analysis of plasmon propagation along a chain of metal nanospheres using the generalized multipole...

Analysis of plasmon propagation along a chain of metalnanospheres using the generalized multipole technique

S. Mohsen Raeis Zadeh Bajestani,* Mahmoud Shahabadi, and Nahid Talebi

Photonics Research Lab, Center of Excellence on Applied Electromagnetics Systems, School of Electrical and ComputerEngineering, University of Tehran, North Kargar Avenue, Tehran, Iran

*Corresponding author: [email protected]

Received December 10, 2010; revised February 9, 2011; accepted February 14, 2011;posted February 17, 2011 (Doc. ID 139383); published March 30, 2011

We compute the dispersion diagram of an infinite chain of silver nanospheres. The Drude model is used to definethe permittivity of nanospheres, and the generalized multipole technique (GMT) is applied to solve theMaxwell’s equation and, thus, to analyze the plasmon excitation. The obtained dispersion diagram using theGMT is compared with the result of the dipolar interacting model as well as the quasistatic model. Results ofthe finite element method are also presented to verify the accuracy of our results. Finally, a finite chain of metalnanospheres is examined for its scattering and propagation length of the guidedmodes. © 2011Optical Society ofAmerica

OCIS codes: 130.2790, 260.2030, 240.6680, 000.4430.

1. INTRODUCTIONMetal nanoparticle (MNP) waveguides are plasmonic wave-guides which may, for instance, be composed of nanospheresof radius a separated by a distance L. Coupling between ex-cited plasmons of two neighboring spheres, as verified by [1],can generate a surface plasmon polariton (SPP) propagationmode. The possibility of operating below the diffraction limitand the existence of a number of assembly methods such asthose proposed by [2,3] make the MNP chain waveguide a can-didate for miniaturizing photonic devices.

Determination of the dispersion diagram of such a wave-guide structure is of great importance in obtaining modefrequencies, the group velocity of different modes, and thepropagation length of the SPP modes. So far, methods suchas the finite difference time domain (FDTD) method [1], finiteelement method (FEM) [4], quasistatic approximation [5], anddipolar interacting model [6] have been applied to determinethe dispersion diagram of an MNP chain. In the FDTD, specialtreatment is required at the interface of the metal nanostruc-ture and the surrounding medium [7]. Moreover, FDTD isbased on simplified material models such as the Drude orLorentz model for developing an iterative computational algo-rithm. As explained in [8], one can use a combination of theLorentzian line shape functions to enhance the accuracy ofthe permittivity model at the cost of increasing computationalcomplexity. Dipolar models replace each nanosphere by apoint dipole, which then interacts with other dipoles retard-edly. This method is based on a simplified model, which de-monstrates the guiding mechanism in an MNP chain to someextent [6,9,10]; however, it is unable to characterize higher or-der modes. Also, the accuracy of the method depends on thespacing between adjacent spheres. The quasistatic method [5]uses a simplified model, so that its results for the transversemode deviate from those of the dipolar model. Finding an ef-ficient method of analysis free from the above-mentioned lim-itations is of great importance. Here, we use the generalizedmultipole technique (GMT).

The GMT is a semianalytical method appropriate for analyz-ing linear and isotropic materials. It can be categorized as afrequency-domain method applicable to problems with arbi-trary permittivity and permeability. The method has success-fully been applied to the analysis of structures with metalliccomponents [11,12]. In this paper, the dispersion diagram ofan infinite chain of MNPs is computed using the GMT. Lightpropagation along a finite chain of MNPs is also investigatedwith the help of this method.

2. GENERALIZED MULTIPOLE TECHNIQUEFORMULATION FOR THE ANALYSIS OF 3DNANOSTRUCTURESIn the GMT, the electromagnetic field in every region of thestructure is expressed as a linear combination of the fieldsproduced by a group of multipoles located outside the region.Because of the geometry of the problem and for ease of con-vergence, spherical multipoles are used to expand the fields inall regions. The radial components of the electromagneticfield of a spherical multipole located at the origin is as follows:

Enmr ¼ nðnþ 1Þ

jωϵr2 Pmn ðcos θÞHð2Þ

n ðkrÞejmϕ− n ≤ m ≤ n; ð1Þ

Hnmr ¼ nðnþ 1Þ

jωμr2 Pmn ðcos θÞHð2Þ

n ðkrÞejmϕ− n ≤ m ≤ n: ð2Þ

Equations (1) and (2) represent the r component of the TM-to-r and TE-to-r polarizations, respectively. In Eqs. (1) and (2),Pmn are the associate Legendre polynomials and Hð2Þ

n are theSchelkunoff–Hankel functions of the second kind [13]. Notethat we have assumed a time-harmonic of the form expðjωtÞ.Other components of the electromagnetic field can be com-puted from the above components. In order to compute thecontribution of each multipole, the field components are to

Raeis Zadeh Bajestani et al. Vol. 28, No. 4 / April 2011 / J. Opt. Soc. Am. B 937

0740-3224/11/040937-07$15.00/0 © 2011 Optical Society of America

be obtained in a coordinate system whose origin coincideswith the location of the multipole. For example, the radialcomponent of the electric field is obtained as follows:

Er0 ¼nðnþ 1Þjωϵr02 Pm

n ðcos θ0ÞHð2Þn ðkr0Þejmϕ0

; ð3Þ

where the primed variables are related to the shifted coordi-nates. When a cluster of multipoles determines the field insidea specific region, their various field components must besummed up appropriately. Because the spherical unit vectorschange in the shifted coordinate system, one should obtain thefield components of each multipole within a common refer-ence coordinate system. In the reference coordinate system,summation can be carried out simply by superposing the cor-responding components.

One of the key points in using the GMT method is the loca-tion of the multipoles. A good multipole setting increases theaccuracy of the results and also can ease the convergence ofthe numerical calculation. Different procedures for a two-dimensional (2D) multipole setting have been proposed [14].Some of them can also serve in three-dimensional (3D)cases. However, there are some differences between 2Dand 3D problems, which restrict the applicability of someof the proposed 2D methods. The major difference betweenthese two cases can be attributed to the 2 degrees of freedomfor 3D multipoles. Here, increasing the order of the associatedLegendre function increases the unknown coefficients consid-erably. Thus, we should keep the order of multipoles as low aspossible. In each region, where the higher order multipolesare needed, we use a cluster of nearby multipoles to expandthe neighboring fields. Furthermore, for spherical objects, weavoid placing multipoles near the boundaries, except for spe-cial cases such as very close spheres. Note that for a limitednumber of multipoles, placing a large number of them near theboundary generally deteriorates the accuracy of field compu-tation. In the special case of field enhancement calculationbetween the two spheres, we have to make use of additionalmultipoles close to the boundary where the spheres arecloser.

3. MODAL ANALYSIS OF ANINFINITE CHAINIn order to analyze a periodic structure along the z axis with aperiod of L, the periodic boundary conditions must be satis-fied on z ¼ −L=2 and z ¼ L=2. The common GMT procedurefor finding modes of a periodic structure involves exciting thestructure properly and then searching the real frequency axisto find the extremum values of a cost function such as error orfield intensity [12]. These extrema correspond to the mode fre-quency of the different modes. A typical form of multipole set-ting, shown in Fig. 1, provides conditions under which bothtypes of boundary conditions (periodicity of fields and conti-nuity of tangential components) can be satisfied at once. Theexcitation specified in Fig. 1 is a spherical dipole placed insidethe cell. Because of the concentration of the field componentsnear the spheres, there is no need to distribute the multipolesoutside the z ¼ −L=2 and z ¼ L=2 boundaries. Furthermore,because the convergence of the method is satisfactory, thereis no need to truncate the z ¼ −L=2 and z ¼ L=2 boundaries aswas done in [11]. The scattered field in domain D2 and thetotal field in domain D1 are as follows:

~ED2s ¼

XMn2

k¼1

X

mn

ðCnm~EnmTEr

ð~r − ~rkÞ þ Dnm~EnmTMr

ð~r − ~rkÞÞ; ð4Þ

~ED1t ¼

XMn1

k¼1

X

mn

ðC0nm

~EnmTEr

ð~r − ~rkÞ þ D0nm

~EnmTMr

ð~r − ~rkÞÞ; ð5Þ

where Mn1 is the number of multipoles that produce the fieldinside the sphere and Mn2 is the number of multipoles thatproduce the field in region D2. By ~Enm

TMrð~r − ~rkÞ and ~Enm

TErð~r −

~rkÞ we denote the electric fields produced by the TM-to-r andTE-to-r multipoles located at ~rk. The magnetic field in the twodomains has a similar expression as the electric field. Tangen-tial components of the fields on the sphere boundary can becomputed using these expressions [13].

Inserting a group of multipoles outside the boundary willincrease the unknown coefficients and consequently the com-putation time. The problem associated with the increasednumber of multipoles can be solved by another type of multi-pole setting in which the periodic boundary condition is satis-fied intrinsically. Figure 2 shows the location of multipoles

Fig. 1. (Color online) Multipole setting for analyzing a periodic structure.

938 J. Opt. Soc. Am. B / Vol. 28, No. 4 / April 2011 Raeis Zadeh Bajestani et al.

inside the nanospheres along with their complex amplitudes.To ensure satisfaction of the periodic boundary condition,one can expand the scattered field in domain D3 of Fig. 2as follows:

~Eðr; θ;ϕÞ ¼X∞

p¼−∞

e−jpKLX

nm;k

ðCnm~EnmTEr

ð~r − ~rk − pLzÞ

þ Dnm~EnmTMr

ð~r − ~rk − pLzÞÞ; ð6Þ

where K is the wave vector along the z direction and ~ETMrð~r −

~rknm − pLzÞ is the electric field produced by a multipolelocated at ~rnm þ pLz. The summation on p is due to the infi-nite chain of MNPs. Because of the infinity of chain and thesimilarity of the field in the first and second coordinate sys-tems of the structure, the total field in the second coordinatesystem (Fig. 2) can be expressed as

~E0ðr0; θ0;ϕ0Þ ¼X∞

p¼−∞

e−jpKLX

nm;k

ðCnmejKL ~EnmTEr

ð ~r0 − ~r0k − pLzÞ

þ DnmejKL ~EnmTMr

ð ~r0 − ~r0k − pLzÞÞ¼ ejKL ~Eðr; θ;ϕÞ: ð7Þ

It can be seen that the periodic boundary condition is ful-filled. Because the proposed multipole setting forces the per-iodic boundary condition, excitation should also fulfill theperiodic boundary condition. Thus, we used an array of sphe-rical dipoles with a period of L to excite the structure. Theexcitation field can be formulated as follows:

~Eðr; θ;ϕÞ ¼X∞

p¼−∞

e−jpKLð~E1;0TEr

ð~r − ~rS − pLzÞ

þ ~E1;0TMr

ð~r − ~rS − pLzÞÞ; ð8Þ

where rS is the location of the dipole inside the cell.

The only remaining step is to evaluate the unknown com-plex amplitudes of multipoles numerically. They can be com-puted by discretizing the surface of region D4 and satisfyingthe continuity of the tangential components of fields on theboundary. The following matrix equation can serve in findingthe multipole coefficients:

:::þ ½A�2½C�e2jKL þ ½A�1½C�ejKL þ ½A�0½C� þ ½A�−1½C�e−jKL

þ ½A�−2½C�e−2jKL þ :::

¼ ½A�out½C�out þ ½B�exc; ð9Þ

where ½C� is the matrix that represents the multipole coeffi-cients located inside region D4 and ½B�exc represents the trans-verse components of the excitation fields on the discretizedsurface. The dimensions of matrix ½A�

−n are q ×M , whereM is the number of complex magnitudes of the multipolesand q is the number of boundary divisions. Matrix ½A�

−n relatesthe unknown coefficients of the multipoles inside the spherelocated at nL to the tangential components of their fields onthe discretized boundary. Matrices ½C�out and ½A�out have simi-lar definitions for multipoles that produce an electromagneticfield inside region D4. Equation (9) can be summarized by

X∞

p¼−∞

e−jpKL½A�−p½C� ¼ ½A�out½C�out þ ½B�exc: ð10Þ

The infinite series should be truncated at a reasonable limitbefore running a convergence test to check the accuracy ofthe numerical results. Using the above matrix equation, onecan compute the coefficients of the multipoles and conse-quently the field in all regions.

This kind of multipole setting keeps the number of un-known coefficients as low as this number for a single sphereproblem; however, a summation for all multipoles must bedone to evaluate the field inside the z ¼ −L=2 and z ¼ L=2boundaries. Practically, depending on the distance between

Fig. 2. (Color online) Multipole setting that automatically satisfies the periodic boundary condition.


two neighboring spheres and on wavelength, one method mayperform better.

4. CONVERGENCE TEST FOR THE GMTGenerally speaking, solving a problem using the GMT involvesfinding the unknown coefficients of the assigned multipoles inorder to satisfy specific boundary conditions. After discretiz-ing the boundary, the unknown coefficients can be obtainedfrom a matrix equation of the form

AðωÞCðωÞ ¼ BðωÞ; ð11Þ

where BðωÞ represents the excitation field at ω, CðωÞ is thematrix of the unknown coefficients, and AðωÞ is a matrix thatrelates the unknown coefficients to the specific componentsof the fields on the boundaries. For example, a matrix repre-sentation for computing the field scattered off a perfect elec-tric conducting (PEC) sphere can be written as

APECðωÞCPECðωÞ þ BexcðωÞ ¼ 0; ð12Þ

where BexcðωÞ contains the tangential components of the ex-citation electric field on the surface of the sphere, CPECðωÞ is amatrix representation of multipole amplitude for the multi-poles located inside the sphere, and APECðωÞ relates eachmultipole to the tangential component of its electric fieldon the sphere. In order to find a valid solution using theGMT, the number of matching points on the boundariesshould exceed by far the number of unknown coefficients.Moreover, the error defined by

error ¼ ∥AðωÞCðωÞ − BðωÞ∥∥CðωÞ∥ ð13Þ

must be independent of the number of matching points. Oncethe above conditions are fulfilled, the number and expansionorders of the multipoles can be increased to check if the con-vergence is achieved. In this paper, the number of matchingpoints is at least twice the number of unknown coefficients.For the numerical results presented in this paper, the energyconvergence criterion of

Rv ϵ=2ðj~Eðnþ1Þj2 − j~Enj2Þ þ μ=2ðj ~Hðnþ1Þj2 − j ~Hnj2Þdv

Rv ϵ=2j~Enj2 þ μ=2j ~Hnj2dv

< 0:05

ð14Þ

is used to examine the accuracy of the obtained results. Thevolume v is defined as the region ða < r < aþ L=2Þ in whichthe field is highly intense.

5. DISPERSION DIAGRAM OF AN INFINITEMNP CHAINWe used the above-mentioned method to find the propagationmodes of an infinite chain of MNPs. The radius of each nano-sphere is a ¼ 25nm, and the center spacing of two nearbyspheres is L ¼ 75nm. We used the Drude model, that is,

ϵðωÞ ¼ 1 −ω2p

ωðωþ jνÞ ð15Þ

to model the optical response of the Ag nanospheres atℏωp ¼ 6:18 eV. Figure 3 shows the computed band diagramof the metal nanospheres for ν ¼ ωp=100. We truncated thesummation in Eq. (10) to 21 terms, and the number of multi-poles is set in such a way that the relative error remains lessthan 1%. The field distribution of the two first modes of thestructure is illustrated in Fig. 4. The two first modes ofthe structure can be associated to similar dipolar modes

Fig. 3. (Color online) Dispersion diagram of infinite chain of MNPs(ω0 ¼ ωp=

ffiffiffi3

p). Open circles show higher order modes of the structure

obtained using the GMT method.

Fig. 4. (Color online) Distribution of the magnetic field (Hy) on the x–z plane for KL ¼ π at t ¼ 0.


known as the transverse and longitudinal modes. The fielddistribution for the transverse mode resembles that of dipoleswhose directions are perpendicular to the direction of propa-gation, while the longitudinal mode field resembles that of

dipoles whose orientations are in the direction of propagation.The field distribution of the higher order modes shown inFig. 5 corresponds to the higher order modes of the sphericalmultipoles.

Fig. 5. (Color online) Distribution of the magnetic field for higher order modes on the x–z plane for KL ¼ π at t ¼ 0.

Fig. 6. (Color online) Computed transverse and longitudinal dispersion diagrams using the GMT, FEM, dipolar model, and quasistaticapproximation. The results of the dipolar model and quasistatic approximation are from [6].


In order to verify the results obtained using the GMT, weinvestigate the structure using the FEM. We used the periodicboundary condition for applying the FEM method to one cellof the structure. The typical procedure to find the dispersiondiagram using FEM involves setting the wave vector’s vectorand searching the real frequency axis to obtain the mode fre-quencies from a linear equation. Because the metal is highlydispersive and the automatic search for finding the dispersioncurve cannot converge, we manually searched the real fre-quency axis to find several points of the dispersion diagram.First, we set the wave vector’s vector to KL ¼ π and then

searched the real frequency axis to find different modes ofstructure. After that, we changed the wave vector, and usingthe previously obtained values as the starting points, we spe-cified the complete dispersion diagram for the two first modesof the structure.

Figures 6(a) and 6(b) shows the computed dispersion dia-gram for the first two modes of lossless metals using the qua-sistatic approximation, dipolar model, GMT, and FEM. It canbe seen that the presented result of the GMT method and theFEM agree well; however, the results of the dipolar model andquasistatic approximation reported in [6] deviate from the

Fig. 7. (Color online) Scattering from a finite chain of nanospheres for λ ¼ 350nm. As can be seen, the electromagnetic field is decaying.

Fig. 8. (Color online) Plasmon propagation along a finite chain of nanospheres at λ ¼ 390nm. (a) Intensity of the normalized H. (b)–(e) Corre-sponding values of Hx for λ ¼ 390nm at t ¼ 0, T=2, T , and 3T=2, respectively.


GMT and FEM results. The main reason for the deviation is theeffect of higher order multipoles, which are neglected in thedipolar model.

Let us now investigate a finite chain of only six nanospheresat two different frequencies. The corresponding wavelengthsare specified in Fig. 3. As shown in Fig. 3, one of the frequen-cies is related to the band diagram of the MNP chain, and theother is an arbitrary frequency not belonging to the dispersiondiagram. We used a plane wave, which is polarized in the zdirection with kz ¼ 2π=λ0 to excite the structure. Figures 7and 8 show the scattering from finite chain for λ0 ¼ 350nm(corresponding to a stop band) and λ0 ¼ 390nm (correspond-ing to a propagating mode). It can be seen that forλ0 ¼ 350 nm, the intensity of the magnetic field is decreasing;however, for λ0 ¼ 390nm the magnetic field distribution forfour different instances of time indicate a propagating electro-magnetic wave.

6. CONCLUSIONThe dispersion diagram of an infinite chain of nanosphereshas been computed using the GMT, and the results are verifiedusing the FEM. The results obtained using the GMT emphasizethe role of the higher order multipoles for computing the dis-persion diagram of an infinite chain. In other words, the effectof higher order multipoles is significant as far as the disper-sion diagram is concerned. Moreover, the GMT can be moreefficient than the FEM, because its computation time was afraction of that of the FEM and also the required memoryfor the GMT calculation is by far less than the required mem-ory for the FEM. The typical computation time for a singlepoint of the dispersion diagram of Fig. 6 was 35 s for theGMT running on an eight-core Intel 2:33GHz CPU PC usingone of its cores, whereas this time for the FEM was 3 minon the same computer.

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analysis of plasmon propagation along a chain of metal nanospheres using the generalized multipole...

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