Manifestation of planar and bulk chirality mixture in plasmonicÎ

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Manifestation of planar and bulk chirality mixture in plasmonic #shaped nanostructures caused by symmetry breaking defects} Aline Pham, Quanbo Jiang, Airong Zhao, Joel Bellessa, Cyriaque Genet, and Aurélien Drezet ACS Photonics, Just Accepted Manuscript • DOI: 10.1021/acsphotonics.7b00666 • Publication Date (Web): 12 Sep 2017 Downloaded from http://pubs.acs.org on September 13, 2017

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Manifestation of planar and bulk chirality mixture in plasmonic Λ-shaped nanostructures caused by symmetry breaking defects Aline Pham,

†

Quanbo Jiang,

†

Airong Zhao,

†

Joel Bellessa,

Aurélien Drezet

†Institut ‡Institut

‡

Cyriaque Genet,

¶

and

∗,†

NEEL, CNRS-Université Grenoble Alpes, F-38000 Grenoble, France

Lumière Matière, CNRS- Université de Lyon, 69622 Villeurbanne cedex, France

¶ISIS,

UMR 7006, CNRS-Université de Strasbourg, 67000 Strasbourg, France

E-mail: [email protected]

Abstract We report on the coexistence of planar and bulk chiral eects in plasmonic Λ-shaped nanostructure arrays arising from symmetry breaking defects. The manifestation of bi(2D) and three-(3D) dimensional chiral eects are revealed by means of polarization tomography and conrmed by symmetry considerations of the experimental Jones matrix. Notably, investigating the antisymmetric and symmetric parts of the Jones matrix points out the contribution of 2D and 3D chirality in the polarization conversion induced by the system whose eigenpolarizations attest of the coexistence of planar and bulk chirality. Furthermore, we introduce a generalization of the microscopic model of Kuhn, yielding to a physical picture of the origins of the observed planar chirality, circular birefringence and dichroism, theoretically prohibited in symmetric Λ-shaped nanostructures.

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Three-dimensional chirality, also called bulk chirality, refers to the general concept of chirality dened as the lack of any mirror symmetry of a system. Its study is of crucial importance as chiral geometry exists in a wide range of constituents in living organisms for which opposite congurations can produce distinct physiological processes. Because chirality gives rise to polarization eects such as optical activity (O.A.) and circular dichroism, they are investigated for enantiomer discrimination. With the perspective of increasing the sensitivity of these optical eects, chiral plasmonic nanostructures play a key role since they have been reported to enhance the interactions between light and chiral molecules due to strong local elds and the remarkable sensitivity of surface plasmons (SPs) to their local environments. 1 In addition, they have also been investigated for their potential applications in light manipulation at the nanoscale, 25 negative refractive index, 6 asymmetric transmission 79 and optical vortex generation. 10 While optical rotation has been intensively studied in 3D chiral media, only few fundamental works have been devoted to genuine 2D chiral systems. 1115 Two-dimensional chirality also known as planar chirality, is a particular case of chirality encountered in 2D. It arises from planar optical systems that cannot be put into coincidence with their mirror image unless lifted from the plane. Specically, it has been shown that 2D chiral systems give rise to distinct optical chiral signatures from 3D chiral structures. Indeed, in genuine planar chiral structures, the sense of twist is reversed upon light path reversal. This means that optical eects, induced by 2D chiral medium, manifest themselves with an opposite handedness if we reverse the illumination direction. Therefore, O.A. which follows the Lorentz reciprocity theorem, is fordidden in planar chiral structures. In constrast, the handedness of 3D chiral structures is maintained upon light path reversal and in agreement with the reciprocity principle, such media exhibit O.A.. 16,17 Consequently, the observation of optical rotation in plasmonic structures, 18 such as in gammadions 1922 implies that 3D symmetry breakings must occur in the system. For example, they can arise due to the presence of a substrate or small dissymmetry defects occurring during the fabrication process. 23 2


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Figure 1: (a-b) Scanning electron microscope (SEM) images of Λ-shaped nanoapertures forming the chiral plasmonic structures: (a) Dimensions of the left arm are L = 194 nm and w = 51 nm, and right arm are L = 197 nm and w = 41 nm. (b) Dimensions of the left arm are L = 155 nm and w = 27 nm, and right arm are L = 115 nm and w = 25 nm. The angle between the two arms for (a) and (b) is 90˚ and d = 150 nm. (c) SEM image of an array of 2 × 10 unit cells made of (b) with horizontal and vertical periodicity of 600 nm and 300 nm, respectively. (d) Tomography polarization setup: L i focusing lens, Pi polarizers, Hi half-wave plates, Qi quarter-wave plates, O i microscope objectives in the illumination ( i = 1) and collection (i = 2) sides. DP and F P refer to the direct and Fourier plane, respectively. In the present study, we report on the unprecedented observation of a mixture of planar and bulk chirality eects as a result of asymmetric arms in Λ-shaped plasmonic nanostructures (Figure 1(a-c)). Such systems have been widely investigated for their capability to manipulate the direction, the phase and polarization of light. 2426 Here, the symmetry breaks are due to reproducible shape imperfections produced during the focused ion beam milling of the nanostructures. We show that our method, which relies on a full polarization tomography 27 combined with Jones matrix symmetry analysis, 28,29 provides crucial information on hidden symmetry properties of the sample, in turn about the quality of the fabricated nanostructures. Foremost, whereas previous works have been focusing on the study of either 2D or 3D chirality, here we demonstrate how our approach leads to the identication of planar and bulk chirality both existing in a single optical system. In particular, a systematic decomposition of the Jones matrix J into its antisymmetric JAS and symmetric JS parts enables to reveal that each nanostructure (Figure 1(a),(b)) gives rise to distinct chiral optical responses whose 3D and 2D chirality contribution can be quantied. Indeed, as discussed 3


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in, 15 the most general form of the Jones matrix characterizing a chiral optical system, can be expressed as a sum of a Jones matrix associated with O.A. JO.A and planar chirality J2D . Because O.A. follows the Lorentz reciprocity principle, 15 one should retrieve the initial state of polarization (SOP) after reversing the light path through the optical active medium. As a result, JAS is connected to JO.A which writes: 



 0 γ JO.A. =  , −γ 0

(1)

where γ is a complex valued number. JO.A takes the form of a rotation matrix of gyromagnetic factor | γ |. Oppositely, JS can be identied with strictly planar chirality 7,12,13,15 if it fullls the requirements imposed by the Jones matrix describing 2D chirality J2D . The handedness of a planar chiral structure is reversed as the light path is inverted prohibiting the initial SOP retrieval thus O.A. in 2D chiral systems. If the later takes the following general form: 



A B  J2D =   C D

(2)

where A, B , C , D are complex valued numbers, then JS must satisfy the three conditions below:

=(

B = C,

(3)

A 6= D,

(4)

2B ) 6= 0, A−D

(5)

with = refering to the imaginary part. By means of symmetry considerations via JAS and JS , we thus show that our plasmonic nanostructures exhibit a mixture of planar and bulk

chirality which will be further characterized with the determination of the eigenpolarizations. In addition, we provide an intuitive understanding of the chiral eect generation which 4


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connects to a macroscopic Jones matrix description, a microscopic picture of the Λ-shaped apertures based on a generalized Kuhn model. 30 Specically, 2D and 3D broken symmetries between the two arms are taken in consideration and asymmetric dipole coupling is discussed. It is then shown to be well adapted for the description of O.A. mixed with planar chirality as exhibited by our chiral plasmonic nanostructures. Figure 1(a) and (b) show the Λ-shaped apertures forming the two plasmonic structures investigated in this work. These unit cells are periodically arranged in an 2 ×10 array as depicted in Figure 1(c). The nanostructures are fabricated by means of focused ion beam milling on a 200 nm opaque gold layer evaporated on a glass plate. The sample is illuminated by a weakly focused laser beam ( λ = 633 nm) at normal incidence via a rst microscope objective (40×, numerical aperture (NA) = 0.75). The transmitted light is collected by a CCD camera via a second microscope objective (oil immersion, 100×, NA= 1.45). A Fourier lens (L4 in Figure 1(d)) is used in order to image the back focal plane of the second objective, thus to image the scattered light in the momentum space [ kx , ky ]. By carrying out a complete polarization tomography, our aim is to determine the 4 × 4 Mueller matrix M , 27 which fully characterizes the SOP conversion of an optical system. Any incident waves associated with a Stokes vector Sin = [S0 S1 S2 S3 ]Tin is transformed after propagating through the system to Sout = M Sin (superscript T refers to the transpose operator). We recall that the four Stokes parameters are dened as follows: 27 S0 = IX + IY , S1 = IX − IY , S2 = IP − IM ˆ, and S3 = IR − IL , with Ii (i = X, Y, P, M, R, L) the intensity projected on the linear X ˆ (-45◦ ), and circular polarization states R ˆ and L ˆ (See cartesian basis in Yˆ , Pˆ (+45◦ ), M

Figure 1(a)). To retrieve the Mueller matrix, the SOP are prepared and analyzed in the six polarization states, via two sets of half Hi , quarter waveplates Qi and polarizers Pi placed on the illumination (i = 1) and collection (i = 2) sides. In the present work, instead of displaying 4 × 6 images corresponding to the four Stokes parameters mapped for six incident SOP, a

more straightforward and intuitive representation based on polarization ellipses is preferred. The ellipticity ε, orientation θ and helicity h are derived using the Stokes parameters such as 5


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Figure 2: Experimental Fourier plane images of the zero-order diraction beam recorded for the array made of rectangular apertures as depicted in inset (Scale bar: 1 µm): L =200 nm, w =50 nm, the vertical and horizontal pitches are 600 nm. The white arrows indicate the SOP of the incident beam. Polarization ellipses are superimposed on the images: black and red colors refers to +1 and -1 helicity, respectively. Scale bar: k0 N A/8 sin(2ε) = S3 /S0 , tan(2θ) = S2 /S1 and h = sign(S3 ). In addition, we point out that because

the depolarization eects of our systems, namely the losses in coherence polarization, are negligible, we can restrict our discussion to the Jones matrix J in the following. In this case, the latter can be directly derived from the Mueller matrix 27 (See Supporting Information). We remind that the Jones matrix relates the input to the output eld such as Eout = JEin . Finally, we will show that crucial information on chiral optical eects can be deduced from symmetry considerations of J . Let us rst treat the theoretical case of an ideally symmetric unit cell, made of two identical rectangular slits lying on the sample plane. The system being non chiral, we do not expect any chiral optical eects. Indeed, each slit (denoted as 1 and 2) excited by an impinging eld Ein induces a dipole moment p1 and p2 , respectively. They are dened as: p1 = α1 (ˆ n.Ein )ˆ n + β1 (m.E ˆ in )m, ˆ

(6)

p2 = α2 (m.E ˆ in )m ˆ + β2 (ˆ n.Ein )ˆ n,

(7)

6


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where nˆ =

√1 (1; 1) 2

and m ˆ =

. Each slit composing the Λ-shaped aperture is then

√1 (−1; 1) 2

associated with a Jones matrix: 







 α1 + β1 −α1 + β1  α2 + β2 α2 − β2  J1 =   ; J2 =  , −α1 + β1 α1 + β1 α2 − β2 α2 + β2

(8)

with αi and βi (i = 1, 2) accounting for the aspect ratio of the rectangular slit thus for the relative contributions of the major and minor-axis dipoles. 3 In case of a perfectly symmetric Λ-shaped aperture, we have α1 = α2 = α and β1 = β2 = β . We deduce from the total eld Eout ∝ p1 + p2 (here, we take α = 1 for convenience), the Jones matrix associated with a

symmetric Λ-shaped aperture at k = 0: 



1 0  J = (1 + β)  . 0 1

(9)

We thus expect that the incident SOP propagates through the system without any polarization changes. It is worth mentioning that our experimental approach allows the determination of the complex value of β , that is the relative weight of the minor dipole amplitude with respect to the major dipole. Recently, we demonstrated that the aspect ratio of the slit is critical for the design of nanostructures with optimized plasmonic coupling and directivity eciency. 3,31 Notably, we used leakage radiation microscopy on a thin metal lm 32,33 in order to quantify the contribution of each dipole in the SP generation. However, only partial information on the relative weight of the minor axis dipole, that is its real part, was accessed. Here, by implementing a polarization tomography on a reference array made of rectangular slits (see inset in Figure 2), we show that we can retrieve the complete complex value of β . On Figure 2, we verify that the array mainly acts as a linear polarizer with a direction that is normal to the major axis of the slits. While we ascribe the vertical polarization to the residual transmission of the incident light in Figure 2(b), we also detect a slight polarization conversion and rotation that can be quantied with the determination of the Mueller matrix. 7


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It is given by: 

 −0.168 −0.010  −0.129 −0.042  . −0.127 0.251    −0.290 −0.166

0.940  1.000   0.926 0.984  M =  0.148 0.153   −0.038 −0.042

(10)

s

from which we nd a global degree of depolarization D(M ) = 0.99 with D(M ) =

2 T r(M T M ) − M00 ≤ 2 3M00

1. It then indicates that the light is not depolarized during its propagation through our

optical systems, allowing us to restrict our study to the Jones matrix (See Supporting Information):   J =

 −0.077 + 0.014i , 0.078 − 0.021i −0.076 − 0.140i 1.000

(11)

which is decomposed as J = JAS + JS with JAS = (J − J T )/2 the antisymmetric and JS = (J + J T )/2 symmetric part:  0 −0.078 + 0.017i  JAS =  , −(−0.078 + 0.017i) 0     1.000 0.001 − 0.004i  1 0   JS =  ≈ . 0.001 − 0.004i −0.076 − 0.140i 0 β 

(12)

(13)

We observe that the array does not respond as a perfect linear polarizer. On one hand, from the analysis of JAS , it is seen that the system already features residual O.A. This attests of a 3D break of symmetry that can be imputed to experimental errors or to the presence of the substrate, which must be taken in consideration in the chiral responses of the Λ-shaped system. Importantly, it has been shown 23 that during the sample fabrication, nanostruc8


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Figure 3: Simulated Fourier plane images of the zero-order diraction beam recorded for the array made of perfectly symmetric Λ-shaped apertures in which the experimental value of β is used. The white arrows indicate the SOP of the incident beam. Polarization ellipses are superimposed on the images: black and red colors refers to +1 and -1 helicity, respectively. Scale bar: k0 N A/16. tures can be fabricated with irregularities and defects which can also substantially alter the resulting signal by breaking the symmetry of the nanoapertures. Although the imperfections weakly aect the single rectangular slit response, we will show that they dominate in the chiral optical properties arising from the Λ-shaped structure, due to the 3D asymmetry between the two rectangular apertures forming the Λ. On the other hand, we verify that JS does not relate to planar chirality ( =(2B/(A − D)) = −0.003), as expected given the

geometry of the unit cell, but inform on the minor axis dipole contribution. In order to simulate the optical response of an array made of ideally symmetric and planar Λ-shaped apertures, we then deduce from JS the complex value of β = −0.076 − 0.140i (See

eq 13) and insert it in our multi-dipole based simulation, similar as the approach developed in. 3 Here, the transmitted eld is computed considering a TE and TM eld description. Note that the diraction and the eects of the imaging systems are also included in the analysis. 34 The simulation results are displayed in Figure 3. In the central part of the Fourier plane images (kx , ky ≈ 0), we observe that the outgoing SOP is conserved after propagation

9


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Figure 4: Experimental Fourier plane images of the zero-order diraction beam recorded for the array made of Λ-shaped apertures in Figure 1(a). The white arrows indicate the SOP of the incident beam. Polarization ellipses are superimposed on the images: black and red colors refers to +1 and -1 helicity, respectively. Scale bar: k0 N A/16. through the sample which is in accordance with eq 14. However for kx , ky 6= 0, we observe a deviation of the SOP which is not rigorously identical to the incident polarization. For example in Figure 3(a-b), we note a conversion from linear to elliptical polarization for increasing kx , ky . Indeed, this is due to the spacing distance d between the two dipoles which induces a phase dierence for k 6= 0 and can lead up to circular conversion if kx d/2 = π/4. For example, if we consider an excitation polarized along x ˆ , the outgoing eld is given by Eout ∝ (1 + β) cos(kx d/2)ˆ x + i(1 − β) sin(kx d/2)ˆ y resulting in the elliptical polarization transformation seen in Figure 3(a) for increasing kx , ky . Let us now compare these simulation data with the measurements presented in Figure 4. They correspond to the response of the plasmonic system made of the Λ-shaped apertures depicted in Figure 1(a). It displays the zero-order diraction beam resulting from constructive interferences between the two lines of Λ-shaped apertures. Qualitatively, while the output SOP overall follow the trends previously discussed, we observe however that the linear SOP ˆ and M ˆ are converted into elliptical polarizations (See Figure 4(c), (d)). To understand P

the origin of this circular dichroism and perform quantitative comparison with the perfectly

10


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symmetric case, let us now analyze the symmetry of the experimental Jones matrix:   J =

 1

0.150 + 0.019i , 0.144 − 0.013i 0.949 + 0.594i

(14)

decomposed into: 

JAS

   0 0.003 + 0.016i 1.000 0.147 + 0.003i   =  , JS =   . (15) −(0.003 + 0.016i) 0 0.147 + 0.003i 0.949 + 0.594i

As discussed previously, it has been shown that genuine 2D chiral structures give rise to chiral optical eects that dier from O.A. Identifying JAS with JO.A , we nd that the manifestation of the optical rotation induced by our sample is weak. This is veried on Figure 4, where the orientation of the transmitted linear SOP overall remains unchanged, meaning that our sample exhibits negligible 3D chirality. The observed circular dichroism then arises from the planar chirality of the structure. Because the experimental JS matrix is found to fulll the above 2D chirality criteria with =(2B/(A − D)) = 0.246, we can infer that the polarization conversion seen in Figure 4 mainly results from the planar chirality of the nanostructures. This is conrmed by geometrical measurements on the apertures which reveal that the two arms dier in width by 11% (by 0.8% in length). Indeed, despite the progress in FIB milling techniques, this study demonstrates that structural imperfections, such as edges distortions and asymmetric arms dimensions, can yield to highly 2D chiral responses. 23,35 We will now show that our technique can also be implemented to reveal hidden symmetry breaking in 3D. Let us now study the sample displayed in Figure 1 (b),(c) which will be shown to exhibit both signicant planar chirality and O.A. As previously, a polarization tomography is performed on the sample and the experimental results are shown in Figure 5. We observed that the outgoing SOP now strongly diers from the data expected for the ideal symmetrical Λ-shaped aperture (Figure 3). More precisely, it is theoretically predicted that the output

11


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Figure 5: Fourier plane images of the zero-order diraction beam recorded for the array made of asymmetrical Λ-shaped apertures in Figure 1(b). The white arrows indicate the SOP of the incident beam. Polarization ellipses are superimposed on the images. Red and black ellipses refers to a helicity of +1 (left handedness) and -1 (right handedness), respectively. Scale bar: k0 N A/16. signal conserves the same SOP as the incident beam. However, Figure 5(a)-(c) show that the output SOP is considerably transformed after propagation through the sample with a striking optical dichroism while the simulations (Figure 3 (a)-(c)) predict linearly polarized beams. In Figure 5(f), the output eld is also measured with a strong linear polarization component, in contradiction with the circular polarization state expected in Figure3(f). The experimental Mueller matrix leads to the following Jones matrix: 

 1 −0.387 − 1.348i  J = , −0.368 + 0.557i −0.008 + 2.393i

(16)

which can be rewritten in its antisymmetric and symmetric forms: 

JAS

   0 −(0.009 + 0.953i) 1 −0.378 − 0.395i   =  , JS =   .(17) 0.009 + 0.953i 0 −0.378 − 0.395i −0.008 + 2.393i

On one hand, the analysis of JAS indicates that the sample highly induces O.A. with a gyromagnetic factor γ = 0.953, which is a clear signature of bulk chirality in the sample. The origin of the 3D symmetry breaking is not fully known yet but preliminary studies on the inuence of the milling loops used during the fabrication process is shown to induce 3D 12


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asymmetry.

Figure 6: SEM images of Λ-shaped aperture crosscuts FIB-milled with 6 loops (a) and 12 loops (b). Scale bar value: 200 nm. Indeed, in order to investigate the origin of observed the bulk chirality, we reproduce two arrays of Λ-shaped apertures under the same fabrication conditions as for the nanostructures depicted in Figure 1(a) and (b). In part, while the weakly asymmetric sample (Figure 1(a)) was FIB milled using 6 loops, the highly asymmetric sample (Figure 1(b)) was fabricated using 12 loops. We present on the Figure 6, the SEM images of the crosscuts performed on these samples. Whereas no signicant symmetry breaking is visible in Figure 6(a), we clearly observe a height dierence between the two arms forming the Λ in Figure 6(b) contributing to the bulk chirality exhibited in our sample. Although the origin of the 3D break of symmetry is not fully identied, further investigation are undergoing to clarify the inuence of the numbers of loops in the FIB milling on the symmetry breaking. On the other hand, we also nd that JS fullls the planar chirality conditions (eq 3-5) with =(2B/(A − D)) = −0.193, thereby clearly stating that we have 2D and 3D chiral eects both existing in the nanostructure. Alike planar chiral systems investigated in the past, 12 the present structure is also reported to induce optical rotation eects (that is 3D chiral eect) despite an apparent dominant 2D chiral geometry. However, thanks to a thorough analysis that systematically decomposes the optical response into 2D and 3D chirality, each contribution can now be claried. Moreover, while we can easily detect planar chirality from SEM image observation, assessing 3D symmetry property often requires destructive 13


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method, such as crosscuts, to be evidenced. Hence, we show that our approach is a relevant non-destructive technique to disclose hidden bulk chirality in structures that predominantly appear 2D chiral. In addition, to further characterize this mixed chirality, we investigate the eigenpolarizations which are representative of the chiral structure. We remind that they dene the SOP that remain unchanged after propagation through the system. They are determined by solving the general eigenvalue problem: 

    A B  Ex  Ex  JE = λE ⇒    = λ  C D Ey Ey

(18)

with the associated eigenvalues expressed as: λ± =

p 1 (A + D) ± (A − D)2 + 4BC . 2

(19)

The eigenvectors associated with each eigenvalue can be derived by solving eq 18 for the components Ex ,Ey . They are given by: 



1  B  E± = p  . |B|2 + |λ± − A|2 λ± − A

(20)

In Figure 7(a) and (b), we present the eigenpolarizations determined from the experimental Jones matrices (eq 14,16). In the case of the weakly chiral sample (Figure 7(a)), the eigenpolarizations are represented by two ellipses of similar ellipticities, rotated by 90˚ and having the same helicity, which is in agreement with our theoretical expectations for a non-optical active system. This is supported by previous studies, 12 13 reporting similar co-rotating elliptical eigenpolarizations as a result of planar chirality. We recall that in the case of a purely optically active medium, namely described by JO.A. (eq 1), the eigenpolarizations are readily √

given by circular polarizations with opposite handedness: E± = (ˆx ∓ iˆy)/ 2. Consequently, we expect that a system featuring 3D chirality to possess counter-rotating eigenvectors. In 14


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Figure 7: Eigenpolarizations measured for (a) the weakly chiral plasmonic system and (b) for the strongly chiral aperture. The sample in consideration is displayed in inset. ±1 refers to the helicity.(c) Representation of the Kuhn model. contrast, in the most general case of genuine 2D chirality, the eigenpolarizations are described by two co-rotating elliptical polarizations, with dierent orientations and ellipticities. Additionally, a system that exhibits planar chirality must obey to the conditions required by J2D (eq 3-5) which impose the non-orthogonality of the eigenpolarizations. Indeed, as detailed 2|B|eiφ ) ∝ sin(φ) 6= 0 (for |α| and α = A − D = |α|), hence requires that φ 6= mπ

in the Supporting Information, we show that eq 5 implies that =( calculations convenience, B = |B|eiφ

(with m ∈ Z). Oppositely, we nd that the condition for orthogonal eigenpolarization states E∗+ .E− = 0 imposes φ = nπ (with n ∈ Z) which is in contradiction with the planar chiral-

ity criteria. therefore, a structure featuring genuine 2D chirality cannot possess orthogonal eigenpolarizations. This is clearly evidenced in Figure 7(b) where the eigenpolarizations corresponding to the highly chiral sample appear strongly non-orthogonal with two elliptical polarizations with distinct ellipticities and orientations. We also retrieve that 2D chirality is mixed with 3D chirality in that nanostructure, as demonstrated by the two counter-rotating eigenpolarizations. Consequently, this study has demonstrated how the eigenpolarizations transform in case of a medium exhibiting a mixture of planar and bulk chirality. Now, in order to get a microscopic interpretation of these polarization eects, we develop a coupled dipole model which generalizes the Kuhn model 36 to two dierent oscillators in a complex medium. In its original version, which is widely used to explain O.A. in molecular 15


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media, the molecule is represented by two identical harmonic oscillators, orthogonal and vertically separated by a distance h along the optical axis z (See Figure 7(c)). An incident light excites and sets in motion the oscillators which are in interaction with each other. Here, we extend the model to two dierent and coupled dipoles associated with distinct Jones matrix J1 and J2 , previously introduced. In turn, the system of coupled motion equations writes: f1 = J1 Ein + J1 V eikz h p f2 p

(21)

f2 = J2 Ein + J2 V e−ikz h p f1 p

(22)

which can be expressed in a bloc matrix form: 

 I  

−J2 V T e−ikz h

ikz h

−J1 V e I







f1  J1 E  p   =   f2 p J2 E

(23)

f1 = p1 eikz h/2 where I is the identity matrix,rV the coupling matrix between the two dipoles, p

ω√ ω2 ε − kx2 − ky2 ≈ ε and ε the dielectric function of the 2 c c f1 + p f2 = J E the generalized Jones matrix : medium. We deduce from the total eld p f2 = p2 e−ikz h/2 , kz = and p

J = (I − J1 V J2 V T )−1 J1 + J1 V (I − J2 V T J1 V )−1 J2

(24)

+J2 V T (I − J1 V J2 V T )−1 J1 + (I − J2 V T J1 V )−1 J2 .

(25)

In isotropic media such as considered in the Kuhn model, the interaction matrix is often deduced from the eld radiated by the dipole 1 on the dipole 2 which writes: E 1→2 = −

(p1 .r12 ) 1 (p − 3 r12 ) 1 3 2 4πε0 r12 r12

(26)

where r12 is the distance between the centers of mass of the dipoles 1 and 2. Hence, the

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interaction matrix veries V = V T and is dened as:  V =−







1 3 x12 x12 x12 y12  V11 V12  ) = (I −    . 3 2 4πε0 r12 r12 x12 y12 y12 y12 V12 V22

(27)

In this conguration of reciprocal mutual interactions, we model the system as two dierent oscillators (α1 = 1, α2 = α) , innitely thin so their minor-axis contribution can be neglected (β1 = β2 = 0). In the (O, x0 , y 0 ) frame used for calculation convenience, they are described by: 











1 0  0 0   0 V12  J1 =   ; J2 =  ;V =  . V12 0 0 0 0 α

(28)

By inserting these quantities in eq 25, we deduce the values for the antisymmetric and symmetric parts of the Jones Matrix: 

   0 iαV12 sin(kz h)  1 αV12 cos(kz h)  J = + . −iαV12 sin(kz h) 0 αV12 cos(kz h) α

(29)

As predicted by the classical Kuhn model, we retrieve that the antisymmetric matrix is associated with O.A. with a gyrotropic power of αV12 sin(kz h). One veries that the necessary condition to observe O.A. is a non zero distance h between the oscillators meaning that a 3D break of symmetry is required. As for the symmetric part, it is related to planar chirality if the two oscillators are dierent ( α 6= 1) and if the following condition holds: =(

2αV12 cos(kz h) ) 6= 0, 1−α

(30)

criteria that can be fullled, for example, if the terms α, V12 , or kz are complex values, outlining the inherent connection between losses in the medium and planar chirality . 7,13,3739 Precisely, =(α), =(V12 ) or =(kz ) characterize dissipation in metals and more generally in 17


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real systems interacting with the surroundings. This is particularly relevant in our systems consisting of nano-apertures in gold lm in which losses and more specically the excitation of SPs can be determinant. For example, the essential role of SPs in the manifestation of planar chirality in 2D spiral plasmonic antenna has been addressed in our previous work. 13 Here, our theoretical prediction accounts for the necessity of losses in 2D chiral property, in agreement with broken time invariance. Indeed, while coupling with the environment, that necessarily involves losses, breaks the time-reversibility symmetry, it is shown 13,15 that a time-invariant system (i.e. non-lossy) imposes unitarity of its Jones matrix. However, this property is contradictory with a genuine chiral system described by J2D which is shown to be non-unitary (see demonstration in supporting information), conrming that dissipation necessarily occurs in a planar system exhibiting 2D chiral response. Therefore, our model emphasizes the fundamental role of dissipation in planar chirality. Remarkably, even within this simple framework of symmetric dipole interaction, the Kuhn model nonetheless yields to a physical interpretation of the Jones matrix, hence remains as a powerful tool to intuitively understand the origins of both bulk and planar chirality. Noteworthy, while this physical model points out the critical role played by 3D chiral defects in the O.A. response, a theoretical treatment with V 6= V T is necessary to account for the break of symmetry due to the presence of a substrate or asymmetric dipole interactions. Indeed, the Jones matrix provided in eq 25 describes a very general case in which the two dipoles can interact not only via dipole-dipole interactions in an isotropic medium but also through other coupling channels occurring in complex media which does not necessarily satisfy symmetric dipole coupling (V = V T ). For example, let us consider the case of dipole interactions via SPs channels. SPs produced by the dipole 1 located near an air-metal interface propagate and excite the dipole 2 which is at a distance h below the dipole 1. It becomes then clear that the coupling symmetry can be broken with the presence of a substrate. If positioned near the substrate, the dipole 2 will interact with the dipole 1 through SPs that are now created at the metal-substrate interface resulting into a radiation that is dierent from the eld 18


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induced by the dipole 1. In turn, we expect asymmetric coupling eects between the dipoles so that V 6= V T has to be taken in consideration. Furthermore, it should be noted that while eq 25 captures the main physics underlying the polarization transformation mechanisms, our model deserves to be completed to account for the unique eects of SPs intended in chiral plasmonic metamaterials. Central to their behavior is the excitation of localized SP at the hole's ridges and their tunneling through the hollow aperture which have been shown to play an essential role in the transmission and polarization of light emerging from the metallic array. 40 Therefore, we expect future theoretical study including waveguided and evanescent modes through the holes as developed in, 41 to report on the crucial impact of SPs in 2D and 3D chirality in our chiral nanostructures. To conclude, we demonstrate both qualitatively and quantitatively signicant optical chiral eects, such as the presence of both 2D and 3D chiral eects, arising from structural defects in plasmonic nanostructures. Importantly, decomposition into antisymmetric and symmetric matrices of the experimental Jones matrix allows us to identify the contribution of O.A. from genuine planar chirality, therefore to infer on the nature (2D or 3D) of the symmetry breaking in Λ-shaped apertures. Additionally, a generalization of the Kuhn model connects the microscopic origins of the polarization eects induced by bulk and planar chiral media with the macroscopic Jones matrices predicted by symmetry considerations. With the increasing complexity and interest in chiral plasmonic structures, 42,43 it becomes then crucial to fully monitor and characterize their optical responses. 14 Prospective investigations combining the presented methodology with spectroscopic polarimetry 44 are anticipated to provide crucial information on spectral preponderance of 2D and 3D chirality. We then expect our method to be directly applicable for disclosing underlying symmetry properties of complex plasmonic structures that couple both planar and bulk chirality.

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Acknowledgement This work was supported by Agence Nationale de la Recherche (ANR), France, through the SINPHONIE (ANR-12-NANO-0019) and PLACORE (ANR-13-BS10-0007) grants. Cyriaque Genet also thanks the ANR Equipex Union (ANR-10-EQPX-52-01). The Ph.D. grant of A. Pham by the Ministère de l'enseignement et la recherche, scientique, and of Q. Jiang by the Région Rhône-Alpes is gratefully acknowledged. We thank J.-F. Motte and G. Julie, from NANOFAB facility in Neel Institute, for sample fabrication.

Supporting Information Available Derivation details of the Mueller-Jones matrix, relationship between losses and genuine planar chirality, demonstration of non-orthogonality of the eigenpolarizations in genuine 2D chiral medium.

This material is available free of charge via the Internet at http:

//pubs.acs.org/ .

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Graphical TOC Entry

For Table of Contents Use Only Manifestation of planar and bulk chirality mixture in plasmonic Λ-shaped nanostructures caused by symmetry breaking defects. Aline Pham, Quanbo Jiang, Airong Zhao, Joel Bellessa, Cyriaque Genet, Aurélien Drezet. The present Table of Content (TOC) Graphic depicts SEM images of the Λ-shaped apertures investigated in the present study along with their associated eigenpolarization states, characteristics of the 2D and 3D chiral eects induced by the systems.

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