Optical Monitoring of the Magnetoelectric Coupling in Individual

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Optical monitoring of the magneto-electric coupling in individual plasmonic scatterers Julien Proust, Nicolas Bonod, Johan Grand, and Bruno Gallas ACS Photonics, Just Accepted Manuscript • DOI: 10.1021/acsphotonics.6b00041 • Publication Date (Web): 14 Jul 2016 Downloaded from http://pubs.acs.org on July 16, 2016

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Optical monitoring of the magneto-electric coupling in individual plasmonic scatterers Julien Proust,

†Aix-Marseille

†

Nicolas Bonod,

†

Johan Grand,

‡

and Bruno Gallas

∗, ¶

Université, Centrale Marseille, CNRS, Institut Fresnel, UMR 7249, Campus de St. Jérôme, 13397 Marseille, France

‡Univ.

Paris Diderot, Sorbonne Paris Cité, ITODYS, UMR CNRS 7086, 75013, Paris, France

¶Sorbonne

Universités, UPMC Univ Paris 06, CNRS-UMR 7588, Institut des NanoSciences de Paris, F-75005, Paris, France

E-mail: [email protected]

Abstract Engineering the shape of metallic nanostructures is of crucial interest to tailor the electric and magnetic elds at a deep subwavelength scale. Individual nanoscatterers behave as optical atoms with strong electric and magnetic responses that can be tuned via the composition and the shape of the nano-object. U -shaped scatterers are assumed

to exhibit strong magneto-electric coupling yielding local chiral elds. However, these magneto-optical couplings have never been measured at the scale of a single meta-atom. In this paper, we report transmission measurements performed on individual U -shaped metallic scatterers using dierent linear polarization conditions. A point multipole model, with modes up to the electric quadrupole, including magneto-electric coupling, provides an excellent description of the polarization dependent optical response of the single scatterer for dierent directions of propagation. The comparison of this model 1

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with the optical measurements allows to quantify the magnitude of the dierent multipoles and more importantly of the magneto-electric coupling over the visible and near infrared spectrum. Keywords: Magneto-electric coupling , meta-atoms, polarizability, plasmonics, multipole theory

2


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Metallic nanoscatterers can resonantly interact with the incoming light thanks to the excitation of localized surface plasmon resonances.

1

The electric and magnetic near eld

intensities can be drastically enhanced with important applications to improve the sensitivity of single molecule detectors using Raman spectroscopy or photoluminescence.

2,3

Many

organic molecules are chiral, i.e. they do not coincide with their mirror image. As a consequence, they interact preferentially with one particular circularly polarized light, left or right. To describe this interaction, a chirality density waves characterized by an electric eld in a medium of permittivity

o ,

~ E

C

has been dened for electromagnetic

and a magnetic induction

the chirality density is given by

~. B

At a frequency

~ ∗ .B) ~ .4 C = − 2o ωIm(E

interaction of electromagnetic waves with chiral molecules scales with

C

ω

The

and it would be

benecial to detection to be able to create local optical elds with large values of chirality density.

3

For that purpose, a lot of plasmonic nanostructures have been designed to enhance

the chirality density of the electromagnetic elds. structures,

59

Among the wide variety of plasmonic

U -shaped scatterers have attracted a large attention because they exhibit surface

plasmon modes referred to the literature as electric and magnetic depending on the symmetries of the local currents in the scatterers.

10

The optical response of a single

U -shaped

scatterer can be given by a polarizability tensor which quanties the coupling of the electromagnetic eld with the polarizability moments of the scatterer. The polarizability tensor contains o-diagonal elements quantifying the interaction between the electric (respectively magnetic) eld and magnetic (respectively electric) moment. This latter interaction is referred as magneto-electric coupling and is generally associated with chirality.

U -shaped

resonators have proven to present a strong magneto-electric coupling.

Arrays of

1116

This

should yield local elds with large chirality density at the magnetic resonance in the vicinity of the resonators. ers

18,19

17

However, in arrays of scatterers, the near eld coupling among scatter-

and the presence of the lattice modies the response of the assembly of resonators.The

inuence of the lattice may be observed by the presence of Rayleigh lines, of the resonances and on their positions

12

20

on the strength

or may even induce transfers of polarizations even

3


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beyond the Rayleigh lines

20

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which may be mistaken for chirality.

21,22

Determining the macroscopic response of metamaterials is still mostly empirical and there is a strong need for establishing experimentally the link between the optical response of individual scatterers and that of arrays of scatterers. eect of lattice on the response of arrays of

Previous works have investigated the

U -shaped scatterers assuming that diluted arrays

may provide the response of isolated scatterers.

23

However, high order modes raise at short

wavelengths and their radiations are likely to be superposed to Rayleigh anomalies.

24

If such

scatterers were to be used as antennae to create large local chirality densities for detecting chiral molecules, it would then be dicult to disentangle the chiral responses originating in the individual scatterers from the coupling among resonators and the chiral molecules themselves. To address these challenges, the scattering properties of individual meta-atoms must be measured with a tight control over the polarization of the incident and scattered elds. The optical response of single scatterers can be described by a polarizability tensor relating the electric and magnetic moments to the exciting electric and magnetic elds. versatile control of the incident polarization used to excite proposed through the use of Laguerre-Gauss beams.

27

U -shaped

9,13,25,26

A

scatterers has been

However, this technique requires a

precise control of the scatterer position inside the highly focused beam, it yields strong eld gradients and a modeling step remains necessary to determine the polarizability tensor of the scatterer. In this work, we combine experimental transmission measurements and numerical modeling to evidence the presence of a magneto-electric coupling in single

U -shaped

scatterers. This allows us to present the spectral feature of the magneto-electric coupling. The plasmonic meta-atoms investigated here and sketched in Figure 1(a) were fabricated with electron beam lithography following a classical lift-o method. A 130 nm thick layer of PMMA was spin coated on a clean glass substrate (BK7) and a thin layer of aluminum was thermally evaporated on the PMMA to avoid any charging eect during the exposure with electrons.

After exposure and before the development of the resist, this conductive layer

was removed in a KOH bath. After development, a 4 nm chromium layer was evaporated to

4


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ensure the adhesion of the 42 nm gold layer on the glass substrate. The resist was then liftedo with acetone, and the substrate subsequently rinsed in ethanol to reveal the scatterers. Figure 1(b) displays a typical

U -shaped

scatterer imaged by Scanning Electron

Microscopy (SEM). The resonator was placed in the center of a reticule. 50

µm

U -shaped

The distance of

between the tips of the reticle ensured that each resonator was isolated from any

interaction with the reticule. Optical spectroscopy on individual scatterers was performed with a home-made confocal microscope (Figure 1(c)).

Incident light illumination was obtained by 250 W Quartz

Tungsten Halogen (Oriel QTH) injected in a 105

µm

core ber through a silica lens. The

light coming from the ber was then polarized using a Glan-Thompson linear polarizer and focused from the back of the sample with a

×10 Cassegrain lens (Thorlabs) with a numerical

aperture of 0.15 to obtain a uniform illumination. The light scattered by the resonator was collected by a

×100

microscope objective, numerical aperture of 0.7, with a long working

distance (20mm) (Mitutoyo innity corrected, NIR) to allow for measurements with a sample tilted by

30o

with respect to the optic axis of the set-up. The collected light was then ana-

lyzed using a second Glan-Thompson linear polarizer and focused on the spectrometer with an optical ber with a core diameter of 100

µm.

The light collected by the ISOPLANE spec-

trometer (Princeton Instruments) was dispersed and projected on a liquid nitrogen cooled InGaAs linear IR-CCD array (Pylon) for measurements from 600 nm to 1600 nm. A separator was used to image the surface of the sample surface on a CCD camera and to check that the ber's core and the surface were conjugated through the collection / detection part of the set-up (Figure 1(d)). The intensity spectra recorded on the

U -shaped

scatterers were

corrected by the background noise of the spectrometer. They were normalized by the transmission of the substrate alone taken far enough from the

U -shaped

scatterer to prevent any

inuence from the resonator but to keep the same illumination and collection conditions. The measurements were then presented as absolute extinction coecient by correcting the measurements by :

(i) the dimension of the collection spot, assuming its size limited by

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diraction for wavelengths above 1 below 1

µm,

µm

and (ii) the surface of the

Page 6 of 27

and by the bers core dimension for wavelengths

U -shaped

scatterer as measured on the SEM images.

The measurements recorded between crossed-polarizers were normalized to the transmission spectra of the lamp through the substrate with parallel polarizers with the same acquisition conditions.

We always checked that the transmission through the substrate between

crossed-polarizers was below the background noise of the detector in the same acquisition conditions.

Figure 1: (a) Sketch of the single scatterer. The

x and y

U -shaped

scatterer. (b) SEM images of the area containing a

axes of the Cartesian referential are presented in the image. (c)

Scheme of the polarization resolved confocal microscope. (d) Optical image of the surface where the measurement area is visualized with the red spot of an alignment laser at 650 nm and the diraction limited image of the scatterer can be seen as a dark spot. The scale bar as been estimated from the dimension of the scatterer.

MODEL The independent determination of the polarizability tensor elements requires many measurement congurations, some of them being not experimentally realistic.

12,28

Moreover, the

spectral overlap between the electric and the magnetic modes, the possible presence of multipolar contributions and the use of a large numerical aperture objective also prevent from extracting directly the values of the magneto-electric coupling tensor from angle dependent

6


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transmission measurements with circularly polarized light.

12,28

These experimental issues can

be avoided by combining experimental and numerical results in the analysis of the light transmission. In the following, we describe the model of radiation of point-like multipoles used to analyze the measurements and the particular properties expected for

U -shaped

scatterers.

In non-centrosymmetric resonators, the electric and magnetic dipole moments,

p~

and

m ~,

are likely to depend on the incident elds as well as on their gradients through :

25,26

pi

em inc ee inc Hj + aijk ∇k Ejinc + · · · Ej + αij = αij (1)

mm inc me inc Hj + bijk ∇k Ejinc + · · · Ej + αij mi = αij where

αee , αmm

and

αem

are the electric, magnetic and magneto-electric polarizability ten-

sors, respectively and (i,j,k)=(x,y,z). The terms

aijk

and

bijk

quantify the dependence on the

gradients of the elds. However, some simplications are possible. The scatterer complies with the Onsager principle of reciprocity. sors and

me em . = −αji αij

28

As a consequence

αee , αmm

are symmetric ten-

The dependence of the electric and magnetic dipole moments on the

gradients of the elds in Equations 1 can be taken into account by correcting the electric, magnetic and magneto-electric cross-coupling polarizability tensor's elements. tion does not add new elements to

αee , αmm

angle of incidence in the existing elements.

and

αem

16

This correc-

but introduces a dependence on the

16

In addition, an electric quadrupolar contribution to the polarizability tensor has also been proposed.

16,2830

The quadrupolar moment

Q

is dened as a traceless symmetric tensor.

31

It is excited by the electric eld and its gradients. Using the symmetries of the derivative of the electric eld, it is convenient to dene a vector containing the derivatives of the electric eld as :

~ = [∂x Ex , ∂y Ey , ∂z Ez , (∂y Ex + ∂x Ey )/2, (∂y Ez + ∂z Ey )/2, (∂x Ez + ∂z Ex )/2] E and the quadrupolar moment

Q as : 28

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Q = [Qxx , Qyy , Qzz , Qxy , Qyz , Qxz ]

(2)

In this case, the quadrupolar polarizability of the scatterer is described by a 6 ×12 tensor that relates the induced moments to the incident elds and their derivatives as follows :



Q = αQe αQm

where

αQe

and

αQm

~ inc

28



 E    inc  ~ αQ  H     ~ inc E

(3)

are 6×3 tensors that relate the quadrupolar moment to the incident

electric and magnetic elds, respectively. The 6 ×6 tensor

αQ

describes the dependence of

the quadrupolar moment on the gradients of the electric eld. Expression 3 implies the determination of many elements that can not be easily accessed experimentally.

Again, some simplications can be performed.

We rst address the de-

pendence on the electric eld and its gradients, described in Equation 3 by

αQe

and

αQ ,

respectively. It can be assumed that the part of the quadrupolar moment driven directly by incident electric eld, described by electric eld along the y -axis, described by

αQ ,

αQe ,

Eyinc .

depends mainly on the projection of the incident

The dependence on the gradient of the electric eld,

can be addressed with a calculation similar to that developed in Ref. 16

provided that we use only the elements

Q αijkl

detailed in Ref. 28. By considering the prop-

agation in dierent propagation planes, additional relations can be obtained from Equation 3 for angle

Qxx . θ

If we consider a TE-polarized plane wave with amplitude

from the z -axis and an angle

φ

E inc

propagating at an

from the x -axis in the (x,y ) plane, one obtains :

Qe Q Q − αxxyy )sin(θ)sin(φ) Eyinc . Qxx = αxxy + ik(αxxxx

8


(4)

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with

k the wavenumber.

In the case of TM-polarized plane wave one obtains :

Qe Q Q Q Q Qxx = αxxy + ik(αxxxx − αxxyy )sin(θ)sin(φ) Eyinc + ik(αxxxx − αxxzz )sin(θ)Ezinc .

Similar relations are obtained for

Qe αyyy

and

Qe αzzy .

(5)

We can assume that the component of

the electric eld along the z -axis can be neglected because of the low angle of incidence used, then Equations 4 and 5 show that corrected expression

α eQe

Qxx can be cast with respect to Eyinc only by introducing the

dened by

Qe Qe exxy Eyinc , with α Qxx = α exxy

being equal to the expression

of the right hand side of Equation 4. Following previously published results, we also expect the presence of the element on the y -axis as well.

28,32

Qe αxyy

in

αQe .

It is excited by the projection of the electric eld

This element is also modied by correcting

the elds. However, there is no particular constraint on the value of value will be noted

Qe α exyy

αQe

for the gradient of

Qe αxyy

and its corrected

without writing explicitly its dependence on the gradients. We have

used the numerical results of Ref. 17 to estimate the relative orders of magnitude of the elements. The element

Q αxxxx

should have the largest value and, as a result, the corrected

Q α eiijj

α eQe

remains mostly a traceless tensor. Finally, the dependence of the quadrupolar moment on the gradient of the electric eld does no longer need to be described explicitly in Equation 3. Concerning the dependence on the incident magnetic eld, described by to be very weak and will be neglected in the following.

αQm ,

it is likely

28

Finally, by using the corrected polarizabilities, the electric and magnetic moments and the electric quadrupolar moment can be cast with respect to the incident elds only and the problem reduces to :

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     inc ee em ~ e α e  E   p~   α    = ~ inc −(e αem )T α emm H m ~     Qe Q α e xxy   xx       Qe Qyy  −e     αxxy /2 inc   Ey =  Q  −e  Qe α /2  zz   xxy      Qe α exyy Qxy where

Qe Q Q Qe Qe )sin(θ)sin(φ) and α exyy −αxxyy +ik(αxxxx = αxxy α exxy

(6a)

(6b)

are the amplitudes of the electric

quadrupolar moment corrected for the gradient of the electric eld. From Equations 6, it is possible to calculate the dipolar and quadrupolar moment strengths for any known x - or y -polarized incident electromagnetic elds

~ inc E

and

~ inc . H

The elds scattered in the radiation domain in the direction

~n

were calculated using

a point-like multipole model to predict the results expected for some useful measurement conditions.

The magnetic eld radiated by an electric dipole moment

dipole moment

~ rad H m

and an electric quadrupolar moment

~ rad H q

are :

~ rad , H e

a magnetic

31

ck 2 eikr rad ~ He = (~n × p~), 4π r 2 ikr ~ rad = k e (~n × m) H ~ × ~n, m 4π r 3 ikr ~ rad = −ick e (~n × (Q.~n)), H q 24π r r the distance from p ~ = µo /o (H ~ × ~n). For E with

the center of the scatterer.

(7a)

(7b)

(7c)

The electric eld is then given by

a given incident polarization, the resulting moments

p~, m ~

and

Q

were obtained using Equations 6 and they were inserted in Equations 7 to yield the radiated elds. The elds were summed coherently to yield the total elds

~ tot H

and

~ tot E

which

were subsequently projected on the x - and y -directions to yield the polarization dependent intensities measured. The time-averaged radiated intensity in the direction of the detector

10


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is

~ tot ×H ~ ∗ ]. 1/2.Re[~n.E tot

The intensity and polarization of the elds scattered can thereby be

calculated in any direction for any illumination. To compare the eld calculations to transmission measurements, the substrate was taken into account by modifying the illumination conditions owing to the dierent coecients of transmission between x - and y - polarized light at oblique incidence and owing to the internal angle in the substrate being dierent from the observation direction. The redirection of the radiation pattern above the substrate was taken into account in our calculations.

33

Owing to the low index of the substrate, the modications

of the strength and position of the resonances due to non-local eects associated with the substrate were not taken into account.

34

Figure 2: Normalized intensity scattered in the forward direction between crossed polarizers as a function of the angle of incidence and wavelength for an incident light TE polarized (top Figures) and TM polarized (bottom Figures). The insets depict the illumination conditions. Each Figure is separated in two panels: (left) light propagating in the ( x,z ) plane at an angle

α

from z, (right) light propagating in the ( y,z ) plane at an angle

β

from z. Dierent

contributions are investigated : (a) magneto-electric coupling only without substrate, (b) magneto-electric coupling only with substrate and (c) magneto-electric coupling and electric quadrupolar with substrate.

The positions and amplitudes of the moments were chosen

arbitrarily for illustration purpose. The white dotted lines in (c) indicate the measurement conditions.

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Figures 2(a-c) present the intensity scattered in the direction of the beam propagation (forward direction) between crossed polarizers obtained numerically. The eects of the substrate, of the magneto-electric coupling and of the electric quadrupolar moment are presented as a function of the angle of incidence, wavelength and direction of propagation of the incident light. The substrate was taken into account in two ways in the calculations : rst, the intensity exciting the scatterer through the substrate depends on the polarization and second, the radiation pattern is modied by the coherent superposition of the elds directly scattererd and the elds scattered after undergoing reection at the surface. eect is also polarization dependent.

33

This latter

The spectral dependence of the dierent tensor ele-

ments was supposed to have a Lorentzian lineshape. In presence of magneto-electric coupling only (Figures 2(a) and (b)), light intensity is observed between crossed polarizers for light propagating in the ( y,z ) plane (right panels). Without substrate (Figure 2(a)), the intensity is the same from x - to y -polarization and from y - to x -polarization. Indeed, the polarization eigenmodes are no longer linearly polarized states for propagation at oblique incidence.

1214

A consequence of the electromagnetic reciprocity theorem, in the absence of quadrupolar moment, is that we should expect the transfers of polarizations from x - to y -polarization and from y - to x -polarization to be equal in amplitude.

15,35

However, the presence of the

quadrupolar moment may modify the symmetry between the transfers of polarization. We calculated the electric eld radiated in the forward direction by a single scatterer in vacuum using the assumption that the polarizability tensor of Equation 6(a) had the same elements as proposed in previous works. the components of

~n

12,13,15,16,28

In that case of light propagating in the ( y,z ) plane,

are (0, cos(θ ), sin(θ )).

The electric eld scattered in a polarization

direction orthogonal to the direction of the incident polarization is given by :

12


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rad Etot,xy

=

ik2 em sin(2θ)(−e αyz 8πco

+

kc Qe eikr α e ) and 6 xyy r

rad Etot,yx =

ik2 em eikr sin(2θ)e αyz (8) 8πco r

It should be noted that in the case of an illumination from the substrate (Figure 2(b)), the intensity scattered depends on the polarization case even without the contribution of the electric quadrupolar moment. introduces an asymmetry between

The element

rad Etot,xy

and

Qe α exyy

rad Etot,yx

of the electric quadrupolar moment

in addition to that introduced by the

substrate, as evidenced by Equation 8. In the presence of an electric quadrupole, a new band is observed in Figure 2(c) for an incident light propagating in the ( x,z ) plane and y -polarized. This is supported by the total electric eld scattered in the forward direction by a single scatterer in vacuum calculated as for Equation 8 and leading to :

rad Etot,yx

=0

and

3

ikr

ik rad Qe e Etot,xy = − 32π sin(2θ)e αxxy r o

. (9)

We show here that these two contributions can be determined independently by performing measurements of the light scattered between crossed polarizers at oblique incidence along dierent propagation directions. We can notice that, at normal incidence, there is no signal between crossed polarizers associated with the magneto-electric coupling or the quadrupolar moment, and the scattered elds are only related to

rad Etot,ii

=

α eee

by :

k2 α eee , (10) 4πo ii

13


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with (i)=(x,y).

RESULTS AND DISCUSSION The relative contributions of the tensor elements were obtained by tting the transmittance measurements with the model described in Section MODEL using the elds scattered by a superposition of electric and magnetic point-like dipoles and quadrupoles.

16,28,36,37

The spec-

tral dependence of each tensor element was assumed to feature a Lorentzian lineshape. To determine the dipole moments, the amplitudes, positions and broadening of the Lorentzian lineshapes were tuned to match the experimental transmission spectral features at normal incidence between parallel polarizers. At normal incidence, the contributions of the electric and magnetic modes are decoupled and the extinction cross-sections are related only to the electric dipole polarizability tensor elements by

ii Cext = 4πkIm[e αiiee ], (i = (x, y)).

The simu-

lated and monitored spectra are plotted in Figures 3(a-b). The resonances of the individual plasmonic scatterers were identied by polarization dependent measurements performed at normal incidence. When the resonator is placed on the focused beam (the illumination area is much larger than the scatterer dimensions), the optical theorem states that the decrease of the transmitted intensity is proportional to the extinction cross-section onator.

38

Cext

of the res-

This theorem applies in our experiments since the beam waist is approximately

equal to 1

µm.

Consequently, the transmission measurements were converted to extinction

cross-sections after correcting the illuminated area by the scatterers geometric sections. The measured extinction cross-sections of a single scatterer extracted from the transmittance measurements are displayed in Figure 3. For light polarized parallel to the bottom arm of the scatterer, absorption bands were observed near

800

nm and

1370

i.e.

along the y -axis, two

nm (Figure 3(a)). They are asso-

ciated with the fundamental and secondary modes of the resonator, as veried numerically (not shown here). These modes are characterized by instantaneous currents circulating in

14


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opposite directions in the lateral arms of the resonator.

These currents yield a net mag-

netic dipole moment in the center of the resonator and are referred in the literature as magnetic modes.

10,16

For light polarized perpendicularly to the bottom arm,

the x -axis, one absorption band is observed at

990

nm in Figure 3(b)).

i.e.

along

The latter is as-

sociated with the excitation of the electric mode of the resonator. It can be observed that the relative amplitudes of the resonances decrease when increasing their order.

However,

in the near infrared/visible spectrum, losses in gold also contribute to the damping of the higher order modes.

The values of extinction cross-section at the fundamental magnetic

mode reached approximately

Cext = 0.70µm2

which amounts to more than 12 times the

geometric area of the resonator. The extinction cross-section of dierent scatterers can be roughly compared by normalizing their values with their volume,

VU

was

0.24 × 0.23 × 0.042 = 2.4 10−3 µm3 ,

In our experiments,

i.e. 2.44 times larger than that used in Ref. 38.

Cext

For the same volume of scatterer, our value of agreement with previously reported values.

VU .

would amount to

0.29µm2 ,

in excellent

13,38

Light scattered at normal incidence between crossed-polarizers is presented in Figure 4(a). As expected, no signal can be signicantly detected in this case. In contrast, the measurement of light scattered between cross-polarizers at oblique incidence is a clear indication of a magneto-electric coupling as evidenced in Equation 8. To verify the presence of magnetoelectric coupling in the polarizability tensor of the single scatterer, scattering measurements were performed in cross-polarized conditions at oblique incidence in the plane of incidence (y,z ) in the case of an incident electric eld polarized along the (TM) incident polarization) or the

x

(Transverse Magnetic

y axis (Transverse Electric (TE) incident polarization).

contrast to Figure 4(a), for light propagating at

In

θ = 30◦ in the plane (y,z ) in TM polarization,

two transmission bands are clearly observed near

1370

nm and

800

nm,

i.e.

at the positions

of the magnetic modes (Figure 4(b)). The experimental observation of light scattered between crossed-polarizers at oblique incidence clearly establishes the presence of the magneto-electric coupling at the level of a

15


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(a)

0.8

2

CextYY (µm )

0.6 200 nm

0.4 0.2

(b)

0.4 0.8 600

800

0.3 0.6

1000

1200

1400

1600

1000 1200 1400 1400 Wavelength (nm) Wavelength (nm)

1600 1600

Wavelength (nm)

2

XX CC extabs (µm )

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 16 of 27

200 nm

0.2 0.4 0.1 0.2

600

800

Figure 3: Extinction cross-section of a single resonator measured at normal incidence with (a) incident light polarized parallel to the bottom arm and (b) incident light polarized perpendicular to the bottom arm. The solid curves were obtained using the point-like multipoles model.

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single meta-atom. This result is important since chiral scatterers such as

U -shaped resonators

are widely used in metamaterials and it shows that a part of the polarization transfer in this type of metamaterial originates in the shape of the resonator inducing magneto-electric coupling.

39

This contribution must be distinguished from non-local eects observed in arrays

of symmetric resonators.

21,22

The dierence in the scattered intensities from x to y and

y to x originates from (i) the dierent excitation conditions between x - and y -polarized

incident light impinging onto the resonator from the substrate in our experimental set-up as conrmed numerically in Figure 2 and from (ii) the presence of an element

Qe α exyy

in the

polarizability tensor of the quadrupolar moment. Two scattered bands were also observed between crossed-polarizers near

1250

nm and

720

nm (see Figure 4(c)) for light propagating

at oblique incidence in the plane containing the lateral arms of the resonator (plane ( x,z )). They correspond to light scattered with a polarization along the x -axis for an incident light polarized along the y -axis. In contrast, no scattered light could be observed in the reverse polarization case. This suggests the presence of a quadrupolar moment in the ( x,y ) plane most likely excited by the electric eld at the second magnetic mode.

32

The asymmetry

between the scattered signals has also been observed numerically without a substrate for

U-

shaped scatterers which conrms its origin in a quadrupolar moment and not in the presence of dipole images in the substrate.

16,28

The contributions of the magneto-dielectric coupling and quadrupolar moment in the polarizability tensor of the single resonator are quantied by the point multipole model previously applied to analyze the normal incidence measurements. The measurements obtained between crossed-polarizers are added in the tting procedure to determine the contribution of the magneto-electric coupling and quadrupolar moments. The polarizability tensor elements were scaled to scatter the same power for the same units of strength in the polarizability tensor.

17

The experimental conditions did not allow the decorrelation between

the relative contributions of the electric and magnetic dipole moments which scale as the square of the sine of the angle of incidence.

15

As a consequence, we decided to keep only

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Figure 4: Spectra of scattered light through a single

Page 18 of 27

U -shaped

resonator measured between

crossed polarizers with polarizer parallel to the bottom arm ( x,y ) and polarizer perpendicular to the bottom arm ( y,x ). (a) Measurements at normal incidence, (b) with an angle of ◦ ◦ incidence of 30 from z to y, and (c) with an angle of incidence of 30 from z to x. The green curves were obtained using the point multipoles model described in the text.

18


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the electric dipole moments. In contrast, the measurements between crossed polarizers for dierent propagation directions and angles of incidence allowed to decorrelate completely the relative contributions of the magneto-electric coupling and quadrupolar moments as shown in Figure 2.

λ = 1370nm

λ = 800nm 0.93 ±0.02 0.40 ±0.02

ee ~ α

em ~ α

me ~ α

0.93 ±0.02 0.39 ±0.05

me ~ α

4.46 ±0.02

ee ~ α

mm ~ α

Log(|α|) mm ~ α

em ~ α >0.38 ±0.12

>0.02 ±0.25

0.15 ±0.05

Qe ~ α

0.27 ±0.12

0.31 ±0.02

Qm ~ α

Qm ~ α

Qe ~ α

0.24 ±0.05

(b)

(a)

Figure 5: Absolute values (logarithmic scale) of the elements of the polarizability tensor of a single scatterer (Equation 3) at the positions of (a) the second (800 nm) and (b) rst (1370 nm) magnetic modes. The values are expressed in units of volume of resonator and normalized relative to the largest value. The values at their peak resonance are indicated mm is the value estimated (see text). The grayed parts by the numbers. The element α ezz correspond to the electric polarizability elements, not determined in this work, for the electric eld of the incident light polarized along z.

The magnitudes of the elements of the polarizability tensor at the positions of the magnetic resonances, obtained after tting of the experimental transmission spectra, are presented in Figure 5. They are expressed in units of volume of resonator that the maximum of the electric quadrupolar element ically 0.26 eV. The quadrupolar elements

Qe α exyy

Qe α exxy

VU .

It must be noted

was found at larger energy, typ-

contained only a high energy contribution

in agreement with calculations obtained with the multipolar theory.

32

This representation

illustrates the elements of the polarizability tensor involved in the optical response of the isolated scatterer. We can observe some discrepancies between modeled and measured values, in Figures 4. The error bars were obtained by determining the best model tting sequentially to only one particular incident polarization, either TE or TM. They are indicative of the ac-

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Page 20 of 27

curacy of the values obtained rather than that of the tting procedure. We can only estimate lower bounds for the magnitude of the magnetic dipole contributions using the constraint imposed by the balance of energy in the quasi-static approximation :

√ αee .αmm ≥ αem . 13

We nd that for the rst magnetic mode, the magnetic dipole could be very small with

mm α ezz ≥ 0.018 VU .

For the second magnetic mode, it should be at least of the same order of

magnitude as the magneto-electric coupling with

mm α ezz ≥ 0.38 VU .

These values are those

used in Figure 5. However it is also interesting to compare the spectral dependence of the magneto-electric coupling and its relative contribution to the resonant modes. Figure 6 presents the spectral dependence of the magneto-electric coupling contribution

em α eyz

and of the electric dipole contribution

ee . α eyy

Figure 6: Absolute values of the elements of the polarizability tensor (a) a function of the wavelength.

em α eyz

and (b)

ee α eyy

as

The mode at 1370nm contains a sizeable proportion of electric dipole contribution with

ee α eyy = 1.05 10−2 µm3

while the magneto-electric coupling amounts to

em α eyz = 0.94 10−3 µm3 .

The position of the magneto-electric coupling is also shifted towards shorter wavelength as compared to the position of the electric dipole. We remind that these tensor elements are values corrected for their dependence on the derivative on the electric eld. The value for the electric dipole polarizability compares favorably with the values already published. For the mode near 800 nm

ee α eyy

and

em α eyz

amount to

1.5 10−3 µm3

and

13,38

= 0.94 10−3 µm3 ,

respectively. We had also observed a large contribution of the magneto-electric coupling in

20


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arrays of

U -shaped

resonators for this mode.

15

The relative magnitudes displayed in Figure

5 agree with the values found for the superpolarizability tensor of dolmen-like scatterers at a resonance at

665

nm, although in that case the quadrupolar mode was driven through near-

eld excitation by the top part of the scatterer.

28

It is possible that the magneto-electric

coupling depends also strongly on the gradients of the elds at shorter wavelength and that the correction introduced in our polarizability tensors accounts for a sizable proportion of the value of

em α eyz

in this case. Measurements for many dierent angles of incidence may help

to answer this point.

CONCLUSION In conclusion, by analyzing experimental transmission measurements we have evidenced and quantied the presence of magneto-electric coupling in single

U -shaped

scatterers. We have

shown that the magneto-electric coupling contribution increased with the mode order relatively to that of the electric dipole. Measurements along dierent propagation directions evidenced independently the presence of an additional electric quadrupole. The measurements have been analyzed using a multipolar point-like model. This model allowed to quantify the magnitudes of the dierent contributions, except for the magnetic dipole which could only be estimated with our measurements conguration. We have shown that magneto-electric coupling and electric quadrupole contributed with the same order of magnitude and that these two eects have to be taken into account to describe properly the polarization properties of the radiated elds of

U -shaped

scatterers. These results bring new insights in the

potential presence of large near-eld chirality density in the vicinity of individual scatterers which may be used as antennae to enhance the response in chiroptical spectroscopy.

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Page 22 of 27

Acknowledgement Work at Institut Fresnel has been carried out thanks to the support of the A*MIDEX project (n

◦

ANR-11-IDEX-0001-02) funded by the Investissements d'Avenir French Government pro-

gram managed by the French National Research Agency (ANR).

References (1) Enoch, S.; Bonod, N. Plasmonics : From Basics to Advanced Topics , Springer, Berlin, 2012

(2) Novotny, L.; van Hulst, N. Antennas For Light. Nat. Photon.

2011, 5, 83-90

(3) Hendry, E.; Carpy, T.; Johnston, J.; Popland, M.; Mikhaylovskiy, R. V.; Lapthorn, A. J.; Kelly, S. M.; Barron, L. D.; Gadegaard, N.; Kadodwala, M. Ultrasensitive Detection and Characterization of Biomolecules Using Superchiral Fields. Nat. Nanotechnol.

2010,

5, 783-787

(4) Andrews, D.L.; Matt M.C. Measures of Chirality and Angular Momentum in the Electromagnetic Field. Opt Lett

2012 37

3009-3011

(5) Prosvirnin, S. L.; Zheludev, N. I. Analysis of Polarization Transformations by a Planar Chiral Array of Complex-Shaped Particles. Journal of Optics A: Pure and Applied Optics

2009, 11, 074002 (6) Wegener, M.; Zheludev, N. I. Articial Chiral Materials. Journal of Optics A: Pure and Applied Optics

2009, 11, 070201

(7) Fan, Z.; Govorov, A. O. Plasmonic Circular Dichroism of Chiral Metal Nanoparticle Assemblies. Nano lett.

2010, 10, 2580-2587

22


Page 23 of 27

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

ACS Photonics

(8) Kuzyk, A.; Schreiber, R.; Fan, Z.; Pardatscher, G.; Roller, E.-M.; Hogele, A.; Simmel, F. C.; Govorov, A. O.; Liedl, T. DNA-based Self-Assembly of Chiral Plasmonic Nanostructures With Tailored Optical Response. Nature

2012, 483, 311-314

(9) Hentschel, M.; Schaferling, M.; Weiss, T.; Liu, N.; Giessen, H. Three-Dimensional Chiral Plasmonic Oligomers. Nano lett.

2012, 12, 2542-2547

(10) Rockstuhl, C.; Zentgraf, T.; Pshenay-Severin, E.; Petschulat, J.; Chipouline, A.; Kuhl, J.; Pertsch, T.; Giessen, H.; Lederer, F. The Origin of Magnetic Polarizability In Metamaterials At Optical Frequencies - An Electrodynamic Approach. Opt. Express

2007,

15, 8871-8883

(11) Sihvola, A.H.; Juntunen, J.O.; Eratuuli, P. Macroscopic Electromagnetic Properties of Bi-Anisotropic Mixtures. IEEE Trans. Ant. Propag.

1996, 44, 836-842

(12) Sersic, I.; Frimmer, M.; Verhagen, E.; Koenderink, A.F. Electric and Magnetic Dipole Coupling in Near-Infrared Split-Ring Metamaterial Arrays. Phys. Rev. Lett.

2009, 103,

213902

(13) Sersic, I.; Tuambilangana, C.; Kampfrath, T.; Koenderink, A.F. Magnetoelectric Point Scattering Theory for Metamaterial Scatterers. Phys. Rev. B

2011, 83, 245102

(14) Ren, M.; Plum, E.; Xu, J.; Zheludev, N.I. Giant Nonlinear Optical Activity In a Plasmonic Metamaterial. Nat. Comm.

2012, 1805, 1-6

(15) Guth, N.; Gallas, B.; Rivory, J.; Grand, J.; Ourir, A.; Guida, G.; Abdeddaim, R.; Jouvaud, C.; de Rosny, J. Optical Properties of Metamaterials:

Inuence of Electric

Multipoles, Magnetoelectric Coupling, and Spatial Dispersion. Phys. Rev. B

2012,

85,

115138

(16) Varault, S.; Rolly, B.; Boudarham, G.; Demesy, G.; Stout, B.; Bonod, N. Multipolar

23


ACS Photonics

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 24 of 27

Eects On The Dipolar Polarizability Of Magneto-Electric Antennas. Opt. Express

2013,

21, 16444-16454

(17) Alizadeh, M.H.; Reinhard, B.M. Plasmonically Enhanced Chiral Optical Fields and Forces in Achiral Split Ring Resonators. ACS Phot.

2015, 2, 361-368

(18) Menzel, C.; Paul, T.; Rockstuhl, C.; Pertsch, T.; Tretyakov, S.; Lederer, F. Validity Of Eective Material Parameters For Optical Fishnet Metamaterials. Phys. Rev. B

2010,

81, 035320

(19) Agranovich, V.M.; Gartstein, Y.N. Electrodynamics Of Metamaterials And The Landau And Lifshitz Approach To The Magnetic Permeability. Metamaterials

2009, 3, 1-9

(20) Brakstad, T.; Kildemo, M.; Ghadyani, Z.; Simonsen, I. Dispersion of Polarization Coupling, Localized and Collective Plasmon Modes in a Metallic Photonic Crystal Mapped by Mueller Matrix Ellipsometry. Opt. Express

2015 23

22800-22815

(21) Gompf, B.; Braun, J.; Weiss, T.; Giessen, H.; Dressel, M. Periodic Nanostructures: Spatial Dispersion Mimics Chirality. Phys. Rev. Lett.

2011, 106, 185501

(22) Berrier, A.; Gompf, B.; Fu, L.; Weiss, T.; Schweizer, H. Optical Anisotropies Of SingleMeander Plasmonic Metasurfaces Analyzed By Mueller Matrix Spectroscopy, Phys. Rev. B

2014, 89, 195434

(23) Sersic, I.; van der Haar, M.A.; Arango, F.B., Koenderink, A.F. Ubiquity of Optical Activity in Planar Metamaterial Scatterers. Phys. Rev. Lett.

2012 108

223903

(24) Lunnemann, P; Sersic,I.; Koenderink, A.F. Optical Properties of Two-Dimensional Magnetoelectric Point Scattering Lattices. Phys. Rev. B

2013 88

245109

(25) I. V. Lindell, A. Sihvola, and S. Tretyakov, Electromagnetic Waves in Chiral and BiIsotropic Media (Artech House, 1994)

24


Page 25 of 27

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

ACS Photonics

(26) R. E. Raab and O. L. de Lange, Multipole Theory in Electromagnetism:

Classical,

Quantum, and Symmetry Aspects, with Applications (Oxford University , 2005)

(27) Banzer, P.; Peschel, U.; Quabis, S.; Leuchs, G. On the Experimental Investigation of the Electric and Magnetic Response of a Single Nano-Structure, Opt. Express

2010, 18,

10905-10923

(28) Arango, F. B.; Coenen, T.; Koenderink,A. F. Underpinning Hybridization Intuition for Complex Nanoantennas by Magnetoelectric Quadrupolar Polarizability Retrieval. ACS Phot.

2014, 1, 444-453

(29) Cho, D.J.; Wang, F.; Zhang, X.; Shen, Y.R. Contribution of The Electric Quadrupole Resonance In Optical Metamaterials. Phys. Rev. B

2008, 78, 121101(R)

(30) Kruk, S.S.; Decker, M.; Staude, I.; Schlecht, S.; Greppmair, M.; Neshev, D.N., Kivshar, Y.S. Spin-Polarized Photon Emission by Resonant Multipolar Nanoantennas. ACS Phot.

2014 1, 1218-1223 (31) Classical Electrodynamics , Third Edition, J.D. Jackson, John Wiley and Sons Editor, New York, 1999

(32) Muhlig, S.; Menzel, C.; Rockstuhl C.; Lederer F. Multipole Analysis of Meta-Atoms. Metamaterials

2011, 5, 64-73

(33) Lukosz, W.; Kunz, R.E. Light Emission by Magnetic and Electric Dipoles Close to a Plane Dielectric Interface. III. Radiation Patterns of Dipoles With Arbitrary Orientation. J. Opt. Soc. Am.

1979, 69, 1495-1503

(34) Haarmans, M.T.; Bedeaux, D. The Polarizability and The Optical Properties of Lattices And Random Distributions of Small Metal Spheres on a Substrate. Thin Solid Films

1993, 224, 117-131

25


ACS Photonics

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(35) Li, L. Symmetries of cross-polarization diraction coecients of gratings. J. Opt. Soc. Am. A

2000, 17, 881-887

(36) Petschulat, J.; Yang, J.; Menzel, C.; Rockstuhl, C.; Chipouline, A.; Lalanne, P.; Tuennermann, A.; Lederer, F.; Pertsch, T. Understanding The Electric and Magnetic Response of Isolated Metaatoms by Means of a Multipolar Field Decomposition. Opt. Express

2010, 18

14454-14466

(37) Grahn, P.; Schevchenko, A.; Kaivola, M. Electromagnetic Multipole Theory for Optical Nanomaterials. New J. Phys.

2012, 14, 093033

(38) Husnik, M.; Klein, M.W.; Feth, N.; Konig, M.; Niegemann, J.; Busch, K.; Linden, S.; Wegener, M. Absolute Extinction Cross-Section of Individual Magnetic Split-Ring Resonators. Nat. Photon.

2008, 2, 614-617

(39) Saadoun, M.M.I.; Engheta, N. A Reciprocal Phase Shifter Using Novel Pseudochiral or

Ω

Medium. Microwave Opt. Technol. Lett.

1992, 5, 184-188

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Page 27 of 27

Graphical TOC Entry Magneto-electric coupling

Single U-shaped scatterer TE

400x10

-6

300

θ Transmittance

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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b

200 100 0 -100

a

b a 1000

1200

1400

1600

Wavelength (nm)

27


Optical Monitoring of the Magnetoelectric Coupling in Individual

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