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Physical limitations on metamaterials: restrictions on scattering and absorption over a
frequency interval
Sohl, Christian; Gustafsson, Mats; Kristensson, Gerhard
2007
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Citation for published version (APA):
Sohl, C., Gustafsson, M., & Kristensson, G. (2007).
Physical limitations on metamaterials: restrictions on
scattering and absorption over a frequency interval
. (Technical Report LUTEDX/(TEAT-7154)/1-11/(2007)).
[Publisher information missing].
Total number of authors:
3
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Electromagnetic Theory
Department of Electrical and Information Technology
Lund University
Sweden
CODEN:LUTEDX/(TEAT-7154)/1-11/(2007)
Revision No. 2: September 2007
Physical limitations on metamaterials:
Restrictions on scattering and
absorption over a frequency interval
Christian Sohl, Mats Gustafsson, and Gerhard Kristensson
Christian Sohl, Mats Gustafsson, and Gerhard Kristensson
{Christian.Sohl,Mats.Gustafsson,Gerhard.Kristensson}@es.lth.se
Department of Electrical and Information Technology
Electromagnetic Theory
P.O. Box 118
SE-221 00 Lund
Sweden
Editor: Gerhard Kristensson
c
°
Christian Sohl
et al.
, Lund, July 20, 2007
1
Abstract
A limitation on the extinction cross section, valid for all scatterers satisfying
some basic physical assumptions, is investigated. The physical limitation is
obtained from the holomorphic properties of the forward scattering dyadic.
The analysis focuses on the consequences for materials with negative permit-
tivity and permeability,
i.e.
, metamaterials. From a broadband point of view,
the limitations imply that there is no fundamental dierence between metama-
terials and ordinary materials with respect to scattering and absorption. The
analysis is illustrated by three numerical examples of metamaterials modeled
by temporal dispersion.
1 Introduction
Since the investigation of negative refractive index materials by V. G. Veselago in
Ref. 14, there has been an enormous theoretical and experimental interest in the
possibilities of such materials. These materials are often referred to as metamateri-
als, even though a metamaterial in general is a much broader concept of a structured
material, and not necessarily composed of materials with negative permittivity and
permeability values. Negative refractive index materials seem not to occur naturally,
and if they can be manufactured, they possess extravagant properties promising for
various physical applications, see Refs. 9 and 11, and references therein.
The scattering properties of obstacles consisting of metamaterials have been of
considerable scientic interest during the last decade. Mostly canonical geometries,
such as the spheres, see
e.g.
, Ref. 10, have been employed, and the design of scatterers
that both increases and decreases the scattering properties have been reported.
The analysis presented in this paper shows that, from a broadband point of
view, the scattering and absorption properties of any material (not just metamate-
rials) that satisfy basic physical assumptions, are limited by the static electric and
magnetic behavior of the composed materials. In particular, we show that, when
these limitations are applied to low-frequency resonances on metamaterials, large
scattering eects have to be traded for bandwidth. Specically, the lower the reso-
nance frequency, the higher its
Q
-value. For a single frequency, metamaterials may
possess exceptional characteristics, but, since bandwidth is essential, it is impor-
tant to study metamaterials over a frequency interval, and with physically realistic
dispersion models.
The results presented in this paper are independent of how the material that
constitutes the scatterer is constructed or produced. This broad range of material
models is a consequence of the fact that the analysis is solely based on the principles
of energy conservation and causality applied to a set of linear and time-translational
invariant constitutive relations.
The present paper is a direct application of the theory for broadband scattering
introduced in Ref. 12. In addition to material modeling, the theory has also been
applied successfully to physical limitations on arbitrary antennas in Refs. 1 and 3.
The underlying mathematical description for broadband scattering is motivated by
2
the study of causality and dispersion relations in the scattering theory of waves and
particles in Refs. 7 and 8.
2 Derivation of the integrated extinction
Consider a localized and bounded scatterer
V ⊂ R
3
of arbitrary shape. The dy-
namics of the material in
V
is modeled by the Maxwell equations with general
heterogeneous and anisotropic constitutive relations. The constitutive relations are
expressed in terms of the electric and magnetic susceptibility dyadics,
χ
e
and
χ
m
,
respectively. Due to the heterogeneous character of
χ
e
and
χ
m
,
V
can be interpreted
both as a single scatterer and as a set of multiple scatterers. The present analysis
includes the perfectly conducting material model as well as general temporal disper-
sion with or without a conductivity term. The analysis can also be extended with
minor changes to bianisotropic materials with the same conclusions drawn.
The direct scattering problem addressed in this paper is Fourier-synthesized
plane wave scattering by
V
. Due to the linearity of the Maxwell equations, it is
sucient to consider monochromatic plane waves with time dependence
e
−iωt
. The
incident wave is assumed to impinge in the
ˆ
k
-direction with an electric eld
E
i
de-
pending only on the dierence
τ = c
0
t −
ˆ
k · x
, where
x
denotes the space variable.
Introduce the far eld amplitude
F
via
E
s
= F (c
0
t − x,
ˆ
x)/x + O(x
−2
)
as
x → ∞
,
where
E
s
represents the scattered electric eld. Under the assumption that the
constitutive relations of
V
are linear and time-translational invariant,
F
is given by
the convolution
F (τ,
ˆ
x) =
Z
∞
−∞
S
t
(τ − τ
0
,
ˆ
k,
ˆ
x) · E
i
(τ
0
) dτ
0
.
Here,
S
t
is assumed to be primitive causal in the forward direction,
i.e.
,
S
t
(τ,
ˆ
k,
ˆ
k) =
0
for
τ < 0
, see Ref. 8. Furthermore, introduce the forward scattering dyadic
S
as
the Fourier transform of
S
t
evaluated in the forward direction,
i.e.
,
S(k,
ˆ
k) =
Z
∞
0
−
S
t
(τ,
ˆ
k,
ˆ
k)e
ikτ
dτ,
(2.1)
where
k = ω/c
0
. The extension of (2.1) to complex-valued
k
with
Im k > 0
improves
the convergence of the integral and implies that
S
is holomorphic in the upper half of
the complex
k
-plane. Recall that the cross symmetry relation
S(k,
ˆ
k) = S
∗
(−k
∗
,
ˆ
k)
is a direct consequence of such an extension.
Introduce
E
0
as the Fourier amplitude of the incident wave, and let
ˆ
p
e
= E
0
/|E
0
|
and
ˆ
p
m
=
ˆ
k ×
ˆ
p
e
denote the associated electric and magnetic polarizations, respec-
tively. Recall that
E
0
is subject to the constraint of transverse wave propagation,
i.e.
,
E
0
·
ˆ
k = 0
. Under the assumption that
ˆ
p
e
and
ˆ
p
m
are independent of
k
, it
follows from the analysis above that also
%(k) =
ˆ
p
∗
e
· S(k,
ˆ
k) ·
ˆ
p
e
/k
2
is holomorphic
for
Im k > 0
. Cauchy's integral theorem applied to
%
then yields, see Ref. 12,
%(iε) =
Z
π
0
%(iε − εe
iφ
)
2π
dφ +
Z
π
0
%(iε + Re
iφ
)
2π
dφ +
Z
ε<|k|<R
%(k + iε)
2πik
dk.
(2.2)
