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.105.013901 PACS numbers: 42.25.�p, 78.66.Bz

Understanding the interaction between optical radiationand individual subwavelength objects is of fundamentalimportance for the study of optical physics, and has prac-tical significance for applications such as imaging, cloak-ing, biomedicine, and optical antennas [1–10]. Thestrength of this interaction is characterized by the scatter-ing and the absorption cross sections, defined as Csct �Psct=I0, Cabs � Pabs=I0. Here, I0 is the intensity of anincident plane wave, Psct and Pabs are the amount of powerscattered or absorbed by the particle, respectively. Whenthe subwavelength object is a single atom in a three-dimensional (3D) vacuum, one can rigorously prove thatits maximum scattering cross section is ð2lþ 1Þ�2=2� atthe atomic resonant frequency, where l is the total angularmomentum of the atomic transition involved. This limitbecomes 3�2=2� for a typical electric dipole transition[11]. Theoretically, in two dimensions, one can similarlyprove that the maximum cross section of an atom cannotexceed 2�=�. These limits in 3D or 2D, for reasonsapparent 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 crosssection is typically less than such single-channel limit[2,4,6,8–10]. But there have in fact been a few reportswhere the cross section goes somewhat beyond such a limit[12].

In this Letter, we provide a theoretical discussion on thelimit of the scattering and absorption cross sections of asubwavelength structure. The theory indicates that, on theone hand, most of the nanostructures indeed should havetheir maximum cross section subject to the single-channellimit, and on the other hand, in plasmonic nanoparticles ornanowires, there is in fact a substantial opportunity tosignificantly overcome this limit. As a numerical demon-stration, we present a subwavelength plasmonic structurewhere 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 inscattering theory. For simplicity, we will illustrate the mainphysics by considering the two-dimensional case where theobstacle is uniform in the z direction. Consider an obstacle

located at the origin, surrounded by air. When a TM planewave, defined by a wave vector k, and with its magneticfield polarized along the z direction, impinges on theobstacle, the total field in the air region outside the scat-terer is

Htotal ¼ H0�eik�r þ X

1

m¼�1imSmH

ð1Þm ðk�Þeim�

�: (1)

Here, we use the convention that the field varies in time asexpð�i!tÞ. (�, �) are the polar coordinates centered at theorigin. Hð1Þm is the mth order Hankel function of the firstkind, and represents an outgoing wave in the far field. Themth term in the summation thus defines the scattered wavein themth angular momentum channel. Sm is referred to asthe scattering coefficient. With the choice of H0 ¼1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiWattmeter

!"02

q, jSmj2 represents the scattered power, normal-

ized by the unit of power which is Wattmeter in 2D.

The scattering coefficient Sm is in fact strongly con-strained by energy conservation consideration. To seethis, we express Eq. (1) alternatively as a summation ofincoming and outgoing waves in each channel:

Htotal ¼X1

m¼�1H0½hþmHð2Þm ðk�Þ þ h�mHð1Þm ðk�Þ�eim� (2)

where Hð2Þm is the mth order Hankel function of the secondkind that represents the incoming wave. hþm ¼ im=2 andh�m ¼ imðSm þ 1=2Þ are the incoming and outgoing waveamplitudes, respectively. With our choice of H0, jhþm j2 andjh�m j2 represent the power carried by incoming and out-going waves in the mth channel. Moreover, for systemswith cylindrical symmetry, the angular momentum is con-served. Consequently, by energy conservation, the reflec-tion coefficient defined as

Rm � h�m

hþm¼ 1þ 2Sm (3)

is subject to

jRmj � 1: (4)

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0031-9007=10=105(1)=013901(4) 013901-1 � 2010 The American Physical Society

http://dx.doi.org/10.1103/PhysRevLett.105.013901

For such a scatterer, the total scattered power isPsct ¼

PmjSmj2, while the total absorbed power is Pabs ¼P

m14 ð1� jRmj2Þ. On the other hand, the incident wave has

an intensity I0 ¼ �2� Wattmeter . As a result, the total scatteringand absorption cross sections become:

Csct ¼X1

m¼�1

2�

�

��������Rm � 1

2

��������2� X

1

m¼�1Csct;m

Cabs ¼X1

m¼�1

�

2�ð1� jRmj2Þ �

X1m¼�1

Cabs;m:

(5)

Using Eq. (4), one can immediately infer that the maximalcontribution of a single channel to the scattering andabsorption cross section is 2�� and

�2� , respectively. The

strongest scattering occurs when Rm ¼ �1, while thestrongest absorption occurs when Rm ¼ 0.

For subwavelength particles, one can maximize thescattering cross section of a single channel by introducinga resonance. In the presence of a resonance in the mthchannel, the scattering process can be modeled by thetemporal coupled-mode equations [13,14]:

dc

dt¼ ð�i!0 � �0 � �Þcþ i

ffiffiffiffiffiffi2�

phþ

h� ¼ hþ þ i ffiffiffiffiffiffi2�p c;(6)

where c is an amplitude and normalized such that jcj2corresponds to the energy of the resonance, !0 is theresonant frequency, �0 is the intrinsic loss rate due tomaterial absorption, and � is the external leakage ratedue to the coupling of the resonance to the outgoingwave. All variables in Eq. (6) are specific to the mthchannel. For notation simplicity we suppress the subscriptm in Eq. (6). In deriving Eq. (6), we assume that, awayfrom the resonant frequency, the scattering from the par-ticle is negligible, hence h� � hþ when c � 0 [14]. For anincident wave at a frequency !, the reflection coefficientcan be obtained from Eq. (6):

Rm ¼ ið!0 �!Þ þ �0 � �ið!0 �!Þ þ �0 þ � : (7)

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

Csct;m ¼ 2���2

ð!�!0Þ2 þ ð�0 þ �Þ2(8a)

Cabs;m ¼ 2����0

ð!�!0Þ2 þ ð�0 þ �Þ2: (8b)

The scattering cross section thus exhibits a Lorentzian

spectral line shape, reaching a maximum of 2���2

ð�0þ�Þ2 atthe resonant frequency !0. In the strong overcouplinglimit, i.e. � � �0, at the resonant frequency the scatteringcross section of a single channel can approach the maximalvalue of 2�� .

The analysis above, especially Eq. (5), reproduces theresults of standard scattering theory [15]. We repeat these

analysis here, however, to emphasize that while there is arigorous upper limit on cross sections in each individualscattering channel, there is no general theoretical con-straint on the total cross section for an object with a givengeometric dimension. Rather, in principle, arbitrarily largetotal cross sections can be reached, provided that onemaximizes contributions from a sufficiently large numberof channels.For subwavelength objects, those angular momentum

channels that do not support a resonance typically havevery small contributions to the total scattering cross sec-tion. Therefore, if resonance is present in only one angularmomentum channel, the total scattering cross section be-comes constrained by the single-channel limit (2�=� in2D, and ð2lþ 1Þ�2=2� in 3D), as observed by a largenumber of studies [2,4,6,8–10]. In contrast, we will showthat one can in fact significantly overcome such a single-channel limit, by creating resonances in large numbers ofchannels, by ensuring that these resonances all operate inthe strong overcoupling limit, and by aligning their reso-nant frequencies. Below, we will refer to a subwavelengthobject having a scattering cross section that far exceeds thesingle-channel limit as a superscatterer.To design a superscatterer we consider the nanorod

schematically shown in the inset of Fig. 1, which consistsof multiple concentric layers of dielectric and plasmonicmaterials. The plasmonic material is described by theDrude model with " ¼ 1�!2p=ð!2 þ i�d!Þ. The dielec-tric material has " ¼ 12:96. Similar structures have beenexperimentally explored before for other purposes [7,9].We consider a lossless structure first by setting �d ¼ 0.

The property of the cylindrical structure as shown in Fig. 1

ρ1

ρ2

ρ3

0.25 0.252 0.254 0.256 0.2580

2

4

6

8

ω/ωp

Sca

tterin

g cr

oss−

sect

ion

(2λ/

π) totalm=±1m=±2m=±3m=±4

FIG. 1 (color online). Total scattering cross section of a nano-rod in the lossless case, and the contributions from individualchannels. The m ¼ �1 channel is near resonant in the plottedfrequency range. Its resonant line shape is not apparent since itsresonant linewidth is much larger than the frequency rangeplotted here. (Inset) Schematic of the nanorod. The dark grayand light gray areas correspond to a plasmonic material and adielectric material, respectively. The geometry parameters are�1 ¼ 0:3485�p, �2 ¼ 0:5623�p, and �3 ¼ 0:6370�p, where�p ¼ 2�c=!p, with c being the speed of light in vacuum.

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can be understood by first analyzing a corresponding pla-nar structure (Inset of Fig. 2). For the TM wave, thecorresponding planar structure supports three photonicbands (solid lines in Fig. 2). Two of them are associatedwith the dielectric layer. When sandwiched between twosemi-infinite metal regions, a thin dielectric layer supportsthe lower and upper bands (dashed lines in Fig. 2) [16].With a proper choice of the dielectric layer thickness, theupper band can be made flat [17]. In the planar structureconsidered here, since one of the metal regions has a finitethickness, this upper band anticrosses with the surfaceplasmon band at the metal-air interface (dotted line inFig. 2), resulting in the band structure shown as solid linesin Fig. 2.

Starting from modes in the corresponding planar struc-ture, the resonances of the nanorod can be understoodusing 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�r0 ¼ 2�m; (9)where r0 ¼ ð�1 þ �2Þ=2. Thus, the existence of a flat bandin the planar structure should directly translate into neardegeneracy in terms of frequencies between nanorod reso-nances at different angular momentum channels.

We plot the resonant frequencies and the leakage rates ofthe nanorod as a function of jmj=r0 in Fig. 2. We indeedobserve resonances between m ¼ �4 and m ¼ þ4, alllying close to the frequency of 0:2542!p where the planar

structure has a flat band. The positions of these resonancesin general agree quite well with Eq. (9), with a betteragreement observed for larger angular momentum chan-

nels [18]. Also, the leakage rates of these resonancesdecease 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 sectionreaches a peak value of 7:94ð2�=�Þ, which is far beyondthe single-channel limit, even though the scatterer just hasa subwavelength diameter of 0:32�. Such a large totalscattering cross section is a result of the near degeneracyof resonances in multiple channels. The contribution fromeach channel between m ¼ �1 and m ¼ �4 has aLorentzian line shape that peaks with a value of 2�=�, ata 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 of0:2542!p, when a plane wave, with unity amplitude, illu-

minates the nanorod. The nanorod leaves a large ‘‘shadow’’behind it. The size of the shadow is much larger than thediameter of the rod. The presence of the rod also leads tosignificant 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. Auniform plasmonic cylinder of the same size has a muchsmaller scattering cross section [Fig. 3(b)].We now consider the lossy case. For the damping con-

stant in the Drude model, we use �d ¼ �bulk þ A VF=lrin order to take into account both the bulk and the surfacescattering effects [20]. Here �bulk ¼ 0:002!p is appropri-ate for the bulk silver at the room temperature [21], A � 1[20], VF ¼ 7:37 10�4�p!p is the Fermi velocity forsilver [22], and lr is the electron mean free path. In ourstructure, for the metallic core, we take lr ¼ �1 and hence�d ¼ 0:004!p; for the metallic shell, we take lr ¼ �3 ��2 and hence �d ¼ 0:012!p. The total scattering crosssection at the peak frequency of 0:2542!p is 1:92ð2�=�Þ[Fig. 4(a)], which is still about 2 times the single-channellimit. The contributions to scattering are mostly from them ¼ �1, �2 channels. In the higher angular momentumchannels, the loss rate dominates over the radiation leakagerate. These resonances are therefore no longer in the over-

0 0.5 1 1.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

ω/ω

p

β / kp, or |m|/(r

0k

p)

a b

airair

FIG. 2. A planar structure. The dark and light gray regionscorrespond to metal and dielectric material, respectively. a ¼�2 � �1, b ¼ �3 � �2. (Main Figure) The solid lines are thedispersion curves for the structure shown in the inset. The dashedand 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 resonancein the scatterer at different angular momentum channels labeledby m. The center and the height of each bar correspond to theresonant frequency and the leakage rate, respectively.

−9 0 9−6

−3

0

3

6

x (λ)

y (λ

)

−9 0 9

−1

−0.5

0

0.5

1

x (λ)

(a) (b)

FIG. 3 (color online). Real part of the total field distributionand the Poynting vector lines at the frequency of 0:2542!p(a) for the nanorod shown in Fig. 1, (b) when the rod is replacedby a same-size uniform plasmonic cylinder. Here the amplitudeof an incident plane wave is unity. The white circle at the centerindicates the size of the rod.

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coupled limit and do not contribute significantly to thescattering. Since the radiation leakage rate is generallysmaller for the higher angular momentum channel[8,15,19], the presence of loss has a more significant effectin the higher angular momentum channels in general.Nevertheless, our results show that a superscatterer canbe designed even in the presence of realistic loss. Fig-ure 4(b) plots the total field distribution and the Poyntingvector lines at the frequency of 0:2542!p for the lossy

case. All the visual signature of the superscattering effect isstill quite evident.

For the same structure, the absorption cross section has apeak of 0:73ð2�=�Þ at the frequency of 0:2542!p. Thusthis 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 ismaximized at the critical coupling when � ¼ �0 [cf.Eq. (8b)]. Based our theory, therefore, we can seek to eithermaximize or minimize the total absorption cross section aswell.

We remark that in the literature, sometimes the scatter-ing efficiency, which is defined as the ratio of the scatteringcross section over the geometrical cross section, rather thanthe scattering cross section itself, is optimized. An exampleof a subwavelength object that has a very large scatteringefficiency is in fact an atom, since an atom has a scatteringcross section that is at the wavelength scale on resonance[11], and a very small geometrical cross section. One mayregard subwavelength particles as electromagnetic ‘‘meta-atoms’’. From this consideration, it is very unlikely thatsuch a ‘‘meta-atom’’ can outperform a real atom in terms ofscattering efficiency. Instead, our results here indicate thatsuch a ‘‘meta-atom’’ can have an electromagnetic crosssection beyond what an atom can achieve, and thus repre-sents a fundamentally new capability.

We end by contrasting our superscatterer with some ofthe related works. In Refs. [23,24], the concept of trans-formation optics has been used to drastically enhance thescattering cross section of a particle. To implement these

concepts, however, it requires a material whose both per-mittivity and permeability are highly inhomogeneous andanisotropic. In contrast, our design requires only conven-tional plasmonic and dielectric materials. Also, it is wellknown that the cross section of an antenna can be drasti-cally enhanced by making the antenna directional [25]. Ourapproach is fundamentally different—our designed struc-ture is isotropic due to the cylindrical symmetry. Finally,we have in fact generalized the idea to three dimensionsand designed subwavelength spherical particles with scat-tering cross section significantly higher than the single-channel limit of ð2lþ 1Þ�2=2� [26].The authors acknowledge the support of the Inter-

connect Focus Center, funded under the Focus CenterResearch Program (FCRP), a Semiconductor ResearchCorporation entity, and the DOE Grant No. DE-FG07ER46426.

[1] S. Nie et al., Science 275, 1102 (1997).[2] J. B. Jackson et al., Proc. Natl. Acad. Sci. U.S.A. 101,

17 930 (2004).[3] L. R. Hirsch et al., Proc. Natl. Acad. Sci. U.S.A. 100,

13 549 (2003); A. Barhoumi et al., Chem. Phys. Lett. 482,171 (2009).

[4] J. Aizpurua et al., Phys. Rev. Lett. 90, 057401 (2003).[5] A. Alù and N. Engheta, Phys. Rev. Lett. 100, 113901

(2008).[6] J. A. Schuller et al., Nat. Photon. 3, 658 (2009).[7] F. Hao et al., Nano Lett. 8, 3983 (2008).[8] M. I. Tribelsky and B. S. Lykyanchuk, Phys. Rev. Lett. 97,

263902 (2006).[9] R. Bardhan et al., J. Phys. Chem. C 114, 7378 (2010).[10] A. E. Miroshnichenko, Phys. Rev. A 80, 013808 (2009).[11] C. J. Foot, Atomic Physics (Oxford University Press,

New York, 2005), pp. 140–142.[12] K. L. Kelly et al., J. Phys. Chem. B 107, 668 (2003); P. K.

Jain et al., ibid. 110, 7238 (2006).[13] R. E. Hamam et al., Phys. Rev. A 75, 053801 (2007).[14] Z. Ruan et al., J. Phys. Chem. C 114, 7324 (2010).[15] H. C. van de Hulst, Light Scattering by Small Particles

(Dover, New York, 1981).[16] E. N. Economou, Phys. Rev. 182, 539 (1969); H. Shin and

S. Fan, Phys. Rev. Lett. 96, 073907 (2006).[17] H. Shin et al., Appl. Phys. Lett. 89, 151102 (2006).[18] P. B. Catrysse et al., Appl. Phys. Lett. 94, 231111 (2009).[19] Z. Jacob et al., Opt. Express 14, 8247 (2006).[20] R. D. Averitt, D. Sarkar, and N. J. HalasPhys. Rev. Lett.

78, 4217 (1997).[21] M. Ordal et al., Appl. Opt. 22, 1099 (1983).[22] C. Kittel and P. McEuen, Introduction to Solid State

Physics (Wiley, New York, 1996).[23] T. Yang et al., Opt. Express 16, 18 545 (2008).[24] W. Wee et al., New J. Phys. 11, 073033 (2009).[25] C. A. Balanis, Antenna Theory: Analysis and Design

(Wiley, New York, 1996), 2nd ed.[26] Z. Ruan and S. Fan, (unpublished).

0.25 0.252 0.254 0.256 0.2580

1

2

3

4

ω/ωp

Sca

tterin

g cr

oss−

sect

ion

(2λ/

π) totalm=±1m=±2m=±3m=±4

−9 0 9−6

−3

0

3

6

x (λ)

y (λ

)

−1

−0.5

0

0.5

1(a) (b)

FIG. 4 (color online). (a) Total scattering cross section of thenanorod for the lossy case of realistic silver, and the contribu-tions from each individual channel. The m ¼ �1 channel is nearresonant in the plotted frequency range. Its resonant line shape isnot apparent since its resonant linewidth is much larger than thefrequency range plotted here. (b) Field distributions and thePoynting vector lines at the frequency 0:2542!p.

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http://dx.doi.org/10.1126/science.275.5303.1102http://dx.doi.org/10.1073/pnas.0408319102http://dx.doi.org/10.1073/pnas.0408319102http://dx.doi.org/10.1073/pnas.2232479100http://dx.doi.org/10.1073/pnas.2232479100http://dx.doi.org/10.1016/j.cplett.2009.09.076http://dx.doi.org/10.1016/j.cplett.2009.09.076http://dx.doi.org/10.1103/PhysRevLett.90.057401http://dx.doi.org/10.1103/PhysRevLett.100.113901http://dx.doi.org/10.1103/PhysRevLett.100.113901http://dx.doi.org/10.1038/nphoton.2009.188http://dx.doi.org/10.1021/nl802509rhttp://dx.doi.org/10.1103/PhysRevLett.97.263902http://dx.doi.org/10.1103/PhysRevLett.97.263902http://dx.doi.org/10.1021/jp9095387http://dx.doi.org/10.1103/PhysRevA.80.013808http://dx.doi.org/10.1021/jp026731yhttp://dx.doi.org/10.1021/jp057170ohttp://dx.doi.org/10.1103/PhysRevA.75.053801http://dx.doi.org/10.1021/jp9089722http://dx.doi.org/10.1103/PhysRev.182.539http://dx.doi.org/10.1103/PhysRevLett.96.073907http://dx.doi.org/10.1063/1.2360187http://dx.doi.org/10.1063/1.3148692http://dx.doi.org/10.1364/OE.14.008247http://dx.doi.org/10.1103/PhysRevLett.78.4217http://dx.doi.org/10.1103/PhysRevLett.78.4217http://dx.doi.org/10.1364/AO.22.001099http://dx.doi.org/10.1364/OE.16.018545http://dx.doi.org/10.1088/1367-2630/11/7/073033