Fano resonances in nanoscale plasmonic systems: a parameter-free modeling approach.
ABSTRACT The interaction between plasmonic resonances, sharp modes, and light in nanoscale plasmonic systems often leads to Fano interference effects. This occurs because the plasmonic excitations are usually spectrally broad and the characteristic narrow asymmetric Fano line-shape results upon interaction with spectrally sharper modes. By considering the plasmonic resonance in the Fano model, as opposed to previous flat continuum approaches, here we show that a simple and exact expression for the line-shape can be found. This allows the role of the width and energy of the plasmonic resonance to be properly understood. As examples, we show how Fano resonances measured on an array of gold nanoantennas covered with PMMA, as well as the hybridization of dark with bright plasmons in nanocavities, are well reproduced with a simple exact formula and without any fitting parameters.
- SourceAvailable from: Nicholas P. Hylton[show abstract] [hide abstract]
ABSTRACT: We illustrate the important trade-off between far-field scattering effects, which have the potential to provide increased optical path length over broad bands, and parasitic absorption due to the excitation of localized surface plasmon resonances in metal nanoparticle arrays. Via detailed comparison of photocurrent enhancements given by Au, Ag and Al nanostructures on thin-film GaAs devices we reveal that parasitic losses can be mitigated through a careful choice of scattering medium. Absorption at the plasmon resonance in Au and Ag structures occurs in the visible spectrum, impairing device performance. In contrast, exploiting Al nanoparticle arrays results in a blue shift of the resonance, enabling the first demonstration of truly broadband plasmon enhanced photocurrent and a 22% integrated efficiency enhancement.Scientific Reports 10/2013; 3:2874. · 2.93 Impact Factor
- [show abstract] [hide abstract]
ABSTRACT: We demonstrate that double-layered stacks of gold and insulator nanoparticles arranged on a flat gold surface dramatically enhance the sensitivity in absorption infrared microscopy. Through morphological variations of the nanoparticles, the frequency of the plasmon resonances can be tuned to match the frequency of the molecular vibration in the mid-infrared region. The results show that the nanostructures enhance the absorption signal of the molecules by a factor of up to ∼2.2 × 10(6), while preserving their characteristic line-shape remarkably well.Nanoscale 11/2013; · 6.23 Impact Factor
Article: Nonlinear Plasmonic Nanorulers.[show abstract] [hide abstract]
ABSTRACT: The evaluation of distances as small as few nanometers using optical waves is a very challenging task that can pave the way for the development of new applications in biotechnology and nanotechnology. In this article, we propose a new measurement method based on the control of the nonlinear optical response of plasmonic nanostructures by means of Fano resonances. It is shown that Fano resonances resulting from the coupling between a bright mode and a dark mode at the fundamental wavelength enable unprecedented and direct manipulation of the nonlinear electromagnetic sources at the nanoscale. In the case of second harmonic generation from gold nanodolmens, the different nonlinear sources distributions induced by the different coupling regimes are clearly revealed in the far-field distribution. Hence, the configuration of the nanostructure can be accurately determined in 3-dimensions by recording the wave scattered at the second harmonic wavelength. Indeed, the conformation of the different elements building the system is encoded in the nonlinear far-field distribution, making second harmonic generation a promising tool for reading 3-dimensions plasmonic nanorulers. Furthemore, it is shown that 3-dimensions plasmonic nanorulers can be implemented with simpler geometries than in the linear regime whilst providing complete information on the structure conformation, including the top nanobar position and orientation.ACS Nano 04/2014; · 12.06 Impact Factor
rXXXX American Chemical Society
dx.doi.org/10.1021/nl201207n|Nano Lett. XXXX, XXX, 000–000
Fano Resonances in Nanoscale Plasmonic Systems: A Parameter-Free
Vincenzo Giannini,* Yan Francescato, Hemmel Amrania, Chris C. Phillips, and Stefan A. Maier
Department of Physics, Imperial College London, London SW7 2AZ, United Kingdom
S Supporting Information
of designing new materials with a controlled response to light at
the nanoscale.1?6Fano interferences are also involved in the
unprecedented signal enhancement occurring when vibrational
excitations of a molecule are coupled with a plasmon resonance
in surface-enhanced IR absorption (SEIRA) spectroscopy7and
nances in atoms, molecules, and quantum dots. More generally,
scale are described by the interference of two different excitation
pathways. In particular, if in one path a discrete state (DS) is
excited and in the other a continuum state (CS), Fano resonances
a large number of areas of physics and supplies a simple and
physical situations. A defining characteristic of these resonances
is their asymmetric line profile due to the close coexistence of
destructive and constructive interferences.12Indeed, when de-
structive and constructive phenomena take place at close energy
positions, very sharp resonances are observed. Fano published
his first results in 19358concerning the autoionization of noble
gases, and a more general and incisive work appeared in 1961.9
Majorana arrived at similar conclusions in 1931, working on the
interaction between discrete and continuum channels.
Important examples of Fano interferences are found in
the ionization of atoms,8Raman scattering,15molecular
spectroscopy,16,17quantum transport,18optical absorption in
quantum wells19and light propagation in photonic devices;20,21
he spectral shaping of plasmon resonances in metal nano-
structures via controlled plasmon hybridization and Fano
there is a very rich literature on this topic, partially surveyed in
recent reviews by Miroshnichenko, Luk’yanchuk and Nie.10,12,22
also possible to observe this phenomenon.1?5,7,10,11,23,24In this
case, the excitations are achieved through the interference of a
plasmon resonance (PR), which acts as the CS, with a DS.
Depending on the situation, the DS can be the electronic
excitation of a molecule,4,7,24the excitation of a diffraction
channel (e.g., gratings, hole arrays, or plasmonic crystals),25?29
the excitation of a dark plasmon mode1,2,5or the excitation of a
guided mode.20Curiously, the first reported Fano resonance, by
measured asymmetric resonances studying thelight reflection by
metallic gratings; however, since at that time no satisfying
explanation was found, these were named Wood’s anomalies.
The first understanding of the profiles was given by Fano, who
considered the excitation of surface modes by such a grating.25
Although Fano resonances have been known for along time, still
there is a need to fill the gap between theory and experimental
the approach to study these asymmetric resonances is to fit them
with a Fano profile,23to apply a scattering matrix method con-
sidering all the channelsasdiscrete levels,20or bymeans of classical
phenomenological models consisting of coupled oscillators.31
predict any information about the real physical process nor do
they answer several questions; for example, (i) How does the
energy width of the PR affect the profile? (ii) How does the
April 11, 2011
May 25, 2011
ABSTRACT: The interaction between plasmonic resonances,
to Fano interference effects. This occurs because the plasmonic
excitations are usually spectrally broad and the characteristic
narrow asymmetric Fano line-shape results upon interaction with
in the Fano model, as opposed to previous flat continuum
approaches, here we show that a simple and exact expression for
the line-shape can be found. This allows the role of the width and
energy of the plasmonic resonance to be properly understood. As
dark with bright plasmons in nanocavities, are well reproduced with a simple exact formula and without any fitting parameters.
KEYWORDS: Fano resonances, plasmonics, localized surface plasmons, plasmon hybridization, nanoantennas, Fano theory
dx.doi.org/10.1021/nl201207n |Nano Lett. XXXX, XXX, 000–000
DS? (iii) Why does the dip in the Fano resonance have a lower
energy than the peak or vice versa? (iv) What is the effect of the
coupling strength between the DS and PR?
In this Letter, we consider the interaction of a plasmonic
resonance (acting as the CS) with a discrete resonance, in the
framework of the Fano model9,32(see Figure 1). The result is a
new mixed state that accounts for both excitation paths. We
perform exact calculations of the probability of exciting that
mixed state, and we obtain a simple analytic description of Fano
resonances mediated by PRs. Remarkably, this formulation
enables also the possibility of studying the interaction of DSs
with sharp plasmonic resonances. As an example, we compare
of the PMMA at 1730 cm?1interferes with the localized surface
plasmon resonance (LSPR), giving rise to a Fano line-shape. We
parameters. As a second example, we apply our theory to an all-
plasmonic system sustaining bright and dark plasmon modes.
The shape of the resonance caused by the coupling of a DS
with a flat CS has the following form,9
σðEÞ ¼ðE þ qÞ2
Here q is the shape parameter that determines the asymmetry of
the profile and is expressed as the excitation probability ratio
between the discrete and the continuum state. E is the reduced
energy, which depends on the energy of the incident photons E,
on the energy of the DS, Ed, and its width, Γd; E is defined as
2(E ? Ed)/Γd.23Three special cases of the Fano formula 1 are
is small and the profile is determined by the transition through
the DS; this results in a Lorentzian line-shape (see Figure 1a,
top) and describes, for example, the Breit?Wigner resonance
common in atomic and nuclear scattering.10When q = 0, a
symmetric antiresonance arises, (see Figure 1a, middle) known
(see Figure 1a, bottom) is obtained that is typical in many-body
systems and known as Feshbach resonance.10It has been shown
recently that all three Fano resonance cases can be achieved in
plasmonics, where the formula 1 is widely used in order to
describe these processes.12
Note that in eq 1 no feature of the PR is included. Here we
Hamiltonian H0, which has a DS |dæ, with eigenvalue Ed, and a CS
|cæ, with a continuum spectra of eigenvalues E. Following the
prediagonalized states method of Fano, we assume that the DS |dæ
is coupled to the continuum state |cæ by a coupling Hamiltonian
V (see Figure 1b), and the matrix elements of H0and V are
ÆdjH0jdæ ¼ Ed¼ 0
ÆcjH0jc0æ ¼ EδðE? E0Þ
ÆcjVjdæ ¼ v
|cæ, |c0æ and E, E0are different states and their respective energies
within the continuum. In eq 2c, the coupling of the discrete state
to the continuum is given by ν(L (E))1/2, i.e. it is determined by
theplasmonicline-shapeL (E)andbythecoupling factorv.The
plasmonic line-shape L (E) is a Lorentzian with energy position
EPand width ΓPgiven by the PR
L ðEÞ ¼
ÆdjVjdæ ¼ ÆcjVjc0æ ¼ 0
We first solve the eigenvalue problem H |Ψæ = E|Ψæ, where
H = H0þ V, and Ψ is the new mixed quasi-CS. Second, we
consider an incident photon in the DS |iæ that is coupled by the
Hamiltonian W to the states |dæ and |cæ (see Figure 1b)
ÆijWjdæ ¼ w
ÆijWjcæ ¼ g
where w and g are the coupling factors.
(see Supporting Information) and by calculating the probability
that a photon in state |iæ excites a quasi-CS |Ψæ, that is, |Æi|
W|Ψæ|2. If we normalize the latter result to the probability of
exciting the continuum in the absence of the DS, that is, the PR,
the same result of eq 1 is obtained, but now q and E are also
linked to the PR as
ΓdðEÞ=2þE ? EP
Figure 1. (a) Different Fano line-shapes obtained varying the value of the
plasmonic continuum state; an incident state |iæ excites a quasi-continuum
state obtained from the interaction of a plasmonic resonance, |cæ, with a dis-
dx.doi.org/10.1021/nl201207n |Nano Lett. XXXX, XXX, 000–000
whereΓd(E)=2πν2L (E)is related totheDS energy width. Itis
simple to see in fact, that Γd(E) coincides with the decay rate of
seen from eqs 2c and 3.
Equations 6a?6c describe the interaction of a PR with a
discrete state. In this case q and E are no longer constant; they
depend on the coupling factors, on the PR width ΓP, and on
the plasmon energy EP. This explains why the Fano resonance
exhibits different degrees of asymmetry depending on the
For example, in Figure 2 we plot the profiles obtained with
eqs 6a?6c in the most common situation, that is, when the
plasmon width ΓPis much bigger than Γd. Two main cases are
considered. In the first, the coupling factor to the DS, w, is very
small compared with the other coupling factors v and g (w = 0,
see Figure 2 panels a,c,e); in the second case, we consider w . g
(see Figure 2 panels b,d,f). Furthermore, we take in account the
three possible situations between the plasmon energy and the
and EP> Ed(Figure 2e,f).
by the plasmon resonance, which is modified by the coupling to
the DS. This happens because the DS is being mainly excited
indirectly through the plasmonic state (see Figure 1b). Also, it is
important to note that the minimum of the Fano resonance
plasmon states; in the situation EP= Ed(Figure 2c) a symmetric
dip is obtained.
These kind of resonances are particular to several physical
of dark with bright plasmons1,2,5or coupling of molecular
vibrational excitations with broadbandPRs.4,7,24Actually, in all
these cases a weak interaction of the incident photons with the
DS is present. A similar behavior is found when w is not zero but
still small, except for an asymmetry observed in the case EP= Ed
We have a different scenario when w . g, that is, when the
interaction with the quasi-continuum mainly goes through the
DS (see Figure 2b,d,f). In this case, the main resonance is due to
the DS, particularly in the cases EP< Ed(Figure 2b) and it
weakens when the plasmon resonance moves toward higher
energies (Figure 2f). Strikingly, now we can see in Figure 2f
that the minimum is at lower energy then two maxima. This
represents an important difference from Figure 2e that can be
helpful in order to understand in which coupling regime we are
in. Between the two extreme cases, w = 0 and w . g, fall the
phenomena of extraordinary transmission in metallic hole
arrays,26excitation of PRs in metal gratings,25or lattice reso-
nances in plasmonic crystals.27?29
position of the minimum in the Fano resonance, that occurs
itissimpletoseethat theminimumisobtained whenE=?vw/g
(note that w has dimensions of an energy, while, due to the
continuum normalization, it is v2and g2that have dimensions of
an energy). Interestingly, the energy of the minimum position
does not depend on the PR, but only on the coupling factors.
Next, we show that eqs 6a?6c are indeed valuable tools for
predicting Fano resonances in plasmonic systems without the
needfor brute-force numerical electromagnetic modeling. Let us
consider a PR formed in a Au nanoantenna coupled to a vibra-
tional resonance of an adjacent molecule. Figure 3 shows the
tuning of the Fano line-shape via shifting the PR position with
respect to this vibrational resonance. The studied system (see
lithography deposited on barium fluoride (BaF2) andspincoated
Figure 2. Fano resonances calculated from eqs 6a?6c for two different
Figure 3. (a) Scanning electron microscope top view image of an array
eqs 6a?6c) extinction of an array of nanoantennas as a function of the
frequency ω and the nanoantenna length L. The incident light is
polarized parallel to the long axis of the antennas.
dx.doi.org/10.1021/nl201207n |Nano Lett. XXXX, XXX, 000–000
with PMMA. The latter exhibits a strong and distinct resonance
at 1730 cm?1arising from the stretching of the CdO bond. The
nanoantennas are 20 nm thick, 100 nm wide, and their lengths
vary from 1 to 2 μm, spaced laterally by 1.2 μm and separated by
50 nm from each other along their main axis (see Figure 3a,b).
In Figure 3c, we show the measured extinction spectra, that is,
antennas length, the LSPR is moving from higher to lower
energies, crossing the DS due to PMMA absorption. For each
curve in Figure 3c, we can clearly see the main resonance
corresponding with the LSPR and the asymmetric feature due
to the Fano interference. Note that the incident electric field is
polarized along the main antennas axis.
These results can be theoretically reproduced by means of
eqs 6a?6c. From the experiments it is possible to obtain ΓPand
EP(extinction obtained without PMMA), as well as Γdand Ed
(from the PMMA absorption). The coupling factor g is given by
the probability of exciting the plasmon resonance with a plane
wave, in other words by the plasmon resonance width, that is,
g2∼ ΓP/2π. Considering that the plasmon resonance is much
with a good approximation via Fermi’s golden rule, as v2∼ Γd/
considering that in the experiment we are in the linear regime of
low-power excitation, implying w ∼ v.33Hence, we have all the
ingredients for eqs 6a?6c and the result is shown in Figure 3d;
we can appreciate a very good agreement without any fitting
parameters, while it is very common to use eq 6a and search a
constant q that gives a reasonable fit.
In Figure3c wecan seethat theminimumdoes notreachzero
as in Figure 2, this is because in the experiment not all the
molecules in the sample are interacting with the nanoantennas,
but actually only the molecules present in the gap.11Also, the
molecules can decay in different states that are not plasmons or
photons (i.e., heat). This results in the presence of a component
in the transmitted light that does not interfere.
We now apply our formalism to an all-plasmonic system with
interfering bright and dark plasmon modes. We choose a dol-
Figure 4b, extinction cross section), and for symmetry reasons11
it ispossible toexcite this resonance onlywhenthe incident light
is not perpendicular to the antenna, otherwise we have a
completely dark resonance (full black line in Figure 4b). The
short nanoantenna alone now is of a dimension so that a dipole
(first order, bright) resonance is excited almost at the same
wavelength as the aforementioned dark resonance (dotted red
line in Figure 4b). Now we consider a composite structure with
both kinds of nanoantennas arranged in a dolmen configuration
as in Figure 4a. The incident field is normal to the structure and
polarized along theshorternanoantenna,that is,onlythedipolar
resonance can be excited. In Figure 4b, we show the extinction
cross section (full blue curve) of that structure, calculated with a
finite-difference time-domain method (FDTD). We can see that
the interaction of the dark and bright mode results in a Fano
resonance.2,34Note that the incident field cannot excite the dark
resonance directly, but only through the dipolar resonance
which, via near-field coupling, excites the dark mode. The Fano
interference can now be studied with our model, where the dark
mode plays the role of the discrete state. In order to do so we
assume that the coupling factor between an incident photon and
resonance is g2∼ ΓP/2π . The widths ΓPand Γdare obtained
from the plasmonic resonances (Figure 4b). The last parameter
25? tilt, respectively). (c) Extinction cross sections of the dolmen structure as obtained from FDTD (full blue line) and from eqs 6a?6c with different
coupling factors (interrupted lines); completely dark resonance, i.e., w = 0, with coupling through near-field similar to bright resonance excitation
(i.e., v = (Γd/2π)1/2= v0, dotted green line) or tuned to show better agreement (v = 1.25v0, dot-dashed black line), small direct excitation of the dark
resonance (w = g2/35 and v = 1.72v0, dashed red line).
dx.doi.org/10.1021/nl201207n |Nano Lett. XXXX, XXX, 000–000
mode. Clearly this parameter depends on the details of the
geometry, particularly the distance between the antennas, and
therefore should be obtained by a detailed study varying these
parameters. In a first attempt, we will simply assume a behavior
similar to the bright resonance, that is, v2∼ Γd/2π. The result is
shown in Figure 4c (dotted green line); even though the
approximation is rough we can see an overall shape that is very
close to the FDTD simulation. Therefore our theoretical model
of the observed spectral response in the coupled system.
A further improvement is possible by tuning the unknown
1.25(Γd/2π)1/2= 1.25v0). Now the resonances are more sepa-
rated and move toward the result obtained by FDTD. However,
the strength of the second resonance.
The minima does not reach zero in the FDTD simulation due
to the photons absorbed in the structures during the Fano
interference. As in the previous case (Figure 3), the “lost”
photons can be considered as an additive background term in
the eq 6a. Also, notethatthe slightstrengthdifference of the two
resonances can be taken into account if w differs from zero, that
is, the dark resonance is not completely dark due to the
asymmetry of the dolmen nanostructure. Following these con-
siderations, we can see in Figure 4c (dashed red line) that a very
good agreement is reached when v = 1.72v0and w = g2/35. Note
also that w is very small compared with the other coupling
constants, as expected being the discrete state a dark resonance.
when the width of the two resonances are comparable the Fano
model describes plasmon hybridization theory.36This means
that an analytical description of plasmon hybridization model is
dark resonances from the measured experimental Fano inter-
In conclusion, we have shown that considering a plasmonic
resonance in the Fano model a simple and exact analytic relation
can be found. This allows the role of the plasmonic mode in the
Fano interference to be understood in a way that properly takes
into account the width and the position of the plasmonic
resonance and of the sharp mode. The description is general
and can also describe the case where the resonances have similar
widths, that is, not just when the continuum state is much
broader than the discrete one. This theory is particularly useful
for understanding experimental results in plasmonics.
We have demonstrated that through the coupling factors to
the discrete or to the continuum state a full description of many
is obtained, particularly for the interaction of bright modes with
dark modes orforthe interactionof emitters withnanoantennas.
Important additional examples include the extraordinary trans-
mission of light in hole arrays and the lattice resonances in
quick and parameter-free prediction of Fano interferences in
vided. This material is available free of charge via the Internet
Additional information pro-
This work was sponsored by the Engineering and Physical
Sciences Research Council (EPSRC) and from Leverhulme
Trust. V.G. acknowledges funding from the EU through the
spincoating, Dr. Rob Airey for the sample fabrication, and Ana
O’Loghlen for the stimulating discussions.
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