Bloch basis is a set of generalized plane waves $$\label{eqn:Psi-eq-E0-H0-exp}
\Psi_{m \alpha} = \{ \mathbf{E}_{m \alpha}, \mathbf{H}_{m \alpha} \} = 
\{ \mathbf{E}_{0 m \alpha}, \mathbf{H}_{0m \alpha} \} \exp(i {\mathbf{q}}_{m \alpha} \cdot {\mathbf{r}})$$ which satisfy Maxwell’s equations in a homogeneous but possibly anisotropic medium; subscript ‘0’ indicates the field amplitudes to be determined. Further technical details of the procedure can be found in [@Tsukerman-Markel14; @Tsukerman-PLA17]. The final result is as follows. First, the coarse-level wave vector for each plane wave is taken to be the same as its counterpart for the corresponding Bloch wave, which is already reflected in our notation above , . Secondly, the amplitudes $\{ \mathbf{E}_{0 m \alpha}, \mathbf{H}_{0m \alpha} \}$ of each plane wave are the *boundary* average of the *tangential* components of the respective fine-scale Bloch wave: $$\label{eqn:EH0-eq-face-avrg-eh}
{\mathbf{E}}_{0m \alpha} \,=\, {\mathcal{A}}_{m} ^{\, \tau} \tilde{{\mathbf{e}}}_{m \alpha}, ~~~
{\mathbf{H}}_{0m \alpha} \,=\, {\mathcal{A}}_{m}^{\, \tau} \tilde{{\mathbf{h}}}_{m \alpha}$$ The averaging operator ${\mathcal{A}}_{m}^{\, \tau}$ for tangential components of a generic vector field $\mathbf{f}$ is defined, in the case of an orthorhombic cell $\mathbb{C}_{m}$, as $$\label{eqn:bdry-avrging-operator-tau}
({\mathcal{A}}_m^{\, \tau})_{\gamma} \, \mathbf{f} \,\equiv\, 
\frac{
\int_{\partial \mathbb{C}_{m}} f_{\gamma} \, |{\hat{\bf n}}\times \hat{{\mathbf{r}}}_{\gamma}| \, dS
}
{
\int_{\partial \mathbb{C}_{m}} |{\hat{\bf n}}\times \hat{{\mathbf{r}}}_{\gamma}| \, dS
}
, 