Complex dispersion relation of a double chainof lossy metal nanoparticles
Massimiliano Guasoni* and Matteo Conforti
CNISM and Dipartimento di Ingegneria dell’Informazione, Università di Brescia, Via Branze 38, 25123 Brescia, Italy*Corresponding author: [email protected]
Received October 25, 2010; revised February 4, 2011; accepted February 10, 2011;posted March 2, 2011 (Doc. ID 137148); published April 11, 2011
1. INTRODUCTIONMetal optics is becoming a valid route for the miniaturizationof photonic circuits [1,2]. Plasmonic waveguides can be rea-lized by exploiting straight metal–dielectric interfaces [3],slots [4], wedges [5], grooves [6], or linear chains of closelyspaced nanoparticles [7–19].
The coupled-dipole approximation is usually used in orderto calculate the dispersion relation of a linear chain of nano-particles, where the spheres are treated as point dipoles with acertain polarizability αðωÞ. Recently, Conforti and Guasonihave developed a method to calculate the dispersion relationof a lossy linear chain analytically by means of the Mie theoryand numerically by a proper formulation of the finite elementmethod (FEM) for the Maxwell equations [19].
At optical frequency, the linear chain is characterized by atrade-off between the field concentration and propagationlosses: the more the field is concentrated around the nano-structure, the more the imaginary part of the mode wave vec-tor is high. A solution to this trade-off can be offered by pairingtogether two chains of nanoparticles [20,21]. This geometrycan offer high field confinement between the chains, withoutgiving rise to excessively high propagation losses. This struc-ture has been studied recently by exploiting concepts of op-tical nanocircuits, and two propagating modes have beenidentified [21].
In this paper we derive the dispersion relation of the firstsix modes for couples of nanoparticle chains by exploiting theMie scattering method.
We calculate the complex band diagram by numerically sol-ving the dispersion relation for lossy particles, by fixing a realfrequency and finding a complex wave vector. We find a com-plex dispersion relation where the real part of the propagationconstant is modified by the losses.
To conclude, we compare the results of the Mie model withthe exact Bloch mode dispersion calculated by means of a re-vised FEM formulation of the Maxwell equations, in which, asexplained in [19], frequency is a parameter and the strong dis-persion of the metal is easily taken into account.
The paper is organized as follows. In Sections 2 and 3 wefind the complex dispersion relation of a double chain ofnanoparticles following the Mie scattering approach. We showthat, differently from the case of a uniform single chain [19],the dispersion relations cannot be expressed as a linear com-bination of polylogarithms but they need to be calculated as alinear combination of Lerch functions. The treatment can alsobe applied to calculate the dispersion relations of a nonuni-form single chain, in which the center-to-center distancesbetween spheres are not constant but are alternated.
In Section 4 we analyze the dispersion relation in a case ofinterest, that is, two parallel chains of nanoparticles. We putforth evidence that six nondegenerate modes exist differentlyfrom the case of the single chain, where only two nondegene-rate modes exist [19]. We also highlight that, in four of the sixmodes, there is a coupling interaction between the transversemode of a chain and the longitudinal mode of the other chain(and vice versa). In the end, we also show that the usefultransmission band of the double chain can be much greaterthan the band of the single chain (until five times, [19]). InSection 5 we compare the results of the finite element simula-tion with those of the analytical approach. Section 6 containsthe concluding remarks.
The use of Lerch functions, the possibility of treating thenonuniform single chain, the demonstration of the existenceof six modes, and the increased useful band are the mainachievements of this work with respect to the previous onesregarding the uniform single chain [19].
2. DISPERSION RELATION OF A DOUBLECHAIN OF NANOPARTICLES: MIETHEORY APPROACHIn this section we exploit the generalized Mie theory ofGerardy and Ausloos [22] to study the properties of a doublechain of nanospheres. The nanospheres in both chains haveradius R, center-to-center spacing d, dielectric constant ϵs,and they are embedded in an infinite matrix with the dielectricconstant ϵm. The two chains, that we assume, without loss ofgenerality, to be disposed on the x–y plane, aligned along the
M. Guasoni and M. Conforti Vol. 28, No. 5 / May 2011 / J. Opt. Soc. Am. B 1019
x axis, are separated by a distance L on the y axis and a dis-placement s on the x axis (see Fig. 1).
As explained in [22], the total electric field is a linear com-bination of the vector spherical harmonics (VSH) of the firstand third kind ~m1
lmðnÞ, ~n1lmðnÞ, ~m3
lmðnÞ, and ~n3lmðnÞ centered in
the nth sphere, where l sweeps from 1 to infinity, while msweeps from −l to þl. The VSH of the third kind is generatedby the scattering of the VSH of the first kind, and they arelinked to each other by means of the scattering coefficientsΓl and Δl. When the radius R is sufficiently smaller thanthe wavelength of the input field, only the coefficient Δ1 issignificant, so that a great simplification of the treatment oc-curs because only the vector functions ~n1
1−1, ~n110, ~n
111, ~n
31−1, ~n
310,
and ~n311 need to be considered (see [19]). By following the
treatment described in [19], it is then possible to write forthe nth sphere:
bi;1mðnÞ ¼ Δ−11 d1mðnÞ −
Xv≠n
X1q¼−1
T1q1mðv; nÞd1qðvÞ; ð1Þ
where m ¼ f−1; 0; 1g and bi;1mðnÞ are the coefficients of thelinear combination related to the vector functions ~n1
1mðnÞ thatrepresent the incident field, while d1mðnÞ are the coefficientsrelated to ~n3
1mðnÞ that represent the scattered field. T1q1mðv; nÞis the coupling coefficient between ~n1
1mðnÞ and ~n31qðvÞ in the
nth frame. Once the incident field is known (i.e., the coeffi-cients bi;1mðnÞ for any sphere), the scattered field can be cal-culated by solving the system (1) for the coefficients d1mðnÞ.The coupling coefficients can be easily calculated by means ofsimple analytical formulas [23], and when the spheres arelocated on the x–y plane, we have T101−1ðv; nÞ ¼T1011ðv; nÞ ¼ T1−110ðv; nÞ ¼ T1110ðv; nÞ ¼ 0, so that the func-tions ~n1
10 (~n310) are decoupled from ~n3
1−1 and ~n311 (~n1
1−1 and~n111, respectively). Let us now distinguish the even spheres,
which are in the right chain, from the odd spheres, whichare in the left chain (see Fig. 1). Then it is possible to write,for the coefficients d10:
bi;10ð2nÞ ¼ Δ−11 d10ð2nÞ −
Xv≠n
T1010ð2v; 2nÞd10ð2vÞ
−
Xv
T1010ð2vþ 1; 2nÞd10ð2vþ 1Þ; ð2Þ
bi;10ð2nþ 1Þ ¼ Δ−11 d10ð2nþ 1Þ −
Xv
T1010ð2v; 2nþ 1Þd10ð2vÞ
−
Xv≠n
T1010ð2vþ 1; 2nþ 1Þd10ð2vþ 1Þ: ð3Þ
Let us now denote with E and O the quantities related tothe even and odd spheres, respectively. We adopt the nota-tion: bi;10ð2nÞ ¼ bi;10EðnÞ, bi;10ð2nþ 1Þ ¼ bi;10OðnÞ, d10ð2nÞ ¼d10EðnÞ, d10ð2nþ 1Þ ¼ d10OðnÞ, −T1010ð2v; 2nÞ ¼ −T1010ð2vþ1; 2n þ 1Þ ¼ U1010ðv; nÞ, −T1010ð2v þ 1; 2nÞ ¼ E1010ðv; nÞ,−T1010ð2v; 2nþ 1Þ ¼ O1010ðv; nÞ. The spacing d being constantin the two chains U1010ðv; nÞ ¼ U1010ðv − nÞ and similarly forthe other coupling coefficients, (v; n) can be substituted by(v − n), so that Eqs. (2) and (3) are conveniently rewritten as
bi;10EðnÞ ¼ Δ−11 d10EðnÞ þ
Xv≠n
U1010ðv − nÞd10EðvÞ
þ þXv
E1010ðv − nÞd10OðvÞ; ð4Þ
bi;10OðnÞ ¼ Δ−11 d10OðnÞ þ
Xv
O1010ðv − nÞd10EðvÞ
þ þXv≠n
U1010ðv − nÞd10OðvÞ: ð5Þ
By defining U1010ð0Þ ¼ Δ−11 , it is possible to rewrite Eqs. (4)
Finally, Eqs. (6) and (7) can be rewritten in the spatial fre-quency domain by using the discrete time Fourier transform(DTFT):
MA
�d10EðkÞd10OðkÞ
�¼
�bi;10EðkÞbi;10OðkÞ
�; ð8Þ
where
MA ¼�U1010ðkÞ E1010ðkÞO1010ðkÞ U1010ðkÞ
�; ð9Þ
where the accent^denotes the DTFT.All the treatment can be repeated for the coefficients bi;1−1,
bi;11, d1−1 and d11: let us call bi;1mð2nÞ ¼ bi;1mEðnÞ,bi;1mð2nþ 1Þ ¼ bi;1mOðnÞ, d1mð2nÞ ¼ d1mEðnÞ, d1mð2nþ 1Þ ¼d1mOðnÞ, −T1l1mð2v;2nÞ¼−T1l1mð2vþ1;2nþ1Þ¼U1l1mðv−nÞ,−T1l1mð2vþ1;2nÞ¼E1l1mðv−nÞ, −T1l1mð2v;2nþ1Þ¼O1l1mðv;nÞ,with l and m ¼ −1; 1, then considering that U1−11−1ð0Þ ¼U1111ð0Þ ¼ Δ−1
1 and U1−111ð0Þ ¼ U111−1ð0Þ ¼ 0, and using theDTFT, we can write
Fig. 1. The system analyzed in this paper consists of two chains ofspheres over the x–y plane. The chains have distance L, displacements along the x axis, and the spheres (of radius R) in the chains have acenter-to-center distance d. Odd spheres stay in the left chain, andeven spheres stay in the right chain. Note as an example, the angleβð1Þ between the spheres 0 and 3.
1020 J. Opt. Soc. Am. B / Vol. 28, No. 5 / May 2011 M. Guasoni and M. Conforti
Following the treatment in [23], it is possible to find analy-tical formulas for the coupling coefficients:
U1010ðnÞ ¼ i32eidU ðnÞ
dUðnÞ−32eidU ðnÞ
dUðnÞ2− i
32eidU ðnÞ
dUðnÞ3; n ≠ 0
U1111ðnÞ ¼ i34eidU ðnÞ
dUðnÞþ 34eidU ðnÞ
dUðnÞ2þ i
34eidU ðnÞ
dUðnÞ3; n ≠ 0
U1−111ðnÞ ¼ −i34eidU ðnÞ
dUðnÞþ 94eidU ðnÞ
dUðnÞ2þ i
94eidU ðnÞ
dUðnÞ3; n ≠ 0
E1010ðnÞ ¼ i32eidEðnÞ
dEðnÞ−32eidEðnÞ
dEðnÞ2− i
32eidEðnÞ
dEðnÞ3;
E1111ðnÞ ¼ i34eidEðnÞ
dEðnÞþ 34eidEðnÞ
dEðnÞ2þ i
34eidEðnÞ
dEðnÞ3;
E1−111ðnÞ ¼ −i34eid
0EðnÞ
dEðnÞþ 94eid
0EðnÞ
dEðnÞ2þ i
94eid
0EðnÞ
dEðnÞ3;
E111−1ðnÞ ¼ −i34eid
00EðnÞ
dEðnÞþ 94eid
00EðnÞ
dEðnÞ2þ i
94eid
00EðnÞ
dEðnÞ3; ð12Þ
where dUðnÞ is the distance between the 2m sphere and the2ðmþ nÞ sphere in the right chain [or equivalently betweenthe 2mþ 1 sphere and the 2ðmþ nÞ þ 1 sphere in the leftchain] normalized respect to the wave vector kM in the matrix[that is, dUðnÞ ¼ kMdjnj]. Similarly, dEðnÞ indicates the nor-malized distance between the 2m sphere in the right chain andthe 2ðmþ nÞ þ 1 sphere in the left chain, that is, dEðnÞ ¼kM
βðnÞ is the angle between these two last spheres (see Fig. 1),and d00EðnÞ ¼ dEðnÞ þ 2βðnÞ. Moreover, the subsequent equal-ities hold true: U111−1ðnÞ ¼ U1−111ðnÞ, U1111ðnÞ ¼ U1−11−1ðnÞ,E1111ðnÞ ¼ E1−11−1ðnÞ, O1111ðnÞ ¼ O1−11−1ðnÞ, and O1l1mðnÞ ¼E1l1mð−nÞ (l and m ¼ f−1; 1g).
As consequence, in the matrix MB (11) U111−1ðkÞ ¼U1−111ðkÞ, U1111ðkÞ ¼ U1−11−1ðkÞ, E1111ðkÞ ¼ E1−11−1ðkÞ,O1111ðkÞ ¼ O1−11−1ðkÞ, and O1l1mðkÞ ¼ E1l1mð−kÞ (l andm ¼ f−1; 1g).
In order to find the propagating modes of the double chain,we proceed as follows: we fix a value of frequency ωand impose a vanishing input field [i.e., bi;1−1ðnÞ ¼bi;10ðnÞ ¼ bi;11ðnÞ ¼ 0]. We then search for values of wave vec-tor k that give nontrivial solutions of systems (9) and (10), byimposing DetðMAðkÞÞ ¼ 0 and DetðMBðkÞÞ ¼ 0. These valuesof k represent the propagation constants of the modes nor-malized with respect to the period d of the system. Thecorresponding coefficients dlmEðnÞ and dlmOðnÞ (l and
m ¼ f−1; 0; 1g) are of the form DðkÞeikn [DðkÞ constant inde-pendent of n]. In the next section, we discuss the search for kin the complex plane.
3. DERIVATION OF THE COMPLEXDISPERSION RELATIONSThe system being periodic, the real part of k falls between 0and π. It is possible to show that under the condition of k realand kMd < k < π (i.e., under the light line), the DTFTs in MA
and MB, except U1010ðkÞ and U1111ðkÞ, have a vanishing realpart, and ReðU1010ðkÞ −Δ−1
1 Þ ¼ ReðU1111ðkÞ −Δ−11 Þ ¼ 1. If
the system is lossless, it is also verified that ReðΔ−11 Þ ¼ −1
[19], implying ReðU1010ðkÞÞ ¼ ReðU1111ðkÞÞ ¼ 0. To sum up,in a lossless system, the DTFTs in MA and MB have a vanish-ing real part if kMd < k < π, making Im½DetðMAðkÞÞ� ¼Im½DetðMBðkÞÞ� ¼ 0; then, it is possible to find modes that pro-pagates without damping by looking for real k between kMdand π that solve Re½DetðMAðkÞÞ� ¼ 0 or Re½DetðMBðkÞÞ� ¼ 0. Itis worth noting that, k being real, these last two equations arewell posed, because the summations in the DTFTs convergeand can be evaluated by truncating them to a finite numberof terms.
When the system is lossy ReðΔ−11 Þ ≠ −1, then ReðU1010ðkÞÞ ≠
0 and ReðU1111ðkÞÞ ≠ 0. In this case Im½DetðMAðkÞÞ� ≠ 0 andIm½DetðMBðkÞÞ� ≠ 0 even for k under the light line. As a con-sequence, DetðMAðkÞÞ ¼ 0 and DetðMBðkÞÞ ¼ 0 have no realsolutions, and solutions have to be found in the complexplane. This means that the modes propagate necessarily withdamping. In this case, being k complex, the DTFTs of MAðkÞand MBðkÞ diverge, making the problem ill posed.
In [19] it is shown that U1l1mðkÞ (for l and m ¼ f−1; 0; 1g)can be rewritten as the sum of the polylogarithms and thenevaluated by exploiting the analytic continuation, as sug-gested by Citrin [14], so that the difficulties entailed by thepresence of a complex k can be overcome. The other DTFTscannot be expressed directly as a sum of the polylogarithmsbecause of the term
can manage to write them as functions that admit an analyticcontinuation, in order to avoid the divergence problem. Letus then take, first of all, the case of L ¼ 0, so thatdEðnÞ ¼ kM jdnþ sj, and let us consider, as example, the DTFTE1010ðkÞ:
E1010ðkÞ ¼ E1010ð0Þ
þX3v¼1
av
�X∞n¼1
eikM ðdnþsÞ−ikn
kvM ðdnþ sÞv þX∞n¼1
eikM ðdn−sÞþikn
kvMðdn − sÞv�;
ð13Þ
where a1 ¼ ð3i=2Þ, a2 ¼ −ð3=2Þ, and a3 ¼ −ð3i=2Þ. The sum-mations in n can be expressed by means of the Lerch func-tions [24] that, similarly to polylogarithms, admit ananalytic continuation. In fact, the following relation holdstrue:
X∞n¼1
eikM ðdn∓sÞ�ikn
kvMðdn∓sÞv ¼ ðkMdÞ−veikM ðd∓sÞ�ik
× LerchðeikMd�ik; v; 1∓ðs=dÞÞ: ð14ÞThen E1010ðkÞ can be written as a linear combination of the
Lerch functions, and, similarly, all the other DTFTs in MAðkÞ
M. Guasoni and M. Conforti Vol. 28, No. 5 / May 2011 / J. Opt. Soc. Am. B 1021
and MBðkÞ, so that the analytic continuation is possible for allof them. In this way, the divergence problem in the evaluationof DetðMAðkÞÞ and DetðMBðkÞÞ is avoided, and complex solu-tions k can be found in the case L ¼ 0. This case correspondsto a nonuniform linear chain of spheres whose center-to-cen-ter distance is not constant but is, alternately, s and (d − s).
When L ≠ 0, it is not possible to write the DTFTs as a com-bination of the Lerch functions. In this case we expand themin a Taylor series in L around L ¼ 0. In particular, let us con-sider the infinite summations in n of the DTFTs, that possesthe form
P∞
n¼1 f ðnÞ, with
f ðnÞ ¼ eikMffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL2þðdn∓sÞ2
(v ¼ 1; 2; 3). We note that any term f ðnÞ can be expanded inthe Taylor series f T ðnÞ about 0 only if jLj < jðdn∓sÞj. Let usthen take the first N for which jLj < jðdN∓sÞj and rewrite thesums in the following way:
X∞n¼1
f ðnÞ ¼XN−1
n¼1
f ðnÞ þX∞n¼N
f ðnÞ ≈XN−1
n¼1
f ðnÞ þX∞n¼N
f T ðnÞ ¼
¼X∞n¼1
f T ðnÞ þXN−1
n¼1
ðf ðnÞ − f T ðnÞÞ;
ð15Þ
where the approximation f T ðnÞ ≈ f ðnÞ is done for n ≥ N . Theterm
PN−1n¼1ðf ðnÞ − f T ðnÞÞ does not diverge because it is made
over a finite number of elements, and the termP
∞
n¼1 f T ðnÞ canbe expressed as a linear combination of the Lerch functions.To summarize, we succeeded in approximating any infinitesummation in the DTFTs as a sum between a linear combina-tion of the Lerch functions [
P∞
n¼1 f T ðnÞ] and the finite termPN−1n¼1ðf ðnÞ − f T ðnÞÞ: in this way we can exploit the analytic
continuation.Let us note that the approximations can be improved at will
by considering a sufficiently high number of terms in the Tay-lor expansion. Let us note that the proposed methodology canbe extended by taking into account for a higher order VHS(but at the price of a major complexity) so that spheres whoseradius is not much smaller than the wavelength could be takeninto account.
Note also that, even if the developed approach is valid forstrictly spherical particles, a modified version of the Mie the-ory exists for spheroidal particles [25,26], and our methodol-ogy can be applied by exploiting this theory. The derivation ofour method for chains of spheroidal particles follows themethod described for chains composed of spherical particles.
We conclude by noting that the derived theory can also re-present the core of a method to find the modes of other guid-ing structures, such as plasmonic photonic crystal waveguidesor arrays.
4. DESIGN AND ANALYSIS OF A DOUBLECHAIN WITHOUT DISPLACEMENTIn this section, we calculate the dispersion curves for a case ofinterest by considering two parallel chains of silver spheresembedded in glass (ϵm ¼ 2:25), with null displacements ¼ 0, period d ¼ 75nm, distance between the chainsL ¼ 75nm, and radius R ¼ 25nm. The dielectric constant
ϵsðωÞ of the spheres is calculated by means of a fitting modelbased on values tabulated by Johnson and Christy in [27]. Wedecided to avoid the popular Drude model, because it is notaccurate in the visible range, especially when concerningabsorption.
It is well known (see, for example, [19,14] or [10]) that thesingle chain has two transverse (T1 and T2) and one longitu-dinal (L) modes. The two transverse modes are degenerate,because the second is simply a ninety-degree rotation ofthe first around the axis of the chain.
The modes T1 and T2 are found by solving U1010ðkÞ ¼ 0,and the corresponding electric fields can be written, respec-tively, as
Pn ~n
310ðnÞeikn and
Pnð~n3
11ðnÞ − ~n31−1ðnÞÞeikn, whereas
L is found by solving U1111ðkÞ þ U111−1ðkÞ ¼ 0, and the corre-sponding electric field is
Pnð~n3
11ðnÞ þ ~n31−1ðnÞÞeikn.
In the double chain, because the functions ~n110 are de-
coupled from ~n31−1 and ~n3
11, the mode T1 of the left (right)chain is decoupled from the modes T2 and L of the right (left)chain. In this way, we find that two modes of the double chainare generated by the coupling of the modes T1 of the isolatedchains. These modes are found by solving DetðMAðkÞÞ ¼ 0.Having assumed s ¼ 0, we find that E1010ðkÞ ¼ O1010ðkÞ, sothat DetðMAðkÞÞ ¼ 0 implies U1010ðkÞ � E1010ðkÞ ¼ 0, andd10EðnÞ ¼ �d10OðnÞ. Then the solution of U1010ðkÞ þE1010ðkÞ ¼ 0 corresponds to a transverse mode T1i in whichthe two T1 modes of the single chains propagate in phase,whereas the solution of U1010ðkÞ − E1010ðkÞ ¼ 0 correspondsto a transverse mode T1a in which the two T1 modes ofthe single chains propagate in the antiphase.
The mode T2 of the single left (right) chain is not decoupledfrom the mode L of the right (left) chain, and vice versa, so weexpect that in the double chain, some modes exist that arecombinations of T2 and L. They can be found by solvingDetðMBðkÞÞ ¼ 0. Let us note that, s being equal to 0,E1111ðkÞ ¼ O1111ðkÞ, E1−111ðkÞ ¼ O111−1ðkÞ, and E111−1ðkÞ ¼O1−111ðkÞ. It is easy to prove that, in this case, the matrixMBðkÞ has eigenvectors ½d11EðkÞ; d1−1EðkÞ; d11OðkÞ; d1−1OðkÞ�of the form ½a; b; b; a� or ½a; b;−b;−a�. In the first case, wecan write
which means that the electric field given by the left chain is
Xn
aþ b2
½~n311ðnÞ þ ~n3
1−1ðnÞ�eikn
þXn
a − b2
½~n311ðnÞ − ~n3
1−1ðnÞ�eikn; ð16Þ
while the electric field given by the right chain is
Xn
aþ b2
½~n311ðnÞ þ ~n3
1−1ðnÞ�eikn
−
Xn
a − b2
½~n311ðnÞ − ~n3
1−1ðnÞ�eikn: ð17Þ
Then in the left chain, the linear combination of L and T2 isða=2þ b=2ÞLþ ða=2 − b=2ÞT2, while in the right chain, it isða=2þ b=2ÞL − ða=2 − b=2ÞT2 (mode L in phase and T2 inantiphase).
1022 J. Opt. Soc. Am. B / Vol. 28, No. 5 / May 2011 M. Guasoni and M. Conforti
Similar arguments hold true when the eigenvector is of theform ½a; b;−b;−a�: in this case, in the left chain the linear com-bination of L and T2 is −ða=2þ b=2ÞLþ ða=2 − b=2ÞT2, whilein the right chain, it is ða=2þ b=2ÞLþ ða=2 − b=2ÞT2 (mode Lin antiphase and T2 in phase). The coefficients (a=2þ b=2)and (a=2 − b=2) give more or less weight to the modes Land T2 in the linear combination. We expect that, if for a fixedfrequency ω only the mode T2 propagates in the single chain(1:3 < ωd < 1:65 in Fig. 3), then in the double chain, it is domi-nant with respect to L [ða=2þ b=2Þ ≪ ða=2 − b=2Þ], so we candefine two modes where T2 is dominant over L, in which T2 isin phase and L is in antiphase (T2iLa) or vice versa (T2aLi), onthe basis of what we stated above. Conversely, if for a fixedpulsation ω only the mode L propagates in the single chain(ωd > 1:65 in Fig. 5), then L is dominant over T2 in the doublechain, so we can define two modes (LiT2a) and (LaT2i) in
which the two L modes of the single chains are in phase orantiphase, respectively (and vice versa for the T2 modes).
In Figs. 2–5, we plot the dispersion curves [ReðkÞ] and thelosses of the modes [ImðkÞ] in the single chain and in the dou-ble chain. Because the system is lossy, in order to calculatethe DTFTs of MAðkÞ and MBðkÞ, we exploited the proceduredescribed in the previous section, using Eq. (15), stopping thecorrespondent Taylor series at the eighth order. Moreover, weanalyzed the eigenvectors of MBðkÞ (the coefficients(a=2� b=2) in order to understand if the components T2
and L are in phase or in antiphase and if T2 is dominant overL or vice versa.
We note that at low frequencies (ωd < 1:65), we have themodes T1i and T2iLa (T2 is dominant), because in this range,only T1 and T2 are allowed in the single chain. Indeed, thedispersion curve of T2aLi lies in between those of T2 and Lof the single chain, because it is in the range of frequencies(1:56 < ωd < 1:72) in which they both exist in the single
Fig. 2. General view of the real and imaginary part of kðωÞ for thetheoretically predicted modes in the double chain: bold black dashedcurve, mode T1i; bold black curve, T2iLa; thin black curve, T1a; thinblack dashed curve, T2aLi together with LiT2a; black dotted curve,LaT2i. The real part represents the dispersion curve of the mode;the imaginary part represents its losses.
Fig. 3. Real and imaginary part of kðωÞ for the mode T1i (thin blackcurve, theoretical results; triangles, FEM results) and T2iLa (boldblack curve, theoretical results; circles, FEM results). The blackdashed–dotted curve is relative to modes T1 and T2 in the singlechain.
Fig. 4. Real and imaginary part of kðωÞ for the mode T1a (bold blackcurve, theoretical results; circles, FEM results). The black dashed–dotted curve is relative to modes T1 and T2 in the single chain.
Fig. 5. Real and imaginary part of kðωÞ for the mode T2aLi and LiT2a(thin black curve, theoretical results; triangles, FEM results) andLaT2i (bold black curve, theoretical results; circles, FEM results).The black dashed–dotted curve is relative to mode L in the singlechain. The curve relative to T2aLi flows into that of LiT2a startingfrom ωd ≈ 1:72.
M. Guasoni and M. Conforti Vol. 28, No. 5 / May 2011 / J. Opt. Soc. Am. B 1023
chain, so they have nearly the same weight in the linear com-bination that generates T2aLi; the curve of T2aLi flows intothat of LiT2a for frequencies ωd > 1:72, for which L start tobe dominant over T2.
We can also observe that losses are acceptable only formodes T2iLa and T1i and at low frequencies; this is due tothe fact that for the other modes and at high frequencies,the electric field is almost concentrated around the metalspheres, increasing the absorption.
Let us conclude by noting that in the double chain, the use-ful transmission band is greater than in the single chain. If werequire a decay length of 5 μm and a transverse mode decay at1=e of 50nm, the sum of the useful bands of modes T1i
(1:34 < ωd < 1:45) and T2iLa (1:24 < ωd < 1:34) in the dou-ble chain is nearly five times greater than the useful bandof T1 in the single chain (1:44 < ωd < 1:48).
5. FINITE ELEMENT SIMULATIONSWe calculated the dispersion curves (ReðkÞ) and the losses[ImðkÞ] of the modes in the double chain by solving Maxwell’sequations with the FEM, in order to asses the validity of thetheoretical results. When calculating the Bloch modes, it isusual to fix the wave vector and to find the frequency by sol-ving a linear eigenvalue problem. In our case, because the me-tal is dispersive, this requires an iterative cycle. Moreover, thewave vectors are complex due to material losses, making theiterative search even harder. In order to avoid these difficul-ties, we used a different approach, reformulating the probleminto a quadratic eigenvalue problem where the frequency isfixed and the wave vector is searched. The description of thisformulation is reported in [19].
As can be seen in Figs. 3–5 the curves of modes T1i, T1a,and T2iLa calculated with the FEM match quite perfectly tothose predicted both in the real and imaginary parts. As faras the other modes are concerned, we notice that the FEMresults are the greater of nearly 10% for the real part; evengreater errors are present in the imaginary part. This is dueto the fact that the curves of these modes stay in a rangeof high frequencies, for which the influence of high-orderVSH is not neglectable, so that our model cannot describe ac-curately the dispersion relations. Let us note that, as pre-dicted, losses are acceptable only at low frequencies formodes T2iLa and T1i, making them useful for transmission,and we also note that all the modes have a band foldingfor which it is impossible to reach a zero group velocity[19]. Figure 6 shows the magnetic field of mode T1 in the sin-gle chain and of modes T1i and T1a in the double chain overthe y–z plane: only the component Hx is shown because Hy
and Hz are vanishing over this plane. It is clear that T1i is thesum of the two T1 modes of the single chains that propagate inphase, as expected, while T1a is the sum of the two T1 modesthat propagate in antiphase.
In Fig. 7, the magnetic field of modes T2 and L in the singlechain and of modes T2iLa, T2aLi, and LiT2a in the doublechain is shown over the y–z plane. As was stated in the pre-vious section, mode T2iLa is the combination of the two T2
modes of the single chains in phase and of the two L modesin antiphase, and most of the energy is carried by the T2
modes. Figure 7c fully confirms this scenario: the componentsHy and Hz, which are absent in the mode T2 of the singlechain, are due to the presence of the L modes, but they are
negligible with respect to Hx, which is the only nonvanishingcomponent of T2.
Indeed, Fig. 7e shows that the situation is reversed with re-spect to Fig. 7c: for mode LiT2a, the main modes in combina-tion are the two Lmodes in phase, so that componentsHy andHz are much greater thanHx, which is absent in mode L of thesingle chain, and in this case it is due to the presence of the T2
modes. In Fig. 7d, mode T2aLi is shown: as observed in theprevious section, in this case the modes L and T2 have nearlythe same weight in the combination, which is confirmed fromthe fact that Hx is comparable with Hy and Hz.
On the basis of the obtained results, it is clear that the mainlimitation of the proposed theoretical model compared to afull numerical solution is that it is accurate when the particlesare sufficiently small with respect to the wavelength, in orderto excite only the first-order spherical harmonic. If this con-dition is not fulfilled, some errors are present (of the order of10% in our simulations). Nevertheless, we highlight the factthat very often, the useful modes stay in the band in which
Fig. 6. (Color online) Field Hx of modes T1 in the single chain(a, ωd ¼ 1:50) and of modes T1i (b, ωd ¼ 1:50) and T1a in the doublechain (c, ωd ¼ 1:59).
Fig. 7. (Color online) Fields Hx, Hy and Hz of modes T2(a, ωd ¼ 1:50) and L (b, ωd ¼ 1:70) in the single chain and of modesT2iLa (c, ωd ¼ 1:34), T2aLi (d, ωd ¼ 1:70) and LiT2a (e, ωd ¼ 1:76) inthe double chain.
1024 J. Opt. Soc. Am. B / Vol. 28, No. 5 / May 2011 M. Guasoni and M. Conforti
the above condition is respected and that the proposed meth-odology is several orders of magnitude faster and requiresnegligible memory consumption with respect to a full numer-ical solution.
6. CONCLUSIONSIn this paper we derived the complex dispersion relation of adouble chain of lossy spheres by using the Mie theory and con-sidering the interaction between spherical vector harmonicsof the first order, which is a very good approximation whenthe radius R of the spheres is sufficiently smaller then the wa-velength. The search of the roots of the dispersion relationsrequires the evaluation of some DTFTs for which divergenceproblems arise due to the complex nature of the solution k. Inorder to avoid these problems, the DTFTs have been approxi-mated by means of the Lerch functions, for which analyticcontinuation is possible. Moreover, the pursued analysis givesthe exact solution for the case of two chains with vanishingdistance L, which corresponds to a single nonuniform binarychain. As an example, a practical case of two parallel chainswithout displacement has been analyzed: six modes are pre-sent, which correspond to the coupling interaction betweenthe transverse and longitudinal modes of the single chains.We have shown that in this interaction, the weight of the trans-verse component can be very different with respect to theweight of the longitudinal component and that only twomodes have a degree of loss that allows for a propagation overa reasonably long distance. The main goal achieved is that thetotal useful band is nearly five times that of the single chain.To conclude, all the results has been compared with the finiteelement solutions of the Maxwell equations.
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