Are optical anti-reflection coatings reciprocal? Consider a simple anti-reflection coating, say a quarter-wave layer of MgF${}_2$ on glass.  Normal incidence.   Is the reflectivity going from air to glass the same as the reflectivity going from glass to air?
 A: At normal incidence the reflectivity from glass to air is the same as that of air to glass.
Phase wise the layer is reciprocal; a quarter wave to produce deconstructive interference. As a result one might expect the layer to be thus reciprocal.
However, for perfect deconstructive interference, not only the phase, but also the intensity must match. Therefore for an optimised coating with first layer reflectivity $R_1$ and second layer reflectivity $R_2$ we want:
$R_1 = R_2 (1-R_1^2) \sum_n -1^n R_1^n R_2^n$
$R_1 = R_2 (1-R_1^2)/(1+R_1 R_2)$
As a result we have $R_1 \ne R_2$. I.e. optimising for the two directions will result in different coefficients. Propagation will be different (non-reciprocal) due to the summation of higher order reflectivities being different for non equal reflectivities.
If the reflectivities were equal then the coating would be truly reciprocal. In practice; however, the choice of dielectric refractive index is not perfectly precise, not to mention that would be temperature dependent and have a reversible and irreversible shift with UV exposure. In general they can be expected to be non-reciprocal even when designed to be reciprocal.
