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dx.doi.org/10.1021/nl201207n|Nano Lett. XXXX, XXX, 000–000

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pubs.acs.org/NanoLett

Fano Resonances in Nanoscale Plasmonic Systems: A Parameter-Free

Modeling Approach

Vincenzo Giannini,* Yan Francescato, Hemmel Amrania, Chris C. Phillips, and Stefan A. Maier

Department of Physics, Imperial College London, London SW7 2AZ, United Kingdom

b

S Supporting Information

T

interferencesisreceivinggrowingattention,duetothepossibility

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

inmanyother interactionsofplasmonicsystemswithsharpreso-

nances in atoms, molecules, and quantum dots. More generally,

indeedaconsiderablenumberofphenomenaoccurringonthenano-

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

mayarise.8?12Fanotheoryhenceexplainstheresultsobtainedin

a large number of areas of physics and supplies a simple and

powerfultoolthatallowsonetounderstanddeeplyseveralcommon

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

roleofselectionrulesforthenonradiativedecayoftwoelectronic

excitationsinatomicspectra;13,14inhispapers,heconsideredthe

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

Recently,ithasbeenshownthatinplasmonicnanostructuresitis

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

Woodin1902,belongsnowadaystothefieldofplasmonics.30He

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

resultsobtainedwhenplasmonicresonancesareinvolved.Generally,

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

Allthesemethodscangivegoodresults,but,theyneitheraddnor

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

Received:

Revised:

April 11, 2011

May 25, 2011

ABSTRACT: The interaction between plasmonic resonances,

sharpmodes,andlightinnanoscaleplasmonicsystemsoftenleads

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

spectrallysharpermodes.Byconsideringtheplasmonicresonance

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,weshowhowFanoresonancesmeasuredonanarrayofgoldnanoantennascoveredwithPMMA,aswellasthehybridizationof

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

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asymmetrydependontheenergyseparationbetweenthePRand

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

theoryandexperimentfortheextinctioncrosssectionofanarray

ofgoldnanoantennascoveredwithPMMA.Theabsorptionpeak

of the PMMA at 1730 cm?1interferes with the localized surface

plasmon resonance (LSPR), giving rise to a Fano line-shape. We

reproducetheseresultswithasimpleformulaandwithoutanyfitting

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

E2þ 1

ð1Þ

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

obtainedwhenqf¥,q=0,andwhenqisdifferentfrom0or¥.

Inthefirstcase,theprobabilityofdirectlyexcitingthecontinuum

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

asBreit?Wignerdip;10whenqisfinite,anasymmetricline-shape

(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

overcomethislimitationbyconsideringthatthecouplingtotheCSis

governedbytheexcitationofaPR.Wefirstconsidertheunperturbed

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

ð2aÞ

ð2dÞ

ÆcjVjdæ ¼ v

ffiffiffiffiffiffiffiffiffiffiffi

L ðEÞ

p

ð2cÞ

wherewechosethediscreteenergystateastheorigin,i.e.,Ed=0.

|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

1

1þ

E? EP

ΓP=2

??2

ð3Þ

Also,weassumethatalltheothermatrixelementsofVarezero,thatis

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

ð4Þ

ð5aÞ

ÆijWjcæ ¼ g

ffiffiffiffiffiffiffiffiffiffiffi

L ðEÞ

p

ð5bÞ

where w and g are the coupling factors.

TheFanoprofileisrecoveredbysolvingthepreviousproblem

(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

jÆijWjΨæj2

jÆijWjcæj2¼ðE þqÞ2

E2þ 1

ð6aÞ

q ¼

vw=g

ΓdðEÞ=2þE ? EP

E

ΓdðEÞ=2?E? EP

ΓP=2

ð6bÞ

E ¼

ΓP=2

ð6cÞ

Figure 1. (a) Different Fano line-shapes obtained varying the value of the

asymmetryparameterqwhenthecontinuumisflat.(b) Fanoprocesswitha

plasmonic continuum state; an incident state |iæ excites a quasi-continuum

state obtained from the interaction of a plasmonic resonance, |cæ, with a dis-

cretestate|dæ.Theinteractionisdescribedbythecouplingfactorsw,g,andv.

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whereΓd(E)=2πν2L (E)is related totheDS energy width. Itis

simple to see in fact, that Γd(E) coincides with the decay rate of

theDSwhenthecontinuumisflatandwhenitispossibletoapply

Fermi’sgoldenrule(i.e.,VissmallcomparedwithH0),ascanbe

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

physical situation.

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

discretelevel,thatis,EP<Ed(Figure2a,b),EP=Ed(Figure2c,d)

and EP> Ed(Figure 2e,f).

Whenwissmallweseealine-profilethatismainlydetermined

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

alwaysliesbetweenthetwomaximaarisingfromthediscreteand

plasmon states; in the situation EP= Ed(Figure 2c) a symmetric

dip is obtained.

These kind of resonances are particular to several physical

situations,forexample,Fanoresonancesduetothehybridization

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

(not shown).

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

Wenotethatthisformalismalsoallowsthevalueoftheenergy

position of the minimum in the Fano resonance, that occurs

whenq=?E (seeeqs6a?6c),tobeobtained.Fromeqs6a?6c,

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

Figure3b)isanarrayofgoldnanoantennasfabricatedbye-beam

lithography deposited on barium fluoride (BaF2) andspincoated

Figure 2. Fano resonances calculated from eqs 6a?6c for two different

couplingregimes,(a,c,e)w=0;(b,d,f)w.g;whenaDSwithenergyEd

interactswithacontinuumplasmonstatewithenergyEP(ΓP=10Γd)for

differentrelativepositions.(a,b)EP<Ed,(c,d)EP=Ed,and(e,f)EP>Ed.

Figure 3. (a) Scanning electron microscope top view image of an array

ofgoldnanoantennas.(b)Schematicrepresentationofthearraycovered

withPMMAonaBaF2substrate.(c)Measuredand(d)calculated(with

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.

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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,

[1?transmittance],ofthevarioussamples.Increasingthenano-

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

broaderthanthePMMAresonance,thecouplingfactorvisgiven

with a good approximation via Fermi’s golden rule, as v2∼ Γd/

2π,whilewcanbeobtainedbymeansoftheEinsteincoefficients,

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-

men-typestructure2,34consistingofashortandtwolongergold35

nanoantennas(seeFigure4a).Thelongnanoantennaalonehasa

second-orderresonancearoundλ=800nm(dashedgreenlinein

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

thedarkmode,w,iszeroandthecouplingtothedipolarplasmon

resonance is g2∼ ΓP/2π . The widths ΓPand Γdare obtained

from the plasmonic resonances (Figure 4b). The last parameter

requiredisthecouplingfactorvbetweenthedarkandthedipolar

Figure4. (a)Dolmenstructurecomposedoftwo200nmparallelgoldantennasclosetoa80nmperpendicularone.Theelectricfieldispolarizedalong

theshorterantenna.(b)ExtinctioncrosssectionscalculatedbyFDTDforsingleantennaswithdimensions80?40?40nmand200?40?40nm;the

shorterantenna(dottedredline)presentsadipolarmodeclosetothedarkresonance ofthelongerantenna(fullblackanddashedgreenlinewith0and

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).

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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

allowsustoquicklyandquiteaccuratelypredictthemainfeatures

of the observed spectral response in the coupled system.

A further improvement is possible by tuning the unknown

parameterv;thisisshowninFigure4c(dot-dashedblackline,v=

1.25(Γd/2π)1/2= 1.25v0). Now the resonances are more sepa-

rated and move toward the result obtained by FDTD. However,

therearestillsomediscrepanciesintheminimumpositionandin

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.

Itisimportanttoremarkthatthepreviousexampleshowsthat

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

providedbyourtheoryaswell.Lastly,notethatitisnowpossible

toobtainthewidthandposition ofthenoninteractingbrightand

dark resonances from the measured experimental Fano inter-

ference spectrum.

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

situationswhereaFanoresonancecanbeobservedinplasmonics

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

plasmoniccrystals.Thetheoryoutlinedinthisworknowallowsa

quick and parameter-free prediction of Fano interferences in

such systems.

’ASSOCIATED CONTENT

b

S

Supporting Information.

vided. This material is available free of charge via the Internet

at http://pubs.acs.org.

Additional information pro-

’AUTHOR INFORMATION

Corresponding Author

*E-mail: v.giannini@imperial.ac.uk.

’ACKNOWLEDGMENT

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

MarieCurieIEFprogram.Thanks toHongYoonforthePMMA

spincoating, Dr. Rob Airey for the sample fabrication, and Ana

O’Loghlen for the stimulating discussions.

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