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 ( Left ) i ( θ ) = |〈 i | Ψ out 〉| 2 = |〈 i | ˆ J L | θ 〉| $^{2}$. Similarly we also write I ( Right ) i ( θ ) = |〈 i | ˆ J R | θ 〉| $^{2}$. From equation (10) we deduce: where we used | i $^{′}$〉 = ˆ Π − $^{1}$| i 〉 = ˆ Π | i 〉 and | − θ 〉 = ˆ Π | θ 〉 . We consequently have: Such symmetries are clearly visible in Fig. 2 and correspond to a direct signature of optical chirality in the planar systems considered. 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 Poincar´ e sphere of unit radius in the space X$_{1}$, X$_{2}$, X$_{3}$ . The transmitted output state after 13 〈 i $^{′}$| ˆ J L | θ 〉 = 〈 i | ˆ J R | − θ 〉 , (11) I ( Left ) total ( θ ) = I ( Right ) total ( − θ ) , I ( Left ) x,y ( θ ) = I ( Right ) x,y ( − θ ) , I ( Left ) ± 45 ◦ ( θ ) = I ( Right ) ∓ 45 ◦ ( − θ ) , I ( Left ) L,R ( θ ) = I ( Right ) R,L ( − θ ) . (12) S$_{0}$ = I$_{x}$ + I$_{y}$, S$_{1}$ = I$_{x}$ − I$_{y}$ S$_{2}$ = I$_{+45}$ ◦ − I$_{−}$$_{45}$ ◦ , S$_{3}$ = I L − I R , (13) We remind that the Stokes parameters associated with a polarization state of light | Ψ 〉 are defined by VIII. APPENDIX E: PLANAR CHIRALITY ON THE POINCAR ´ E SPHERE [IN- TERPRETING FIGURE 3]