Comparison investigation of near- and far-field properties

for plasmon resonance of silver nanosphere dimers

Lei Liu, Bin Wang, Xuewei Cao, Xiaoxuan Xu, Yufang Wang *

Department of Physics, Nankai University, Tianjin 300071, China

Received 2 October 2010; received in revised form 6 May 2011; accepted 7 June 2011

Available online 14 June 2011

Abstract

Near- and far-field plasmon resonance properties of silver nanospheres dimers are calculated and compared based on the

Generalized Multiparticle Mie-solution method. Greater differences between near- and far-field plasmon resonances are found than

those of silver single nanoparticle, while the dimer of strongest near-field electric field amplitude and the dimer of highest far-field

extinction coefficient almost have the same geometry parameter value under the illumination with fixed wavelength and polarization.

# 2011 Elsevier B.V. All rights reserved.

Keywords: Surface plasmons; Near-field mircroscopy; Mie theory; Multiple scattering; Surface enhanced Raman scattering

www.elsevier.com/locate/photonics

Available online at www.sciencedirect.com

Photonics and Nanostructures – Fundamentals and Applications 10 (2012) 16–24

1. Introduction

Accompanying advances in surface-enhancedspectro-

scopy, including surface-enhanced Raman Spectroscopy

(SERS) [1] and surface-enhanced fluorescence [2],

electromagnetic scattering of metallic nanoparticles

or nanostructures has attracted more and more

attentions. It is desirable to efficiently devise and

fabricate robust substrate with a large surface-

averaged electrical field enhancement (Eave) or a

large maximum point enhancement (Emax), so-called

hot-spot, both of which are most intimately tied with

the local field quantities of the scatter. However, far-

field scattering properties, including extinction,

scattering and absorption cross section (Cext, Csca,

Cabs), are conventional quantities characterizing the

plasmon resonance property of metallic nanoparticles

or nanostructures. For individual metallic nanosphere

* Corresponding author.

E-mail address: yfwang@nankai.edu.cn (Y. Wang).

1569-4410/$ – see front matter # 2011 Elsevier B.V. All rights reserved.

doi:10.1016/j.photonics.2011.06.004

[3,4] and nanorod [5], exact electromagnetic calcula-

tions have showed that the plasmon resonance

deduced from the near-field enhancement can be

significantly red-shifted from the resonance in far-

field scattering. And recently, a theoretical model

based on mechanism of damped harmonic oscillator

has also well explained this shift of single metallic

nanosphere, nanorod and nanoshell caused by radia-

tion damping [6]. It is then spontaneously to ask what

is the feature of the shift between these two classes of

resonance properties in a metallic nanoparticles

cluster system, and what influence does dimer

geometry has on this shift, where coupled damped

harmonic oscillators sometimes result in chaotic

behavior.

Among the various nanoparticle geometries have

been developed, such as ellipsoids, rods, cubes, rice,

stars, shells and sphere chains [7–13], dimer of

nanospheres perhaps has been most widely theoretically

considered [14] and experimentally verified to be

suitable for near-field enhancement [15,16]. Here we

choose silver nanosphere dimer, which is the most

L. Liu et al. / Photonics and Nanostructures – Fundamentals and Applications 10 (2012) 16–24 17

Fig. 1. Configuration of silver nanosphere dimers. Two silver spheres

of radius r, are separated by a top-to-top distance d. The incident light

propagates perpendicular to this plane and we have considered two

polarization conditions: in longitudinal polarization, incident light is

polarized parallel to the dashed line; in transversal polarization,

incident light is polarized perpendicular to the dashed line.

typical and simplest cluster system, to compare the

near- and far-field properties for plasmon resonance of

particles cluster system.

In order to precisely calculate both near- and far-field

properties of metallic dimers, Generalized Multiparticle

Mie-solution (GMM) [17,18] method is employed. In

GMM method, the external plane wave incident on one

particle is superimposed by scattering fields of all

neighboring particles, so this method is very suitable for

the calculation of light scattering by a cluster of spheres.

By varying polarization condition, radius of component

sphere and dimer distance (top-to-top distance between

the two spheres), we explore the dependence of

differences for plasmon resonance between the near-

and far-field properties on these configurational

characteristics, and attempt to give a hint at geometry

optimization of nanoparticle applied in surface-

enhanced spectroscopy in the future.

2. Calculation method

Since generally used simulation tools, like finite

element method [19], finite differential time domain

method [20] and discrete dipole approximation method

[21], are all sensitive to the finesse of the mesh or

discretization, they are less convincible when a high

accuracy of the electrical field distribution just on the

scatter’s surface is required. We adopted GMM method

in this paper, which effectively give an analytical

solution to electromagnetic multisphere scattering.

Simply speaking, to calculate far-field scattering

properties, GMM method first expands the incident

plane wave in displaced coordinate systems which

originated at the component spheres’ centers respec-

tively, next partial scattered field of individual spheres

will be calculated in these sub-coordinate system, then

through expansion base function (vector spherical

wave function) translation, total scattered field

obtained in an unified coordinate system and all kinds

of far-field cross section and coefficients can be figured

out just like classical Mie-scattering; to calculate the

electrical field distribution on each component sphere’s

surface, partial fields are also calculated in sub-

coordinate system at first, followed by summed up.

Correctness of the GMM method has been thoroughly

verified by comparison to either experiments [22] or

other numerical results [23].

The configuration of our calculation is presented in

Fig. 1. Two silver spheres of radius (r) are separated by a

top-to-top distance (d). The incident light propagates

perpendicular to this plane and two polarization

conditions, longitudinal and transversal, are considered.

To make sure the convergence of our calculation

results, truncation numbers were chosen big enough so

that the error tolerance for determining single-sphere

field expansion was 1.0e�20 and the error tolerance for

the iterative solution process to solve the interacting

scattering coefficients was 1.0e�10. The former one

insured that our results satisfied Wiscombe’s criterion.

Referring to the local field quantities, another factor that

will influence the results’ credibility was the number of

points we used to calculate the average (Eave) or search

for the maximum electric field amplitude (Emax) on the

surface of both dimer spheres. Convergence tests were

carried out with different numbers of grid points equal-

spaced distributed on each sphere until relative errors

were smaller then 2.0e�4, which allowed us a precision

below 1 nm when judging plasmon resonance peaks’

positions. As a typical case, 2 � 65 � 63 points were

averaged for one Eave value. The complex permittivity

of silver is given by Lorentz–Drude model [24].

3. Result and discussion

To get a general ideal of the dependence of plasmon

resonance on the geometrical characteristics, we first

focus on the d-induced resonance peaks shift for dimers.

As a representative case, Fig. 2 shows normalized Cext,

Csca, Cabs, Eave and Emax spectra of a dimer (r = 60 nm,

d = 1 nm) in longitudinal polarization (a) and in

transversal polarization (b). The primary resonance

peak around 700 nm in longitudinal polarization is

indicated by shaded region in (a). While in a transversal

polarization, there is only one resonance peak around

380 nm in visible region.

From Fig. 2(a) and (b), it is clear that differences of

the resonance wavelength between all these properties,

L. Liu et al. / Photonics and Nanostructures – Fundamentals and Applications 10 (2012) 16–2418

Fig. 2. Normalized Cext, Csca, Cabs, Eave and Emax spectra of a dimer (r = 60 nm, d = 1 nm) in longitudinal polarization (a) and in transversal

polarization (b). Shaded region is added in (a) to indicate the primary resonance peak around 700 nm in longitudinal polarization. With a transversal

polarization, there is only one resonance peak around 380 nm in visible region.

including Cext, Csca, Cabs, Eave and Emax, occur for the

dimer with d = 1 nm in logitudinal and transversal

polarizations. This point is almost universal in a

metallic nanosphere dimer system of any geometry size,

which will be more evident in Sections 3.1 and 3.2.

Fig. 3. Cext, Eave and Emax spectra for dimers of the same radius (r = 60 nm),

in (a–c), or with a transversal polarization in (d–f), respectively. In addition,

square symbols in (a), (b) and (c) respectively.

In Fig. 3, comparisons of Cext, Eave and Emax spectra

between the dimers of the same radius (r = 60 nm), but

different d (d = 1, 2, 4, 6, 8, 10 nm) are plotted

respectively with a longitudinal polarization in (a–c), or

with a transversal polarization in (d–f). In addition, Cext,

but different d (d = 1, 2, 4, 6, 8, 10 nm) with a longitudinal polarization

Cext, Eave and Emax of a single sphere (r = 60 nm) are also plotted with

L. Liu et al. / Photonics and Nanostructures – Fundamentals and Applications 10 (2012) 16–24 19

Eave and Emax of a single sphere (r = 60 nm) are added

with square symbols in (a), (b) and (c) respectively.

As expected, the peak wavelength of Cext redshift for

longitudinal polarization when decreasing the spacing

between two spheres in (a), whereas blueshift for the

transversal polarization in (d), which is in agreement

with experimental observations [25,26]. In both

polarizations, similar results appear for the resonance

peaks of Eave (b, e) and Emax (c, f). Finally, based on the

calculated results given in Fig. 3, we can conclude that

both near- and far-field properties for plasmon

resonance of the metallic dimers are much more

sensitive to the spacing between two spheres when the

incident light is polarized along the dimer axis, while

there are only tiny displacements of the resonance peaks

with transversal polarization. So, we will consider only

longitudinal polarization in the following discussion.

One point needs to be noted is that, comparing to a

single sphere, the electromagnetic interaction in

transversal polarization, which is not parallel to the

line connecting the centers of two spheres of the dimer

[26], shift the plasmon resonance peak. The resonance

peak wavelength is 376 nm for a single sphere, 439 nm

for the dimer. And this shift has small dependence on

the dimer distance when two spheres are quite close (in

this case, d � 10 nm).

3.1. Comparison of near- and far-field resonance

peaks

After we discussed the dependences of the resonance

peaks on the spacing between two spheres, now we pay

attention to the differences between the near- and far-

field properties. On one side, as shown in Fig. 2(a),

when two spheres get very close, the primary resonance

peak (�700 nm) of Eave is on the long-wavelength side

of that of Cext, and the primary peak of Emax is on the

short-wavelength side of that of Cext; On the other side,

according to Ross and Lee’s work [3], differences

between resonance of Cext, Eave and Emax of single silver

nanosphere also exist (see lines with square symbols in

Fig. 3(a–c)), which should be the extremity case when

the spacing between these two spheres is large enough.

Combining these two facts, we may reasonably expect

that it is not easy to find an optimum wavelength

exciting near- and far-field resonance simultaneously

for any specific dimer geometry.

In order to illustrate this point we give a

representative example in Fig. 4, where two silver

spheres of 60 nm radius are separated by a distance of

30 nm. Normalized Cext, Eave, and Emax spectra of this

dimer are shown in Fig. 4(a). Three vertical lines

respectively indicate the maximum Eave wavelength

(381 nm), the Cext primary peak wavelength

(507 nm), and the Eave primary peak wavelength

(573 nm); electric field amplitude on the surface of

this dimer in the back scattering direction (up-half

panel) and in the forward scattering direction (down-

half panel) are depicted with free space wavelength

(b) 381 nm, (c) 507 nm and (d) 573 nm, all in the

same scale. Obviously, Fig. 4(b) has the highest

average brightness, while the most brilliant spot

appears in Fig. 4(d).

3.2. Comparison of near- and far-field resonance

peaks shift

Since distinctions for resonance of near- and far-field

are general, scattering of various dimers with different r

and d are calculated and compared, mainly focused on

the difference between the primary resonance peak

position of Cext and that of Eave. The main calculated

results are given in Fig. 5.

In Fig. 5(a), the wavelength differences between the

Eave primary peak and Cext primary peak,

DpeakðlEave primary peak � lCext primary peakÞ, of dimers with

different r (24.8–91.5 nm), are plotted versus distance d

in unit of the component sphere’s diameter. There are

some common features in these curves: (i) Regardless

of the radius, each curve can be roughly divided into

three sections. When the distance between two spheres

is smaller than the radius (d < r), Dpeak increases as the

distance increases. When distance is in the range of the

radius and the diameter (r < d < 2r), Dpeak keeps

relatively great. And if distance continually expands,

Dpeak decreases and eventually approaches the Dpeak

value of single nanosphere. (ii) Sphere of larger radius

results in larger Dpeak when the two spheres are far away

from each other, which is reasonable for retardation

effects and consequently the radiation damping would

be larger in a larger single particle, and is in consistency

with the previous study [3]. (iii) The second point is not

only appropriate for large spacing, actually for any

distance, larger radius leads to larger resonance

wavelength difference. This makes sense when con-

sidering the radiation damping of whole cluster is

basically originated from the radiation damping of

composed nanospheres.

Fig. 5(a) also reflected that dimers of small spheres

have different Dpeak curve shape compared with dimers

of large spheres (see r = 60.0 and r = 71.8). The

maximum values of Dpeak for radius-fixed but dis-

tance-changed dimers Dmax, and dðDpeak;maxÞ the

distance at which the radius-fixed dimer has Dmax are

L. Liu et al. / Photonics and Nanostructures – Fundamentals and Applications 10 (2012) 16–2420

Fig. 4. (a) Comparison between the normalized Cext, Eave and Emax spectra of a dimer (r = 60 nm, d = 30 nm). Three vertical lines indicate the

maximum Eave wavelength (381 nm), the Cext primary peak wavelength (507 nm), and the Eave primary peak wavelength (573 nm), respectively;

electric field amplitude on the surface of this dimer in the back scattering direction (up-half panel) and in the forward scattering direction (down-half

panel) with free space wavelength (b) 381 nm, (c) 507 nm and (d) 573 nm, all in the same scale. A double-headed arrow points to the brightest spots

in (d).

extracted from Fig. 5(a), and plotted versus the sphere’s

radius in Fig. 5(b) and (c), respectively.

In contrast with single silver sphere, greater

differences (more than three times) between the Eave

primary peak and Cext primary peak may appear in

dimers according to Fig. 5(b). Such huge distinction

indicates that we should carefully use light of

wavelength at the Cext resonance to illuminate the

metallic dimer substrate, when what we really wanted is

a maximum near-field enhancement. To give an extent

of Dpeak for a specific dimer, piecewise fittings are

plotted with solid lines in (b) and (c), and fitting

parameters are listed in Table 1. With these approximate

formulas, we can roughly predict how much of the

maximum shift and the distance when this max-shift

occur for silver dimers, which would do great help when

we need to choose a working wavelength in surface-

enhanced spectroscopy.

3.3. Geometry optimization

Although difference of hundreds of nanometer for

primary resonance peak wavelength may present

between near- and far-field in metallic dimers, the

L. Liu et al. / Photonics and Nanostructures – Fundamentals and Applications 10 (2012) 16–24 21

Fig. 5. DpeakðlEave primary peak � lCext primary peakÞ, the wavelength difference between Eave primary peak and Cext primary peak, are plotted in (a) as a

function of dimer distance (d: 0–6 � r), different symbols representing different radii (r: 20–90 nm); The maximum values of Dpeak for radius-fixed

but distance-changed dimers, Dmax, are plotted against radius in (b), and compared with those of the single nanospheres. dðDpeak;maxÞ, the distance at

which the radius-fixed dimer has the Dmax, are plotted versus radius in (c). Solids curves in (b) and (c) represent curve fittings presented in Table 1.

results given in Section 3.2 do not reflect the absolute

height of peaks in scattering spectra. Even a specific

wavelength is the resonance wavelength for dimer A

with one geometry size and it is not the best choice for

dimer B with another geometry size, scattering property

value of A is not definitely higher then B. That is to say

dimer B still behaves better than A under this

wavelength illumination. When we are selecting

optimum dimer geometry, which has the highest near-

or far-field scattering property value, in surface-

enhanced applications, both resonance peak position

Table 1

Fits for Dmax and dðDpeak;maxÞ as a function of radius r (nm).

Structure Properties Curve fit equation

Dimer

sphere

Dmax 927.7–925.5/(1 + (r/102.0)4.834) r � 80

6.237 � r � 286.9 r > 80

dðDpeak;maxÞ 117.3–94.03/(1 + (r/63.36)16.70) r � 80

1.4 � r + 3 r > 80

Single

sphere

Dpeak 200.6–197.9/(1 + (r/85.14)5.810) r � 80

3.455 � r � 192.2 r > 80

and peak height should be considered. As typical cases,

the volume-equivalent extinction coefficient Qvext

(Qext ¼ Cext=rv, where rv is the volume-equivalent

sphere radius) and Eave of dimers with radius between 1

and 100 nm and distance between 1 and 20 nm are

presented in Fig. 6, with widely used wavelength (a, b)

514 nm, (c, d) 633 nm, (e, f) 785 nm respectively. In

addition, dashed contour lines of value Qvext = 10 are

plotted in (a), (c) and (e), and contour lines of value

Eave = 4 V/m are plotted in (b), (d) and (f).

As shown in Fig. 6, Qvext has similar pattern as Emax

in our calculated parameter region, and they share some

common features. When the wavelength of the incident

light increases, radii of the resonant dimers also

increase, accompanying a lower maximum value and

a broader region of resonant dimers geometry para-

meters. However, comparing the area enclosed by

contour lines under these three illumination conditions,

it is clear that region of superior resonant dimer

geometry parameters, for both Qvext and Eave, decreases

as the wavelength of the incident light increases.

Besides these similarities, Qvext and Eave have almost

L. Liu et al. / Photonics and Nanostructures – Fundamentals and Applications 10 (2012) 16–2422

Fig. 6. Qvext and Eave of dimers with radius between 1 and 100 nm and distance between 1 and 20 nm with wavelength (a) and (b) 514 nm, (c) and (d)

633 nm, (e) and (f) 785 nm respectively. Dimers with geometry parameters in region enclosed by dashed line have a value of Qvext larger than 10 in

(a), (c) and (e), while similarly dimers in the enclosed region in (b), (d) and (f) have a value of Eave larger than 4 V/m.

L. Liu et al. / Photonics and Nanostructures – Fundamentals and Applications 10 (2012) 16–24 23

the same optimum geometry in all three incident light

conditions, with difference of only a few nanometers.

This result convinced us to optimize dimer geometry in

SERS with Qvext, for it is more complicated and time-

consuming to do this with local field electrical field

intensity.

4. Summary

Using GMM method, near- and far-field plasmon

resonance properties of silver nanosphere dimers are

calculated and compared with various dimer geometry

parameters. Plasmon resonance is insensitive to the

geometry features when incident light polarized

perpendicular to the dimer axis in neither far- nor

near-field. For any specified dimer system, larger

difference between the two classes of properties may

occur compared to single metal sphere in longitudinal

polarization, which indicates that exciting wavelength

for dimers utilized in surface-enhanced spectroscopy

should be more carefully chosen. Fitted formulas are

also given to judge the possible maximum peak position

shift. For three specified wavelength, optimum dimer

geometry for near-field property (Eave) has almost the

same size as that for far-field properties (Qvext).

Acknowledgements

This work is supported by the National Science

Foundation of China (60878025, 11074131), the

Doctoral Foundation of Education Ministry of China

(200800551053), the Fundamental Research Funds for

the Central Universities (10055-010-65010981,

65010821), the Major Research Plan of National

Natural Science Foundation of China (90923038). Y.

Wang would like to thank the Research Fund for the

Doctoral Program of Higher Education of China

(20100031110004). B. Wang would appreciate the

Open Research Fund of State Key Laboratory of

Precision Measuring Technology and Instruments and

Open Research Fund of Key Laboratory of Semicon-

ductor Materials Science of Institute of Semiconductors

CAS (KLSMS-1009).

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