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Theory of supercoupling, squeezing wave energy, and field confinement in narrow channels
and tight bends using
near-zero metamaterials
Mário G. Silveirinha
1,2,
*
and Nader Engheta
1,†
1
Department of Electrical and Systems Engineering, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
2
Department of Electrical Engineering, Instituto de Telecomunicações, Universidade de Coimbra, 3030 Coimbra, Portugal
共Received 17 May 2007; published 10 December 2007
兲
In this work, we investigate the detailed theory of the supercoupling, anomalous tunneling effect, and field
confinement originally identified by Silveirinha and Engheta 关Phys. Rev. Lett. 97, 157403 共2006兲兴, where we
demonstrated the possibility of using materials with permittivity near zero to drastically improve the trans-
mission of electromagnetic energy through a narrow irregular channel with very subwavelength transverse
cross section. Here, we present additional physical insights, describe applications of the tunneling effect in
relevant waveguide scenarios 共e.g., the “perfect” or “super” waveguide coupling兲, and study the effect of metal
losses in the metallic walls and the possibility of using near-zero materials to confine energy in a subwave-
length cavity with gigantic field enhancement. In addition, we systematically study the propagation of electro-
magnetic waves through narrow channels filled with anisotropic near-zero materials. It is demonstrated that
these materials may have interesting potentials, and that for some particular geometries, the reflectivity of the
channel is independent of the specific dimensions or parameters of near-zero transition. We also describe
several realistic metamaterial implementations of the studied problems, based on standard metallic waveguides,
microstrip line configurations, and wire media.
DOI: 10.1103/PhysRevB.76.245109 PACS number共s兲: 78.66.Sq, 52.40.Db, 52.40.Fd, 42.70.Qs
I. INTRODUCTION
In recent years, there has been a growing interest in the
development of technologies or approaches that potentially
allow confining and guiding electromagnetic energy with
mode sizes below the diffraction limit. This may have key
applications in several fields such as telecommunications
共e.g., realization of compact cavities or waveguides
1,2
兲, im-
aging with subwavelength resolution,
3,4
devices with in-
creased storage capacity, delivery and concentration 共nano-
focusing兲 of the optical radiation energy on the nanoscale,
5–7
and realization of compact optical resonators.
8
Most of these
proposals rely on the excitation of surface plasmon polari-
tons supported by metallic structures with negative permit-
tivity. In our recent work,
9
we proposed a different paradigm
to break the diffraction limit and squeeze light through chan-
nels and bends with subwavelength cross section. We theo-
retically demonstrated that if a narrow channel is filled with
a near-zero 共ENZ兲 material, then, in the lossless limit, its
reflectivity only depends on the volume of the channel, being
independent of its specific geometry and of the transverse
cross section 共relatively to the direction of propagation兲.
Moreover, in a counterintuitive way, our results establish that
the transmission through the ENZ channel is improved when
the transverse cross section is made more and more narrow.
These properties suggest that ENZ materials may have inter-
esting potentials in efficiently transmitting energy through
subwavelength regions and effectively providing “supercou-
pling” between two ports and/or waveguides.
In Ref. 10, we have further shown that by loading the
ENZ material with dielectric or metallic particles, it is pos-
sible to tailor the magnetic permeability of the material. In
this way, by suitably designing the inclusions, it is possible
to tune the permeability of the ENZ filling material, without
changing its electric properties. In particular, it may be fea-
sible to design a matched zero-index metamaterial having
both permittivity and permeability near zero, and thus im-
proved transmission characteristics. A remarkable property
of such matched zero-index materials is that the way they
interact with electromagnetic waves is independent of the
granularity of the composite material or of the specific lattice
arrangement.
The objective of the present work is to study in more
detail the theory of the supercoupling, field confinement, and
tunneling effect reported in Ref. 9, giving physical insights
of this phenomenon and describing its emergence and poten-
tial applications in different configurations and scenarios.
Namely, here we investigate the group velocity of the wave
as it travels through the ENZ channel, evaluate the effect of
losses in the metallic walls, and demonstrate the possibility
of concentrating and confining the electromagnetic fields in a
very small subwavelength air cavity with enormous electric
field enhancement. Moreover, we propose a realistic
共metamaterial兲 emulation of the studied propagation sce-
narios at microwaves using standard “empty” metallic
waveguides or alternatively using a microstrip line configu-
ration. We report results obtained with a full wave electro-
magnetic simulator that demonstrate the emergence of a
similar tunneling and supercoupling effect in these very re-
alistic, and arguably simple, setups.
In addition, following the ideas of Ref. 9, we exploit the
possibility of using anisotropic ENZ materials to create an
analogous tunneling effect. In fact, as referred to in Ref. 9,
while at infrared and optical frequencies 共isotropic兲 materials
with ⬇0 may be readily available in nature 共e.g., some
metals,
11
semiconductors,
12
or polar dielectrics
13
兲, at micro-
waves these materials are not readily accessible 共an excep-
tion is the electron gas, of which the ionosphere is a well-
known example at radiowaves兲. Nevertheless, these materials
may, in principle, be fabricated as artificial microstructured
materials. However, nowadays the fabrication of isotropic
ENZ materials is still relatively more challenging due to the
PHYSICAL REVIEW B 76, 245109 共2007兲
1098-0121/2007/76共24兲/245109共17兲 ©2007 The American Physical Society245109-1
complexity of the required isotropic microstructure of the
material. To circumvent this problem, in Ref. 9 we suggested
using anisotropic ENZ materials—which are comparatively
simpler to synthesize—in order to obtain the same tunneling
effect. In this work, we further develop these concepts and
develop a detailed theory for the propagation of electromag-
netic waves through narrow channels filled with anisotropic
ENZ materials. We also discuss the design of wire-medium-
based implementations of these anisotropic materials at mi-
crowaves.
This paper is organized as follows. In Sec. II, we investi-
gate the supercoupling properties of metallic channels filled
with isotropic ENZ materials. The effect of metallic and di-
electric losses is discussed, as well as possible applications
共e.g., for waveguide coupling兲 and specific features of the
tunneling phenomenon. It is also described how to emulate
the studied propagation scenarios at microwaves using real-
istic three-dimensional 共3D兲 configurations based on the con-
cept of artificial plasma. In Sec. III, we study the propagation
of waves through anisotropic ENZ materials. It is shown that
for some configurations, the scattering parameters may be
independent of the specific geometry of the structure. The
design of anisotropic ENZ slabs using wire media is dis-
cussed in detail, enlightening unique features and character-
istics of such metamaterials in propagation scenarios of in-
terest. Finally, in Sec. IV, the conclusions are drawn.
The time variation of the electromagnetic fields is as-
sumed of the form e
−i
t
, where
is the angular frequency
and i=
冑
−1.
II. SUPERCOUPLING AND SQUEEZING ENERGY
THROUGH ENZ ISOTROPIC CHANNELS
In this section, we present physical insights related with
the propagation of electromagnetic waves through metallic
channels filled with ENZ isotropic materials. Important as-
pects such as the group velocity and the effect of losses in
the metallic walls are analyzed. The possibility of concen-
trating the electric field in a small air cavity with gigantic
enhancement is suggested.
A. Overview and physical insights of the tunneling effect
Here, we briefly review the main results derived in Ref. 9
and describe the emergence of the tunneling effect in several
propagation scenarios. In Ref. 9, we studied a generic two-
dimensional problem 共with geometry invariant along the z
direction兲, and we assumed that the polarization of the fields
is such that H =H
z
共x ,y兲u
ˆ
z
. We studied the properties of the
electromagnetic fields inside a material with ⬇0. It was
demonstrated that in order that the electric field is finite in-
side the ENZ material, it is necessary 共in the lossless limit兲
that H
z
=const inside the material. We used this fundamental
result to characterize the transmission of energy through a
generic ENZ transition in a waveguide scenario. More spe-
cifically, we examined a configuration in which two parallel-
plate waveguides are interfaced by a ENZ channel of arbi-
trary shape. We found out that when a transverse
electromagnetic mode 共TEM兲 impinges on the ENZ channel,
the reflection coefficient 共for the magnetic field兲 is given by
关in the ⬇0 lossless limit and assuming that the walls of the
metallic waveguides are perfectly electric conducting 共PEC兲
materials兴
=
共a
1
− a
2
兲 + ik
0
r,p
A
p
共a
1
+ a
2
兲 − ik
0
r,p
A
p
, 共1兲
where k
0
=
/ c is the free-space wave number, a
1
and a
2
define the spacing between the metallic plates of the input
and output waveguides, respectively,
r,p
is the relative per-
meability of the ENZ material, and A
p
is the area of the cross
section of the ENZ channel 共contained in the x-y plane兲. The
transmission coefficient is given by T =1+
. The geometry
of the problem is depicted in Fig. 1 for a very specific ENZ
channel in which the transition is shaped as a “U,” but we
underline here that Eq. 共1兲 is valid independent of the precise
geometry of the ENZ transition. As discussed in Ref. 9 Eq.
共1兲 demonstrates that in the case a
1
=a
2
⬅a, it may be pos-
sible to squeeze more and more energy through the channel,
as the transverse section of the channel 共relative to the direc-
tion of propagation兲 is made more and more tight, i.e., as
A
p
/ a is made increasingly small. Such effect was demon-
strated in Ref. 9 for the case of a waveguide with a 180°
bend.
Next, we further explore some alternative possibilities,
namely, we present results of full wave simulations com-
puted with
CST MICROWAVE STUDIO™,
14
which demonstrate
the emergence of the same tunneling effect in the geometry
of Fig. 1 共“U-shaped” transition channel兲. We will assume
that a
1
=a
2
⬅a since this situation favors the transmission of
energy through the channel. It is clear that for such U-shaped
channel, the area of the cross section is A
p
=共L
1
+L
2
兲a
+La
ch
. Thus, for a and L arbitrarily fixed, it is evident that
共in the lossless limit兲 we can make A
p
/ a arbitrarily small
共and consequently make the reflection coefficient approach
zero兲 by reducing more and more the transverse section of
the channel a
ch
and by making the transition regions L
1
and
L
2
more and more thin. Thus, even though the impedance
contrast between the free-space region and the ENZ material
is infinite, the wave may effectively tunnel through the nar-
row channel with high transmissivity. Note that the previous
a
ch
0
L
1
L
2
L
Metallic
walls
a
1
a
2
E
inc
H
inc
x
y
z
x=0
x’=0
FIG. 1. 共Color online兲 Geometry of the two-dimensional prob-
lem: Two parallel-plate metallic waveguides are interfaced by a
U-shaped channel filled with a ENZ material. The incident wave is
the fundamental TEM mode. The structure is uniform along the z
direction.
MÁRIO G. SILVEIRINHA AND NADER ENGHETA PHYSICAL REVIEW B 76, 245109 共2007兲
245109-2
discussion holds independently of the electrical size of the
channel, i.e., of the value of k
0
L.
In practice, in a realistic physical system, such an effect is
limited by finite losses in the ENZ material and/or by dielec-
tric breakdown. In Fig. 2, we illustrate the effect of losses in
the ENZ material. The simulations were obtained for a struc-
ture with L
1
=L
2
=a
ch
=0.1a and L =1.0a. The ENZ material
is characterized by a Drude-type model with relative permit-
tivity =1−
p
2
/
共
+i⌫兲, where
p
is the plasma frequency
and ⌫ is the collision frequency 共rad/s兲. In the simulations,
we have taken
p
a/ c =
/ 2. Note that at
=
p
, the permit-
tivity of the channel is given by ⬇i⌫/
p
. In this simula-
tion, the effect of losses in the metallic walls was neglected
and will be discussed later in the paper.
The results of Fig. 2 confirm that at
=
p
, the wave does,
in fact, tunnel through the narrow channel, especially when
dielectric losses are small, and that the transmission is still
quite significant for moderate losses. In Fig. 2, we also show
共dashed line兲 the transmission coefficient when the U-shaped
channel is empty 共i.e., filled with air兲. It is seen that at
=
p
, the wave is unable to propagate around the obstacle and
is strongly reflected at the interface. It is worth noting that
the transmissivity of the unfilled channel can be quite signifi-
cant around
/
p
⬇1.5. This happens due to a geometrical
resonance characteristic of the U-shaped geometry, i.e., for
certain very specific frequencies related with the very precise
values of L
1
, L
2
, L, and a
ch
, it may be possible to squeeze
energy through the U-shaped channel, even though it is filled
with air 共Fabry-Perot-type transmission兲. However, such an
effect is conceptually very different from the effect that can
be obtained using a ENZ material. In fact, for a channel filled
with a ENZ material, the energy can, in principle, be
squeezed through the channel independent of its specific ge-
ometry 共e.g., the exact electrical length of the channel兲.
Moreover, Eq. 共1兲 predicts that provided A
p
/ a is kept small,
the transmitted power is nearly unchanged, even if the pre-
cise shape of the channel is radically modified. Thus, in that
regard, ENZ materials are indeed unique solutions to squeeze
energy through obstructed paths because they effectively cre-
ate a zero-order resonance 共electrical length of the channel is
zero兲 that enhances the transmission of energy, independent
of the exact geometry of the transition. We also note that
below the plasma frequency 共in particular, for Re兵其⬇−1,
i.e., around
/
p
⬇0.7兲, the transmission can also be greatly
enhanced due to the excitation of “quasistatic” localized
resonances 共local plasmon resonance兲 characteristic of sys-
tems with objects with negative permittivity.
15
These reso-
nances are further analyzed in the Appendix.
It is also interesting to characterize the behavior of the
ENZ material in a waveguide configuration with a 90° bend.
The geometry is illustrated in the upper panel of Fig. 3. The
incoming wave propagates along the negative y direction.
The distance between the parallel metallic plates is a 共in both
the input and output waveguides兲. The transition between the
two waveguides is filled with two thin perpendicular 共and
connected兲 ENZ layers with thickness a −a
imp
. In the simu-
lations, we assumed that a
imp
=0.9a and that the plasma fre-
quency of the ENZ material is such that
p
a/ c =3
/ 4. In the
lower panel of Fig. 3, we plot the S
21
parameter 共transmis-
sion coefficient兲 as a function of frequency. It is seen that
around
=
p
, the wave transmission is greatly enhanced as
compared to the case in which the transition is unfilled
共empty channel兲. The upper panel of Fig. 3 depicts the real
part of Poynting vector lines, clearly showing that the ENZ
material forces the Poynting vector lines to bend and follow
the path defined by the shape of the ENZ transition.
At this point, it is interesting to discuss why ENZ mate-
rials may, in fact, help enhancing transmission through nar-
0.25 0.5 0.75 1 1.25 1.5 1.75
0.2
0.4
0.6
0.8
1
Empty
channel
Normalized frequency,
p
/
p
=0.001
/
p
=0.05
/
p
=0.01
FIG. 2. 共Color online兲 Amplitude of the transmission coefficient
共S
21
parameter兲 as a function of normalized frequency for the
U-shaped ENZ transition depicted in Fig. 1 and different values of
the losses ⌫ /
p
. The dashed line represents the transmission coef-
ficient when the U-shaped transition is filled with air 共empty
channel兲.
2
2
2.0 10
1.2 10
7.2 10
4.2 10
2.5 10
1.4 10
6.7
2.5
0.0
2
/Wm
0.2 0.4 0.6
0.8
1 1.2
0.2
0.4
0.6
0.8
1
Normalized frequency,
p
a
Empty
channel
/
p
=0.001
/
p
=0.05
/
p
=0.01
Waveguide
walls
a
imp
x
y
z
FIG. 3. 共Color online兲 Upper panel: Real part of the Poynting
vector lines in a waveguide with a 90° bend filled with a ENZ
material at
=
p
and negligible losses. Lower panel: Amplitude of
the transmission coefficient 共S
21
parameter兲 as a function of normal-
ized frequency and different values of ⌫ /
p
. The dashed line rep-
resents the transmission coefficient when the 90° bend is filled with
air 共empty channel兲.
THEORY OF SUPERCOUPLING, SQUEEZING WAVE… PHYSICAL REVIEW B 76, 245109 共2007兲
245109-3
