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10 IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS, VOL. 1, 2002
An Idea for Thin Subwavelength Cavity
Resonators Using Metamaterials With Negative
Permittivity and Permeability
Nader Engheta, Fellow, IEEE
Abstract—In this letter, we present and analyze theoretically
some ideas for thin one-dimensional (1-D) cavity resonators in
which a combination of a conventional dielectric material and a
metamaterial possessing negative permittivity and permeability
has been inserted. In this analysis, it is shown that a slab of
metamaterial with negative permittivity and permeability can
act as a phase compensator/conjugator and, thus, by combining
such a slab with another slab made of a conventional dielectric
material one can, in principle, have a 1-D cavity resonator whose
dispersion relation may not depend on the sum of thicknesses of
the interior materials filling this cavity, but instead it depends on
the ratio of these thicknesses. In other words, one can, in principle,
conceptualize a 1-D cavity resonator with the total thickness far
less than the conventional
/2. Mathematical steps and physical
intuitions relevant to this problem are presented.
Index Terms—Cavity resonator, metamaterials, negative index
of refraction, negative permeability, negative permittivity, phase
compensator, phase conjugation.
I. INTRODUCTION
I
N the past several decades, the electromagnetic (EM)
properties of complex media have been the subject of
research study for many research groups [1]–[16]. Several
types of EM complex media such as chiral materials, omega
media, bianisotropic media, local, and nonlocal media to name
a few, have been studied. Recently, the idea of composite
materials in which both permittivity and permeability possess
negative values at certain frequencies has gained considerable
attention [17]–[21]. In 1967, Veselago theoretically investi-
gated plane wave propagation in a material whose permittivity
and permeability were assumed to be simultaneously negative
[22]. His theoretical study showed that for a monochromatic
uniform plane wave in such a medium, the direction of the
Poynting vector is antiparallel with the direction of phase
velocity, contrary to the case of plane wave propagation in
conventional simple media. Recently, Smith
et al. constructed
such a composite medium for the microwave regime and
demonstrated experimentally the presence of anomalous re-
fraction in this medium [17], [18], [20], [21]. It is also worth
noting that previous theoretical study of EM wave interaction
with omega media reveals the possibility of having negative
Manuscript received December 20, 2001; revised February 11, 2002. This
work was presented in part at the International Conference on Electromagnetics
in Advanced Applications (ICEAA), Torino, Italy, September 10–14, 2001.
The author is with the Department of Electrical Engineering, University of
Pennsylvania, Philadelphia, PA 19104 USA (e-mail: engheta@ee.upenn.edu).
Digital Object Identifier 10.1109/LAWP.2002.802576
permittivity and permeability in omega media for a certain
range of frequencies [23]. For metamaterials with negative
permittivity and permeability, several names and terminologies
have been suggested such as “left-handed” media [17]–[22],
media with negative refractive index [17]–[22] “backward
media” (BW media) [24], “double negative metamaterials”
[25], [26], to name a few. The anomalous refraction at the
boundary of such media and the fact that for a plane wave
the direction of the Poynting vector is antiparallel with the
direction of phase velocity, provide us with features that can
be advantageous in design of novel devices and components.
Recently, we introduced and presented in a symposium [27]
one of our ideas for a compact cavity resonator. Here in this
letter, we describe the details of this idea and the mathematical
steps behind our analysis.
A. Metamaterials With Negative
and as Phase
Compensators/Conjugators
When a lossless metamaterial possesses negative real permit-
tivity and permeability at certain frequencies, the index of re-
fraction in such a medium attains real values. So as theoreti-
cally predicted by Veselago, the EM wave can propagate in such
a medium [22]. However, for a monochromatic uniform plane
wave in such a medium the phase velocity is in the opposite di-
rection of the Poynting vector.
Consider a slab of conventional lossless material with real
permittivity
, real permeability , and the index of
refraction
, where and are the per-
mittivity and permeability of the free space. Here,
is taken
to be a positive real quantity. The slab is infinitely extent in the
- plane and has the thickness along the axis. We tem-
porarily assume that the intrinsic impedance of the dielectric
material
is the same as that of the outside re-
gion
, i.e., , but its refractive index is
different from that of outside, i.e.,
. (We will soon re-
move the first part of this assumption.) Let us assume that a
monochromatic uniform plane wave is normally incident on this
slab. The wave propagates through the slab without any reflec-
tion (because for now we are still assuming
). As this
wave traverses this slab, the phase at the end of the slab is obvi-
ously different from the phase at the beginning of the slab by the
amount
, where . Now, consider a slab of
a lossless metamaterial with negative real permittivity and per-
meability, i.e.,
and at certain frequencies. For
this slab, the index of refraction is a real quantity denoted by
1536-1225/02$17.00 © 2002 IEEE
ENGHETA: THIN SUBWAVELENGTH CAVITY RESONATORS WITH NEGATIVE PERMITTIVITY AND PERMEABILITY 11
Fig. 1. A two-layer structure in which the left layer is assumed to be a
conventional lossless dielectric material with
" >
0
and
>
0
and the
right layer is taken to be a lossless metamaterial with negative permittivity and
permeability. In the first layer, the direction of Poynting vector (
S
) is parallel
with the direction of phase velocity or wave vector (
k
), whereas in the second
layer, these two directions are antiparallel. With proper choice of ratio of
d
and
d
, one can have the phase of the wave at the left (entrance) interface to be
the same as the phase at the right (exit) interface, essentially with no constraint
on the total thickness of the structure.
. It is important to note that here we do
not need to specify any sign for the operation of the square root
appearing in the expression of
. We only need to state that
is a real quantity for the lossless metamaterial with and
for a given frequency. (Here, can, for example, be
taken to be a positive real quantity.) This slab is also infinitely
extent in the
- plane, but has a thickness of in the direc-
tion. For now, we again assume that the intrinsic impedance of
this metamaterial
is also the same as that of out-
side region, i.e.,
. We put this slab right next to the first
slab (Fig. 1). As the plane wave exits the first slab, it enters the
slab of metamaterial and finally it leaves this second slab. The
direction of powerflow(i.e., the Poynting vector) in the firstslab
should be the same as that in the second one, because the power
of the incident wave enters the first slab (without any reflection
at the first interface), traverses the first slab, exits the second in-
terface, enters the second slab and traverses it, and finally leaves
the second slab. In the first slab, the direction of the Poynting
vector is parallel with the direction of phase velocity; however,
in the second slab these two vectors are antiparallel (see Fig. 1).
Therefore, the wave vector
is in the opposite direc-
tion of the wave vector
. As a result, the phase at the
end of the second slab is different from the phase at the begin-
ning of it by the amount
. (As was mentioned above,
here is taken to be positive.) So the total phase difference
between the front and back faces of this two-layer structure is
. Therefore, whatever phase difference is de-
veloped by traversing the first slab, it can be decreased and even
cancelled by traversing the second slab. If the ratio of
and
is chosen to be , then the total phase difference
between the front and back faces of this two-layer structure be-
comes zero. (The total phase difference is not 2
,4 ,or6 .
Fig. 2. An idea for a compact, subwavelength, thin cavity resonator. The
two-layer structure discussed in Fig. 1 is sandwiched between the two
reflectors. Our analysis shows that with the proper choice of ratio of
d
over
d
, one can have a resonant cavity in which the ratio of
d
and
d
is the main
constraint, not the sum of thicknesses
d
+
d
.
But instead it is zero!) So indeed the slab of metamaterial with
and at given frequencies can act as the phase
compensator in this structure. This also resembles the process
of phase conjugation. It is important to note that such phase can-
cellation in this geometry does not depend on the sum of thick-
nesses
; rather it depends on the ratio of and . So,
in principle,
can be any value as long as satis-
fies the above condition. Therefore, even though this two-layer
structure is present, the wave traversing this structure would not
experience the phase difference. This feature can lead to several
interesting ideas in design of some devices and components.
B. Compact Subwavelength 1-D Cavity Resonators Using
Metamaterials With
and
What we described above can be used to conceptualize an ex-
citing possibility of designing a compact 1-D cavity resonator.
We can take the above two-layer structure and put two per-
fect reflectors (e.g., two perfectly conducting plates) at the two
open surfaces of this structure (Fig. 2). Here, we generalize
the problem by assuming that the intrinsic impedances of the
first layer (conventional material) and the second layer (meta-
material with
and at specific frequencies) are
not taken to be the same as
. So here, in general, we have
and . We are now interested to
solvefor solutions of Maxwell equations in this cavityresonator.
C. Formulation of the Problem
We use the Cartesian coordinate system
, where the
plane
is taken to be at the perfectly conducting plate lo-
cated at the left face of the conventional material slab shown
in Fig. 2. The other perfectly conducting plate is placed at
, which is the right face of the metamaterial slab. Since
this is assumed to be a 1-D cavity resonator, all quantities are in-
dependent of the
and coordinates. The time dependence for
the monochromatic solutions is taken to be
. Without
12 IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS, VOL. 1, 2002
loss of generality, we take the electric and magnetic field vec-
tors to be oriented along the
and direction, respectively. In
the region
, where the material is a conventional
lossless material, the electric and magnetic fields can be written
as
(1)
and in the region
, where the metamaterial
with
and is located, the fields are written as
(2)
where subscripts “1” and “2” denote the quantities in the regions
“1” and “2,” which are the conventional slab and the metama-
terial slab, respectively. It is worth emphasizing that although
in the above equations the values of indexes of refraction are
taken to be positive quantities (and, hence, no ambiguities are
introduced through the sign of the operation of square root for
), as will be seen shortly the choice of
the sign for
and will be irrelevant in the final results.
The choice of the solutions presented in (1) and (2) guarantees
the satisfaction of the boundary conditions at the perfectly con-
ducting plates at
and . To satisfy the boundary
conditions at the interface between the two slabs we should have
(3)
which leads to
(4)
In order to have a nontrivial solution, i.e., to have
and
, the determinant in (4) must vanish. That is
(5)
which can be simplified to
(6)
In the above dispersion relation, the quantities
, , , ,
and
are all generally frequency dependent. It is important to
note that the choice of sign for
and does not affect this
dispersion relation. Either choice of sign (positive or negative
sign) for
and will leave (6) unchanged. That is why we
specifically mentioned earlier that we did not need to introduce
any ambiguity regarding the choice of sign for
in our anal-
ysis here. Since the first layer is assumed to be made of a loss-
less conventional material, its permeability
is a positive real
quantity. The second layer is taken to be a lossless metamaterial
with
and . Therefore, we can write and
. Substituting these expressions in (6), we obtain
(7)
This implies that for a given frequency
,if , ,
,and ,a nontrivial one-dimensional (1-D) solution
for this cavity is obtained when the thicknesses
and satisfy
the relation
(8)
This relation does not show any constraint on the sum of thick-
nesses of
and . It rather deals with the ratio of tangent of
these thicknesses (with multiplicative constants). So, in prin-
ciple,
and can conceptually be as thin or as thick as oth-
erwise needed as long as the above ratio is satisfied. If we as-
sume that
, and are chosen such that the small-argument
approximation can be used for the tangent function, the above
relation can be simplified as
(9)
This relation shows even more clearly how
and should
be related in order to have a nontrivial 1-D solution with fre-
quency
for this cavity. So conceptually, what is constrained
here is
, not . Therefore, in principle, one can have
a thin subwavelength cavity resonator for a given frequency, if
at this frequency the second layer acts a metamaterial with neg-
ative permittivity and permeability and the ratio
satisfies
the above condition. For example, for frequency of 2 GHz, if a
metamaterial with negative permittivity of
and negative
permeability of
can be constructed as the second slab
and if the conventional material slab is assumed to be air with
and ,then and and,thus, the required ratio of
over should be . If, in principle, this metama-
terial slab can be made thin for this frequency, e.g.,
/10,
where
is the free-space wavelength of operation, then the air
slab should be made with thickness
/20. Thus, the total
thickness of such a thin cavity would be
/20,
which for this example of 2-GHz frequency of operation would
be 2.25 cm! This is, of course, thinner than the conventional air
cavity size of
/2, which would be 7.5 cm for 2 GHz.
The electric and magnetic field expressions for the nontrivial
solutions in this 1-D cavity are given as
(10)
where
and .
It is worth noting that if both layers 1 and 2 had been made
of two conventional lossless dielectric materials, the form of
the dispersion relation in (6) would have remained unchanged.
