Multipole Interference in the Second-Harmonic Optical Radiation from Gold Nanoparticles
Sami Kujala,*Brian K. Canfield, and Martti Kauranen
Optics Laboratory, Institute of Physics, Tampere University of Technology, PO Box 692, FI-33101 Tampere, Finland
Yuri Svirko and Jari Turunen
Department of Physics and Mathematics, University of Joensuu, P.O. Box 111, FI-80101 Joensuu, Finland
(Received 5 December 2006; published 18 April 2007)
We provide experimental evidence of higher multipole (magnetic dipole and electric quadrupole)
radiation in second-harmonic (SH) generation from arrays of metal nanoparticles. Fundamental differ-
ences in the radiative properties of electric dipoles and higher multipoles yield opposite interference
effects observed in the SH intensities measured in the reflected and transmitted directions. These
interference effects clearly depend on the polarization of the fundamental field, directly indicating the
importance of multipole effects in the nonlinear response. We estimate that higher multipoles contribute
up to 20% of the total emitted SH field amplitude for certain polarization configurations.
DOI: 10.1103/PhysRevLett.98.167403PACS numbers: 78.67.Bf, 42.25.Ja, 42.65.Ky
Optical responses from metal nanoparticles arise from
the plasmonic oscillations of conduction electrons . The
plasmon resonances depend on the size and shape of the
particles, their dielectric environment, and on their mutual
ordering. These resonances can lead to strong local elec-
tromagnetic fields in the vicinity of the particles. Such
strong local fields may enhance especially the nonlinear
optical responses of nanostructures [2–4]. Nanoscale gra-
dients in the local material properties and fields may also
enable higher multipoles such as magnetic dipoles, electric
quadrupoles, etc. to contribute to the optical responses.
However, the precise role of different multipolar orders
in the nonlinear responsesof nanoscale particles is far from
Usually, the optical responses of particles that are small
compared to the wavelength can be described in the frame-
work of electric-dipole approximation. However, when the
particle size approaches thewavelength, the dipolar picture
may no longer provide a complete description, and higher
multipolar interactions should also be considered. When
discussing multipoles, it should be noted that there are two
types of multipoles: multipoles arising from the light-
matter interaction Hamiltonian , corresponding to mi-
croscopic multipole moments, and multipoles related to
Mie scattering theory . Standard Mie theory is based on
dipolar interaction, and multipoles arise from the size and
retardation effects. However, both microscopic and effec-
tive multipoles lead to similar radiation patterns in the far
The contribution of multipoles in the linear optical
response of metal nanoparticles has been discussed in the
literature [6,7]. Krenn et al. reported experimental evi-
dence of multipolar plasmon resonances from elongated
silver nanoparticles. By changing the nanoparticle length
they were able to tailor the optical responses and observed
several spectrally separate multipole resonances .
Interest in the nonlinear properties of nanoparticles is
steadily increasing [9–12]. However, studies concerning
the role of multipole contributions in the nonlinear re-
sponse are scarce.
A first-principles microscopic theory of the nonlinear
properties of nanoscale particles is still lacking, although
there are several phenomenological treatises [13–17] on
the subject of second-harmonic (SH) scattering, where
both microscopic multipole moments and nonlinear Mie
scattering are included with the result that both contribu-
tions are important. Experimental evidence of these ef-
fects has been reported . Nappa et al. observed strong
size-dependent retardation effects in SH scattering from
20–80 nm gold and silver spheres [19,20]. The responses
from the smallest particles could be explained using the
simple dipole picture, but the responses of larger particles
had both dipolar and quadrupolar contributions. The quad-
rupolar contributions were attributed to retardation ef-
fects arising from the nonlocal excitation of surface
Recently, Klein andco-workers studied second-
harmonic generation (SHG) from a metamaterial consist-
ing of split-ring resonators . The SH response was
greatest when a magnetic resonance was excited. The
response was explained by the fundamental field exciting
the magnetic resonance and driving the velocity of elec-
trons, whose coupling through the Lorentz force gave rise
to the SH response and led to dipolar radiation. In the
suggested mechanism, the magnetic excitation acts as a
sort of local field effect. However, the fundamental cou-
plings between the radiated fields and the particles occur
through dipolar mechanisms.
In this Letter, we provide experimental evidence that
higher multipole radiation accounts for a significant frac-
tion of SHG from arrays of metal nanoparticles. Our
experimental method is based on the differences between
the fundamental radiative properties of electric dipoles as
opposed to magnetic dipoles and electric quadrupoles ,
thereby making the evidence direct. We estimate that for
the samples studied, the higher multipoles can contribute
PRL 98, 167403 (2007)
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© 2007 The American Physical Society
up to 20% of the total emitted second-harmonic field
Multipole sources can be recognized by their far-field
angular emission patterns. However, in the present work,
we measure coherent SH signals, which dominate the
response of the surfacelike samples used and give rise to
strong signals only in the transmitted and reflected direc-
tions. The basis of our measurements is therefore illus-
trated in a simplified way in Fig. 1, where we compare the
radiated far fields of a single electric dipole, a single
magnetic dipole, and an effective quadrupole formed
from a pair of electric dipoles at a given instant in time.
The radiation pattern of a dipole dictates that the direction
of the electric field E does not change when the measure-
ment direction is reversed. However, for both the magnetic
dipole and the quadrupole, the direction of E in the radi-
ated electromagnetic wave does change if we reverse the
direction from which we measure the electric field.
Therefore, for the case of coherent and directional SHG,
the radiative properties of the various multipoles in the
transmitted and reflected directions lead to opposing inter-
ference effects in the two directions. Moreover, we may
expect the strength of the various types of sources to
depend on the polarization of the driving field. The re-
sponse will then exhibit polarization-dependent interfer-
ence effects, which allows us to distinguish the different
multipolar contributions to the overall SHG response, even
when the absolute signal levels cannot be calibrated.
In our experiments, we used a sample consisting of an
array of L-shaped metal nanoparticles, prepared using
electron-beam lithography . The linewidth of the L’s
is ?100 nm, the arms are ?200 nm long, and the gold
layer is 20 nm thick. The nanoparticles are also covered
with a 20 nm protective layer of glass. The nanoparticles
are arranged in a regular array on a glass substrate, with an
array spacing of 400 nm. The period is subwavelength for
both the fundamental and SH wavelengths. However, the
period is larger than the SH wavelength in the substrate.
Therefore, some SH light can be emitted into the substrate,
but at such a large angle that it cannot escape the substrate,
which leaves only the zeroth diffraction order to contribute
to the propagating SH signals. In addition, our technique is
not based on absolute signal levels and is therefore not
compromised by the emission into the substrate. The active
area of the sample is 1 ? 1 mm2.
The symmetry of an ideal L suggests a set of in-plane
coordinate axes, where the x axis coincides with the only
in-plane mirror symmetry axis that bisects the arms of the
L (cf. Fig. 2). The sample exhibits strong dichroism ,
with the x and y polarizations having well-defined plas-
monic resonances at the wavelengths of ?1050 nm and
?1500 nm, respectively. Note that x polarization is near
resonant with our laser wavelength, 1060 nm.
Our experimental setup is shown schematically in Fig. 2.
A train offemtosecond pulses from a Nd:glass laser system
(Time-Bandwith ProductsGLX-200;200fspulse duration,
350 mW average power, 82 MHz repetition rate) is
chopped with an optical chopper and weakly focused
electric dipole (p), magnetic dipole (m), and quadrupole (Q).
Simple schematic of the radiative properties of an
FIG. 2 (color online).
QWP ? quarter wave plate;
analyzer; PMT ? photo-multiplier tube. Inset: the experimental geometry.
Experimental setup. C ? optical chopper; L ? lens, f ? 300 mm; P ? polarizer; HWP ? half wave plate;
VISF ? long-wavelength pass filter;
S ? the sample;
IRF ? short-wavelength pass filter;
PRL 98, 167403 (2007)
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onto the sample with a spot size of ?200 ?m. The state of
polarization of the fundamental beam is controlled with a
half wave plate (HWP) and a quarter wave plate (QWP).
The HWP is used to set the azimuth angle of the initial
linear polarization, and the QWP, mounted in a computer-
controlled motorized rotation stage, modulates the polar-
ization state continuously. The inset in Fig. 2 depicts the
experimental geometry. The sample is tilted slightly off
normal with respect to the fundamental beam (? < 2?) due
to experimental constraints in reflection.
The generated SH light is then polarized with an
s-directed (normal to the plane of incidence) analyzer.
Detecting only s polarization guarantees that differences
in transmission and reflection cannot arise from interfer-
ence between x- and z-directed dipole sources, which
would interfere differently for p-polarized (in the plane
ofincidence) detection . We then detect the s-polarized
SH beam as a function of the QWP angle with a sensitive
photomultiplier tube connected to a lock-in amplifier refer-
encing the chopper frequency.
The detected polarization line shapes in transmission
and reflection are shown in Fig. 3. To analyze the line
shape, we fit the data to an equation relating the intensity
of the SH signal to the polarization components of the
fundamental field :
p?!? ? gE2
s?!? ? hEp?!?Es?!?j2:
This model is the most general one, where the complex
expansion coefficients f, g, h describe the dependence of
the SH response on the various quadratic combinations of
the fundamental field. The results of Fig. 3 have been
normalized to the maximum intensity in order to accom-
modate differences in the light collection efficiency of the
detection arms for the transmitted and reflected SH beams.
This has no influence on the interpretation of our results,
because the technique does not rely on absolute signal
intensities; rather, only relative differences in the polariza-
tion dependence are important.
A significant relative phase shift is immediately appar-
ent between the two data sets. The excellent quality of the
fits of the measurement data to Eq. (1) shows that the phase
shift is not a measurement artifact, but a real, substantive
effect. The line shapes are very sensitive to small changes
in the expansion coefficients and it is therefore impossible
tofit, e.g.,the transmission line shapeusingthe coefficients
from the reflection line shape. This implies that the higher
multipolar contributions to the SH response are consider-
able, but the response is still most likely dominated by a
To exclude simple alignment issues as explanations of
ourresults,weperformed several tests.The influence ofthe
z (normal) component of the fields was checked by repeat-
ing the measurements at an incidence angle of approxi-
mately 2?. No change in the line shape features was found,
implying that the SH response does not couple strongly to
the zcomponent forshallowincidence angles. Tocheck the
sensitivity of the line shape features to the alignment of the
analyzer, the analyzer was tilted ?1?away from s polar-
ization. Only a slight decrease in the overall magnitude of
the SH intensity was observed. In our previous studies, we
have found that similar samples can exhibit optical activity
. To rule out the possibility of any polarization effects
at the SH wavelength, we used a potassium titanyl phos-
phate crystal to frequency double the fundamental beam
and illuminated the sample with SH light. By comparing
the incident and transmitted polarization states, it was
found that the sample negligibly alters the state of polar-
ization at the SH wavelength.
Because the angle of incidence is small and the results
do not depend on the z component of the fields, the s and p
polarizations are essentially equivalent to x and y polar-
izations. The expansion coefficients f, g, h can then be
interpreted as the components of the macroscopic non-
linear response tensor  Axyy, Axxx, and Axxy? Axyx,
respectively. This tensor obeys the electric-dipole-type
selection rules for a given experimental geometry, i.e.,
for reflected and transmitted directions separately. To ana-
lyze the results further, we assume that the largest coeffi-
cient g ? Axxx, which is allowed for the ideal symmetry of
the structure and is resonant at the fundamental wave-
length, is predominantly of electric-dipole origin and nor-
malize it to unity for both directions. The differences in the
other two coefficients for the transmitted and reflected
directions can then be taken as the measure of the impor-
tance of higher multipole contributions. Moreprecisely,we
assume that the g coefficient is of purely dipolar origin and
radiates symmetrically in the reflected and transmitted
directions, while f ? Axyyand h ? Axxyeach consist of
Axyy? Asxyy? Aas
–90 –450 45 90
QWP Angle [deg.]
reflection SH data. Solid and dashed lines are fits to Eq. (1) in
transmission and reflection geometries, respectively.
Measured line shapes. 4: transmission SH data; ?:
TABLE I. Results of the fits to Eq. (1)
0:66 ? 0:58i
0:51 ? 0:13i
0:37 ? 0:67i
0:37 ? 0:26i
PRL 98, 167403 (2007)
20 APRIL 2007
Axxx? Asxxx; Download full-text
Axxy? Asxxy? Aas
where superscripts ‘‘s’’ and ‘‘as’’ refer to parts that trans-
form symmetrically and antisymmetrically between trans-
mission and reflection. Recalling Fig.1,the symmetric part
is identified as originating from the dipolar response, and
the antisymmetric part describes the multipolar contribu-
tion to the response. The fitted values of the coefficients
and their symmetric and antisymmetric parts are shown in
Table I. The antisymmetric parts are approximately 20% of
the symmetric parts, indicating that the multipolar pro-
cesses are indeed a non-negligible portion of the total SH
response. Note also that largest relative multipolar part is
associated with the coefficient h ? Axxy, which arises from
the chiral symmetry breaking of the actual sample .
The above analysis shows that higher multipole and
electric-dipole contributions to the coherent SH signal
can be separated by comparing emission in the transmitted
and reflected directions. We believe that both magnetic and
quadrupolar emission are equally likely candidates for the
higher multipole contribution. Specifically, the x-polarized
signal detected along ?z directions of the laboratory frame
(see Fig. 1) can equally originate from the magnetic dipole
myoriented along the y direction and/or component Qxzof
the electric quadrupole tensor.
In conclusion, we have provided experimental evidence
of multipolar interference in the SH response from a regu-
lar array of gold nanoparticles. The evidence was obtained
by comparing the SH responses in the reflected and trans-
mitted directions as a function of the polarization of the
incident fundamental beam. The differences in the polar-
ization dependence were interpreted in terms of the differ-
ent radiative properties of electric dipoles and higher
multipoles in the two directions, leading to opposite inter-
ference effects. The higher multipole part was estimated to
contribute up to ?20% of the total SH field amplitude,
depending on the polarization of the fundamental field.
This implies that the SH radiation from gold nanoparticles
includes contributions beyond the electric-dipole approxi-
mation that selectively interfere with the dipolar response.
We believe that the higher multipoles arise from nanoscale
gradients in the local fields and material properties. The
local fields depend on the plasmonic resonances of the
individual particles, their defects, and the interparticle
coupling through the array structure. For the present sam-
ple the resonances arise mainly from the individual parti-
cles, with the array playing a lesser role .
This work was supported by the Academy of Finland
(No. 102018 and No. 209806). S.K. acknowledges the fi-
nancial support from the Graduate School of the Tampere
University of Technology. We gratefully acknowledge
Konstantins Jefimovs for preparing the sample used.
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