## Abstract

The relation between the near–field and far–field properties of plasmonic nanostructures that exhibit Fano resonances is investigated in detail. We show that specific features visible in the asymmetric lineshape far–field response of such structures originate from particular polarization distributions in their near–field. In particular we extract the central frequency and width of plasmonic Fano resonances and show that they cannot be directly found from far–field spectra. We also address the effect of the modes coupling onto the frequency, width, asymmetry and modulation depth of the Fano resonance. The methodology described in this article should be useful to analyze and design a broad variety of Fano plasmonic systems with tailored near–field and far–field spectral properties.

© 2011 OSA

## 1. Introduction

In 1977, Ugo Fano recounted the circumstances which prompted him to develop an original formalism to describe asymmetric spectral lineshapes: “Late in 1960, R.L. Platzman called to my attention a strikingly asymmetric line profile in an unpublished spectrum of energy transfers in electron collisions by Lassettre and coworkers. This spectrum appeared analogous to those I had interpreted in 1935. My reply to Platzman provided the opportunity for a modernized formulation of the analytical treatment” [1]. This seminal work, published in 1961 [2], has generated a very strong interest, first in the atomic and molecular spectroscopy scientific communities, where asymmetric spectral lines can be observed in experiments where the discrete excited state of an atom or molecule interacts with a continuum.

More recently, advances in nanofabrication techniques have given rise to a new family of structures where Fano resonances can be observed experimentally at optical frequencies. These structures rely on plasmonic materials and we refer the reader to Refs. [3, 4] which provide detailed reviews of the current state–of–the–art. This new family of Fano resonant structures can provide new insights into the mechanisms leading to asymmetric lineshapes. Indeed, the “atoms” constituting such structures have typical extensions in the tens of nanometers. Therefore, it is possible to associate with each Fano resonance a specific near–field distribution and a polarization distribution in these “atoms” [5–12].

In this paper, we specifically investigate the link between the near–field distribution and the spectral response of a plasmonic system exhibiting Fano resonances. In particular, we show that specific features visible in the far–field response of the structure originate from specific field and polarization distributions in the near–field.

## 2. Results and discussion

Fano resonances are built from the interference between a continuum of radiative
waves and a non–radiative (i.e. dark) mode that are spectrally and spatially
overlapping [3]. The radiative
waves can for example be constructed from a radiative (bright) mode of a first
plasmonic nanostructure, whose reflectance exhibits the following symmetric
lorentzian lineshape as a function of the frequency *ω*:

*a*is the maximum amplitude of the resonance,

*ω*the resonance frequency and

_{s}*W*its spectral width for

_{s}*W*≪

_{s}*ω*. Contrary to a quantum mechanical formulation, the electromagnetic modes’ eigenvalues are expressed in eV

_{s}^{2}and their resonance shape are expressed as function of

*ω*

^{2}(see for instance Eq. (1) in Ref. [13]). However, for convenience we discuss from now on the central frequency and width of the resonance in eV. The first nanostructure supporting the lorentzian resonance must then be placed in close interaction with a second one supporting the dark mode. Alternatively, a single nanosystem can also produce a Fano resonance, provided that it supports two modes that can hybridize into a bright and a dark mode [3, 6, 14, 15]. The interference of these bright and dark modes leads to a Fano–like asymmetric resonance whose lineshape is given by [13]:

*ω*is the central spectral position,

_{a}*W*the spectral width for

_{a}*W*≪

_{a}*ω*,

_{a}*q*the asymmetry parameter introduced by Fano in his quantum theory [2], and

*b*the modulation damping parameter appearing with intrinsic losses. Note that Eq. (2) is a generalization of the original Fano formula for systems with losses, such as plasmonic materials [13]. For the particular case

*q*= 0, the lineshape of Eq. (2) becomes the one of an antiresonance centered around

*ω*with minimum value

_{a}*b*and width

*W*. As the absolute value of the asymmetry parameter

_{a}*q*increases, the lineshape of Eq. (2) becomes comparable to a lorentzian profile (

*q*= ±∞). Equation (2) with constant parameters is valid in a small frequency interval around the frequency of a single resonant dark mode, and assumes that the dark mode’s spectral width is smaller than the bright mode’s spectral width [13]. The reflectance of the entire system is given by the product of the symmetric reflectance spectrum of the bright mode

*R*[Eq. (1)] modulated by

_{b}*σ*[Eq. (2)]: Let us now discuss the validity of Eq. (3). If two dark modes interact with the same continuum but their respective asymmetric lineshapes do not spectrally overlap, Eq. (3) becomes the product of the background resonance

*R*with two independant lineshapes

_{b}*σ*and

*σ*′. For the case of two bright modes, where one does not interact with the dark mode, the additional bright mode alters the total reflectance

*R*. The approach shown here is therefore valid only for a single bright and a single dark mode that are spectrally and spatially overlapping, and requires a preliminary insight into the eigenmode spectrum of the structure, as shown in Figure 2 of Ref. [13].

We now discuss how the near–field interactions within different plasmonic
nanostructures building the Fano system relate to the far–field response
given by Eq. (3). To this end, we
consider the reflectance of a dolmen–type plasmonic nanostructure consisting
of three metallic beams arranged as shown in Fig.
1(a) [16, 17]. The two parallel beams support a quadrupolar
dark mode, while the third perpendicular beam supports a dipolar bright mode. The
bright mode has a large spectral width due to radiative damping, and can be
therefore considered as a continuum of radiative waves. The interference between the
two modes leads to a Fano–type resonance. At frequency
*ω _{s}*, the bright mode of the single beam
is resonantly excited. The dark mode is perturbed by the coupling to the bright mode
and resonates at a frequency

*ω*slightly detuned from its original resonance frequency, and the width of this resonance is determined by both the modes coupling and intrinsic losses [13]. The movie in Fig. 1 shows the evolution of the system as a function of the illumination energy. The dolmens are arranged in vacuum in a two–dimensional array with period 500 nm. The material is chosen to satisfy a Drude model with plasma frequency

_{a}*ω*= 1.37 × 10

_{p}^{16}s

^{−1}and damping

*γ*= 1.23 × 10

^{14}s

^{−1}, which corresponds to the dielectric permittivity of gold. The black dashed line in Fig. 1(b) shows the reflectance spectrum of the array under normal illumination, calculated using the surface integral method [18, 19]. This spectrum is then fitted with Eq. (3), where the constrain

*a*< 1 is imposed in order to fulfill energy conservation. The fit parameters are

*a*= 1.00,

*ω*= 1.29 eV,

_{s}*W*= 0.12 eV,

_{s}*ω*= 1.08 eV,

_{a}*W*= 0.03 eV,

_{a}*q*= −0.96,

*b*= 0.34, and the corresponding curve is drawn as a solid red line in Fig. 1(b). In Fig. 2(a), the same geometry and illumination conditions are considered, but for a single dolmen instead of an array. Due to retardation effect, the quadrupolar mode can be excited from the far–field at grazing incidence [17], and by reciprocity cannot be considered as completely dark. In the geometry of Fig. 1 and Fig. 2(b), positioning the structures in a subwavelength array suppresses radiation at a non normal angle and ensures that the quadrupolar mode is a true dark mode. As a result, the fit of Fig. 2(a) (with parameters

*a*= 2.63,

*ω*= 1.33 eV,

_{s}*W*= 0.10 eV,

_{s}*ω*= 0.98 eV,

_{a}*W*= 0.05 eV,

_{a}*q*= −1.83,

*b*= 1.80) is not as satisfying as the fit of Fig. 2(b) (with parameters

*a*= 1.00,

*ω*= 1.29 eV,

_{s}*W*= 0.03 eV,

_{s}*ω*= 1.00 eV,

_{a}*W*= 0.03 eV,

_{a}*q*= −3.88,

*b*= 3.75) or the fit of Fig. 1(b). In a periodic array, near–field interactions between nearest neighbors have to be taken into account, as will be seen later.

For calculating the spectrum in Fig. 1(c), the
electric field is sampled at each frequency on a 5 nm homogeneous grid of points at
1 nm from the nanostructures surface. The surface integral method used in this work
calculates the field semi–analytically at positions arbitrarily close to
surfaces and therefore does not suffer from numerical inaccuracies [19–21]. The maximal value of the electric intensity for each
frequency is reported in Fig. 1(c). The
maximal near–field enhancement is found at 1.08 eV, which corresponds
exactly to the resonance frequency *ω _{a}* found from
the fit in Fig. 1(b). The points of maximal
near-field enhancement are located at (

*x,z*) = (±80,0)nm, i.e. at the two top interior corners of the beam pair [Fig. 1(d)]. Figures 1(d) to (f), which show the normalized intensity enhancement and the corresponding polarization charges through the structure, indicate that the quadrupolar mode of the double beam structure is resonantly excited at

*ω*. The second peak in Fig. 1(c) corresponds to the excitation of the bright mode. The maximum intensity enhancement is found at 1.29 eV, which also corresponds exactly to its resonance frequency

_{a}*ω*found from the fit in Fig. 1(b). The maximum intensity enhancement observed at

_{s}*ω*is about 2.3 times the maximum intensity enhancement observed for the excitation of the bright mode at

_{a}*ω*. This significant near–field enhancement originates from the fact that the dark mode does not suffer from radiative losses and is therefore able to store a larger amount of electromagnetic energy than the bright mode. Figure 1(e) also shows that the phase of the instantaneous electric field switches by

_{s}*π*around

*ω*. Two pathways have to be considered in the excitation from the far–field: the direct excitation of the bright mode and the excitation of the dark mode through its coupling to the bright mode. These two pathways interfere to produce the asymmetric lineshape observed in Fig. 1(b) around

_{a}*ω*(Eq. (22) of [13]). Its degree of asymmetry is described by the

_{a}*q*parameter, while the modulation damping by intrinsic losses is described by the

*b*parameter. The latter prevents the reflectance spectrum to reach zero values and can be understood as the ratio of the energy that is lost in heat to the metallic structure, to the energy that is transferred from the continuum to the dark mode (see Section IV of [13] for a closed form expression of these parameters for the classical model of two coupled oscillators). In the minimum of the resonance spectrum (around 1.1 eV), the Fano interference opens a narrow transparency window for the incoming light. At this frequency, the maximum intensity enhancement is only about 1.9 times the maximum intensity enhancement observed for the excitation of the bright mode at

*ω*. This shows that locating the central resonance frequency

_{s}*ω*is a critical point for applications such as refractive index sensing or lasing [3, 22, 23].

_{a}This interplay between near–field and far–field is also observed in
Fig. 3 for a metallic double grating
structure. In this nanosystem the hybridization of the plasmon modes of each
nanowire leads to the formation of a bonding antisymmetric mode and an antibonding
symmetric mode which are respectively non-radiative and radiative [14]. These two modes are supported by
the same nanostructure and therefore cannot be assigned to different parts of the
nanosystem, as was the case for the dolmen structure in Fig. 1. However, it is still possible to perform the fit
to Eq. (3) and extract the background
resonance from the bright mode [blue thin solid curve in Fig. 3(a)]. The fit parameters are
*a* = 0.84, *ω _{s}*
= 1.88 eV,

*W*= 0.37 eV,

_{s}*ω*= 1.38 eV,

_{a}*W*= 0.13 eV,

_{a}*q*= 0.66, and

*b*= 0.27. The electric field is sampled at each frequency on a 1 nm homogeneous grid of positions at 1 nm from the nanostructures surface. The maximal electric field intensity for each frequency is then reported in Fig. 3(b). The maximal intensity in the spectrum of Fig. 3(b) is fitted with a lorentzian profile similar to Eq. (1). Its central frequency is found at approximately 1.39 eV and has a width of 0.06 eV. These values agree well with the values of

*ω*and

_{a}*W*found from the fit of Fig. 3(a). At this frequency, the maximum intensity enhancement is about 16 times the one observed for the excitation of the bright mode at

_{a}*ω*. This lets us conclude that the dark mode is resonantly excited at this frequency. The near-field distribution in Fig. 3(c) also indicates a quadrupolar charges configuration in the structure at

_{s}*ω*. The points of maximal near-field enhancement are located at (

_{a}*x,z*) = (±50,14)nm, i.e. at the two bottom corners of the top nanoparticle. Let us emphasize that it is not possible to determine

*a priori*the resonance frequency

*ω*from a Fano resonance spectrum such as the black dashed curve in Fig. 1(b) or in Fig. 3(a), since its shape is determined by many parameters and

_{a}*ω*does not correspond to a specific point in the spectrum (for instance a local minimum or maximum). However, the fit with Eq. (3) is able to extract

_{a}*ω*easily, without detailed knowledge of the near–field interactions.

_{a}We now turn our attention to the influence of the respective spectral positions of
the bright and dark modes. In Fig. 4, the
length of the two parallel beams is shortened in order to blue shift the dark
mode’s resonance to frequencies above
*ω _{s}*. The same procedure as in Fig. 1 is repeated for this structure. The fit parameters
are now

*a*= 1.00,

*ω*= 1.25 eV,

_{s}*W*= 0.12 eV,

_{s}*ω*= 1.55 eV,

_{a}*W*= 0.02 eV,

_{a}*q*= 1.30, and

*b*= 0.15. The asymmetric resonance appears now on the other shoulder of the bright mode’s resonance, compare Fig. 1(b) and Fig. 4(b). The shift in the modes frequency is a function of their spectral detuning. As a result, the resonance frequency of the bright mode red shifts from 1.38 eV to 1.25 eV. Since the asymmetry parameter depends on the frequency difference ${\omega}_{a}^{2}-{\omega}_{s}^{2}$ [13], it reverses its sign from Fig. 1 to Fig. 4, which now makes the resonance appearing as a dip followed by a peak for higher frequencies. The fit with Eq. (3) also enables extracting the resonance frequency

*ω*in this configuration. The resonance is now at a higher frequency than the dip’s frequency, because the asymmetry parameter is positive. The absolute value of

_{a}*q*is larger than in the case of Fig. 1, which pushes the resonance frequency towards the second reflectance peak (Fig. 1 in Ref. [2]). The electric field is sampled at each frequency on a 5 nm homogeneous grid at 1 nm from the nanostructures surface. The maximal electric field intensity for each frequency is then reported in Fig. 4(c). The largest field enhancement, corresponding to the resonant excitation of the dark mode, appears at approximately 1.57 eV. The points of maximal near-field enhancement are located at (

*x,z*) = (±161,100)nm, i.e. at the two exterior bottom corners of the beam pair. A phase shift of the dark mode similar to the case of Fig. 1 occurs in a frequency region around

*ω*[Fig. 4(e)]. The first peak in Fig. 4(c) is attributed to the bright mode’s excitation and is centered around 1.25 eV, in agreement with the value

_{a}*ω*= 1.25 eV found from the fit.

_{s}It has been observed [16, 17] and later theoretically
demonstrated [13] that the
coupling between the bright and dark modes plays an important role for controlling
the line shape of Fano–like resonances. This dominant effect of coupling is
described by the modulation damping parameter *b* giving the ratio
between the power lost in the metallic structure and the power transferred from the
bright mode to the dark mode. The near–field intensity enhancement maps in
Figs. 1 and 4 indicate that the most efficient way to change the
coupling is by tuning the gap size *g* between the single beam and
the pair of parallel beams, Fig. 5. For high
coupling, in the spectral minimum in the reflectance response, the Fano interference
opens a narrow transparency window for the incoming light, whose magnitude depends
on the *b* parameter. As *g* increases, the coupling
between the two modes decreases, the *b* parameter drastically
increases and the resonance modulation depth is reduced. As a result, the
reflectance of the system approaches that of the bright mode alone [Fig. 5(a)]. This effect appears only in
the presence of intrinsic losses and goes along with a reduction of the absolute
value of the asymmetry parameter (from −1.68 for *g*
= 15 nm to −0.21 for *g* = 50 nm). As the
coupling decreases, the resonance becomes more symmetric and its frequency shifts
towards the reflectance dip. In addition, the resonance frequency blue shifts, and
converges to the non–perturbed resonance frequency of the dark mode
[13]. The coupling of the
dark mode to the bright mode opens a radiative decay channel for the dark mode,
which in turns widens the Fano resonance. Figure
5(b) shows the intensity enhancement in the structure at the resonance
frequency *ω _{a}* extracted from a fit with Eq. (3); this frequency is indicated by
a red dot in panel (a). The correlation between near–field and
far–field is also verified in this case: as the two modes decouple, the
power transfer between the two parallel beams and the third becomes less effective.
This results in a decrease of the field intensity in the two parallel beams, leading
finally to field enhancement solely at the extremities of the single beam, when both
structures are decoupled and the Fano resonance has disappeared,

*g*≃ 60 nm (Fig. 5). As mentioned previously, the structures are placed in an array to ensure that the quadrupolar mode is a true dark mode. Although its subwavelength character does not produce grating effects, this configuration is accompanied by strong near–field interactions between nearest neighbors. The gap size of

*g*=60 nm in Fig. 5 leading to a complete vanishing of the Fano resonance corresponds to a configuration for which the dipolar beam is placed at the same distance from its two adjacent beam pairs, i.e. to a configuration for which the array has a

*y*= 0 symmetry plane. Moving the dipolar bar from this position induces the symmetry breaking required for the excitation of the dark mode, which explains the high sensitivity of the resonance lineshape to the gap size

*g*.

## 3. Conclusion

We have investigated the relation between the near–field and far–field properties of plasmonic nanostructures that exhibit Fano resonances. By fitting reflectance spectra for different geometries with a general equation derived in Ref. [13] we have been able to determine the central frequency, width and modulation of the Fano resonance, and to further retrieve the shape of the bright mode in these systems. We have shown that the central frequency of the asymmetric modulation corresponds to the maximum near–field intensity enhancement in the structure, but does not correspond to a specific point of the reflectance spectrum (for instance a local minimum or maximum). We have also related the far–field spectrum to the near–field configuration of the bright and dark modes, and addressed the effect of their coupling to the frequency, width, asymmetry and modulation depth of the Fano resonance. The methodology described in this article should by useful to analyze and design a broad variety of Fano plasmonic systems with tailored near–field and far–field spectral properties.

## Acknowledgments

Funding from the CSEM and the CCMX–Fanosense project as well as stimulating discussions with M. Schnieper and A. Stuck are gratefully acknowledged.

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