IOP PUBLISHING JOURNAL OF OPTICS A: PURE AND APPLIED OPTICS

J. Opt. A: Pure Appl. Opt. 9 (2007) S294–S300 doi:10.1088/1464-4258/9/9/S03

Peculiarities of light scattering bynanoparticles and nanowires near plasmonresonance frequencies in weaklydissipating materialsB S Luk‘yanchuk1, M I Tribelsky2, V Ternovsky3, Z B Wang4,M H Hong1, L P Shi1 and T C Chong1

1 Data Storage Institute, Agency for Science, Technology and Research, 117608, Singapore2 Moscow State Institute of Radioengineering, Electronics and Automation (TechnicalUniversity), 78 Vernadskiy Avenue, Moscow 119454, Russia3 M V Lomonosov Moscow State University, Faculty of Computing Mathematics andCybernetics, MSU, Vorobjovy Gory, Moscow 119899, Russia4 School of Mechanical, Aerospace and Civil Engineering, University of Manchester,Manchester M60 1QD, UK

E-mail: Boris L@dsi.a-star.edu.sg

Received 3 February 2007, accepted for publication 12 April 2007Published 22 August 2007Online at stacks.iop.org/JOptA/9/S294

AbstractLight scattering by a small spherical particle and nanowire with lowdissipation rates are discussed according to the Mie theory (and similarsolution for the cylinder). It is shown that near plasmon (polariton) resonancefrequencies one can see non-Rayleigh anomalous light scattering with quite acomplicated near-field energy flux.

Keywords: plasmon polariton resonance, anomalous light scattering,near-field energy flux

(Some figures in this article are in colour only in the electronic version)

1. Introduction

Light scattering by small particles is an important problem formodern applications in plasmonics and nanotechnologies [1].However the majority of research on plasmonics was done formaterials with rather a strong dissipation at plasmon (polariton)resonance frequencies, e.g. gold, platinum, etc. In this caselight scattering by nanostructures can be analysed at the dipoleapproximation (the Rayleigh scattering), i.e. a point dipole forspheres and a linear dipole for nanowires. At the same timeit is well known that all transverse electromagnetic modes forthe particle have finite lifetimes because of radiative damping,see [2]. The Rayleigh scattering is valid provided the radiativedamping is negligible compared to dissipative losses [3–5].Meanwhile a few publications devoted to the study of theopposite limit [6–13] clearly show that light scattering in thiscase is characterized by very unusual properties. In this paper

we refer to certain new aspects of light scattering in materialswith weak dissipation rates.

2. Optical resonances for volume and surface modes

Though light scattering by a spherical particle is one ofthe most fundamental problems of classical electrodynamics,the general physical understanding of the problem has notchanged much since the publication of its exact solution byMie in 1908 [14]. As for light scattering by a particlewhose size is much smaller than the wavelength of incidentlight, its understanding up to now is based upon the approachdeveloped by Lord Rayleigh in 1871 [15]. According tothe approach a small particle should emit electromagneticradiation as an oscillating electric dipole. The point to bemade is that this simple description has quite a general andvery important exception, when the scattering process has

1464-4258/07/090294+07$30.00 © 2007 IOP Publishing Ltd Printed in the UK S294

Light scattering by nanoparticles and nanowires near plasmon resonance frequencies

Figure 1. Amplitudes of the first three optical resonances a�(electric) and b� (magnetic) for nondissipative material ε′′ = 0, atn = 1.5. Optical resonances are reached at the points whereRe a� = 1 or Re b� = 1.

very little in common with the Rayleigh scattering, and theextinction (scattering) cross section differs from that givenby the Rayleigh approximation in orders of magnitude. Thisexception corresponds to the low dissipation limit and will bediscussed below.

The formula for the Rayleigh approximation can beeasily found from the general Mie solution. According tothis solution, the extinction, scattering and absorption crosssections are given by the expression σ = πa2 Q, where relatedefficiencies Q are presented as follows [3–5]:

Qext = 2

q2

∞∑

�=1

(2�+ 1)Re (a� + b�),

Qsca = 2

q2

∞∑

�=1

(2�+ 1){|a�|2 + |b�|2

},

Qabs = Qext − Qsca.

(1)

Here we consider light scattering by a nonmagnetic (μ =1) spherical particle of radius a immersed in a transparentmedium with purely real positive refractive index nm. Thequantity q = nmaω/c represents the size parameter (q � 1for a small particle); here c is the speed of light in vacuumand ω stands for the incident light frequency. The scatteringamplitudes a� (electric) and b� (magnetic) are defined by theMie formulae; it is convenient to write them in the followingway:

a� = �(a)�

�(a)� + i�(a)�

, b� = �(b)�

�(b)� + i�(b)�

, (2)

�(a)� = nψ ′

� (q)ψ� (nq)− ψ� (q) ψ′� (nq) ,

�(a)� = nχ ′� (q)ψ� (nq)− ψ ′

� (nq) χ� (q) ,(3)

�(b)� = nψ ′

� (nq) ψ� (q)− ψ� (nq)ψ ′� (q) ,

�(b)� = nχ� (q)ψ′� (nq)− ψ� (nq) χ ′

� (q) .(4)

Here ψ�(z) = √πz/2 · J�+ 1

2(z), χ�(z) = √

πz/2 ·N�+ 1

2(z), where J�(z) and N�(z) are the Bessel and the

Neumann functions, respectively. The strokes in formulae (3)and (4) indicate differentiation over the entire argument ofthe corresponding functions, i.e. ψ ′

�(z) ≡ dψ�(z)/dz, etc;n = √

ε = n + iκ is a relative complex refractive index, whereε stands for relative dielectric permittivity: ε = εp/εm; indexes‘p’ and ‘m’ indicate the particle and media, respectively. Weconsider that both real and imaginary parts of the relativerefractive index are positive quantities.

The scattering amplitudes a� and b� depend on parameterq and the real and imaginary parts of ε = ε′ + iε′′. Forfixed ε amplitudes a� and b� oscillate versus size parameterq. They reach maximal values at some points (the so-calledoptical resonances [3–5]). For the case of positive ε′ > 0and nondissipative media these resonances were studied innumerous papers, see e.g. [16], due to their important rolein radiation pressure, optical levitation, etc. One can seeimmediately from equation (2) that, for nondissipative media,maximal values of amplitudes are a� = 1 and b� = 1.They are reached at the points, where �(a)� (q, ε) = 0 and�(b)� (q, ε) = 0, respectively. These equations present thetrajectories of optical resonances on the {q, ε} plane. At ε′ > 0the optical resonances are related to excitation of volume wavesin the spherical cavity. It is important that for any reasonablevalues of n = Re

√ε these resonances arise at rather large

values of the size parameter q > 1 and for this case resonancesof electric and magnetic amplitudes are overlapped, see infigure 1.

At ε < −1 other branches of optical resonances related toexcitation of surface electromagnetic waves arise. At q → 0these resonances occur at ε = ε� = −(�+ 1)/�. Thebranches of volume and surface Mie resonances converge atsome negative values of ε, e.g. at ε ≈ −5 and q ≈ 1.2 fordipole resonance � = 1, see in figure 2.

Expanding the Bessel and Neumann functions in powerseries, it is easy to find that at small q

�(a)� ≈ q2�+1 (�+ 1)

[(2�+ 1)!!]2 n�(n2 − 1

),

�(a)� ≈ n��

2�+ 1

[n2 + �+ 1

�− q2

2

(n2 − 1

)

×(

n2

2�+ 3+ �+ 1

� (2�− 1)

)],

�(b)� ≈ − nq2

2�+ 1�(a)� ,

�(b)� ≈ −n�+1

[1 + 1 − n2

2 (2�+ 1)q2

].

(5)

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B S Luk‘yanchuk et al

Figure 2. Trajectories of the five first optical resonances a� (solid)and b� (dashed) for nondissipative material ε′′ = 0, at negative ε.

Far from the resonances � � �. In this case the term with� = 1 (dipole scattering) plays the dominant role. Also forsmall particles one can neglect magnetic amplitudes comparedto the electric ones because of their additional smallness in q.It yields the classical Rayleigh formula:

Qsca ≈ 8

3

∣∣∣∣ε − 1

ε + 2

∣∣∣∣2

q4. (6)

The Rayleigh scattering approximation for a small particleis applicable for all cases far from the optical resonances.Close to the resonances it should be modified, provided thedissipation is small enough (the right-hand side of equation (6)just diverges at ε = −2). Note, there are two possibilities toachieve an optical resonance for small particles. The first wayis related to large values of ε. For example, at ε = 400 opticaldipole magnetic resonance occurs at q ≈ 0.157 and dipoleelectric resonance at q ≈ 0.385. The other way is to work withnegative ε in the vicinity of plasmon (polariton) resonances,where ε ≈ −(�+ 1)/�. In this case optical resonances areassociated with electric amplitudes solely and correspond toresonant excitation of surface plasmon (polariton) modes.

Pronounced peculiarities of light scattering by smallweakly dissipative particles near the plasmon resonancefrequencies differ from the Rayleigh case so dramaticallythat it allows us to name such a scattering ‘anomalousscattering’ [10, 11, 13]. In anomalous scattering far fieldone can see the so-called ‘inverse hierarchy of opticalresonances’ [6, 10, 13]. Namely, at the resonance frequenciesω = ω� the corresponding electric amplitude a� = 1 whileb� is negligibly small. Then, as it follows from equation (1)Q(�)

sca = 2(2�+ 1)/q2, where Q(�)sca stands for the resonance

partial efficiency. As in the vicinity of the resonances the netefficiency is overwhelmingly determined by the correspondingpartial one the expression Q(�)

sca = 2(2�+ 1)/q2 means theresonance scattering cross section increases with an increasein order of the resonance �. Thus, the cross section at thequadrupole resonance is 5/3 of that at the dipole resonance,etc. However to observe this ‘inverse hierarchy’ at least thenecessary condition ε′′ � 1 should be satisfied. Usuallyexperiments are carried out with small particles of gold, silver,mercury and platinum [17]. For all these metals the condition

of weak dissipation at the resonance frequencies, ε′′(ω�) � 1,does not hold. A possible candidate for manifestation of theanomalous scattering may be an additively coloured crystalof KCl with colloidal potassium particles as scatterers [6].Another possible example discussed in [10] is an aluminiumparticle in vacuum. The third example is Na (also as colloidalparticles in crystals of NaCl with stoichiometric excess ofsodium). According to [17] the three materials have weakdissipation rates (about ε′′ ≈ 0.1) at the frequencies ofsurface plasmon excitation, i.e. at λ ≈ 125–140 nm for Al,λ ≈ 310–380 nm for Na and λ ≈ 500–550 nm for K.Our calculations with experimental values for the dielectricfunction show that for Al particles with a = 30 nm theratio of the extinction cross-sections at quadrupole and dipoleresonances is about 1.19 [10]. Naturally, it is smaller than 5/3for a nondissipative particle, but much greater than that forthe Rayleigh approximation. In our calculations [10] we tookinto account the size effect with the help of renormalizationof collision frequency of free electrons due to their collisionswith the particle surface [18], γ → γ∞ + vF/a. The data forFermi velocity vF for this renormalization was also taken fromthe experiment [19].

Optical plasmon resonances for weakly dissipativematerials are extremely sharp. In the case of the Rayleighscattering the width of the resonance line is directly relatedto ε′′ and vanishes at ε′′ → 0. In contrast to that the exactMie solution at ε′′ = 0 near plasmon resonance frequenciesyields the usual Lorentzian contour with a certain characteristicwidth γ�. To show this let us consider the case of a metallicparticle whose dielectric permittivity is described by the Drudeformula:

ε = n2 = 1 − ω2p

ω2 + γ 2+ i

γ

ω

ω2p

ω2 + γ 2. (7)

Here, as usual, ωp denotes the plasma frequency, while γ isthe frequency of electron collisions. Inserting equation (7)in equation (6) in the Raleigh case one obtains a Lorentzianscattering contour,

Q(Ra)sca = 8

3

ω4sp

(ω2 − ω2

sp

)2 + ω2γ 2

q4, (8)

where ωsp = ωp/√

3 stands for frequency of the dipolesurface plasmon resonance at q → 0. As is seen fromequation (8), the resonance width is directly connected withthe parameter responsible for dissipation. The Drude formulacan be written in a similar way in the absence of dissipation(γ = 0). The expression for the partial dipole scatteringefficiency following from equation (1) is Qsca ≈ 6|a1|2/q2.Here a1 = �(a)

1 /(�(a)1 + i�(a)1 ), where �(a)

1 and �(a)1 at q � 1are determined from equation (5). We should remember thatthe plasmon resonance frequencies are defined by the condition�(a)1 = 0, and therefore, in the nearest vicinity of ωsp,�(a)1 ≈i√

2(ω2 − ω2sp)/ω

2sp and �(a)

1 ≈ −2i√

2q3/3. It yields thefollowing Lorentzian profile:

Qsca = 8

3

ω4sp

(ω2 − ω2

sp

)2 + 49 q6ω4

sp

q4. (9)

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Light scattering by nanoparticles and nanowires near plasmon resonance frequencies

Figure 3. The nondissipative limit for a spherical particle with radiusa. Frequency dependence of Qext for three values of q . Notedifferent scales of vertical axis on different panels. Calculationsaccording to the exact Mie solution; ε = 1 − ω2

p/ω2, where

ωsp = ωp/√

3 stands for the frequency of the dipole surface plasmonresonance at q → 0 (ωp is the plasma frequency).

Comparing equations (8) and (9), it is easy to see that the roleof the dissipation parameter in equation (9) is played by thequantity

γeff = 2

3ωspq3 = 2

3

ω4spa3

c3. (10)

This damping is related to the finite plasmon lifetime τp =γ −1

eff caused by the radiative losses. The effects of a finiteplasmon lifetime have already been discussed in the literature.It has been attributed to dissipative (non-radiative) losses andfor a small particle estimated as τp ≈ a/vF, see [18]. Incontrast, in our case, the finiteness of the lifetime is attributedto radiative (nondissipative) losses due to the transformationsof the localized plasmons into scattering light [2, 6, 13] andcorresponding time sharply increases with a decrease in theparticle size: τp ∝ a−3. Formula (10) represents the ‘naturalwidth’ of the dipole resonance related to this transformation.In the general case the natural width of the arbitrary resonanceis given by the following expression [13]:

γ� = (�+ 1)q2�+1

[�(2�− 1)!!]2(dε/dω)�, (11)

where derivative (dε/dω)� is taken at the correspondingplasmon resonance frequency ω = ω�. Note an extremelysharp decrease in γ� with both a decrease in q and an increasein �, see in figure 3.

However this fascinating effect is strongly suppressedby dissipation. The necessary conditions for the anomalousscattering to come into being may be found from the Mietheory, taking into account the dissipation factor ε′′ in thedenominator of the scattering amplitude. This considerationleads to the applicability condition [6, 13]

ε′′ (ω�) � q2�+1

� [(2�− 1)!!]2 . (12)

When this condition is fulfilled the anomalous scattering isdominant. In the opposite case the Rayleigh scattering isrestored. This condition clearly explained numerical resultsfound in [10]. For example, it follows from equation (12) that

Figure 4. Polar scattering diagrams for the scattering of linearlypolarized (red) and nonpolarized (blue) light in the vicinity ofquadrupole resonance for the nondissipative small particle withq = 0.1 and different values of the dielectric permittivity (shown inthe pictures).

with any small ε′′ the anomalous scattering is suppressed forvery small particles. Thus, under real experimental conditionsanomalous scattering can be realized just in some intermediaterange of size parameters and only up to a certain order of theresonances: � < �max.

Another peculiarity of weakly dissipative material is theextra high sensitivity of the angular distribution of scatteringlight near plasmon resonance frequency, see figure 4. Onecan compare this picture with figure 10.14 in [3], wherecalculations are done for a small gold particle. From figure 4follows that very small variation in the incident light frequencychanges the scattering diagram from forward scattering tobackward scattering. Note that for a small perfectly reflectedsphere the ratio of forward and backward scattered intensitiesis 1:9, see problem 2 to § 92 in [20].

For the Rayleigh scattering all components of the scatteredfields vanish at q → 0. In contrast, the Mie theory fornondissipative materials near plasmon resonance frequenciesyields singularities and divergent fields, proportional to q−�−2

for E (�)r,θ,ϕ components of the electric field and proportional to

q−�−1 for H (�)θ,ϕ components. This divergence is stabilized at

ε′′ �= 0, however the inverse size dependence may result invery large enhancement rates of the fields achieved at a smallq.

3. Near-field structure of the energy flux

Though the discussed far-field effects are already quiteunusual, the most appealing manifestation of the anomalousscattering takes place in the near field. The key point is that thedramatic changes in both the modulus and phase of complexamplitude a� in the vicinity of plasmon resonances yield thecorresponding dramatic changes in the near-field structure. Forthe dipole mode at the dipole resonance point (a1 = 1) theexact Mie solution yields the following equation for the fieldlines at x–z plane: dρ/dθ = ρSr/Sθ , where ρ = r/a,and Sr and Sθ are corresponding spherical components of the

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B S Luk‘yanchuk et al

Figure 5. The Poynting vector field in the vicinity of quadrupoleresonance at q = 0.3 and ε = −1.553 . . .. Various singular points(red circles) are marked with Arabic numerals. The red linesrepresent the separatrixes; the blue line denotes the particle surface.Points 1–4 indicate the centres of four localized plasmons(polaritons) on the surface of the particle. Within the particle withweak dissipation these points correspond to centres of the ‘opticalwhirlpools’ discussed in [9]. Points 5–8 and 11 are saddles. Note twooptical vortices (points 9 and 10).

Poynting vector:

Sr = cos θ

2q3ρ3(2q3ρ3 − 3q2ρ2 cos K + (3 − 3q2ρ2) sin K ),

K = qρ(1 − cos θ),

Sθ = sin θ

2q4ρ4(−2q4ρ4 − 9(1 − qρ) cos θ

− 3q2ρ2(2 − qρ cos θ) cos K

+ 3qρ(2 + q2ρ2 cos θ) sin K ).

(13)

At the same time an analogous equation for the field lines at theRayleigh scattering [21] contains singularities at ε′′ → 0. Weshould add also that even small deviations of ω from the exactresonance values make the single-partial-mode approximationinsufficient to describe the near-field distribution [7].

While the far-field effects are restricted by stronginequality (12) the near-field distribution is affected by theanomalous scattering up to much larger dissipation rates. Forexample, for a particle with q = 0.3 complete restorationof the Rayleigh scattering happens only at ε′′ > 0.6 [7].Modifications of the Poynting vector field and a bifurcationdiagram in the vicinity of the dipole resonance have alreadybeen discussed in [7, 11, 12]. Here we present a particularexample of the Poynting vector field in the vicinity of thequadrupole resonance at ε′′ = 0, see figure 5. The Poyntingvector field is shown in the xz plane for incident electric fieldpolarized along the x axis, and a wave propagating along the zaxis. All the singular points are lying in the near field, i.e. allthese field peculiarities have the characteristic scale muchsmaller than the wavelength. It provides a unique opportunityto control optically field distribution in the nanoscale region.

The radiation losses related to transformation of theplasmon into the propagating electromagnetic radiation can beclearly seen in a 3D picture of the near-field distribution of thePoynting vector in figure 6.

Figure 6. The Poynting vector lines in the vicinity of a sphericalparticle for the nondissipative case (incident plane wave with electricfield polarized along the x axis comes from z = −∞). Left insertshows 2D field in the xz plane. Points 1 and 2 are saddles. Thick redlines indicate separatrixes in the xz plane. Field lines in 2D picturedemonstrate circular energy flux around centres (points 3 and 4). Inthe 3D plot one can see the energy flow outward the particle(helicoidally shaped field lines). It illustrates the radiative losses ofenergy, general directions of which are shown by arrows on thebottom xy projection plane.

4. Anomalous light scattering by nanowires

Effects related to the radiative damping are important also fornanowires with surface plasmon (polaritons). This also leadsto deviation of extinction and scattering characteristics fromthe Rayleigh approximations for a linear dipole, e.g. for thescattering efficiency [4]

Q(Ra)sca = π2

4

(ε − 1

ε + 1

)2

q3. (14)

We used a tilde to distinguish cylindrical geometry.Equation (14) is written in normalized dimensionless units; tofind the dimensional cross sections σsca, one should multiplyQsca by the geometrical cross section σgeom. For a cylinderσgeom = 2aL , where a is the radius of the wire and L a isthe length of the cylinder. Quantities q, ε and n have the samemeaning as before.

Scattering of light by an infinite cylinder also has the exactsolution, similar to the Mie solution for a sphere, see e.g. [4].The simplest form this solution has is for the normal incidenceof radiation and TE-mode. In this case the scattering efficiencyis expressed in terms of coefficient a� only [4]:

Qsca = 2

q

∞∑

n =−∞|an|2, where a� = ��

�� + i��and

�� = n J� (nq) J ′� (q)− J ′

� (nq) J� (q) , (15)

�� = n J� (nq) N ′� (q)− J ′

� (nq) N� (q) . (16)

Here J�(ρ) and N�(q) are the Bessel and Neumann functions.Coefficients a� are symmetric: a−� = a�, In contrast to aspherical particle, where the Mie expansion begins with thedipole term (� = 1), the cylinder contains the monopole term(� = 0) also.

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Light scattering by nanoparticles and nanowires near plasmon resonance frequencies

Figure 7. Trajectories of the first resonances (� = 0, 1, 2, 3, 4) foundfrom the equation ��(ε, q) = 0. For the dipole resonance (� = 1) thesurface plasmon branch merges with the volume optical one (forwhich dq(ε)/dε > 0) at q ≈ 0.7 and ε ≈ −1.3. The trajectories ofall other resonances with � > 1 demonstrate similar behaviour.

Employing the well-known expansion for the Besselfunctions, [22]:

J� (z) =( z

2

)� ∞∑

p=0

(− z2

4

)p

p!� (�+ p + 1), (17)

we can obtain the following term in the numerator of a� for thecase of small size parameter q � 1 [12]

�� ≈

⎧⎪⎪⎨

⎪⎪⎩

n(n2 − 1

)

16q3, if � = 0

n�−1(n2 − 1

)

22��! (�− 1)! q2�−1, if � > 0.

(18)

The corresponding expansion of �� in powers of small qcontains a product of convergent Bessel functions J�(q) todivergent Neumann functions N�(q). As a result the leadingterm in this expansion is as follows:

�� ≈

⎧⎪⎪⎨

⎪⎪⎩

2n

πq+ . . . , if � = 0

n�−1(n2 + 1

)

πq+ . . . , if � > 0,

(19)

where ellipses indicate dropped higher order in q terms. Theresonances at q → 0 correspond to ε = −1 for all modes with� � 1 in contrast to a spherical particle where all the modescorrespond to different resonant frequencies: ε = −(�+ 1)/�.Such degeneracy is removed at finite q when every mode hasits individual resonance conditions, see below.

Far from the resonance frequencies the inequality |��| |��| holds, thus, a� ≈ −i��/��, which yields

a0 ≈ −iπ

32(ε − 1)q4, a� ≈ −i

π

22��!(�− 1)!ε − 1

ε + 1q2�,

for � > 0. (20)

One can see that the dominant term at small q is representedby the dipole partial mode with � = 1, thus Qsca ≈4|a1|2/q and we arrive at the Rayleigh formula equation (14).

Figure 8. Spectral dependences of extinction efficiencies for aninfinite cylinder (TE-mode) with different values of the sizeparameter: q = 0.1 (a), 0.5 (b) and 1 (c). Three curves in each plotcorrespond to different values of dissipation parameter γ/ωp = 0,3 × 10−3 and 10−2 in the Drude formula equation (7). Fornondissipative material γ = 0 and q = 0.5 and 1.0 the threesequential resonances are seen clearly.

However the exact resonances correspond to the situation when��(ε, q) = 0. Thus, accurate description of the resonancesrequires accounting for the �� contribution in the resonantdenominators. It results in finite and real values a� = 1at the resonant frequencies even in the absence of the usualdissipation, i.e. at Im ε = 0. For this ‘nondissipative’ casewith −1 < ε < 0 all the amplitudes tend to zero at q → 0.With a larger size parameter these amplitudes demonstrateresonances similar to the Mie resonances for a sphere. Forsmall q resonances arise at ε < −1. They occur just formodes with � � 1 and do not happen for the monopole mode� = 0, see equation (19). These resonances are extremelysharp. The trajectories of the resonances are determined byequation ��(ε, q) = 0. To find this equation at small q itsuffices to take into account the term proportional to q2 only.As a result we arrive at the equations

ε + 1 ≈ q2

8(ε − 1)

[2 + ε − 4 log

qC

2

], for � = 1,

ε + 1 ≈ q2

4(ε − 1)

[1

�− 1+ ε

�+ 1

], for � > 1,

(21)where log C ≡ γ ∼= 0.577 is Euler’s constant.

The trajectories of several sequential resonances on theplane of parameters {q, ε} are shown in figure 7. At small qthese trajectories are described by equation (21) and tend tolimit ε = −1 at q → 0. The amplitudes a� are equal oneeverywhere along the corresponding trajectories. Resonancesat small q � 1 correspond to localized surface plasmon(polariton) modes. At certain values of ε they converge withvolume resonances similar to that for the discussed opticalresonances for a sphere, cf figure 2. At small q � 1 surfaceplasmon resonances produce an anomalous light scatteringeffect, see in figure 8.

In the case of the nanowire the near-field structure of theenergy flux turns out to be quite sensitive to fine detuningof frequency of the incident light from the exact resonantfrequencies. Numerous modifications of the Poynting vectorfield were discussed in detail in [12].

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B S Luk‘yanchuk et al

5. Conclusions

Various applications of the anomalous scattering in nanotech-nologies and related fields may be associated with (i) enormousamplification of the incident electromagnetic field in the nearfield; (ii) controllable changes of the near-field structure withchanges of the incident light frequency; (iii) comparable inten-sity of the resonant electromagnetic field at different resonantfrequencies of the incident light, corresponding to different or-ders of resonance, accompanied by quite a different field dis-tribution for each order of the resonance. All this opens newprospects for optical manipulation in the field structure in thenanoscale region.

Acknowledgments

We are very grateful to S I Anisimov, L P Pitaevskiy andN Arnold for discussions and critical comments. This workwas partially supported by Russian Basic Research Foundation(grants 04-02-17225 and 04-02-16972).

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