Vacuum acceleration of electrons in a dynamic laser pulse

A planar laser pulse propagating in vacuum can exhibit an extremely large ponderomotive force. This force, however, cannot impart net energy to an electron: As the pulse overtakes the electron, the initial impulse from its rising edge is completely undone by an equal and opposite impulse from its trailing edge. Here we show that planarlike “flying focus” pulses can break this symmetry, imparting relativistic energies to electrons. The intensity peak of a flying focus—a moving focal point resulting from a chirped laser pulse focused by a chromatic lens—can travel at any subluminal velocity, forward or backward. As a result, an electron can gain enough momentum in the rising edge of the intensity peak to outrun and avoid the trailing edge. Accelerating the intensity peak can further boost the momentum gain. Theory and simulations demonstrate that these dynamic intensity peaks can backwards accelerate electrons to the MeV energies required for radiation and electron diffraction probes of high energy density materials.

Figure 1

(a) A typical luminal intensity peak in vacuum. The electron, shown as a red dot, experiences equal and opposite ponderomotive accelerations on the leading and falling edges of the pulse, respectively, and gains no net energy. (b) A positively chirped flying focus with a subluminal intensity peak. After forward acceleration in the leading edge of the intensity peak, the electron outruns the peak and retains the energy it gained. (c) A negatively chirped flying focus with a subluminal intensity peak that travels in the opposite direction of the pulse. After backward acceleration in the leading edge, the electron outruns the intensity peak and retains the energy it gained.

Figure 2

Final momentum of an electron accelerated in a backward-propagating flying focus intensity peak. Below the cutoff vector potential ($a_c$), an electron acquires a velocity insufficient to outrun the intensity peak. Above the cutoff, an accelerated electron can outrun the intensity peak, and the final momentum is independent of $a_0$.

Figure 3

Trajectories of an electron (black dashed lines) and intensity peak (contours) in the Lorentz frame of the intensity peak, (a) and (b), and the lab frame, (c) and (d). In (a) the normalized vector potential exceeds the cutoff, and the ponderomotive potential reflects the electron, resulting in net momentum gain in the lab frame (c). In (b) the ponderomotive potential is too weak to reflect the electron and it returns to rest in the lab frame (d).

Figure 4

Path of an electron (black dashed line) and a trajectory-locked intensity peak (contours) in the Lorentz frame associated with the initial velocity of the intensity peak. The intensity peak is accelerated to keep the electron at the initial cutoff vector potential as its momentum increases.

Figure 5

(a) Comparison of the momentum gained by an electron in trajectory-locked and constant velocity intensity peaks. An electron exits the constant velocity peak and reaches its asymptotic momentum, $2{\beta }_{I0}{\gamma }_{I0}^2$, at time $T_E$. In the trajectory-locked peak, the electron reaches a momentum $2{\beta }_{I0}{\gamma }_{I0}^2$ at the breakeven time $T_{BE}$ and continues to accelerate. (b) An electron in a trajectory-locked peak reaches a momentum $2{\beta }_{I0}{\gamma }_{I0}^2$ in a fraction of the time it would take in the constant velocity peak over a wide range of parameters. (c) At the constant velocity exit time, an electron in a trajectory-locked peak typically reaches a higher momentum than one in a constant velocity peak.