Surface plasmon resonant scattering in metal-coateddielectric nanocylinders

Peiwen Meng n, Kiyotoshi Yasumoto, Yunfei LiuCollege of Information Science and Technology, Nanjing Forestry University, Nanjing 210037, China

a r t i c l e i n f o

Article history:Received 26 April 2014Received in revised form4 June 2014Accepted 24 June 2014Available online 7 July 2014

Keywords:Near fieldSurface plasmon resonanceLocalized natureScattering cross-section

a b s t r a c t

The scattering of TE polarized plane wave by metal-coated dielectric nanocylinders is investigated with aparticular emphasis on the enhancement of the near fields. If the wavelength of illumination is properlychosen, two unique near field distributions can be excited through the surface plasmon resonances. Theenhanced near fields are localized along the inner or outer interface of the coating metal, beingdependent on the wavelengths. It is shown that the scattering cross-section of the nanocylinders is alsoenhanced when the illuminating field resonates to the surface plasmons of the structures.

& 2014 Elsevier B.V. All rights reserved.

1. Introduction

With the rapid development of nanoscience and nanotechnol-ogy, the interaction of light with nanoscale objects remains as animportant topic in recent years [1,2] because of their wide range ofapplications to the optical sensors, imaging, and integrated devices.The literature on the interaction of light with nanoscale objects ingeneral is of course much broader. The study on the interaction canbe organized in many different ways, being dependent on thedimensionality of the objects and the exciting source. We shallfocus here our consideration on the cylindrical structures, which areusually called as nanocylinders or nanowires.

The visible-light absorption by silicon nanocylinders was reportedin [3]. This study explained the optical ignition phenomena innanoscale objects. The anomalous light scattering and the peculia-rities of energy flux around a thin nanowire under surface plasmonresonances were discussed in [4]. The surface plasmons characterizethe unique response in collective motions of electrons on a metal–dielectric interface [5,6], which is allowed when the permittivity ofthe metal is negative for the wavelength of excitation. The full waveanalysis of the dispersion relation and field distribution of surfaceplasmon plaritons on metal cylinders with a dielectric core werepresented [7] by taking into account the retardation. An experimen-tal technique to determine the dispersion relation of the surface

plasmon polaritons on Ag and Au nanowires was reported in [8]. Thebasic concept underlying the existence of surface plasmons inmetallic structures was reviewed in [9].

The enhanced surface plasmon resonance in noble metallicnanocylinder systems at optical frequencies is expected to be apromising issue for realizing excellent scatterers and absorbers ofvisible light. The plasmon resonances of interacting silver nano-cylinders were analyzed [10] for both non-touching and intersect-ing configurations and the enhanced scattering through theplasmon resonant coupling was discussed using the finite elementanalysis. The surface plasmon resonances and near-field propertiesof nanocylinders with a cap-shaped or shell-shaped metal coatingwere numerically investigated in [11] using the finite-elementmethod. The anomalous light scattering by a nanocylinder nearplasmon resonance wavelengths was demonstrated in [12]. Theenhanced backscattering by metallic nanocylinder arrays nearsurface plasmon resonances was discussed in [13]. Recently, muchattention has been paid to the light scattering from nanocylinderswith a metal–dielectric layered structure. The superscattering[14,15] and cloaking [15] of light in core–shell nanocylinders wereinvestigated in detail.

In this paper, the scattering of TE polarized plane wave by metal-coated dielectric nanocylinders is investigated with a particularemphasis on the enhancement of the near fields. There exist twokinds of surface plasmons in the metal-coated nanocylinders. One isthe surface plasmons localized along the inner interface of themetal layer and the other is localized along the outer interface. Ifthe wavelength of illumination is properly chosen, the illuminating

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Optics Communications

http://dx.doi.org/10.1016/j.optcom.2014.06.0520030-4018/& 2014 Elsevier B.V. All rights reserved.

n Corresponding author. Tel./fax: þ86 25 8542 7693.E-mail address: dorismpw@163.com (P. Meng).

Optics Communications 332 (2014) 18–24

field resonates to each of surface plasmons and the unique nearfield distributions are excited along the cylinders. Numerical resultsdemonstrate that the enhanced near fields are localized around twointerfaces of the coating metal layer and closely related to thesurface plasmon resonances. It is also shown that the spectralresponses of the scattering cross-section of the metal-coatednanocylinders take a peak value at the wavelengths of surfaceplasmon resonances.

2. Formulation of the problem

The cross-section of coaxial cylindrical structures to be con-sidered here is shown in Fig. 1. The coaxial cylinder with outerradius r1consists of a circular dielectric core with radius r2 and ametal coating layer of thickness r1�r2. The coaxial cylinder isinfinitely long in the z direction and placed in free space. Thematerial constants of the outer free space, coating metal, anddielectric core are denoted byðε0;μ0Þ,ðεM ;μ0Þ, and ðε;μ0Þ, respec-tively. Fig. 1 shows the configurations of (a) a single coaxialcylinder and (b) two identical coaxial cylinders separated by adistance dalong the x axis. The cylindrical structures are illumi-nated by a plane wave of unit amplitude which propagatesnormally to the cylinder axis. The angle of incidence of the planewave is φ0 with respect to the x axis. The scattering problem istwo-dimensional and hence the electric and magnetic fields aredecomposed into TE-wave . Since we are interested in the scatter-ing problem related to the plasmon resonances, we focus ourinvestigation on the scattering of TE wave with ðHz; Ex; EyÞcom-ponent.

Let us consider first the scattering by a single coaxial cylindershown in Fig. 1(a). The reflection and transmission of the standingand outgoing cylindrical waves on the two cylindrical interfacesðρ¼ r1 and ρ¼ r2Þ are solved separately in the referencedcoordinate system ðρ;φÞ in x–o–y. This leads to the reflectionand transmission matrices for cylindrical harmonic waves at eachof the interfaces, which are concatenated to obtain the generalizedreflection and transmission matrices [16,17] over two cylindricalinterfaces. If we denote by the column vectors ðΦ0; ΦM ; ΦDÞ andðΨ0; ΨMÞ the set of basis functions for the standing cylindricalwaves and outgoing cylindrical waves in respective regions, thesolutions to the Hz field in three regions of Fig. 1(a) are obtained asfollows:

Hz ¼ΦT0 UpþΨT

0 UTUp for r1oρ ð1Þ

Hz ¼ΦTM UBUpþΨT

M UCUp for r2oρor1 ð2Þ

Hz ¼ΦT UDUp for 0oρor2 ð3Þwith

Φ0 ¼ ½Jmðk0ρÞeimφ�; Ψ0 ¼ ½Hð1Þm ðk0ρÞeimφ�

ΦM ¼ ½JmðkMρÞeimφ�; ΨM ¼ ½Hð1Þm ðkMρÞeimφ�

Φ¼ ½JmðkρÞeimφ�

9>>=>>;

ð4Þ

p¼ ½pm�; pm ¼ imeimφ0 ð01rφ0r1801Þ ð5Þ

k0 ¼ωffiffiffiffiffiffiffiffiffiffiε0μ0

p; kM ¼ω

ffiffiffiffiffiffiffiffiffiffiffiffiεMμ0

p; k¼ω

ffiffiffiffiffiffiffiffiεμ0

p ð6ÞwhereJm ðm¼ 0; 71; 72;⋯Þ is the m-th order Bessel function,Hð1Þ

m is the m-th order Hankel function of the first kind, pm denotesthe amplitude coefficient of the incident plane wave expressed bythe cylindrical harmonic expansion. The first and second terms onthe right hand side of Eq. (1) represent the incident and scatteredwaves, respectively. In Eqs. (1)–(3), T, B, C, and Dare diagonalmatrices which are given as follows:

T¼ ½Tmδmn�; Tm ¼ R12;mþηM2F12;m

2ð1�R21;mR23;mÞ�1R23;m ð7Þ

B¼ ½Bmδmn�; Bm ¼ ð1�R21;mR23;mÞ�1F21;m ð8Þ

C¼ ½Cmδmn�; Cm ¼ ð1�R21;mR23;mÞ�1R23;mF21;m ð9Þ

D¼ ½Dmδmn�; Dm ¼ ð1�R21;mR23;mÞ�1F32;mF21;m ð10Þwith

R21;m ¼ �ηMHð1Þm ðuMÞHð1Þ0

m ðu0Þ�Hð1Þ0m ðuMÞHð1Þ

m ðu0ÞηMJmðuMÞHð1Þ0

m ðu0Þ� J0mðuMÞHð1Þm ðu0Þ

ð11Þ

F12;m ¼ i2=ðπηMÞ

½ηMJmðuMÞHð1Þ0m ðu0Þ� J0mðuMÞHð1Þ

m ðu0Þ�ð12Þ

R12;m ¼ � ηMJmðuMÞJm0 ðu0Þ� J0mðuMÞJmðu0Þ

ηMJmðuMÞHð1Þ0m ðu0Þ� J0mðuMÞHð1Þ

m ðu0Þð13Þ

F21;m ¼ η2MF12;m ð14Þ

R23;m ¼ � ξMJmðwMÞJ0mðwÞ� J0mðwMÞJmðwÞξMH

ð1Þm ðwMÞJ0mðwÞ�Hð1Þ0

m ðwMÞJmðwÞð15Þ

F32;m ¼ � i2=ðπξMÞ

ξMHð1Þm ðwMÞJ0mwÞ�Hð1Þ0

m ðwMÞJmðwÞð16Þ

uM ¼ kMr1; u0 ¼ k0r1; wM ¼ kMr2; w¼ kr2 ð17Þ

ηM ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiεM=ε0

p; ξM ¼

ffiffiffiffiffiffiffiffiffiffiffiεM=ε

pð18Þ

where the matrix T given by Eq. (7) defines the T-matrix for thesingle coaxial cylinder.

For the two-cylinder system shown in Fig. 1(b), we have to takeinto account the multiple interactions of the fields scattered fromindividual cylinders. The interactions can be calculated by usingthe T-matrix T given by Eq. (7) and the translation matrices [18]

Fig. 1. Cross-section of metal-coated circular nanocylinders illuminated by a TE plane wave which is incident normally to the cylinder axis; (a) single cylinder system and(b) two cylinders system.

P. Meng et al. / Optics Communications 332 (2014) 18–24 19

for cylindrical wave functions. Employing two local cylindricalcoordinate systems ðρ1;φ1Þ and ðρ2;φ2Þ whose origins are locatedat each center of two coaxial cylinders, after straightforwardmanipulations, the Hz fields in five different regions of the two-cylinder system are derived as follows:

Hz ¼ΦT0 UpþΨT

0;1 Ua1þΨT0;2 Ua2 for r1oρ1 and r1oρ2

ð19Þ

Hz;j ¼ΦTM;j UBUqjþΨT

M;j UCUqjðj¼ 1;2Þ forr2oρjor1 ð20Þ

Hz;j ¼ΦTj UDUqjðj¼ 1;2Þ for 0rρjor2 ð21Þ

with

Ψ0;j ¼ ½Hð1Þm ðk0ρjÞeimφj �

ΦM;j ¼ ½JmðkMρjÞeimφj �;ΨM;j ¼ ½Hð1Þm ðkMρjÞeimφj �

Φj ¼ ½JmðkρjÞeimφj �ðj¼ 1;2Þ

9>>>=>>>;

ð22Þ

a1 ¼ ðI�Tα12Tα21Þ�1ðe� ik0d cos φ0=2Iþeik0d cos φ0=2Tα12ÞTUp ð23Þ

a2 ¼ ðI�Tα12Tα21Þ�1ðeik0d cos φ0=2Iþe� ik0d cos φ0=2Tα12ÞTUp ð24Þ

α12 ¼ ½α12;mn�; α12;mn ¼Hð1Þm�nðk0d=2Þ ð25Þ

α21 ¼ ½α21;mn�; α21;mn ¼ ð�1Þm�nHð1Þm�nðk0d=2Þ ð26Þ

q1 ¼ e� ik0d cos φ0=2pþα12 Ua2 ð27Þ

q2 ¼ eik0d cos φ0=2pþα21 Ua1 ð28Þwhere α12ðα21Þ is the translation matrix [18] which trans-formsΨ0;2ðΨ0;1Þto Φ0;1ðΦ0;2Þ according to the Graf's additiontheorem [19].

The optical properties of metal coated nanocylinder system canbe quantified by their scattering cross-section σsca and absorptioncross-section σabs, which are calculated in terms of the far fieldcomponent of the scattered field. Following the definition [20] ofthe scattering and absorption cross-sections for a two-dimensionalstructure, σsca and σabs of the single coaxial cylinder shown inFig. 1(a) are given as follows:

σsca ¼ 4k0

∑1

m ¼ �1Tmj j2 ð29Þ

σabs ¼ � 4k0

∑1

m ¼ �1Re Tmf gþ Tmj j2h i

ð30Þ

where Tm is defined by Eq. (7). For the two coaxial cylinder system,on the other hand, we have to transform two scattered fieldcomponents expressed in the local coordinate systems ðρ1;φ1Þ andðρ2;φ2Þ to those in the referenced coordinate system ðρ;φÞ. Usingthe translation matrices β01 and β02 which relate Ψ0;1 and Ψ0;2 toΨ0, the scattered field component in Eq. (19) is rewritten asfollows:

Hsz ¼ΨT

0;1 Ua1þΨT0;2 Ua2 ¼ΨT

0 UTUp for r1þd=2oρ ð31Þwith

Τ¼ β01ðI�Tα12Tα21Þ�1ðe� ik0d cos φ0=2Iþeik0d cos φ0=2Tα12ÞTþβ02ðI�Tα12Tα21Þ�1ðeik0d cos φ0=2Iþe� ik0d cos φ0=2Tα12ÞT ð32Þ

β01ðm;nÞ ¼ ð�1Þm�nJm�nðk0d=2Þ; β02ðm;nÞ ¼ Jm�nðk0d=2Þ ð33Þwhere the matrix T represents the aggregate T-matrix for the twocoaxial cylinder system. Using Eq. (31), the scattering and absorp-tion cross-sections of the two coaxial cylinder system are given asfollow:

σsca ¼ 4k0

∑1

m ¼ �1∑1

n ¼ �1Tmnpn��� ���2 ð34Þ

σabs ¼ � 4k0

∑1

m ¼ �1∑1

n ¼ �1Re ð� iÞmeimφ0Tmnpn

n oþ Tmnpn

��� ���2� �

ð35Þwhere Tmn denotes the ðm;nÞ element of the aggregate T-matrixgiven by Eq. (32) and pn is defined by Eq. (5).

3. Material parameters of coating metal

It is known that the permittivity of a metal in optical regiontakes a complex value with a negative real part and a smallimaginary part. The value of the real part of the permittivitystrongly depends on the wavelength. The proper evaluation ofεMðλÞ for the coating metal layer is crucial in the present analysis.We employ here the Drude–Lorentz model [21] which assumesεMðλÞ in the following form:

εMðωÞε0

¼ ε1�ω2

p;D

ωðωþ iνDÞ�ΔL

ω2p;L

ω2�ω2p;Lþ iνLω

ð36Þ

where ω¼ 2πc=λ, ε1 is the relative permittivity in ω-1, ωp isthe plasma frequency, ν is the collision frequency, ΔL is theweighting factor for Lorentz model, and the subscripts D and Lare referred to Drude model and Lorentz model, respectively. Inthe numerical examples in what follow, we assume Ag for themetal whose parameters are documented [21] in Table 1. Fig. 2shows the relative permittivity of Ag as functions of the wave-length which were calculated using Eq. (36) with the parametersspecified in Table 1. The black and blue lines represent the realpart and imaginary part of the permittivity, respectively. It isclearly seen that as the wavelength increases, the real part of therelative permittivity decreases rather quickly and takes negativevalues, while the imaginary part is relatively small in magnitudeand hardly depends on the wavelength.

4. Near field properties of metal coated nanocylinders

Eqs. (1)–(3) and (19)–(21) were used to calculate the near fielddistributions of Hz for several configurations of metal-coatednanocylinders. In order to show a unique feature of the metalcoated structure, we calculate first the near fields in three differentconfigurations of single cylinder. Fig. 3 shows a comparison ofthree near field patterns obtained for (a) a pure dielectric cylinderwith r1 ¼ 40 nm and ε=ε0 ¼ 6:5, (b) a pure metal (Ag) cylinderwith r1 ¼ 40 nm, and (c) a metal (Ag)-coated dielectric cylin-der with r1 ¼ 40 nm; r2 ¼ 20 nm, and ε=ε0 ¼ 6:5, respectively.We assumed the incidence of TE plane wave with λ¼ 610 nmandφ0 ¼ 0 3 . When the wavelength is λ¼ 610 nm, Eq. (31) andparameter values of Table 1 lead to εM=ε0 ¼ �14:467þ i0:988 forthe dielectric constant of the metal (Ag). The circles depicted bywhite lines indicate the boundary surfaces of the single and coaxialcylinders. The near field distributions of Fig. 3(a) and 3(b) areconventional, in which any enhancement of a localized field ishardly observed. In contrast, the near field of the metal-coatedcoaxial cylinder shows a unique and interesting feature as shownin Fig. 3(c). The incident TE wave penetrates through the metal

Table 1Typical parameters for Ag [21]. All frequencies are given in THz.

ε1 ωp,D/2π ωp,L/2π vD/2π vL/2π ΔL

3.91 13.420 6.870 84 12.340 0.76

P. Meng et al. / Optics Communications 332 (2014) 18–2420

layer of thickness r1 � r2 ¼ 20 nm and excites a strong fieldinside the coaxial cylinder. The excited field is localized in twosides of the cylinder faced to the propagation direction of theincident wave and along the interface between the dielectric coreand the coating metal (Ag) layer. From the field profile of Fig. 3(c) itfollows that the surface plasmon resonance at the inner interfaceof the metal layer occurs when the wavelength isλ¼ 610 nm.

The near field distribution of Hzj jfor the Ag-coated dielectricnanocylinder with r1 ¼ 40 nm, r2 ¼ 20 nm, and ε=ε0 ¼ 6:5 is shownin Fig. 4 for another wavelength λ ¼ 320 nm. The correspondingεMðλÞ of Ag for λ ¼ 320 nmis εM=ε0 ¼ �1:197 þ i0:553. We cansee that the near field pattern in Fig. 4 is quite different from that ofFig. 3(c) obtained for λ ¼ 610 nm. A strong field is excited only inthe illuminated side of the cylinder and along the interface betweenthe metal layer and the outer free space. From the field profile itfollows that the surface plasmon supported by the interface betweenthe metal and free space resonates to the incident TE plane wave ataround λ¼ 320 nm. It is noted that the coaxial metal layer isbounded by the free space and the dielectric core for ρZr1 andρrr2, respectively, and hence there may exist two different wave-lengths of the surface plasmon resonances. A qualitative investigationbased on a planar interface model [5] suggests that the resonancewavelength of the surface plasmon for the interface between a metaland free space is shorter than that for the interface between a metaland a dielectric medium. The results of Figs. 3(c) and 4 are consistentwith this qualitative investigation. Two typical wavelengths λ ¼320 nm and λ ¼ 610 nm mentioned above correspond to thosegiving two peaks in the spectral response of the scattering cross-section of the Ag-coated single cylinder, as will be discussedlater.

In order to discuss the localized nature of the enhanced fielddistributions shown in both Figs. 3(c) and 4 and their interactionthrough the plasmon resonant coupling, we have analyzed thescattered field by a two-cylinder system. Two-identical Ag-coateddielectric nanocylinders with r1 ¼ 40 nm; r2 ¼ 20 nm; andε=ε0 ¼ 6:5 is placed in free space in parallel to each other. Thedistance between the centers of two cylinders is d¼ 100 nm.

Fig. 5 shows the near field distributions of Hzj j for two differentwavelengths. The incident plane wave with φ 0 ¼ 0 3 propagates inthe direction parallel to the array of two cylinders. Since theillumination to the second cylinder is blocked by the first cylinderlocated in the left-hand side, the major response is governed bythe first cylinder. For both wavelengths of 320 nm and 610 nm, thefield patterns are almost same as those shown in Figs. 3(c) and 4,though the peak value of the field intensity is slightly increased inthe presence of the second cylinder. Fig. 6 shows the near fielddistributions of Hzj jfor the two-cylinder system when the incidentplane wave propagates in the direction ðφ0 ¼ 90 3 Þ perpendicularto the array of two cylinders. Since the two cylinders are equallyilluminated by the incident plane wave, the localized fields similarto those for a single cylinder shown in Figs. 3(c) and 4 are excitedin each of cylinder. There is no noticeable interference betweenthe near fields scattered from the individual cylinders.

The near field distributions for the wavelengths aroundλ¼ 610 nm keep the profiles similar to those shown in Figs. 5(b) and 6(b) even if the wavelength changes. This is because theresonated field associated with the surface plasmon on the inner

Fig. 2. Relative permittivity of Ag based on Drude–Lorentz model [21] as functionsof wavelength.

Fig. 3. Comparison of near field distributions of Hzj j for three different configurations of single nanocylinder; (a) a pure dielectric cylinder with r1 ¼ 40 nm and ε=ε0 ¼ 6:5,(b) a pure metal (Ag) cylinder with r1 ¼ 40 nm and εM=ε0 ¼ � 14:467 þ i0:988, and (c) a metal (Ag)-coated dielectric cylinder with r1 ¼ 40 nm, r2 ¼ 20 nm, ε=ε0 ¼ 6:5,and εM=ε0 ¼ � 14:467 þ i0:988. The wavelength is λ¼ 610 nm.(For interpretation of the references to color in this figure legend, the reader is referred to the web versionof this article.)

Fig. 4. Near field distribution of Hzj j for the metal (Ag)-coated nanocylinder withr1 ¼ 40 nm;r2 ¼ 20 nm;ε=ε0 ¼ 6:5; andεM=ε0 ¼ �1:197 þ i0:553. The wavelengthis λ ¼ 320 nm.

P. Meng et al. / Optics Communications 332 (2014) 18–24 21

interface of the metal is well isolated by the metal shell structure.In contrast, the resonated fields associated with the surfaceplasmon on the outer interface of the metal may directly couplethrough the free space region as shown in Fig. 6(a). This couplingresults in a noticeable change of the near field distribution asfunctions of the wavelength around λ¼ 320 nm. Fig. 7 shows thenear field distributions at the wavelength λ¼ 360 nm obtained for(a) φ 0 ¼ 0 3 and (b) φ0 ¼ 90 3 with other parameters same asthose in Fig. 5. The spectral response of the scattering cross-section of the Ag-coated two cylinder system has a peak atλ¼ 360 nm for both incidences of φ 0 ¼ 0 3 and φ0 ¼ 90 3 , as will

be shown later. From Fig. 7, we can see that the incident TE planewave with λ¼ 360 nm resonates to the coupled surface plasmonmodes supported in the gap between two nanocylinders.

5. Far field properties of metal-coated dielectric nanocylinders

For full understanding of light interaction with nanocylinders,it is also required to analyze the scattering and absorption cross-sections as a function of the wavelength, which have beenexpressed in Eqs. (29), (30), (34), and (35). The scattering and

Fig. 5. Near field distributions of Hzj j for two-identical metal (Ag)-coated nanocylinders withr1 ¼ 40 nm, r2 ¼ 20 nm, and ε=ε0 ¼ 6:5. The distance between the centers of twocylinders is d¼ 100 nm and the incident angle of plane wave isφ 0 ¼ 0 3 . Two different wavelengths are considered; (a)λ ¼ 320 nm,εM=ε0 ¼ �1:197 þ i0:553 and(b)λ¼ 610 nm, εM=ε0 ¼ � 14:467 þ i0:988.

Fig. 6. Near field distributions of Hzj j for two-identical metal (Ag)-coated nanocylinders; (a)λ ¼ 320 nm and (b)λ ¼ 610 nm. The incident angle of plane wave isφ0 ¼ 90 3 . Other parameters are the same as those in Fig. 5.

Fig. 7. Near field distributions of Hzj j for two-identical metal (Ag)-coated nanocylinders at λ ¼ 360 nm; (a) φ0 ¼ 0 3 and (b) φ0 ¼ 90 3 . Other parameters are the same asthose in Fig. 5.

P. Meng et al. / Optics Communications 332 (2014) 18–2422

absorption cross-sections of a single Ag-coated nanocylinder areplotted in Fig. 8 for the wavelength range from 250 nm to1000 nm. It is seen that the scattering cross-section (the red line)has two peaks at λ¼ 320 nm and λ¼ 610 nm, respectively. Thesetwo wavelengths correspond to those for which the near fields areenhanced as shown in Figs. 3(c) and 4. The absorption cross-section (the blue line) has also two peaks at λ¼ 308nmandλ¼ 601 nm, which are slightly blue-shifted from two peaksin the scattering cross-section. Fig. 8 together with Figs. 3(c) and 4explain clearly the mechanism of light scattering and absorptionby a metal-coated nanocylinder. The incident wave resonates tothe surface plasmon modes under special wavelengths andenhances the near field as shown in Figs. 3(c) and 4. The enhancednear field excites a harmonic collective motion of electrons locatednear the boundary surfaces of the metal layer. The harmonicmotion of electrons absorbs the incident wave energy throughtheir collisions and also act as a source for the scattered field. Thusthe surface plasmon resonances increase resonantly the scatteringand absorption cross-sections of the metal-coated nanocylinder.The small differences in the wavelengths for which the scatteringcross-section and the absorption cross-section have peak valuesmay be considered as a result of use of the Drude–Lorentz modelfor metals which combines two different dispersion model ofelectrons as given in Eq. (36).

We also examined the influence of the dielectric constant ε=ε0of the core material on the spectral responses of the scattering andabsorption cross-sections. From the numerical results obtained bychanging ε=ε0 we observed that the spectral profile in the shorterwavelength band shown in Fig. 8 remains almost stationary for thechange of ε=ε0, while the longer wavelength band is directlyinfluenced by the dielectric core medium and the peak of theresonance is red-shifted with increasing dielectric constant.

The scattering and absorption cross-sections of a metal-coatedtwo nanocylinders system are plotted in Fig. 9 as functions of thewavelength for two different illumination directions. The spectralresponse of the cross-sections shows that for both illuminations,the two cylinders system has also two main resonance wave-lengths like the single cylinder system shown in Fig. 8. However,the details of the responses in the shorter wavelength bandaround λ¼ 320 nm are quite different from those of correspondingsingle cylinder case. For both illuminations, the wavelength atwhich the scattering cross-section takes a peak is largely red-shifted to 360 nm from320 nm of the single cylinder case. As wasshown in Fig. 7, this red-shift is attributed to the resonance ofincident wave to the coupled surface plasmon modes supported inthe gap between two cylinders. The spectral responses for bothilluminations in the longer wavelength band around λ¼ 610 nm

are similar to that of the single cylinder case, except that themagnitudes of the cross-sections are increased by some amountdue to the interaction of metallic two cylinders.

Finally, we study the influence of the separation distance d on thescattering cross-section σ sca of two metal-coated nanocylinders.We consider the case of incidence at φ0 ¼ 90 3 because it provides astrong plasmon resonant interaction [10] for small separation dis-tances. The scattering cross-section are shown in Fig. 10 for fourdifferent separation distances d¼ 80 ; 90 ; 100 and 110 nm. Notethat the case of d¼ 80 nm corresponds to touching cylinders. We cansee that even if the separation distance is changed, the two-cylindersystem keeps the properties that the spectral response has two mainresonance wavelength bands. The shorter wavelength band located inλ¼ 300 nm to λ¼ 450 nm corresponds to the resonance to thesurface plasmon mode on the inner interface of the metal, whereas

Fig. 8. Scattering cross-section σ sca and absorption cross-section σ abs of metal-coated single nanocylinder as functions of the wavelength λ of the incident planewave. The parameters are the same as those in Fig. 4.(For interpretation of thereferences to color in this figure legend, the reader is referred to the web version ofthis article.)

Fig. 9. Scattering cross-section σ sca and absorption cross-section σ abs of metal-coated two nanocylinders system. Two different incident angles of plane wave areconsidered; (a) φ0 ¼ 0 3 and (b)φ0 ¼ 90 3 . Other parameters are the same as those inFig. 5.

Fig. 10. Scattering cross-section σ sca of metal-coated two nanocylinders with fourdifferent separation distances d¼ 80; 90; 100; and 110 nm. The incident angle ofplane wave is φ0 ¼ 90 3 . Other parameters are the same as those in Fig. 5.

P. Meng et al. / Optics Communications 332 (2014) 18–24 23

the longer wavelength band located in λ¼ 600 nm to λ¼ 750 nmcorresponds to the resonance to the surface plasmon mode on theouter interface of the metal. It is interesting to note that for the shorterwavelength band, the behavior of the spectral response as a functionof the separation distance d is quite similar to those [10] obtained for atwo pure metal (Ag) cylinder system. The spectral responses in thelonger wavelength band for d¼ 90 ; 100, and 110 nm are similar tothat of the single cylinder, whereas the response for d¼ 80 nm isnoticeably modified. The wavelength of the resonance peak is largelyred-shifted to λ¼ 700 nm and the peak value is much larger thanother three cases. To get a physical insight into this spectral response,we calculated the near field distribution for d¼ 80 nm andλ¼ 700 nm. The result is shown in Fig. 11 and compared with thefield distribution for d¼ 90 nm and λ¼ 630 nm. From Fig. 11 itfollows that the spectral response for d¼ 80 nm is attributed to theresonance of the incident wave to a new surface plasmon mode whichare supported on the touching two cylinder system. When twocylinders are separated, even if the separation distance is sufficientsmall, the surface plasmons on the inner metal interfaces of twocylinders is well isolated as shown in Figs. 11(a) and 6(b). For theconfiguration of the touching cylinder with d¼ 80 nm, the twosurface plasmons on the inner metal interfaces are strongly coupledand a new surface plasmon mode is formed in the touching region.

6. Conclusion

The scattering of TE plane wave by metal-coated dielectricnanocylinders has been analyzed by using the cylindrical waveexpansion and the T-matrix of a circular cylinder. It was shownthat two unique near field distributions localized along thenanocylinders are excited when the wavelength of illuminationresonates to surface plasmons supported on the inner and outerinterfaces of the metal layer. For full understanding of lightinteraction with nanocylinders, the scattering and absorptioncross-sections have also been analyzed as functions of the excitedwavelength. It was shown that the spectral responses of the

scattering and absorption cross-sections have peaks at the wave-lengths near the surface plasmon resonances. For a two-nanocylinder system, there exists the plasmon resonant couplingwhich induces a new plasmon field in the gap region between twocylinders. The strong coupling is attained for a particular wave-length when the separation distance of the cylinders is specified. Itwas shown that the two-nanocylinder system is illuminated withthe particular wavelength, the strong near field are excited in thegap region through the coupled plasmon resonance and thescattering cross-section is resonantly increased at this wavelength.

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Fig. 11. Near field distributions of Hzj j for two-identical metal (Ag)-coated nanocylinders for the normal incidence ðφ0 ¼ 90 3 Þ of plane wave; (a) d¼ 90 nm; λ¼ 630 nm and(b) d¼ 80 nm;λ¼ 700 nm where each wavelength corresponds to the resonance wavelength for the given separation distanced. Other parameters are the same as those inFig. 5.

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