The proposed numerical method, "FLAME-slab," solves electromagnetic wave scattering problems for aperiodic slab structures by exploiting short-range regularities in these structures. The computational procedure involves special difference schemes with high accuracy even on coarse grids. These schemes are based on Trefftz approximations, utilizing functions that locally satisfy the governing differential equations, as is done in the Flexible Local Approximation Method (FLAME). Radiation boundary conditions are implemented via Fourier expansions in the air surrounding the slab. When applied to ensembles of slab structures with identical short-range features, such as amorphous or quasicrystalline lattices, the method is significantly more efficient, both in runtime and in memory consumption, than traditional approaches. This efficiency is due to the fact that the Trefftz functions need to be computed only once for the whole ensemble. *

Approximations by Trefftz functions are rapidly gaining popularity in the numerical solution of boundary value problems of mathematical physics. By definition, these functions satisfy locally, in weak form, the underlying differential equations of the problem, which often results in high-order or even exponential accuracy with respect to the size of the basis set. We highlight two separate examples in applied electromagnetics and photonics: (i) homogenization of periodic structures, and (ii) numerical simulation of electromagnetic waves in slab geometries. Extensive numerical evidence and theoretical considerations show that Trefftz approximations can be applied much more broadly than is traditionally done: they are effective not only in physically homogeneous regions but also in complex inhomogeneous ones. Two mechanisms underlying the high accuracy of Trefftz approximations in such complex cases are pointed out. The first one is related to trigonometric interpolation and the second one -somewhat surprisingly -to well-posedness of random matrices.

Efficient nanoscale light sources are sought after for applications such as sensing, imaging, and the development of photonic circuits. In particular, free electron light sources have gained much attention due to their ability to tune and direct light emission. Here, we show that radiation from free electrons passing through a 100 nm wide nanohole can reach as high as 90% of the theoretical limit. This is accomplished through the introduction of a circular nanoridge around the hole to form a structure we call the nanowell. The power radiated from the nanowell exceeds that of a regular nanohole by over 100 times and that of nanoholes surrounded by other features, such as bullseyes, by similar enhancement factors. Upon varying the structural parameters of the nanowell, the peak output wavelength can be tuned over a broad frequency range from the visible to the near-infrared. This reveals a route to extracting power from free electrons via material nanopatterning.

We demonstrate, using time domain finite difference simulations, that Smith-Purcell radiation can be precisely tailored. Spectral and spatial emission shaping is achieved by optimizing a grating parameterized by aperiodic sequence, which outperforms standard optimization schemes.