Scissor-cross ionization injection in laser wakefield accelerators

The laser wake field accelerator (LWFA) proposed by Tajima and Dawson has attracted many attentions due to its orders-of-magnitude higher acceleration gradient than that of the conventional radio-frequency accelerators Tajima and Dawson (1979). Great breakthroughs have been made in the past few years. For example, an electron beam with energy of 7.8 GeV has been generated in 20 cm Gonsalves et al. (2019). The electron beams with energy spread in the sub-percentage level have been produced using density-tailored plasma Ke et al. (2021). The 24-hour stable LWFA has been achieved by decoding sources of energy drift and jitter Maier et al. (2020). Improving the output beam quality parameters including the energy spread, the beam charge, the emittance, the energy stability and so on has been a long-term goal in this society for the high-demanding applications such as plasma based light sources and colliders Leemans and Esarey (2009); Assmann et al. (2020); Wang et al. (2021); Zeng and Seto (2021).

To optimize the output beam quality, many controlled injection schemes have been proposed such as pulse collision injections Fubiani et al. (2004); Davoine et al. (2009); Golovin et al. (2018), density gradient injections Brantov et al. (2008); Geddes et al. (2008); Schmid et al. (2010), ponderomotive injections Umstadter et al. (1996); Zeng et al. (2020), external magnetic field injections Vieira et al. (2011, 2012); Bulanov et al. (2013); Rassou et al. (2015), ionization injections Chen et al. (2006, 2012); McGuffey et al. (2010); Pak et al. (2010) and so on. The ionization injection is to release electrons inside the pseudo-potential well of a wakefield by high-order ionization of the dopant species (high-Z elements such as Nitrogen, Oxygen, Neon or Argon). The injection amount can be adjusted by changing the density ratio of the dopant to the background plasma which is pre-ionized from low-ionization-threshold species such as Hydrogen and Helium. The ionization injection has the advantage of high reproducibility, but its energy spread is usually large. To reduce the energy spread, one may use the self-dechirping effect Ke et al. (2021); Wang et al. (2016), and/or reduce the electrons injection length Zeng et al. (2014); Mirzaie et al. (2015). For example, an electron beam with slice energy spread of $13\ \rm keV$ and charge of $0.4\ \rm pC$ can be produced by ionization injection of counter-propagating laser pulse Wan et al. (2016). An electron beam with slice energy spread of $12\ \rm keV$ and charge of $5\ \rm pC$ can be produced by two colliding lasers propagating in the transverse direction Li et al. (2013). The scheme of beat frequency ionization injection using frequency tripled laser was proposed for generating low energy spread electron beams Zeng et al. (2015, 2016). However, this scheme has experimental difficulty due to lacking of high-efficiency frequency tripler.

In this work, we propose a new scheme to trigger electron ionization injection by a frequency doubled pulse colliding with the driver pulse at an acute angle as illustrated in Fig. 1. A driver laser pulse drives a plasma wake (not shown in the figure) and collides with a trigger laser pulse at an angle $\theta$. During the collision, a higher superimposed electric field is generated, triggering the ionization of electrons of the inner shell of the dopant species. The driving laser itself can not ionize the inner shell of the dopant. The ionization injection only occurs when two laser pulses overlap, thus the injection is localized and the energy spread of the produced electron beam is limited. The frequency-doubled trigger laser has largely reduced ponderomotive force compared to a fundamental-frequency trigger laser with the same ionization electric field strength, thus its disturbance to the main wakefield is limited. The trajectories of the driver and trigger lasers are similar to the two blades of a scissor, thus we call this scheme the scissor-cross ionization injection.

The snapshots of the example two-dimensional (2D) particle-in-cell (PIC) simulation using the code WarpX Vay et al. (2021) are shown in Fig. 2. In the plots, $n_{0}$ is the unperturbed plasma density, $\rho_{e}$ is the density of background electrons, $\rho_{d}$ is the density of electrons ionized from the dopant species, $E_{L}$ is the electric field of the laser pulses, and $e$ is the elementary charge. In the simulation, the background plasma is fully pre-ionized gas (e.g. hydrogen or helium) and the dopant is neon pre-ionized to +8 charge state. Because the ionization threshold of $\rm Ne^{8+}$ is $\sim 16.5\ \rm TV/m$, for the driver pulse with normalized vector potential amplitude $a_{0}<4$, it can be guaranteed that $\rm Ne^{8+}$ is not further ionized by the driver pulse, where $a_{0}\approx 8.5\times 10^{-10}\lambda\left[{\rm\mu m}\right]\sqrt{I_{0}\left[{\rm W/cm^{2}}\right]}=E_{0}\left[{\rm TV/m}\right]\cdot\lambda\left[{\rm\mu m}\right]/3.2$ is the normalized vector potential amplitude, $I_{0}$ is the laser intensity and $E_{0}$ is the peak electric field strength of the laser. Meanwhile, by choosing $a_{0}>3$, we can ensure the self-guiding of the driver laser pulse for a sufficient longer electron acceleration distance without using a parabolic plasma channel Lu et al. (2007); Benedetti et al. (2012). Practically, the driver pulse has normalized vector potential amplitude of $a_{0}=3.24$ and wavelength of $800\ \rm nm$, while the trigger pulse has normalized vector potential amplitude of $a_{1}=1.62$ and wavelength of $400\ \rm nm$. Both the two pulses have spot radius of $r_{0}=r_{1}=15\ \rm\mu m$ and pulse duration of $30\ \rm fs$. Their focusing is synchronized spatially and temporally to trigger the inner-shell ionization of the dopant species. After the intersection finishes, the ionization injection does not occur anymore, because the electric field strength cannot reach the inner-shell ionization threshold of the dopant species even if the self-focusing of the driver laser pulse occurs. That is to say, ionization injection only occurs when two laser beams overlap, which ensures a limited injection distance.

To study the effect of different collision angles, we have performed simulations with $\theta$ varying from $10^{\circ}$ to $150^{\circ}$. The snapshots and the quality of the injected beams are shown in Fig. 3 and Fig. 4, respectively. These simulations use pure Neon element, which provides both the pre-ionized outer shell to form the background plasma and the bounded inner shell for ionization injection. As one can see, the beam quality changes with $\theta$. For $\theta\lesssim 30^{\circ}$, the disturbance of the trigger pulse to the main wakefield is small, but the length of the region that the two pulses overlap, and thus the injection length which can be estimated by $r_{0}/\tan\theta$, is relatively large. For $\theta\gtrsim 60^{\circ}$, the injection length can be $\sim 10\ \rm\mu m$, but there is a significant disturbance of the trigger pulse to the main wakefield which degrades the injected beam quality. Moreover, for $\theta\lesssim 30^{\circ}$, the injection mechanism is purely ionization injection, while for $\theta\gtrsim 60^{\circ}$, the injection mechanism is gradually switched to colliding pulse injection as shown in Fig. 4 (a). The phase space distribution of the output beam for the case $\theta=30^{\circ}$ is plotted in Fig. 4 (b), which shows a beam with a mean energy of $578\ \rm MeV$, a root-mean-square (RMS) energy spread of $0.7\%$, and a normalized emittance of $4.7\ \rm mm\cdot mrad$.

Obviously, the strength of the superimposed electric field of the driver and trigger does not depend on their sizes. However, the sizes of the pulses determine the injection quantity and also the injection difficulty in a real experiment. We discuss the effect of changing the trigger size in the following. Assume the wakefield is a spherical bubble, the pseudo-potential in the bubble can be estimated by $\psi\approx\omega_{p}^{2}(r^{2}_{b}(\zeta)-r^{2})/4c^{2}$, where $r_{b}$ is the bubble radius, $r$ is the distance to the central axis, $\zeta=z-ct$ is the co-moving coordinate, $c$ is the speed of light in vacuum, $\omega_{p}=c\sqrt{4\pi r_{e}n_{0}}$ is the plasma frequency and $r_{e}$ is the classical electron radius Kostyukov et al. (2004); Lu et al. (2006). The condition of trapping for an electron is $\Delta\psi\lessapprox-1$ if the electron has negligible initial momentum Zeng et al. (2020). In a plasma with the density $n_{0}=1.63\times 10^{18}\ \rm cm^{-3}$, the laser with $a_{0}=3.24$ and the matched spot size $r_{0}=3.6c/\omega_{p}=15\ \rm\mu m$ drives a wakefield with its distribution of pseudo-potential illustrated in Fig. 5 (a). The electrons been ionized in the dashed red circle satisfy the trapping condition and can be captured by the wakefield. A larger spot size and longer pulse duration of the trigger can decrease the influence of the time delay jitter of the two pulses to the output electron beam energy as one can see in Fig. 5 (b). This is because the time delay jitter of a smaller trigger laser more significantly influences the injection phase of the electron beam, which determines the acceleration field strength exerted on the beam.

We have also performed fully three-dimensional (3D) simulations to verify our scheme. Because the self-focusing effect of the laser in plasmas is stronger in a 3D case than in a 2D case, the former 2D simulation parameters are not suitable for the 3D simulations. The new parameters for a 3D simulation are the following. The driver and trigger pulses have normalized vector potential amplitude of $a_{0}=2.74$ and $a_{1}=1.37$, respectively, and they have the same focal waist radius of $r_{0}=r_{1}=20\ \rm\mu m$. They are both polarized in the $y$ direction and collide at $\theta=8^{\circ}$. We do not choose $\theta=30^{\circ}$ as in Fig. 4(b) because a larger $\theta$ significantly increases the computational cost to an unaffordable level. The pre-ionized background plasma density is $n_{0}=1.36\times 10^{18}\ \rm cm^{-3}$ and the density of $\rm Ne^{8+}$ is $n_{0}/8$. The simulation shows that the maximum electric field strength after occurring of self-focusing is $15\ \rm TV/m$ which is sufficiently smaller than the ionization threshold of $\rm Ne^{8+}$ (which is around $17\ \rm TV/m$), thus the driver pulse itself does not trigger ionization injection. The simulation has the moving window size of ($100\ \rm\mu m$, $100\ \rm\mu m$, $50\ \rm\mu m$) and the cell number of (768, 128, 2432) for ($x$, $y$, $z$) directions, respectively. The results are shown in Fig. 6. The electron beam is injected by our scheme at $t=1.0\ \rm ps$ with a small absolute energy spread (Fig. 6 (a)) and mainly experiencing positive chirping during the acceleration (Fig. 6 (b)). At the acceleration distance of about $4\ \rm mm$ (Fig. 6 (c)), the electron beam enters the negative chirping region due to the weakening of the driver, thus the initial small absolute energy spread is retrieved (phase space is shown in Fig. 6 (d)). The beam after the acceleration distance of $4\ \rm mm$ has the charge of $Q=40\ \rm pC$, the peak current of $I_{\rm peak}=6.22\ \rm kA$, the mean energy of $\rm\left<E\right>=500\ \rm MeV$, the RMS energy spread of $\rm\sigma_{E}/\left<E\right>=1.6\%$, and the normalized emittance of $\varepsilon_{x}=1.11\ \rm mm\cdot mrad$, $\varepsilon_{y}=7.84\ \rm mm\cdot mrad$ for the two transverse directions, respectively.

In conclusion, we have introduced the scissor-cross ionization injection scheme in LWFA, which uses the intense electric field generated at the moment when the trigger pulse overlaps with the driver laser pulse to ionize the inner shell electrons of the dopant species and to trigger the ionization injection. Both 2D and 3D simulations show that this scheme produces high quality electron beams with the energy spread of the order of 1%. The ionization injection is limited to a small region with the length of $\sim r_{0}/\tan\theta$ which is typically $\lesssim 100\ \rm\mu m$, where $r_{0}$ is the spot size of the driver pulse and $\theta$ is the collision angle. It is theoretically possible to further decrease the energy spread to per-mille level by slightly increasing $\theta$ and/or by increasing the acceleration distance with a preformed plasma channel. In experimental implementation of this scheme, the trigger pulse can be replaced by a laser with any frequency, as long as the superimposed electric field exceeds the inner-shell ionization threshold of the dopant species when the two pulses collide. However, with a certain peak electric field strength, a lower-frequency laser has larger ponderomotive force which may degrade the injection quality. We mainly use the frequency doubled trigger in our discussion, for its reasonable experimental difficulty and relatively small ponderomotive disturbance to the main wakefield.