Physical limitations on metamaterials: restrictions on scattering and absorption over a frequency interval

TL;DR: In this article, a limitation on the extinction cross section, valid for all scatterers satisfying some basic physical assumptions, is investigated, based on the holomorphic properties of the forward scattering dyadic.

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 permittivity and permeability, i.e. metamaterials. From a broadband point of view, the limitations imply that there is no fundamental difference between metamaterials and ordinary materials with respect to scattering and absorption. The analysis is illustrated by three numerical examples of metamaterials modelled by temporal dispersion.

Summary

1 Introduction

• The analysis presented in this paper shows that, from a broadband point of view, the scattering and absorption properties of any material (not just metamaterials) that satisfy basic physical assumptions, are limited by the static electric and magnetic behavior of the composed materials.
• Speci cally, the lower the resonance frequency, the higher its Q-value.
• 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.

2 Derivation of the integrated extinction

• The dynamics 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, χe and χm, respectively.
• The present analysis includes the perfectly conducting material model as well as general temporal dispersion with or without a conductivity term.
• For heterogeneous structures, the long wavelength susceptibilities χe(0) and χm(0) also depend on the space variable x. (2.6).
• For non-spherical particles, (2.6) provides a neat veri cation of (2.5) without specifying the orientation of V with respect to the incident wave, see Sec. 4.1.

3 Bounds on scattering and absorption

• (3.1) The right hand side of (3.1) is independent of any material properties, depending only on the geometry and the orientation of V with respect to the incident wave.
• In fact, it is well known that passive materials must be temporally dispersive since the Kramers-Kronig relations imply that χe(0) and χm(0) element-wise are non-negative in the absence of a conductivity term, see Ref. 5. Recall that the Kramers-Kronig relations are a direct consequence of primitive causality, see Ref. 8.
• The Drude model is often preferred over the Lorentz model since it provides a wider bandwidth over which the eigenvalues of χe and χm attain values less than −1.
• Based on the arguments above, it is uninteresting from the point of view of (2.5) and (3.3) which temporal dispersion model is used to characterize metamaterials as long as the model satis es primitive causality.

4 Numerical synthesis of metamaterials

• Numerical results for three temporally dispersive scatterers are discussed in terms of the physical limitations in Sec. 3.
• The examples are chosen to provide a ctitious numerical synthesis of metamaterials.
• For convenience, the examples are restricted to isotropic material properties, i.e., χe = χeI and χm = χmI, where I denotes the unit dyadic.
• A similar example for the Lorentz dispersive cylinder is given in Ref. 12.

4.2 The Drude dispersive strati ed sphere

• The extinction cross section σext for a strati ed sphere with two layers of equal volume is depicted in Fig.
• Furthermore, ζ ∈ [0, 1] denotes the quotient between the inner and the outer radii.
• The integrated extinction of each box is equal to 248.0 cm3 and coincides with the integrated extinction for any other curve in the gure.
• The strati ed sphere is temporally dispersive with electric susceptibility χe given by the Drude model (4.2).
• Fig. 3a is a close-up of the peaks at 0.96 GHz and 1.4 GHz with the associated box-shaped limitations.

5 Conclusions

• For a single frequency, metamaterials may possess extraordinary properties, but with respect to any bandwidth such materials are no di erent from any other naturally formed substances as long as causality is obeyed.
• The present analysis includes materials modeled by anisotropy and heterogeneity, and can be extended to general bianisotropic materials as well.
• The introduction of chirality does not contribute to the integrated extinction since all chiral e ects vanish in the long wavelength limit.
• It is believed that there are more physical quantities that apply to the theory for broadband scattering in Ref. 12.

Figures

Figure 1: The averaged extinction cross section σ̄ext as function of the frequency in GHz for a prolate spheroid with semi-axis ratio ξ = 1/2. Note the normalization with πa2, where a = 1 cm denotes the radius of the volume-equivalent sphere.

Figure 3: The extinction cross section σext as function of the frequency in GHz for a non-magnetic strati ed sphere which attain negative values of the permittivity. Note the normalization with the geometrical cross section πa2, where a = 1 cm denotes the outer radius of the sphere.

Figure 2: The extinction cross section σext as function of the frequency in GHz for a strati ed sphere which attains simultaneously negative values of the permittivity and the permeability. Note the normalization with the geometrical cross section πa2, where a = 1 cm denotes the outer radius of the sphere.

Full text

LUND UNIVERSITY
PO Box 117
221 00 Lund
+46 46-222 00 00
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].
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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 dierence 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 scientic 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 eects have to be traded for bandwidth. Specically, 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
sucient 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 dierence
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)

Frequently asked questions

Q1. What are the contributions in "Physical limitations on metamaterials: restrictions on scattering and absorption over a frequency interval" ?

In this paper, it was shown that, from a broadband point of view, the scattering and absorption properties of any material that satisfy basic physical assumptions, are limited by the static electric and magnetic behavior of the composed materials. 

Q2. What is the right hand side of (2.5)?

Furthermore,4 the right hand side of (2.5) depends solely on the long wavelength limit or static response of V , while the left hand side is a dynamic quantity which unites the scattering and absorption properties of V . 

Q3. What is the underlying mathematical description for broadband scattering?

The underlying mathematical description for broadband scattering is motivated by2 the study of causality and dispersion relations in the scattering theory of waves and particles in Refs. 7 and 8.Consider a localized and bounded scatterer V ⊂ R3 of arbitrary shape. 

Q4. Why can't V be interpreted as a single scatterer?

Due to the heterogeneous character of χe and χm, V can be interpreted both as a single scatterer and as a set of multiple scatterers. 

Q5. What is the extinction of the scatterer in Fig. 3?

Since the strati ed sphere in Fig. 3 has the same electric long wavelength response as the scatterer in Fig. 2 but in addition is non-magnetic, it follows from (4.4) thatthe integrated extinction of the scatterer in Fig. 3 is half the integrated extinction of the scatterer in Fig. 2, i.e., 4π3a3 or 124.0 cm3. 

Q6. What is the importance of studying metamaterials over a frequency interval?

For a single frequency, metamaterials may possess exceptional characteristics, but, since bandwidth is essential, it is important to study metamaterials over a frequency interval, and with physically realistic dispersion models. 

Q7. What are the physical limitations of metamaterials?

For a single frequency, metamaterials may possess extraordinary physical properties, but over any bandwidth they are with respect to scattering and absorption not di erent from materials with the eigenvalues of χe and χm non-negative. 

Q8. What is the extinction cross section of (2.5)?

Since the extinction cross section σext by denition is non-negative, the left hand side of (2.5) can be estimated from below as|Λ| inf λ∈Λσ(λ) ≤ ∫Λσ(λ) dλ ≤ ∫ ∞0σext(λ) dλ, (3.2)where Λ ⊂ [0,∞) denotes an arbitrary wavelength interval with absolute bandwidth |Λ|. 

Q9. What is the main reason for the broad range of material models?

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. 

Q10. What is the extinction cross section for a prolate spheroid?

For a prolate spheroid with semi-axis ratio ξ = 1/2, the depolarizing factors are approximately given by L1(1/2) = L2(1/2) = 0.4132 and L3(1/2) = 0.1736, see Ref. 12. 

Q11. What is the dyadic amplitude of the incident wave?

Introduce E0 as the Fourier amplitude of the incident wave, and let p̂e = E0/|E0| and p̂m = k̂ × p̂e denote the associated electric and magnetic polarizations, respectively. 

Q12. What is the integrated extinction of the stratied sphere?

The integrated extinction of each box is equal to 248.0 cm3 and coincides with the integrated extinction for any other curve in the gure. 

Q13. What is the extinction cross section of (3.2)?

Two popular models for temporal dispersion for metamaterials are the Drude and Lorentz models, see (4.2) and Ref. 8, respectively.