Growth and Phase Velocity of Self-Modulated Beam-Driven Plasma Waves

A long, relativistic particle beam propagating in an overdense plasma is subject to the self-modulation instability. This instability is analyzed and the growth rate is calculated, including the phase relation. The phase velocity of the wake is shown to be significantly less than the beam velocity. These results indicate that the energy gain of a plasma accelerator driven by a self-modulated beam will be severely limited by dephasing. In the long-beam, strongly coupled regime, dephasing is reached in a homogeneous plasma in less than four $e$ foldings, independent of beam-plasma parameters.

Figure 1

Beam radius modulation $r_b/r_0$ vs $k_p\zeta$ with beam-plasma parameters $n_b/n_0=0.008$ (proton beam), $\gamma =480$, and $k_p{r}_0=1$ (and $r_{b0}=r_0$), obtained from numerical solution of Eq. (5), at $k_{p}z=25000$ (blue curve) and $k_{p}z=28000$ (red curve). Dashed curves are the envelope of the asymptotic linear solution Eq. (10).

Figure 2

Normalized phase velocity of accelerating wakefield ${\gamma }_p[\nu (k_b/k_p{)}^2|\widehat{\zeta }|/\gamma {]}^{1/4}$ vs normalized propagation distance $(\nu |\widehat{\zeta }|{)}^{1/2}\widehat{z}$ in the long-beam regime: solid (black) curve is the series solution Eq. (12), dashed (red) curve is the asymptotic solution Eq. (14), and dots (blue) are from the numerical solution of the envelope equation Eq. (5).