procedure is, by far, the highest. The nonlocal procedure includes two additional DoF: the convolution integrals of the tangential components of the electric and magnetic fields. []{data-label="fig:error-R-vs-angle-ExA-PRB2013"}](error_R_vs_lambda_Ex_A_PRB2013 "fig:"){width="0.85\linewidth"} ![Example A of a layered medium from [@Markel-Tsukerman-PRB2013; @Tsukerman-Markel14]. Absolute error in $R$ (left) and $T$ (right) vs. $a / \lambda$; non-asymptotic and nonlocal homogenization. The lattice cell contains three layers of widths $a/4$, $a/2$ and $a/4$, with scalar permittivities $\epsilon_1$, $\epsilon_2$, and $\epsilon_1$, respectively. ($\epsilon_1 = 4 + 0.1i$ and $\epsilon_2 = 1$.) Fine-level basis: $2n_{\mathrm{dir}}$ Bloch modes traveling at $n_{\mathrm{dir}} = 7$ different angles in $(-\pi/2, \pi/2)$; $n_{\mathrm{dir}} = 7.$ The kernel width parameter $\tau_0 = a$. The accuracy of the nonlocal procedure is, by far, the highest. The nonlocal procedure includes two additional DoF: the convolution integrals of the tangential components of the electric and magnetic fields. []{data-label="fig:error-R-vs-angle-ExA-PRB2013"}](error_T_vs_lambda_Ex_A_PRB2013 "fig:"){width="0.85\linewidth"}

As demonstrated in [@Tsukerman-PLA17], the homogenization accuracy can be further improved by including, in addition to the $EH$ amplitudes, integral DoF of the form $$\label{eqn:D-eq-Eps-star-E}
{\mathbf{D}}({\mathbf{r}}) \,=\, \int_{{\mathrm{\Omega}}} \mathcal{E}({\mathbf{r}}, {\mathbf{r}}') \, {\mathbf{E}}({\mathbf{r}}') \, d {\mathrm{\Omega}}$$ where $\mathcal{E}$ is a convolution kernel depending only on the coordinates tangential to the boundary of the sample: $$\mathcal{E}({\mathbf{r}}, {\mathbf{r}}') = 
\mathcal{E}({\hat{\bf n}}\times {\mathbf{r}}, {\hat{\bf 