Space-charge effects in ultrahigh current electron bunches generated by laser-plasma accelerators

Recent advances in laser-plasma accelerators, including the generation of GeV-scale electron bunches, enable applications such as driving a compact free-electron laser (FEL). Significant reduction in size of the FEL is facilitated by the expected ultrahigh peak beam currents (10–100 kA) generated in laser-plasma accelerators. At low electron energies such peak currents are expected to cause space-charge effects such as bunch expansion and induced energy variations along the bunch, potentially hindering the FEL process. In this paper we discuss a self-consistent approach to modeling space-charge effects for the regime of laser-plasma-accelerated ultracompact electron bunches at low or moderate energies. Analytical treatments are considered as well as point-to-point particle simulations, including the beam transport from the laser-plasma accelerator through focusing devices and the undulator. In contradiction to non-self-consistent analyses (i.e., neglecting bunch evolution), which predict a linearly growing energy chirp, we have found the energy chirp reaches a maximum and decreases thereafter. The impact of the space-charge induced chirp on FEL performance is discussed and possible solutions are presented.

Figure 1

Functional dependence of $I=\int dR{e}^{R^2/4}\Gamma (0,R^2/4)/4$.

Figure 2

The function ${\Lambda }^{\prime }$ [Eq. (10)] versus $\zeta /{\sigma }_z$ for $b=0.1$ (dotted curve), $b=0.01$ (solid curve), and $b=0.001$ (dashed curve).

Figure 3

The functions $\Lambda$ and ${\Lambda }^{\prime }$ versus $\zeta /{\sigma }_z$ for $b=0.14$.

Figure 4

Comparison between $u^{\prime }$ [Eq. (13)] (solid curves) and $U^{\prime }/Nm{c}^2$ [Eq. (11)] with $M=3000$ (dashed curves) as a function of (a) energy, (b) initial longitudinal length ${\sigma }_{z,0}$, and (c) initial transverse size ${\sigma }_{r,0}$. Beam parameters are $Q=0.42\mathrm{nC}$, ${\sigma }_{z,0}=1.3\mu \mathrm{m}$ in (a) and (c), ${\sigma }_{r,0}=42\mu \mathrm{m}$ in (a) and (b), and $\gamma =300$ in (b) and (c).

Figure 5

The ratio of ${\beta }_z^{*}$, derived using Eq. (15) with parameters taken from the numerical calculations in both lab and rest frame runs, and ${\beta }_z^{\prime }$ from the numerical calculation in the rest frame. The parameters were chosen for the longitudinal bunch expansion regime, for which Eq. (15) must be applied.

Figure 6

Beam energy chirps $⟨\gamma {⟩}^{-1}d⟨\gamma ⟩/d\zeta$ (relative chirp per micron) at the beam centroid ($\zeta =0$) versus $z$ for different electron bunch energies calculated using GPT. The beam parameters were ${\sigma }_z=1\mu \mathrm{m}$, ${\sigma }_x=30\mu \mathrm{m}$, and $I=50\mathrm{kA}$. (a) The chirp predicted by GPT (dash-dotted curve), the frozen beam approximation with $\gamma =300$ for a Gaussian bunch Eq. (9) (solid line), the results of Ref. 35 (dotted line), and Ref. 34 (dashed line). (b) Self-consistent GPT simulations of the space-charge induced chirp assuming $\gamma =300$, 450, 600, 750, and 900 versus propagation distance.

Figure 7

Debunching ${\sigma }_z(z=1\mathrm{m})/{\sigma }_z(z=0)$ as a function of electron energy for the beam parameters: ${\sigma }_z=1\mu \mathrm{m}$, ${\sigma }_x=30\mu \mathrm{m}$, and $I=50\mathrm{kA}$.

Figure 8

Electron beam energy distribution at (a) $z=6\mathrm{cm}$ and (b) $z=1.3\mathrm{m}$ after the laser-plasma accelerator. The initial beam parameters at the exit of the laser-plasma accelerator were ${\sigma }_z=1\mu \mathrm{m}$, ${\sigma }_x=1\mu \mathrm{m}$, $\gamma m{c}^2=150\mathrm{MeV}$, and $I=50\mathrm{kA}$. A focusing optic is placed at $z=8\mathrm{cm}$.