  We present, discuss and validate an adapted S-matrix formalism for an
efficient, simplified treatment of stacked homogeneous periodically structured
metasurfaces operated under normally incident plane wave excitation. The
proposed formalism can be applied to any material system, arbitrarily shaped
metaatoms, at any frequency and with arbitrary subwavelength periods.
Circumventing the introduction of any kind of effective parameters we directly
use the S-parameters of the individual metasurfaces to calculate the response
of an arbitrary stack. In fact, the S-parameters are the complex parameters of
choice fully characterizing the homogeneous metasurfaces, in particular with
respect to its polarization manipulating properties. Just as effective material
parameters like the permittivity and the permeability or wave parameters like
the propagation constant and the impedance, the stacking based upon S-matrices
can be applied as long as the individual layers are decoupled with respect to
their near-fields. This requirement eventually sets the limits for using the
optical properties of the individual layers to calculate the response of the
stacked system - this being the conceptual aim for any homogeneous metasurface
or metamaterial layer and therefore the essence of what is eventually possible
with homogeneous metasurfaces. As simple and appealing this approach is, as
powerful it is as well: Combining structured metasurface with each other as
well as with isotropic, anisotropic or chiral homogeneous layers is possible by
simple semi-analytical S-matrix multiplication. Hence, complex stacks and
resonators can be set up, accurately treated and optimized with respect to
their dispersive polarization sensitive optical functionality without the need
for further rigorous full-wave simulations.
