Energy loss of a high charge bunched electron beam in plasma: Simulations, scaling, and accelerating wakefields

The energy loss and gain of a beam in the nonlinear, “blowout” regime of the plasma wakefield accelerator, which features ultrahigh accelerating fields, linear transverse focusing forces, and nonlinear plasma motion, has been asserted, through previous observations in simulations, to scale linearly with beam charge. Additionally, from a recent analysis by Barov et al., it has been concluded that for an infinitesimally short beam, the energy loss is indeed predicted to scale linearly with beam charge for arbitrarily large beam charge. This scaling is predicted to hold despite the onset of a relativistic, nonlinear response by the plasma, when the number of beam particles occupying a cubic plasma skin depth exceeds that of plasma electrons within the same volume. This paper is intended to explore the deviations from linear energy loss using 2D particle-in-cell simulations that arise in the case of experimentally relevant finite length beams. The peak accelerating field in the plasma wave excited behind the finite-length beam is also examined, with the artifact of wave spiking adding to the apparent persistence of linear scaling of the peak field amplitude into the nonlinear regime. At large enough normalized charge, the linear scaling of both decelerating and accelerating fields collapses, with serious consequences for plasma wave excitation efficiency. Using the results of parametric particle-in-cell studies, the implications of these results for observing severe deviations from linear scaling in present and planned experiments are discussed.

Figure 1

The longitudinal field profile given by PIC simulation for PWFA excitation in the blowout regime. The decelerating field, peak accelerating field, and a defined “useful field,” which avoids the narrow spike region, are indicated in the drawing.

Figure 2

Configuration space from MAGIC cylindrically symmetric PIC simulation with $\tilde{Q}\cong 20$, $k_p{\sigma }_z=0.11$, $k_{p}a=0.2$, and beam center at $z=1.33\mathrm{cm}$. Color code indicates electron positions that have relativistic positive momenta $p_z/m_{e}c>1$ in black, with all other plasma electrons colored red. The initially accelerated plasma electrons are just ahead of the blowout region, where radial motion moves the electrons away from the beam channel.

Figure 3

OOPIC simulation of the $\tilde{Q}=200$ case, drive beam having uniform distribution of width $\tilde{a}=0.2$ and length ${\tilde{l}}_z=0.1$. False color (a) plasma longitudinal current density (color map key on right indicates current density in $\mathrm{A}/{\mathrm{m}}^2$) and (b) plasma electron density (color map key on right, ambient level is at 15, indicated in blue).

Figure 4

Current densities just behind driving beam with $\tilde{Q}=0.2$, uniform distribution of width $\tilde{a}=0.2$, and length ${\tilde{l}}_z=0.1$. Analytical prediction ${\tilde{J}}_z=\frac{1}{2}{\tilde{J}}_r^2$ from theory also shown.

Figure 5

Current densities just behind driving beam with $\tilde{Q}=0.2$, uniform beam of width $\tilde{a}=0.2$, length ${\tilde{l}}_z=0.1$. Analytical prediction ${\tilde{J}}_z=\frac{1}{2}{\tilde{J}}_r^2$ also shown.

Figure 6

Current densities just behind driving beam with $\tilde{Q}=20$, uniform distribution of width $\tilde{a}=0.2$, and length ${\tilde{l}}_z=0.1$. Note that the currents have moved away from the beam distribution ( $\tilde{r}>\tilde{a}=0.2$) significantly during the beam passage.

Figure 7

The average normalized energy loss rate of ${\tilde{F}}_{\mathrm{dec}}=e|⟨E_z⟩|/m_{e}c{\omega }_p$ of an electron beam with ${\tilde{l}}_z=0.1,\tilde{a}=0.2$, as a function of $\tilde{Q}$, from linear theory (solid bold line) and self-consistent PIC simulation (circles), the peak accelerating field behind the beam, ${\tilde{F}}_{\max }=e|E_{z,\max }|/m_{e}c{\omega }_p$, from linear theory (solid fine line) and PIC simulation (squares), and the useful field for acceleration (diamonds), defined by the geometry in Fig. 1.

Figure 8

The average normalized energy loss rate of ${\tilde{F}}_{\mathrm{dec}}=e|⟨E_z⟩|/m_{e}c{\omega }_p$ of a Gaussian-current electron beam with $k_p{\sigma }_z=1.1,\tilde{a}=0.2$, as a function of $\tilde{Q}$, from linear theory (solid bold line) and self-consistent PIC simulation (circles), the peak accelerating field behind the beam, ${\tilde{F}}_{\max }=e|E_{z,\max }|/m_{e}c{\omega }_p$, from linear theory (solid fine line) and PIC simulation (squares), and the useful field for acceleration (diamonds).

Figure 9

Surface plot of $E_z$ in OOPIC PWFA simulation with $\tilde{Q}=20$, (a) long-beam geometry, as in Fig. 8, and (b) ultrashort beam geometry, as in Fig. 7.

Figure 10

The average normalized energy loss rate of ${\tilde{F}}_{\mathrm{dec}}=e|⟨E_z⟩|/m_{e}c{\omega }_p$ of a Gaussian-current electron beam with $k_p{\sigma }_z=0.1$, using experimental scaling, $\tilde{a}=0.2\sqrt{\tilde{Q}/2}$, as a function of $\tilde{Q}$, from linear theory (solid bold line) and self-consistent PIC simulation (circles), the peak accelerating field behind the beam, ${\tilde{F}}_{\max }=e|E_{z,\max }|/m_{e}c{\omega }_p$, from linear theory (solid fine line) and PIC simulation (squares), and the useful field for acceleration (diamonds).