layered medium from [@Markel-Tsukerman-PRB2013; @Tsukerman-Markel14]. The real part of $R$ (left) and $T$ (right) vs. the sine of the angle of incidence; 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 reflection and transmission coefficients from nonlocal homogenization are visually indistinguishable from the exact ones. The nonlocal procedure includes two additional DoF: the convolution integrals of the tangential components of the electric and magnetic fields.[]{data-label="fig:Re-RT-vs-angle-ExA-PRB2013"}](Re_T_vs_angle_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 