Attosecond control of electron beams at dielectric and absorbing membranes

Ultrashort electron pulses are crucial for time-resolved electron diffraction and microscopy of the fundamental light-matter interaction. In this work, we study experimentally and theoretically the generation and characterization of attosecond electron pulses by optical-field-driven compression and streaking at dielectric or absorbing interaction elements. The achievable acceleration and deflection gradient depends on the laser-electron angle, the laser's electric and magnetic field directions, and the foil orientation. Electric and magnetic fields have similar contributions to the final effect and both need to be considered. Experiments and theory agree well and reveal the optimum conditions for highly efficient, velocity-matched electron-field interactions in the longitudinal or transverse direction. We find that metallic membranes are optimum for light-electron control at mid-infrared or terahertz wavelengths, but dielectric membranes are excellent in the visible and near-infrared regimes and are therefore ideal for the formation of attosecond electron pulses.

Figure 1

Concept of attosecond control of free electrons at dielectric or absorbing membranes. Without a membrane (green), the electron momentum (black dotted curve) returns to the initial value ($p_0$) after interaction with a laser field. In contrast, the electron acquires a finite momentum $\mathrm{\Delta }p$ (red curve) in the presence of a membrane, mainly due to a phase shift of the oscillations caused by the refractive index and thin-film interferences. The actual momentum shift is a vectorial quantity that depends on various angles and polarizations, as investigated in this work.

Figure 2

Theory for laser streaking and acceleration or deceleration of free electrons with dielectric foils. (a) Angle definitions. (b) Time-frozen snapshot of a $p$-polarized electric field around a 60-nm-thin Si membrane. There is a phase shift (dotted lines) between the crests in front and behind the foil. (c) Peak deflection and peak acceleration for a 60-nm-thick Si foil. (d) Peak deflection and peak acceleration for a 50-nm-thick $\mathrm{S}{\mathrm{i}}_3{\mathrm{N}}_4$ foil. The incident laser field has a wavelength of 1030 nm and a peak field strength of 1 GV/m and 3.3 T. White dotted lines represent angle combinations that provide velocity matching (V.M.) or Brewster's angle (B.A.). The white dashed line (Exp) denotes the conditions used in most of the experiments. (e) Mean deflection as a function of refractive index. (f) Mean deflection as a function of membrane thickness. (g) Mean deflection as a function of laser wavelength.

Figure 3

Contribution of the effects outside and inside of the membrane. (a) Acceleration approximated by assuming no effect inside the foil. (b) Acceleration with full calculation. (c) Deflection with no effect inside the foil. (d) Deflection with full calculation. We assume a $p$-polarized incident field of 1 GV/m at 1030-nm wavelength and a 60-nm-thin Si membrane. There is almost no difference; see Sec. 2e.

Figure 4

Experimental results on attosecond sideways deflection. (a) Experimental setup. $\lambda /2$, half-wave plate; BBO, $\beta$-barium-borate crystal for second-harmonic generation; yellow, photocathode. (b) Photographs of the freestanding membranes (dark-field microscopy). Scale bar, 100 µm. (c) Electron beam profile. (d) Electron beam profile with a deflecting laser field. Blue dotted line, simulation. (e) Polarization dependence of the deflection along $x$ (red dots) and $y$ (blue squares) in comparison to theory (dotted black line). (f) Measured deflection (symbols) as a function of the incident peak field strength in vacuum for different foil materials in comparison to theory (dotted lines). (g) Dependence of deflection rate on foil thickness $d$ and refractive index $n$. (h) Measured deflection amplitude (red circles) for a Si membrane as a function of the foil angle in comparison to theory (solid line). (i) Measured deflection amplitude (blue squares) for a 50-nm-thick $\mathrm{S}{\mathrm{i}}_3{\mathrm{N}}_4$ membrane in comparison to theory (solid line).

Figure 5

Attosecond deflection of Bragg spots. The images have a size of $\pm 0.3\times \pm 0.3\mathrm{mra}{\mathrm{d}}^2$ and show Bragg diffraction from a single-crystalline Si membrane at $\left[-1/\sqrt{2},\sqrt{2},1/\sqrt{2}\right]$ orientation. The white stripes denote cutout regions without signal. The Miller indices are shown in each panel. The incident field strength is 0.27 GV/m; the white arrow denotes the polarization direction.

Figure 6

Attosecond electron pulses. (a) Experimental setup. $\lambda /2$, half-wave plate; L, lens; PB, polarizing beam splitter; BS, beam splitter; CCD, auxiliary camera. (b) Time evolution of the electron density with propagation distance. (c) Snapshots of simulated two-dimensional electron densities. (d) Laser interferogram on the CCD. (e) Interferometric stability of the optical setup. (f) Measured deflectograms at compression strengths of 30 MV/m, 60 MV/m, and 90 MV/m. (g) Temporal structure of compressed electron pulses at 60 MV/m. The solid curve is the result of global fitting. Circles and squares represent temporal shapes directly given by the deflectogram (see Sec. 4). (h) Measured pulse duration as a function of compression field amplitude. Dotted curves are for eye guiding. (i) Space-time couplings and origin of the asymmetry in some data. (j) Deflectogram simulated by including some minor time-dependent deflection at the compression foil.

Figure 7

Comparison of metallic, absorbing, and dielectric foils. (a)–(c) Time-frozen snapshots of $p$-polarized electric fields around 100-nm-thick aluminum, graphite, and $\mathrm{S}{\mathrm{i}}_3{\mathrm{N}}_4$ membranes located at $z=0$. (d)–(f) Peak acceleration for 70-keV electrons. (g)–(i) Peak deflection. The white dotted curves represent configurations with velocity matching. (j) Mean deflection as a function of thickness for measured damage thresholds. (k) Mean deflection as a function of laser wavelength, also for measured damage thresholds. (l) Reflectivity of materials as a function of the real and imaginary part of the refractive index $n+i\kappa$. The reflectivity is given for 1030-nm light and normal incidence.

Figure 8

Effect of magnetic fields. (a)–(c) Deflection $\mathrm{\Delta }{p}_x$ with $p$-polarized light and a 60-nm-thick Si foil. (a) Electric part only. (b) Magnetic part only. (c) Complete effect. Dotted curves V.M. represent the electron-laser velocity-matching condition. (d)–(f) Deflection $\mathrm{\Delta }{p}_y$ with $s$-polarized light. (d) Electric part only. (e) Magnetic part only. (f) Complete effect. All simulations are for 60-nm-thick Si and 1 GV/m laser fields at 1030-nm wavelength.

Figure 9

Calculation scheme for obtaining electromagnetic fields inside and outside of a dielectric membrane (green). See Appendix pp1 for definitions of the position and wave vectors $\mathrm{\Delta }\mathbf{x}$, $k$ and the relative field amplitudes r, t.

Figure 10

Velocity mismatch and pulse front tilt. (a) Definition of tilt of compressed electrons (see Appendix pp5). (b) Velocity-mismatching effect as a function of laser and foil angles. (c) Tilt angle. Dotted line, velocity-matching condition; dashed line, condition in most of the reported experiments. (d) Tilt changes by converging or diverging beams.