Explain materials with 4-fold symmetry having same reflectance when shone with LCP and RCP This is my first post here.
I am currently reading "Optical planar chiral metamaterial designs for strong circular dichroism and polarization rotation" by Do-Hoon Kwon, Pingjuan L. Werner, and Douglas H. Werner.
This paragraph mentions that
"Strong circular dichroism
For maximum CD, the fitness to be maximized is defined as
f =|A+−A−|=|T+−T−|, (1)
where A± are the absorbances for the RCP and LCP incident waves, which are given by A ± = 1−R± −T±. The second equality in (1) follows from R+ = R−, which can be obtained from the reciprocity theorem for structures having four-fold rotational symmetry and a normally incident plane wave [7]."
Can somebody explain to me what the reciprocity theorem is, and how it affects the reflectance of structures with 4-fold symmetry.
 A: The reciprocity theorem Kwon is referring to is often called the Lorentz reciprocity theorem (and sometimes de Hoop reciprocity among other names) and refers to the interchange of an electromagnetic source and signal resulting in equivalent measurements.
More accurately, they are actually referring to a special consequence of reciprocity in chiral planar metamaterials of $C_4$ symmetry, discussed in their source [here] (in equations 13-15 of Part IV.A, specifically)1. For a derivation of why this is the case, I would refer one to that paper (and also to [Li, 2000], who they cite), but heuristically what they are saying is that since we can relate diffracted light to incident light between two points, reflection coefficient relationships between polarizations are implied. Specifically $C_4$ symmetry is not necessary, but the sample must have a certain level of symmetry ($C_2$, which a $C_4$ material also has) in multilayer materials, which is what Kwon is concerned with. For multilayer materials, the absence of such symmetry prevents an equivalence between specific configurations that would imply equal reflectance of orthogonal polarizations. For a  detailed discussion, section 4 of [Li, 2000] is your best bet.
One might notice that the authors of your source are measuring transmittances in their experiment, so they want to ensure that the fact that they are not directly measuring reflectance does not matter as a fitness metric of their level of circular dichroism (CD). Since CD is defined as a difference in absorbances, not transmittances, it would be awful if what the authors interpret as high levels of CD were actually differential reflectances that they neglected to measure!
