Coherent regime of Kapitza-Dirac effect with electrons

TL;DR

The study demonstrates elastic coherent Kapitza-Dirac diffraction of fast electrons in vacuum by a standing optical wave inside a scanning electron microscope, using electrons with kinetic energies of $20~\mathrm{keV}$ and $30~\mathrm{keV}$ (de Broglie wavelengths of approximately $8.5~\mathrm{pm}$ and $7~\mathrm{pm}$ respectively) interacting with a standing wave of wavelength $\lambda=1030~\mathrm{nm}$. The optical standing wave is generated by two counterpropagating pulses, and the electron pulses are photoemitted and synchronized with the laser to enable precise time and space overlap, with CBED resolving individual photon sidebands separated by $2\hbar k$ where $k=2\pi/\lambda$. In the coherent regime, the electron-wave populations follow the Jacobi-Anger expansion and the amplitudes are governed by $J_n(\beta)$ with coupling parameter $\beta$, leading to coherent, reversible oscillations among diffraction orders as $\beta$ increases; the incoherent regime shows broadening without oscillations due to a spread in $\beta$. This work establishes a Raman–Nath-type, elastic Kapitza–Dirac interaction for high-energy electrons in vacuum and demonstrates a potential coherent beam splitter or phase plate for electron microscopes, enabling time-resolved and light–matter–coupled nanoscale interferometry.

Abstract

Electron matter waves coherently diffract when passing through a periodic structure of light formed by two interfering light waves. In this so-called Kapitza-Dirac effect, the electron momentum changes due to absorption and emission of photons via stimulated Compton scattering. Until now, the effect has only been observed with low energy electrons due to the small momentum of a visible photon compared to the momentum of high energy electron leading to diffraction angles of 10^(-4) rad or smaller. We report on the observation of the Kapitza-Dirac effect in a scanning electron microscope using high energy (20 keV and 30 keV) electrons with de-Broglie wavelengths of 9 pm and 7 pm, respectively. The photon sidebands in the electron transverse momentum spectrum are detected in the convergent beam diffraction geometry using spatial filtering. As the coupling strength between the electrons and the light field increases, the sideband populations exhibit coherent, reversible oscillations among diffraction orders. The effect can serve as a coherent electron beam-splitter or a phase-plate in various types of electron microscopes.

Figures (3)

• Figure 1: (a) Layout of the experimental setup used for demonstration of Kapitza-Dirac effect with fast electrons based on scanning transmission electron microscopy combined with convergent beam electron diffraction (STEM-CBED). Pulsed convergent electron beam diffracts at an optical standing wave placed upstream the beam focus. A nanoslit with the width of 70 nm is placed to the electron beam focus to spatially filter a narrow distribution of electron transverse momentum. The electron diffraction pattern is measured by scanning the electron beam and detecting the electrons transmitted through the slit.
• Figure 2: Raman-Nath (diffraction) regime of Kapitza-Dirac effect with electron kinetic energy of 20 keV. (a) Electron transverse scattering patterns as a function of the average laser power and maximum coupling constant $\beta_{\text{max}}$. (b) Normalized measured electron scattering patterns for selected values of $\beta_{\text{max}}$ (points) compared with the theory (dashed curves). Data are vertically shifted for clarity.
• Figure 3: Coherent Raman-Nath (diffraction) regime of Kapitza-Dirac effect measured with electron kinetic energy of 30 keV.(a) Measured and (b) calculated electron transverse scattering patterns as a function of the average laser power and coupling constant $\beta_{\text{max}}$. (c) Normalized measured electron scattering patterns for selected values of $\beta_{\text{max}}$ corresponding to the dashed lines in (a) (points) compared to theoretical simulations (dashed curves). Data are vertically shifted for clarity. (d) Measured populations of the individual diffraction orders (black squares corresponds to sideband $n=1$, red circles $n=2$, blue triangles $n=3$ and green triangles $n=4$) as a function of the laser power and coupling constant $\beta_{\text{max}}$ compared to the theoretical data (dashed curves). The population of each peak with $n\neq0$ is calculated as an average of populations in peaks $+n$ and $-n$.