Observation of the Relativistic Reversal of the Ponderomotive Potential

The secular dynamics of a nonrelativistic charged particle in an electromagnetic wave can be described by the ponderomotive potential. Although ponderomotive electron-laser interactions at relativistic velocities are important for emerging technologies from laser-based particle accelerators to laser-enhanced electron microscopy, the effects of special relativity on the interaction have only been studied theoretically. Here, we use a transmission electron microscope to measure the position-dependent phase shift imparted to a relativistic electron wave function when it traverses a standing laser wave. The kinetic energy of the electrons is varied between 80 and 300 keV, and the laser standing wave has a continuous-wave intensity of $175{\mathrm{GW}/\mathrm{cm}}^2$. In contrast to the nonrelativistic case, we demonstrate that the phase shift depends on both the electron velocity and the wave polarization, confirming the predictions of a quasiclassical theory of the interaction. Remarkably, if the electron’s speed is greater than $1/\sqrt{2}$ of the speed of light, the phase shift at the electric field nodes of the wave can exceed that at the antinodes. In this case there exists a polarization such that the phase shifts at the nodes and antinodes are equal, and the electron does not experience Kapitza-Dirac diffraction. Our results thus provide new capabilities for coherent electron beam manipulation.

Figure 1

(a) Schematic. An electron beam intersects a standing laser wave (electric field shown in magenta, magnetic field shown in cyan) formed between the two mirrors (blue cylinders) of a Fabry-Pérot optical cavity. The dimensions of the cavity are not shown to scale. The polarization axis of the standing wave (magenta arrows) makes an angle $\theta$ with the electron beam axis. The polarimeter is tilted by an angle $\xi$ relative to the electron beam axis. (b) Phase modulation detection scheme. The electron beam crosses the standing laser wave after passing through a focus. The interaction with the standing wave imprints a spatial phase modulation on the electron beam which is converted to intensity modulation as the electron beam propagates to the image plane. (c) Ronchigram of the standing laser wave. The direct electron detection camera records the number of electrons landing on each of its pixels. In the image plane, the standing wave structure of the phase modulation manifests as a series of bright and dark fringes.

Figure 2

(a) Relative phase modulation depth as a function of polarization angle. Measurements of the relative phase modulation depth (dots) are plotted along with fitted theory curves (lines) as a function of the laser wave polarization angle $\theta$ for several values of the nominal electron energy $K$ (equivalently, electron speed $\beta$). The fitted theory (solid lines) is described by Eq. (7). (b) Relative phase modulation depth as a function of electron speed. Fit values for the relative phase modulation depth at $\theta =0$ are shown as a function of ${\beta }^2$ for each of the nine electron energies examined (black dots). The theoretical dependence on ${\beta }^2$ predicted by Eq. (7) is shown in red. (c) Relativistic reversal. Ronchigram standing wave fringes are shown as a function of polarization angle $\theta$ and position along the laser beam axis $x$ for the $K=300\mathrm{keV}$ data set used in panel (a). The bright (red) and dark (blue) fringes reverse positions around the relativistic reversal angle ${\theta }_r$ (horizontal black line). The vertical lines are a guide to the eye.