Modeling of three-dimensional betatron oscillation and radiation reaction in plasma accelerators

Betatron oscillation is a commonly known phenomenon in laser or beam-driven plasma wakefield accelerators. In the conventional model, the plasma wake provides a linear focusing force to a relativistic electron, and the electron oscillates in one transverse direction with the betatron frequency proportional to $1/\sqrt{\gamma }$, where $\gamma$ is the Lorentz factor of the electron. In this work, we extend this model to three-dimensional by considering the oscillation in two transverse and one longitudinal directions. The long-term equations, with motion in the betatron time scale averaged out, are obtained and compared with the original equations by numerical methods. In addition to the longitudinal and transverse damping due to radiation reaction which has been found before, we show phenomena including the longitudinal phase drift oscillation, betatron phase shift, and betatron polarization change based on our long-term equations. This work can be highly valuable for future plasma-based high-energy accelerators and colliders.

Figure 1

The numerical comparison of the LTEs and the original equations solved by PTracker in the betatron phase shift dominant regime. (a) $S_x$ and $S_y$ change with the period $\pi /\mathrm{\Omega }=3.89\times {10}^5$ according to Eq. (37), but $S_x+S_y$ is a constant. (b) $\gamma$ has oscillation frequencies of $2{\omega }_{\beta }\approx 0.14$ due to the BO and ${\omega }_{⟨\zeta ⟩}\approx 3.11\times {10}^{-3}$ due to the drift oscillation of $⟨\zeta ⟩$. (c) The gray curve shows $xy$ obtained from PT, and the black curve shows its low-pass filtered result, which is compared with the LTE solution according to Eq. (39). (d) The gray curve shows $x·\cos \phi$ obtained from PT, and the black curve shows its low-pass filtered result, which is compared with the LTE solution according to Eq. (38).

Figure 2

The numerical comparison of the LTEs and the original equations solved by PTracker in the transition between the betatron phase shift dominant and the RR dominant regimes. (a) $S_x+S_y$ is initially approximately a constant but decreases later. (b) $\gamma$ increases due to the acceleration field and passes the regime transition at $\gamma ={(k_p{r}_e)}^{-2/5}={10}^4$. (c) and (d) show the same treatments as in Figs. 1 and 1.

Figure 3

The numerical comparison of the LTEs and the original equations solved by PTracker in the RR dominant regime. $\mathrm{\Delta }\mathrm{\Phi }=\pi /2$ and $S_x>S_y$, thus the major axis of the particle trajectory ellipse is on the $x$ axis. (a) $R=S_y/S_x$ decreases with time due to Eq. (33), thus the ellipse is getting thinner. (b) $\gamma$ increases due to the acceleration field. (c) and (d) show the same treatments as in Figs. 1 and 1.

Figure 4

The numerical comparison of the LTEs and the original equations solved by PTracker in the RR dominant regime. $S_x=S_y$, thus the particle trajectory in the $x\text{−}y$ plane is an oblique ellipse. (a) $S_x$ and $S_y$ decrease at the same rate. (b) $\gamma$ initially decreases with time because the longitudinal RR damping is stronger than the acceleration. Later the RR damping becomes weaker due to the decrease of $S_x$ and $S_y$, and $\gamma$ increases. (c) and (d) show the same treatments as in Figs. 1 and 1, but the oscillation amplitudes are divided so that the changes of $\mathrm{\Delta }\mathrm{\Phi }$ and ${\mathrm{\Phi }}_x$ are clearer. $\mathrm{\Delta }\mathrm{\Phi }$ is between $\pi /2$ and $\pi$, thus $\mathrm{\Delta }\mathrm{\Phi }$ gradually approaches $\pi$.