Optical chirality without optical activity: How surface plasmons give a twist to light

Light interacts differently with left and right handed three dimensional chiral objects, like helices, and this leads to the phenomenon known as optical activity. Here, by applying a polarization tomography, we show experimentally, for the first time in the visible domain, that chirality has a different optical manifestation for twisted planar nanostructured metallic objects acting as isolated chiral metaobjects. Our analysis demonstrate how surface plasmons, which are lossy bidimensional electromagnetic waves propagating on top of the structure, can delocalize light information in the just precise way for giving rise to this subtle effect.


I. INTRODUCTION
Since the historical work of Arago [1] and Pasteur [2], chirality (the handedness of nature) has generally been associated with optical activity, that is the rotation of the plane of polarisation of light passing through a medium lacking mirror symmetry [3,4]. Optical activity is nowadays a very powerful probes of structural chirality in varieties of system. However, two-dimensional chiral structures, such as planar molecules, were not expected to display any chiral characteristics since simply turning the object around leads to the opposite handedness (we remind that a planar structure is chiral if it can not be brought into congruence with its mirror image unless it is lifted from the plane). This fundamental notion was recently challenged in a pioneering study where it was shown that chirality has a distinct signature from optical activity when electromagnetic waves interact with a 2D chiral structure and that the handedness can be recognized [5]. While the experimental demonstration was achieved in the giga-Hertz (mm) range for extended 2D structures, the question remained whether this could be achieved in the optical range since the laws of optics are not simply scalable when downsizing to the nanometer level. Here we report genuine optical planar chirality for a single subwavelength hole surrounded by left and right handed Archimedian spirals milled in a metallic film. Key to this finding is the involvement of surface plasmons, lossy electromagnetic waves at the metal surfaces, and the associated planar spatial dispersion [6,7]. Our results reveal how, in a stringent and unusual way, this optical phenomenon connects concepts of chirality, reciprocity and broken time symmetry.
Importantly, and in contrast to the usual three dimensional (3D) chiral medium (like quartz and its helicoidal structure [3,19]), planar chiral structures change their observed handedness when the direction of light is reversed through the system [9,20]. This challenged Lorentz principle of reciprocity [4] (which is known to hold for any linear non magneto-optical media) and stirred up considerable debate [9,10,12,21] which came to the conclusion that optical activity cannot be a purely 2D effect and always requires a small dissymmetry between the two sides of the system [12,[16][17][18]. Nevertheless Zheludev and colleagues did demonstrate in the GHz spectrum that a pure 2D chiral structure lacking rotational symmetry can have an optical signature which is distinct from optical activity [5]. They went on to predict that it should be possible to observe the same phenomena in the optical range by scaling down their fish-scale structure and playing on localized plasmons [22]. Following a different strategy, we show here that SP waves propagating on a 2D metal chiral grating resonantly excited by light provide an elegant solution to generate planar optical chirality in the visible.

II. EXPERIMENTS AND RESULTS
This is a challenging issue as it leads to two fundamental points which are apparently incompatible. On the one hand, finding such a 2D chiral effect in the optical domain is not equivalent to a simple rescaling of the problem from the GHz to the visible part of the spectrum. Indeed, losses in metal become predominant at the nanometer scale so that the penetration length of light through any chiral structure will become comparable to the thickness of the structure. In-depth spatial dispersion along the propagation direction of light will hence be induced, corresponding to the usual 3D optical activity [11][12][13][14][15][16][17][18]. One thus expects optical activity, through the losses, to be a more favorable channel than 2D optical chirality. On the other hand, losses (i.e., broken time invariance at the macroscopic scale) are necessary to guarantee planar chiral behavior [5,22]. With this in mind, we chose to make single SP structures such as a single hole in an optically thick metal film surrounded by an Archimedian spirals ( Figure 1) which can provide all the necessary ingredients for observing 2D optical chirality. It is a 2D structure lacking point symmetry, that is rotational and mirror invariances. At the same time, it resonates due to coupling to surface plasmons which, as lossy waves, represent a natural way for delocalizing information along a planar interface, moving in-depth losses to the surface. Importantly, the thickness of the metal film optically decouples both interface [23], and consequently only the structured chiral side is involved in the 2D optical chiral effect reported here. Finally, the structures gives rise to enhanced transmission [7] enabling high optical throughput for all the characterization measurements.
Using focus ion beam (FIB), we milled in an opaque gold film a clockwise (right R) or anticlockwise (left L) Archimedian spiral grooves around a central subwavelength hole. The polar equation (ρ, θ) of the left handed Archimedian spiral is ρ = P · θ/(2π), and the right handed enantiomeric spiral is obtained by reflection across the y axis (see Fig. 1). The geometrical parameter P is the radial grating period and we take its value equal to the SP wavelength λ SP P ≃ 760 nm (for an excitation at λ ≃ 780 nm). We recorded optical transmission spectra at normal incidence with unpolarized light for both isolated structures ( Fig. 1). As it can be seen, both enantiomers behave like resonant antennas with quasi identical transmission properties. This resonant behaviour is a direct indication of the SP excitation by the grating similarly to what is observed for circular antennas [24].
To observe and fully characterize the optical signature of planar chirality we perform a full polarization tomography [25,26] In the case of planar chiral structures displaying 2D chiral activity, they have the following form [5,22]: where A, B and C are complex valued numbers such that |B| = |C|. This inequality account for chirality. Being non diagonal, these matrices correspond to polarization converter elements with no rotational invariance around the z axis ( Fig. 1). They are thus fundamentally different from Jones matrices associated with optical activity, e.g., gammadions. Importantly the conditions |B| = |C| implies the non unitarity of J th. L,R which means that reversing the light path through the chiral structures is not equivalent to reversing the time. From equation (1) we deduce the associated theoretical forms for the Mueller matrices (see appendix C) which are used to fit J L and J R from experimental results.
After normalization by A we deduce These matrices indeed satisfy the chirality criteria of equation (1) within the ∼ 1% uncertainty evaluated from the degree of purity of the Mueller matrice of the empty setup.

III. DISCUSSION AND CONCLUSION
To illustrate the polarization conversion properties of our chiral structures, we compare in between the measurements and the theoretical predictions deduced from the Jones matri-ces (see appendix D) is clearly seen, together with the mirror symmetries between the two enantiomers. Importantly, these symmetries also imply that for unpolarized light, and in complete consistency with Fig. 1, the total intensity transmitted by the structures is independent of the chosen enantiomer. Furthermore, the conversion of polarization is well (geometrically) illustrated by using the Poincaré sphere representation [25]. Indeed, as shown in Fig. 3 between the experiment and the prediction of equations (1,2) shows the sensitivity of the polarization tomography method and the high reliability of the FIB fabrication.
The degree of optical 2D chirality is quantified by diagonalizing J L th. and J R th. . L,R (R, R) = A, has far reaching consequences, as pointed out in reference [5]. It implies that a 2D plasmonic spiral mimics a Faraday medium when we reverse the light path and this even if the system, unlike a true Faraday medium, obeys rigorously to the principle of reciprocity [4,5] (inversely, one can show that equation (1) results from both this requirement and the absence of mirror symmetry). It means that a photon coming from the second side will probe a structure of opposite chirality. After going through the structure and retracing back the light path with a mirror normal to the axis, the polarization state will be different at the end of journey from the initial one. This would be impossible for an optically active medium and is solely due to planar chirality. To summarize, our results therefore demonstrate that 2D chirality is possible in the visible domain in the absence of optical activity and add another element to the promising plasmonic toolkit.

IV. APPENDIX A: POLARIZATION TOMOGRAPHY SETUP.
We apply a procedure similar to the one considered in [26,27] in order to record the Mueller matrix: a collimated laser beam at λ = 785 nm is focussed normally on the struc-ture by using an objective L 1 (×50, numerical aperture=0.55). The transmitted light is collected and recollimated by using a second objective L 2 (×40, numerical aperture=0.6).
The input and output states of polarization are respectively prepared and analyzed in the collimated part of the light path by using polarizers, half wave plates and quarter waveplates. A sketch of the setup is provided below (see Fig. 4).
The Mueller matrix is built by applying an experimental algorithm equivalent to the one described in [25]. More precisely, in order to write down the full Mueller matrix, we measured here 6 × 6 intensity projections corresponding to the 6 unit vectors |x , |y , | + 45 • , | − 45 • , |L , and |R for the input and the output polarizations. Actually only 16 measures are needed to determine M [25]. Our actual procedure is thus more than sufficient to obtain M.
The isotropy of the setup was first checked by measuring the Mueller matrix M glass with a glass substrate. Up to a normalization constant, we deduced that M glass is practically identical to the identity matrix I with individuals elements deviating by no more than 0.02.
More precisely, the optical depolarization (i. e, the losses in polarization coherence) can be precisely quantified through the degree of purity of the Mueller matrix defined by [25] We have F M L exp. ≃ 0.967 and F M R exp. ≃ 0.939.
We must also note that the normalization used here neglects a small additional coefficient of proportionality |M L exp. The precise form of the theoretical Mueller matrice M L th. deduced from equation (1) is Together with equation (3)  The best fit we obtained (see equation (2)) are: From theory we can deduce that F M L,R th. = 1 (i.e., after normalization by M th. Let |Ψ in = E x |x + E y |y and |Ψ out = E ′ x |x + E ′ y |y be respectively the incident and transmitted electric fields when we consider the left handed planar chiral structure. We have |Ψ out =Ĵ L |Ψ in (8) whereĴ L is the operator associated with the Jones matrix J L . The mathematical definition of planar chirality is that whatever the mirror symmetry operationΠ in the plane X-Y we haveĴΠ −ΠĴ = 0. It equivalently states thatΠĴΠ −1 =Ĵ . If we consider for example the mirror reflection through the Y axis (see Fig. 1 which agrees with equation (1) and constitutes an other optical definition of chirality.
The previous equations are used in order to interpret the results of Fig. 3 of the main article.
The input state considered in Fig. 2 is a linearly polarized light |θ = sin (θ)|x + cos (θ)|y (the angle is measured relatively to the Y axis) and the transmitted intensity projected along a direction of analysis |i (i.e, |x , |y , | + 45 • , | − 45 • , |L , and |R ) is written From equation (10) we deduce: where we used |i ′ =Π −1 |i =Π|i and | − θ =Π|θ . We consequently have: total (−θ), x,y (θ) = I (Right) x,y (−θ), Such symmetries are clearly visible in Fig. 2 and correspond to a direct signature of optical chirality in the planar systems considered. We remind that the Stokes parameters associated with a polarization state of light |Ψ are defined by where I i are projection measurement along the direction i, i.e, I i = | i|Ψ | 2 . The Stokes vector X is a convenient representation of such a state. We have X = X 1 x 1 + X 2 x 2 + X 3 x 3 with X 1 = S 1 /S 0 , X 2 = S 2 /S 0 , X 3 = S 3 /S 0 and with (x 1 , x 2 , x 3 ) a cartesian orthogonal and normalized vector basis.
The coherent input state satisfies the normalization [25] |X| = 1, that is the vector draw a Poincaré sphere of unit radius in the space X 1 , X 2 , X 3 . The transmitted output state after interaction with the left or right handed structure is defined by the relation The output state defines a Stokes vector X L,R such that |X L,R | ≤ 1. A typical value for this radius is given by F (M L,R ).
If the input state is linearly polarized the input Stokes vector is: and draw a circle ( in ) along the equator contained in the plane X 1 , X 2 of the unit radius Poincaré sphere. Using equation (14) the output Stokes vector is now a function of θ: X L,R (θ) drawing a closed curve ( L,R ) (see Fig. 3) which is the image, through the Mueller matrix transformation, of the equator circle ( in ) above mentioned. Importantly, since the Mueller matrix M given by equation (5) represents a linear relation connecting X in to X out , we conclude that the image of the incident polarization state contained in the equator plane X 1 , X 2 through M must also be contained in a plane in the space X 1 , X 2 , X 3 .
To analyze this point more in details we consider the normalized Vector product n L,R = (X L,R (0) − X L,R (2π/3)) × (X L,R (0) − X L,R (π/2)) |(X L,R (0) − X L,R (2π/3)) × (X L,R (0) − X L,R (π/2))| (16) and we write it with |U L,R | 2 + |V L,R | 2 + |W L,R | 2 = 1. It represents a typical normal to the closed curve Actually, if each curve ( L,R ) is contained in a (different) plane P L,R we must have n L,R · (X L,R (θ) − X L,R (0)) = 0 (19) for every θ. This was indeed checked numerically up to a precision of 10 −11 . It was also checked that |X L,R (θ)| = 1 up to the same precision. This proves that each curve ( L,R ) must be a circle. The equations of the two planes P L,R are given by n L,R · (X − X L,R (0)) = 0 where X is the Stokes vector associated with a running point belonging to each plane. We