  The paper extends and enhances in several ways the recently proposed
homogenization theory of metamaterials [J. Opt. Soc. Am. B 28, 577 (2011)]. The
theory is based on a direct analysis of fields in the lattice cells rather than
on an indirect retrieval of material parameters from transmission / reflection
data. The theory is minimalistic, with only two fundamental premises at its
core: (i) the coarse-grained fields satisfy Maxwell's equations and boundary
conditions exactly; and (ii) the material tensor is a linear relationship
between the pairs of coarse-grained fields. There are no heuristic assumptions
and no artificial averaging rules. Nontrivial magnetic behavior, if present, is
a logical consequence of the theory. The method yields not only all 36 standard
material parameters, but also additional ones quantifying spatial dispersion
rigorously. The approximations involved are clearly identified. A tutorial
example and an application to a resonant structure with high-permittivity
inclusions are given.
