2702 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 50, NO. 12, DECEMBER 2002
Planar Negative Refractive Index Media Using
Periodically L–C Loaded Transmission Lines
George V. Eleftheriades, Senior Member, IEEE, Ashwin K. Iyer, Student Member, IEEE, and Peter C. Kremer
1
Abstract—Recent demonstrations of negative refraction
utilize three-dimensional collections of discrete periodic
scatterers to synthesize artificial dielectrics with simultaneously
negative permittivity and permeability. In this paper, we propose
an alternate perspective on the design and function of such
materials that exploits the well-known L–C distributed network
representation of homogeneous dielectrics. In the conventional
low-pass topology, the quantities L and C represent a positive
equivalent permeability and permittivity, respectively. However,
in the dual configuration, in which the positions of L and C are
simply interchanged, these equivalent material parameters
assume simultaneously negative values. Two-dimensional
periodic versions of these dual networks are used to demonstrate
negative refraction and focusing; phenomena that are
manifestations of the fact that such media support a propagating
fundamental back-ward harmonic. We hereby present the
characteristics of these artificial transmission-line media and
propose a suitable means of implementing them in planar form.
We then present circuit and full-wave field simulations
illustrating negative refraction and focusing, and the first
experimental verification of focusing using such an
implementation.
Index Terms— Artificial dielectrics, backward waves, focusing,
left-handed media (LHM), metamaterials, negative permeability,
negative permittivity, negative refractive index, periodic
structures.
I. I
NTRODUCTION
N THE LATE 1960s, Veselago proposed that materials with
simultaneously negative permittivity and permeability are
physically permissible and possess a negative index of refraction [1].
Veselago termed these Left-Handed Media (LHM), because the
vectors E, H, and k would form a left-handed triplet instead of a
right-handed triplet, as is the case in conventional, Right-Handed
Media (RHM). His conceptual exploration of this phenomenon
revealed that, through negative refraction, planar slabs of such media
would cause light or electromagnetic radiation to focus in on itself.
Although it has been known for some time that arrays of thin metallic
wires, by virtue of their collective plasma-like behaviour can produce
an effectively negative dielectric permittivity, it was not clear as to
Manuscript received April 4, 2002; revised August 7, 2002. This work was
supported by the Natural Sciences and Engineering Research Council of
Canada.
The authors are with The Edward S. Rogers Sr. Department of Electrical
and Computer Engineering, University of Toronto, Toronto, ON, Canada M5S
3G4 (e-mail gelefth@waves.utoronto.ca; iyer@waves.utoronto.ca;
claus@waves.utoronto.ca).
Digital Object Identifier 10.1109/TMTT.2002.805197
how to produce a simultaneously negative permeability. The recent
development of the Split-Ring Resonator (SRR) by Pendry et al. [2]
was successful in this effort. Subsequently, three-dimensional (3-D)
electromagnetic artificial dielectrics (metamaterials), consisting of an
array of resonant cells, each comprised of thin wire strips and split
ring-resonators, were developed to synthesize the simultaneously
negative permittivity and permeability required to produce a negative
refractive index, and, indeed, successfully demonstrated reversed
refraction [3], [4]. However, the metamaterials presented in [3] and
[4] are bulky 3-D constructions, which are difficult to adapt for
RF/microwave device and circuit applications. They consist of
loosely coupled unit cells that rely explicitly on the split-ring
resonance to synthesize a negative magnetic permeability.
Consequently, the structures can achieve a negative index of
refraction only within a narrow bandwidth. Furthermore, when
applied to wireless devices at RF frequencies the split ring-resonators
have to be scaled to larger dimensions, which, in turn would make
such devices less compact. It should also be noted that, to date, there
has been no experimental verification of the property of focusing
inherent to LHM.
This paper offers a fresh perspective on the operation of LHM
that enables the design of a new class of metamaterials to synthesize a
negative refractive index. The proposed structures go beyond the
wire/SRR composites of [2]–[4] in that they do not rely on SRRs to
synthesize the material parameters, thus leading to dramatically
increased operating bandwidths. Moreover, their unit cells are
connected through a transmission-line network and they may,
therefore, be equipped with lumped elements, which permit them to
be compact at frequencies where the SRR cannot be compact. The
flexibility gained through the use of either discrete or printed
elements enables the proposed planar metamaterials to be scalable
from the megahertz to the tens of gigahertz range. In addition, by
utilizing varactors instead of capacitors, the effective material
properties can be dynamically tuned. Furthermore, the proposed
media are planar and inherently support two-dimensional (2-D) wave
propagation. Therefore, these new metamaterials are well suited for
RF/microwave device and circuit applications.
The remainder of this paper is divided into four sections. Section II
provides an introduction to the theoretical foundations describing left-
handed behaviour in L–C loaded transmission lines, drawing
distinctions between the distributed and periodic cases. Section III
outlines a method of designing practical periodic L–C loaded
transmission lines to synthesize left-handed properties. Section IV
presents microwave circuit and full-wave field simulations using
these designs. Section V presents the design of planar LHM and the
first experimental verification of focusing in LHM.
I
0018-9480/02$17.00 © 2002 IEEE
2703 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 50, NO. 12, DECEMBER 2002
Fig. 1: Unit cell for a 2-D distributed L-C network.
II. THEORY
A. Distributed Network Approach
It is well known that dielectric properties like permittivity and
permeability can be modelled using distributed L–C networks. The
standard case is the use of periodic versions of these distributed L–C
networks to represent free space so that the characteristic impedance of
the network is equal to the free-space wave impedance (377Ω). In this
case, the per-unit-length capacitance and inductance can be directly
related to the free-space permittivity and permeability, respectively. To
illustrate how these material parameters may be effectively derived
from a network of generic distributed series impedances and shunt
admittances, consider the unit cell pictured in Fig. 1. The 2-D
telegrapher’s equations representing the distributed structure of Fig. 1
can be expressed as
Zi
x
v
Zi
z
v
x
y
Z
y
−=
∂
∂
−=
∂
∂
(1)
and
.Yv
x
i
z
i
y
x
z
−=
∂
∂
+
∂
∂
(2)
Combining (1) and (2) yields
ZYv
z
v
x
v
y
yy
−±==+
∂
∂
+
∂
∂
ββ
0
2
2
2
2
2
(3)
where
β
is the propagation constant.
It is now appropriate to map field components to the voltages and
currents in the medium. In a thin homogeneous isotropic medium, a
quasi-static transverse magnetic (TM
y
) solution (i.e., assuming a weak
y-variation) maps v
y
to E
y
(by means of the definition of potential
difference) i
x
to H
z
(by means of Ampere’s law) and i
z
to –H
x
and, thus,
(1) and (2) correspond to the following field equations:
xS
y
zS
y
Hj
z
E
Hj
x
E
ωµωµ
+=
∂
∂
−=
∂
∂
(4)
and
ys
z
x
Ej
x
H
z
H
ωε
+=
∂
∂
−
∂
∂
(5)
yielding the effective material parameters as
ω
µωµ
j
Z
Zj
SS
=⇒=
(6)
and
.
ω
εωε
j
Y
Yj
SS
=⇒=
(7)
For the case of free space, we require
µ
S
=
µ
0
and
ε
S
=
ε
0
, and we
choose
Z = j
ωµ
0
and Y = j
ωε
0
. This implies a medium in a low-pass
topology with
L =
µ
0
(H/m) and C =
ε
0
(F/m), both of which are
positive real quantities. The corresponding propagation constant
reduces to that of a standard transmission line (in this case, air-filled),
00
εµωωβ
==−= LCZY
(8)
The resulting phase and group velocities at low frequencies are parallel
and given by
g
v
LC
v =
∂
∂
====
−1
00
11
ω
β
εµ
β
ω
φ
. (9)
That the phase and group velocities are both positive results from the
choice of the positive root in (8). The choice is arbitrary since it serves
only to select one of two solutions related through a space-reversal or,
equivalently, one of the two branches of the corresponding dispersion
diagram. That is, in RHM, the phase lags in the direction of the
positive group velocity (which, in this case, is parallel to the Poynting
vector), a fact that is invariant to the sign selected for the root. In the
case of RHM, (9) implies that the refractive index is positive. Indeed,
the refractive index can be defined as
00
εµ
φ
LC
v
c
n ==
(10)
which is positive and equal to unity for the case of free space.
Furthermore, the characteristic impedance of the network is exactly
equal to the free space wave impedance, as expected, and as follows:
.377
0
0
00
Ω≅===
ε
µ
η
C
L
Z
(11)
Naturally, this concept need not be limited to free space; rather, it is
applicable to any other homogeneous isotropic dielectric, since we are
free to specify the choice of
L and C. This prompts us to examine the
scope of distributed
L–C networks for reproducing unique material
parameters, including those of LHM. Specifically, Veselago’s
postulation of a negative permittivity and permeability prompts the
question of whether the
L and C parameters in a network
representation can also be made negative. Naturally, from an
impedance perspective, imposing a negative
L and C essentially
exchanges their inductive and capacitive roles, so that the series
inductor becomes a series capacitor, and the shunt capacitor becomes a
shunt inductor. The emerging dual structure is easily recognized as
having the topology of a high-pass filter network. The resulting
equivalent permittivity and permeability can be shown to be negative,
using arguments similar to those of the free space case. Consider such
a dual high-pass topology, with distributed
2704 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 50, NO. 12, DECEMBER 2002
series capacitance
C′ (F⋅m) and shunt inductance L′ (H⋅m). Using (6)
and (7), we arrive at the following results:
C
Cj
Zj
SS
′
−=⇒
′
==
2
11
ω
µ
ω
ωµ
(12)
L
Lj
Yj
SS
′
−=⇒
′
==
2
11
ω
ε
ω
ωε
, (13)
which show that the equivalent material parameters are indeed
negative for a dual
L–C network. The corresponding propagation
constant is related inversely to the frequency,
C
L
ZY
′′
−=−−=
ω
β
1
. (14)
In this case, the phase and group velocities are antiparallel, and are
given by
CLvCLv
g
′′
+=
∂
∂
=
′′
−==
−
2
1
2
,
ω
ω
β
ω
β
ω
φ
(15)
where the choice of the negative root in (14) has assured a positive
group velocity (in this case also parallel to the Poynting vector).
This implies that, in LHM, the phase
leads in the direction of the
group velocity, or power flow, and the index of refraction should
accordingly be negative. Indeed, the corresponding refractive index
is negative as follows:
.
1
00
00
2
εµ
εµ
εµω
φ
SS
CL
v
c
n =
′′
−==
(16)
This result further justifies the interpretation of (12) and (13) as
equivalent effective material parameters. It is also clear that the
relationship between the network characteristic impedance and
equivalent wave impedance is preserved according to the following
equation:
S
S
C
L
Z
ε
µ
η
==
′
′
=
0
(17)
It should be noted that, although the parameters themselves are
negative, their spectral derivatives remain positive due to the
negative, inverse-square dependence on frequency, so that the total
stored time-averaged energy [5], expressed by the following:
22
)()(
HE
ω
ω
ε
ω
ω
µ
∂
∂
+
∂
∂
=
SS
W
(18)
remains positive, and causality is not violated.
The results indicated by (15) are familiar because they seem to
describe the well-known phenomenon of the backward wave, in
which the Poynting and wave vectors are oppositely directed [6].
Indeed, Veselago has alluded to the presence of backward waves in
his definition of left handedness [1]. Thus, the above development
seems to suggest that there exists a multitude of materials capable
of demonstrating negative refraction phenomena solely as a
consequence of the fact that they support backward waves.
Fig. 2: Phase matching at the interface between a right-handed medium M1
and a generic medium M2.
We shall now use these backward waves to assert that the
proposed dual
L–C networks do demonstrate negative refraction.
Consider a two-medium interface, where medium 1 (M1) exhibits
a positive index of refraction and medium 2 (M2) is regarded as
generic for the moment. The complete arrangement is depicted in
Fig. 2. A plane wave originating in M1 and impinging on the
interface with wave vector k
1
establishes a refracted wave in M2
with wave vector k
2
such that the tangential wave-vector
components k
1t
and k
2t
are equal. This phase matching is a
generalized form of Snell’s Law, and that it specifies constraints
only on the tangential components of the wave vectors presents
some interesting consequences. There is, in fact, the freedom of
two possibilities for the normal component of k
2
: the first and usual
case, in which k
2
is directed away from the interface, and the
second case, usually reserved for reflected waves, in which k
2
is
directed towards the interface. These two cases are represented as
Case 1 and Case 2 in Fig. 2. We may further invoke the constraints
imposed by the conservation of energy, so that normal components
of the Poynting vectors S
1
and S
2
remain in the positive x-direction
through both media.
It is clear that Case 1 depicts the familiar situation in which M2
is a conventional positive-index medium (depicted as denser than
M1 in Fig. 2). However, if M2 is a medium supporting propagating
backward waves (LHM), then it is, indeed, true that the wavevector
k
2
must be directed opposite to the Poynting Vector S
2
, so that
power is propagated along the direction of phase advance.
Therefore, refraction in media that support backward waves must
be described by the second case, for which it may be argued that
power is refracted through an effectively negative angle, which,
indeed, implies a negative index of refraction.
According to [1] and [7], electromagnetic waves from a point
source located inside a RHM can be focused by a planar LHM
slab, as depicted in Fig. 3. Geometrical optics indicates that, for a
planar interface, the focal point is generally dependent on the
incident angle. For the case of negative-index optics, some
elementary manipulations of Snell’s Law at the first interface
depicted in Fig. 3 show that the distance of the internal focal point
from the interface is given by the following:
()
(
)
()
inc
refr
RHM
LHM
refrinc
n
n
sf
θ
θ
θθ
cos
cos
,
11
⋅⋅=
(19)
where
s
1
is the distance of the source from the interface, as depicted
in Fig. 3, and
θ
inc
and
θ
refr
are the absolute values of the angles of
incidence and refraction, respectively. From the
2705 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 50, NO. 12, DECEMBER 2002
Fig. 3: Internal and external focusing using a LHM slab of thickness d.
second interface, the distance of the external focal point can be
given by equation (20),
()
()
−⋅= 1
,
,
1
12
refrinc
refrinc
f
d
sf
θθ
θθ
(20)
where
d is the slab thickness. Consequently, both the internal and
external focal points generally depend on the angle of incidence,
suggesting inherent spherical aberration. However, it is clear that
they may be made angle-independent in the special case for which
the relative refractive index is –1. In this case,
θ
inc
and
θ
refr
become
equal, and the internal and external focal points become unique.
This observation seems to corroborate the findings of [8], which
state that, unless the relative refractive index is –1, only a paraxial
focus can be defined.
B. L-C Loaded Transmission Line Network Approach
The previous discussion has confirmed that distributed L–C
networks possessing a high-pass topology support backward
waves; a property that enables these media to demonstrate
negative refraction and focusing phenomena, and therefore,
permits us to refer to them as left handed (in Veselago’s
terminology). However, any practical implementation of these
structures at RF/microwave frequencies must be a periodic one,
and must therefore possess a non-vanishing cell dimensionality.
Herein, we consider a practical planar design that periodically
loads a host transmission line network with discrete reactive
elements. The finite-length transmission lines provide the needed
dimensionality, but at the same time perturb the homogeneity of
the distributed system to the extent that the previously described
treatment requires modification.
A rigorous understanding of the behaviour of the proposed
LHM structures can be gained through their dispersion
characteristics. Only the one-dimensional (1-D) case is treated
here for simplicity, but the results can be easily extended to two
dimensions. The desired dispersion relation can be obtained
through standard periodic analysis based upon the 1-D,
symmetric unit cell of Fig. 4. Here,
Z and Y represent discrete
loading elements since the cell dimension
d is explicitly provided
for by the transmission line segments (
Z
0
, k).
In the previous discussion on distributed
L–C networks, the
frequency response permitted propagation at all frequencies.
However, when these finite-dimension cells are periodically
cascaded, the corresponding dispersion diagram develops a band
structure. The propagation constant
β
can be determined
Fig. 4: Unit cell for the one-dimensional periodic L-C network featuring the
host transmission line medium (Z
0
,k).
using the standard procedure for 1-D periodic analysis of microwave
networks [9], [10]. The resulting dispersion relation is
θ
θ
θβ
sin
22
cos
2
coscos
00
2
+++=
Y
Y
Z
Z
j
ZY
d
(21)
where we have used
θ
=kd. It is clear from (21) that when the period of
the structure is infinitesimally small, i.e., cos
θ
→ 1 and sin
θ
→ 0, the
effects of the transmission lines disappear and the medium becomes
continuous. This corresponds to the case of a high-pass filter, when
operating well within the filter passband. Retaining only the first two
terms in the Taylor expansion of the resulting expression, we may
determine a first-order approximation for the phase shift per unit cell
according to
()
ZYd
ZY
d
−−=⇒+≅−
β
β
2
1
2
1
2
, (22)
yielding the previously obtained result for the distributed
L–C network
as given by (14). This result describes the phase shift incurred by
propagation through the periodic structure over a distance
d. Making
the high-pass substitutions
00
1
,
1
Lj
Y
Cj
Z
ωω
==
(23)
with
L
0
and C
0
being the discrete loading elements, (22) gives
.
1
00
CL
ZYd
ω
β
−=−−=
(24)
The reader will recall that, while (14) regards
L′ and C′ to be
distributed parameters (with units henry–m and farad–m, respectively),
the equivalence between (14) and (24) is established if the cell
dimensionality
d in (24) is absorbed into L
0
and C
0
. It should also be
noted that, once again, the negative square root is taken in (24) to
ensure a positive group velocity as given by (15). The corresponding
effective index of refraction is given by
dCL
d
ZY
dk
d
n
0000
2
000
1
εµω
εµω
β
−=
−−
==
(25)
where, again, the equivalence between (16) and (25) is established
through the cell dimension
d.
The above development has assumed a unit cell with small
dimensions, so that the loading elements can still be treated as
distributed parameters. However, in general, the effect of the discrete,
periodic loading of the host transmission line medium cannot be
neglected. In the full dispersion relation given by (21),
2706 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 50, NO. 12, DECEMBER 2002
Fig. 5: Dispersion relation for periodically L–C loaded LHM using L
0
=13.2nH,
C
0
=0.5pF, and
θ
=0.25 radians at 1.5GHz, indicating Bragg frequency f
b
and
stopband limits f
C,1
and f
C,2
.
it is evident that
β
can become periodically complex, forbidding
propagation at these frequencies. Choosing
L
0
=13.2nH, C
0
=0.5pF,
and
θ
=0.25 radians at 1.5GHz, the full dispersion relation of (21) is
depicted in Fig. 5. As shown, passbands are observed, separated by
a finite stopband. The lowest cut-off (Bragg) frequency associated
with this dispersion curve (indicated in Fig. 5 by a dashed line) is
approximately determined from the condition
β
d=
π
to be
.
4
1
00
CL
f
b
π
=
(26)
Employing a higher level of precision in the approximation of
(21) than (22) permits the analytic determination of the stopband
limits
.
222
1cos
00
2
θ
θ
β
+++−≅
Y
Y
Z
Z
j
ZY
d
(27)
Making the substitutions of (23), with
θ
=kd=
ω
d/v (where v is the
phase velocity in the host medium), and setting
β
=0 in (27), the
solution of the resulting quadratic equation in
ω
yields the desired
cut-off frequencies (see Fig. 5) as follows:
dYL
v
f
dZC
v
f
cc
00
2,
00
1,
2
1
,
2
1
ππ
==
(28)
Note that both cutoff frequencies in (28) tend to infinity as the cell
dimensionality d approaches zero, thus arbitrarily increasing the
bandwidth of the first LH passband (see Fig. 5). Furthermore,
closing this stopband by equating these two cutoff frequencies
yields the matching condition
0
0
0
C
L
Z =
(29)
which suggests that the width of the stopband may be controlled by
adjusting the mismatch between the characteristic impedances of
the host medium and the loading. It is clear from the lowest
passband of this dispersion diagram (the left-hand-side branch in Fig.
5) that the phase and group velocities are oppositely directed,
implying that this periodic version of the previous distributed
structure also supports backward waves. Furthermore, as was
explained earlier, the bandwidth of this region can be controlled and
substantially widened by selecting a sufficiently small periodicity d.
The reader will also recognize from Fig. 5 that successive passbands
alternately exhibit left or right handedness (using Veselago’s
terminology) since the concavity of the dispersion surface changes
with frequency. Although all periodic structures support an infinite
number of forward- and backward-wave spatial harmonics, [6], [9],
the dispersion diagram of Fig. 5 shows that the periodic dual L–C
network supports a backward-wave fundamental spatial harmonic
(extending over the region –
π
<
β
d < 0).
III. D
ESIGN
In [11], a procedure was described for the design of high-pass L–C
networks to demonstrate negative refraction. This procedure will be
presented in detail herein.
The demonstration of negative refraction requires the realization of
an interface similar to that depicted in Fig. 2, so it is necessary to
design both LHM and RHM using periodic L–C arrays. The final
design therefore consists of two large periodic arrays of 2-D L–C unit
cells, each as shown in Fig. 1. One array is constructed using unit cells
configured in a low-pass topology to simulate free-space conditions,
and the second array is designed according to the proposed dual L–C
topology. To prevent reflections at the interface, the characteristic
impedances of both media are designed to be equal.
As a starting point to the design, the dimensionless periodic L–C
unit cell described by (22) may be employed. However, it is clear from
(24) and (25) that, since the reactive elements included in the unit cells
are discrete elements, the propagation constant and, therefore, the
refractive index, is dependent on the periodicity d. Consequently, it is
not possible to specify an index of refraction unless the manner of
distribution of the discrete elements over d is specified. However, this
constraint may be alleviated by specifying a relative index of refraction
between the two media, or equivalently the ratio of the phase incurred
in each medium through the period d,
β
LHM
d/
β
RHM
d. The arrays may
therefore be designed by means of circuit simulations using discrete
elements, since all dependence on dimension is removed in the ratio.
Choosing a relative refractive index using the ratio
β
LHM
d/
β
RHM
d
and a suitable operating frequency, the LHM and RHM capacitors and
inductors are specified according to equations (24) and (17), and will
hereinafter be referred to as (C
LHM
, L
LHM
) and (C
RHM
, L
RHM
),
respectively. The corresponding cut-off frequencies are then
determined and verified to enclose the chosen operating frequency.
Once this starting point is established, it is necessary to account for
the effect of the finite-length transmission lines comprising the host
medium. The lengths must be chosen to be small enough that the
distributed parameters of the lines do not supersede the effects of the
loading elements. The process of compensation for the presence of
lines will now be described in detail.
The presence of the discrete, periodic loading renders the results of
(24) and (25) ineffective in predicting the behaviour of the L–C loaded
transmission line medium. It is, therefore, neces-
