  Metamaterial homogenization theories usually start with crude approximations
that are valid in certain limits in zero order, such as small frequencies, wave
vectors and material fill fractions. In some cases they remain surprisingly
robust exceeding their initial assumptions, such as the well-established
Maxwell-Garnett theory for elliptical inclusions that can produce reliable
results for fill fractions far above its theoretical limitations. We here
present a rigorous solution of Maxwell's equations in binary periodic materials
employing a combined Greens-Galerkin procedure to obtain a low-dimensional
eigenproblem for the evanescent Floquet eigenmodes of the material. In its
general form, our method provides an accurate solution of the multi-valued
complex Floquet bandstructure, which currently cannot be obtained with
established solvers. It is thus shown to be valid in regimes where
homogenization theories naturally break down. For small frequencies and wave
numbers in lowest order, our method simplifies to the Maxwell-Garnett result
for 2D cylinder and 3D sphere packings. It therefore provides the missing
explanation why Maxwell-Garnett works well up to extremely high fill fractions
of approximately $50\%$ depending on the base materials, provided the
inclusions are arranged on an isotropic lattice.
