We describe a simple metamaterial homogenization procedure that calculates a limited set of constitutive parameters
that are functions of both the frequency and the wavenumber. This procedure is based on replacing
the inclusions of the metamaterial with current sheets excited by the local electromagnetic fields.
We observe critical coupling to surface phonon-polaritons in silicon carbide by attenuated total reflection of
mid-infrared radiation. Reflectance measurements demonstrate critical coupling by a double-scan of wavelength
and incidence angle. Critical coupling occurs when prism coupling loss is equal to losses in silicon carbide and
the substrate, resulting in maximal electric field enhancement.
We present a numerical algorithm for extracting all 36 linear constitutive parameters of a metamaterial crystal as
a function of frequency and wavenumber based on driving a metamaterial with electric and magnetic charge and
current. We demonstrate how spatial dispersion can result in bianisotropy in a centrosymmetric crystal. Several
tests are performed on a 2D metamaterial crystal to validate these "current driven" constitutive parameters.
Finally, we show how our method can be used to study spatial dispersion by studying constitutive parameters
for small k.