Resonances of nanocylinders with gap defects

J. Merle Elson a,*, Klaus Halterman a, Surendra Singh b

a Physics and Computational Sciences, Research and Engineering Sciences,

Naval Air Warfare Center, China Lake, CA 93555, USAb University of Tulsa, Electrical Engineering Department, Tulsa, OK 74104, USA

Received 30 August 2005; accepted 29 January 2006

Available online 20 February 2006

Abstract

We have investigated the plasmonic resonance characteristics of canonical circular and square cylinders, with gap defects, that

are illuminated by a plane wave. The circular and square cylinders have vee shaped gaps and constant width gaps, respectively. The

electric and magnetic fields are obtained by solving the Lippmann–Schwinger equation from which we compute the far-field

scattering cross-section and near-field local electromagnetic energy density.

Numerical results are given for numerous wavelength and gap dimensions to qualitatively present the effects of gap defects on

the scattering cross-section and local electromagnetic energy density.

Published by Elsevier B.V.

Keywords: Nanocylinders; Defect modes; Plasma resonance; Scattering cross section

www.elsevier.com/locate/photonics

Photonics and Nanostructures – Fundamentals and Applications 4 (2006) 94–102

1. Introduction

The concept of the surface plasmon, reported

theoretically by Ritchie [1], was crucial in explaining

experimental observations in optical scattering [2].

These studies were concerned with surface plasmons

that could propagate along a planar interface but were

localized in the direction perpendicular to the interface.

In the case of metallic particles that are much smaller

than the wavelength of incident light, the surface

plasmon resonance can also occur that can yield intense

field localization with a spatial extent that is much

smaller than the classical diffraction limit. In recent

years, such surface plasma resonance effects have

received considerable attention by the research com-

munity. This is in part due to potential applications in

plasmon-based optical devices, such as optical wave-

* Corresponding author. Tel.: +1 505 281 1812.

E-mail address: jmende@copper.net (J.M. Elson).

1569-4410/$ – see front matter. Published by Elsevier B.V.

doi:10.1016/j.photonics.2006.01.004

guides [3,4] and bio-sensors [5]. In addition to

theoretical results indicating the existence of nanopar-

ticle plasmon resonance, experimental evidence has also

revealed the optical field enhancement in the vicinity of

the nanoparticles [6]. Metals such as gold, silver and

aluminum are capable of supporting surface plasmon

modes due to the abundance of conduction electrons.

The plasmon resonance is responsible for field

enhancement leading to a large scattering cross-section

(SCS) at specific wavelengths for different shapes of

metallic nanoparticles. The number of resonances and

the wavelength at which these resonances occur are

strongly dependent on the shape of the nanoparticle. For

instance, a round cylinder exhibits a single resonance

while a triangular cylinder of the same metal may exhibit

five or more distinct resonances [7–10]. The optical

behavior of an array of nanocylinders of circular cross-

section has also been reported based on a finite

difference time domain (FDTD) approach [11], and

an experimental work reported the response of a three

dimensional array of microscopic wires [12]. A

J.M. Elson et al. / Photonics and Nanostructures – Fundamentals and Applications 4 (2006) 94–102 95

theoretical and experimental study of the response of

gold nanoparticles and nanoparticle dimmers to a near

filed excitation has been utilized in detecting slight

asymmetries in dimmers comprising of identical

nanoparticles [13]. Recently, a transparent photonic

band has been described for an all-evanescent metal-

dielectric multilayer structure [14,15].

Recently, the authors have reported on plasmonic

resonances and electromagnetic forces between two

silver nanowires of square cross section [17]. In the

present work, we use a similar integral equation

formulation to compute the scattering cross-section

and local electromagnetic energy density distribution

for three types of cross sections of nanowires versus

wavelength and various gap defect configurations.

These shapes provide interesting results on the plasma

resonance characteristics. The first shape we consider is

a circular cylinder with a vee gap having various angles.

The circular cylinder is considered in both hollow and

solid forms as shown in Fig. 1a and b. We also consider a

square hollow cylinder with a square core and a gap of

constant width as shown in Fig. 1c. The physical

dimension of the cylinder cross section is on the order of

60 nm.

This paper is organized as follows: Section 2 briefly

describes the mathematical equations used for the

computation of the electromagnetic field and a

discussion of the numerical approach. In Section 3

we illustrate as discuss aspects of the numerical results

obtained for the scattering cross-section and local

electromagnetic energy density.

Fig. 1. Schematics of canonical cylinders with gap defects. Also shown is the

cylinder in (c) has the simple square grid pattern. The circular shapes (a) and

have a dot that indicates the location of the area centroid and the grid eleme

annuli. The cylinders are illuminated from above with the electric field po

direction. The exterior dimensions are d2 ¼ 67:70 nm and d3 ¼ 60 nm. The

by angle u and the square cylinder gap is described by width d.

2. Theory and numerical implementation

2.1. Maxwell’s equations

We consider infinitely long square or circular

cylinders that have gap defects in the cylindrical

symmetry. The cylinder has permittivity eðvÞ and is

embedded in vacuum. Schematic cross-sections of the

cylinders that are considered here are given in Fig. 1.

We illuminate a cylinder with a plane wave electric

and magnetic field (Einc, Hinc) having exp ð�ivtÞ time

dependence, wave vector kinc ¼ ð0;�k0Þ and

k0 ¼ v=c ¼ 2p=l. The incident beam is polarized

with the electric vector perpendicular to the axis

of the cylinder. Solutions to the Maxwell equations

for the total electric and magnetic fields are given

by

EðrÞ ¼ EincðrÞ þ k20BS

dr0Ge

$ðr; r0Þ � eðr0ÞEðr0Þ

� 1

2eðrÞEðrÞ (1a)

HðrÞ ¼ HincðrÞ � ik0

ZS

dr0Gm

$ðr; r0Þ � eðr0ÞEðr0Þ

(1b)

where Eq. (1a) is a principal value integral. Since the

z-direction is invariant (cylinder axis) and r ¼ ðx; yÞ,the integration is over the cross-sectional area of

the cylinder, S, in the x-y plane. The eðr0Þ � eðvÞ � 1

when r0 is within S and eðr0Þ ¼ 0 otherwise. The

basic pattern used for the computational grid discretization. The square

(b) have grid elements that are segments of annuli. All grid elements

nts for the circular geometry have slightly different area for different

larized in the x direction, and the incident wave vector is in the �y

interior dimensions are d1 ¼ d4 ¼ 20 nm. The vee gaps are described

J.M. Elson et al. / Photonics and Nanostructures – Fundamentals and Applications 4 (2006) 94–10296

electric Ge

$and magnetic Gm

$Green tensors are

well known for 2D geometry and they are described

elsewhere [16,17].

Numerical integration of Eqs. (1a) and (1b) is

straightforward by dividing the cross-section S into a

grid of k ¼ 1!K area segments, each having area Ak

with area centroid rk. With this, we write Eq. (1a) in a

form suitable for matrix formalism as

EðrkÞ ¼ EincðrkÞ

þ k20

XK

k0¼1

�ð1� dk;k0 ÞAk0Ge

$ðrk; rk0 Þeðrk0 ÞEðrk0 Þ

þ dk;k0

�M$ðrkÞ �

1

2eðrkÞ I

$�� EðrkÞ

�

M$ðrkÞ ¼ B

Akdr0Ge

$ðrk; rk þ r0Þeðrk þ r0Þ (2b)

where the M$

integral is the self term, I$

is the identity

matrix, and dk;k0 is the Kronecker delta. To give an

approximate value to M$

, we analytically integrate

Eq. (2b) over a circular area of radius Rk where

pR2k ¼ Ak. The electric field solution integral equation

as written in Eq. (2a) is now in the form of a linear

system of equations and can be solved by the BiConju-

gate Gradient Method. Using this method has the dis-

tinct advantage that minimal matrix storage is required

as only a row or column is required at any given time.

The magnetic field solution is obtained from Eq. (1b).

2.2. Grid elements for cylinder cross-section

In the case of cylinders with square cross-section S,

as in Fig. 1c, a simple grid that is used in this work is

typically a series of squares with constant area Ak and

their area centroids are at the center of each grid square.

However, when the cross-section S is circular as in

Fig. 1a and b, square grids are a poor choice. In this

case, we devise a grid pattern that takes advantage of

the circular geometry. The radial direction is divided

into n ¼ 1!N annuli each having spatial resolution

of thickness dr ¼ rn � rn�1. It is easy to show that

the n th annulus will have a radial area centroid

rcn ¼ ð2=3Þðr3

n � r3n�1Þ=ðr2

n � r2n�1Þ. The spatial resolu-

tion in the azimuthal direction for the n th annulus is

written as dnf ¼ rc

nDfn, which is generally different for

each annulus. The Dfn is the azimuthal resolution in

angle for the n th annulus. Then, using the radial

resolution dr as a starting value, the azimuthal angle

resolution in the n th annulus is initially written as

Dfn ¼ dr=rcn and this nominal value may need a slight

adjustment. Since the total azimuthal angle along the

annulus is c ¼ ð2p� ugÞ, we examine the ratio

c=Dfn ¼ m and then adjust Dfn!Dfn such that m

becomes the nearest integer. Each grid segment in the n

th annulus will then have area ðr2n � r2

n�1ÞDfn=2 with

approximately equal spatial resolution in the radial dr

and azimuthal dnf ¼ rc

nDfn directions. It follows that for

the cylinders with circular geometry, the area of the k th

grid element in the n th annulus has the form

Ak ¼ ðr2n � r2

n�1ÞDfn=2. This value is constant for grid

elements within any annulus, but generally is slightly

different when the area of grid elements between

different annuli are compared.

2.3. Scattering and local electromagnetic energy

density

Evaluation of plasmonic resonance for the cylinder

geometry considered here can be analyzed by comput-

ing the far-field scattered radiation or the near-field

electromagnetic energy density in and around the

cylinder. The far-field expressions yield the angular

distribution of scattered radiation and the scattering

cross-section. The local electromagnetic field energy

density yields the spatial distribution of field energy. In

all numerical data to follow, the incident plane wave has

wave vector in the �y direction and the electric field is

polarized in the x direction. For this case the incident

power as given by the Poynting vector is just S0 ¼ c=8p.

2.3.1. Differential scattering and scattering

cross-section

Considering Eqs. (1a) and (1b) in the far-field

where jrj� jr0j and k0jrj� 1, we compute the radial

component of the Poynting vector SðuÞ for propagation

in the u direction. It is straightforward to show that for

the scattered fields the differential scattered power per

unit polar angle is

1

S0

dSðuÞdu¼ ð2pÞ4

l3jPxcos u � Pysin uj2 (3)

where

P j ¼Z

S

dr0eðr0Þ4p

E jðr0Þ and j ¼ x or y: (4)

The polar angle u is measured relative to the y axis and

Px and Py are the net polarization components of the

cylinder. This expression for scattering has the l�3 pre-

factor corresponding to the classic Rayleigh wavelength

dependence for 2D geometry. In the case of plasmon

resonance, however, the scattering strengths as given by

Px and Py depend strongly on wavelength. Integration of

J.M. Elson et al. / Photonics and Nanostructures – Fundamentals and Applications 4 (2006) 94–102 97

Eq. (3) over all u yields the total amount of energy

scattered by the cylinder as the SCS which is written

SCS ¼ 1

S0

Z 2p

0

dSðuÞdu

du ¼ pð2pÞ4

l3½jPxj2 þ jPyj2�: (5)

2.3.2. Local electromagnetic energy density

We compute the time-averaged local electromag-

netic energy density UðrÞ [18] of the electromagnetic

field at point r as

UðrÞ ¼ 1

8p

�Re

�dðveðvÞÞ

dv

�jEðrÞj2 þ jHðrÞj2

�(6)

where the dðveðvÞÞ=dv reduces to unity when r is

outside a cylinder cross-section.

Fig. 2. Plots of SCS and optical parameters versus wavelength for a (a) hollo

cylinder with a constant width gap, and (d) material parameters. The dimensi

various angles u (degrees) and the constant width gap also has various widths

u ¼ 0� or d ¼ 0 nm. The cylinder material used in this work is Ag, and show

Re½eðlÞ � l@eðlÞ=@l� with l as derived from numerical data [19]. The electr

the �y direction.

3. Numerical results and discussion

The dimensions of the cylinders considered here are

given in Fig. 1. The nominal resolution of the numerical

integration is 0.5 nm for the circular cylinders and

0.75 nm for the square cylinders. These values are

sufficient to yield convergence in the solution to

Eq. (2a).

3.1. Scattering cross-section and optical

parameters

The calculated SCS data is shown in Fig. 2a–c. Also

shown in Fig. 2d is the permittivity eðlÞ for Ag. These

data are obtained from Palik [19] to which we fit a

w cylinder with a vee gap, (b) solid cylinder with a vee gap, (c) square

ons of the cylinders are given in Fig. 1. As indicated, the vee gaps have

d(nm). Also shown for reference is the case without a gap as noted by

n in (d) is the variation of the permittivity eðlÞ and Re½@ðveðvÞÞ=@v� ¼ic field is polarized in the x direction, and the incident wave vector is in

J.M. Elson et al. / Photonics and Nanostructures – Fundamentals and Applications 4 (2006) 94–10298

polynomial. From this, we can easily obtain the

derivative term needed with Eq. (6) and these data

are also shown.

For the circular cylinders in Fig. 2a and b and smaller

gap angles, the dominant resonance is in the vicinity of

340 nm. In this range and as seen in Fig. 2d, the

ReðeÞ9 � 1 and this coincides with the onset of the

coherent surface plasma resonance oscillation of the free

electron density for a semi-infinite planar interface. As

the gap angle increases, shape effects become increas-

ingly important such that highly localized corner modes

can dominate. This is especially noticeable in Fig. 2b for

u ¼ 45� ! 120�. This is clearly seen later in the contour

plots of the local electromagnetic energy density UðrÞ. In

Fig. 2c the SCS is shown for the square cylinder for

several gap widths d, the peaks are in the vicinity of

400 nm. Finally, we show in Fig. 2d the coefficient

Re½eðlÞ � l@eðlÞ=@l� that is relevant to the UðrÞ. It is

seen that for the dispersive material considered here, this

derivative coefficient contributes significantly to the local

electromagnetic energy density.

3.2. Hollow core circular cylinder

In Figs. 3–5 , we show contour plots of the local

electromagnetic energy density for a hollow cylinder at

wavelengths of 340, 410, and 480 nm where the gap

angles range from 0� to 180�. All plots are shown on the

same log scale and when comparing all three sets of

plots, it is evident that the l ¼ 340 nm case generally

Fig. 3. Contour plot for l ¼ 340 nm of the local electromagnetic energy de

within the core and vee gap is free space and the darker band around the p

cylinder.

has an energy density distribution where the collective

resonance exists throughout the cylinder volume.

Looking at the SCS curves in Fig. 2a for hollow

cylinders with various gaps, this happens to coincide

with the peaks around l ¼ 340 nm. Also at this

wavelength, the SCS curves are monotonically decreas-

ing with increasing gap angle and this is due to a

steadily decreasing value of induced polarization jPxj2(see Eq. (3)) at this wavelength. This indicates that in

spite of the collective oscillation of the electron plasma,

the appearance of a gap breaks the circular symmetry,

reduces the net polarizability, and hence the far-field

SCS. In Fig. 2a and at a wavelength range well beyond

340 nm, additional SCS peaks are seen at gap angles of

90� (l ¼ 485 nm) and 120� (l ¼ 465 nm). Comparing

these data with the comparable plots in Fig. 5, the more

intense resonant nature and induced polarization that is

responsible for the two satellite peaks becomes

apparent. In looking at the u ¼ 60� curve in Fig. 2a,

it also appears that another peak might exist beyond the

l ¼ 510 nm limit. Finally, we note in Fig. 2a the rather

weak SCS peaks around l ¼ 410 nm that is a resonance

mode associated with a canonical hollow circular

cylinder. These peaks are noticeable for gap angles of

30� or less. As the gap angle increases, this mode

weakens rapidly. In the first plot of Fig. 4, the mode

structure is evident and the next two plots (u ¼ 15� and

30�) show that the mode structure is completely

changed but there is still plasmonic activity that leads

to an effective induced polarization Px. The SCS

nsity versus position for a hollow cylinder with a vee gap. The region

erimeter of the cylinder is also a free space region just exterior to the

J.M. Elson et al. / Photonics and Nanostructures – Fundamentals and Applications 4 (2006) 94–102 99

Fig. 4. Contour plot for l ¼ 410 nm of the local electromagnetic energy density versus position for a hollow cylinder with a vee gap. The region

within the core and vee gap is free space and the darker band around the perimeter of the cylinder is also a free space region just exterior to the

cylinder. The non-zero gap introduces the line pattern which has left-right symmetry.

continues to diminish until the u ¼ 180� plot where

increased SCS is seen in Fig. 2a and plasmonic activity

in Fig. 4.

3.3. Solid core circular cylinder

We now consider the case of a solid circular cylinder

with vee gap defects as shown in Figs. 6–8 . These data

are all plotted on the same log scale and are analogous to

Figs. 3–5 except that the hollow core is absent.

Fig. 5. Contour plot for l ¼ 480 nm of the local electromagnetic energy de

within the core and vee gap is free space and the darker band around the pe

cylinder. Similar to Fig. 4, the non-zero gap introduces the line pattern wh

Comparing Fig. 2b with a, it is clear that there is

now much more activity in terms of a redshift in the SCS

with increasing gap angles. There is significant redshift

for gap angles and wavelengths of u ¼ 45� at

l> 510 nm, u ¼ 60� at l ¼ 485 nm, u ¼ 90� at

l ¼ 450 nm, u ¼ 120� at l ¼ 425 nm and u ¼ 180�

at l ¼ 400 nm. This activity correlates very well with

the contour plot data shown in Figs. 6–8. Looking at

Fig. 6 and analogous with Fig. 3, all cylinder

configurations appear to have a collective plasma

nsity versus position for a hollow cylinder with vee gaps. The region

rimeter of the cylinder is also a free space region just exterior to the

ich has left-right symmetry.

J.M. Elson et al. / Photonics and Nanostructures – Fundamentals and Applications 4 (2006) 94–102100

Fig. 6. Contour plot for l ¼ 340 nm of the local electromagnetic energy density versus position for a solid cylinder with a vee gap. The region

within the vee gap is free space and the darker band around the perimeter of the cylinder is also a free space region just exterior to the cylinder.

resonance throughout the cylinder volume with some

plasmonic activity appearing around the vee gap as u

increases. Also, in Fig. 2b at l ¼ 340 nm, the SCS

again has common peaks around l ¼ 340 nm that

monotonically decrease with increasing gap angle.

Looking next at Fig. 7 where l ¼ 410 nm, there appears

to be shape-dependent resonance modes that gradually

increase in activity and are most intense for

u ¼ 90� ! 180�. This is consistent with the correspond-

ing SCS peaks in Fig. 2b that are peaked in the 400 to

450 nm range. Finally, looking at Fig. 8, the most

intense resonance seems to be for u ¼ 60� and u ¼ 90�

Fig. 7. Contour plot for l ¼ 410 nm of the local electromagnetic energy d

within the vee gap is free space and the darker band around the perimeter o

that is again consistent with increased SCS shown in

Fig. 2b. Although the redshift associated with the u ¼45� curve in Fig. 2b is beyond the l ¼ 510 nm limit,

there is a clear indication in Fig. 8 of elevated plasmonic

activity in the vee.

The last plot in Fig. 8 for u ¼ 180� shows some

activity at the center of the flat surface. This is an

artifact of the manner in which the grid pattern is

constructed as indicated in Fig. 1b. In creating a vee gap

within a solid cylinder, the vee region near the origin

does not necessarily represent an ideal vee shape.

Magnifying the grid points associated with the center

ensity versus position for a solid cylinder with a vee gap. The region

f the cylinder is also a free space region just exterior to the cylinder.

J.M. Elson et al. / Photonics and Nanostructures – Fundamentals and Applications 4 (2006) 94–102 101

Fig. 8. Contour plot for l ¼ 480 nm of the local electromagnetic energy density versus position for a solid cylinder with a vee gap. The region

within the vee gap is free space and the darker band around the perimeter of the cylinder is also the free space region just exterior to the cylinder.

Fig. 9. Contour plots of local electromagnetic energy density versus position for a hollow square cylinder with a 0, 2, 8 and 16 nm gaps. The

wavelengths are 340, 410 and 480 nm. The region within the core and gap is free space and the darker band around the perimeter of a square is the

free space region just exterior to the cylinder.

J.M. Elson et al. / Photonics and Nanostructures – Fundamentals and Applications 4 (2006) 94–102102

region and along the flat surface of the u ¼ 180� plot

shows that the surface deviates from planarity near the

origin by as much as 0.02 nm peak-to-peak. This has the

effect of introducing artifacts that can be interpreted

physically as surface irregularity or material inhomo-

geneity. The resonance structure is very sensitive to

such perturbations resulting in ‘‘hot spots’’ that generate

additional scattering currents. While the u ¼ 180� plot

has the most noticeable surface irregularity, all vee

shapes near the origin are not represented as a perfect

vee shape, but the deviation is small compared to the

0.02 nm value mentioned above. This imperfect

representation of a vee gap might illustrate the

unavoidable surface roughness that may arise naturally

or during fabrication. Finally, this artifact of grid

construction does not occur in the hollow cylinder or the

square cylinder to follow.

3.4. Hollow core square cylinder

The SCS in Fig. 2c data associated with the hollow

square cylinder has similarities and differences com-

pared to the SCS curves for the circular cylinder cases in

Fig. 2a and b. As shown in Fig. 2c, the curves now show

common SCS peaks around l ¼ 400 nm and again a

monotonic decrease in SCS as the gap width increases.

The contour plots are are given on a common log scale

as shown in Fig. 9 for wavelengths l ¼ 340, 410, and

480 nm and gap widths d ¼ 0, 2, 8, and 16 nm.

Consistent with the examples given for the hollow and

solid circular cylinder cases, the plots for l ¼ 340 nm

clearly show a plasmonic resonance energy density that

is distributed throughout the cylinder for all four gap

widths. In this case, however, Fig. 2c shows the SCS at

this wavelength to be quite small. This is easily

explained by the fact that Px is relatively small for this

wavelength and geometrical shape as the integrated

polarization tends to cancel over the cylinder cross-

section. The Py is negligible since the incident beam is

polarized in the x direction. Conversely, the SCS for the

next set of plots at l ¼ 410 nm is near maximum and

this coincides with the jPxj2. For the l ¼ 480 nm case,

the jPxj has again decreased monotonically with gap

width leading to SCS curves that decline with

increasing wavelength. The l ¼ 410 and 480 nm cases

in Figs. 4, 5, 7–9 all show ripples or lines in their energy

density contours that coincide with breaking the

symmetry of the canonical circular or square shape.

It is interesting in Fig. 9 for l ¼ 340 nm that there is

little field strength in the gap region for the canonical

square shape mode dominates even with a small gap.

For l ¼ 410 and 480 nm, there appears much more

field activity with in the gap, especially for the

d ¼ 2 nm case. In spite of this increased gap activity,

the Px is not increased within our wavelength range of

computation such that anomalous SCS peaks occur.

This is also the case for the hollow circular cylinder for

small gap angles u (not shown).

Acknowledgments

Support for J.M.E. and K.H. was provided by the

Office of Naval Research In-House Independent

Laboratory Research (ILIR) Program funds and by a

grant of HPC computing resources from the Arctic

Region Supercomputing Center at the University of

Alaska Fairbanks as part of the Department of Defense

High Performance Computing Modernization Program.

Support for S.S. was provided by the ONR/ASEE

Summer Faculty Program.

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