Multiple Multipole Method with Automatic Multipole Setting Applied to the Simulation of Surface Plasmons in Metallic Nanostructures
ABSTRACT this paper is classical. Hence macroscopic Maxwell equations and boundary conditions have been applied, as well as bulk permittivity functions to model surface plasmons in nanoscopic systems
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ABSTRACT: Optical detection and spectroscopy of single molecules and single nanoparticles have been achieved at room temperature with the use of surface-enhanced Raman scattering. Individual silver colloidal nanoparticles were screened from a large heterogeneous population for special size-dependent properties and were then used to amplify the spectroscopic signatures of adsorbed molecules. For single rhodamine 6G molecules adsorbed on the selected nanoparticles, the intrinsic Raman enhancement factors were on the order of 10(14) to 10(15), much larger than the ensemble-averaged values derived from conventional measurements. This enormous enhancement leads to vibrational Raman signals that are more intense and more stable than single-molecule fluorescence.Science 03/1997; 275(5303):1102-6. · 31.20 Impact Factor
Article: Environment effects on surface-plasmon spectra in gold-island films potential for sensing applications.[show abstract] [hide abstract]
ABSTRACT: The effects of the local dielectric environment on the surface-plasmon resonances of annealed gold-island films as a potential for sensing applications are studied experimentally and modeled theoretically. Gold-island films were annealed at 600 degrees C to produce spheroidal shape particles that exhibit well-resolved resonances in polarized, angle-resolved, absorption spectra. These resonances are shifted in different amounts by the depolarization effect of the surrounding medium (liquids with various refraction indices). Cross-section calculations based on nonretarded, single-particle, dielectric interaction for these various configurations are presented and are found to be in good agreement with the experimental observations. The results show an interesting potential for biosensing or environmental monitoring applications.Applied Optics 01/1999; 37(34):8030-7. · 1.41 Impact Factor
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ABSTRACT: A new optical sensor that uses local plasmon resonance is proposed. A peak that is due to the local plasmon resonance appears in the absorption spectrum of a gold colloid suspension in the visible region, and its height and wavelength depend on the refractive index of the suspension. These properties are used for optical sensors. We used gold colloid monolayers in which colloidal gold particles a few tens of nanometers in diameter were immobilized upon a glass slide by a functional organic coupling agent. We measured the absorption spectra of the the gold colloid monolayers, which were immersed in liquid samples or coated with thin films. We observed increases of both the resonance wavelength and the absorbance as the refractive indices of the sample liquids or the thickness of the coated films increased. The proportional constants of the resonance wavelength to the film thickness were 3.6 and 5.7 for a 13.9- and a 20.2-nm gold colloid monolayer, respectively.Optics Letters 04/2000; 25(6):372-4. · 3.40 Impact Factor
Multiple multipole method with automatic
multipole setting applied to the simulation
of surface plasmons in metallic nanostructures
Laboratory for Electromagnetic Fields and Microwave Electronics, Swiss Federal Institute of Technology,
ETH-Zentrum, Gloriastrasse 35, CH-8092, Zurich, Switzerland, and Swiss Center for Electronics
and Microtechnology, Badenerstrasse 569, CH-8048, Zurich, Switzerland
Daniel Erni, Christian Hafner, and Ru ¨diger Vahldieck
Laboratory for Electromagnetic Fields and Microwave Electronics, Swiss Federal Institute of Technology,
ETH-Zentrum, Gloriastrasse 35, CH-8092, Zurich, Switzerland
Received March 5, 2001; revised manuscript received June 14, 2001; accepted June 18, 2001
Highly accurate computations of surface plasmons in metallic nanostructures with various geometries are pre-
sented. Calculations for cylinders with irregular cross section, coupled structures, and periodic gratings are
shown.These systems exhibit a resonant behavior with complex field distribution and strong field enhance-
ment, and therefore their computation requires a very accurate numerical method.
tiple multipole (MMP) method, together with an automatic multipole setting (AMS) procedure, is well suited
for these computations. An AMS technique for the two-dimensional MMP method is presented.
the global topology of each domain boundary to generate a distribution of numerically independent multipole
expansions.This technique greatly facilitates the MMP modeling. © 2002 Optical Society of America
OCIS codes: 000.4430, 240.6680, 260.3910, 350.3950, 290.4020, 290.5850, 290.4210, 050.1950, 240.0310,
It is shown that the mul-
It relies on
Studying the interaction of light with nanosized struc-
tures is important both theoretically and from a techno-
logical point of view. In particular, metallic nanostruc-
turedobjectsshow an interesting
electromagnetic field may excite collective oscillations of
the object’s free electrons, and, for a certain frequency
range of the exciting field, a complex resonant behavior
can occur with strong near-field enhancement and local-
ization.This resonant phenomenon is governed by the
dielectric function ?(?) of the metallic object and by its
For example, spherical particles small in
comparison with the exciting wavelength exhibit a main
resonance for R??(?)? ? ?2?background.
nance, the fields are quite localized on the surface, and
they are therefore known as surface modes or surface
Because of their field distribution, surface plasmons
are very sensitive to surface properties, and for that rea-
son they have been investigated in relation to surface-
enhanced Raman scattering5
Plasmon coupling along a chain of particles has been pro-
posed for guiding energy in the subwavelength scale,8–11
and primitive routing devices12,13have been considered in
this context as well.The field-confinement properties of
metallic media have stimulated studies concerning sur-
face plasmon propagation in nanowires14–17and scanning
near-field optical microscopy.18–22
transmission through thin metallic films patterned with a
At the reso-
and optical sensing.6,7
subwavelength periodic array of holes23,24has been at-
tributed to the coupling of plasmons at both sides of the
microscopy25,26and polarizing optical filters.27,28
Excluding those cases in which the electrostatic
approximation29can be made, the computation of the
resonances’ spectrum requires the consideration of retar-
dation effects, and hence the resolution of the full vecto-
rial Maxwell equations is needed.
are available only for simple geometries, and therefore
numerical methods have to be used in general.
resonances, very strong enhancement of the electromag-
netic field is achieved, and the field distribution may
present a fairly complex structure, which demands an ac-
curate modeling scheme. Techniques such as the dis-
crete dipole approximation, the T-matrix method, or
Green’s dyadic technique, have been applied for the com-
putation of near-field optical problems.
these and other methods can be found in Refs. 30 and 31.
These numerical methods have allowed the computation
of plasmon resonances for complex geometrical configura-
tions such as isolated particles of irregular shapes,32par-
ticles on top of a substrate,33,34
particles,20,35cylinders of irregular cross sections,36,37and
Numerical techniques have difficulties
with large field gradients, and to overcome this issue, a
Green’s dyadic method with finite elements has been
Nevertheless, most of the usual techniques
are specific for one type of geometrical configuration, or
An overview of
Moreno et al.
Vol. 19, No. 1/January 2002/J. Opt. Soc. Am. A101
0740-3232/2002/010101-11$15.00© 2002 Optical Society of America
they are not flexible enough to be extended for the com-
putation of all the above-mentioned geometrical arrange-
ments. Therefore we propose the computation of surface
plasmons in structures with a wide range of geometrical
configurations using one single method:
multipole (MMP) method.
The MMP method40,41belongs to a group of techniques
that are collectively known as the generalized multipole
The GMT expands the fields as a lin-
ear superposition of basis functions.
GMT is the high degree of freedom in the selection of
those functions, in particular the multipolar functions.
This feature allows one to achieve a very high accuracy,
but it makes the modeling more difficult.
cally, the positioning of the multipolar function origins is
not an easy task, and, for this reason, attempts have been
made to develop procedures to locate the multipoles (or
auxiliary sources) in a systematic way.
some semiheuristic rules were presented for the MMP
method,43the method of auxiliary sources,44the multifila-
ment current model,45,46and for a particular kind of the
MMP method that employs only two-dimensional (2D)
Based on this type of rule, several algo-
rithms for automatic positioning of the auxiliary sources
were proposed: Leuchtmann48,49presented strategies for
2D electrostatics, Regli50and Tudziers51produced meth-
ods for three-dimensional electrodynamics, and Hafner52
outlined a semiautomatic procedure for 2D electrodynam-
ics. Lacking an established name, we refer to this kind of
algorithm as automatic multipole setting (AMS) tech-
niques.None of the mentioned algorithms is suited for
complex geometries, and therefore we present here a very
fast AMS procedure especially devised for 2D electrody-
namics, which simplifies enormously the MMP modeling
of systems with complex boundaries.
After summarizing the most relevant features of the
MMP method in Section 2, we present our AMS procedure
in Section 3. In Section 4, these techniques are applied
to compute surface modes in metallic nanostructures for
various geometrical configurations.
drawn in Section 5.
A peculiarity of the
Conclusions will be
2. MULTIPLE MULTIPOLE METHOD
The MMP method is a numerical technique for perform-
ing electrodynamic field calculations.
for systems with piecewise homogeneous, isotropic, and
linear material media, and it works essentially as follows.
The region where the fields are to be computed is divided
into domains Di, where the material parameters ?i,?i
are constant. The field ?Diin every Di[?Didenotes a
generic field potential out of which the electric (E) and
magnetic (H) fields can be extracted] is expanded as a lin-
ear superposition of N known analytical solutions ?k
the Maxwell equations in the corresponding domains:
It was developed
field and ?exc
denotes the approximation to the actual
Direpresents the exciting field (Fig. 1). To de-
termine the weight ak
must impose the boundary conditions on the fields at the
interfaces ?Dijof the domains.
is claimed to be semianalytical because ?approx
cally satisfies the Maxwell differential equations in every
Di, while the algebraic boundary conditions are approxi-
mately fulfilled at every ?Dij.
method, since only the boundaries have to be discretized,
resulting in a lower computational effort.
feature of the MMP method is the possibility of estimat-
ing the quality of the solution found.
fields are computed by minimization of the errors in the
fulfillment of the boundary conditions. The evaluation of
the residual errors at the interfaces ?Dijallows one to es-
timate the local accuracy53of the solution.
The basis functions employed for the expansion of the
field in each domain Dihave to be analytical solutions of
the Maxwell equations but are otherwise completely arbi-
trary.This is one of the advantages of the method that
gives a high degree of flexibility. For instance, if the ana-
lytical solution of a problem is known, the solution of a
perturbation of this problem can be obtained by including
the solution of the unperturbed problem among the basis
functions.In this way, all known information about the
solution of the problem can be incorporated by selecting
appropriate basis functions, and very accurate solutions
can be found.
As the name of the method suggests, the multipolar
functions are the most used basis functions.
next the reason for this choice, which is based on their
useful physical and mathematical properties.
tipolar functions can be found by applying appropriate
differential operators to a solution of the scalar Helmholtz
equation separated in spherical coordinates.54
multipolar solutions—whose radial dependency is essen-
tially given by Hankel functions—represent harmonic
monopolar, dipolar, (and so on) sources, and they are sin-
gular at the point where they are located.
electrodynamics problem with these multipolar sources is
physically intuitive:The radiation falling upon an inter-
face ?Dijbetween two different media experiences a scat-
tering process in which energy is reradiated toward both
media.The radiation toward domain Diis modeled by a
set of multipolar sources inside domain Dj(they are often
Diof every basis function ?k
Hence the MMP method
It is also a boundary
As said above, the
modeling of a scattering problem.
functions are represented by circles for the inner domain Djand
by ? symbols for the outer domain Di.
Multipole expansions for the multiple multipole (MMP)
The origins of the multipolar
102J. Opt. Soc. Am. A/Vol. 19, No. 1/January 2002 Moreno et al.
located close to the interface) (Fig. 1).
sembles the classical method of images with multipolar
functions acting as ‘‘images’’ of the incident fields.
other favorable property of the multipolar functions is
their asymptotic behavior:
ing sources, they satisfy the radiation conditions at infin-
ity and so no special care is needed for open domains.
The choice of position, number, and multipolar order of
the sources depends on the complexity of the problem,
where an important role is played by the exciting electro-
magnetic field, the geometry of the interfaces, and the
material constants. In the previous paragraph, we men-
tioned some physically intuitive considerations that can
now be reformulated from an algebraic point of view:
The choice of the sources has to be made in such a way
that the boundary conditions for the fields at both sides of
the interface can be accurately satisfied.
that the radial and angular dependencies of the multipo-
lar functions have to be able to capture all the complexity
of the actual field. In the MMP method, this is achieved
in two steps. First, every radiating source consists of a
‘‘cluster’’ of multipolar functions, all of them at the same
point and including several multipolar orders, as is also
done in Mie theory.This is called a multipole expansion.
Second, to avoid the convergence problems of Mie-like
theories for nonspherical geometry, not only one but sev-
eral multipole expansions at different positions are em-
ployed for modeling the fields.
The use of multiple multipole expansions with different
origins gives a high flexibility and helps to model complex
fields.However, this flexibility has to be used with cau-
tion. The different multipolar orders in a multipole ex-
pansion are linearly independent, but two multipole ex-
pansions located at different points are in principle not
tions are imposed on a non-numerically-independent se-
ries expansion, ill-conditioned matrix equations may be
generated that produce useless results if they are not
properly handled. The mathematical properties of the
multipolar functions help to mitigate this inconvenience:
These functions decay rapidly with distance from the
source origin, and therefore multipole expansions distant
from each other are effectively independent.
other hand, multipole expansions that are close to each
other may cause numerical problems.
we will present in Section 3 a procedure that simplifies
the modeling by automatically selecting the location of
the multipole expansions in such a way that numerical
dependencies are avoided.
promise between sparse multipole distributions, where
multipoles are numerically independent but not accurate
enough, and dense multipole distributions, which permit
high accuracy at the cost of ill-conditioned matrices.
The idea re-
Since they represent radiat-
Thus the name of the
When the boundary condi-
For this reason,
This procedure finds a com-
3. AUTOMATIC MULTIPOLE SETTING FOR
The selection of the basis functions is considered the most
difficult task in the MMP method.
selection of the basis functions is neither possible nor de-
sired. In fact, the possibility of choosing arbitrary basis
A totally automatic
functions is one of the fundamentals of the GMT and re-
flects the user’s knowledge of the problem’s underlying
physics.Nevertheless, multipole expansions are em-
ployed in the modeling of any given problem, and their
properties can be exploited to define a suitable multipole
arrangement with a certain degree of automation.
We describe now the principles on which a new AMS
procedure valid for cylindrical structures (2D electrody-
namics) is based.A motivation for the kind of informa-
tion that is used as input for the method is presented
first.As mentioned in Section 2, the location of the mul-
tipole expansions should in principle depend on (1) the ex-
citing field ?exc
tion, we consider that the exciting field is acting only in
domain Di), (2) the geometry of the domains (which in the
2D case is given by the interface curve ?ij? ?Dij), and
(3) their material properties ?i,?j.
working with the GMT do indeed use the information
pute the analytical continuation of the exciting field in the
domain Dj(the so-called nonphysical domain). This ana-
lytical continuation has singularities inside Dj, which are
the locations for the auxiliary sources.
ations, with an irregular interface ?ijand arbitrary exci-
tation, finding the singularities would require a large
computational effort in itself.
geometry of the boundary ?ij(together with the wave-
lengths ?Di, ?Djof the fields in Di, Dj) as input informa-
tion for our AMS method.
that these are usually the most relevant factors.
ing only this input information, we were able to develop a
very fast AMS algorithm.
decision is that we have designed this AMS technique to
be applied in structural optimization of integrated optical
devices. In these optimizations, many field computations
have to be performed while changing only the geometry of
The method is based on three ideas.
pected that the complexity of the fields is higher where
the interface presents a complex geometry.
cause near these areas, the field may be focused or guided
or may reach a resonant regime.
where the field is potentially complex need a dense distri-
bution of multipole expansions in order to represent the
field adequately.Second, a large total number of multi-
pole expansions is computationally expensive, and hence
this number should be kept to a minimum.
quence, the density of multipoles should be low in those
regions where the field is expected to be regular.
into account these two points, we propose a distribution of
multipole expansions adapted to the field complexity,
which, in turn, has to be extracted from the geometry of
the interface curve ?ij. The third point is that, in any
case, the multipole expansions have to be located while
avoiding numerical dependencies among them. This pre-
cludes the problem of ill-conditioned matrices explained
in Section 2.
These ideas are implemented by following certain
simple semiheuristic rules.
pressed in terms of the concept of area of maximum
influence57of a multipole expansion, which we recall
Di(for the sake of simplicity in this descrip-
Dito find the position of the sources. They com-
In practical situ-
Therefore we use only the
Our experience has shown
Another motivation for this
First, it is ex-
This is be-
Therefore these areas
Some of these rules are ex-
Moreno et al.
Vol. 19, No. 1/January 2002/J. Opt. Soc. Am. A103
pansion is (in two dimensions) a circle of radius R cen-
tered at the source [Fig. 2(b)].
the nearest part of the boundary.
boundary conditions are imposed, every multipole expan-
sion has a maximum influence in a segment of ?ij, and it
has almost no influence in those parts of the boundary ly-
ing far from it and closer to other multipole expansions.
Typically, R ? ?d, where d is the minimum distance
from the source to the boundary and ? ? 1.2–1.4.58
With this concept, we can enunciate the following rules.
The first two rules ensure that the multipole expansion
distribution is able to model the complexities of the field:
The area of maximum influence of a multipole ex-
This source ‘‘illuminates’’
Therefore, when the
concave side of a part of ?ij.
multipole expansion to the boundary must be d ? ?,
where ? is the local radius of curvature of ?ij[Fig. 2(a)].
Rule 2 (Ref. 57). Every part of the boundary ?ijhas to
be inside the area of maximum influence of some multi-
pole expansion [Fig. 2(b)].
Let a multipole expansion be located in the
The distance d from the
Rule 1—together with rule 2—helps to obtain a denser
distribution of multipoles where the curvature is high.
This rule is also related to a guideline59used in the
method of auxiliary sources.
ployed in Ref. 59 can be applied to justify rule 1.
content of rule 2 is clear: It prevents the boundary con-
ditions from not being accurately fulfilled in every part of
the interface. The next rule avoids numerical dependen-
cies among multipole expansions:
The same argument em-
Rule 3 (Ref. 60).
sions whose minimum distances to the boundary are
Let there be two multipole expan-
dk, dl and whose radii of maximum influence are
Rk, Rl, respectively. If their areas of maximum influ-
ence overlap on the boundary, the distance sklbetween
the multipoles must be skl? max(Rk,Rl) ? ?max(dk,dl)
The considerations that led to rule 3 (Ref. 61) necessitate
that the distances from the multipoles to the boundary al-
ways be smaller than the wavelength in the domain
where they act. A last obvious rule is the following:
Rule 4 (Ref. 62).
main Dihas to be located outside this domain.
A multipole expansion acting in do-
Without describing in full detail our AMS algorithm,
we present now an appropriate way to incorporate the
previous three ideas and four rules. Figure 3 shows how,
given an interface curve ?ij, the locations of the multipole
expansions for domain Diare defined. First, an auxiliary
curve ?ioutside Di(see rule 4) and running ‘‘parallel’’ to
?ijis constructed, and then, starting from one end of ?i,
multipolar sources are laid along it.
multipoles (k, k ? 1) may be neither too close (see rule 3)
nor too far (see rule 2) from each other.
rule 1 is satisfied and to minimize the total number of
multipole expansions, we construct the auxiliary curve ?i
by defining the distance d???between ?ijand ?ias a frac-
tion of the local radius of curvature ? of the interface
(but attending to the condition d???? ?Di), i.e., d???
? min(??,??Di), with ?,? ? (0,1).
curve ?iruns very close to the interface where the radius
of curvature is small and recedes where the radius of cur-
vature is large.With such a curve ?iand by using rules
2 and 3 to distribute the multipole expansions, we define
a tight distribution of multipoles near the irregularities of
the boundary, whereas a less dense and farther located
multipole distribution arises where the interface is flatter.
Values that proved to be adequate are ? ? 0.25,63
? ? 0.5, and sk,k?1? max(Rk,Rk?1) ? ?max(dk,dk?1);
i.e., consecutive multipoles are located as close as rule 3
Note that the construction of ?idepends only on the lo-
cal radius of curvature of the interface.
said that ?iis constructed with a local algorithm.
To enforce that
In this manner, the
Hence it can be
the local radius of curvature), (b) rule 2 [the area of maximum
influence of multipole k is shown as well (nonshaded circle)], (c)
Rules for positioning the multipoles (?):(a) rule 1 (? is
the outer domain Diare located on the inner auxiliary curve ?i.
This curve is constructed by using the local radius of curvature of
?ij. The distance between two consecutive multipoles k and
k ? 1 is determined by using rule 3.
Automatic multipole setting (AMS):multipoles (?) for
104J. Opt. Soc. Am. A/Vol. 19, No. 1/January 2002 Moreno et al.
useful for a large number of geometries, the construction
of the parallel curve ?ihas to be refined with a set of
modifications to cope with some problems.
when the boundary has a straight segment, the radius of
curvature becomes infinite and the construction of ?ifails
[Fig. 4(a)]. To avoid this problem, for a flat section of ?ij,
we construct the curve ?iby interpolation from the previ-
ous and next sections of ?i.
when the flat section is long in comparison with the ra-
dius of curvature of the previous and following sections:
The interpolation produces an unnecessarily large num-
ber of multipole expansions [Fig. 4(b)].
use a special kind of interpolation in which ?irecedes
from ?ijnear the middle of the flat segment.
modifications, the construction of the curve ?iis not local
anymore, because it uses information about extended
parts of ?ijto define the distance between both curves.
But it does not take into account the global topology of the
interface. Other refinements have been incorporated to
make the method truly global.
is described in Figs. 4(c) and 4(d).
necks, it may happen that a multipole is located outside
the allowed region (against rule 4) [Fig. 4(c)].
lem is avoided by deforming the curve ?iappropriately.
As shown in Fig. 4(d), it may also happen that multipoles
come too close to each other (against rule 3).
vented by detecting and replacing them by a single mul-
tipole. Our implementation includes several such refine-
geometries. It has free parameters (such as ?, ?, and ?)
A different problem occurs
To avoid this, we
The effect of two of them
For domains with
This is pre-
that can be tuned in specific cases to produce multipole
distributions with a higher or a lower density.
of the resulting multipole distributions with our algo-
rithm are presented in Fig. 5.
The presented method computes the position of the
multipole expansions. Their maximum multipolar order
can be constant for every multipole expansion, or it can be
variable. A scheme that proved to give good results is to
correlate the maximum order of each multipolar expan-
sion with the inverse of the distance between expansion
and interface in such a way that sources close to the in-
terface have higher multipolar order.
thin curves depict the auxiliary curves ?i.
resents the distribution of multipoles generated with the basic
procedure without refinements, and the right column does so
with the corresponding refinements.
Refinements of the basic procedure for the AMS.The
The left column rep-
bal topology of the domains, (b) the corners were rounded as in
Ref. 37, and (c) the radii of the circumferences are r ? 25 nm,
and the distance between the centers is l ? 48 nm.
were rounded with a radius r ˆ ? 0.25 nm.
Distributions of multipoles generated with our AMS
(a) Note that the procedure takes into account the glo-
Moreno et al.
Vol. 19, No. 1/January 2002/J. Opt. Soc. Am. A105
It can be asked if the multipole distributions generated
with this AMS algorithm are optimal or not, although it is
not completely clear what ‘‘optimal’’ means in the context
of the AMS.Theoretically, it would be possible to find
more economical distributions of multipoles producing
equal or more accurate results, but in practice this re-
quires a tremendous effort in most cases. The big advan-
tage of the method is that, in complex problems where
several hundreds of multipole expansions are used, it
simplifies enormously the modeling effort.
tion problems, an AMS algorithm is mandatory:
many electrodynamic calculations have to be solved, and
setting the multipoles by hand is not possible.
PLASMONS FOR VARIOUS GEOMETRIES
As already stated earlier,4the whole treatment presented
in this paper is classical. Hence macroscopic Maxwell
equations and boundary conditions have been applied, as
well as bulk permittivity functions to model surface plas-
mons in nanoscopic systems.
It is worth mentioning that the applicability of such a
classical macroscopic approach may be questioned when
size ranges of the underlying metallic structure tend to-
ward the nanometer scale.
ticle size, fundamental processes give rise to spatial dis-
persion; namely, the electron scattering at confined
particle boundaries will affect the mean free path of con-
duction electrons, resulting in wavelength shift and
broadening ofthe surface-plasmon
Proper quantum surface effects arise within remarkably
lower length scales around the Fermi wavelength of the
conduction electron gas, indicating a fundamental break-
down of all Mie-like theories. Nevertheless, balancing ef-
fects between the quantum-spillout phenomenon (causing
redshift) and the reduced screening of the Coulomb inter-
action (causing blueshift) can result in a surprising agree-
ment with classical predictions involving bulk dielectric
functions,66even for the size ranges addressed by the
various geometries in our examples.
In the framework of metallic nanoparticles, finite-size
dependencies originating from quantum effects are still
under discussion.Our interest in surface-plasmon
modes is mainly driven by the large field variations of this
resonant state, defining, hence, an attractive test case for
the validation of computational electromagnetics codes.
All computations in this section have been performed
with MaX-1 (Ref. 67), which contains the latest imple-
mentation of the MMP method, including an AMS proce-
dure following the principles described in Section 3.
COMPUTATION OF SURFACE
Thus, with decreasing par-
A. Cylinders with Irregular Cross Section
Analytical solutions for scattering on cylindrical struc-
tures are available only for cylinders with circular or el-
liptical cross section. For circular cylinders whose diam-
eter is small compared with the wavelength, the main
resonance occurs for R??(?)? ? ?1?background.
to determine detailed features of the plasmon resonances
of a cylinder with arbitrary cross section—such as, e.g.,
the enhancement of the electromagnetic field—a numeri-
cal computation is required.
In this subsection, we show
calculations for two different geometries in order to as-
sess the accuracy of our method.
In the first computation, the scattering of a plane wave
on a silver cylinder with the cross section depicted in Fig.
5(c) is presented.The background medium is vacuum,
and the dielectric constant of silver for the incident wave-
length (? ? 340nm) is ?Ag? ?1.16 ? i0.30. This wave-
length corresponds to one of the surface modes of this
The wave vector of the exciting field is
k ? ?k?ex, and the electric field vector is contained in the
XY plane. Figure 6 mirrors the complexity of the electric
field near a small-radius-of-curvature part of the inter-
face.This justifies our main assumption for the AMS
procedure: A dense multipole distribution is needed near
the geometrical irregularities of the interface.
computation, the highest relative error along the inter-
face was 0.48%, and the average relative error along the
interface was 0.012%, which demonstrate the high accu-
racy of the solution.
We show a second scattering computation in order to
compare our results with reference data obtained with
Green’s tensor technique with finite elements.39
7, it can be seen that the agreement between both tech-
niques (including the maximum values of the electric
field) is excellent.The cross-section geometry is plotted
in Fig. 5(b). The wave vector of the incident plane wave
is contained in the XY plane, and its direction is perpen-
dicular to the triangle’s hypotenuse, coming from the bot-
tom left of the figure (see Fig. 7 insets). The electric field
vector is contained in the XY plane. For such a silver cyl-
inder in vacuum, the lowest and highest plasmon reso-
? ? 331nm
For these wavelengths, the dielectric con-
stants are ?Ag(? ? 331nm) ? ?0.61 ? i0.28 and ?Ag(?
? 456nm) ? ?7.34 ? i0.23.
plitude of the electric field (normalized to the incident am-
plitude) along the vertical segment with x ? ?5 nm and
y ? ?10nm, 20nm?.The average relative error along
the boundary in the MMP computations was 0.013% for
the lower wavelength and 0.22% for the higher one.
? ? 456nm,
Figure 7 depicts the am-
cusp (see the inset).
complex field pattern can be observed, which justifies a dense
distribution of multipoles near the cusps.
Detail of the electric field distribution around the upper
The lateral size of the figure is 12 nm.A
106 J. Opt. Soc. Am. A/Vol. 19, No. 1/January 2002Moreno et al.
B. Circular Cylinder near an Interface
Proximity between particles modifies their respective sur-
This effect is of great interest in
surface-enhanced Raman scattering because higher field
enhancements can be achieved.
presents a rich spectrum of plasmon resonances by itself,
and the spectrum resulting from the interaction with an-
other object is even more complex.
ficult to interpret this spectrum (especially if one takes
into account that plasmons have complex resonance fre-
quencies and therefore the modes broaden and overlap).
For this reason, we restrict our study to the interaction of
a simple structure (cylinder with circular cross section)
with a substrate.
The simulated system is an infinitely long silver cylin-
der of radius r ? 25nm, near the interface between two
different media.This interface is the XZ plane, and the
cylinder’s axis is parallel to the Z axis (Fig. 8 insets).
The medium above the interface is vacuum, whereas
below the interface, ? ? 2.25.
model was used to represent the dielectric constant of
An irregular particle
Hence it may be dif-
For simplicity, a Drude
?Ag??? ? 1 ?
2???1 ? i2????, (2)
with ? ? 1.45 ? 10?14s and ?p? 1.32 ? 1016s?1.
?Ag(?) function suffices for our purposes, but for more re-
alistic results the experimental values of the dielectric
constant should be used. The excitation is a plane wave
with k ? ??k?ey, and the electric field vector is con-
tained in the XY plane. To study the influence of the cou-
pling strength on the spectrum, we performed the compu-
tations for various distances between the interface and
the cylinder. The distance between the interface and the
cylinder’s closest point to the interface is h (positive if the
cylinder lies in vacuum). In Fig. 8, the scattering cross
section as a function of the exciting field frequency is plot-
ted for several values of h.
For h ? ??, a single maximum is found, which shifts
according to the background medium.
the electric field is enhanced by a factor 246 in vacuum
(h ? ??) and by a factor 167 in a dielectric (h ? ??).
For h ? ?5 nm, the spectrum presents a bump (?0
? 1.26PHz) in addition to two distinct maxima (?1
? 1.3785PHz, ?2
mode have a four-fold pattern, whereas the second mode
The amplitude of
The field patterns for
The field lines of the first
using two different techniques.
(normalized to the incident amplitude) along the dotted line (see
the insets). The circles represent Green’s tensor technique with
finite elements, and the curves represent the MMP with the
AMS. (a) ? ? 331 nm and (b) ? ? 456 nm.
Comparison of the electric field amplitude computed by
The plots show the amplitude
(a) cylinder above the interface and (b) cylinder below the inter-
face. The parameter h denotes the distance between the cylin-
der and the interface.
Scattering cross section as a function of the frequency:
Moreno et al.
Vol. 19, No. 1/January 2002/J. Opt. Soc. Am. A107
has a six-fold pattern (Fig. 9).
again four-folded (i.e., the same as when there is no inter-
face) and small enhancement (147) occurs, whereas for
more complex structure is revealed with more modes
hancements (up to 1658 for ?3
a similar behavior to that before: ?1
ten-folded (see Fig. 10). The enhancement is again small
even more complex for cylinders under the interface [see
In all the computations, the maximum relative error
along the interface was smaller than 0.2%, and the aver-
age relative error was smaller than 0.02%.
?5, the pattern is
?5the enhancement factor is 498. For h ? ?1 nm, a
?1? 1.195PHz, ?1
?1? 1.4255PHz, ?4
?1? 1.3295PHz, ?2
?1? 1.448PHz) and stronger en-
?1). The field lines present
?1four-folded and ?4
?1(only 241), which is four-folded. The spectrum is
C. Periodic Grating
The MMP method allows an efficient computation of peri-
For this purpose, a special kind of
boundary is implemented to define the (fictitious) bound-
aries of the system’s unit cell.
procedure to model a periodic grating. A straightforward
modification to the AMS technique is required:
fields have to be expanded only inside the unit cell, and
therefore, for boundaries defining the unit cell, only mul-
tipoles lying outside the cell have to be generated.
We have modeled a thin silver film (same dielectric con-
stant as that in Subsection 4.B) in vacuum. The metallic
film has a thickness of 112 nm and lies in the XZ plane.
It is corrugated on both sides of the film (the corrugations
are parallel to the Z axis, and there is no relative phase
shift between the upper and lower gratings (see Fig. 12
inset below)).The period of the upper corrugation is 150
nm, and that of the lower corrugation is 300 nm. The ge-
ometry of the grooves consists of Gaussian-shaped dips
inside the film:
We have used our AMS
x ? x0
where the height H of the dips is 50.5 nm and the widths
w are 10 nm at the upper corrugation and 12 nm at the
lower one.The excitation is a plane wave with
k ? ??k?ey, and the electric field vector is contained in
the XY plane.
For the frequencies considered in Figs. 11 and 12, the
penetration depth is ??c/(2?p)?13nm, and, in principle,
no transmission should be expected.
pling of the plasmons in the upper grating with those of
assisted resonant tunneling of light, which has been in-
vestigated for complete23,70and incomplete28perforation
of the film.This is clearly seen in Fig. 11, where the in-
tensities of the zero-order transmitted and reflected
waves are plotted and a power transmission of almost
80% occurs for a resonant frequency.
pected phenomenon is apparent in Fig. 12, where the
first-order intensities transmitted and reflected by the
structure (which has 300-nm periodicity) are plotted.
Electric field for the two highest plasmon resonances
?5) when h ? ?5 nm.
tion for the four highest plasmon resonances (?1
when h ? ?1 nm.
Diagram of the electric field lines and charge distribu-
108 J. Opt. Soc. Am. A/Vol. 19, No. 1/January 2002 Moreno et al.
For a frequency of 1.08 PHz (corresponding to a wave-
length of 278 nm), the first order is a propagating wave.
But, for this wavelength, the periodicity of the upper cor-
rugation (150 nm) should not allow any transmitted or re-
flected propagating order at all (except the zero-order).
Again, resonant plasmon coupling with the lower grating
is responsible for this effect.
In all the computations regarding the metallic grating,
the maximum relative error along the interface was
smaller than 0.34%, and the average relative error was
smaller than 0.02%.
We have presented an automatic multipole setting (AMS)
procedure for the two-dimensional multiple multipole
(MMP) method.Taking into account the geometry of the
simulated system, this AMS technique generates a distri-
bution of multipole expansions adapted to the geometry of
the domains used in the MMP model. The AMS proce-
dure can be applied for the modeling of any electromag-
netics problem with cylindrical symmetry.
computations of surface modes in metallic nanostructures
relevant for various applications have been presented.
Highly accurate solutions have been achieved where the
resonant behavior of these devices does not pose difficul-
ties for the method. The technique is currently being ex-
tended to three-dimensional geometries.
this technique already for other problems such as optimi-
zation of nonperiodic gratings and computation of guided
modes for cylindrical structures with irregular cross sec-
tions, which will be presented later.
In this paper,
We have used
The authors thank J. P. Kottmann from the Nanotechnol-
ogy group in the Laboratory for Electromagnetic Fields
and Microwave Electronics at the Swiss Federal Institute
of Technology for discussions and for sharing some data
shown in Subsection 4.A.This work was supported by
the Swiss National Science Foundation and by the Swiss
Center for Electronics and Microtechnology.
Address correspondence to Esteban Moreno at the loca-
tion on the title page or by e-mail, email@example.com.
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