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1576 J. Opt. Soc. Am. B/Vol. 27, No. 8 /August 2010 M. Conforti and M. Guasoni

Dispersive properties of linear chains of lossymetal nanoparticles

Matteo Conforti* and Massimiliano Guasoni

CNISM and Dipartimento di Ingegeria dell’ Informazione, Università di Brescia, Via Branze 38, 25123 Brescia, Italy*Corresponding author: matteo.conforti@ing.unibs.it

Received April 30, 2010; accepted June 8, 2010;posted June 15, 2010 (Doc. ID 127831); published July 16, 2010

We study the propagation characteristics of optical signals in waveguides composed of linear periodic arrange-ments of metallic nanoparticles embedded in a dielectric host. We find the complex Bloch band diagram for theguided modes including material losses by employing Mie scattering theory as well as coupled dipole approxi-mations. The results of the model are validated through finite element solution of the Maxwell’s equations.© 2010 Optical Society of America

OCIS codes: 260.3910, 130.2790, 240.6680, 260.2030, 000.4430.

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. INTRODUCTIONince the pioneering paper by Quinten et al. [1] the sub-

ect of light propagation in linear chains of metal nano-articles has attracted a lot of research efforts [2–12].his interest is mainly motivated by the wide range of po-ential applications that sweep from the realization of bio-ogical nanosensors [13] to sub-wavelength imaging [14],nd to the merging of electronic circuits to photonic de-ices [15].

All the relevant properties of a guiding structure arembedded in the dispersion relation k=k���, i.e., the func-ional relation between the wavevector of the propagatingode and the operating frequency. Several methods to

alculate the dispersion relation for linear chains of nano-articles were based on the coupled dipole approximationCDA), where the electromagnetic field is supposed to behe sum of the fields emitted by the spheres, treated asoint dipoles with a certain polarizability ����. The firsttudies considered only nearest neighbor interaction andeglected retardation effects, i.e., they considered thetatic polarizability of the particles and only the near-eld of the dipole [2,3]. Later studies showed that retar-ation effects are fundamental and cannot be neglected inhe computation of dispersion relation [4]. The inclusionf these aspects leads to a dispersion equation expresseds an infinite series that diverges when the unavoidableosses of the metal are taken into account. Analytical con-inuation techniques were employed to express the dipoleum in terms of polylogarithmic functions [8], whoseroperties were studied in detail in [10]. In all these stud-es, however, the effect of losses was neglected or treatedt the first order by perturbation techniques. The complexispersion relation fully taking into account the losses inetal was solved by Koenderink and Polman [9] by fixingreal wavevector and finding a complex frequency. Thisethod is, however, unphysical when dealing with aaveguide and gives bad results when the group velocity

0740-3224/10/081576-7/$15.00 © 2

f the propagating mode becomes small or when the decayate is considerable.

In this paper we derive the dispersion relation foranoparticle chains by exploiting the Mie scatteringethod. This dispersion relation is exact when all modes

re considered. In a certain range of parameters only therst-order spherical vector harmonics need to be consid-red, and simple expressions for longitudinal and trans-erse modes are obtained. Interestingly enough, we showhat the dispersion relation reduces to the usual one ob-ained by the CDA, provided that the polarizability of thepheres is calculated by Mie scattering coefficients.

Next we calculate the complex band diagram by nu-erically solving the dispersion relation for lossy par-

icles, by fixing a real frequency and finding a complexavevector. We find a complex dispersion relation that

trongly differs from previous studies [9,10]: losses modifyven the real part of the propagation constant (an effectot captured by first order perturbation) and prevent thexistence of resonator modes [7] characterized by vanish-ng group velocity.

To conclude we compare the results of the Mie modelith the exact Bloch mode dispersion calculated by finitelement solution of the Maxwell equations. We revisit thenite element method formulation for the calculation ofhree-dimensional periodic crystal bands. The methodields a quadratic eigenvalue equation in the Blochavevector modulus. In this case frequency is a param-ter so that the strong dispersion of the metal is easilyaken into account.

The paper is organized as follows. In Section 2 we findhe complex dispersion relation of linear chains of nano-articles following the Mie scattering approach. In Sec-ion 3 we show the complex band diagram for a case of in-erest, highlighting the difference with previous results.n Section 4 we develop the proper formulation of finitelement and compare the results of finite element simu-

010 Optical Society of America

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M. Conforti and M. Guasoni Vol. 27, No. 8 /August 2010 /J. Opt. Soc. Am. B 1577

ation and analytical approach. Section 5 contains theoncluding remarks.

. DISPERSION RELATION OFANOPARTICLE CHAINS: MIE THEORYPPROACH

n this section we develop a model to analyze the proper-ies of a periodic linear chain of nanospheres based on theeneralized Mie theory of Gerardy and Ausloos for a clus-er of spheres [16]. The nanospheres have radius R,enter-to-center spacing d, dielectric constant �s, and arembedded in an infinite matrix with dielectric constantm. Mie theory states that any field can be expressed as ainear combination of the vector spherical harmonicsVSHs) of the first and third kinds m� lm

1 , m� lm3 , n� lm

1 , and n� lm3

ecause they constitute a complete and orthogonal set ofector basis functions [16–18]. Longitudinal waves de-cribed by functions l�lm are neglected, because we assumehat the dielectric constants of the spheres and the matrixre isotropic.We can expand any arbitrary incident electromagnetic

eld (E� i, H� i) in the matrix as linear combination of VSHsf the first kind m� lm

1 �n� and n� lm1 �n� centered in the nth

phere,

E� i = �lm

�ai,lm�n�m� lm1 �n� + bi,lm�n�n� lm

1 �n��,

H� i =kM

i�0��lm

�bi,lm�n�m� lm1 �n� + ai,lm�n�n� lm

1 �n��, �1�

here n refers to the nth sphere in the chain. The index lweeps from 1 to infinity, while m sweeps from −l to +l,nd ai,lm�n� and bi,lm�n� are the coefficients of the linearombination. kM=���m /c is the wavevector in the matrix,0 is the magnetic permeability of vacuum, and � is thengular pulsation of the input field.Starting from here and for the rest of the paper we de-

elop our calculation based on the electric field only, as its usually done [16]. In the frame centered in the nthphere the total electric field E� �n� can be expressed as theum of the incident field [that is a linear combination ofhe VSHs of the first kind m� lm

1 �n� and n� lm1 �n�] and of the

cattered field [that is a linear combination of VSHs of thehird kind m� lm

3 �n� and n� lm3 �n�] [16],

E� �n� = �lm

�alm�n�m� lm1 �n� + blm�n�n� lm

1 �n� + clm�n�m� lm3 �n�

+ dlm�n�n� lm3 �n��. �2�

Functions m� lm1 �n� and n� lm

1 �n� are excited both by the in-ut and scattered fields of the other spheres that are aum of functions of the third kind m� lm

3 �v� and n� lm3 �v� with

�n. For this reason the coefficients alm�n� and blm�n� inq. (2) differ from ai,lm�n� and bi,lm�n� in Eq. (1), relative

o the input field only. In general, any functions m� lm1 �n�

nd n� lm1 �n� produce scattered vector functions, that is, re-

pectively, m� lm3 �n� and n� lm

3 �n�. Coefficients alm�n�, blm�n�,lm�n�, and dlm�n� are linked by means of the scatteringoefficients of the Mie theory [16],

clm�n� = �l�n�alm�n�,

dlm�n� = �l�n�blm�n�, �3�

here �l�n� and �l�n� are the scattering coefficients re-ated to the nth sphere. In the case described in this pa-er, all the spheres have the same radius and dielectriconstant so that the scattering coefficients are equal forll the spheres and do not depend on n. In this case gen-ralized scattering coefficients �m and �n reduce to thecattering coefficients of the single isolated sphere (usu-lly called am and bm in the literature [18] that differ fromm and bm in our notation).The coefficients alm�n� are the sum of ai,lm�n� due to the

nput field and of all the contributions Tpqlm�v ,n�cpq�v�nd Cpqlm�v ,n�dpq�v� due, respectively, to the scatteredunctions m� pq

3 �v� and n� pq3 �v� of all the other spheres in the

atrix so that

alm�n� = clm�n�/�l = ai,lm�n� + �v�n

�pq

�Tpqlm�v,n�cpq�v�

+ Cpqlm�v,n�dpq�v��. �4�

ere Tpqlm�v ,n� is the coupling coefficient between m� lm1 �n�

nd m� pq3 �v� in the frame centered in the nth sphere or,

quivalently, the coupling coefficient between n� lm1 �n� and

� pq3 �v�. Cpqlm�v ,n� is the coupling coefficient between

� lm1 �n� and m� pq

3 �v� in the frame centered in the nth sphere,r, equivalently, the coupling coefficient between m� lm

1 �n�nd n� pq

3 �v�. Similar arguments hold true for coefficientslm�n� so that it is possible to write

blm�n� = dlm�n�/�l = bi,lm�n� + �v�n

�pq

�Cpqlm�v,n�cpq�v�

+ Tpqlm�v,n�dpq�v��. �5�

Equations (4) and (5) constitute a system that allowsne to calculate the coefficients cpq�n� and dpq�n� from thenowledge of the input field [coefficients ai,pq�n� andi,pq�n�]. By inserting cpq�n� and dpq�n� in Eqs. (2) and (3)he total field in the matrix can be exactly calculated.

It is often possible to simplify the system (4) and (5) byonsidering only the scattering coefficients that are sig-ificantly different from zero. A drastic simplification ofhe treatment is possible when only the first coefficient �1s significant: this usually happens when the radius R isufficiently smaller than the wavelength of the inputeld. For the rest of the paper we actually work underhis assumption. In this way only functions n� 1−1

1 , n� 101 , and

� 111 are interacting. In fact by setting �l=0 and �l+1=0 forny l�1, the coefficients cpq and dpq in Eqs. (4) and (5)re equal to zero, except d1−1, d10, and d11, whose valuesre now given by Eq. (5) with m=−1,0,1,

bi,1m�n� = d1m�n�/�1 − �v�n

�q=−1

1

T1q1m�v,n�d1q�v�. �6�

If the chain of nanospheres is located on the x–y plane,hen the functions n� 10

1 are decoupled from n� 1−11 and n� 11

1 ,ecause it can be shown (for example, using resultsn [19]) that T101−1�v ,n�=T1011�v ,n�=T1−110�v ,n�T �v ,n�=0. Moreover, being the spheres equidistant,

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1578 J. Opt. Soc. Am. B/Vol. 27, No. 8 /August 2010 M. Conforti and M. Guasoni

1010�v ,n�=T1010�v−n�; of course also for the other cou-ling coefficients �n ,v� can be substituted by �v−n�. Fromq. (6) it is then possible to write, for the coefficients d10,

bi,10�n� = �1−1d10�n� − �

v�nT1010�v − n�d10�v�

= U1010�n� � d10�n�, �7�

here the symbol � denotes the discrete convolution ande put U1010�0�=�1

−1 and U1010�n�=−T1010�n� for n�0.easoning in a similar way, from Eq. (6) we can obtain for

he coefficients d1−1 and d11 as follows:

bi,1−1�n� = U1−11−1�n� � d1−1�n� + U111−1�n� � d11�n�,

bi,11�n� = U1−111�n� � d1−1�n� + U1111�n� � d11�n�, �8�

here U1−11−1�0�=U1111�0�=�1−1, U111−1�0�=U1−111�0�=0,

nd for any n�0 U1−11−1�n�=−T1−11−1�n�, U111−1�n�−T111−1�n�, U1−111�n�=−T1−111�n�, and U1111�n�−T1111�n�. Moreover, from [19] it is possible to see that1−11−1�n�=T1111�n� and T111−1�n�=T1−111�n� so that1−11−1�n�=U1111�n� and U111−1�n�=U1−111�n�.As far as the calculation of the coupling coefficients is

oncerned, a lot of efforts have been done in order to cal-ulate them in a straightforward and fast way; in thisense, an excellent result has been reached in [19], wheree can find simple analytical formulas. Following the

reatment it is possible to show that for n�0:

T1010�n� = − i3

2

eikM�d·n�

kM�d · n�+

3

2

eikM�d·n�

�kM�d · n��2 + i3

2

eikM�d·n�

�kM�d · n��3 ,

T1−111�n� = i3

4

eikM�d·n�

kM�d · n�−

9

4

eikM�d·n�

�kM�d · n��2 − i9

4

eikM�d·n�

�kM�d · n��3 ,

T1111�n� = − i3

4

eikM�d·n�

kM�d · n�−

3

4

eikM�d·n�

�kM�d · n��2 − i3

4

eikM�d·n�

�kM�d · n��3 .

�9�

Equation (7) can be rewritten in the “spatial frequency”or wavevector) domain by using the discrete time Fourierransform so that

bi,10�k� = U1010�k�d10�k�, �10�

here bi,10�k�, U1010�k�, and d10�k� are, respectively, theourier transforms of the sequences bi,10�n�, U1010�n�, and10�n�. It is then straightforward to calculate d10�n� as a

inear filtering of the input coefficients bi,10�n� by meansf the transfer function 1/U1010�k�.

We can check for the existence of a self-sustainingode of the chain by forcing a vanishing input field [i.e.,

i,10�n�=0] and by looking for those k for which d10�n�0, that is, to find those k’s that make U1010�k�=0. Alsoq. (8) can be rewritten in the frequency domain,

b �k� = U �k�d �k� + U �k�d �k�,

i,1−1 1111 1−1 1−111 11

bi,11�k� = U1−111�k�d1−1�k� + U1111�n�d11�k�, �11�

here the relations U1−11−1�k�=U1111�k� and U1−111�k�U111−1�k� are used. Once again, the coefficients d1−1�n�nd d11�n� are easily calculable as a linear filtering of thenput coefficients bi,1−1�n� and bi,11�n�. As before we forcehe input coefficients bi,1−1�n� and bi,11�n� to zero in ordero find the modes of the chain. In this case the system (11)as nontrivial solutions d1−1�k� and d11�k� only if

ˆ1111�k�±U111−1�k�=0 that corresponds to d11�k�± d1−1�k�, i.e., d11�n�= ±d1−1�n�.We look for k that solves U1010�k�=0 in order to find the

rst mode of the chain; being U1010�0�=�1−1 and using Eq.

9) for T1010�n�, we can write

U1010�k� = �1−1 − �

n=1

�

T1010�n�e−ikn − �n=1

�

T1010�− n�e+ikn = 0,

�12�

hat is (from the definition of polylogarithm [20]),

0 = �1−1 + i

3

2

Li1�ei�kMd−k�� + Li1�ei�kMd+k��

kMd

−3

2

Li2�ei�kMd−k�� + Li2�ei�kMd+k��

�kMd�2

− i3

2

Li3�ei�kMd−k�� + Li3�ei�kMd+k��

�kMd�3 , �13�

here Lip�x� is the polylogarithm function of order p.ˆ

1010�k� is symmetric and periodic of in k so that we canimit the search of the real part of k between 0 and thats coherent with the fact that the system is periodic. If k ishe solution of Eq. (13), then the propagation constant ofhe mode is =k /d, and d10�n�=eikn so that the electricnd magnetic fields of the mode are

E� = �n=−�

�

n� 103 �n�eikn,

H� =kM

i�0� �n=−�

�

m� 103 �n�eikn. �14�

long the direction of alignment (x axis) functions n� 103

osses only the z component, making the electric fieldransverse with respect to the direction of propagation.

The second and third modes of the chain are found byolving U1111�k�±U1−111�k�=0. Being U1111�0�=�1

−1,1−111�0�=0 and using Eq. (9) for T1111�n� and T1−111�n�, it

s possible to recast the second mode equation in this way,

U1111�k� + U1−111�k� = 0, �15�

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M. Conforti and M. Guasoni Vol. 27, No. 8 /August 2010 /J. Opt. Soc. Am. B 1579

0 = �1−1 + 3

Li2�ei�kMd−k�� + Li2�ei�kMd+k��

�kMd�2

+ i3Li3�ei�kMd−k�� + Li3�ei�kMd+k��

�kMd�3 . �16�

The solution of Eq. (16) �0�Re�k�� imposes d1−1�k�d11�k�, then d1−1�n�=d11�n� so that the electric and mag-etic fields of the mode are

E� = �n=−�

�

�n� 1−13 �n� + n� 11

3 �n��eikn,

H� =kM

i�0� �n=−�

�

�m� 1−13 �n� + m� 11

3 �n��eikn. �17�

long the direction of alignment (x axis) functions n� 103

osses only the x component, making the electric field lon-itudinal with respect to the direction of propagation.

The equation for the third mode is U1111�k�−U1−111�k�0 and has the same formulation of Eq. (13): the first and

hird modes are degenerate. This fact is not surprisingince the third mode is simply a 90° rotation of the firstne around the axis along which the chain is lined up.his makes the two modes degenerate because of the cy-

indrical rotational symmetry of the system.Equations (13) and (16) have to be solved in the com-

lex plane. If both spheres and the matrix have purelyeal dielectric constant, then the system is lossless and itan be shown that ��1�2=−Re��1�: this implies that �1

−1

�1� / ��1�2=−1− i Im��1� /Re��1� so that Re��1

−1�=−1. Itan be also shown that Re�U1010�k�−�1

−1�=Re�U1111�k�U1−111�k�−�1

−1�=1 for any k real with kMd�k�. As aonsequence, under this last condition Re�U1010�k��Re�U1111�k�+U1−111�k��=0 when the system is lossless.

t is then possible to look for real k solutions between kMdnd (that is, under the light line) by solvingm�U1010�k��=0 and Im�U1111�k�+U1−111�k��=0. The corre-ponding modes propagate without damping.

When the system is lossy, the condition Re�U1010�k��1

−1�=Re�U1111�k�+U1−111�k�−�1−1�=1 remains verified

kMd�k��, but Re��1−1��−1 so there does not exist a

eal k that solves Re�U1010�k��=0 or Re�U1111�k�U1−111�k��=0: solutions k have to be found in the com-lex plane, as expected.Equations (13) and (16) can be related to the dispersion

elation for the transverse and longitudinal modes ob-ained by the CDA [4,9]. In fact Eqs. (13) and (16) can beewritten as

1 +����

4d3�m�0�T��,� = 0, �18�

1 − 2����

4d3�m�0�L��,� = 0, �19�

here =k /d is the wavevector of the mode, ����i3c3� ���4� � / ��� 3�3� is the “exact” polarizability of

1 m 0 M

he spheres calculated by Mie scattering coefficients [21],nd �L�� ,� and �T�� ,� are sums of polylogarithmicunctions and depend on � through kM���. This correspon-ence tells us that (i) the CDA is exact when the polariz-bility of spheres is calculated from Mie scattering coeffi-ients and the particle radius (separation) is small (large)ith respect to wavelength in the matrix and (ii) we canasily obtain more accurate dispersion relation for biggerarticles by including more VSHs in Eqs. (4) and (5).We want to underline that obtaining the CDA from the

xact Mie theory shows exactly what approximationsave been done and indicates the way to improve the ac-uracy of the treatment when required. In fact in previousorks the CDA was obtained by simply assuming that the

pheres behave as point dipoles with a certain polarizabil-ty, and the mathematical formulation of the scatteredelds is simply assumed. By means of Mie theory we wereble to calculate the field scattered by the nanoparticlesnd the dispersion relation to a degree of accuracy that isxed by the number of the considered VSHs. In the fol-

owing numerical examples we will show that the lowestrder approximation furnish good enough results formall spheres, with size comparable to that usually ex-loited in real experiments.

. COMPLEX DISPERSION RELATIONn this section we solve Eqs. (18) and (19) for a case of in-erest by considering silver nanoparticles embedded in alass substrate. In contrast to previous studies we solveqs. (18) and (19) for a complex propagation constant atfixed real value of frequency �. This is the natural way

o proceed when studying a waveguide: in fact in thehysical situation an electromagnetic field at a fixed realrequency impinges on the waveguide and excites someuided modes that propagate with a possibly complexropagation constant. The imaginary part of fixes theecay length of the guided modes along the structure. Theual way to proceed (fixing a real and finding a complex) was used in previous studies [9] to circumvent someathematical problems. However, this way to proceed is

uitable for chains of finite lengths (i.e., resonators),here we expect to excite a vibrating mode along all the

hains, characterized by a real , and to see its decay inime fixed by the imaginary part of frequency. When themaginary parts of and � are small, and away from zeroroup velocity points, the two methods give approxi-ately the same results. However, in the proximity of

anishing group velocity or in the presence of strongosses (as in the case of metals at optical frequencies) thewo methods give totally different results, as we shall seen this section.

We consider a particle with radius of R=25 nm and aenter-to-center spacing of d=75 nm and we take the di-lectric constant of glass �m=nm

2 =2.25. For what concernshe metal, we used a fitting model based on the dielectriconstant of silver tabulated in [22]. We decided not to usehe popular Drude model since it is not very accurate inhe visible range, especially for what concerns absorption.

Figure 1 (Fig. 2) shows the frequency of the guidedodes as a function of the real (imaginary) part of the

ropagation constant. Let us start our discussion by con-

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1580 J. Opt. Soc. Am. B/Vol. 27, No. 8 /August 2010 M. Conforti and M. Guasoni

idering first dispersion curves calculated neglectingosses (dotted curves): we have one longitudinal mode �L�nd two transverse modes (T1 and T2). Modes L and T1re the usual longitudinal and transverse modes dis-ussed previously in the literature [4] and have purelyeal propagation constant under the light line =�nm /c.bove the light line, also in the absence of losses, theropagation constant becomes complex. The dispersionurve we find is very different from previous studies, ase are solving dispersion relation for complex at a fixed

eal �. For example, considering the longitudinal mode L,e can see that above the light line the real part of theropagation constant describes approximately an arc ofircumference that intersects light line in two points (nor-alized frequencies around 0.17 and 0.13). In the analo-

ous case studied in [9], the real part of dispersion curveescribes approximately a parabola with vertex centeredn =0, implying only an intersection with light line.hese differences are explained by the fact that above the

ight line the losses are very high and modify also the realart of propagation constant. In view of these aspects wean assert that above the light line, calculating the dis-ersion curve by fixing real wavevector leads to totallyrong results.The complex band diagram shows a second transverseode T2 with negative group velocity that has complex

ropagation constant (i.e., decays during propagation)

0.2 0.25 0.3 0.35 0.4 0.45 0.5

0.13

0.14

0.15

0.16

0.17

0.18

0.19

0.2

Re{ β d / (2 π) }

ωd

/(2

πc)

T1

T2

L

ig. 1. (Color online) Real part of the roots ��� of the disper-ion relations (18) and (19). Dotted curves, lossless metal; circlesnd crosses, lossy metal.

−0.6 −0.4 −0.2 0 0.2

0.13

0.14

0.15

0.16

0.17

0.18

0.19

0.2

ωd

/(2

πc

)

Im { β d / (2 π ) }

T2

T1

L

ig. 2. (Color online) Imaginary part of the roots ��� of the dis-ersion relations (18) and (19). Dotted curves, lossless metal;ircles and crosses, lossy metal. The horizontal line �d / �2c�0.169 indicates the zero group velocity of the transverse modein the lossless case.

1

ven in the absence of losses also below the light line. Athe border of the first Brillouin zone �d / �2�=0.5 theropagation constant has a fixed real part �Re�d / �2�0.5� and a varying imaginary part. This mode has beenverlooked in previous studies, as it is not possible to findy fixing a real wavevector. Physically it corresponds ton evanescent mode that has a complex propagation con-tant and decays oscillating during propagation. It indi-ates the presence of a frequency stop band due to the pe-iodic nature of the waveguide.

When losses are considered the dispersion curveshange dramatically (circles and crosses in Figs. 1 and 2).first general feature is that zero group velocity is not al-

owed and the curves Re���� bend in order to avoidoints where � Re� /��=0. For example the longitudinalode displays zero group velocity at the edge of the Bril-

ouin zone in the absence of losses. When losses are in-luded the real part of propagation constant turns back-ard at around Re�d / �2�0.48.The real part of the dispersion curve for transverseode T1 does not depart too much from the lossless curve

or normalized frequencies below 0.165 and the imaginaryart is very low. For higher frequencies, up to 0.17, theosses increase and the real part of is smaller than theossless case and reaches the maximum value of 0.35 for aormalized frequency approximately equal to 0.17, whichorresponds to null group velocity for lossless particles.ncreasing further the frequency causes a fold of theurve that follows now the (lossless) T2 mode and inter-ects again with the light line at the point�d / �2c� ,kd / �2��= �0.26,0.1735�. This behavior is remi-iscent of the surface plasmon mode at the straight inter-ace between a lossy metal and a dielectric [23].

As far as mode T2 is considered, the effect of losses is tomoothen the real part of the dispersion curve eliminat-ng the edge at the junction of the two branches of theossless T2 curve [at ��d / �2c� ,kd / �2��= �0.17,0.5�]. Thenfluence of material losses in this case is quite smallince the imaginary part of propagation constant is highlso in the lossless case. This discussion shows that, whenreal metal is considered, the losses are so high that the

ffects on the dispersion curves cannot be treated by firstrder perturbation, as it is evident from the big influencef absorption also in the real part of propagation constant.

. FINITE ELEMENT SIMULATIONSn order to asses the validity and to check the accuracy ofhe model described in the previous sections, we solvedhe Maxwell’s equations with a finite element method.he usual way to calculate the Bloch modes of a periodictructure is to fix the wavevector and to find the fre-uency by solving a linear eigenvalue problem. Since theetal is strongly dispersive, this entails an iterative cycle

hat can require several iterations; it is possible to evalu-te only a mode at once and the iterations can even notonverge. In order to avoid these complications we refor-ulate the problem into a quadratic eigenvalue problem

or the wavevector at a fixed frequency. While the qua-ratic eigenvalue equation is nonlinear, it is a more trac-able problem than the general nonlinear case above andan yield solutions more efficiently. In addition, this for-

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M. Conforti and M. Guasoni Vol. 27, No. 8 /August 2010 /J. Opt. Soc. Am. B 1581

ulation inherently yields bands of purely imaginary-nd complex-wavevector Bloch modes, which may be par-icularly hard to obtain with nonlinear search routines. Aimilar formulation was exploited previously for the cal-ulation of complex Bloch bands of a two-dimensionallasmonic crystal [24].Starting from the Maxwell’s equations in frequency do-ain, we eliminate the electric field and write the vectorave equation for the magnetic field in the domain R3, representing the unit cell of the structure,

� � �−1 � � H = ��

c �2

H in . �20�

he unit cell is a volume of size Lx ,Ly ,Lz, where Lxd is fixed by the periodicity of the chain along x, and Ly,z are arbitrary. A good choice of these two lengths (forumerical purpose) implies that the mode amplitude athe cell boundary must be decayed to a negligible value.

Due to the periodicity of the dielectric constant �, wean write H=Heik·x, where k= �kx ,ky ,kz� is the wavevec-or and H is periodic, with basic cell the domain . By in-erting this ansatz in Eq. (20) and following the Galerkinrocedure [25], we can write the variational formulationf the problem,

find � � R such that ∃ 0 ” H � V:

��

�−1��+ ik� � H · ��+ ik� � �

= ��

c �2�

H · �, ∀ � � V, �21�

here � : →R3 are test functions chosen from an appro-riate functional space V (details can be found in [26,27]nd references therein). After some algebra and by ex-loiting the vector identities �a�b� ·c=a · �b�c� and �ab� · �a�c�= �a�2�b ·c�− �a ·b��a ·c�, we can rewrite the

ariational formulation of Eq. (21) for the wavenumber,

ind � C such that ∃ 0 ” H � V:

�

�−1�� � H� · �� � �� − ��

c �2�

H · �

= − i��

�−1k · �H � � � ��

−�

�−1k · �� � � � H��− 2�

�−1�H · � − �k · H��k · ���, ∀ � � V, �22�

here =�kx2+ky

2+kz2 is the modulus of wavevector and k

s its unit vector.This integro-differential equation may be transformed

nto matrix format by following the usual finite elementethod discretization [25]: the domain is divided into

everal tetrahedral subdomains (elements) in which lo-ally supported expansion functions are defined, H is ex-

anded in terms of such functions within each element,nd � is taken to be each one of the local expansion func-ions inside each element. Then the following matrix ei-envalue equation in results:

�A −�2

c2 B�x = i�C − D�x + 2Fx.

ere x is a vector containing the coefficients of the expan-ion of H, and matrices A to F may be individually relatedo each integral in Eq. (22) by inspection. Explicit expres-ions for the matrices in this case can be found in [25].he eigenvalue equation may be solved at a fixed fre-uency � [and thus fixed ����] and k-vector direction. Theigenvalue itself is the k-vector amplitude . The mostommon way of solving the quadratic eigenvalue problems by linearization, which results in a (linear) systemwice the original size. We implemented formulation (22)ith a commercial finite element software [28].In the following we show the numerical results for the

wo low-loss modes T1 and L. Figure 3 shows the real andmaginary parts of the dispersion curve T1

��� calculatedy solving dispersion relation (18) and by finite elements.e can see that the agreement between the two methods

f solution is almost perfect, both for the real and imagi-ary parts of wavevector: finite element simulations con-rm the band folding and the impossibility of reachingero group velocity. In fact, at least for the positive sloperanch of the dispersion curve, we can affirm that theosses are inversely proportional to the group velocity andx a lower limit to an attainable group velocity. Onceeached this lower limit, the band folds back and theosses increase dramatically. By looking at the imaginaryart of the mode wavevector, we can estimate a usefulropagation bandwidth from normalized frequencies inhe interval (0.155, 0165), corresponding to a wavelengthange of (450 nm, 480 nm). The insets show three differ-nt slices (x=0,z=0,y=0 planes) of the norm of the mag-etic field for the transverse mode at �d / �2c�=0.16. Theoughnut shape visible in the x–y plane implies an elec-ric dipole mode sitting in the y–z plane, transverse to theirection of alignment of the spheres (x axis), as expected.

ig. 3. (Color online) Transverse mode T1. Real and imaginaryarts of the roots ��� of the dispersion relations (18) (solid line)nd results of finite element simulation (circles). Insets showhree slices (x=0,z=0,y=0 planes) of the norm of the magneticeld for the transverse mode at �d / �2c�=0.16.

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1

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2

2

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2

2

2

2

2

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1582 J. Opt. Soc. Am. B/Vol. 27, No. 8 /August 2010 M. Conforti and M. Guasoni

Figure 4 shows the real and imaginary parts of the dis-ersion curve L��� calculated by solving dispersion rela-ion (19) and by finite elements. In this case the finite el-ment solution shows an appreciable deviation from theDA model. For the real part we can see that the simu-

ated propagation constant is always greater than thenalytical one by a factor of 10%. The imaginary partgrees better for normalized frequencies of up to 0.186,nd then begins to deviate consistently. These effects cane ascribed to the fact that for high frequencies we cannoteglect the influence of multi-pole modes. The insets showhree different slices (x=0,z=0,y=0 planes) of the normf the magnetic field for the transverse mode atd / �2c�=0.183. The doughnut shape visible in the y–zlane implies an electric dipole mode sitting along the di-ection of alignment of the spheres (x axis), as expected.

. CONCLUSIONSn this paper we derived the exact dispersion relation foranoparticle chains by exploiting the Mie scatteringethod. In a certain range of parameters only the first-

rder spherical vector harmonics need to be considered,nd simple expressions for longitudinal and transverseodes are obtained that we showed to reduce to the usual

nes obtained by the CDA (provided that the polarizabil-ty of the spheres is calculated by Mie scattering coeffi-ients). We found a complex dispersion curve thattrongly differs from previous studies: losses modify evenhe real part of the propagation constant and prevent thexistence of resonator modes characterized by vanishingroup velocity. The results of the Mie model agree veryell with the exact Bloch mode dispersion calculated bynite element solution of Maxwell’s equations.

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