Active steering of cathodoluminescence through a generalized Smith-Purcell effect

TL;DR

The paper tackles active control of cathodoluminescence (CL) by unifying the Smith-Purcell (SP) effect with finite, nonuniform metasurface arrays. It derives a generalized SP condition, $\sin\theta_{n\ell} = \frac{1}{\beta} - \left(n - \frac{\ell}{N}\right)\frac{\lambda}{a}$, revealing new radiative channels indexed by $(n,\ell)$ that arise when array elements have nonidentical dipolar responses. By engineering the Fourier content of the array’s dipoles, the authors demonstrate active steering of CL through harmonic p-j distributions and validate this with VO$_2$ disk and graphene ribbon platforms, achieving targeted emission angles in the THz–visible range. The framework enables programmable, electron-driven light sources and points to scalable extensions to 2D arrays and holographic, ultrafast photonic devices using tunable materials and inverse-design approaches.

Abstract

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.

Figures (8)

• Figure 1: Generalized Smith-Purcell emission.(a) Artistic sketch of the considered system, composed of a periodic array of scatterers interacting with an electron beam moving parallel to the array. (b) Simplified scheme of the sketch in (a), where we indicate the period $a$ of the array, the electron (e$^-$) velocity $v$ and impact parameter $b$. We further indicate the adopted coordinate system and the polar and azimuthal angles $\theta$ and $\phi$ for a general position vector ${\bf r}$. All scatterers are taken as point dipoles with dipole moments indicated by red arrows. (c) Generalized Smith-Purcell (GSP) emission condition in the $\phi=0$ plane, given by Eq. \ref{['eq:SPcondition']}, for an electron with velocity $v=0.1\,c$ passing near an array with $N=51$ elements and period indicated by the legend. The SP order $n$ for each curve is marked next to it, while all remaining values of $n$ lie outside the plot region for the considered values of $a/\lambda$. The gray areas indicate the regions where $|\sin \theta_{n\ell}|> 1$ and therefore constructive interference is kinematically forbidden. Dots inside (outside) this region are marked as unfilled (filled), and the corresponding emission angle $\theta_{n\ell}$, when allowed, is denoted along the right axis. (d,e) Schematic illustrations of the available GSP emission channels for each of the color-coordinated arrays in (c). Each arrow corresponds to a specific $n,\ell$ pair (see labels) and is represented outgoing at the $\theta_{n\ell}$ angle. In (d), the $\ell=0$ case is highlighted by a thicker arrow, denoting the conventional SP emission of a uniform array.
• Figure 2: Cathodoluminescence steering with non-uniform arrays.(a) GSP condition in Eq. \ref{['eq:SPcondition']} for $n=1$ and $\ell=0-10$ in an array with $N=51$ elements and period $a/\lambda = 0.090$, excited by an electron beam with velocity $v=0.1\,c$ ($\approx 2.6$ keV). Modes $\ell=1-9$ lie within the region $|\sin \theta_{1,\ell}|\leq 1$ and are marked by filled circles, whereas modes $\ell=0$ and $\ell=10$ lie outside that region and are marked with empty circles. Each mode is targeted by the same-color induced dipole moment distribution in (b) that selects the mode $\ell=\xi$. (b) Induced dipole moment distributions following Eq. \ref{['eq:pjharmonic']} (with oscillation amplitude $A=1$) for the color-coded values of $\xi$ indicated by the legend, as a function of array element index $j$. Each distribution of induced dipole moment $|{\bf p}_j^0|/p_0$ oscillates around $1$ between $0$ and $2$ with a frequency $\xi$. (c) Angle-resolved cathodoluminescence far-field emission amplitude $|{\bf f}(\theta)|$ polar plot, normalized to $f_0=Nk^2p_0$, as a function of the $\theta$ angle in the $\phi=0$ plane, for an electron passing above an array whose induced dipole moments are given by the same-color curve in (b) and polarized along $x$ (${\bf p}_j^0 \parallel \hat{\bf x}$). The colored dashed lines mark the position of the target angles $\theta_{1,\xi}$ for $\xi=1-9$ and the shaded gray area represents the condition $|{\bf f}(\theta)|/f_0 > \cos(\theta)$. (d) Peak emission angle $\theta_{\rm max}$ (right axis) and corresponding peak far-field amplitude $|{\bf f}(\theta_{\rm max})|/f_0$ (left axis) as a function of the target emission frequency $\theta$ for $\xi=1-9$ (see top axis). The dashed gray curve represents the function $|{\bf f}(\theta)|/f_0=\cos(\theta)/2$ and the yellow dashed line corresponds to the condition $\theta_{\rm max}=\theta$.
• Figure 3: Active tuning of CL emission.(a) Scheme of an array of VO$_2$ disks with diameter $D=250~$nm and thickness $t=2$ nm, separated from their nearest neighbors by a center-to-center distance $a=450~$nm, with an electron passing parallel to the array at a distance $b=10$ nm and with velocity $v=0.1\,c$. The array is illuminated by a pump beam (in red) that is spatially engineered to deliver a fluence $F_j$ at the $j$th array element. (b) Induced dipole moment at a wavelength $\lambda=2\pi c/\omega=5.0~\mathrm{\mu m}$ (top) along different array elements, following Eq. \ref{['eq:pjharmonic']} with $\xi$ values as indicated by the labels, $A=1$, and $p_0=5.4\times 10^{-3} (eD/\omega)$, as a function of element $j$, for the color-coordinated fluence distributions (bottom), ranging between $F_{\rm min}=0.25~\mathrm{J/cm^2}$ and $F_{\rm max}=0.55~\mathrm{J/cm^2}$. (c) Far-field emission distribution for the same-color array distributions in (b), with $f_0=2.7\times 10^{-2}(e/D\omega)$. (d--f) Same as (a--c), but for an array of graphene ribbons with width $W=100$ nm whose $j$th element's Fermi level is set to $E_{{\rm F}j}$ (as depicted by the Dirac cones) between $E_{\rm F,min}=0.1$ eV and $E_{\rm F,max}=0.5$ eV, with a period $a=615~\mathrm{nm}$ and evaluated at a wavelength $\lambda=6.8~\mathrm{\mu m}$. In (e), the dipole moment corresponds to the $q=0$ component, with $p_0=4.48(eW/\omega)$. In (f), we have $f_0=1.95/(e/W\omega)$.
• Figure 4: Cathodoluminescence steering with large non-uniform arrays.(a) GSP condition in Eq. \ref{['eq:SPcondition']} for $n=1$ and $\ell=1-20$ in an array with $N=101$ elements and period $a/\lambda = 0.090$, excited by an electron beam with velocity $v=0.1c$ ($\approx 2.6$ keV). Modes $\ell=2-19$ lie within the region $|\sin \theta_{1,\ell}|\leq 1$ and are marked by filled circles, whereas modes $\ell=1$ and $\ell=20$ lie outside the same region and are marked with empty circles. (b) Angle-resolved cathodoluminescence far-field emission amplitude $|{\bf f}(\theta)|$ polar plot, normalized to $f_0=Nk^2p_0$, as a function of $\theta$ angle in the $\phi=0$ plane, for an electron passing over an array whose induced dipole moments are given by Eq. \ref{['eq:pjharmonic']} with $\xi$ as indicated in the legend and polarized along $x$ (${\bf p}_j^0 \parallel \hat{\bf x}$). The colored dashed lines mark the position of the target angles $\theta_{1,\xi}$ for $\xi=2-19$ and the shaded gray area represents the condition $|{\bf f}(\theta)|/f_0 > \cos(\theta)$. (c) Peak emission angle $\theta_{\rm max}$ (right axis) and corresponding peak far-field amplitude $|{\bf f}(\theta_{\rm max})|/f_0$ (left axis) as a function of the target emission frequency $\theta$ for $\xi=2-19$ (see top axis). The dashed gray curve represents the function $|{\bf f}(\theta)|/f_0=\cos(\theta)/2$ and the yellow dashed line represents the condition $\theta_{\rm max}=\theta$. (d,e) Same as (b,c), respectively, but for dipoles polarized along $z$ (${\bf p}_j^0 \parallel \hat{\bf z}$). In (d), the shaded gray area represents the condition $|{\bf f}(\theta)|/f_0 > \sin(\theta)$. In (e), the dashed gray curve represents the function $|{\bf f}(\theta)|/f_0=\sin(\theta)/2$.
• Figure 5: Properties of the CL far-field emission profile.(a) Induced dipole distribution profile following Eq. \ref{['eq:pjharmonic']} with $\xi=4$, with the parameter $A$ ranging from 0 to 1 (see legend). (b) CL angle-resolved far-field distribution $|{\bf f}(\theta)|$ at $\phi=0$ for each of the color-coded distributions in (a). (c,d) Same as (a,b), respectively, but for induced dipole profiles following Eq. \ref{['eq:pjharmonic']} with $A=1$ for $\xi=3$, $\xi=7$, and the averaged superposition of both profiles, as shown in the legend.