Generating multi-GeV electron bunches using single stage laser wakefield acceleration in a 3D nonlinear regime

The extraordinary ability of space-charge waves in plasmas to accelerate charged particles at gradients that are orders of magnitude greater than in current accelerators has been well documented. We develop a phenomenological framework for laser wakefield acceleration (LWFA) in the 3D nonlinear regime, in which the plasma electrons are expelled by the radiation pressure of a short pulse laser, leading to nearly complete blowout. Our theory provides a recipe for designing a LWFA for given laser and plasma parameters and estimates the number and the energy of the accelerated electrons whether self-injected or externally injected. These formulas apply for self-guided as well as externally guided pulses (e.g. by plasma channels). We demonstrate our results by presenting a sample particle-in-cell (PIC) simulation of a $30\mathrm{fs}$, 200 TW laser interacting with a 0.75 cm long plasma with density $1.5\times {10}^{18}{\mathrm{cm}}^{-3}$ to produce an ultrashort (10 fs) monoenergetic bunch of self-injected electrons at 1.5 GeV with 0.3 nC of charge. For future higher-energy accelerator applications, we propose a parameter space, which is distinct from that described by Gordienko and Pukhov [Phys. Plasmas 12, 043109 (2005)] in that it involves lower plasma densities and wider spot sizes while keeping the intensity relatively constant. We find that this helps increase the output electron beam energy while keeping the efficiency high.

Figure 1

A sequence of 2-dimensional slices $(x-z)$ reveals the evolution of the accelerating structure (electron density, blue) and the laser pulse (orange). Each plot is a rectangular of size $z=101.7\mu \mathrm{m}$ (longitudinal direction, $z$) and $x=129.3\mu \mathrm{m}$ (transverse direction, $x$). A broken white circle is superimposed on each plot to show the shape of the blown-out region. When the front of the laser has propagated a distance (a) $z=0.3\mathrm{mm}$, the matched laser pulse has clearly excited a wakefield. Apart from some local modification due to beam loading effects, as seen in (b) this wakefield remains robust even as the laser beam propagates though the plasma a distance of 7.5 mm [as seen in (c) and (d)] or 5 Rayleigh lengths. After the laser beam has propagated 2 mm [as seen in (b)] into the plasma, one can clearly see self-trapped electrons in the first accelerating bucket. The radial and longitudinal localization of the self-trapped bunch is evident in part (c). After 7.5 mm the acceleration process terminates as the depleted laser pulse starts diffracting.

Figure 2

Four plots depicting the Wigner transform of the $x$ component of the laser electric field in the middle of the simulation box and the relative permittivity as functions of $z$. The Wigner transform of a function $\phi (z)$ is defined as [39] $W_{\phi }(z,k)={\int }_{-\infty }^{\infty }{e}^{-ik{z}^{\prime }}\phi (z+\frac{z^{\prime }}{2}){\phi }^{*}(z-\frac{z^{\prime }}{2})d{z}^{\prime }$ and the relative permittivity valid under the quasistatic approximation is ${\epsilon }_r={\eta }^2=1-\frac{n_p}{n_c}\frac{1}{1+\psi }$. In each plot, the vertical axis on the left corresponds to the local wave number over the initial laser wave number and the vertical axis on the right to ${\epsilon }_r$. The four plots represent results from four different computer simulations after 0.18 cm of laser propagation into a plasma with $n_p=1.5\times {10}^{18}{\mathrm{cm}}^{-3}$. (a) A 3D simulation of a 30 fs laser with $a_0=4$ and matched profile. (b) A 2D simulation of a 50 fs laser with $a_0=4$ and matched profile. (c) A 2D simulation of a 50 fs laser with $a_0=10$ and matched profile. (d) A 2D simulation of a 50 fs laser with $a_0=1$. Plot (c) is typical for the ultrarelativistic blowout regime while plot (d) is typical for the linear regime. Plots (a) and (b), which result from simulations with $a_0=4$, are more similar to the ultrarelativistic blowout case.

Figure 3

(a) A lineout of the wakefield along the $z$ axis after 0.3 mm shows that within the first bucket the slope of the wakefield is nearly constant and equal to $e{E}_z/(mc{\omega }_p)\simeq \xi /2$, where $\xi =[k_p(ct-z)]$. After 5.7 mm of propagation (b) the wakefield has been modified by beam loading (flattening of the wake between $400–450c/{\omega }_p$). This is corroborated by the $p_z$ vs $z$ plot that is superimposed on the lineout of the wakefield. Panels (a) and (b) reveal that the acceleration mechanism is extremely stable during the simulation. The energy spectrum after 7.5 mm (c) exhibits an isolated spike of 0.3 nC at 1.5 GeV with energy spread $\Delta \gamma /\gamma =3.8\%$ corresponding to the first bucket and a second spike of 50 pC at 700 MeV with energy spread $\Delta \gamma /\gamma =1.5\%$ corresponding to the second bucket.

Figure 4

The normalized emittance $[{\epsilon }_N{]}_i=\pi \sqrt{⟨\Delta p_i^2⟩⟨\Delta x_i^2⟩-⟨\Delta p_i\Delta x_i{⟩}^2}$ (where $\Delta p_i$ is normalized as indicated by the figure and the emittance is in units of $\Delta x_i$) and is the approximate area in phase space $p_i{x}_i$. For the panels above which correspond to the first bunch, this formula yields $[{\epsilon }_N{]}_x\simeq 35\pi \mathrm{mm}\mathrm{rad}$ and $[{\epsilon }_N{]}_y\simeq 29\pi \mathrm{mm}\mathrm{rad}$. An upper limit for the emittance can be found by multiplying the typical divergences shown in the figure; this method leads to an overestimation which for this case is about $25\%$. For the second bunch of accelerated electrons (not shown in this figure), the emittances are significantly lower: $[{\epsilon }_N{]}_x\simeq 10\pi \mathrm{mm}\mathrm{rad}$ and $[{\epsilon }_N{]}_y\simeq 11\pi \mathrm{mm}\mathrm{rad}$.

Figure 5

$E[\mathrm{GeV}]$ vs power $P[\mathrm{TW}]$ and density $n/n_c$ from Eq. (6): The blue lines of constant power show the strong dependence of the energy of the self-trapped electrons to the density. The black points correspond to (a) experiment [10], (b) experiment [11] which uses a channel for guiding, (c) experiment and 3D PIC simulation [12], (d) 3D PIC simulation [7] which uses a channel for guiding, (e) A 3D PIC simulation in 6, and (f) 3D PIC simulation presented in this article. Each of these points is very close to 1 of the blue lines indicating agreement with our scaling law.

Figure 6

Panel (a) shows the laser electric field with orange on top of the electron density with blue after 0.4 mm of propagation. The laser front is chosen sharper than the back because it has been found through simulations that this leads to more stable propagation. The blowout region is well formed. Panel (b) shows that particles at the rear of the ion channel have already reached velocities higher than the laser velocity and therefore are trapped. The region in the $z$ direction plotted in panel (b) corresponds to the $z$ region between the broken lines in panel (a).