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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

[11]. 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

[12].

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-

terer is

?

m¼?1

Here, we use the convention that the field varies in time as

expð?i!tÞ. (?, ?) are the polar coordinates centered at the

origin. Hð1Þ

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

1

meter

2

, 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:

Htotal¼ H0

eik?rþ

X

1

imSmHð1Þ

mðk?Þeim?

?

:

(1)

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Watt

!"0

q

meterin 2D.

Htotal¼

X

1

m¼?1

H0½hþ

mHð2Þ

mðk?Þ þ h?

mHð1Þ

mðk?Þ?eim?

(2)

where Hð2Þ

kind that represents the incoming wave. hþ

h?

amplitudes, respectively. With our choice of H0, jhþ

jh?

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

mj2and

mj2represent the power carried by incoming and out-

Rm?h?

m

hþ

m

¼ 1 þ 2Sm

(3)

is subject to

jRmj ? 1:

(4)

PRL 105, 013901 (2010)

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0031-9007=10=105(1)=013901(4)013901-1

? 2010 The American Physical Society

Page 2

For such a scatterer, the total scattered power is

Psct¼P

m

an intensity I0¼?

and absorption cross sections become:

????????

Cabs¼

m¼?1

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

2?

P

1

Watt

meter. As a result, the total scattering

Csct¼

X

X

1

m¼?1

1

2?

?

Rm? 1

2

????????

2?

X

X

1

m¼?1

1

Csct;m

?

2?ð1 ? jRmj2Þ ?

m¼?1

Cabs;m:

(5)

?and

?

2?, respectively. The

dc

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 [14]. For an

incident wave at a frequency !, the reflection coefficient

can be obtained from Eq. (6):

ffiffiffiffiffiffi

2?

p

hþ

ffiffiffiffiffiffi

2?

p

c;

(6)

Rm¼ið!0? !Þ þ ?0? ?

ið!0? !Þ þ ?0þ ?:

(7)

Substituting Eq. (7) to (5), we have

Csct;m¼2?

Cabs;m¼2?

?

?2

ð! ? !0Þ2þ ð?0þ ?Þ2

??0

ð! ? !0Þ2þ ð?0þ ?Þ2:

(8a)

?

(8b)

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

value of2?

?.

The analysis above, especially Eq. (5), reproduces the

results of standard scattering theory [15]. We repeat these

?

?2

ð?0þ?Þ2 at

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

of channels.

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-

ρ1

ρ2

ρ3

0.250.2520.2540.2560.258

0

2

4

6

8

ω/ωp

Scattering cross−section (2λ/π)

total

m=±1

m=±2

m=±3

m=±4

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.

013901-2

Total scattering cross section of a nano-

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Page 3

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) [16].

With a proper choice of the dielectric layer thickness, the

upper band can be made flat [17]. 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:

?2?r0¼ 2?m;

(9)

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 [18]. 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-

minatesthenanorod.The nanorodleavesalarge‘‘shadow’’

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

[Fig. 3(a)].

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 [20]. Here ?bulk¼ 0:002!pis appropri-

ate for the bulk silver at the room temperature [21], A ? 1

[20], VF¼ 7:37 ? 10?4?p!pis the Fermi velocity for

silver [22], 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-

0 0.5

β / kp, or |m|/(r0kp)

1 1.5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

ω/ωp

a b

air air

FIG. 2.

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

−909

−6

−3

0

3

6

x (λ)

y (λ)

−909

−1

−0.5

0

0.5

1

x (λ)

(a)

(b)

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

PRL 105, 013901 (2010)

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Page 4

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

well.

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

[11], and a very small geometrical cross section. One may

regard subwavelength particles as electromagnetic ‘‘meta-

atoms’’. From this consideration, it is very unlikely that

such a‘‘meta-atom’’canoutperformarealatomintermsof

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-

callyenhancedbymakingtheantennadirectional[25].Our

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? [26].

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

07ER46426.

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(Wiley, New York, 1996), 2nd ed.

[26] Z. Ruan and S. Fan, (unpublished).

0.250.2520.2540.2560.258

0

1

2

3

4

ω/ωp

Scattering cross−section (2λ/π)

total

m=±1

m=±2

m=±3

m=±4

−909

−6

−3

0

3

6

x (λ)

y (λ)

−1

−0.5

0

0.5

1

(a)

(b)

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

PRL 105, 013901 (2010)

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