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PRL 100, 023903 (2008)
PHYSICAL REVIEW LETTERS
week ending
18 JANUARY 2008
Experimental Demonstration of Electromagnetic Tunneling Through
an Epsilon-Near-Zero Metamaterial at Microwave Frequencies
Ruopeng Liu
Department of Electrical and Computer Engineering, Duke University, Durham, North Carolina 27708, USA
Qiang Cheng
State Key Laboratory of Millimeter Waves, Southeast University, Nanjing 210096, China
Thomas Hand and Jack J. Mock
Department of Electrical and Computer Engineering, Duke University, Durham, North Carolina 27708, USA
Tie Jun Cui*
State Key Laboratory of Millimeter Waves, Southeast University, Nanjing 210096, China
Steven A. Cummer and David R. Smith†
Center for Metamaterials and Integrated Plasmonics, Department of Electrical and Computer Engineering, Duke University,
Durham, North Carolina 27708, USA
(Received 23 March 2007; revised manuscript received 4 November 2007; published 18 January 2008)
Silveirinha and Engheta have recently proposed that electromagnetic waves can tunnel through a
material with an electric permittivity () near zero (ENZ). An ENZ material of arbitrary geometry can thus
serve as a perfect coupler between incoming and outgoing waveguides with identical cross-sectional area,
so long as one dimension of the ENZ is electrically small. In this Letter we present an experimental
demonstration of microwave tunneling between two planar waveguides separated by a thin ENZ channel.
The ENZ channel consists of a planar waveguide in which complementary split ring resonators are
patterned on the lower surface. A tunneling passband is found in transmission measurements, while a twodimensional spatial map of the electric field distribution reveals a uniform phase variation across the
channel —both measurements in agreement with theory and numerical simulations.
DOI: 10.1103/PhysRevLett.100.023903
PACS numbers: 41.20.Jb, 42.25.Bs, 78.20.Ci, 84.40.Az
The available palette of electromagnetic properties has
expanded considerably over the past several years due to
the development of artificially structured materials, or
metamaterials. In particular, metamaterials whose relative
constitutive parameters —such as the electric permittivity
or the magnetic permeability —are less than unity
have proven to exhibit a remarkable influence on wave
propagation properties, and have thus become a vigorous
research topic [1–3].
Much attention has recently focused on structures for
which the real part of one or both of the constitutive
parameters approaches zero. These structures have been
used to form interesting devices such as highly directive
antennas [4] and compact resonators [5]. Most recently,
Silveirinha and Engheta [6] have proposed that a material
whose electric permittivity is near zero—or an epsilonnear-zero (ENZ) medium —can form the basis for a perfect
coupler, coupling guided electromagnetic waves through a
channel with arbitrary cross section. While it might at first
be thought that the inherent impedance mismatch between
an ENZ medium and any other medium should degrade the
transmission efficiency, remarkably this turns out not to be
the case so long as at least one of the physical dimensions
of the ENZ material channel is electrically small.
0031-9007=08=100(2)=023903(4)
In this Letter, we make use of planar complementary
split ring resonators (CSRRs) patterned in one of the
ground planes of a planar waveguide to form the electromagnetic equivalent of an ENZ. The CSRR structure was
proposed by Falcone et al. [7], who showed by use of the
Babinet principle that the CSRR has an electric resonance
that couples to an external electric field directed along the
normal of the CSRR surface. It was further shown explicitly that a volume bounded by a CSRR surface behaves
identically to a volume containing a resonantly dispersive
dielectric. Variants of the wire medium could also potentially be used to form the ENZ medium [8], though spatial
dispersion and effects due to the finite wire length can
cause significant complication [9].
The geometry considered here to demonstrate the tunneling effect is chosen to be compatible with our planar
waveguide experimental apparatus (Fig. 1) previously described [10]. Three distinct waveguide sections are formed,
distinguished by the differing gap heights between the
upper and lower metal planes. There is a gap of 11 mm
between the upper and lower conducting plates that serve
as the input and output waveguides. The narrowed tunneling channel with patterned CSRRs in the lower surface has
a gap of 1 mm between the plates. The planar waveguide is
023903-1
© 2008 The American Physical Society
PRL 100, 023903 (2008)
week ending
18 JANUARY 2008
PHYSICAL REVIEW LETTERS
R
FIG. 1 (color online). Experimental setup, in which h 11 mm, hw 10 mm, d 18:6 mm (16.6 mm for CSRR regime), and w 200 mm. Lower figure is the sample inside
chamber.
bounded on either side by layers of absorbing material,
which approximate magnetic boundary conditions and also
reduce reflection at the periphery. The waveguide and
channel thus support waves that are nearly transverse
electromagnetic (TEM) in character. Assuming the CSRR
region can be treated as a homogenized medium, the entire
configuration is well approximated as two dimensional,
with the average field distribution having little variation
along the width.
There are three impedances that can be defined for the
waveguide geometry used here [11]: the intrinsic, wave,
and characteristic impedances. The wave impedances corresponding to TEM waves in the three waveguide regions
are equivalent irrespective of the waveguide shape, all
equal to Z1 . However, the characteristic impedance is
defined by the equivalent voltage and equivalent current
of the transmission line [11], which are dependent on the
cross sections of the waveguides. There is thus a severe
characteristic impedance mismatch at the two interfaces
between the planar waveguides and the narrow channel
that inhibits the transmission of waves from the left waveguide region to the right. A characteristic impedance model
for the specific geometry considered here has been described in detail in [12], where a transmission-line model
is derived showing that the narrow waveguide region can
be replaced by a region having an impedance Z2 d=bZ1 . d=b is the ratio of the planar waveguide and
channel heights. In addition, there is a shunt admittance
Y jB at the interface between the mismatched waveguides, which becomes quite large (and therefore unimportant) when the waveguides differ significantly in height.
Because Z2 Z1 , there is no coupling between the input
and output waveguides, except possibly when a resonance
condition is met and a Fabry-Perot oscillation occurs. If the
channel is now loaded with an ENZ material, the characteristic impedance of the channel region is raised to the point
where the three waveguide regions are matched and perfect
transmission once again should occur.
The experimental configuration studied here corresponds to one of the two cases considered by Silveirinha
and Engheta [6]. In the first case and tend to zero
simultaneously, while in the second is near zero and the
area of channel is assumed electrically small. Our experimental setup belongs to the latter case, whose mechanism
is illustrated by calculating the reflectance based on the
simplified model described in [12]. We find
R12 1 ei2k2z d 1 R212 ei2k2z d
(1)
in which R12 Z2 ==iB Z1 =Z2 ==iB Z1 and Z2 ==iB iBZ2 =iB Z2 . R12 is the reflection coefficient between the planar waveguide and the
channel, d is the effective length of the channel, and k2z
is the wave vector inside the channel. Z1 and Z2 are the
effective wave impedances outside and inside the narrow
channel, respectively, whose ratio Z1 =Z2 corresponds to
the height ratio 11=1 [12] in the absence of the patterned
CSRRs. When and are simultaneously near zero, the
characteristic impedance of the zero index material may
p
take on the finite value lim;!0 = , which may differ
from the impedance of adjacent regions. However, since
k ! 0, the reflection coefficient vanishes, indicating the
tunneling of the wave across the channel [6]. In the present
configuration, since Z2 =Z1 approaches zero, the reflection
coefficient no longer vanishes in a simple manner. Instead,
when the ENZ medium possesses a small but finite value of
permittivity, Z2 =Z1 may approach unity, and the tunneling
effect is restored.
The equivalence between an ENZ metamaterial and the
CSRR structure shown in Fig. 2 can be established by
performing a numerical retrieval of the effective constitutive parameters for the channel. To obtain the retrieved
permittivity shown in Fig. 2, the scattering (S) parameters
are simulated using Ansoft HFSS, a commercial full-wave
electromagnetic solver [13–16]. Figure 2(b) shows the
simulation setup used to retrieve the effective constitutive
parameters of the CSRR channel. The polarization of the
incident TEM wave is constrained by the use of perfect
magnetic conducting (PMC) boundaries on the sides of the
computational domain. The separation between the metal
plates in the channel region is h 1 mm, while the vacuum region has a height of d 11 mm with a perfect
electric conducting (PEC) boundary on the lower surface.
Radiation boundaries are assigned below the ports, as
shown in the figure. The two ports are positioned far
away from the CSRR structure to avoid near-field cross
coupling between the ports and the CSRRs; however, the
phase reference planes for the retrieval are chosen (or
deembedded) to be just on either side of the channel.
The simulated reflection and transmission coefficients as
a function of frequency for the channel with and without
the ENZ metamaterial are shown in Fig. 4. These results
are compared with the simplified model presented in
p
Eq. (1), where Z2 Z1 =11 eff;r with the effective
length d chosen as 13 mm. An approximate analytical
expression for B obtained by a conformal mapping procedure is presented in [12]. The transmission and reflection
coefficients predicted by the analytical model are plotted in
Fig. 4, where they can be seen to be in very good agreement
with those simulated, supporting the interpretation of the
transmission peak as an indication of tunneling. In addition, Fig. 3(a) shows the Poynting vector distribution at
023903-2
PRL 100, 023903 (2008)
PHYSICAL REVIEW LETTERS
week ending
18 JANUARY 2008
FIG. 3 (color online). Poynting vector and medium model:
(a) Poynting vector. (b) Simplified model.
FIG. 2 (color online). Retrieval results, dimensions of CSRRs,
and simulation setup. (a) Extracted permittivity and CSRRs’
dimensions, in which a 3:333 mm, b 3 mm, c d 0:3 mm, and f 1:667 mm. (b) Simulation configuration for
CSRR unit cell. d 11 mm, h 1 mm, and L 23:333 mm.
(c) Configuration of tunneling effect simulation.
8.8 GHz, revealing the squeezing of the waves through the
narrow channel.
To validate experimentally the ENZ properties of the
CSRR region, a channel patterned with CSRRs was fabricated and its scattering compared to an unpatterned control channel. Both the CSRR and control channels were
formed from copper-clad FR4 circuit board (0.2 mm thick),
fabricated with dimensions 18:6 200 mm2 (shown in
Fig. 1). The array of CSRR elements (shown in Fig. 2) was
patterned on the circuit board using standard photolithography. A total of 200 CSRRs (5 in the propagation direction, 40 in the transverse direction) were used to form the
effective ENZ metamaterial. The CSRR or control substrates were then placed on a Styrofoam support, with
dimensions 18:6 10 200 mm3 . Copper tape was
used to cover the sides of the Styrofoam and carefully
placed to ensure that the copper-clad substrates would
make good electrical contact with the bottom plate of the
waveguide.
To obtain a base level of transmission, the control sample was positioned inside the planar waveguide halfway
between the two ports and a transmission measurement
taken. The results are shown in Fig. 4 and compared with
both the analytical model and the HFSS simulation. The
passband of the control slab, due to the resonance condition, was found to occur at 7 GHz—shifted from the
passband found in simulation. The consistent shift between
measurement and simulation likely reflects the differences
between the experimental environment and the simulation
model (e.g., finite slab width and fluctuation of channel
height). The measurement and simulation, however, are in
excellent qualitative agreement. By uniformly shifting the
frequency scale, the measured and simulated curves are
almost identical (note the two scales indicated on the top
and bottom axes).
With the control slab replaced by the CSRR channel, the
measured transmitted power (shown in Fig. 4) reveals a
passband near 7.9 GHz (8.8 GHz from the simulation),
which is identical with retrieval prediction. This passband
is absent when the control slab is present, demonstrating
that electromagnetic tunneling takes place at approximately the frequency where the effective permittivity of
the ENZ region approaches zero.
Phase sensitive maps of the spatial electric field distribution throughout the channel region were constructed
[10]. The electric field magnitude was mapped inside a
FIG. 4 (color online). Experimental, theoretical, and simulated
transmissions for the tunneling and control samples.
023903-3
PRL 100, 023903 (2008)
PHYSICAL REVIEW LETTERS
FIG. 5 (color online). Two-dimensional mapper results at
8.04 GHz. (a) Field distribution of tunneling sample. (b) Field
distribution of control.
180 180 mm2 square region for both the copper control and CSRR channels. Figure 5 shows the mapped fields
for both configurations taken at 8.04 GHz (where the
effective permittivity of the CSRR structure is approximately zero). The field is normalized by the average field
strength. Yet it is clear that the CSRR channel allows
transmission of energy to the second port, measured to
be 5 dB, whereas only 14 dB of field energy propagates to port 2 when the control slab is presented. Note the
uniform phase variation across the channel at the tunneling
frequency, f 8:04 GHz.
A linear plot of phase versus position (shown in Fig. 6)
further illustrates the tunneling of energy through the ENZ
channel versus the Fabry-Perot–like resonant scattering.
The latter mechanism of transmission has been studied in
detail by Hibbins et al. [17]. As can be seen in Fig. 6, a
strong phase variation exists across the control channel at
f 7 GHz, corresponding to the passband that is seen in
Fig. 3. The clearly distinguished phase advance within the
channel implies that the propagation constant is nonzero.
The large transmittance results from a resonance condition
related to the length of the channel. By contrast, the spatial
phase variation for the CSRR channel is shown in Fig. 6 for
f 8:04 GHz (the passband of the CSRR loaded channel), where we see that the phase advance across the
channel at this frequency is negligible.
FIG. 6 (color online). Phase shift for 5 unit-cell tunneling
sample at 8.04 GHz and control at 7 GHz.
week ending
18 JANUARY 2008
While the CSRR waveguide used in these experiments
does not form a volumetric metamaterial, we have nevertheless shown that the planar waveguide channel can be
treated equivalently as having a well-defined resonant
permittivity, with zero value at a frequency of 8.04 GHz.
Furthermore, the set of transmission and mapping measurements we have presented demonstrates that the tunneling observed through the channel is consistent with the
behavior of an -near-zero medium. The measurements
confirm that ‘‘squeezed waves’’ will tunnel without phase
shift through extremely narrow ENZ channels. ENZ materials may thus be used as highly efficient couplers with
broad application in microwave and THz devices.
This work was supported by the Air Force Office of
Scientific Research through a Multiple University
Research Initiative, Contract No. FA9550-06-1-0279.
T. J. C. acknowledges support from the National Basic
Research Program (973) of China under Grant
No. 2004CB719802 and the National Science Foundation
of China under Grant No. 60671015.
*[email protected]
†
[email protected]
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023903-4
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