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.
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