total electric field ${\mathbf{e}}({\mathbf{r}})$ can be written as a superposition of the incident and scattered fields, viz, $$\label{eqn:ei_es}
{\mathbf{e}}({\mathbf{r}}) = {\mathbf{e}}_i({\mathbf{r}}) + {\mathbf{e}}_s({\mathbf{r}}) \ .$$ Outside the material, the reflected and transmitted fields form the scattered field: $$\label{eqn:es-eq-er-et}
{\mathbf{e}}_s({\mathbf{r}}) = \left\{
\begin{array}{ll}
{\mathbf{e}}_r({\mathbf{r}}) \ , & z < 0 \ , \\
{\mathbf{e}}_t({\mathbf{r}}) \ , & z > L \ .
\end{array} \right.$$ The scattered field *inside* the material is also formally defined by .

It is natural to approximate fine-level fields via a basis set of Bloch waves traveling in different directions: $$\label{eqn:eh-Bloch}
{\mathbf{e}}_{m \alpha}({\mathbf{r}}) = \tilde{{\mathbf{e}}}_{m \alpha}({\mathbf{r}}) \exp(i {\mathbf{q}}_{m \alpha} \cdot {\mathbf{r}}) \ , \ \
{\mathbf{h}}_{m \alpha} = \tilde{{\mathbf{h}}}_{m \alpha}({\mathbf{r}}) \exp(i {\mathbf{q}}_{m \alpha}
\cdot {\mathbf{r}}) \ ,$$ where index $\alpha$ labels both the wave vector and the polarization state of the Bloch wave in a lattice cell $m$; $\tilde{{\mathbf{e}}}_{m \alpha}({\bf r})$, $\tilde{{\mathbf{h}}}_{m \alpha}({\bf r})$ are the respective lattice-periodic factors. As the notation indicates, the basis is defined cell-wise; different bases in different lattice cells could be used. This makes the homogenization problem tractable and reducible to a single cell, rather than global and encompassing the whole sample.

On the coarse scale, a natural counterpart of the fine-scale 