Superscattering of Light from Subwavelength Nanostructures
Zhichao Ruan and Shanhui Fan
Ginzton Laboratory, Department of Electrical Engineering, Stanford University, Stanford, California 94305
(Received 9 March 2010; revised manuscript received 2 June 2010; published 28 June 2010)
We provide a theoretical discussion of the scattering cross section of individual subwavelength
structures. We show that, in principle, an arbitrarily large total cross section can be achieved, provided
that one maximizes contributions from a sufficiently large number of channels. As a numerical
demonstration, we present a subwavelength nanorod with a plasmonic-dielectric-plasmonic layer struc-
ture, where the scattering cross section far exceeds the single-channel limit, even in the presence of loss.
DOI: 10.1103/PhysRevLett.105.013901PACS numbers: 42.25.?p, 78.66.Bz
Understanding the interaction between optical radiation
and individual subwavelength objects is of fundamental
importance for the study of optical physics, and has prac-
tical significance for applications such as imaging, cloak-
ing, biomedicine, and optical antennas [1–10]. The
strength of this interaction is characterized by the scatter-
ing and the absorption cross sections, defined as Csct?
Psct=I0, Cabs? Pabs=I0. Here, I0is the intensity of an
incident plane wave, Psctand Pabsare the amount of power
scattered or absorbed by the particle, respectively. When
the subwavelength object is a single atom in a three-
dimensional (3D) vacuum, one can rigorously prove that
its maximum scattering cross section is ð2l þ 1Þ?2=2? at
the atomic resonant frequency, where l is the total angular
momentum of the atomic transition involved. This limit
becomes 3?2=2? for a typical electric dipole transition
. Theoretically, in two dimensions, one can similarly
prove that the maximum cross section of an atom cannot
exceed 2?=?. These limits in 3D or 2D, for reasons
apparent from standard scattering theory that we will re-
iterate in this Letter, will be referred to as the single-
channel limit. For subwavelength nanoparticles and nano-
wires, a common observation has been that the cross
section is typically less than such single-channel limit
[2,4,6,8–10]. But there have in fact been a few reports
where the cross section goes somewhat beyond such a limit
In this Letter, we provide a theoretical discussion on the
limit of the scattering and absorption cross sections of a
subwavelength structure. The theory indicates that, on the
one hand, most of the nanostructures indeed should have
their maximum cross section subject to the single-channel
limit, and on the other hand, in plasmonic nanoparticles or
nanowires, there is in fact a substantial opportunity to
significantly overcome this limit. As a numerical demon-
stration, we present a subwavelength plasmonic structure
where the scattering cross section is far beyond the single-
channel limit, even in the presence of loss.
We start by briefly summarizing the relevant aspects in
scattering theory. For simplicity, we will illustrate the main
physics by consideringthe two-dimensional casewhere the
obstacle is uniform in the z direction. Consider an obstacle
located at the origin, surrounded by air. When a TM plane
wave, defined by a wave vector k, and with its magnetic
field polarized along the z direction, impinges on the
obstacle, the total field in the air region outside the scat-
Here, we use the convention that the field varies in time as
expð?i!tÞ. (?, ?) are the polar coordinates centered at the
m is the mth order Hankel function of the first
kind, and represents an outgoing wave in the far field. The
mth term in the summation thus defines the scattered wave
in the mth angular momentum channel. Smis referred to as
the scattering coefficient. With the choice of H0¼
, jSmj2represents the scattered power, normal-
ized by the unit of power which isWatt
The scattering coefficient Smis in fact strongly con-
strained by energy conservation consideration. To see
this, we express Eq. (1) alternatively as a summation of
incoming and outgoing waves in each channel:
mðk?Þ þ h?
kind that represents the incoming wave. hþ
amplitudes, respectively. With our choice of H0, jhþ
going waves in the mth channel. Moreover, for systems
with cylindrical symmetry, the angular momentum is con-
served. Consequently, by energy conservation, the reflec-
tion coefficient defined as
m is the mth order Hankel function of the second
m¼ im=2 and
m¼ imðSmþ 1=2Þ are the incoming and outgoing wave
mj2represent the power carried by incoming and out-
¼ 1 þ 2Sm
is subject to
jRmj ? 1:
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2 JULY 2010
? 2010 The American Physical Society
For such a scatterer, the total scattered power is
an intensity I0¼?
and absorption cross sections become:
Using Eq. (4), one can immediately infer that the maximal
contribution of a single channel to the scattering and
absorption cross section is2?
strongest scattering occurs when Rm¼ ?1, while the
strongest absorption occurs when Rm¼ 0.
For subwavelength particles, one can maximize the
scattering cross section of a single channel by introducing
a resonance. In the presence of a resonance in the mth
channel, the scattering process can be modeled by the
temporal coupled-mode equations [13,14]:
mjSmj2, while the total absorbed power is Pabs¼
4ð1 ? jRmj2Þ. On the other hand, the incident wave has
meter. As a result, the total scattering
2?ð1 ? jRmj2Þ ?
2?, respectively. The
dt¼ ð?i!0? ?0? ?Þc þ i
h?¼ hþþ i
where c is an amplitude and normalized such that jcj2
corresponds to the energy of the resonance, !0is the
resonant frequency, ?0is the intrinsic loss rate due to
material absorption, and ? is the external leakage rate
due to the coupling of the resonance to the outgoing
wave. All variables in Eq. (6) are specific to the mth
channel. For notation simplicity we suppress the subscript
m in Eq. (6). In deriving Eq. (6), we assume that, away
from the resonant frequency, the scattering from the par-
ticle is negligible, hence h?? hþwhen c ? 0 . For an
incident wave at a frequency !, the reflection coefficient
can be obtained from Eq. (6):
Rm¼ið!0? !Þ þ ?0? ?
ið!0? !Þ þ ?0þ ?:
Substituting Eq. (7) to (5), we have
ð! ? !0Þ2þ ð?0þ ?Þ2
ð! ? !0Þ2þ ð?0þ ?Þ2:
The scattering cross section thus exhibits a Lorentzian
spectral line shape, reaching a maximum of2?
the resonant frequency !0. In the strong overcoupling
limit, i.e. ? ? ?0, at the resonant frequency the scattering
cross section of a single channel can approach the maximal
The analysis above, especially Eq. (5), reproduces the
results of standard scattering theory . We repeat these
analysis here, however, to emphasize that while there is a
rigorous upper limit on cross sections in each individual
scattering channel, there is no general theoretical con-
straint on the total cross section for an object with a given
geometric dimension. Rather, in principle, arbitrarily large
total cross sections can be reached, provided that one
maximizes contributions from a sufficiently large number
For subwavelength objects, those angular momentum
channels that do not support a resonance typically have
very small contributions to the total scattering cross sec-
tion. Therefore, if resonance is present in only one angular
momentum channel, the total scattering cross section be-
comes constrained by the single-channel limit (2?=? in
2D, and ð2l þ 1Þ?2=2? in 3D), as observed by a large
number of studies [2,4,6,8–10]. In contrast, we will show
that one can in fact significantly overcome such a single-
channel limit, by creating resonances in large numbers of
channels, by ensuring that these resonances all operate in
the strong overcoupling limit, and by aligning their reso-
nant frequencies. Below, we will refer to a subwavelength
object having a scattering cross section that far exceeds the
single-channel limit as a superscatterer.
To design a superscatterer we consider the nanorod
schematically shown in the inset of Fig. 1, which consists
of multiple concentric layers of dielectric and plasmonic
materials. The plasmonic material is described by the
Drude model with " ¼ 1 ? !2
tric material has " ¼ 12:96. Similar structures have been
experimentally explored before for other purposes [7,9].
We consider a lossless structure first by setting ?d¼ 0.
The property of the cylindrical structure as shown in Fig. 1
p=ð!2þ i?d!Þ. The dielec-
Scattering cross−section (2λ/π)
FIG. 1 (color online).
rod in the lossless case, and the contributions from individual
channels. The m ¼ ?1 channel is near resonant in the plotted
frequency range. Its resonant line shape is not apparent since its
resonant linewidth is much larger than the frequency range
plotted here. (Inset) Schematic of the nanorod. The dark gray
and light gray areas correspond to a plasmonic material and a
dielectric material, respectively. The geometry parameters are
?1¼ 0:3485?p, ?2¼ 0:5623?p, and ?3¼ 0:6370?p, where
?p¼ 2?c=!p, with c being the speed of light in vacuum.
Total scattering cross section of a nano-
PRL 105, 013901 (2010)
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can be understood by first analyzing a corresponding pla-
nar structure (Inset of Fig. 2). For the TM wave, the
corresponding planar structure supports three photonic
bands (solid lines in Fig. 2). Two of them are associated
with the dielectric layer. When sandwiched between two
semi-infinite metal regions, a thin dielectric layer supports
the lower and upper bands (dashed lines in Fig. 2) .
With a proper choice of the dielectric layer thickness, the
upper band can be made flat . In the planar structure
considered here, since one of the metal regions has a finite
thickness, this upper band anticrosses with the surface
plasmon band at the metal-air interface (dotted line in
Fig. 2), resulting in the band structure shown as solid lines
in Fig. 2.
Starting from modes in the corresponding planar struc-
ture, the resonances of the nanorod can be understood
using the whispering gallery condition [14,18]. The nano-
rod supports the mth order resonance at a frequency !,
when the propagation constant ? of a mode in the corre-
sponding planar structure satisfies:
where r0¼ ð?1þ ?2Þ=2. Thus, the existence of a flat band
in the planar structure should directly translate into near
degeneracy in terms of frequencies between nanorod reso-
nances at different angular momentum channels.
We plot the resonant frequencies and the leakage rates of
the nanorod as a function of jmj=r0in Fig. 2. We indeed
observe resonances between m ¼ ?4 and m ¼ þ4, all
lying close to the frequency of 0:2542!pwhere the planar
structure has a flat band. The positions of these resonances
in general agree quite well with Eq. (9), with a better
agreement observed for larger angular momentum chan-
nels . Also, the leakage rates of these resonances
decease as one increases the angular momentum, in con-
sistency with Refs. [8,15,19].
The spectra for the scattering cross section of the nano-
rod are shown in Fig. 1. The total scattering cross section
reaches a peak value of 7:94ð2?=?Þ, which is far beyond
the single-channel limit, even though the scatterer just has
a subwavelength diameter of 0:32?. Such a large total
scattering cross section is a result of the near degeneracy
of resonances in multiple channels. The contribution from
each channel between m ¼ ?1 and m ¼ ?4 has a
Lorentzian line shape that peaks with a value of 2?=?, at
a frequency around 0:2542!p(Fig. 1).
Figure 3(a) plots the real part of the total field distribu-
tion and the Poynting vector lines at the frequency of
0:2542!p, when a plane wave, with unity amplitude, illu-
behind it. The size of the shadow is much larger than the
diameter of the rod. The presence of the rod also leads to
significant redistribution of the power flow around the rod
We emphasize that the superscattering effect is not an
automatic outcome with the use of plasmonic material. A
uniform plasmonic cylinder of the same size has a much
smaller scattering cross section [Fig. 3(b)].
We now consider the lossy case. For the damping con-
stant in the Drude model, we use ?d¼ ?bulkþ A ? VF=lr
in order to take into account both the bulk and the surface
scattering effects . Here ?bulk¼ 0:002!pis appropri-
ate for the bulk silver at the room temperature , A ? 1
, VF¼ 7:37 ? 10?4?p!pis the Fermi velocity for
silver , and lris the electron mean free path. In our
structure, for the metallic core, we take lr¼ ?1and hence
?d¼ 0:004!p; for the metallic shell, we take lr¼ ?3?
?2and hence ?d¼ 0:012!p. The total scattering cross
section at the peak frequency of 0:2542!pis 1:92ð2?=?Þ
[Fig. 4(a)], which is still about 2 times the single-channel
limit. The contributions to scattering are mostly from the
m ¼ ?1, ?2 channels. In the higher angular momentum
channels, the loss rate dominates overthe radiation leakage
rate. These resonances are therefore no longer in the over-
β / kp, or |m|/(r0kp)
correspond to metal and dielectric material, respectively. a ¼
?2? ?1, b ¼ ?3? ?2. (Main Figure) The solid lines are the
dispersion curves for the structure shown in the inset. The dashed
and dotted lines represent the dispersion curves for a correspond-
ing metal-dielectric-metal structure, and the metal-air interface,
respectively. The bars represent the properties of the resonance
in the scatterer at different angular momentum channels labeled
by m. The center and the height of each bar correspond to the
resonant frequency and the leakage rate, respectively.
A planar structure. The dark and light gray regions
FIG. 3 (color online).
and the Poynting vector lines at the frequency of 0:2542!p
(a) for the nanorod shown in Fig. 1, (b) when the rod is replaced
by a same-size uniform plasmonic cylinder. Here the amplitude
of an incident plane wave is unity. The white circle at the center
indicates the size of the rod.
Real part of the total field distribution
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2 JULY 2010
coupled limit and do not contribute significantly to the
scattering. Since the radiation leakage rate is generally
smaller for the higher angular momentum channel
[8,15,19], the presence of loss has a more significant effect
in the higher angular momentum channels in general.
Nevertheless, our results show that a superscatterer can
be designed even in the presence of realistic loss. Fig-
ure 4(b) plots the total field distribution and the Poynting
vector lines at the frequency of 0:2542!pfor the lossy
case. All thevisual signature of the superscattering effect is
still quite evident.
For the same structure, the absorption cross section has a
peak of 0:73ð2?=?Þ at the frequency of 0:2542!p. Thus
this structure has a scattering cross section that signifi-
cantly dominates over its absorption cross section. In gen-
eral, for a given channel, the absorption cross section is
maximized at the critical coupling when ? ¼ ?0 [cf.
Eq. (8b)]. Basedour theory,therefore, we canseek toeither
maximize or minimize the total absorption cross section as
We remark that in the literature, sometimes the scatter-
ing efficiency, which is defined as the ratio of the scattering
cross sectionover thegeometrical crosssection, rather than
the scattering crosssection itself,is optimized. Anexample
of a subwavelength object that has a very large scattering
efficiency is in fact an atom, since an atom has a scattering
cross section that is at the wavelength scale on resonance
, and a very small geometrical cross section. One may
regard subwavelength particles as electromagnetic ‘‘meta-
atoms’’. From this consideration, it is very unlikely that
scattering efficiency. Instead, our results here indicate that
such a ‘‘meta-atom’’ can have an electromagnetic cross
section beyond what an atom can achieve, and thus repre-
sents a fundamentally new capability.
We end by contrasting our superscatterer with some of
the related works. In Refs. [23,24], the concept of trans-
formation optics has been used to drastically enhance the
scattering cross section of a particle. To implement these
concepts, however, it requires a material whose both per-
mittivity and permeability are highly inhomogeneous and
anisotropic. In contrast, our design requires only conven-
tional plasmonic and dielectric materials. Also, it is well
known that the cross section of an antenna can be drasti-
approach is fundamentally different—our designed struc-
ture is isotropic due to the cylindrical symmetry. Finally,
we have in fact generalized the idea to three dimensions
and designed subwavelength spherical particles with scat-
tering cross section significantly higher than the single-
channel limit of ð2l þ 1Þ?2=2? .
The authors acknowledge the support of the Inter-
connect Focus Center, funded under the Focus Center
Research Program (FCRP), a Semiconductor Research
Corporation entity, and the DOE Grant No. DE-FG
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Scattering cross−section (2λ/π)
FIG. 4 (color online).
nanorod for the lossy case of realistic silver, and the contribu-
tions from each individual channel. The m ¼ ?1 channel is near
resonant in the plotted frequency range. Its resonant line shape is
not apparent since its resonant linewidth is much larger than the
frequency range plotted here. (b) Field distributions and the
Poynting vector lines at the frequency 0:2542!p.
(a) Total scattering cross section of the
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