The proposed homogenization theory—a major extension of the recently published methodology [J. Opt. Soc. Am. B 28, 577 (2011)]—yields an extended second-order material tensor consolidating the usual 36 local material parameters and additional ones that rigorously quantify nonlocality. The local part of the tensor relates the mean values of pairs of coarse-grained fields, while the nonlocal part relates the mean values to variations of the fields. The theory is based on a direct analysis of fields in a lattice cell rather than on an indirect retrieval of material parameters from transmission/reflection data. There are no heuristic assumptions and no artificial averaging rules. Nontrivial magnetic behavior, if present, is a logical consequence of the theory. The approximations involved and the respective errors are clearly identified. Illustrative examples of resonant structures with high-permittivity inclusions are given.
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