denoted with small letters ${\mathbf{e}}$, ${\mathbf{d}}$, ${\mathbf{h}}$ and ${\mathbf{b}}$. In general, their variation in space is rapid and consistent with the microstructure of metamaterial cells. *Coarse-level* fields ${\mathbf{E}}$, ${\mathbf{D}}$, ${\mathbf{H}}$, ${\mathbf{B}}$ vary on a characteristic scale greater that the cell size. They represent some smoothed (averaged) versions of the fine-level fields and are auxiliary mathematical constructions rather than measurable physical quantities. The coarse-level fields are sought to satisfy Maxwell’s equations *and all interface boundary conditions* as accurately as possible.

Importantly, effective magnetic properties of metamaterials cannot be determined from the bulk behavior alone *as a matter of principle*. This is due, in particular, to the fact that the Maxwell equation $\nabla \times {\mathbf{H}}= -i k_0 {\mathbf{D}}$ is invariant with respect to an arbitrary simultaneous rescaling of vectors **H** and **D**. Loosely speaking, bulk behavior defines the dispersion relation only, while magnetic characteristics depend on the boundary impedance as well.

The fine-level fields satisfy macroscopic Maxwell’s equations of the form $$\label{eq:Maxwell_exact}
   \nabla \times {\mathbf{h}}({\mathbf{r}}) = -i k_0 \tilde{\varepsilon}({\mathbf{r}}) \, {\mathbf{e}}({\mathbf{r}}) \ , \ \ ~~
   \nabla \times {\mathbf{e}}({\mathbf{r}}) = i k_0 \, {\mathbf{h}}({\mathbf{r}})$$ everywhere in space, supplemented by the usual radiation boundary conditions at infinity. Outside the slab, the most general 