Local Field Asymmetry Drives
Second-Harmonic Generation in
Brian K. Canfield,* Hannu Husu, Janne Laukkanen,†Benfeng Bai,†
Markku Kuittinen,†Jari Turunen,†and Martti Kauranen
Institute of Physics, Optics Laboratory, Tampere UniVersity of Technology,
P.O. Box 692, FI-33101 Tampere, Finland
Received January 17, 2007; Revised Manuscript Received March 16, 2007
We demonstrate that second-harmonic generation (SHG) from arrays of noncentrosymmetric T-shaped gold nanodimers with a nanogap arises
from asymmetry in the local fundamental field distribution and is not related strictly to nanogap size. Calculations show that the local field
contains orthogonal polarization components not present in the exciting field, which yield the dominant SHG response. The strongest SHG
responses occur through the local surface susceptibility of the particles for a fundamental field distributed asymmetrically at the particle
perimeters. Weak responses result from more symmetric distributions despite high field enhancement in the nanogap. Nearly constant field
enhancement persists for relatively large nanogap sizes.
A great deal of the recent interest in the optical responses
of metal nanoparticles, nanoapertures in metal films, and
metamaterials focuses on enhancing local electromagnetic
fields to facilitate light-matter interactions. Random and
fractal metal clusters have been predicted to lead to giant
enhancements of the local electric field.1-3Strong local fields
are particularly important for nonlinear optical processes,
such as surface-enhanced Raman scattering and second-
harmonic generation (SHG),4-7which scale with a high
power of the field.8Indeed, field enhancement by rough metal
surfaces has enabled the sensing of single molecules by
surface-enhanced Raman scattering.9Controllable local-field
enhancement would also benefit other photonics applications
such as nanoscale antennae that operate at optical wave-
lengths,10,11nanoscale lenses for subwavelength focusing and
photolithography,12two-photon microscopy,13and even
magnetically resonant metamaterials.14,15
Enormous enhancement factors of 103-106compared to
the fundamental electric field at a flat metal surface have
been calculated for closely spaced “designer” metal nano-
structures such as chains of self-similar metal spheres, disks,
and truncated tetrahedral prisms.16-18These large values were
predicted for gaps of just a few nm (“nanogaps”) between
nanoparticles. However, very small features such as particle
separations of less than 10 nm are exceptionally difficult to
fabricate lithographically,19and particle boundaries are often
less than ideal.20Actual nanogaps thus tend to be both
substantially larger and less well-defined than their theoretical
counterparts. As a consequence, experimentally obtained
enhancement factors are often far less impressive, generally
only 1-2 orders of magnitude.21-23
Symmetry of the nanostructure and polarization of the field
also play very important roles in field enhancement. For
nanodimer systems formed by stripes,11tip-to-tip triangle
“bowties”,24intersecting cylinders19,25and even cylindrical
apertures in a metal film,26high enhancement is observed
only when the incident polarization is aligned parallel to the
interparticle axis. Illuminating the sample with the perpen-
dicular polarization yields significantly less enhancement.
The special role of this axial direction has also been
emphasized in the less symmetric case of enhanced SHG
from the tip of a near-field optical microscope.27
In most cases, the effects exhibiting the highest enhance-
ments have depended mainly on the intensity of the local
field. However, additional phase and symmetry consider-
ations dictate that second-order nonlinear optical processes
(such as SHG) require noncentrosymmetry.8Therefore,
strong fields alone may not be sufficient for SHG if the
sample is centrosymmetric. Several of the two-dimensional
lithographic designer structures on a substrate appear cen-
trosymmetric when investigated at normal incidence,11,19,25,26
although structures with lower in-plane symmetry have also
been presented.16,28,29A chain of self-similar spheres, on the
other hand, forms a noncentrosymmetric system and is
predicted to enhance SHG by a (nonresonant) factor of ∼104,
but only for very small sphere separations of a few nm.30
We provide here experimental evidence supported by
numerical calculations that the intensity of SHG from arrays
* Corresponding author. E-mail: firstname.lastname@example.org.
†Department of Physics and Mathematics, University of Joensuu,
P.O. Box 111, FI-80101 Joensuu, Finland.
Vol. 7, No. 5
10.1021/nl0701253 CCC: $37.00
Published on Web 03/31/2007
© 2007 American Chemical Society
of noncentrosymmetric T-shaped gold nanodimers with a
nanogap does not depend strictly on the nanogap size and
having a strong local field in the nanogap. Instead, asym-
metry in the distribution of the induced local field at the
fundamental wavelength and its interaction through the local
surface susceptibility of the nanodimers play key roles. We
applied rigorous diffraction theory to calculate the local field
distributions as a function of nanogap size. The induced local
field exhibits orthogonal polarization components, not present
in the incident exciting field, that yield the dominant SHG
response. The calculations also reveal that the symmetry of
the field distribution varies with nanogap size in a more
complicated manner than simple size dependence. As op-
posed to more symmetric nanodimer systems, field enhance-
ment occurs for both incident field polarizations and, counter
intuitively, persists up to relatively large nanogap sizes of
several tens of nm.
We fabricated several square arrays (0.9 mm per side) of
T-shaped gold nanoparticles on the same fused silica
substrate using electron beam lithography.29The “T” is
formed by properly orienting separate horizontal and vertical
bars. We introduced nanogaps between the bars ranging from
a “barely contacted” 0 nm shown in the scanning electron
micrograph (SEM) image in Figure 1a to a well-separated
40 nm in Figure 1b. For comparison, we also fabricated an
array of fully contacted T’s, as shown in Figure 1c. The
structural design of the T’s suggests a natural X-Y
coordinate system as depicted in Figure 1a; consequently, a
single mirror-symmetry axis lies along Y. While an ideal T
structure would be formed by rectangular bars with square
corners, the actual bars exhibit rounded corners (a “stadium”
shape). All bars were intended to be of equal length
(approximately 250 nm), although difficulties in the fabrica-
tion process (such as beam astigmatism) caused variations
of up to 10% in this value, most commonly in the horizontal
bars. The line width of all bars is 125 nm, the gold thickness
is 20 nm, and the grating spacing in all arrays is 500 nm.
The arrays are also protected by a 20 nm thick layer of fused
Prior to SHG measurements, we measured the extinction
spectra of the arrays for both X- and Y-polarizations using
fiber optic spectrometers and a white light source (Supporting
Information). The main extinction resonances in metal
nanoparticles arise from collective dipolar oscillations of
conduction electrons (usually denoted “plasmons”), and the
peak wavelength depends on the plasmon oscillation length
(i.e., bar length).25,31,32Although some of the bars in the 0
nm array of Figure 1a visually appear to be contacted, the
Y-polarized extinction spectra in Figure 2 clearly indicate
that the bars of this array respond separately and distinctly,
like those of the larger-nanogap arrays. The isolated vertical
bars from larger-nanogap arrays (being all about the same
length) display resonance peaks near 1100 nm, with the 0
nm nanogap (where the vertical bars are slightly shorter)
nearby at 1060 nm. The much longer oscillation path length
in the fully contacted array of Figure 1c, on the other hand,
results in a resonance peak location at a much longer
wavelength of 1530 nm. The X-polarization resonances
exhibit more variation in the peak location (1100 ( 100 nm)
because of differences in bar length resulting from the
fabrication process as described earlier. The smaller reso-
nances below 750 nm are due to the (transverse) linewidths
of the horizontal bars,31but as they lie far from both the
fundamental laser wavelength at 1060 nm and the second-
harmonic wavelength at 530 nm, we neglect them.
The SHG responses of the arrays were measured using a
simple one-beam, normal-incidence geometry (Supporting
Information, Figure S1). Because of normal incidence, the
polarizations of the fundamental and SHG fields can be
expressed in the same X-Y coordinate system as the sample,
and electric-dipole-type selection rules apply.31The mirror
symmetry with respect to the Y axis dictates that the allowed
signals are YYY, YXX, XYX, and XXY, where the two
last indices refer to the polarization of the fundamental beam
(which acts twice) and the first to that of the SHG beam. Of
these, we measure the YYY and YXX signals, which
represent the pure polarization combinations of the output
SHG and input fundamental fields.
The intensities of the SHG responses YXX (circles) and
YYY (triangles) are shown in Figure 3 as a function of
nanogap size, normalized by setting the weakest response
to unity. The SHG responses clearly depend on the size of
the nanogap, but not in smoothly decreasing functions as
might be inferred from theory.30The largest nanogap (40
nm) yields weak SHG responses for both components, as
might be expected from theory that predicts that the nanogap
must be small (a few nm or less) to enhance the SHG
response25,30and from experimental evidence of overlapping
cylindrical apertures.26Another consideration applies as
well: as the nanogap size increases, the whole structure
approaches a centrosymmetric configuration of alternating
horizontally and vertically oriented bars, which would inhibit
an SHG response.8The largest SHG response in Figure 3
noncontacted 0 nm nanogap, (b) 40 nm nanogap, (c) fully contacted.
The scale in (c) is the same for all three SEMs.
SEM images of T-shaped gold nanoparticles. (a)
Figure 2. Y-polarized extinction spectra for different nanogap
sizes. Dots: 0 nm nanogap; solid line: 15 nm nanogap; dashes:
40 nm nanogap. For comparison, the spectrum of fully contacted
T’s is included (dash-dots).
Nano Lett., Vol. 7, No. 5, 2007
occurs for YXX from the 0 nm nanogap and is about a factor
of 50 larger than the response from the 40 nm nanogap array.
This result is also in accordance with theoretical calcula-
tions,25,30although the enhancement observed here is smaller.
However, the most striking features of Figure 3 cannot
be readily explained merely by requiring a small nanogap.
Although both X- and Y-polarizations are approximately
equally resonant at the fundamental, the YYY response is
much weaker than the YXX response for smaller nanogaps.
The YYY response grows with increasing nanogap size up
to 20 nm, as opposed to the decreasing YXX response. Also,
the origin of the drop and recovery of the YXX response in
the 20-30 nm nanogap range is mysterious. To explain these
issues, it is necessary to consider both the symmetry of a
T-particle and the local fundamental electric field distribution
in an array unit cell.
Local electric field distributions at 1060 nm (fundamental
wavelength) and 530 nm (wavelength of second-harmonic)
were calculated in a unit cell using the Fourier modal
method33for both X- and Y-polarized incident fields of unit
magnitude (Supporting Information). The calculations reveal
that the fundamental X-polarized field component localizes
at the ends of the horizontal bar, while the Y-component
concentrates in the nanogap region (cf., Figure 4). However,
at 530 nm, no strong local fields are observed (enhancement
factors are less than 1.7 for X-polarization and 1.3 for
Y-polarization), the distributions remain unaffected by the
presence of the nanogap, and the field never localizes in the
nanogap. Therefore, the SHG response is indeed driven by
localized enhancement of the nearly resonant fundamental
field. Moreover, this result agrees with previous findings that
SHG from nanostructured metal surfaces results from
overlapping eigenmodes of the fundamental and second-
harmonic fields,34especially from where the local configu-
ration is asymmetric.7However, in our case, the local field
effects at the SHG wavelength are weak, and the response
is therefore dominated by the strong and asymmetric local
The strongest fundamental fields occur close to the particle
boundaries, meaning that the SHG responses arise through
the local surface susceptibility of the metal particles. The
total SHG response thus results from integration of the local
response along the particle perimeters (Supporting Informa-
tion). The structure of the local tensor implies that contribu-
tions from the portions of the perimeters possessing opposite
surface normals tend to cancel.8Therefore, the more sym-
metrically the local field is distributed along the perimeters,
the weaker we expect the SHG response to be. Moreover,
although the nanogap region itself is formally noncentrosym-
metric, the responses from the horizontal bar bottom and
the vertical bar endcap have a strong tendency to cancel when
the local field is evenly distributed across the nanogap
(Supporting Information, Figure S2).
The polarization behavior of the field displays intriguing
behavior that, while not readily apparent from the SHG
experiment, nevertheless agrees well and helps explain the
unanticipated results. We note that the induced local funda-
mental field contains orthogonal polarization components not
present in the exciting field. For instance, a purely X-
polarized incident field induces a local Y-component com-
parable in magnitude to the local X-component, as depicted
in Figure 4a and c for 1 and 20 nm nanogaps. For the dimer
with 1 nm nanogap, this Y component localizes around the
horizontal bar but is distributed asymmetrically, with strong
enhancement around the nanogap region and the bar surface.
This induced polarization component, coupled with its
asymmetric distribution, comprises the origin of the dominant
SHG response, YXX, in good agreement with the high
response of the 0 nm nanogap in Figure 3. The sample with
a 20 nm nanogap, on the other hand, shows not only much
weaker enhancement but a more symmetrical distribution,
which then explains the low YXX response at 20 nm in
Figure 3. The calculations also show that the asymmetric
distribution of Figure 4a partially recovers for even larger
(25-35 nm) gap sizes, which explains the secondary YXX
peak in the experimental data.
With the nanogap positioned at the end of the vertical bar,
one might expect the field to localize in the nanogap region
Figure 3. SHG responses. Circles: YXX; triangles: YYY. The
intensities have been normalized so that the smallest response equals
1. The solid and dotted lines are visual guides.
Figure 4. Electric field Y-component distributions for incident X-
and Y-polarizations, as indicated by the column headings: (a,b) 1
nm nanogap; (c, d) 20 nm nanogap.
Nano Lett., Vol. 7, No. 5, 2007 1253
better for incident Y-polarization from simple electromag-
netic considerations. Surprisingly, we find that the strongest
fields in Figure 4b lie not in the nanogap itself but slightly
off to the sides. In fact, the highest field concentrations for
the smallest nanogaps (e15 nm) do not localize strictly
within the narrowest nanogap region for either polarization,
nor do they necessarily occur at the metal surface. Our T
design thus leads to an entirely different situation than the
case of self-similar spheres, where the field was predicted
to localize directly between the spheres in a nanogap of ∼5
In addition, the strong field is evenly distributed between
the horizontal bar bottom and the vertical bar endcap, leading
to suppression of the SHG response.30For our T’s, the
nanogap must be substantially larger (20 nm) before the
Y-component finally localizes within the nanogap, as seen
in Figure 4d. Close inspection reveals that the peak field is
now located closer to vertical bar endcap, yielding the growth
in the SHG response shown in Figure 3 for incident
Y-polarization over the 15-25 nm nanogap range. However,
despite high enhancement, the Y-component distribution for
incident Y-polarization is more symmetric in all cases,
resulting in the overall weaker YYY responses in Figure 3.
For both incident polarizations, the local fundamental field
is enhanced by roughly a factor of 8. Enhancement for
X-polarized input remains relatively constant for the smaller
and larger nanogap ranges, but drops in the middle range of
20-30 nm (Supporting Information, Figure S3). This slight
decrease alone is not enough to explain the very low YXX
response at 20 nm in Figure 3, though. The symmetrical field
distribution shown in Figure 4c must be considered as well.
Y-polarization, on the other hand, increases very slightly up
to 35 nm and decays beyond that point. Thus, strong
fundamental field enhancement persists even for relatively
large nanogaps. Also, the highest SHG response occurs for
YXX although both X- and Y-polarizations exhibit nearly
the same enhancement factor, countering the argument that
enhancement due to small nanogap size only is sufficient to
increase the SHG response.
In summary, we have observed that SHG from T-shaped
gold nanodimers with nanogaps depends more strongly on
polarization and the symmetry of the local fundamental field
distribution than on the nanogap size. It is evident that
asymmetrical nanoparticle designs and field polarization
considerations merit further study, as they may offer exciting
new opportunities for photonic applications of metal nano-
particles. The asymmetry argument runs counter to the
conventional findings of more symmetric systems that only
one fundamental polarization induces enhancement and that
a small nanogap alone is sufficient to enhance the SHG
response. These results also demonstrate that full understand-
ing of the nonlinear optical properties of nanostructures
requires consideration of not only the local field distribution
but also its polarization properties. Furthermore, potentially
large field enhancements may be obtained even for relatively
large nanogap sizes, which could relax the necessity of
exceptionally small features that are difficult to fabricate.
Finally, the results suggest that approaches seeking to
describe the nonlinear response in terms of average quantities
or effective media may be inappropriate.
Acknowledgment. This work was supported by grant
102018 from the Academy of Finland.
Supporting Information Available: Additional discus-
sion of experimental procedures (Figure S1), local field
distribution calculation details, a simple example of how
symmetric local field distribution couples to the local surface
susceptibility in the nanogap region and weakens the SHG
response (Figure S2), local field enhancement factors as a
function of nanogap size (Figure S3), and supporting
references. This material is available free of charge via the
Internet at http://pubs.acs.org.
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