content_image/374885.jpg content_image/374886.jpg
$^{Actually, if each curve (}$∑ L , $_{R}$) is contained in a (different) plane P$_{L}$$_{,}$$_{R}$ we must have 

<img src='content_image/374885.jpg'>

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 write 

<img src='content_image/374886.jpg'>

with D$_{L}$ = − 0 . 0237 and D$_{R}$ = − 0 . 0266. | D$_{L}$$_{,R}$ | represents the distance separating the center $^{of the circle (}$∑ L , $_{R}$) to the origin of the poincar´ e sphere. This proves that the planes are not going through the center of the sphere. It was checked after lengthy calculations that if | B | = | C | in the Jones matrix (see equation (1)) then D = 0. This shows that the property | D$_{L}$$_{,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$_{L}$$_{,}$$_{R}$ = √ (1 − D 2 L $_{,R}$) 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. 

[1] D. -F. M. Arago,“M´ emoire sur une modification remarquable qu’´ eprouvent les rayons lu- mineux dans leur passage ` a travers certains corps diaphanes, et sur quelques autres nouveaux ph´ enom` enes d’optique,” M´ em. Inst. France, Part I 12 (1811). 

[2] L. Pasteur, “M´ emoire sur la relation qui peut exister entre la forme cristalline et la composition chimique, et sur la cause de la polarization rotatoire,” C. R. Acad. Sci. Paris 26 , 535-539 (1848). 

[3] E. Hecht, Optics 2 nd ed. (Addison-Wesley, Massachusetts, 1987). 

[4] L. D. Landau, E. M. Lifshitz, L. P. Pitaevskii, Electrodynamics of continuous media 2 nd ed. (Pergamon, New York, 1984). 

[5] V. A. Fedotov, P. L. Mladyonov, S. L. Prosvirnin, A. V. Rogacheva, Y. Chen and N. I. Zhe- ludev, “Asymmetric propagation of electromagnetic waves through a planar chiral structure,” Phys. Rev. Lett. 97 , 167401 (2006). 

