Collective Effects in Second-Harmonic Generation from Split-Ring-Resonator Arrays
S. Linden,1,2F.B.P. Niesler,1J. Fo ¨rstner,3Y. Grynko,3T. Meier,3and M. Wegener1
1Institute of Nanotechnology, Institute of Applied Physics, and DFG-Center for Functional Nanostructures (CFN),
Karlsruhe Institute of Technology (KIT), 76128 Karlsruhe, Germany
2Physikalisches Institut, University of Bonn, 53115 Bonn, Germany
3Physics Department, University Paderborn, 33098 Paderborn, Germany
(Received 7 February 2012; published 6 July 2012)
Optical experiments on second-harmonic generation from split-ring-resonator square arrays show a
nonmonotonic dependence of the conversion efficiency on the lattice constant. This finding is interpreted
in terms of a competition between dilution effects and linewidth or near-field changes due to interactions
among the individual elements in the array.
DOI: 10.1103/PhysRevLett.109.015502 PACS numbers: 81.05.Xj
Tailored man-made effective materials called metamateri-
als are widely known for supporting unprecedented linear
optical properties, largely based on creating artificial magne-
tism via magnetic split-ring resonators and variations thereof
[1–6]. Nonlinear optical properties of metamaterials are at
ever, they have been studied experimentally to a much lesser
that are orders of magnitude above those of natural substan-
ces. So far, however, experiments lag far behind this ambi-
tious goal. This status is largely due to a lack offundamental
understanding of the underlying physics. For example, even
the microscopic source of the constituent metals’ (e.g., gold
or silver) optical nonlinearity is still the subject of scientific
controversy [17–19]. Nevertheless, the ‘‘hot spots’’ 
known from rough metal films raise hopes that huge en-
hancements might eventually be reproducibly achievable in
results have been obtained regarding high-harmonic genera-
tion from noble gases near bow-tie antennas .
One design strategy could be to create a tiny individual
building block (meta-atom), exhibiting large local-field
enhancement, add a nonlinear constituent to these hot
spots, and then pack these meta-atoms as densely as pos-
sible to form an effective nonlinear metamaterial. In this
Letter, we show experimentally and theoretically that col-
lective effects of the meta-atoms can substantially alter this
picture, leading to optimal behavior at some intermediate
packing density of the meta-atoms. Second-harmonic gen-
eration from split-ring resonators with a fundamental reso-
nance at around 1:4 ?m wavelength serves as an example.
The samples for our experiments have been fabricated
using standard electron-beam lithography and standard
high-vacuum evaporation of gold, followed by a lift-off
procedure. We use split-ring resonators (SRR) as the para-
digmatic building block of metamaterials. The SRR
are arranged on a square lattice with lattice constant a.
To study the effect of packing, or equivalently to study the
second-harmonic generation (SHG) efficiency versus a, a
large set of samples has been fabricated in which we vary
the electron-beam exposure dose for each of the different
lattice constants (a ¼ 280, 300, 325, 350, 400, 450,
500 nm). From this set we pick those arrays that exhibit
a nearly constant resonance wavelength (about 1:4 ?m)
of the fundamental SRR resonance. This choice aims at
easing the interpretation of the nonlinear-optical experi-
ments. It becomes apparent from Fig. 1 that the resulting
FIG. 1 (color online).
of the gold split-ring-resonator (SRR) square arrays used in our
experiments. The gold film thickness is 30 nm, the lattice
constant a is indicated in each case, and the footprint of each
array is 100 ? 100 ?m. The red arrow illustrates the incident
linear polarization of light used throughout this Letter. The
length of the white scale bars is 200 nm. The yellow SRR on
the top illustrates the geometrical parameters used for the
calculations in Fig. 3 for all a.
Selected top-view electron micrographs
PRL 109, 015502 (2012)
6 JULY 2012
? 2012 American Physical Society
size variations of the SRR for the different arrays are
extremely small. This aspect is important because resulting
changes in the individual SRR properties (e.g., damping)
would be an artifact (also see ). All SRR arrays have a
footprint of 100 ? 100 ?m and a gold film thickness of
Fig. 2(a) exhibits the measured linear-optical extinction
spectra (negative decadic logarithm of the intensity trans-
mittance) for normal incidence of light, depicted on a
false-color scale. As expected from the dilution and con-
sistent with previous results, we find a monotonic decay
of the peak extinction of the fundamental resonance
versus lattice constant. The small wiggles in the reso-
nance position are due to small SRR size variations
among the different arrays. The same aspects hold true
for a higher-order resonance centered around 750 nm
wavelength. Panel (c) of Fig. 2 reveals the measured
SHG signal versus center wavelength of the incident laser
pulses that are derived from a tunable optical parametric
amplifier (OPA) and versus lattice constant, again de-
picted on a false-color scale. To eliminate any parasitic
effects due to changes in the pulse duration, pulse energy,
pulse shape, beam divergence etc. when tuning the OPA,
the SHG signals from the SRR arrays are consistently
normalized to the off-resonant SHG from a quartz sur-
face. Details of this setup have been published previously
. In sharp contrast to the linear optical data, the SHG
signals show a nonmonotonic behavior versus lattice con-
stant. The SHG signal in Fig. 2(c) at 1395 nm excitation
wavelength first rises from normalized levels of 1.3 at
280 nm lattice constant to SHG levels of 3 at 400 nm
lattice constant. For yet larger lattice constants, the SHG
rolls off, reaching a level of 0.8 at 500 nm lattice constant.
To rule out any effects from the slightly varying SRR
resonance wavelength, we have taken complete SHG
spectra for each lattice constant. At each lattice constant,
we find the same general nonmonotonic behavior. The
decay of the SHG signal at very large lattice constants is
determined by the decreasing number of oscillators per
area or per volume. After all, zero SRR density will
surely lead to zero SHG from the SRR array. This dilution
corresponds to a scaling of the second-order nonlinear
polarization / 1=a2; hence, the SHG signal intensity
scales / 1=a4. In the opposite limit of very small a, the
SRR eventually touch (which happens at a ¼ 195 nm, see
Fig. 1), the SRR resonance disappears, and both the
extinction and the SHG signal are expected to decrease.
However, as becomes clear from the extinction spectra in
Fig. 2(a), a well-defined SRR resonance is observed for
all lattice constants investigated. Even at the smallest
lattice constant of a ¼ 280 nm, no drop of the extinction
with decreasing a is found. The initial rise of the SHG
signal versus lattice constant for small a must, thus, have
a different origin.
Intuitively, one might be tempted to suspect some sort
of diffractive effect, e.g., brought about by the Wood
(or Rayleigh) anomaly. Fortunately, closely similar
samples have recently been characterized in detail in
linear-optical experiments  (also see ). For normal
FIG. 2 (color online).
transmittance) versus wavelength and versus lattice constant a of the SRR square arrays, plotted on a false-color scale. Two selected
cuts through these data are shown by the white curves. The white dashed horizontal lines are the respective zero levels. (b) Damping
versus lattice constant as obtained from Lorentzian fits to the data in (a). (c) Second-harmonic generation (SHG) signal from the same
SRR arrays versus incident fundamental wavelength of the optical parametric amplifier (OPA) and versus a. The SHG signal is
normalized to a quartz reference and plotted on a false-color scale. A selected cut through these data versus lattice constant a is shown
by the black curve. The white dashed vertical line is the zero level. Note the nonmonotonic behavior of the SHG, whereas the linear
properties in (a) show a monotonic decay with increasing a.
(a) Normal-incidence, linear-optical extinction (negative decadic logarithm of the measured intensity
PRL 109, 015502 (2012)
6 JULY 2012
wavelength, the Wood anomaly occurs at lattice constants
larger than about 900 nm, i.e., diffraction of the incident
light into the SRR plane can be ruled out under the present
conditions. Diffraction of the SHG signal would lead to a
decrease rather than to the observed initial increase in the
(zeroth-order) forward direction but may well contribute to
the expected decay of the SHG signal at larger lattice
constants. Yet, these linear optical experiments  also
revealed a pronounced decrease of the linewidth of the
SRR resonance with increasing lattice constant. For the
present samples, the damping ? as obtained from
Lorentzian fits to the data shown in Fig. 2(a) decreases
from ? ¼ 16 THz at a ¼ 280 nm nearly linearly to ? ¼
10 THz at a ¼ 500 nm (see Fig. 2(b)). This dependence
can be interpreted as being due to a retarded long-range
interaction among the SRR in the array . The linewidth
of the resonance enters sensitively into the second-order
nonlinear-optical susceptibility ?ð2Þ—even in the simple
textbook nonlinear oscillator model . Smaller damping
leads tolarger resonant ?ð2Þ. In addition,theamplitude
of local-field-enhancement effects also increases with de-
creasing damping. The overall SHG intensity therefore
further increases with decreasing damping. Moreover, the
spatial distribution of the local SRR fields also enters
sensitively into the SHG conversion efficiency. Combined
with the trivial dilution effect discussed above, these three
aspects can qualitatively explain the measured nonmono-
tonic behavior of the SHG signal versus lattice constant.
To test this qualitative reasoning quantitatively and to
rule out any experimental artifacts, we have performed
numerical calculations using the discontinuous Galerkin
time-domain method [23,24] for the experimentally
investigated gold split-ring-resonator square arrays. We
describe the optical response of the metal by the
FIG. 3 (color online).
comparison. The geometrical parameters used for the split-ring resonators are shown at the top of Fig. 1. Panel (c) uses the
hydrodynamic Maxwell-Vlasov theory to describe the nonlinear response of the gold split-ring resonators.
Calculations corresponding to the experiment in Fig. 2. The representation is the same, allowing for direct
FIG. 4 (color online).
parameters as in Fig. 3. The square modulus of the electric-field vector, j~Ej2, at the fundamental SRR resonance frequency is shown on
a logarithmic false-color scale. The normalization is the same for all three panels. For clarity, half of the golden SRR in one unit cell is
made transparent. For the lattice constants shown, an increase of the SRR lattice constant a leads to an increase of the strength of the
near and internal fields and, hence, to a larger SHG far-field signal. At yet larger lattice constants, this trend is reversed by the trivial
Near-field distributions for three different lattice constants a as obtained from numerical calculations with
PRL 109, 015502 (2012)
6 JULY 2012
state-of-the-art hydrodynamic Maxwell-Vlasov theory
[25,26]. Its linear limit corresponds to the Drude free-
electron model, for which we have chosen a plasma
frequency !pl¼ 1:33 ? 1016rad=s, a collision frequency
!col¼ 8 ? 1013rad=s,
constant of ?1¼ 9:84. The refractive index of the glass
substrate is taken as n ¼ 1:46. The geometric SRR
parameters (which are the same for all lattice constants
a) are according to the yellow SRR in Fig. 1. Fig. 3(a)
shows calculated linear-optical extinction spectra. These
calculations reproduce the
monotonic decrease of the extinction peak and of the
damping [see Fig. 3(b)] with increasing lattice constant
a. The nonlinear SHG calculations are depicted in
Fig. 3(c). We find a pronounced maximum of the SHG
signal versus lattice constant at about a ¼ 400 nm
throughout the entire spectral resonance. This nonmono-
tonic behavior versus lattice constant nicely reproduces
the experimental findings shown in Fig. 2(c). Thus, the
numerical results strongly support the above qualitative
reasoning in that the SHG signal is strongly influenced
by collective effects via the SRR damping as well as via
the SRR near-field distributions.
Since the detailed microscopic mechanism of the
metal nonlinearity is still subject to debates (see above),
we have also performed calculations using other models
for the nonlinearity. In particular, this includes a simple
generic treatment with an effective instantaneous second-
order nonlinear susceptibility for the gold SRR. The
results (shown in the Supplemental Material  to-
gether with details on the model) confirm the fundamen-
tal nature of the reported nonmonotonic behavior.
However, none of these calculations is able to precisely
reproduce the asymmetry of the spectral SRR resonance
shown in Fig. 2(c).
Fig. 4 illustrates the origin of the collective effects as
already qualitatively discussed above. Indeed, the SRR
near fields within one unit cell at the fundamental reso-
nance frequency depend on the lattice constant. Stronger
near fields lead to stronger internal currents and hence to
larger SHG signals. Once the SRR are separated by more
than the extent of their near-fields, the near fields no
longer increase with increasing a and the SHG signal
eventually decreases due to the trivial dilution effect.
In conclusion, we have observed a nonmonotonic
behavior of the resonant second-order nonlinear conver-
sion efficiency in split-ring-resonator arrays versus pack-
ing density. The corresponding theoretical modeling
indicates that this finding is a rather general phenomenon
that is based on collective effects among the metamate-
rial building blocks and that should occur in many non-
linear metamaterials. Thus, future experiments aiming at
achieving large effective (high-order) optical nonlineari-
ties should keep these collective effects in mind. The
same very likely holds true for experiments aimed at
anda background dielectric
compensating metamaterial losses by parametric gain or
by stimulated emission.
We acknowledge support by the DFG-Center for
Functional Nanostructures (CFN) via subproject A1.5,
the Deutsche Forschungsgemeinschaft (DFG) through
the priority program SPP 1391, the research training group
GRK 1464, and the Emmy-Noether program. The project
METAMAT is supported by the Bundesministerium fu ¨r
Bildung und Forschung (BMBF). The Ph.D. education of
F.B.P. Niesler is embedded in the Karlsruhe School of
Optics & Photonics (KSOP). S. Linden and F.B.P. Niesler
contributed equally to this work.
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