Active steering of cathodoluminescence through a generalized Smith-Purcell effect

Optical metasurfaces can shape the near fields of energetic electrons, enabling Smith-Purcell (SP) emission. We introduce a generalized SP effect relying on finite periodic arrays whose elements possess individually tunable polarizabilities, allowing us to explore higher-order SP radiation. By controlling the amplitude and phase of each of the elements, we show through rigorous theory the ability to create an SP steering device. In particular, we explore the active tuning capabilities of doped graphene, and thermally driven phase-change materials, which we compare with standard passive plasmonic structures made of gold and silver. Our results establish programmable electron-driven light sources and spectroscopic probes spanning the terahertz-to-visible range, advancing tunable metasurfaces for next-generation electron-photon technologies.

💡 Research Summary

The manuscript presents a comprehensive theoretical and numerical study of a “generalized Smith‑Purcell” (GSP) effect that enables active steering of cathodoluminescence (CL) emitted by a swift electron passing over a finite periodic metasurface. Traditional Smith‑Purcell radiation assumes an infinite array of identical scatterers, so that the induced dipoles are uniform and only the conventional SP orders (n) satisfy the phase‑matching condition sin θ = (1/β − n λ/a). The authors relax both assumptions: the array is finite (N ≈ 50–100 elements) and each element j can possess a distinct polarizability α_j, which can be tuned electrically or thermally.

Starting from the self‑consistent dipole equation p_j = α_j E_ext(r_j) + ∑i G{ji} p_i, where G_{ji} is the dipole‑dipole interaction tensor, they introduce a Fourier decomposition of the dipole distribution p_j = ∑_ℓ \tilde p_ℓ e^{2πiℓj/N}. This separates the spatial arrangement (indexed by ℓ) from the electron‑induced phase factor χ_ℓ = 2πℓ/N + k a(1/β − sinθ cosφ). Constructive interference occurs when χ_ℓ = 2πn, leading to the generalized emission condition

 sin θ_{nℓ} = (1/β − (n + ℓ/N)) · (λ/a).

When ℓ = 0 the formula reduces to the classic SP law; any non‑zero ℓ opens an additional emission channel. Thus the number of observable directions is controlled by three parameters: the electron speed β, the lattice constant a relative to the wavelength λ, and the array size N. Crucially, the existence of a given channel depends on the Fourier component \tilde p_ℓ of the induced dipole pattern, which in turn is dictated by the distribution of α_j across the array.

To exploit this, the authors propose simple, experimentally feasible dipole‑amplitude modulations of the form |p_j(ξ)| = p_0