OPTO-ELECTRONICS REVIEW 14(3), 167–177
DOI: 10.2478/s11772-006-0022-0
Metamaterial-based source and scattering enhancements: from
microwave to optical frequencies
Invited papers
R.W. ZIOLKOWSKI*
University of Arizona, Department of Electrical and Computer Engineering, 1230 E. Speedway Blvd.
Tucson, Arizona, 85721-0104 USA
A brief review of metamaterial applications to source and scattering problems in the microwave and optical frequency regimes is given. Issues associated with modelling these systems are discussed. Electrically small radiating and scattering systems are emphasized. Single negative, double negative, and zero-index versions of these metamaterial-based systems are introduced that provide a means to manipulate their efficiency, bandwidth, and directivity characteristics.
Keywords: metamaterials, complex media, computational electromagnetics, electrically small antennas, lasers, scattering.
1. Introduction
Metamaterials (MTMs) are artificial media characterized
by constitutive parameters generally not found in nature.
The “meta” refers to the resulting effective properties
whose electromagnetic responses are “beyond” those of
their constituent materials. It has been discussed how
metamaterials can have counterintuitive properties in their
interactions with electromagnetic waves that can be exploited for use in a wide variety of guiding and radiating
structures [1–5]. In addition to these unusual physical effects, the large interest in metamaterials stems from their
potential for providing media whose permittivity and/or
permeability values can be tailored for specific applications. Unfortunately, the possible variations in the permittivities and permeabilities of materials available in nature,
e.g., their magnitudes, losses, frequency dependencies, and
signs, are limited. Even traditional composite materials
formed from combinations of naturally occurring substances have limited responses. More often than not, one must
build a device or system based upon such a restricted set of
material choices. If there was a means of designing and
building materials with permittivity and permeability values that can be tailored to a specific application, the potential benefit to electromagnetic systems would be enormous.
Materials may be categorized by their constitutive parameters e and µ according to the diagram shown in Fig. 1. If
both the permittivity and permeability have positive real
parts as in the first quadrant of Fig. 1, as most of the materials in nature do, they will be called “double positive (DPS)”
media. In contrast, if both of these quantities are negative, as
in the third quadrant of Fig. 1, they will be called “double-negative (DNG)”. The materials with one negative pa*e-mail:
[email protected]
Opto-Electron. Rev., 14, no. 3, 2006
rameter, quadrants two and four, will be called “single-negative (SNG)”. If the permittivity is negative, as in the second quadrant, these SNG materials will be called “epsilon-negative (ENG)”. The ionospheric plasma layer exhibits
this behaviour at AM radio frequencies while natural plasmonic materials, e.g., noble metals, do at optical frequencies. If the permeability is negative, as in the fourth
quadrant, they will be called “mu-negative (MNG)”. Ferromagnetic materials exhibit this behaviour in the VHF and
UHF regimes. If both e and µ are zero or very close to zero,
the materials will be called “zero index”. This nomenclature
will be used throughout this discussion. We note that the realization of SNG materials may be relatively easier than that of
DNG materials. Therefore, many recent efforts have been
aimed at exploring if SNG media can be used to achieve
some of the applications originally proposed for DNG media.
To date most metamaterials have been realized as composite artificial media formed by periodic arrays of dielectric
or metallic inclusions in a host substrate. The effective me-
Fig. 1. Classification of metamaterials by the real parts of their
constitutive parameters, i.e., their permittivity and permeability.
R.W. Zió³kowski
167
Metamaterial-based source and scattering enhancements: from microwave to optical frequencies
dia that have been used to produce DNG and SNG effects
have unit cells whose characteristic sizes are deffective medium £
l/10 The inclusions act like artificial molecules that produce
local polarization (artificial electric dipole) and magnetization (artificial magnetic dipole) fields; and, hence, they modify the permittivity and permeability of the substrate in
which they are embedded. Both volumetric and planar versions of DNG and SNG metamaterials have been achieved
[3–5]. In contrast, the distances between the inclusions in
electromagnetic bandgap (EBG) structures, which have also
been used to achieve DNG effects [3], are on the order of
dEBG ~l/2. They achieve their properties through Bragg reflections. The effective medium constructs are, thus, generally smaller than their EBG counterparts. This aspect is critical when considering applications that require highly
subwavelength metamaterial regions to achieve the desired
operating behaviours.
Some of the attractive properties of metamaterial-based
antenna systems will be reviewed briefly. Particular emphasis will be given to electrically small metamaterialbased resonant radiating systems that arise by designed
pairings of conventional materials with metamaterials and
that can be used to overcome conventional physical limitations. It will also be shown that reciprocal electrically small
metamaterial-based scattering configurations exist [6] and
have similar behaviours. It will be demonstrated that both
the source and scattering configurations can be engineered
to achieve unusual operating characteristics that may have
significant impacts on microwave and optical technologies.
2. Simulations
Maxwell’s equations allow for a wide variety of interesting
physical phenomena. Many of these occur because of the
properties of the materials involved in the application. For
instance, a change in the index of refraction from one region of space to another is responsible for phenomena such
as refraction and total internal reflection. Periodic media
changes allow or prevent the propagation of certain wave
vectors and frequencies. Wave interactions can stimulate
energy exchanges with active media. Basic phenomena
such as these are responsible for devices such as lenses, dielectric guides, filters, and amplifiers. These devices are
used, for example, in electromagnetic wave systems from
UHF to microwave to millimetre wave to terahertz wave to
optical frequencies.
Electromagnetic devices are modelled by solving
Maxwell’s equations with these material properties properly incorporated into the problem definition. To achieve
negative values of the constitutive parameters e and µ,
metamaterials must be dispersive, i.e., their permittivity
and permeability must be frequency dependent, otherwise
they would not be causal [7]. This dispersive behaviour is
most naturally studied in the time domain. As was shown
originally in Ref. 8, the unusual properties of DNG media
can be modelled effectively by incorporating the Drude
dispersion model into the finite difference time domain
168
(FDTD) method [9–12]. Unlike the first analytical DNG investigations, the use of such a full-wave numerical simulator avoided the need to choose the signs of any of the derived electromagnetic quantities, such as the index of refraction, and allowed the DNG effects to express themselves simply from the fundamental properties of the materials. The FDTD approach has been used to investigate a
variety of material models and the resulting SNG and DNG
electromagnetic effects [13–15]. Many of these studies
have been reproduced and expanded by many researchers
with commercial software, including REMCOM’s XFDTD
[16] and CST’s Microwave Studio [17], which is based
upon the FIT method [18,19].
Frequency domain finite element approaches have also
been used to investigate a variety of metamaterial-based
applications. The material effects can be included on a frequency-by-frequency basis; and, as a result, one must deal
with a different problem at each frequency. In particular,
the commercial code, ANSOFT’s high frequency structure
simulator (HFSS) [20], has been used by a variety of researchers for three dimensional structures. However, as
will be discussed below, the axisymmetric ability of
COMSOL’s Multiphysics software [21] has advantages for
some dispersion studies associated with the electricallysmall antenna systems.
Of course, analytical models have a variety of advantages that allow one to study metamaterial effects and their
applications quickly. Moreover, they provide guidance for
the more general numerical studies. However, as will be
stressed below, there are considerations associated with realistic systems for which analytical results cannot provide
the desired answers.
3. MTM-based efficient electrically-small
systems
The use of metamaterial coatings to enhance the radiation
and matching properties of electrically small electric and
magnetic dipole antennas has been championed in [22–25].
The unconventional properties exhibited by metamateials,
such as the resonances in electrically-small cavities, scatterers, and radiators, occur when they are paired with materials having at least one oppositely-signed constitutive parameter [3]. In such electrically small electromagnetic systems, these “complementary” interfaces allow a squeezing
of the resonant dimensions, the effect being dependent on
specific filling ratios between the SNG and DNG metamaterials and the DPS materials rather than on the total electrical size of the system.
3.1. Antennas
Let a be the radius of the sphere that bounds the antenna
system. Let the driving frequency of the antenna be f0,
l 0 = c f 0 being the corresponding free space wavelength,
where c is the speed of light in vacuum. With
Opto-Electron. Rev., 14, no. 3, 2006
© 2006 COSiW SEP, Warsaw
k 0 = 2p l 0 being the associated wave number, the antenna system is said to be electrically small if k 0 a £ 1.
An electrically small dipole antenna is known to be a
very inefficient radiator. For example, an electrically small
dipole antenna has a very small radiation resistance while
simultaneously having a very large positive reactance; and,
thus, it presents a large impedance mismatch to any realistic power source. As a result, without a matching network,
this antenna is a very inefficient radiator. One can achieve
matching by incorporating a large inductor to obtain a conjugate match (i.e., forcing the total reactance of the system
to be zero) and a quarter-wave transformer to obtain a resistance match. Consequently, if one includes the matching
network as an actual part of the antenna system, its total
size is not electrically small; i.e., k 0 ( a + l 0 4 ) > p 2 > 1. In
effect, the need for the matching network to make it an efficient radiator significantly nullifies the effective miniaturization of the overall antenna system.
There have been many approaches suggested to achieve
efficient electrically small radiators. We have proposed a
metamaterial-based paradigm in Refs. [22–25]. We have
demonstrated, for instance, that enclosing an electrically
small dipole antenna in an electrically small ENG shell leads
to an efficient radiator. The inductive nature of the ENG
shell compensates for the capacitive nature of the electrically
small electric dipole antenna and functions as a distributed
matching element. When the geometrical parameters are
correctly selected, an LC resonance occurs. The dipole and
the entire matching network are contained within the outer
radius of the ENG shell. It was shown in Ref. 6 that this naturally resonant radiating system is reciprocal to the resonant
scattering of a plane wave from the corresponding
ENG-coated DPS sphere. Additionally, by duality, the capacitive nature of an MNG shell can be designed to compensate for the inductive nature of an electrically small magnetic
loop antenna to achieve such a resonant system. A DNG
shell can be used with either radiating element, the size and
material parameters have to be adjusted appropriately for the
additional capacitance or inductance introduced by the other
negative material component.
Several variations on these metamaterial-based shell
systems have been considered [22–25]. These include analytical models of an electrically small (constant current)
electric dipole antenna in single and multilayered ENG-based spherical shell systems and HFSS numerical models of
an electrically small centre-fed electrical dipole in the same
multilayered shell system and a coaxially-fed monopole in
the corresponding hemispherical multilayered shell system,
as well as an electrically small loop antenna in the corresponding MNG-based shell systems. Illustrations of the
single layer ENG shell numerical models are given in Figs.
2(a) and 2(b).
The analytical models are numerically efficient, allowing for many variations of the properties of the systems
they describe in order to explore their resonant properties.
Moreover, they allow a straightforward means to introduce
dispersion into the metamaterial regions. Unfortunately,
Opto-Electron. Rev., 14, no. 3, 2006
Fig. 2. Examples of electrically-small metamaterial-based antenna
systems that have been modelled numerically, (a) electric
dipole-ENG shell and (b) coax-fed monopole-ENG shell.
while the total radiated power can be calculated easily with
these analytical models, they do not provide the corresponding input impedance of the system. On the other
hand, the HFSS models do provide both the total radiated
power and the input impedance. Thus, the HFSS models
provide a complete prediction of the overall efficiency of
the system. Nonetheless, the introduction of dispersion into
the HFSS models is very problematic because of the required number of simulations as well as the requisite fineness of the frequency steps.
Consider the MacLean derivation of the Chu limit of
the quality factor Q of an electrically small antenna system
[26]. It gives
QChu =
1
(ka )
3
+
1
~ (ka ) -3 for ka << 1
ka
.(1)
The Q factor thus increases dramatically as the electrical size of the antenna system decreases. If f +,3 dB and
f -,3 dB represent the frequencies above and below the resonance frequency where the radiated power falls to half its
peak value, the fractional bandwidth, FBW, is related to the
radiation quality factor, QBW, by the relation [27]
R.W. Zió³kowski
FBW =
Df 3 dB
1
,
=
f 0 dB
QBW
(2)
169
Metamaterial-based source and scattering enhancements: from microwave to optical frequencies
where the 3-dB bandwidth Df 3 dB = f +3 dB - f -3 dB = BW .
Therefore, we can calculate the maximum fractional bandwidth based upon the Chu limit as
FBWChu =
1
QChu
» (ka ) 3 for ka << 1.
(3)
Consequently, the fractional bandwidth of an antenna
system decreases dramatically as the electrical size of the
antenna system decreases. Alternatively, it has been shown
by Yaghjian and Best in Ref. 28 that a more general and
accurate approach to calculate the quality factor is from the
input impedance using the formula
QYB =
fR
2 Rinput ( f R )
(¶ f Z input )( f R )
fR
=
[¶ f Rinput ( f R )] 2 + [¶ f X input ( f R )] 2
Rinput ( f R )
, (4)
where f R is the antenna resonance (anti-resonance) frequency, i.e., the frequency at which the input reactance
passes through zero and the derivative of the reactance with
respect to the frequency is positive (negative). For the systems that we have been considering, the enclosing spheres
have radii on the order of 15 mm at 300 MHz, giving
ka = 0.094 and, therefore, QChu = 1214.61. This means
FBWChu = 0.0823. Thus, to calculate the quality factor numerically using the Yaghjian-Best formula in HFSS, one
would need to resolve frequency variations on the order of
1 ´ 10 4 Hz around the centre frequency f 0 = 3 ´ 10 8 Hz to
calculate accurately the requisite derivatives in the expression for QYB. The HFSS models shown in Fig. 2 that we
have been considering with ENG shells already require
discretizations which are on the order of 100,000 to
150,000 tetrahedra on our 8 GB department server. They
require several days of simulation time to obtain the input
impedance as a function of the frequency. These simulations are marginally convergent for systems involving
non-dispersive metamaterial shells. There has simply been
no hope numerically to get a more standard calculation for
dispersive systems using HFSS and the more realistic antennas in three dimensions.
It must be noted that the analytical model of the electrically small electric dipole is driven by a constant current
source. Consequently, the input impedance of the antenna
is a quantity that is not available in these analytical models
and, hence, the quality factor is not directly available from
their solutions. Nonetheless, we have developed an approach based on the radiated power ratio (RPR) calculated
from the analytical solutions, i.e., the ratio of the power radiated by the dipole antenna in the presence of the
metamaterial shell to the value of that antenna radiating in
free space for the same driving current, to estimate the
quality factor when dispersion is present or not. The RPR
values as a function of the frequency are first obtained, and
then the half-RPR values are identified. The bandwidth associated with these values lead to a fractional bandwidth
170
and, hence, to a Q value. These Q values compare very favourably with those obtained with HFSS for the more realistic centre-fed dipole-non-dispersive-ENG spherical shell
and coax-fed monopole-non-dispersive-ENG hemispherical shell systems.
The advantage provided by the COMSOL Multiphysics
software over the HFSS software for this investigation is
simply an issue of numerics, not physics. Both are based
upon the finite element method. However, the COMSOL
package allows for 2D axisymmetric geometries, which the
dipole and monopole antennas in the spherical and hemispherical shells shown in Fig. 2 are. Being able to use 2D
axisymmetric geometries, one can easily use more than
100,000 unknowns in 2D rather than in 3D to calculate the
desired characteristics of these systems. This gives more
than an order of magnitude more resolution in the
numerics, and hence allows us to consider the realistic geometries in the presence of dispersion. In particular, we
have been able use the COMSOL Multiphysics software to
simulate the performance of the realistic centre-fed dipoledispersive-ENG spherical shell and coax-fed monopoledispersive-ENG hemispherical shell systems. Basically, we
establish the convergence of the numerical solution on an
extremely fine mesh and then calculate the input impedances for each epsilon value corresponding to different
driving frequencies. The derivatives in Eq. (4) can then be
calculated numerically with a high degree of accuracy.
With the higher resolution meshes, we can safely vary the
permittivity values (ENG shells) with the source frequency
and be assured that the numerics are a non-issue.
To begin to illustrate these results, consider first a
coax-fed monopole-non-dispersive-ENG hemispherical
shell system. Let the coaxial line have the inner radius rinner
= 1.5 mm and the outer radius router = 3.490343, rinner =
5.2355145 mm, which defines a 75-W feedline. Let the
monopole be the extension of the centre conductor of this
coax line above a perfectly conducting ground plane with a
length l = 5.05 mm. Let the regions inside and outside of
the ENG hemispherical shell be free space. Let the ENG
hemispherical shell have the inner radius r1 = 10 mm and
the outer radius r2 = 19.45 mm. Let the ENG shell be
lossless and non-dispersive with a relative permittivity
value er = –3.0. Let the target frequency of the source be
near f0 = 300 MHz. A port source was used to drive the
75-W coax feed line. The degrees of freedom solved were
110129 with extra fine gridding enforced in the coax-monopole and ENG shell regions.
The COMSOL Multiphysics predicted magnetic field
distribution at the resonance frequency and the input impedance values as a function of the source frequency for
this coax-fed monopole-ENG hemispherical shell system
are shown in Figs. 3 and 4, respectively. The actual resonance frequency for this configuration was determined
from these results to be fR = 309.50 MHz. With no losses
present, the overall efficiencies at the resonance frequency
were 99.96%. Note that these configuration and source val-
Opto-Electron. Rev., 14, no. 3, 2006
© 2006 COSiW SEP, Warsaw
ues give k0a = 0.1168 and QChu = 611.33 at the resonance
frequency. Similar results were obtained with a 50-W
source. With both of these simulation cases successfully
completed, it has then been demonstrated that one can design a metamaterial shell and antenna system to match a
given source. The 75-W value was chosen here simply because the usual choice for a resonant dipole is one that is a
half-wavelength in size and has an input resistance (73 W)
close to this value.
As noted above, the COMSOL Multiphysics mesh for
this example was first designed with a non-dispersive ENG
medium. It was confirmed that the finer resolution mesh allowed us to achieve a frequency resolution of 1×104 Hz
about the resonance frequency and, hence, that the required
derivatives in Eq. (4) could be calculated accurately. Several dispersion models were then considered.
In particular, we have studied various models that have
been presented in the literature as the physical limits on the
dispersion properties of a material. These generally are derived from considerations of the positive-definiteness of
the energy in a dispersive medium or from the Kramers-Krönig relations that relate the real and imaginary parts of
the permittivity and permeability. The latter discourse is often associated with discussions of causality. However, in
our examination of the performance of an electrically small
antenna, we need to emphasize lossless materials since
losses will increase the bandwidth (and, hence, decrease
the Q value) but at the cost of a loss in the efficiency. Thus,
the lossless cases will provide the most stringent limits on
the quality factors. Consequently, we have concentrated on
the limits derived from the average electromagnetic energy
in a dispersive medium, which is generally taken to be [29]
dispersive
Wtotal
=
é1
r
r
òòòdV êë 4 {¶w[we r (w)]}e 0 Ew (r )
V
r r 2ù
1
+ {¶ w[wm r (w)]}m 0 H w (r ) ú
4
û
2
. (5)
The average electromagnetic energy in a dispersionless
medium follows immediately by assuming the relative
permittivity and permeability are independent of the frequency. If one assumes that the difference between the energy in the medium and its value in vacuum for the same
fields is positive definite, Yaghjian and Best [27] obtain
¶ f [ fe r ( f )] - 1 ³ 0.5 f¶ f [e r ( f )] ³ 0,
(6)
while Landau and Lifschitz [29] obtain
¶ f [ fe r ( f )] ³ 1 and f¶ f [e r ( f )] ³ 0,
(7)
with similar constraints on the permeability. On the other
hand, if one simply requires positive definiteness of both
the electric and magnetic energies, one obtains the entropy
condition [5], [30–32]
¶ f [ fe r ( f )] ³ 0.
Fig. 3. Magnitude of the magnetic field distribution for the
non-dispersive 75-W case at the antenna resonance frequency.
Fig. 4. Input impedance for the non-dispersive 75-W case as a
function of the source frequency.
One finds that the Yaghjian and Best criterion is the
most restrictive and basically says that the best one can do
locally is a dispersive medium described by the Drude
model. For the desired er(fR) = –3.0 value associated with
the coax-fed monopole example, these constraints are basically satisfied by the following dispersion models ordered
from the most to the least restrictive
(8)
If one only requires the positive definiteness of the total
energy in an ENG medium, one obtains the “Z limit” [24]
Opto-Electron. Rev., 14, no. 3, 2006
(9)
¶ f [ fe r ( f )] ³ -1.
R.W. Zió³kowski
e 2 r,Drude ( f ) = 1 - 4 ( f R f ) 2
e 2 r,LL ( f ) = 1 - 4 ( f R f )
e 2 r,E ( f ) = 1 - 4 ( f R f ) 3
4
.
(10)
e 2 r,Z ( f ) = 1 - 4 ( f R f )1 2
171
Metamaterial-based source and scattering enhancements: from microwave to optical frequencies
Table 1. COMSOL predicted values of Q/QChu for the dispersion-limit models.
Drude
Landau-Lifschitz
Entropy
Z
Nondispersive
Analytical
1.59
0.80
0.60
0.41
0.010
75-W coax case
1.38
0.70
0.53
0.36
0.015
The ENG shell in the COMSOL Multiphysics model
was then modelled according to these values. Because the
permittivity value is identical by design for each model at
the resonance frequency, the overall efficiency of the antenna system, even with dispersion being present, remains
99.96%. On the other hand, because the slope of the resistance and reactance curves will change with each dispersion model, the resulting Q values will be different for each
of them. The predicted values of Q/QChu using the expression for all of these cases are given in Table 1. The results
for the corresponding analytical resonant infinitesimal dipole-ENG shell system with r2 = 14.9658 mm, which were
obtained with the RPR approach, are also given in Table 1.
The agreement between the analytical RPR-based results
and the numerical input-impedance-based results is very
good. In fact, as indicated in Ref. 24, the Q values obtained
with the RPR approach are slightly higher than those obtained with the input impedance approach. It is found, as in
our previous investigations [23,24], that all of the dispersion limit models, except the Drude (Yaghjian-Best) version, have Q values smaller than the corresponding Chu
limit value. As to which dispersion model is the most correct one, the majority of references favour the Entropy condition, i.e., positive-definiteness independently of both the
electric and magnetic field energies. The design of metamaterials to achieve these various constraints is currently in
progress. It is hoped that one could determine experimentally which model gives the ultimate physical limitation on
the dispersion properties.
As was considered previously in Ref. 24, we have also
investigated the effects of using active ENG regions to decrease the Q values in an attempt to reach those predicted
for the non-dispersive ENG medium. While the passive,
dispersive ENG models produce bandwidths (Q values)
above (below) the Chu limit, they are not as practical as the
ones obtained from the non-dispersive model would be. In
particular, the Lorentz-Lorentz gain medium and the Lorentz-2TDLM gain medium, which are described by models which combine an active Lorentz resonance with either
a passive Lorentz or 2TDLM [33–36] resonance have been
ordered. The permittivities associated with these gain media are
e r , LorentzGrain (w) = 1 +
+
172
c passive w 2p1
2
-w +
2
jwG1 + w 001
, ,
c active w 2p 2
2
-w 2 + jwG 2 + w 00
,2
(11)
e r ,2 TDLM Grain (w) = 1 +
+
- c 2 TDLM w 2
2
-w 2 + jwG1 + w 001
,
c Lorentz w 2p 2
. (12)
2
-w 2 + jwG 2 + w 00
,2
The 2TDLM model allows for the flattest behaviour of
the permittivity after its resonance. For the COMSOL
Multiphysics simulations, the parameters were determined
to achieve as flat a dispersion curve as possible through the
resonance frequencies. The parameters used in the lossless
versions of these models to achieve e r ( f R ) = -30
. were
Lorentz-Lorentz model (analytical), fR = 300 MHz, cpassive
= 1.19108435, w 0 0 , 1 = 0.08w R , c a c t i v e =
–6.50004163851947, w00,2 = 1.8222wR, wp1 = wp2 = wR,
G1 = G2 = 0.0; Lorentz-Lorentz Model (numerical): fR =
309.5 MHz, w00,1 = 0.08wR, cactive = –6.50004163851947,
w00,2 = 1.8222wR, wp1 = wp2 = wR, G1 = G2 = 0.0; and Lorentz-2TDLM model (analytical and numerical): fR = 309.5
MHz, c2TDLM = –3.385, w00,1 = 0.1515wR, cLorentz =
–3.5161592452504, w00,2 = 2.7507wR, wp1 = wp2 = wR, G1
= G2 = 0.0. The COMSOL Multiphysics predicted results
are given in Table 2. As with our previous results, these
values of Q/QChu for the active dispersive ENG hemispherical shell recover the bandwidths predicted for the non-dispersive ENG hemispherical shell. The analytical Lorentz-Lorentz gain medium value is slightly worse than the
numerical ones because the dispersion model for the analytical results was not quite as flat at the lower frequency as
were the models constructed and used for the COMSOL
Multiphysics simulations. These results confirm the possibility of there being an advantage of using active metamaterials for bandwidth considerations in electrically small
antenna systems.
Table 2. COMSOL predicted values of Q/QChu for the
gain medium dispersion models.
Lorentz-Lorentz
gain medium
Lorentz-2TDLM
gain medium
Analytical
0.023
0.010
75-W coax case
0.015
0.015
Unfortunately, realizing an ENG medium with a
metamaterial that is based on finite-sized inclusions at
UHF frequencies has turned out to be very challenging.
Note that the low frequency properties of a bed-of-nails
ENG medium [37–39] occur because the wires are infinite.
Cut wires have significantly higher resonant frequencies.
Opto-Electron. Rev., 14, no. 3, 2006
© 2006 COSiW SEP, Warsaw
On the other hand, ENG media occur naturally at very high
frequencies (plasmonic effects at optical frequencies). In
contrast, an MNG medium is readily obtained at low frequencies (loop-based media or ferrites) while it is difficult
to obtain at high (optical) frequencies. Consequently, a resonant loop antenna-MNG shell system may be easier to realize at UHF frequencies than the electric dipole antenna-ENG shell system. It is also noted that an ENG medium is straightforwardly obtained with a plasma medium
at UHF frequencies. The relative permittivity of a lossy
cold plasma is described by the Drude model. A plasma
density corresponding to that in a fluorescent light bulb
would produce the values considered in the above example
[24]. Both of these SNG configurations, as well as lumped
element-based ENG inclusions, are currently being investigated in order to achieve a UHF proof-of-concept experiment. It must be emphasized that the coax-fed electric
monopole-ENG hemispherical shell antenna systems are
the most likely candidates for initial testing since they provide a means to drive the antenna without disrupting the
shell properties. It has been demonstrated, as expected, that
these metamaterial-based coax-fed monopole hemispherical ENG shell antenna cases recover the performance characteristics of the corresponding electric dipole-ENG spherical shell antenna cases [23–25].
malized by their unity values as functions of the radius of the
outer boundary of the shell are shown in Figs. 6(a) and 6(b).
As shown, the incident plane wave excites the resonant dipole modes for the same configuration that produces the resonant radiating state. The magnetic field distributions for the
radiating and the scattering dipole modes for the resonant
Au2S nanoshells are shown in Figs. 7(a) and 7(b). The additional features that are apparent in the scattering results, but
are absent in the source results, occur because of the presence of higher order resonant modes excited by the plane
wave. The second peak corresponds to a resonant quadrupole mode. The deep null corresponds to a non-scattering
state that is analogous to the transparency effect reported
first in Ref. 41 and discussed more recently as a cloaking effect in Refs. 42, 43, and 44. Note that these higher order
mode effects are present in the source situation when the
source is located away from the origin.
3.3. Laser cavities
The possibility of using the phase compensation properties
of matched DPS-DNG or ENG-MNG bilayers to achieve
thin, sub-wavelength cavity resonators, i.e., the ability of
3.2. Nano-radiators and nano-scatterers
These resonant source enhancements and similar effects for
the reciprocal scattering configurations can be reproduced
in nano-sized particles at optical frequencies. Consider an
optical source, e.g., a quantum dot that is 1.0 nm in size
and radiates as an infinitesimal dipole in the visible light
spectrum at 600 nm in free space. Let this source be embedded in both, a gold-sulfide, Au2S, and a silver-silica,
AgSi, nanoshell. The cores of the Au2S and AgSi nanoshells have, respectively, the relative permittivity values
er,S core = 5.44 and er,Si core = 2.04. Let the inner radius of the
gold and silver shells be, respectively, r1 = 10 nm and r1 =
12 nm. Because of the smallness of these shells, the material properties are size dependent [40]. The analytical solution was iterated with different shell thicknesses and permittivities until the resonant radiating mode was obtained.
With the permittivity of gold for a 5-nm thickness, er(f0) =
–9.2293 – j3.2382, and the permittivity of silver for a 2-nm
thickness er(f0) = –15.0951 – j4.567, the RPR values shown
in Figs. 5(a) and 5(b) were obtained. The peaks of the RPR
values occur for the outer radius of the gold and silver
shells that was, respectively, r2 = 14.8004 nm and r2 =
13.7494 nm. The RPR values for the corresponding
lossless ENG cases and the cases for which the imaginary
part of the permittivities are reduced by a factor of 100 are
also included in those figures. A large enhancement of the
radiated power, even in the presence of the actual, significant loss values was obtained.
The same nanoshells were then illuminated by a linearly
polarized plane wave. The total scattering cross sections norOpto-Electron. Rev., 14, no. 3, 2006
Fig. 5. The RPR values predicted by the analytical model for the
infinitesimal electric dipole driven by a 600-nm source in the
presence of a resonant ENG spherical nanoshell, (a) gold based
nano-radiator and (b) silver based nano-radiator.
R.W. Zió³kowski
173
Metamaterial-based source and scattering enhancements: from microwave to optical frequencies
Fig. 6. The total scattering cross section normalized by the unity
value predicted by the analytical model when a 600-nm plane wave
is incident on a resonant nanometer-sized ENG spherical shell, (a)
gold based nano-scatterer and (b) silver based nano-scatterer.
oppositely-signed index of refraction layers to attain a zero
phase change across the bilayer and, hence, a resonance
condition based on a ratio of the dimensions rather than a
sum, was suggested in Ref. 45. This sub-wavelength resonator concept has been used to advance the concept of an
ultra-thin laser cavity [46].
Consider the one-dimensional laser cavity geometry
shown in Fig. 8. An electric current source, such as a quantum dot, is sandwiched between two metamaterial slabs.
One slab, which has the thickness d1, is terminated with a
perfect reflector. The other slab, which has the thickness
d2, is terminated with a partially-reflecting mirror. The partially reflecting mirror can be defined analytically by simply forcing an amplitude reflection coefficient at the plane
z = d 2 of the form RM = x exp( jd ), where RM = x and d
define, respectively, the magnitude of the reflectivity and
its phase. The power reflection coefficient is simply x 2 so
the capabilities of the mirror can be defined readily. As
shown in Ref. 46, the cavity becomes resonant when the total wave impedance of the cavity becomes zero, which oc-
174
Fig. 7. The resonant dipole mode magnetic field distributions, (a)
Re( HTotal - HSource ) for the resonant gold-based nano-radiator,
and (b) Re( HTotal ) for the gold based nano-scatterer.
curs if the slabs are an ENG-MNG pair with their wave impedances h 1 = -h 2 and their phase lengths
k1d1 = k 2 d 2 or a DPS-DNG pair with h 1 = h 2 and
k1d1 = -k 2 d 2 . Let the source amplitude be I 0 = 2 h 0 so
that the source in free space produces a 1-V/m electric field
propagating away from it in either direction. The source is
assumed to have a free space wavelength l0 = 500 nm.
When the relative permittivity and permeability values are
greater than or equal to one, they are taken to be constants
with respect to any frequency variation. When they are less
than one, they are described by the lossless Drude models
e r (w) = 1 m r (w) = 1 -
w 2pe
w2 .
w 2pm
(13)
w2
The Drude models were preferred to the Lorentz models for this discussion simply because they produce slower
variations in the frequency properties of the slabs in the re-
Opto-Electron. Rev., 14, no. 3, 2006
© 2006 COSiW SEP, Warsaw
Fig. 8. One-dimensional metamaterial-based electrically small
laser cavity.
gions of interest and have no resonant behaviour that could
obscure the resonances which were expected from the configuration of the metamaterial slabs. Finally, Region 3 was
considered to be free space, i.e., e r3 = +1, m r3 = +1.
To illustrate the behaviour of this cavity, the slabs were
first taken to be the ENG-MNG pair with Re[e r1 (w 0 )]
= -0.01, Re[ m r1 (w 0 )] = +10
. , Re[e r 2 (w 0 )] = +0.01, and
Re[ m r 2 (w 0 )] = -10
. . Then, they were taken to be the DPSDNG pair with Re[e r1 (w 0 )] = +0.01, Re[ m r1 (w 0 )] = +10
. ,
Re[e r 2 (w 0 )] = -0.01, and Re[ m r 2 (w 0 )] = -10
. . The mirror
2
reflectivity in both cases was RM = 0.98. The slab thicknesses were d1 = d 2 = l 0 40 so that the overall length of
the cavity was d = d1 + d2 = l0/20 = 25 nm, a tenth smaller
than a normal laser cavity. The predicted magnitudes of the
output electric field values for these cases are shown in Fig.
9(a). The values at the source frequency were approximately 45 MV/m for the DPS-DNG case and 39 MV/m for
the ENG-MNG case. The peak magnitudes, which are even
larger than at the source frequency, occur when the values
of the Drude permittivity pass through zero. Thus, one
finds that even larger output field values could be obtained
with a source frequency tuned to f = 0.995 f0 or f = 1.005
f0 for the specified configurations. This observation leads
one to a zero-index design.
It has been demonstrated that zero-index metamaterials
can be used to achieve high directivity antennas [15,47–52].
Because a signal propagating in a zero-index metamaterial
will stimulate a spatially-static field structure that varies in
time, the phase at any point in a zero-index metamaterial
will have the same constant value once steady state is
reached [47]. This coherent response causes the phase across
the output face of a slab of zero-index metamaterial to be
uniform. Moreover, from Snell’s law, all of the outgoing
wave vectors will be normal to the interface. Consequently,
Opto-Electron. Rev., 14, no. 3, 2006
this effective aperture will produce the highest-directivity
beam. These behaviours have been confirmed with analysis
and supporting FDTD simulations [15,47]. Such highly directive sources have been achieved with electromagnetic
band-gap (EBG) structures [48–52]. Zero-index metamaterials have also been used to achieve a zeroth-order resonance (ZOR) antenna [53,54]. Because the index is zero or
near zero, the wavelength in the medium is effectively infinite and the resonance is independent of the physical dimensions. It is this property that will contribute to the behaviour
of the ultra-thin laser cavity. The physical size of a ZOR antenna can thus be made smaller than a half-wavelength. The
perfectly uniform current distribution associated with the
zero-order mode may also decrease the overall loss resistance of the system and, hence, provide a higher overall efficiency. Experimental verification of these properties has
been accomplished [53,54].
A zero-index metamaterial was then introduced into the
same laser cavity. One slab was defined to have a relative
permittivity equal to zero at the source frequency and a relative permeability equal to one. The other slab was taken to
be free space. The predicted magnitudes of the output electric field values for these cases are shown in Fig. 9(b). The
peak magnitudes were 272.3 GV/m for the Zero-DPS case
and 90.11 MV/m in the DPS-zero case. Even though the
cavity is again ten times smaller than a normal laser cavity,
Fig. 9. Output fields for electrically small, resonant laser cavities.
R.W. Zió³kowski
175
Metamaterial-based source and scattering enhancements: from microwave to optical frequencies
it resonates and produces a correspondingly large output
field at the source frequency.
Another interesting use of near-zero-index metamaterials
has been uncovered with optimization studies of multilayered metamaterial-coated sphere systems [25]. It has been
found that in the same manner in which there are resonant
radiating states for the dipole-ENG shell system, there are
also resonant non-radiating (NR) modes. These NR modes
of the source problem are believed to be reciprocal to the
transparency and cloaking scattering states noted above and
reported in [41–44]. These NR modes occur when the ENG
shell has an effective permittivity approximately equal to
zero. Both resonant radiating and non-radiating modes occur
for the same configuration when the shell is described with a
dispersion model, i.e., if a dipole-ENG shell configuration is
designed to have a resonant radiating state and the dispersion of the ENG shell is described by the Drude model, then
at a higher frequency the real part of the permittivity of the
ENG shell will pass through zero and the non-radiating
mode will express itself. While the exterior field is large for
the resonant radiated mode, the field becomes resonantly
trapped in the ENG shell for the NR mode. Multilayered
spherical shells lead to multi-band resonant radiating and
non-radiating operation [25]. Consequently, multilayered
spherical shells can be designed reciprocally to be transparent at multiple frequencies.
4. Conclusions
The metamaterials research area has evolved into prominence only very recently. Nonetheless, it is already having
a large impact on the international electromagnetics community. Metamaterials have revitalized our interests in
complex media, their exotic properties, their analysis and
numerical modelling, and their potential applications.
There have been large strides in our understanding of their
anomalous behaviours and of their possible utilization in
many electromagnetic applications from the microwave to
the optical regime.
This review has only briefly touched upon some selective research efforts associated with metamaterials and
their antenna applications. Metamaterials, because of their
promise to provide engineerable permittivities and permeabilities, possess interesting properties for the design of
next-generation structures for radiating and scattering applications. Resonances arising in electrically small regions
of space where single and double negative materials are
paired with common double positive materials were shown
to have a great potential for overcoming the limits generally associated with several electromagnetic problems by
providing a means to engineer the overall responses of the
systems. Radiating systems in the UHF regime, as well as
nano-sized radiators and scatterers were presented to emphasize these points. Zero-index metamaterials, media with
permittivities and permeabilities with zero or near zero values, may have strong impact in some applications as illustrated with an electrically small laser cavity example.
176
Metamaterials have been considered as a means to manipulate the size, efficiency, bandwidth, and directivity of
several basic radiating and scattering systems. There are
many researchers whose works were not mentioned, and I
apologize to all of them. Metamaterials and their antenna
applications is a very fertile research area in which there is
much interest. The initial seed physics efforts are only now
beginning to bear some engineering applications fruit.
There remain many challenging and potentially rewarding
problems left to solve; we all look forward to sharing some
of these solutions in the near future.
Acknowledgements
This work was supported in part by DARPA contract number HR0011-05-C-0068.
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