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Journal of Quantitative Spectroscopy &Radiative Transfer

Journal of Quantitative Spectroscopy & Radiative Transfer ] (]]]]) ]]]–]]]

0022-40http://d

n Corr5516, UFrance.Tel.: þ3

E-mV. Tishc

PleasTran

journal homepage: www.elsevier.com/locate/jqsrt

Analysis of plasmon resonances on a metal particle

Saïd Bakhti, Nathalie Destouches, Alexandre V. Tishchenko n

Université de Lyon, F-42023 Saint-Etienne, France; CNRS, UMR 5516, Laboratoire Hubert Curien, Université Jean Monnet, 18 rue Pr. LaurasF-42000 Saint-Etienne, France

a r t i c l e i n f o

Article history:Received 15 October 2013Received in revised form17 January 2014Accepted 21 January 2014

Keywords:Metal nanoparticlesLocalized plasmon resonanceNull-field methodPole searching algorithm

73/$ - see front matter & 2014 Elsevier Ltd.x.doi.org/10.1016/j.jqsrt.2014.01.014

esponding author at: Laboratoire Hubertniversité Jean Monnet, 18 rue Pr. Lauras F-4

3 477915819.ail address: alexandre.tishchenko@univ-st-ethenko).

e cite this article as: Bakhti S, et alsfer (2014), http://dx.doi.org/10.1016

a b s t r a c t

An analytical representation of plasmon resonance modes of a metal particle is developedin the basis of the null-field method and its modal expansion of the particle opticalresponse. This representation allows for the characterization of plasmon modes proper-ties, as their spectral position, bandwidth, amplitude and local field enhancement.Moreover, the derivation of a phenomenological equation governing such resonancesrelates them to open resonator behavior. The resonance bandwidth corresponds to theplasmon life-time, whereas its amplitude is related to the coupling coefficient with theincident excitation. An efficient algorithm is used to compute and characterize theresonance parameters of silver spheroids as function of the particle geometry. The normalmodes present on spheres are split into different azimuthal resonant modes in the case ofspheroids, with amplitude depending on the incident polarization and position dependenton the particle aspect ratio.

& 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Optical properties of metal nanoparticles differ signifi-cantly from that of bulk materials [1] and are characterizedby resonances in the spectral response of the particle whichcontribute to both absorption and scattering. The surfaceplasmons of a particle, associated with collective oscillationsof surface charge density at the metal/dielectric interface, canbe coupled with an electromagnetic excitation in the form ofsurface plasmon polaritons (SPPs) resulting in a multimoderesonant electromagnetic response of the particle [2], whereeach mode corresponds to a plasmon-polariton coupledstate. The modal characteristics (the number of excitedmodes, their spectral position and width) intrinsicallydepend on the particle shape, size and on the opticalproperties of the host medium [3]. In the case of noble

All rights reserved.

Curien, CNRS, UMR2000 Saint-Etienne,

ienne.fr (A.

. Analysis of plasmon/j.jqsrt.2014.01.014i

metal particles, resonances occur in the visible spectrum andtheir excitation at resonant frequencies induces a large near-field enhancement confined at nanoscale. This property andthe high sensitivity of the resonance spectral positionrelative to the particle surrounding refractive index, makesuch particles used in an increasing range of applications likesurface-enhanced Raman spectroscopy [4,5], bio-sensing [6],bio-medicine [7] and nano-photonics [8].

Modeling the optical properties of metal particles appearsto be important for understanding the mechanisms under-lying the SPPs and for the design of plasmonic structures fora specific application. In the particular case of a perfectlyspherical particle, the Mie theory is an efficient tool to studythe SPPs resonances because it provides an exact electro-magnetic solution of the scattering problem in sphericalcoordinates. The more general and realistic case of lightscattering by a non-spherical particle has been consideredfor few decades through many theoretical developments [9].Among the large number of available methods, the null-fieldmethod (NFM) [10–12], also called the extended boundarycondition method (EBCM), is an efficient surface integralequation method. It gives the solution of the scattering

resonances on a metal particle. J Quant Spectrosc Radiat

S. Bakhti et al. / Journal of Quantitative Spectroscopy & Radiative Transfer ] (]]]]) ]]]–]]]2

problem through a transition matrix (T-matrix) relatingincident and scattered waves. Like the Mie formulation, thismethod is particularly well suited to study the modalresponse of localized plasmons. There are differentapproaches in the literature concerning study of SPPs modalcharacteristics, based on the Mie theory for spheres [13,14]and on the surface integral eigenvalue technique [15] or theplasmon hybridization theory [16] for more complex struc-tures, but none of them using the NFM.

This paper is mainly concerned with the description ofsingle particle plasmon modes based on the NFM calculationand using an analytical representation of the resonant part ofoptical response in form of singular functions. Each functioncorresponds to a particular mode and contains all resonantmodal characteristics (position, bandwidth and amplitude).

The proposed approach presents following advantages.First it is based on the null-field method which permits arigorous vector analysis of the electromagnetic problem.Secondly it uses a recent development [17,18] in resonancecharacteristic investigation by polar representation of thesystem resonant optical response. This formulation based onan efficient and accurate algorithm allows for the computa-tion of modal characteristics as well as of plasmon fieldsaround the particle. Finally, a notable physical description ofplasmon resonances is derived, relating them to an openresonator behavior. The case of silver spheres and spheroids,widely used in practical applications, is considered to illus-trate the capabilities of our theoretical approach.

2. Scattering problem statement

Consider an incident monochromatic plane wave withelectric field Eincejkr� jωt, interacting with an isolated andnon-concave particle occupying a volume of space D2 bor-dered by a regular surface S. This particle is defined by itspermittivity ε2, magnetic permeability μ2 and its surface S isexpressed in spherical coordinates by radial function r¼R(θ,φ). The particle is surrounded by non-absorbing medium inexternal space D1 having dielectric permittivity ε1 andmagnetic permeability μ1. Both media are supposed to belinear, homogeneous and isotropic. The incident wave islinearly polarized with E0β and E0α the components of theelectric field parallel and orthogonal to the incidence plane,and directed by β0 (zenith) and α0 (azimuthal) angles inspherical coordinates, respectively. The wavenumber in Di isgiven by ki¼ω(εiμi)1/2, k0 is the wave number in free space.The scattering problem consists in finding both scattered(Esca, Hsca) and internal (Eint, Hint) fields. All consideredelectromagnetic fields are time-harmonic, satisfying Max-well's equations

∇� EðrÞ ¼ jωμiHðrÞ∇�HðrÞ ¼ � jωεiEðrÞ

(rADi ð1Þ

and the boundary conditions on the particle surface

eincðrÞþescaðrÞ ¼ eintðrÞhincðrÞþhscaðrÞ ¼ hintðrÞ

(rAS ð2Þ

where e¼n�E and h¼n�H are the tangent surface fieldswith n the unit vector normal to the surface S.

Please cite this article as: Bakhti S, et al. Analysis of plasmonTransfer (2014), http://dx.doi.org/10.1016/j.jqsrt.2014.01.014i

When illuminating a small particle, some energy is lostfrom the incident light through the scattering (radiation)and/or absorption (heating) process. A way to characterizethis energy transformation is to introduce the scatteringand absorption cross-sections Csca and Cabs, respectively,defined as the power removed from the incident light byscattering or absorption normalized to the incident waveintensity. Extinction cross-section Cext, which is the sum ofthese two quantities, represents the total lost power.

3. The null-field method

This section presents a brief description of the null-fieldmethod following the derivation by Doicu et al. [12] andusing their notations. Fields are expanded on the basis oflocalized spherical vector wave functions M1;3

mnðkrÞ andN1;3

mnðkrÞ with indices 1 and 3 corresponding to regular (atthe origin) and radiating solutions, respectively (seeAppendix A). Electric fields of incident, scattered andinternal waves are written in spherical coordinates

EincðrÞ ¼ ∑1

n ¼ 1∑n

m ¼ �namnM1

mnðk1rÞþbmnN1mnðk1rÞ ð3Þ

EscaðrÞ ¼ ∑1

n ¼ 1∑n

m ¼ �nf mnM

3mnðk1rÞþgmnN

3mnðk1rÞ ð4Þ

EintðrÞ ¼ ∑1

n ¼ 1∑n

m ¼ �ncmnM1

mnðk2rÞþdmnN1mnðk2rÞ ð5Þ

Using this formalism, each field can be viewed as asuperposition of spherical waves Mmn and Nmn, weightedby expansion coefficients, and corresponding respectivelyto TE (i.e. with no radial component) and TM (with radialcomponent) polarization. These waves are also known asmagnetic and electric waves.

Scattering parameters can be expressed by means ofexpansion coefficients. The optical cross-sections are givenusing their definition and orthogonality of spherical functions

Csca ¼ π

k21∑1

n ¼ 1∑n

m ¼ �nðjf mnj2þjgmnj2Þ ð6Þ

Cext ¼ � π

k21∑1

n ¼ 1∑n

m ¼ �nReðf mna

n

mnþgmnbn

mnÞ ð7Þ

where the asterisk denotes the complex conjugate.Determining the scattering expansion coefficients

through the null-field method results from the generalnull-field equation

�EincðrÞ ¼∇�ZSeintðr0Þgðk1; r; r0ÞdSðr0Þ

þ jk1

∇� ∇�ZShintðr0Þgðk1; r; r0ÞdSðr0Þ rAD2

ð8Þand the expression of the scattered field from Huygens'

principle:

EscaðrÞ ¼∇�ZSeintðr0Þgðk1; r; r0ÞdSðr0Þ

þ jk1

∇�∇�ZShintðr0Þgðk1; r; r0ÞdSðr0Þ rAD1 ð9Þ

resonances on a metal particle. J Quant Spectrosc Radiat

Fig. 1. Extinction efficiency of a silver sphere versus the wavelength andthe particle size.

S. Bakhti et al. / Journal of Quantitative Spectroscopy & Radiative Transfer ] (]]]]) ]]]–]]] 3

with the Green function:

gðk; r; r0Þ ¼ ejkjr� r0 j

4πjr�r0j ð10Þ

The dyadic Green function is expressed in terms ofvector wave functions as

gðk; r; r0ÞI

¼ jkπ

∑1

n ¼ 1∑n

m ¼ �n

½M3�mnðkr0ÞM1

mnðkrÞþN3�mnðkr0ÞN1

mnðkrÞ�þ irrotational terms; ror0

½M1�mnðkr0ÞM3

mnðkrÞþN1�mnðkr0ÞN3

mnðkrÞ�þ irrotational terms; r4r0

8>>>><>>>>:

ð11Þ

Considering the null-field Eq. (4), with r restricted on aspherical surface included into the volume of the particleD2, and using the expansions of both incident field andgreen dyadic function together with the orthogonality ofvector spherical wave functions on a spherical surface weget

jk21π

ZS

eintðr0ÞUN3

�mnðk1r0ÞM3

�mnðk1r0Þ

!þ j

ffiffiffiffiffiμ2ε2

rhintðr0Þ

"

UM3

�mnðk1r0ÞN3

�mnðk1r0Þ

!#dSðr0Þ ¼ �

amn

bmn

!ð12Þ

Inserting the spherical expansion of intern field and thedefinition of surface fields, one obtains the matrix relationbetween incident and internal expansion coefficients

Q 31ðk1; k2Þcmn

dmn

!¼ �

amn

bmn

!ð13Þ

where matrix Qpq(ka, kb) is defined in Appendix B.The scattered field expansion coefficients are deter-

mined by Huygens' principle and the vector sphericalwave expansion of surface fields, leading to the matrixrelation:

f mn

gmn

!¼Q 11ðk1; k2Þ

cmn

dmn

!ð14Þ

The transition matrix relating the expansion coeffi-cients of scattered and incident fields is found by combin-ing Eqs. (13) and (14)

f mn

gmn

!¼ �Q 11ðk1; k2Þ½Q 13ðk1; k2Þ��1

amn

bmn

!¼ T

amn

bmn

!

ð15Þ

In the case of axially symmetric particles, all Q and Tmatrices are diagonal with respect to the azimuthalindices, i.e. Tij

mnm0n0 ¼ Tijmnmn0δmm0 . In the particular case of

a spherical particle, the T-matrix is diagonal due to theorthogonality of spherical functions, and non-zero elementsare given analytically

T11mnm0n0 ¼ �bn ¼

jnðk1RÞ~jnðk2RÞ� ~jnðk1RÞjnðk2RÞ~hð1Þn ðk1RÞjnðk2RÞ�hð1Þn ðk1RÞ~jnðk2RÞ

δnn0δmm0

ð16Þ

Please cite this article as: Bakhti S, et al. Analysis of plasmonTransfer (2014), http://dx.doi.org/10.1016/j.jqsrt.2014.01.014i

T22mnm0n0 ¼ �an ¼

k1jnðk1RÞ~jnðk2RÞ�k2~jnðk1RÞjnðk2RÞk2 ~h

ð1Þn ðk1RÞjnðk2RÞ�k1h

ð1Þn ðk1RÞ~jnðk2RÞ

δnn0δmm0

ð17Þ

where R is the sphere radius, an and bn are the Lorentz–Miecoefficients as defined by Bohren and Huffman [19],jn and hð1Þn are the spherical functions defined in Appendix A.

In next sections, the null-field method will be appliedto calculate scattering on silver nanoparticles, embeddedinto a dielectric matrix with permittivity ε1¼2.25ε0. Thepermittivity of silver particles is defined by the modifiedDrude model (see Appendix C). All expansions are trun-cated to order n¼Nmax ensuring satisfying convergence ofresults.

4. Representation of plasmon resonances

The surface plasmon polaritons, corresponding to theincident electromagnetic excitation of the surface plas-mons, are characterized by the resonant optical responseof the particle. From general point of view, several polar-iton states exist, each one corresponding to an eigenmodeof the system and appearing in the form of a resonantband in the spectral response of the particle. As anexample, Fig. 1 shows the extinction efficiency of a silversphere versus the incident wavelength and the particleradius. Different resonant bands appear in the spectrum,with maximum wavelength and bandwidth depending onthe sphere size. The aim of our analysis is to provide ananalytical tool to describe and to characterize each reso-nant mode of a particle.

All the particle response information including reso-nances is contained into the scattering expansion coeffi-cients calculated by the null-field method. Moreover,plasmon resonances are assumed to correspond to TMmodes, and then only gmn expansion coefficients containthe resonance information. The key idea is to express each

resonances on a metal particle. J Quant Spectrosc Radiat

S. Bakhti et al. / Journal of Quantitative Spectroscopy & Radiative Transfer ] (]]]]) ]]]–]]]4

of these coefficients as a singular function of the pulsation

gmnðωÞ ¼pmnωmn

ω�ωmnþ ∑

1

k ¼ 0qmnkðω�ωmnÞk ð18Þ

This representation assumes that each consideredcoefficient exhibits a resonance, characterized by thesingular part with complex pulsation ωmn and amplitudepmn of the plasmon resonance. The real part of ωmn

corresponds to the spectral position of the resonancewhereas the imaginary part equals its half-bandwidth.

An efficient algorithm has been developed [17,18] tocompute accurately both amplitude and complex pulsationby filtering the regular part in Eq. (18). The general idea ofthis method is to perform a Nth order numerical derivationof Eq. (18) with respect to the pulsation with N sufficientlylarge to nullify the regular part, permitting the computa-tion of the plasmon resonance pulsation and amplitude.This numerical derivation is done by discretizing theparticle response using N pulsations around the resonanceposition and by applying Newton's divided differences onthe system obtained by considering Eq. (18) for all points.Finally, resonance parameters are expressed in function ofdiscretized values of pulsation and response.

Then we decompose firstly Eq. (18) in Nþ1 linear equa-tions by taking Nþ1 different values of the pulsation ωi.Multiplying each equation by ωi�ωmn, yields

ðωi�ωmnÞgmnðωiÞ ¼ pmnωmnþ ∑1

k ¼ 0qmnkðωi�ωmnÞkþ1 ð19Þ

Applying now the (Nþ1)th order Newton's divideddifference operator to (19) leads to

∑N

i ¼ 0

ðωi�ωmnÞgmnðωiÞ∏N

j¼ 0ja i

ðωi�ωjÞ� 0 ð20Þ

This equation specifies the resonant pulsation

ωmn ¼∑N

i ¼ 0ωigmnðωiÞ=∏Nj¼ 0ja i

ðωi�ωjÞ

∑Ni ¼ 0gmnðωiÞ=∏N

j¼ 0ja i

ðωi�ωjÞð21Þ

Coming back now with the expression (18) multipliedby ω�ωmn and taking Lagrange's form of the left-hand side

ðω�ωmnÞgmnðωÞ ¼ ∑N

i ¼ 0ðωi�ωmnÞgmnðωiÞ ∏

N

j¼ 0ja i

ωmn�ωj

ωi�ωjð22Þ

it gives the resonant amplitude in the limit ω-ωmn

pmn ¼ ∑N

i ¼ 0

ðωi�ωmnÞωmn

gmnðωiÞ ∏N

j¼ 0ja i

ωmn�ωj

ωi�ωjð23Þ

For each resonant scattering expansion coefficient, bothamplitude and complex pulsation of the correspondingresonance can then be determined by computing only fewvalues of the expansion coefficient around the resonanceposition. Practical application of this algorithm shows thattaking six points is sufficient to ensure a good accuracy ofresults.

Please cite this article as: Bakhti S, et al. Analysis of plasmonTransfer (2014), http://dx.doi.org/10.1016/j.jqsrt.2014.01.014i

Here we can introduce the plasmon expansion coeffi-cients

gpmnðωÞ ¼pmn

ω�ωmnð24Þ

and the plasmon field in the basis of scattered fieldexpansion in term of spherical vector wave functions (4)

Epðr;ωÞ ¼ ∑1

n ¼ 1∑n

m ¼ �ngpmnðωÞN3

mnðk1rÞ ð25Þ

In general, all resonant pulsations ωmn are not distinctfor all couples (m, n), and the plasmon expansion coeffi-cients having the same complex pulsation ωl contribute toa single plasmon mode. The lth plasmon field can then bewritten as

Elpðr;ωÞ ¼

ωl

ω�ωl∑μpμN

3μðk1rÞ ð26Þ

where μ corresponds to all (m, n) indexes providingωmn¼ωl. This last relation will be used to compute theplasmon fields of individual plasmon modes.

The main advantage of this method is its ability toextract the pure resonant response from the overallresponse of a particle, with an efficient filtering of scatter-ing or interference effects. The plasmon fields can becomputed, giving the exact contribution of resonances inthe total near-field.

5. Phenomenological analysis

The description of the resonant response of a metalparticle as a singular function of the pulsation permits toderive a phenomenological approach of the plasmonexcitation.

If we define the plasmon expansion coefficient of the lth

mode as a singular part of the response to an excitation

glðωÞ ¼plωl

ω�ωlf 0ðωÞ ð27Þ

with f0(ω) the excitation function (e.g. the incidentfield) and ωl¼ω1þ jω2the plasmon pulsation, its represen-tation, as well as its derivative, in time domain is giventhrough the Fourier transforms

GlðtÞ ¼12π

Z 1

�1

plωl

ω�ωlf 0ðωÞe� jωt dω ð28Þ

dGlðtÞdt

¼ � 12π

Z 1

�1jω

plωl

ω�ωlf 0ðωÞe� jωt dω ð29Þ

A particular combination of these expressions gives

dGlðtÞdt

þ jωlGlðtÞ ¼ � jωlplF0ðtÞ ð30Þ

Both Gl(t) and F0(t) are time-harmonic functions. If weconsider these functions at resonance position pulsationω1, we obtain

F0ðtÞ ¼ ~F 0ðtÞexpð� jω1tÞ ð31Þ

GlðtÞ ¼ ~GlðtÞexpð� jω1tÞ ð32Þ

dGlðtÞdt

¼ d ~GlðtÞdt

expð� jω1tÞ� jω1~GlðtÞexpð� jω1tÞ ð33Þ

resonances on a metal particle. J Quant Spectrosc Radiat

S. Bakhti et al. / Journal of Quantitative Spectroscopy & Radiative Transfer ] (]]]]) ]]]–]]] 5

with ~F 0ðtÞ and ~GlðtÞ the temporal shapes of F0(t) andGl(t) functions at the resonance pulsation ω1. By equalizing(29) and (33), and inserting (31) and (32), yields aftersimplification to

d ~GlðtÞdt

¼ω2~GlðtÞ� jωlpl ~F 0ðtÞ ð34Þ

The latter relation can be interpreted as the coupledmode equation [18]:

dGðtÞdt

¼ � 1τGðtÞþκG0ðtÞ ð35Þ

describing the response G(t) to an excitation G0(t) of anopen resonator with decay time τ and coupling coefficient κ.Eq. (34) is of major importance to understand the physicalbehavior of plasmon resonances. For a given particle geo-metry, each plasmon mode acts as an open resonator with adecay time, also called life-time, related to the resonancebandwidth τ¼�1/ω2, and a coupling coefficient corre-sponding to the plasmon amplitude κ¼� jωlpl.

The computation of plasmon amplitude and pulsationthen permits to determine phenomenological parametersin (35) and to describe thereby the physical behavior ofeach plasmon resonance mode.

6. Results

We consider first the simplest case of a sphericalparticle. The scattering expansion coefficients correspond-ing to the TM modes are given in function of the Lorentz–Mie coefficients

gmn ¼ �anbmn ð36ÞSince the incident expansion coefficients depend only

on the incident polarization, the plasmon resonance infor-mation are totally contained in an coefficients. Accordingto their expression (17), the different plasmon modes donot depend on azimuthal order m, i.e. the lth resonancemode includes contribution of all azimuthal modes glmwith � lrmr l. In other words, all glm expansion coeffi-cients exhibit the same complex pulsation ωl correspond-ing to the eigen pulsation of the lth plasmon mode. Forcomputations, we consider a z-directed and x-polarizedincident plane wave. In this case, the incident expansioncoefficients are non-null only for |m|¼1 modes; therefore,

Fig. 2. Computed values of (a) the position, (b) the half-bandwidth and (c) the aradius. The first five resonance modes are considered.

Please cite this article as: Bakhti S, et al. Analysis of plasmonTransfer (2014), http://dx.doi.org/10.1016/j.jqsrt.2014.01.014i

only gn,71 scattering coefficients are excited. The algo-rithm described in Section 4 is used to plot resonanceposition, half bandwidth and amplitude of silver spheresfor the (arbitrary) first five plasmon modes versus theparticle size (Fig. 2). When increasing the particle radius,the resonance position (Fig. 2a) of each mode is character-ized by a redshift, more pronounced for modes of lowerorder. Such a redshift is accompanied by a widening of theresonance band (Fig. 2b), corresponding physically to anincrease of the decay time (or the life time) of theplasmon. We can note here that similar results can beobtained by finding the complex pulsations satisfying thedispersion relation of a sphere corresponding to thecomplex zeroes of an coefficient denominator [13,14]:

ε2 ~hð1Þn ðk1RÞjnðk2RÞ�ε1h

ð1Þn ðk1RÞ~jnðk2RÞ ¼ 0 ð37Þ

The other parameter characterizing the resonance is itsamplitude p71l physically related to the coupling coeffi-cient of the resonator, or in other words to the overlappingof the incident excitation with the plasmon mode. Depen-dence of the absolute values of the plasmon modescoupling coefficients |ωlp1l|¼ |ωlp�1l| (Fig. 2c) shows anincrease of resonance amplitudes in the range of consid-ered particle sizes. In the case of the first mode, theamplitude reaches a maximum at sphere radius 56 nmand then decreases. Small spheres, e.g., less than 10 nm inradius, are well-known to exhibit the fundamental plasmonresonance mode only.

In the case of a 40 nm radius silver sphere, only thethree first resonance modes appear in the extinctionspectrum (Fig. 3a), the others do not couple sufficientlywith the incident wave to appear in the spectrum, whereasthe found resonance positions match with the resonancesmaxima. The plasmon field of each resonance mode on theparticle surface is plotted in Fig. 3b–f for the first fiveresonance modes at their resonance position using thefollowing formula derived from Eq. (26):

Elpðr;ReðωlÞÞ ¼ j

ωl

ImðωlÞ∑

m ¼ 1;�1pmlN

3mlðk1rÞ ð38Þ

The lth mode is characterized by 2l local maxima of theplasmon field intensity on the particle surface, reflectingthe corresponding surface charge density oscillations. Thefundamental plasmon mode (l¼1) is known as a dipolar

bsolute value of coupling coefficient for silver spheres versus the particle

resonances on a metal particle. J Quant Spectrosc Radiat

Fig. 3. (a) Extinction efficiency spectrum of a 40 nm in radius silver sphere, where vertical dashed lines indicate the resonance position of first five modescomputed using the extraction algorithm. (b)–(f) The electric field intensity at the surface of the particle computed using Eq. (38) for the first five plasmonmodes at their resonance wavelengths.

S. Bakhti et al. / Journal of Quantitative Spectroscopy & Radiative Transfer ] (]]]]) ]]]–]]]6

mode, because it radiates in the far-field as a dipole. In thesame manner, the next mode (l¼2) is called quadrupolarmode. The positions of local field maxima as well as theirmaximum intensity vary greatly from one mode toanother, with a larger field enhancement for the quad-rupolar mode in this particular example.

Next, we consider silver spheroids with a given inci-dent polarization. Computation of the resonant complexpulsations for scattering expansion coefficients shows thateach plasmon resonance of the lth order is split into lþ1different azimuthal modes when the particle is aspherical.For example, the dipolar mode l¼1 appears as twodifferent modes corresponding to poles in scatteringexpansion coefficients for n¼1, m¼0 and n¼1, |m|¼1.The positions of the split dipolar and quadrupolar modesare plotted in Fig. 4 in the case of prolate and oblateparticles (with a constant volume equal to a sphere of40 nm radius). For dipolar modes, the resonance position

Please cite this article as: Bakhti S, et al. Analysis of plasmonTransfer (2014), http://dx.doi.org/10.1016/j.jqsrt.2014.01.014i

is strongly dependent on the aspect ratio of the particle.In the case of silver particles, this property permits to tunethe resonance positions over the entire visible spectrumby changing the geometrical form of the particle.

Contrary to spheres, where resonant properties areindependent of the incident polarization, each plasmonmode amplitude depends on the incident polarization.Fig. 4c and d depicts the amplitude of the split dipolarand quadrupolar modes versus the incident angle for aprolate particle with a 1.5 aspect ratio. Results show in thecase of a dipolar mode that the overlapping of the incidentwave with each mode is strongly dependent on theincident polarization. Thus, the mode corresponding ton¼1, m¼0 has a minimum excited amplitude for anincident wave polarized along the largest particle dimen-sion and a maximum one along the smallest dimension.This mode is therefore a transverse mode and similarly,the mode corresponding to n¼1, |m|¼1 is a longitudinal

resonances on a metal particle. J Quant Spectrosc Radiat

Fig. 4. Resonant wavelength position of dipolar (a) and quadrupolar (b) modes of spheroidal particles of the fixed volume (corresponding to that of sphereof 40 nm radius) depending on the aspect ratio. Absolute value of coupling coefficient for dipolar (c) and quadrupolar (d) modes of a silver prolate with anaspect ratio of 1.5, versus the incident angle with a parallel polarization schematically represented relative to the particle.

S. Bakhti et al. / Journal of Quantitative Spectroscopy & Radiative Transfer ] (]]]]) ]]]–]]] 7

mode. Hence, for a given particle geometry, varyingincident polarization permits to identify the best couplingpolarization for each mode.

Still considering a prolate particle with an aspect ratio of1.5, the several resonant bands appearing in the extinctionspectrum (Fig. 5a) match with the resonant wavelengthpositions computed for dipolar and quadrupolar modes (otherpeaks in the spectrum correspond to higher resonant modes).As in the case of silver sphere, surface plasmon field of eachmode for a prolate particle with an aspect ratio of 1.1 (forwhich the computation of scattered field on the particlesurface using Eq. (4) is possible) is shown in Fig. 5b–fcalculated by the following relation:

Ejmjnp ðr;ReðωjmjnÞÞ ¼ j

ωjmjnImðωjmjnÞ

∑i ¼ m;�m

pinN3inðk1rÞ ð39Þ

Please cite this article as: Bakhti S, et al. Analysis of plasmonTransfer (2014), http://dx.doi.org/10.1016/j.jqsrt.2014.01.014i

Again, local maxima of the plasmon fields appear onthe particle surface, where the number and the localiza-tion of these maxima depend on the considered mode.

7. Conclusion

An analytical representation of plasmon resonant modes ofmetal particles is developed on the basis of the null-fieldmethod and a modal expansion of the particle opticalresponse. The description of the resonant response by asingular function of the pulsation permits to characterize theplasmon modes and calculate their spectral position, band-width, amplitude as well as the local field enhancement. Thisapproach takes the advantage of an efficient filtering ofthe regular part in the particle response, and leads to

resonances on a metal particle. J Quant Spectrosc Radiat

Fig. 5. (a) Extinction efficiency spectrum of a silver prolate particle (aspect ratio¼1.5) where vertical dashed lines indicate the resonance position ofdipolar and quadrupolar splitted modes computed using the extraction algorithm; (b)–(e) Electric field intensity on the particle surface of silver prolate(aspect ratio¼1.1) for the dipolar and quadrupolar split modes at their resonant wavelengths. All results are given for an incident zenith angle of 451 anda parallel polarization.

S. Bakhti et al. / Journal of Quantitative Spectroscopy & Radiative Transfer ] (]]]]) ]]]–]]]8

a phenomenological equation governing plasmon resonancesand relating them to an open resonator behavior. Thus, theresonance bandwidth corresponds to the plasmon life-time,and its amplitude is related to the coupling of the incidentexcitation to the plasmonmode. An accurate algorithm is usedto compute and characterize resonant parameters of silverspheroids as a function of the particle geometry. The plasmonmode of a sphere is split into different azimuthal resonantmodes in the case of spheroids. The amplitude of these modesvaries with the incident polarization whereas the resonancepositions finely depend on the particle geometry. Plasmonfields are given for different resonance modes with differentlocalizations and intensities of maxima around the particle.This paper provides an efficient numerical tool to design

Please cite this article as: Bakhti S, et al. Analysis of plasmonTransfer (2014), http://dx.doi.org/10.1016/j.jqsrt.2014.01.014i

plasmonic structures for specific applications. We plan toextend it to multiple particle systems in future publications.

Acknowledgments

This work was supported by the LABEX MANUTECH-SISE (ANR-10-LABX-0075) of Université de Lyon, withinthe program “Investissements d’Avenir” (ANR-11-IDEX-0007) operated by the French National Research Agency(ANR). The authors thank the French ANR for its financialsupport in the framework of project UPCOLOR no. JCJC2010 1002 1, and the Rhône-Alpes region for the thesisgrant of S.B.

resonances on a metal particle. J Quant Spectrosc Radiat

S. Bakhti et al. / Journal of Quantitative Spectroscopy & Radiative Transfer ] (]]]]) ]]]–]]] 9

Appendix A

Spherical vector wave functions are solution to thevector wave equation ΔAþk2A¼0 and are expressed interms of spherical functions [20]

M1;3mnðkrÞ ¼

1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2nðnþ1Þ

p z1;3n ðkrÞ jmπmn ðθÞeθ�τmn ðθÞeϕ� �

ejmφ

ðA1Þ

N1;3mnðkrÞ ¼

1k∇�M1;3

mnðkrÞ

¼ 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2nðnþ1Þ

p z1;3n ðkrÞkr nðnþ1ÞPm

n ð cos θÞerþ ~z1;3n ðkrÞ

kr τmn ðθÞeθþ jmπmn ðθÞeφ� �

8<:

9=;ejmφ

ðA2Þwhere z1n is the spherical Bessel function jn (correspondingto a regular solution) and z3n denotes the Hankel sphericalfunctions hð1Þn (radiated waves). ~z1;3n ðkrÞ corresponds to thederivative ½krz1;3n ðkrÞ�0. Normalized associated Legendrefunctions Pm

n are defined as [21]

Pmn ðxÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið2nþ1Þðn�mÞ!

2ðnþmÞ!

sð1�xÞm=2 dm

dxmPnðxÞ ðA3Þ

with Pn(x) the Legendre polynomials. Spherical func-tions πmn and τmn are expressed in terms of Legendrefunctions:

πmn ðθÞ ¼Pmn ð cos θÞsin θ

ðA4Þ

τmn ðθÞ ¼ddθ

Pmn ð cos θÞ ðA5Þ

The incident expansion coefficients in Eq. (4) are [22]

amn ¼ � 4jnffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2nðnþ1Þ

p jmπmn ðβ0ÞEβþτmn ðβ0ÞEα� �

e� jmα0 ðA6Þ

bmn ¼ � 4jnþ1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2nðnþ1Þ

p τmn ðβ0ÞEβ� jmπmn ðβ0ÞEα� �

e� jmα0 ðA7Þ

Appendix B

Matrix elements Qpq(ka, kb) are following:

Q pqðka; kbÞ ¼ðQpqÞ11�mnm0n0 ðQpqÞ12�mnm0n0

ðQpqÞ21�mnm0n0 ðQpqÞ22�mnm0n0

" #ðB1Þ

with

ðQpqÞ11mnm0n0

¼ jk2aπ

ZS

½Mqm0n0 ðkbr0Þ � Np

�mnðkar'Þ�Unðr0Þþ

ffiffiffiffiffiffiffiεbμaεaμb

qNq

m0n0 ðkbr0Þ �Mp�mnðkar0Þ

� �Unðr0Þ

8<:

9=;dSðr0Þ

ðB2Þ

ðQpqÞ12mnm0n0

¼ jk2aπ

ZS

½Nqm0n0 ðkbr0Þ � Np

�mnðkar0Þ�Unðr0Þþ

ffiffiffiffiffiffiffiεbμaεaμb

qMq

m0n0 ðkbr0Þ �Mp�mnðkar0Þ

� �Unðr0Þ

8<:

9=;dSðr0Þ

ðB3Þ

Please cite this article as: Bakhti S, et al. Analysis of plasmonTransfer (2014), http://dx.doi.org/10.1016/j.jqsrt.2014.01.014i

ðQpqÞ21mnm0n0

¼ jk2aπ

ZS

½Mqm0n0 ðkbr0Þ �Mp

�mnðkar0Þ�Unðr0Þþ

ffiffiffiffiffiffiffiεbμaεaμb

qNq

m0n0 ðkbr0Þ � Np�mnðkar0Þ

� �Unðr0Þ

8<:

9=;dSðr0Þ

ðB4Þ

ðQpqÞ22mnm0n0

¼ jk2aπ

ZS

½Nqm0n0 ðkbr0Þ �Mp

�mnðkar0Þ�Unðr0Þþ

ffiffiffiffiffiffiffiεbμaεaμb

qMq

m0n0 ðkbr0Þ � Np�mnðkar0Þ

� �Unðr0Þ

8<:

9=;dSðr0Þ

ðB5Þ

Appendix C

Permittivity of silver particles is defined by the mod-ified Drude model [1]:

εðωÞ ¼ εib�ω2Pε0

ω2þ jωΓðC1Þ

with Γ¼Γ0þAvF/r the modified damping constantwhich takes into account the particle dimensions. Γ0 isthe damping constant of bulk silver (Γ0¼17.6 meV), A is amultiplicative constant (A¼1), vF is the Fermi velocity ofelectrons (vF¼1.39�106 m/s) and r is the radius of theequivalent volume sphere radius of the particle. εib is thecontribution of interband transitions (εib¼3.7ε0, supposedconstant in the visible spectrum), ωp is the plasma pulsa-tion of silver ℏωp ¼ 8:89 eV.

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