Tilted Electron Pulses

We report the all-optical generation and characterization of tilted electron pulses by means of single-cycle terahertz radiation at an electron-transmitting mirror at slanted orientation. Femtosecond electron pulses with a chosen tilt angle are produced at an almost arbitrary target location. The experiments along with theory further reveal that the pulse front tilt in electron optics is directly connected to angular dispersion. Quantum mechanical considerations suggest that this relation is general for particle beams at any degree of coherence. These results indicate that ultrashort electron pulses can be shaped in space and time as versatilely as femtosecond laser pulses, but at ${10}^5$ times finer wavelength and subnanometer imaging resolution.

Figure 1

All-optical generation of tilted electron pulses. (a) Concept and experiment. Electron pulses (blue) pass through a mirror (dark gray) that is irradiated by single-cycle terahertz radiation (red). A traveling interference pattern along the mirror surface converts the electrons into compressed and tilted pulses. (Lower) If the electron beam is focused with the magnetic lens, the tilt angle increases with decreasing beam size, with the opposite sign after its waist. A terahertz-driven resonator (orange) and a phosphor screen (green) provide the necessary metrology to reveal this tilting dynamics by experiment. (b) Definition of angles for the analytical derivations. (c) Amplitude of the longitudinal momentum change as a function of the geometry. (d) Amplitude of the transverse momentum change. (Dotted lines) Angle combinations applied in the experiments; (solid lines) zero-deflection conditions.

Figure 2

Experimental results on tilted electron pulses. (a) Screen images for different mirror slant angles at the zero crossing of the streaking field. Scale bar, 1 mm. The pulse front tilt is evident from the measured lateral arrival time differences (dashed line, time axis). (b) Streaking traces (dots and diamonds) for different $y$ positions within the beam profile and sinusoidal fitting curves (solid). (Inset) Time zero of the streaking traces. (c) Measured arrival time difference (blue dots) and deflection amplitude (red dots). Red line, fit by Eq. (2); squares are excluded. Blue line, prediction by Eq. (5). Error bars indicate one standard deviation. (d) Time-dependent deflection measurements (deflectograms) for different mirror slant angles.

Figure 3

Pulse front tilt vs angular dispersion for electrons, and quantum simulations. (a) Measured angular dispersion in electron pulses as a function of tilt angle (blue) in comparison to this Letter’s conjecture [Eq. (7), magenta]. Error bars indicate one standard deviation and are omitted when smaller than a data point. Unequal scanning ranges of the raw data [Fig. 2] are accounted for by linear interpolation. (b) Quantum simulation of an electron wave packet $\psi (z,y,t)$ with initial position-momentum correlation (left) propagated to the temporal focus $f_{\text{temp}}$ (right). We assume a coherence time of 8 fs, a beam waist of 20 nm, and a linear transverse change of forward momentum, as from the experiment (black arrows). The electron energy is 1 eV in order to make the de Broglie wavelength visible in the results. The phase fronts remain approximately perpendicular to the propagation direction, but the envelope becomes tilted. Angular dispersion is indicated by the black circles and constant throughout the simulation.

Figure 4

Potential applications of tilted electron pulses. (a) Subcycle/attosecond electron diffraction [5]. A crystalline sample (green) is oriented along a zone axis for generating multiple Bragg spots. The laser phase front (violet) has a polarization along the crystal surface. Tilted electron pulses (blue) propagate with sublight speed, but probe the sample in synchrony with the field cycles. (b) Subcycle low-energy electron diffraction (LEED). Laser cycles (violet) illuminate a sample material (green) at an angle due to practical space restrictions [26]. Tilted electron pulses (blue) can probe every part of the surface with the same delay with respect to the field cycles. (c) Subcycle electron microscopy. A transmission electron microscopy grid (TEM grid) can usually not be tilted a lot with respect to the twin lens in a high-resolution microscope. Nevertheless, a combination of tilted electron pulses (blue) and diagonal laser incidence (violet) can provide a uniform subcycle-level delay over the entire illuminated TEM grid area.