## Abstract

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.

©2008 Optical Society of America

## 1. 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 optical properties of materials are not simply scalable when downsizing to the nanometer level. However, theoretical work has been done which suggests ways to overcome this difficulty by using localized plasmon modes excited at the scale of the nanostructures [6]. 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 [7, 8]. Our results reveal how, in a stringent and unusual way, this optical phenomenon connects concepts of chirality, reciprocity and broken time symmetry.

We remind that partly boosted by practical motivations, such as the quest of negative refractive lenses [9] or the possibility to obtain giant optical activity for applications in optoelectronics, there is currently a renewed interest [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19] in the optical activity in artificial photonic media with planar chiral structures. It was shown for instance that planar gammadionic structures, which have by definition no axis of reflection but a four-fold rotational invariance [10, 12], can generate optical activity with giant gyrotropic factors [13, 19, 18, 17]. Importantly, and in contrast to the usual three dimensional (3D) chiral medium (like quartz and its helicoidal structure [3, 20]), planar chiral structures change their observed handedness when the direction of light is reversed through the system [10, 21]. This challenged Lorentz principle of reciprocity [4] (which is known to hold for any linear non magneto-optical media) and stirred up considerable debate [10, 11, 13, 22] 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 [13, 19, 18, 17]. 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 [6]. 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.

## 2. 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 [12, 13, 14, 15, 16, 17, 18, 19]. 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, 6]. It is also important to emphasize that in order to observe a signature of planar chirality the structure considered must necessarily lack the rotational invariance. Indeed, rotational invariance together with chirality necessarily imply the existence of circular birefringence (i.e., gyrotropy and optical activity). This can be proven from group theory arguments and is also consistent with theoretical analysis of transmission through gammadion holes [23, 24, 25] (compare with the analysis in ref. [6]). 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 (Fig. 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 interfaces [26], 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 [8] 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*/(2*π*), 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 _{SPP}*≃760 nm for an excitation at

*λ*≃780 nm (by analogy with what occurs for bull’s eye circular antennas [8] this conditions

*P*≃

*λ*is required in order to obtain a resonant excitation of SPs on the structured metal film interface). 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 [27].

_{SPP}To observe and fully characterize the optical signature of planar chirality we perform a full polarization tomography [28, 29] (see appendix A and Fig. 4) with the aim of determining the 4×4 Mueller matrix 𝓜 associated with each enantiomer. Experimental results 𝓜_{𝓛}
^{exp.}, and 𝓜_{𝓡}
^{exp.} respectively obtained for left and right handed spirals are given in appendixes A and B. Here the important point is that the degree of purity *F* of the Mueller matrices [28] is near unity with *F* (𝓜_{𝓛}
^{exp.})≃0.967 and *F*(𝓜_{𝓡}
^{exp.})≃0.939. This shows that the coherence in polarization is not degraded by the structure and that we can therefore restrict our discussion to Jones matrices [28, 3]. In the convenient left |*L*〉 and right |*R*〉 circular polarization basis, these Jones matrices tie the excitation [*E*
^{in}
_{L},*E*
^{in}
_{R}] to the transmitted [*E*
^{out}
_{L},*E*
^{out}
_{R}] electric fields. In the case of planar chiral structures displaying 2D chiral activity, they have the following form [5, 6]:

where *A*, *B* and *C* are complex valued numbers such that |*B*|≠|*C*| (in references [5, 6] these constants were written *α*,*β* and *γ* respectively). This inequality accounts 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 losses are necessarily present. Since losses implies (macroscopic) time irreversibility, the condition |*B*|≠|*C*| also means that reversing the light path through the chiral structures is not mathematically equivalent to simply reversing the time. This explains why the simple argument based on time invariance given in ref. [13], to exclude planar optical activity, does not here lead to contradiction with our study. From Eq. (1) we deduce the associated theoretical forms for the Mueller matrices 𝓜_{𝓛}
^{th.}, 𝓜_{𝓡}
^{th.} (see appendix C). The determination of the coefficients *A*, *B* and *C* (see Eq. (1)) are obtained by comparing the experimental results 𝓜_{𝓛,𝓡}
^{exp.} with 𝓜_{𝓛,𝓡}
^{th.}. After normalization by *A* we deduce

$${{\mathcal{J}}_{\U0001d4e1}}^{\mathrm{fit}}=\left(\begin{array}{cc}1.000& -0.129+i0.098\\ 0.170+i0.230& 1.000\end{array}\right).$$

These matrices indeed satisfy the chirality criteria of Eq. (1) within the ~1% uncertainty evaluated from the degree of purity of the Mueller matrix of the empty setup.

## 3. Discussion and conclusion

To illustrate the polarization conversion properties of our chiral structures, we compare in Fig. 2 theory and experiment when the input state is linearly polarized and when the output transmitted intensity is analyzed along different orthogonal directions. A good agreement between the measurements and the theoretical predictions deduced from the Jones matrices (see appendix D) is clearly seen, together with the mirror symmetries between the two enantiomers. This agreement shows that our theoretical hypothesis about the form of the matrices 𝓙_{𝓛,𝓡}(see Eq. (1)) is experimentally justified. Importantly, the observed 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 [28]. Indeed, as shown in Fig. 3, the Mueller matrix defines a geometrical transformation which projects the unit Poincaré sphere, drawn by the input Stokes vector, on an output closed surface with typical radius *F*(𝓜^{exp.})≃1 in agreement with the absence of net depolarization as already noticed (from theory, *F* (𝓜^{th.})=1 exactly). Data shown on Fig. 2 are also plotted on this sphere. The input state draws a circle in the equator plane while the output state (for each enantiomer) draws a circle in a different plane, which center is not located at the center of the sphere. This is a direct manifestation of planar chirality (see appendix E). There is clearly an antisymmetrical behaviour between both enantiomers. The good agreement between the experiment and the prediction of Eqs. (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.}. For 𝓙_{𝓛}
^{th.}, the eigenstates are |±_{𝓛}〉=√*B*|*L*〉±√*C*|*R*〉 associated with the eigenvalues
${\lambda}_{\U0001d4db}(\pm )=A\pm \sqrt{\left(B\xb7C\right)}$
. The eigenstates for 𝓙_{𝓡}
^{th.} are obtained by permutation of *B* and *C* with consequently *λ*
_{𝓛}(±)=*λ*
_{𝓡}(±). The scalar product 〈+_{𝓛}|-_{𝓛}
^{th.}=-〈+_{𝓡}|-_{𝓡}〉^{th.}= (|*B*|-|*C*|)/(|*B*|+|*C*|) is the eigenstates Stokes parameter *S*
_{3}/*S*
_{0} and provides a direct measurement of the degree of optical chirality. It also evaluates losses since the non-orthogonality of these two states is related to the necessary non-unitarity of the Jones matrix for planar chirality. We have 〈+_{𝓛}|-_{𝓛}〉^{fit}≃0.255 and 〈+_{𝓡}|-_{𝓡}〉^{fit}≃-0.277, which, within experimental uncertainties, are in good agrement with the theoretical expectations. As shown in the insets of Fig. 2, both eigenstates of each structure (e.g., |±_{𝓛,𝓡}〉) can be represented by two ellipses having the same axis ratio and the same handedness, but rotated 90° relative to each other. In agreement with the theoretical predictions, these polarization ellipses for |±_{𝓛}〉 and |±_{𝓡}〉 are mirror reflections. This behaviour is significantly different from the results obtained with optically active media [14, 15, 13, 19, 18, 17] where the eigenstates associated with a given enantiomer have opposite handedness [4]. This point, which reflects itself in the symmetry property of chiral Jones matrices, namely 𝓙^{th.}
_{𝓛,𝓡}(*L*,*L*)=𝓙^{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 [5, 4] (inversely, one can show that Eq. (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. Finally, it is important to note that even though the structures considered here are fundamentally 3D chiral, given that two sides of the structures are not milled symmetrically, the optical signature reported here is strictly related to manifestations of planar chirality because SPs provide a unique way to generate spatial dispersion on a flat surface with lossy waves making it possible to bypass, in 2D, the limitations imposed in 3D by optical activity. 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.

## 4. Appendix A: Polarization tomography setup.

We apply a procedure similar to the one considered in [29, 30] in order to record the Mueller matrix: a collimated laser beam at *λ*=785 nm is focussed normally on the structure 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 [28]. 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 [28]. 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 [28]
$F\left(\U0001d4dc\right)={\left(\frac{\mathrm{Tr}\left[{\U0001d4dc}^{\u2020}\U0001d4dc\right]-{\U0001d4dc}_{00}^{2}}{3{\U0001d4dc}_{00}^{2}}\right)}^{\frac{1}{2}}\le 1$
Here we measured *F*(𝓜^{glass})= 0.9851. It implies that the light is not depolarized when going through the setup and that consequently we can rely on our measurement procedure for obtaining 𝓜.

Two important points must be noted here: On the one hand we varied the incident illumination spot size on the sample between 2 and 20 *µ*m without affecting the matrix, i. e., without introducing additional depolarisation. In the rest of the experiment on chiral structures we consider the case of a large gaussian spot with FWHM=20 *µ*m in order to illuminate the whole individual spiral (this also makes the setup less sensitive to minute variations in the position of the illumination spot with respect to the structure). On the other hand, it can be observed that in our experiments the polarization in the Airy spot (see Fig. 4b) is homogeneous. This implies that we are actually doing the polarization tomography of the central transmitting hole, i. e., we are dealing only with the SU(2) point symmetry of the Mueller Matrix. This situation clearly contrasts with previous SOP tomography measurements on metallic hole arrays in which the polarization degrees of freedom were mixed with spatial information responsible for SPP-induced depolarization [29].

## 5. Appendix B: Experimental Mueller matrices.

The experimental Mueller matrices deduced from the polarization tomography are after normalization of 𝓜^{exp.}
_{00}:

$${{\U0001d4dc}_{\U0001d4e1}}^{\mathrm{exp}.}=\left(\begin{array}{cccc}1.000& 0.035& 0.111& 0.023\\ 0.027& 0.949& -0.051& 0.246\\ 0.096& -0.034& 0.943& 0.267\\ -0.011& -0.252& -0.277& 0.745\end{array}\right).$$

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 $\mid \frac{{{\U0001d4dc}_{{\U0001d4db}_{\mathrm{ij}}}}^{\mathrm{exp}.}}{{{\U0001d4dc}_{{\U0001d4e1}_{\mathrm{ij}}}}^{\mathrm{exp}.}}\mid \simeq 0.954$ imputed to experimental errors and uncertainties. We also recorded the Mueller matrix of the set up with the glass substrate only. Up to a normalization factor we deduced

which satisfies 𝓜^{glass}≃𝓘 with 𝓘 the identity matrix (ideally one should have 𝓜^{glass}=𝓘). It implies that the optical set up do not induce depolarization and that consequently we can rely on our measurement procedure for obtaining 𝓜.

## 6. Appendix C: Theoretical Mueller matrices.

As shown, in refs. [28, 30] one can easily deduce the mathematical form of the Mueller matrix associated with a given Jones matrix. Every Jones matrix can be related to a Mueller matrix defined by its elements 𝓜_{i, j} (i,j=0,1,2,3) as follows: 𝓜_{i, j}=1/2·Tr[*Jσ _{j}J*

^{†}

*σ*] where Tr[

_{i}*X*] denotes the trace of a matrix

*X*and the

*σ*are the Pauli matrices. Here, the precise form of the theoretical Mueller matrice 𝓜

_{i}_{𝓛}

^{th.}deduced from Eq. (1) is

with 𝓜^{th.}
_{00}=(2|*A*|^{2}+|*B*|^{2}+|*C*|^{2})/2,𝓜^{th.}
_{01}=*Re*[*BA**+*AC**],𝓜^{th.}
_{02}=*Im*[*AB**+*CA**],𝓜^{th.}
_{03}=(|*C*|^{2}-|*B*|^{2})/2, 𝓜^{th.}
_{11}=|*A*|^{2}+*Re*[*B***C*], 𝓜^{th.}
_{12}=*Im*[*B***C*],𝓜^{th.}
_{13}=*Re*[*AC**-*BA**],𝓜^{th.}
_{22}=|*A*|^{2}-*Re*[*B***C*], 𝓜^{th.}
_{23}=*Re*[*CA**-*AB**], 𝓜^{th.}
_{33}=(2|*A*|^{2}-|*B*|^{2}-|*C*|^{2})/2. Similar formula are obtained for 𝓜_{𝓡}
^{th.} after permuting *B* and *C*.

From the previous relations we deduce the useful equations (valid for 𝓜_{𝓛}
^{th.})

$$\frac{C}{A}=\frac{{\U0001d4dc}_{01}^{\mathrm{th}.}-{\U0001d4dc}_{13}^{\mathrm{th}.}}{{\U0001d4dc}_{00}^{\mathrm{th}.}+{\U0001d4dc}_{33}^{\mathrm{th}.}}+i\frac{{\U0001d4dc}_{23}^{\mathrm{th}.}+{\U0001d4dc}_{02}^{\mathrm{th}.}}{{\U0001d4dc}_{00}^{\mathrm{th}.}+{\U0001d4dc}_{33}^{\mathrm{th}.}}.$$

Together with Eq. (3) Eq. (6) allow us to fit *B*/*A* and *C*/*A* if we replace 𝓜^{th.} by 𝓜_{𝓛}
^{exp.} (a similar procedure is applicable to 𝓜_{𝓡}
^{exp.} after permuting *B* and *C*).

The best fit we obtained (see Eq. (2)) are:

$${{\U0001d4dc}_{\U0001d4e1}}^{\mathrm{fit}}=\left(\begin{array}{cccc}1.000& 0.0359& 0.125& 0.026\\ 0.039& 0.949& -0.044& 0.283\\ 0.125& -0.044& 0.948& 0.311\\ -0.026& -0.283& -0.311& 0.897\end{array}\right).$$

From theory we can deduce that *F*(𝓜_{𝓛,𝓡}
^{th.})=1 (i.e., after normalization by 𝓜^{th.}
_{00}). We have thus *F*(𝓜_{𝓛,𝓡}
^{fit})=1

## 7. Appendix D: Symmetries due to chirality [interpreting Fig. 2].

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

where $\widehat{\U0001d4d9}$
_{𝓛} is the operator associated with the Jones matrix 𝓙_{𝓛}. The mathematical definition of planar chirality is that whatever the mirror symmetry operation $\widehat{\Pi}$
in the plane X-Y we have $\widehat{\U0001d4d9}$
$\widehat{\Pi}$
-$\widehat{\Pi}$
$\widehat{\U0001d4d9}$
≠=0. It equivalently states that $\widehat{\Pi}$
$\widehat{\U0001d4d9}$
$\widehat{\Pi}$
^{-1}≠$\widehat{\U0001d4d9}$
. If we consider for example the mirror reflection through the Y axis (see Fig. 1) we have the matrix representation (in the cartesian basis)
$\Pi ={\Pi}^{-1}=\left(\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right)$
and consequently

which agrees with Eq. (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. Indeed from Eqs. (8) and (9) we obtain

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 *I _{i}*

^{(Left)}(

*θ*)=|〈

*i*|Ψ

_{out}〉|

^{2}=|〈

*i*|$\widehat{\U0001d4d9}$

_{𝓛}|

*θ*〉|

^{2}. Similarly we also write

*I*

_{i}^{(Right)}(θ)=|〈

*i*|$\widehat{\U0001d4d9}$

_{𝓡}|

*θ*〉|

^{2}. From Eq. (10) we deduce:

where we used |*i*′〉=$\widehat{\Pi}$
^{-1}|*i*〉=$\widehat{\Pi}$
|*i*〉 and |-*θ*〉=$\widehat{\Pi}$
|*θ*〉. We consequently have:

$${I}_{x,y}^{\left(\mathrm{Left}\right)}\left(\theta \right)={I}_{x,y}^{\left(\mathrm{Right}\right)}\left(-\theta \right),$$

$${I}_{\pm 45\xb0}^{\left(\mathrm{Left}\right)}\left(\theta \right)={I}_{\mp 45\xb0}^{\left(\mathrm{Right}\right)}\left(-\theta \right),$$

$${I}_{L,R}^{\left(\mathrm{Left}\right)}\left(\theta \right)={I}_{R,L}^{\left(\mathrm{Right}\right)}\left(-\theta \right),$$

Such symmetries are clearly visible in Fig. 2 and correspond to a direct signature of optical chirality in the planar systems considered.

## 8. Appendix E: Planar chirality on the Poincaré Sphere [interpreting Fig. 3]

We remind that the Stokes parameters associated with a polarization state of light |Ψ〉 are defined by

$${S}_{2}={I}_{+45\xb0}-{I}_{-45\xb0},\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}{S}_{3}={I}_{L}-{I}_{R},$$

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 [28] |**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 Eq. 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 Eq. (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} (in other words, this curve (∑_{𝓛,𝓡}) is the circle arising from the section of the sphere by a plane).

To analyze this point more in details we consider the normalized Vector product

and we write it

with |*U*
_{𝓛,𝓡}|^{2}+|*V*
_{𝓛,𝓡}|^{2}+|*W*
_{𝓛,𝓡}|^{2}=1. It represents a normal to the closed curve (∑_{𝓛,𝓡}) (the vectorial products of 2 non colinear vectors connecting three points contained in a plane is indeed normal to this plane). From Eqs. 7,14 and 15 we obtain

Actually, if each curve (∑_{𝓛,𝓡}) is contained in a (different) plane *P*
_{𝓛,𝓡} we must have

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 also proves that each curve (∑_{𝓛,𝓡}) must be a circle (indeed the intersection of a plane with a sphere is 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 write

with *D*
_{𝓛}=-0.0237 and *D*
_{𝓡}=-0.0266. |*D*
_{𝓛,𝓡}| represents the distance separating the center of the circle (∑_{𝓛,𝓡}) to the origin of the poincaré sphere. This proves that the planes are not going through the center of the sphere. It was checked after lengthy calculations that if |*B*|=|*C*| in the Jones matrix (see Eq. (1) then *D*=0. This shows that the property |*D*
_{𝓛,𝓡}|≠0 is a characteristic of planar chirality (i.e, the condition |*B*|≠|*C*|). The radius of each circle (∑_{𝓛,𝓡}) is given by
${r}_{\U0001d4db,\U0001d4e1}=\sqrt{\left(1-{D}_{\U0001d4db,R}^{2}\right)}$
and we have *r*
_{𝓛}=0.9997 and *r*
_{𝓛}=0.9996 which are slightly smaller than *r*=1 in agreement with the fact that *P*
_{𝓛,𝓡} are not going through the center of the sphere.

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