Seeding of Self-Modulation Instability of a Long Electron Bunch in a Plasma

We demonstrate experimentally that a relativistic electron bunch shaped with a sharp rising edge drives plasma wakefields with one to seven periods along the bunch as the plasma density is increased. The plasma density is varied in the ${10}^{15}–{10}^{17}{\mathrm{cm}}^{-3}$ range. The wakefields generation is observed after the plasma as a periodic modulation of the correlated energy spectrum of the incoming bunch. We choose a low bunch charge of 50 pC for optimum visibility of the modulation at all plasma densities. The longitudinal wakefields creating the modulation are in the MV/m range and are indirect evidence of the generation of transverse wakefields that can seed the self-modulation instability, although the instability does not grow significantly over the short plasma length (2 cm). We show that the seeding provides a phase reference for the wakefields, a necessary condition for the deterministic external injection of a witness bunch in an accelerator. This electron work supports the concept of similar experiments in the future, e.g., SMI experiments using long bunches of relativistic protons.

Figure 1

(a) Image of the long bunch dispersed in energy a short distance downstream from the mask. (Two low-energy electron bunches not used for these experiments are shown early in time and in the front of the bunch and are to be disregarded.) The experimental traces indicate the sum of the image signal over the vertical and horizontal axes, respectively, and are displayed on the same scale. The energy range selected by the mask is $\mathrm{\Delta }{E}_0$. The bunch length is $L_{\text{beam}}$, and the length scale for the rising edge is $\mathrm{\Delta }L$. (b) Ratio of the normalized longitudinal wakefield amplitude ratio $E_z(\mathrm{\Delta }L)/E_{z0}$ as a function of the length of front rising edge $\mathrm{\Delta }L$. Here, $E_{z0}$ is the result for a step function (when $\mathrm{\Delta }L$ tends to zero). The profile of the bunch’s rising front is modeled as a cosine function over $\mathrm{\Delta }L$ to the value $n_{b0}$. All results are calculated with 2D linear theory.

Figure 2

Energy spectra obtained at various plasma densities. Spectra (a) with $n_e=0$ (no plasma) and (b)–(h) with increasing plasma densities between $2\times {10}^{15}$ and $8\times {10}^{16}{\mathrm{cm}}^{-3}$ obtained in the experiment. The white lines indicate the sum of the images over the vertical image dimension (no dispersion), and the red lines show the positions of the density peaks as identified for Fig. 3. Figure (i) shows a simulated energy spectrum for the case $L_{\text{beam}}/{\lambda }_{\text{pe}}=2$, very similar to the experimental case of (c). The color tables are deliberately chosen different to avoid possible confusion between experimental and simulation results.

Figure 3

Energy of the peaks identified by the red lines on Figs. 2 as a function of plasma density. The yellow zone corresponds to the FWHM of the incoming bunch energy and suggests that peaks are missing in the back, high-energy part of the bunch for $n_e>0.5\times {10}^{16}{\mathrm{cm}}^{-3}$.

Figure 4

(a) Peak accelerating field along the bunch versus propagation distance in the simulation for the electron bunch with 50 pC charge and plasma densities such that $L_{\text{beam}}/{\lambda }_{\text{pe}}=1$ (red), 2 (blue), 3 (green), 4 (black), 5 (purple), 6 (orange), and 7 (cyan line). The initial $E_z(z=0)$ values are 3.7, 3.2, 2.7, 2.2, 1.9, 1.6, $1.3\mathrm{MV}/\mathrm{m}$ for $l=1$ to 7, respectively. (b) Similar figure for the 1 nC bunch for $L_{\text{beam}}/{\lambda }_{\text{pe}}=2$. The vertical dotted green lines indicate the length of the plasma in the experiment $L_p=2\mathrm{cm}$.