Wake excited in plasma by an ultrarelativistic pointlike bunch

We study propagation of a relativistic electron bunch through a cold plasma assuming that the transverse and longitudinal dimensions of the bunch are much smaller than the plasma collisionless skin depth. Treating the bunch as a point charge and assuming that its charge is small, we derive a simplified system of equations for the plasma electrons and show that, through a simple rescaling of variables, the bunch charge can be eliminated from the equations. The equations demonstrate an ion cavity formed behind the driver. They are solved numerically and the scaling of the cavity parameters with the driver charge is obtained. A numerical solution for the case of a positively charged driver is also found.

Figure 1

Ballistic trajectories in the limit $\mathcal{A}\ll 1$. The envelope of the trajectories is shown by the red line. There are no electrons in the region below the envelope.

Figure 2

Electron trajectories in scaled variables $r/\sqrt{\nu }$ and $\xi$ showing a bubble after the point driver.

Figure 3

A small number of selected trajectories showing a transition from an envelope bubble boundary to a single trajectory. The thick blue line shows the ballistic approximation, Eq. (15), for the boundary.

Figure 4

Electron density distributions in radial direction for various values of $\xi$: (1) $\xi =0.5$, (2) $\xi ={\xi }_m=1.57$, (3) $\xi =2.5$, (4) $\xi =3$, (5) $\xi =3.7$.

Figure 5

Comparison of the analytical expression for the bubble boundary (25) and (28) (red dots) with the numerical solution (solid line).

Figure 6

Electric field component $E_{z1}$ (blue line) and the total field $E_z$ (black line) on the axis of the bubble. The plot of $E_z/\nu$ corresponds to $\nu =0.1$. The fields are normalized by parameter $\nu$.

Figure 7

Electron orbits in plasma for a positive driver.

Figure 8

Electron density distributions in radial directions for various values of $\xi$: (1) $\xi =1$, (2) $\xi =2.5$, (3) $\xi =3$, (4) $\xi =4$, (5) $\xi =8$. Note that curve 5 has a maximum outside of the plot area and is seen as two light blue lines (each one indicated by number 5).

Figure 9

Two ballistic electron orbits starting from the initial radial positions $r_0$ and $r_0+d{r}_0$ after interaction with the driver.