normalized electric field distribution for the
lowest-order magnetic and (c) normalized electric field distri-
bution for the lowest-order electric mode in the sphere and
cubic resonators. Dotted white lines indicate field direction.
dex dispersion of materials in the infrared, numerical
computational electromagnetic approaches must be used
for analysis. In Fig. 1 the analytically determined on-
resonance field distribution of an isolated sphere is com-
pared to that of a cubic resonator array calculated using
the commercially available rigorous coupled wave anal-
ysis (RCWA) package, GDCALC. Like the spherical
resonator, the lowest-order mode of a CDR is a mag-
netic dipole (TE011) and the second-lowest mode is an
electric dipole (TM011). However, in contrast to tradi-
tional metallic resonators the electromagnetic responses
of single spherical inclusions are isotropic, and cubic res-
onators represent only a minor perturbation from this
spherical symmetry. Also, whereas significant damping
due to ohmic loss is unavoidable for metallic resonators
in the infrared , the damping of dielectric resonators
can be quite low provided the resonator material lacks
active carriers or phonon modes in the band of inter-
arXiv:1108.4911v1 [physics.optics] 24 Aug 2011
FIG. 2. (color online). (a) Scanning electron micrograph of fabricated CDR. (b) Measured reflection and transmission coeffi-
cients for CDR. Field patterns from Fig. 1 are shown above each corresponding resonance.
est. The scattering cross-sections exhibited by such un-
damped systems are known to be equal to the fundamen-
tal limits imposed by Mie theory .
Thus, identification of candidate low-loss resonator
materials is critical in designing a practical infrared CDR
metamaterial. In addition, the CDR material must pos-
sess a large index of refraction to ensure that the dimen-
sions of the resonator and array spacing are sufficiently
small (non-diffracting) compared to the operating wave-
length. Only two classes of dielectrics exhibit positive
indices of refraction greater than three in the infrared:
highly crystalline polaritonic materials and narrow band-
gap (< 1.5 eV) materials. Metamaterial structures using
polaritonic materials have previously been investigated
, but these materials are less desirable due to their
high loss at the phonon resonance and limited spectral
flexibility. In contrast, narrow band-gap materials ex-
hibit large indices over wide spectral bands, and only
experience significant loss near the band-gap at shorter
wavelengths and on the tail of the free-carrier absorp-
tion at longer wavelengths. Candidate narrow band-gap
materials for infrared CDR designs include silicon ,
germanium, tellurium, and IV-VI compounds containing
lead (such as lead telluride).
Initial investigations of a magnetic active CDR array
in the infrared was carried out using germanium cubes
on a low-index polymer thin-film . Tellurim (Te) was
subsequently selected as a better resonator material due
to its larger index of refraction and low loss at infrared
wavelengths . Because of its trigonal crystal lattice
, Te is naturally anisotropic in single crystal form
with an extraordinary index of refraction of 6.2 at 10 µm
. For CDR applications, a polycrystalline morphol-
ogy is preferable which yields a crystal-averaged index of
refraction of 5.3 at 10 µm, with an extinction coefficient
of less than 10−4. Barium fluoride (BaF2) was se-
lected as the optimal substrate due to its low refractive
index (n ∼ 1.4) and low loss at 10 µm.
Through simulation, a 1.7 µm CDR with a 3.4 µm unit-
cell spacing (1:1 duty-cycle) was chosen to center the re-
flection peak of the magnetic resonance at 10 µm. Prior
to patterning, a 1.7 µm thick film of Te was deposited
on the surface of a 25 mm diameter BaF2 optical flat
via electron-beam evaporation.
and variable angle spectral ellipsometric analysis veri-
fied that the film was predominately polycrystalline, and
ellipsometry analysis yielded a fitted complex index of
refraction of n = 5.02 + 0.04j. The Te film was pat-
terned using electron beam lithography and etched using
a reactive ion etching (RIE) process. A scanning elec-
tron micrograph of the etched pattern is shown in Fig.
2a. The etching process resulted in excellent uniformity
over a 1 cm2area, with only a slight over-etching of the
pattern. The final CDR element was 1.7 µm tall with
a 1.53 x 1.53 µm base and a 10 degree sidewall slope.
The overall process required significantly less steps than
existing three-dimensional metallo-dielectric lithography
 and the feasibility of planarizing the patterned re-
gion for multi-layer fabrication was verified by success-
fully spin-coating a thin-film of polynorbornene on the
surface of the CDR array . Although the CDRs have
an isotropic resonance mode, multi-layer fabrication is
necessary for an angular-independent array response.
Following fabrication, the patterned wafer was char-
acterized using a hemispherical directional reflectometer
(HDR) at an angle of incidence of seven degrees. The
measured collimated transmission and specular reflection
of the array are plotted in Fig. 2b. As expected from the
fabricated topology and measured index of the Te film,
the reflection peak of the magnetic resonance occurred
at 9 µm and the reflection peak of the electric resonance
occurred at 7.5 µm. The resonances are well defined and
occur at wavelengths above the diffraction cut-off limit.
In the spectral region between the two resonances, loss (1
reflection transmission) drops to less than 8%. We note
that no artificial correction factors were used to account
for reflection and absorption losses due to the substrate.
The optical response of the fabricated cube array was
Both x-ray diffraction
and transmission for CDR. (b) Plot of calculated impedance
phase, permittivity, and permeability for the simulated CDR
array. Real values are in red and imaginary values are in blue.
(color online). (a) Plot of simulated reflection
simulated (Fig. 3). The simulated and measured co-
efficients show overall good agreement (Fig. 3a), with
differences primarily arising from the asymmetry and
non-uniformity of the as-fabricated cube. The calculated
surface impedance (Fig. 3b) indicates two distinct reso-
nances with inverted phase delays. Positive phase delay
occurs when the electric field leads the magnetic field in
phase (permeability less than zero) and negative phase
delay occurs when the electric field follows the magnetic
field (permittivity less than zero). Using a standard re-
trieval algorithm  for the planar array, the effective
permittivity and permeability were calculated for these
two regions (inset 3b). In both cases, the extracted pa-
rameters reach values of less than -1, and the loss tangent
of the permeability falls to 0.48 when the real part of per-
meability is equal to -1. We note that the extinction coef-
ficient of the as deposited film is more than two orders of
magnitude larger than the literature value for Te. Thus,
we anticipate that significantly lower-loss metamaterials
will be achievable as the Te loss is minimized.
To further investigate the features of infrared CDRs,
a series of RCWA simulations was run for a 1:1 duty-
cycle array while only varying the refractive index of the
cubes. The solid lines in Fig.
ized wavelength at the points of peak reflectivity (due
4 denote the normal-
FIG. 4. (color online). Design metric for 1:1 CDR metama-
terials. Solid lines correspond to the lowest-order magnetic
(red) and electric (blue) resonances.
defines where peak positive permeability/permittivity occurs
and the bottom line defines where permeability/permittivity
is equal to zero. The indices of several materials at 10 µm are
The top dotted line
to the two primary resonances), versus index of refrac-
tion (restricted to a range of values that is realistic for
known optical materials). These lines also directly corre-
spond to the point where the CDR array switches from
supporting a propagating mode (positive effective per-
meability/permittivity) to a plasma mode (negative ef-
fective permeability/permittivity). The effective perme-
ability and permittivity were also calculated using the
retrieval algorithm  assuming a fixed extinction coef-
ficient of 0.001 for algorithm stability. From these cal-
culated values, the points of peak effective positive per-
meability/permittivity and the zero crossing of the ef-
fective permeability/permittivity curves were determined
and plotted as dotted lines on Fig. 4.
Several critical performance metrics for optical CDRs
can be gleaned from Fig. 4 and related analysis. As
expected, the resonant wavelength of a CDR will scale
linearly with the dimensions and index of refraction of
the cube. This behavior explicitly limits the practicality
of negative permitivity metamaterials in the mid-wave
infrared based around traditional high-index materials
such as silicon and germanium. This limitation is also
more severe in the optical regime when compared to the
rf and THz portions of the spectrum where materials
with indices surpassing thirty exist . These arrays
also exhibit appreciable spatial dispersion  that de-
creases asymptotically with increasing normalized wave-
length (the array period is decreasing relative to the reso-
nant wavelength). Spatial dispersion dominates when the
effective index of the CDR array exceeds its normalized
wavelength and field homogenization breaks down. In
this regime, a photonic crystal band-gap mode is excited
and retrieved optical properties no longer hold physical
meaning. Consequently, the peak effective positive per-
meability/permittivity curve defines the largest index of
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FIG. 5. (color online). Specular (red line) and diffuse (blue
line) transmission for measured CDR.
refraction supported by the array at normal incidence
and the start of the band-gap region. This regime per-
sists until the resonance line is crossed and the behav-
ior of the CDR array becomes dominated by the plasma
mode for which spatial dispersion can largely be ignored.
More exotic behaviors, such as doubly negative parame-
ters, can be realized by mixing two dielectric resonators
with different indices of refraction or different dimensions
in a multi-layer composite.
One of the advantages of characterizing the fabricated
metamaterial with an HDR is that it allowed for exper-
imental validation of the onset of spatial dispersion in
the fabricated CDR. Fig. 5 shows a comparison of the
measured specular and diffuse (diffracted) transmission.
From the figure, the region where the device is dominated
by the photonic crystal band-gap mode corresponds di-
rectly to the appearance of appreciable coupling of in-
cident light into diffracted orders.
manifests near in the center of the metamaterial’s reso-
nance and is stronger for the electric mode, as expected
from theory. Furthermore, this confirms the effective loss
of the CDR is much larger than loss associated with ma-
In this letter, we have described the design, fabrica-
tion, and characterization of a dielectric cubic resonator
metamaterial with electric and magnetic activity in the
mid-infrared. Through theory and simulation, a general-
ized design approach for metamaterial surfaces compris-
ing of cubic resonators at optical frequencies was devel-
oped. This work represents a first step toward the devel-
opment of passive low-loss, multi-layer, isotropic meta-
material devices in the infrared.
This research was supported by the Laboratory Di-
rected Research and Development program at Sandia Na-
tional Laboratories. This work was performed, in part,
at the Center for Integrated Nanotechnologies, a U.S. De-
partment of Energy, Office of Basic Energy Sciences user
facility. Sandia is a multi-program laboratory operated
by Sandia Corporation, a Lockheed Martin Company,
for the U.S. Department of Energy under contract DE-
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