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OPTO−ELECTRONICS REVIEW 18(4), 421–428
DOI: 10.2478/s11772−010−0047−2
Dipole and quadrupole surface plasmon resonance contributions
in formation of near-field images of a gold nanosphere
M. SHOPA*, K. KOLWAS, A. DERKACHOVA, and G. DERKACHOV
Institute of Physics, Polish Academy of Sciences, 32/46 Lotników Ave., 02−668 Warsaw, Poland
Multipolar plasmon optical excitations at spherical gold nanoparticles and their manifestations in the particle images for−
matted in the particle surface proximity are studied. The multipolar plasmon size characteristic: plasmon resonance fre−
quencies and plasmon damping rates were obtained within rigorous size dependent modelling. The realistic, frequency de−
pendent dielectric function of a metal was used. The distribution of light intensity and of electric field radial component at the
flat square scanning plane scattered by a gold sphere of radius 95 nm was acquired. The images resulted from the spatial dis−
tribution of the full mean Poynting vector including near−field radial components of the scattered electromagnetic field.
Monochromatic images at frequencies close to and equal to the plasmon dipole and quadrupole resonance frequencies are
discussed. The changes in images and radial components of the scattered electromagnetic field distribution at the scanning
plane moved away from the particle surface from near−field to far−field region are discussed.
Keywords: near−field imaging, multipolar surface plasmon resonances, Mie theory, optical properties of noble metal
nanospheres, gold nanoparticles, plasmonic nanostructures, NSOM, PSTM, plasmonics.
1. Introduction
The ability of metal/dielectric structures to sustain coherent
electron oscillations known as surface plasmon polaritons
(SPPs) has been recently intensively investigated [1–6]. The
most rapidly growing areas of physics and nanotechnology
focus on plasmonic effects. Thanks to recent fabrication
advances, nanoplasmonic applications have grown in num−
ber and diversity. In particular, the interaction of light with
metal nanoparticles has been an issue of intense research
[7–9].
It has been shown that nanoscale plasmonic structures
greatly enhance local electromagnetic fields in certain re−
gions near their surfaces at specific wavelengths of light,
controlled by the nanostructure geometry. The geometry−
and size−dependent properties of nanoparticles [10–15]
have potential applications in nanophotonics, biophotonics
and biomedicine [16–19], near−field scanning optical mi−
croscopy [20], sensing and spectroscopic measurements
[21,22].
Much of the impetus for excitement connected with
plasmon−related optical phenomena is driven by the disco−
very and development of surface enhanced Raman scatte−
ring (SERS) technique [23–25]. For example, gold nano−
spheres are used as enhancing surfaces and can provide
enhancement up to 1014 intensity magnitude [10–13,26].
These large enhancements are attributed to highly concen−
trated electromagnetic fields associated with strong loca−
lized surface plasmon (SP) resonances.
*e−mail:
[email protected]
Opto−Electron. Rev., 18, no. 4, 2010
Near−field distribution of the scattered light intensity in
the proximity of a plasmonic nanostructure can give us
valuable basic information about optical properties of that
structure, valuable for diverse applications. The near−field's
complicated pattern contains both, the components able to
propagate and components confined to the surface and
damped outside. Our study refers to the near−field imaging
techniques such as near−field scanning optical microscopy
(NSOM) [14,27,28] and photon scanning tunnelling
microscopy (PSTM) allowing to image the structure of
a sample with nanometer resolution, well below the diffrac−
tion limit [15]. However, interpretation of images still poses
some problems in many applications. One of the reasons is
that the electric field at the scanning tip can be highly non−
−homogeneous. An even more fundamental problem is the
question what is the near−field “homogeneous” image itself;
that is how the near−field distribution characterizing the
object looks like and how it reflects the optical plasmonic
properties of that structure.
Our aim is to illustrate the diversity of the near−field
images of a plasmonic spherical particle illuminated by
a homogeneous light field at frequencies close to and equal
to the frequencies of dipole and quadrupole plasmons and to
understand, what the localization of the surface plasmon
means in the case of spherical interface. We chose Au
sphere of radius of 95 nm as an example of a good scattering
plasmonic nanostructure that is characterized by significant
plasmonic radiative rate. Such particle offers both dipole
and quadrupole plasmon resonance frequencies in the visi−
ble range. Let us notice, that in many experiments, particles
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Dipole and quadrupole surface plasmon resonance contributions in formation of near−field images of a gold nanosphere
used are very small in comparison with the light wave−
length. Particles of the radii of up to about 10 nm of well
defined spherical shape and the controlled size can be fabri−
cated more easily than the larger ones. However, the scatte−
ring abilities of much smaller than the light wavelength par−
ticle are negligible in comparison with the absorptive ones.
Therefore, such small particles are not optimal for applica−
tions in which the enhancement of the plasmon near−field
surrounding the particle is needed (for example SERS tech−
nique or expected transport of confined plasmon excitations
from particle to particle at nanometer length scales in
particle arrays).
The near−field images of Au sphere of radius of 95 nm at
resonance dipole and quadrupole frequencies resulted from
the spatial distribution of the light intensity that is propor−
tional to the component of the full mean Poynting vector,
that is normal to the detection scanning plane placed in the
particle proximity. While for such large particles (compared
to the light wavelength), the quasistatic approximation does
not hold, we used the results of the rigorous size dependence
modelling of the inherent particle plasmon size characteris−
tics [29–31]. Using the Mie theory formalism we show, that
the maxima in the total scattering or absorption spectra for
such large particles (which are the manifestations of SP reso−
nances), are blue shifted in respect to the plasmon resonance
positions. The enhancement of the scattered or absorbed
light intensity (arising from the interference of plasmonic
electric and magnetic fields with other nonresonance contri−
butions) is the manifestation of surface plasmons only, the
notion related to TM electromagnetic field waves localized
at the metal−dielectric interface (or to the surface densities
free−electron waves coupled to these fields) [30,31]. That is
why a manifestation of plasmon resonance characterised by
a large radiative plasmon damping rate (spectrally large
maxima in light intensity) can be shifted.
The change in the images with distance between nano−
sphere and detection surface is also discussed. While the
near−field radial components of the electromagnetic field
are expected to play considerable role in the image forma−
tion (as they are coupled with the SP waves at the particle
surface) we completed our consideration by the spatial dis−
tribution of that component over the scanning plane.
2. Formalism and simulation background
2.1. Plasmon size characteristics
Possibility of resonant excitation of the surface plasmon
oscillations (collective oscillations of surface free−electron
densities) and damping of these oscillations we treat as the
inherent property of a conducting sphere, that is abstracted
from the quantity, which can be observed [29,30]. Excita−
tion of SP oscillations is a resonant phenomenon that is pos−
sible when the frequency of the incoming light field appro−
aches the characteristic eigenfrequencies wl¢ ( R), l = 1,2,3,...
of a plasmonic sphere of the radius R with l = 1 for dipole
and l = 2 for the quadrupole plasmon mode. The dynami−
422
Fig. 1. Multipolar (l = 1,2,...) plasmon resonance frequencies for
gold nanosphere as a function of particle radius. The dipole and
quadrupole plasmon resonances for 95−nm gold nanosphere can be
excited with photon energy 1,96 eV and 2,57 eV, respectively.
cally induced surface charge density distributions can be of
diverse complexity. With the increasing sphere surface,
multipolar distributions of higher than dipole distribution
can be induced [31]. If excited, plasmon oscillations are
damped at the corresponding rates wl¢¢( R).
The SP resonance frequencies wl¢ ( R) as a function of size
for gold nanospheres used here are derived in Ref. 30. The
complete plasmon size characteristics result from self−con−
sistent rigorous electromagnetic approach described in more
detail in Refs. 29 and 30. The example of size dependence
of the multipolar plasmon oscillation frequencies wl¢ ( R)
completed by the corresponding damping rates wl¢¢( R) are
shown in Fig. 1 for gold sphere.
As a scattering object we chose Au sphere of the radius
R = 95 nm. For such large particle, the quasistatic approxi−
mation certainly does not work. According to the data in
Fig. 1, a particle of that size is characterized by significant
plasmon radiative rates g rad
( R) = wl¢¢( R) - g 2, l = 1,2,3...,
l
with wl¢¢=1,2 ( R = 95 nm) equal to 0.663 eV and 0.066 eV
correspondingly (see Fig. 1) in both dipole l = 1 and qua−
drupole l = 2 plasmon modes. A particle with such large ra−
diative rates is an efficient radiative antenna able to scatter
the light effectively in proportion to the radiative rate contri−
bution in the total damping rate [31]. Both the dipole wl¢=1
( R = 95 nm) = 1,96 eV and quadrupole wl¢=2 (R = 95 nm) =
2,57 eV plasmon resonance frequencies fall in the visible
range (l = 632.57 nm and 482.43 nm correspondingly) and
are spectrally easy to resolve. Let us notice, that without
knowing plasmon size characteristics, the resonance dipole
and quadrupole plasmon frequency for the sphere of chosen
radius would be unknown, while for such large particles
(compared to the light wavelength) the quasistatic appro−
ximation does not hold and the rigorous size dependence
must be used.
Plasmonic features, that we could derive from plasmon
size characteristics presented in Fig. 1, are reflected in the
spectra of the total absorption and scattering cross−sections
Opto−Electron. Rev., 18, no. 4, 2010
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g = 0,072 eV) were chosen to satisfactorily reproduce the
experimental values of Re[n 2 (w)] (Im[n 2 (w)]) in the fre−
quency ranges 0,8 eV – 5,0 eV [37]. The optical properties
of the dielectric medium outside the sphere e out = n 2out
were assumed to be 1.
2.3. Orthogonal polarization geometries
Fig. 2. Scattering, extinction and absorption cross−sections spectra
for gold nanosphere of R = 95 nm.
calculated for a sphere of R = 95 nm with help of Mie theory
and presented in Fig. 2 [32–34]. However, Mie theory does
not deal with the notion of collective surface density oscilla−
tions (neither the notion of plasmon resonances). The
enhancement of the light intensity (arising from the interfer−
ence of plasmonic electric and magnetic fields with other
nonresonant contributions) is the manifestation of the SP
oscillations. Let us note, that the notion of surface plasmons
is related to the electromagnetic fields localized at the
metal−dielectric interface. These fields are coupled to the
free−electron surface wave densities [30,31].
Let us also notice, that Mie theory is not a handy tool in
studying the size dependence of plasmon resonances that
manifest in some maxima of the light spectra. In addition,
maxima in the total scattering cross−sections can be “blue
shifted” in respect to the multipolar plasmon position derived
from eigenvalue problem and presented in Fig. 1. The fre−
quency shift is bigger if the spectral width of the maximum is
larger. It is due to the dispersion of the metal and complex
interference of scattered fields of many multipolar orders of
TM and TE modes contributing to the maximum [30]. The
broadening of the maxima is approximately proportional to
wl¢¢=1,2 ( R = 95 nm) . Therefore, the assumption of the posi−
tion of the maximum in the scattered light intensity as the
surface plasmon resonance position can be inaccurate.
Contribution of the dipole and quadrupole plasmon reso−
nances to the scattered light intensity and its distribution (so
the particle image), can be conveniently demonstrated by
studying the two orthogonal polarization geometries I ^ and
I || sketched in Fig. 3(a). The chosen orthogonal polarization
geometries allow us to demonstrate the role of the dipole
and the quadrupole plasmon mode contribution in distinc−
tion from plasmon mode contributions of higher multipolar
order [37]. While I ^ and I || are the intensities of the purely
scattered light (no influence of the interference with the EM
field of the incoming light wave propagating in perpendicu−
lar direction), observation of I ^ and I || makes study of the
pure scattering abilities of a plasmonic sphere possible.
Such observation geometries enable the exposure of
a single dipole (l = 1) plasmon contribution and a single
quadrupole (l = 2) plasmon contribution and to separate these
contributions spatially by observing I ^ and I || scattered in−
tensities. Alternatively, when keeping the directions of illu−
mination and observation unchanged, the rotation of linear
polarizers by the angle 90° [Fig. 3(b)] leads to the same ob−
servation opportunity (I ^ ® I Vv and I || ® I Hh ), what is
more convenient from the experimental point of view [37].
The effect of spatial separation of a single dipole and
quadrupole plasmon contributions to the spectra of the scat−
tered light intensities I ^ and I || in the centre of the scanning
plane calculated on the basis of Mie theory is demonstrated
in Fig. 4.
2.2. Input material parameters
In order to take metal's plasmonic properties into account,
we consider the analytic form of the Drude dielectric func−
tion with the effective parameters ein (w) = e D (w) = e ¥ – w2p
/(w2 + igw) that describe optical properties of many metals
quite well within relatively wide frequency range. e ¥ is the
phenomenological parameter describing the contribution of
the bound electrons to the polarizability. w p is the bulk
plasmon frequency and g is the phenomenological electron
damping constant of the bulk material [35,36]. The effective
parameters of the function ein (w)(e ¥ = 9,84, w p = 9,096 eV,
Opto−Electron. Rev., 18, no. 4, 2010
Fig. 3. Scheme of orthogonal polarization geometries showing po−
larization directions of incident and scattered light in respect to an
observation plane.
423
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Dipole and quadrupole surface plasmon resonance contributions in formation of near−field images of a gold nanosphere
Fig. 4. Spectra of the scattered light intensities I ^ and I || for gold
nanosphere of radius 95 nm, with dipole surface plasmon resonance
at w = wl¢= 1( R ) = 1, 96 eV and quadrupole resonance atw = wl¢= 2 ( R )
= 2,57 eV in the orthogonal polarization geometries (Fig. 3).
2.4. Numerical near-field images
Plasmons excited at spherical gold nanoparticles contribute
to the particle near−field distribution and to the particle
images formatted in the particle proximity. Plasmons cause
enhancement of the near−field intensity, but they introduce
also important modification of the spatial field and intensity
distribution. Contribution to the image of the electromag−
netic fields coupled to plasmon oscillations are not straight−
forward, while the spatial distribution of the light field
intensity is a result of constructive and destructive interfe−
rence of all the fields scattered by the particle. The distribu−
tion of electric and magnetic fields of light scattered by
a sphere can be found from Mie theory [35,36].
A measure of the scattered light intensity is the magni−
tude of the Poynting vector S = 1/2Re(E ´ H), averaged
over the time interval which is long compared with the light
wave period 2p w
r$ q$ j$
1
S(q , j , r ) = Re E r Eq Ej = r$S r (q , j , r )
.
2
H r Hq Hj
(1)
+ q$S q (q , j , r ) + j$ S j (q , j , r )
The magnitude of the Poynting vector S represents the
averaged density of the scattered energy flow, the direction
of S corresponds to the direction of energy propagation. The
image plane (Fig. 5) serves as a scattered light intensity
detector that registers photons flux normal to the plane. The
image in the XY plane is formed due to the distribution of
the normal component S n (q , j, r) of the Poynting vector
over this plane
$ (q , j , r )
S n (q , j , r ) = nS
424
(2)
Fig. 5. Scheme of construction of the normal component Sn(q , j , r )
of the averaged Poynting vector over the XY scanning plane at the
distance d.
where n$ is the unit vector normal to the XY plane. If we use
the angles q , j , j1 to parameterize the position (x, y) at the
plane XY [38] according to the notation introduced in
Fig. 5, the normal component S n (q , j , r ) of the averaged
Poynting vector can be represented as
S n (q , j, r) = Sr (q , j, r) cos j1 + Sq (q , j, r) sin q cos j
- Sj (q , j, r) sinj
(3)
In near−field region, the radial components of the elec−
tric and magnetic fields Er and Hr are comparable to the
components Eq , Hq , Ej , Hj , correspondingly. They play
significant role in image formation, while
S n ~ ( Eq Hj - Ej Hq ) cos j1 + ( - Er Hj - Ej Hr )
´ sin q cos j + ( Er Hq - Eq Hr ) sinj
(4)
The amplitudes of the corresponding field components
decay with the distance r as Er , Hr ~ 1 r 2 and Eq ,j , Hq ,j
~ 1 r. In far field region (i.e. for r such that1 / r >> 1 / r 2 ), ra−
dial components of the electric and magnetic fields Er and
Hr are negligible in comparison with the transverse compo−
nents Eq , Hq , Ej , Hj . Therefore, while moving away from
near−field to far field region, the scattered wave gains
transverse character.
3. Results and discussion
We acquired a series of images for gold sphere of the radius
R = 95 nm in both, parallel and perpendicular polarization
geometries. The series illustrate changes in images and Er
distribution with distance of the scattering plane from the
sphere, as well as changes due to the applied frequency of
the illuminating light wave. The angular dimensions of the
scanning plane remain the same as the scanned angles
q , j Î ( -80° , +80° ). Linear dimensions of XY scanning
plane are approximately 1000×1000 nm at the close sphere
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Fig. 6. Numerical images (upper row) and distribution of the radial component of the scattered electric field E r (lower row) at scanning plane
placed at the close proximity of the particle surface d = R. The frequency of incident light w is changing from below to above the dipole
plasmon resonance frequency at wl¢= 1 = 1, 96 eV(see Fig. 1). (a) polarization perpendicular to the scanning plane and (b) polarization parallel
to the scanning plane.
proximity. The actual size of the scanning plane is given in
each image. Polarization of incident light is assumed to be
parallel or perpendicular to the scanning plane (Fig. 3).
While the brightness of images differs by orders of magni−
tude, each image possess its own scale, that allows associate
colours and intensity which is proportional to the normal
component of the full Poynting vector [Eq. (4)] to the scan−
ning plane, as discussed above. The components of the scat−
tered field electromagnetic field contributing to the Poyn−
ting vector are calculated on the basis of Mie theory.
Figure 6 illustrates images at frequency equal to the di−
pole plasmon frequency wl¢=1 ( R = 95 nm) = 1, 96 eV and at
frequencies below and above that frequency. The images in
Vv observation geometry [Fig. 6(a)] are much larger than the
object itself. They are strongly affected by the dipole SP re−
sonance contribution, as one could expect for perpendicular
observation geometry (see also Fig. 4). The important con−
tribution of the near−field radial components of electromag−
netic field is the reason for this effect, as illustrated in Fig.
6(a), lower row. The widespread distribution of the radial
component of the electric field Er above the sphere [lower
row in Fig. 6(a) can be interpreted as a projection of the ex−
cited SP wave at the sphere surface]. The spectral width of
this effect is defined by the corresponding radiative rate in−
cluded in the dipole plasmon rate wl¢¢=1 ( R = 95 nm) = 0,663
eV (Fig. 1) and is much larger than the corresponding rate
wl¢¢=2 ( R = 95 nm) = 0,066 eV for the quadrupole plasmon.
These rates define half−widths at half maximum (HWHM)
of the I Vv spectrum at any point of the plane including the
plane centre (see Fig. 4). However, influence of distribution
on the image formation is not straightforward. The corre−
sponding intensity results from interference of all the
components of the near−field, see Eq. (4).
The images in Hh orthogonal observation geometry
[Fig. 6(b)] are not affected by the surface−localized fields
coupled to the dipole SP, according to the expectation (a
dipole does not radiate in the axis direction). Therefore, in
such observation geometry, the images better reflect the cir−
cular shape of the object and its size.
Figure 7(b) illustrates the images obtained in Hh polari−
zation geometry with intension to demonstrate the role of
quadrupole SP contribution in image formation. The images
in Hh polarization geometry at the quadrupole plasmon fre−
quency wl¢=2 ( R = 95 nm) = 2,57 eV are much brighter than
for neighbouring frequencies used for image simulations; as
expected, the spectral width of the enhancement is much
smaller than for the dipole case, and is given by the corre−
sponding radiative rate wl¢¢=2 ( R = 95 nm) = 0,066 eV. At
Fig. 7. Numerical images (upper row) and distribution of the radial component of the scattered electric field E r (lower row) at scanning plane
placed at the close proximity of the particle surface d = R. The frequency of incident lightwis changing from below to above the quadrupole
plasmon resonance frequency at wl¢= 2 = 2, 57 eV (see Fig. 1). (a) polarization perpendicular to the scanning plane and (b) polarization para−
llel to the scanning plane.
Opto−Electron. Rev., 18, no. 4, 2010
425
M. Shopa
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Dipole and quadrupole surface plasmon resonance contributions in formation of near−field images of a gold nanosphere
Fig. 8. Numerical images (upper row) and distribution of the radial component of the scattered electric field E r (lower row) at the scanning
plane moving away from near−field region to the far field. Distance between the particle surface and the scanning plane is 0 nm, 50 nm,
100 nm, 200 nm, 400 nm, and 800 nm, as seen from left to right. The frequency of incident light wis equal to: (a) the quadrupole plasmon reso−
nance frequency wl¢= 2 = 2, 57 eV for polarization parallel to the scanning plane and (b) the dipole plasmon resonance frequency wl¢= 1 = 1, 96 eV
for polarization perpendicular to the scanning plane.
the quadrupole SP resonance, the radial electric field ampli−
tude Er is strongly enhanced as well [Fig. 7(b), lower row].
Figure 8 illustrates the change in images and in distribu−
tions of the amplitude of radial electric field component at
the scanning plane moved away from the particle surface
from near−field to far−field region. Figure 8(a) reflects the
changes in spatial distributions in Hh geometry at the qua−
drupole plasmon resonance frequency wl¢=2 ( R = 95 nm) =
2,57 eV, while Fig. 8(b) corresponds to Vv polarization ge−
ometry at the dipole plasmon resonance frequency
wl¢=1 ( R = 95 nm) = 1,96 eV.
Square decay of the radial components of the near−field
with distance causes not only the change in the image
brightness, but also the image shape. The lower row in Figs.
8(a) and 8(b) illustrate the changes in the distribution of Er
of the scattered field with the distanced d, correspondingly.
Such distribution reflects the pattern of the plasma waves
excited at the particle surface at the dipole and quadrupole
frequencies. The normal to the interface component of the
electric displacement field Dr must be continuous at the
interface. Resulting jump in Er must be caused by the
induced free electron surface charge density. The asymme−
try of the pattern at the scattering plane (Fig. 8) is caused by
the presence of the illuminating light field (coming from the
left side of the picture) and its weak (due to the chosen
observation geometry) but nonzero contribution to the
patterns.
426
4. Conclusions
Near−field images of a spherical plasmonic particle illumi−
nated by linearly polarized light are deviated from a circular
shape. For given particle size, the shape of the image
changes with the direction of illuminating field polarization
in respect to the direction of observation and depends on the
frequency of the illuminating field. This last tendency is not
only a result of a simple dependence of the metal index of
refraction n(w) on frequency, but is mainly due to the
plasmonic abilities of a particle and is size dependent. SP
features are described by the SP resonance frequencies
wl¢ ( R) and the damping rates wl¢¢( R) with size dependence
due to the radiative damping. These parameters define
plasmonic features of a metallic nanosphere embedded in
a dielectric environment of known index of refraction. As
our analysis show, excitation of SP resonances possess sub−
stantial impact on the particle near−field image; not only its
brightness, but also its shape. The larger the damping rate
wl¢¢( R) is, the broader is the spectral range around the SP res−
onance frequency wl¢ ( R) for these effects. Near−field scat−
tered by a sphere is strongly nonhomogeneous. That causes
dramatic changes in the sphere images with the distance d to
the detector.
The amplitudes of the radial component of the electric
and magnetic field, which decrease in proportion to the
square of the distance from the scatterer, play a significant
Opto−Electron. Rev., 18, no. 4, 2010
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role in building the near−field image of a plasmonic particle,
and cannot be omitted in analysis of the near−field distribu−
tion in the vicinity of nanostructures of different shapes. In
the near−field region, the radial components modify the
length and direction of the Poynting vector.
Our preliminary study is aimed to demonstrate the diver−
sity and complexity of near−field images of a plasmonic
spherical particle illuminated by a light at frequency close to
or equal to the frequency of dipole and quadrupole plasmon
resonance. Our results, though being only a numerical
model, can provide a starting point on what scattered field
we can expect from a noble metal nanosphere. Despite only
the example of gold sphere of the radius R = 95 nm is stud−
ied, the conclusions are general enough to be used for any
other plasmonic particle. Sphere is considerably a simple
object, but has a unique advantage of precise solutions of
interesting electrodynamic problems. Features of localized
surface plasmon waves on a sphere are still not fully under−
stood and revealed. By studying plasmonic sphere features,
we can broaden our knowledge of undergoing electromag−
netic processes connected with SP and SP manifestation in
measurable quantities before moving on to more complex
objects, widely studied nowadays, for example in Refs. 19
and 21. Nevertheless, spherical nanoparticles are widely
used in all kinds of applications, exploiting plasmonic
features of noble metal nanospheres.
Acknowledgements
We would like to acknowledge support of this work by the
Polish Ministry of Science and Higher Education under
Grant No N N202 126837.
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