Filamentation of a relativistic proton bunch in plasma

We show in experiments that a long, underdense, relativistic proton bunch propagating in plasma undergoes the oblique instability, which we observe as filamentation. We determine a threshold value for the ratio between the bunch transverse size and plasma skin depth for the instability to occur. At the threshold, the outcome of the experiment alternates between filamentation and self-modulation instability (evidenced by longitudinal modulation into microbunches). Time-resolved images of the bunch density distribution reveal that filamentation grows to an observable level late along the bunch, confirming the spatiotemporal nature of the instability. We provide a rough estimate of the amplitude of the magnetic field generated in the plasma by the instability and show that the associated magnetic energy increases with plasma density.

Figure 1

Schematic of the AWAKE experimental setup (not to scale). Insets: Typical images of the bunch with neither plasma nor gas in the source. (a), (b) Transverse time-integrated images of the bunch at screens A and B, respectively. (c) Time-resolved image at screen C. Solid black lines in the insets: Transverse projections of the distributions. Red dashed lines: Gaussian fits of the transverse projections. Dashed black line in inset (c): Longitudinal projection.

Figure 2

Transverse, time-integrated, single-event images of the $p^{+}$ bunch after propagation in plasma at screen B (see Fig. 1) with $n_{pe}=9.38\times {10}^{14}{\mathrm{cm}}^{-3}$ and ${\sigma }_{r0}/\delta =2.9$. Black lines: Lineouts along the corresponding dashed gray lines. Color map and lineouts chosen to enhance the visibility of the filaments ($\ge 2\mathrm{nC}/{\mathrm{mm}}^2$).

Figure 3

The ns-scale, time-resolved, single-event image at screen C (see Fig. 1) for the event (b) of Fig. 2. $n_{pe}=9.38\times {10}^{14}{\mathrm{cm}}^{-3}, {\sigma }_{r0}/\delta =2.9$. Front of the bunch at $t<0$. Black dashed and continuous lines: transverse projection for $-0.1<t<-0.06\mathrm{ns}$ (front of the bunch, along the dashed gray line) and $0.24<t<0.28\mathrm{ns}$ (back of the bunch, along the continuous gray line), respectively. Red lines: Transverse projections of Fig. 1, for the same $t$ along the bunch.

Figure 4

(a), (c) Transverse, time-integrated, single-event images of the $p^{+}$ bunch at screen B (see Fig. 1) after propagation in plasma with $n_{pe}=2.25\times {10}^{14}{\mathrm{cm}}^{-3}, {\sigma }_{r0}/\delta =1.5$. Black lines: Lineout along the corresponding dashed gray lines. (b), (d) The ps-scale, time-resolved, single-event images at screen C, corresponding to (a) and (c), respectively. Black lines: On-axis longitudinal distributions obtained by summing counts over $-0.217\le y\le 0.217\mathrm{mm}$. (e) Average power spectra from DFT of on-axis distributions of single-event images [as in (b) and (d)]. Average amplitude of the distributions subtracted to remove the peak at $f=0$. Red line: Average of the power spectra of 20 events showing a single “filament” on screen B [e.g., (a)]. Black line: Average of the power spectra of 23 events showing multiple filaments on screen B [e.g., (c)]. Shaded areas show the extent of the rms variations over the events.

Figure 5

Magnitude of the transverse magnetic field calculated from the images in Figs. 2–2. $n_{pe}=9.38\times {10}^{14}{\mathrm{cm}}^{-3}$ and ${\sigma }_{r0}/\delta =2.9$.

Figure 6

(a) Magnetic energy within the bunch volume as a function of $n_{pe}$. Black symbols: Average of consecutive images; error bars show the rms variation over the events in each dataset. Dashed line: Result of the fit with function $\propto exp\left(n_{pe}^{1/3}\right)$ to the cases where filamentation occurs. Red symbols: Minimum value for each value of $n_{pe}$. (b) Average number of filaments within transverse, time-integrated images at screen B; error bars show the rms variation over the events in each dataset. For $n_{pe}=2.25\times {10}^{14}{\mathrm{cm}}^{-3}$, only events showing multiple filaments (see Fig. 4) are considered.