n}}\times {\mathbf{r}}')$$ A natural (but certainly not unique) choice for this kernel is a Gaussian $$\mathcal{E}({\mathbf{r}}, {\mathbf{r}}') = 
\mathcal{E}_0 \exp(-\tau_0^{-2} \, |{\hat{\bf n}}\times ({\mathbf{r}}- {\mathbf{r}}')|^2 )$$ where the amplitude $\mathcal{E}_0$ and width $\tau_0$ are adjustable parameters, and ${\hat{\bf n}}$ is the unit normal vector.

Since our focus is on the approximation properties of Trefftz functions and not on the homogenization procedure per se, we do not discuss the physics of the problem here, or the merits and demerits of nonlocal vs. local theory.[^1] We also omit further technical details and limit ourselves to just one illustration example.

Shown in Figs. \[fig:Re-RT-vs-angle-ExA-PRB2013\] and \[fig:error-R-vs-angle-ExA-PRB2013\] are the reflection $R$ and transmission $T$ coefficients for electromagnetic waves propagating through a layered slab. These coefficients are defined in a standard way, as the ratio of the complex amplitudes of the reflected/transmitted waves to that of the incident wave. The geometric and physical parameters correspond to Example A of [@Markel-Tsukerman-PRB2013]: the lattice cell of a width $a$ 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 $\epsilon_2 = 1$. The fine-level Trefftz basis contains $2n_{\mathrm{dir}}$ Bloch modes traveling at $n_{\mathrm{dir}} = 