Multi-GeV wakefield acceleration in a plasma-modulated plasma accelerator

We investigate the accelerator stage of a plasma-modulated plasma accelerator (P-MoPA) [Jakobsson et al., Phys. Rev. Lett. 127, 184801 (2021)] using both the paraxial wave equation and particle-in-cell (PIC) simulations. We show that adjusting the laser and plasma parameters of the modulator stage of a P-MoPA allows the temporal profile of pulses within the pulse train to be controlled, which in turn allows the wake amplitude in the accelerator stage to be as much as $72\%$ larger than that generated by a plasma beat-wave accelerator with the same total drive laser energy. Our analysis shows that Rosenbluth-Liu detuning is unimportant in a P-MoPA if the number of pulses in the train is less than $\sim 30$, and that this detuning is also partially counteracted by increased red-shifting, and hence increased pulse spacing, towards the back of the train. An analysis of transverse mode oscillations of the driving pulse train is found to be in good agreement with 2D (Cartesian) PIC simulations. PIC simulations demonstrating energy gains of $\sim 1.5\mathrm{GeV}$ $(\sim 2.5\mathrm{GeV})$ for drive pulse energies of $2.4\mathrm{J}$ $\left(5.0\mathrm{J}\right)$ are presented. Our results suggest that P-MoPAs driven by few-joule, picosecond pulses, such as those provided by high-repetition-rate thin-disk lasers, could accelerate electron bunches to multi-GeV energies at pulse repetition rates in the kilohertz range.

Figure 1

Snapshot at $z=2.1\mathrm{mm}$ of the relative plasma density perturbation $\delta n/n_0$ (blue-green) resonantly excited by a P-MoPA pulse train with normalized intensity ${\left|a\right|}^2$ (red-yellow). The transverse electron density profile of the channel, corresponding to Eq. (7) (green), and the longitudinally averaged transverse intensity profile (purple) of the drive pulses are shown on the end panel of the plot. Results are from a 2D simulation using the PIC code WarpX [18] with $W_{\text{drive}}=1.2\mathrm{J}$, ${\tau }_{\text{drive}}=1\mathrm{ps}$, ${\lambda }_L=1030\mathrm{nm}$, $R=30\mu \mathrm{m}$, $w_0=30\mu \mathrm{m}$, $n_{00}=2.5\times {10}^{17}{\mathrm{cm}}^{-3}$. and modulator parameter $\beta =1.2$ as defined in Eq. (9) (note that for 2D Cartesian geometry, the laser energy $W_{\text{drive}}$ determines the initial $a_0$ of the laser pulse at focus as if it were 3D assuming a cylindrically symmetric Gaussian transverse profile).

Figure 2

(a) Pulse train temporal profiles for various values of the modulator parameter $\beta$ and their respective FWHM durations (dashed). (b) FWHM duration (blue, solid), peak prominence (orange, dashed) and the wakefield amplitude according to 1D linear theory (green, dot-dashed, arb. units) as a function of $\beta$.

Figure 3

Results of a 2D PIC simulation of the propagation of a pulse train in a quasi-square plasma channel. For this simulation, $\xi$ as defined after Eq. (19) at varying propagation distances $z$ in a quasi-square channel of the form given in Eq. (7) with $n_{00}=2.5\times {10}^{17}{\mathrm{cm}}^{-3}$, $\mathrm{\Delta }n=1.26\times {10}^{17}{\mathrm{cm}}^{-3}$, $R=30\mu \mathrm{m}$, $W_{\text{drive}}=2.4\mathrm{J}$, ${\tau }_{\text{drive}}=1\mathrm{ps}$ and modulator parameter $\beta =1.2$. The pulse train at the entrance of the channel initially has a Gaussian transverse profile with spot size $w_0=30\mu \mathrm{m}$. For each plot, the top panel displays the laser intensity profile ${\left|a\right|}^2$ along with the local longitudinal variation of the laser centroid $x_c\left(\xi \right)$ (dashed) and local effective spot size $w_{\text{eff}}\left(\xi \right)$ (solid). The bottom panel displays the wake density perturbation $\delta n/n_{00}$.

Figure 4

Results from the same PIC simulation presented in Fig. 3. (a) The on-axis normalized axial electric field $E_z/E_{\text{wb}}$ (solid, black) and the normalized laser intensity envelope $|E/E_L{|}^2$ with $E_L=m_e{\omega }_{L}c/e$ and its shift in instantaneous frequency $\mathrm{\Delta }\omega /{\omega }_L$ (color scale) after 2 mm (top) and 100 mm (bottom) of propagation. (b) The evolution of the transverse-averaged intensity envelope $\langle |E/E_L{{|}^2\rangle }_{\perp }$. Note that $|E/E_L|\ne |a|$ at large propagation distances due to the significant red-shifting of the trailing pulses.

Figure 5

2D PIC simulation results of accelerator performance at different modulation parameters $\beta$ in a quasisquare channel of the same form used in Fig. 3. The top panels display the normalized laser spectrum $|\widehat{E}\left(\omega \right)|$ evolving with the propagation distance, while the bottom panels show the evolution of the normalized injected bunch electron energy spectrum ${\widehat{Q}}_e\left(W_e\right)$ and maximum on-axis accelerating field $E_{\text{acc}}^{\text{max}}$. Note that the oscillation of the on-axis accelerating field due to spot-size oscillations is a continuous sinusoidal motion which disappears beyond $50\mathrm{mm}$ in (a), (b) and $130\mathrm{mm}$ in (c), and only appears to be jagged in these plots due to subsampling.