Saturation of the Hosing Instability in Quasilinear Plasma Accelerators

The beam hosing instability is analyzed theoretically for a witness beam in the quasilinear regime of plasma accelerators. In this regime, the hosing instability saturates, even for a monoenergetic bunch, at a level much less than standard scalings predict. Analytic expressions are derived for the saturation distance and amplitude and are in agreement with numerical results. Saturation is due to the natural head-to-tail variations in the focusing force, including the self-consistent transverse beam loading.

Figure 1

Comparison of the numerical integration of the hosing equation Eq. (1) in the case of a linear chirp (black curve) with the early-stage solution [Eq. (5); red curve] and the late-stage solution [Eqs. (6) and (7) on the left- and right-hand panels, respectively; blue curve], at different slices along the witness bunch (i.e., different values of ${\kappa }_p\xi$). The vertical dashed line corresponds to $z=L_{\mathrm{sat}}(\xi )$, from Eq. (4). The parameters used here are $k_c^2{n}_{b,0}/n_p=k_{\beta ,0}^2$ and ${\partial }_{\xi }{k}_{\beta }=0.1k_{\beta ,0}{\kappa }_p$ (left-hand panels) or ${\partial }_{\xi }{k}_{\beta }=-0.1k_{\beta ,0}{\kappa }_p$ (right-hand panels).

Figure 2

Evolution of the centroid offset (log-scaled color map) as a function of the head-to-tail coordinate $\xi$ and the propagation distance $z$, for [case (a); left-hand panels] a bunch driver with $r_d=4\mu \mathrm{m}$, ${\ell }_d=3\mu \mathrm{m}$, ${\xi }_d=27\mu \mathrm{m}$, and $n_d=0.7n_p$, and [case (b); right-hand panels] a laser driver with $a_0=0.4$, $\tau =20\mathrm{fs}$, ${\xi }_d=26\mu \mathrm{m}$, $w=24\mu \mathrm{m}$. The top plots show the head-to-tail variations of the betatron frequency $k_{\beta }$ in both cases. The upper color maps show results from the PIC simulations, while the middle and lower color maps show results from the numerical integration of Eq. (1), with $k_{\beta }(\xi )$ either given by Eq. (11) (middle color maps) or considered uniform (lower color maps). The lower plots are lineouts of the color maps, at a fixed head-to-tail distance ($\xi =-10\mu \mathrm{m}$; indicated by a white dotted line on the color maps).