Shot-to-shot electron beam pointing instability in a nonlinear plasma bubble

Shot-to-shot electron beam pointing instability in the plasma bubble, defined here as electron beam pointing jitter (EBJ), is a long-standing problem that limits the potential of the laser wakefield accelerator (LWFA) in a range of demanding applications. In general, EBJ is caused by variations in laser and plasma parameters from shot to shot, although the exact physical mechanism by which EBJ grows in the plasma wave remains unclear. In this work we theoretically investigate the fundamental physics of EBJ inside the plasma bubble and show how the intrinsic betatron oscillation can act as an amplifier to enhance EBJ growth. The analytical formulas for electron trajectory, pointing angle, and EBJ are derived from the basic momentum equation of an electron and verified numerically. It is shown that the shot-to-shot fluctuations of the laser and plasma parameters, such as laser strength, focus, and carrier-envelope phase, as well as the ambient plasma density and profile, lead to EBJ. The evolution of EBJ is dictated by the dynamics of the plasma bubble. Two amplification processes of the betatron oscillation are found in the rapidly evolving bubbles and play important roles in EBJ growth. The first is driven by a linear resonance in the wobbling bubble due to the coupling of the betatron oscillation and the bubble centroid oscillation. The second is a parametric resonance seen in the breathing bubble, where EBJ grows exponentially due to the strong frequency modulation of the betatron oscillation. Their characteristic functions, growth rates, and resonance conditions are deduced analytically and validated numerically. Finally, we also studied how radiation reaction affects EBJ. Our research provides a clear understanding of the basics of EBJ dynamics in LWFA and will help improve the use of LWFA in demanding applications.

Figure 1

(a)–(c) Temporal evolution of the transverse focusing force experienced by the trapped electron beam in the stationary, wobbling, and breathing bubble, respectively. The dashed gray line in each plot shows the centroid trajectory of the corresponding bubble. The expressions of the corresponding force $F_x$, centroid trajectory $x_c\left(t\right),$ and strength $K\left(t\right)$ of the plasma bubble are discussed in Secs. 3, 4, 5. (d) Centroid trajectory and (e) pointing angle of an electron beam in the stationary bubble as discussed in Sec. 3 (dash-dotted blue), in the wobbling bubble as discussed in Sec. 4 (dotted red), and in the breathing bubble as discussed in Sec. 5 (solid black). The dashed green lines present the result in a stationary bubble with the radiation reaction effect considered as discussed in Sec. 6.

Figure 2

Evolution of EBJ introduced due to the shot-to-shot variation of laser strength $a_0$ (a) and plasma density $n_0$ (b). The relative variations in each plot are 20% (dashed green), 10% (dotted red), 5% (dash-dotted black), and 1% (solid blue) with respect to each stable value. (c), (d) $\mathrm{\Delta }{\theta }_J$ as a function of $\delta a_0/a_0$ and $\delta n_0/n_0$ after propagation $t=1000$, respectively. EBJ is calculated by the randomly sampled variations in parameters over 100 shots and by smoothing out the betatron oscillation.

Figure 3

(a) Temporal evolution of pointing angle ${\theta }_x$ calcualted numerically from Eq. (10) (solid red) and theoretically from Eq. (18) (dashed black). (b) Temporal evolution of EBJ $\mathrm{\Delta }{\theta }_h$ calculated from Eq. (19) with (dash-dotted green) or without (solid red) the fast oscillation term ${\mathcal{J}}_h\left(t\right)$. The scaling law is fit according to Eq. (20) (dashed black). The initial injection jitter of the electron is $\mathrm{\Delta }{x}_0=0.3$ and $\mathrm{\Delta }{\mathrm{\Theta }}_0=1\mu \text{rad}$. The other laser and plasma parameters are shot-to-shot stable.

Figure 4

Temporal evolution of EBJ with $1\%$ (solid green), $5\%$ (dash-dotted blue), $10\%$ (dotted red), and $20\%$ (dashed black) shot-to-shot variation of $w_0$ (a) and $n_0$ (b). They are numerically calculated by Eq. (21). The variations in parameters are randomly sampled over 100 shots and the betatron oscillation is smoothed out.

Figure 5

(a) Temporal evolution of pointing angle ${\theta }_p$ from a particular solution calculated from Eq. (21) (solid red) and Eq. (23) (dashed black). (b) Temporal evolution of Betatron frequency ${\omega }_{\beta }$ (solid black) and parametric function $\mathcal{F}(t;{\omega }_{\beta },{\omega }_w)$ (dashed red). The vertical dashed gray line shows the position where the resonant condition ${\omega }_{\beta }\simeq 2{\omega }_w/3$ (dash-dotted green) is met. (c) Distribution of parametric function $\mathcal{F}(t;{\omega }_{\beta },{\omega }_w)$ as a function of ${\omega }_w$ during acceleration. (d) Temporal evolution of particular EBJ $\mathrm{\Delta }{\theta }_{p,w}$ with initial Lorentz factor varying in range $10<{\gamma }_0<100$. The dashed white line shows the resonant position implied by Eq. (32).

Figure 6

Temporal evolution of EBJ in a preformed plasma channel under two conditions: (a) the centroid oscillation solely due to channel guiding calculated from Eq. (38), (39), and (41); and (b) the mixed centroid oscillation due to the CEP effect and channel guiding, calculated from Eq. (44), (46), and (28). The vertical dashed lines indicate the positions where ${\omega }_{\beta }=2{\omega }_w/3$ (dashed blue), and ${\omega }_{\beta }=2{\omega }_{ch}/3$ (dashed gray). $w_0=1.5$ are used. The initial laser jitter is $\mathrm{\Delta }{x}_{l0}=0.3$, and $\mathrm{\Delta }{\theta }_{l0}=1\mu \text{rad}$ over 100 random shots.

Figure 7

(a) Evolution of electron pointing angle in a breathing bubble, calculated with $K_E=1.5$ by Eq. (48) (dash-dotted red) or by Eq. (52) (solid black). The dotted green line shows the case with $K_E=0.3$ and without the driven term. The dashed blue line is calculated from Eq. (59). (b) Evolution of EBJ in a breathing bubble, calculated by Eq. (48) with $K_E=1.5$ (solid black) and $K_E=0.6$ (dash dotted red). The dashed blue line is calculated from Eq. (60). Here the initial EBJ is $\mathrm{\Delta }{\theta }_{\alpha }=0.37\text{m}\text{rad}$. ${\gamma }_0=200$. The other parameters are the same as above.