Excitation of a nonlinear plasma ion wake by intense energy sources with applications to the crunch-in regime

We show the excitation of a nonlinear ion-wake mode by plasma electron modes in the bubble regime driven by intense energy sources, using analytical theory and simulations. The ion wake is shown to be a driven nonlinear ion-acoustic wave in the form of a long-lived cylindrical ion soliton which limits the repetition rate of a plasma-based particle accelerator in the bubble regime. We present the application of this evacuated and radially outwards propagating ion-wake channel with an electron skin-depth scale radius for the “crunch-in” regime of hollow-channel plasma. It is shown that the time-asymmetric focusing force phases in the bubble couple to ion motion significantly differently than in the linear electron mode. The electron compression in the back of the bubble sucks in the ions whereas the space charge within the bubble cavity expels them, driving a cylindrical ion-soliton structure at the bubble radius. Once formed, the soliton is sustained and driven radially outwards by the thermal pressure of the wake energy in electrons. Particle-in-cell simulations are used to study the ion-wake soliton structure, its driven propagation and its use for positron acceleration in the crunch-in regime.

Figure 1

Laser driven nonlinear ion wake at early time ($t=46{\omega }_{\mathrm{pe}}^{-1}=0.17f_{\mathrm{pi}}^{-1}$, where $f_{\mathrm{pi}}$ is the plasma ion frequency) in $m_i=m_p=1836m_e$ plasma. (a) Electron bubble mode in Cartesian coordinates (fixed box) with $\frac{{\omega }_0}{{\omega }_{\mathrm{pe}}}=10$ driven by a matched laser pulse (vector potential $a_0=4$ and frequency ${\omega }_0$) with $R_B\simeq 4\frac{c}{{\omega }_{\mathrm{pe}}}$. (b) Nonlinear ion wake in the form of a cylindrical ion-soliton of radius $\simeq 4\frac{c}{{\omega }_{\mathrm{pe}}}$ excited behind the bubble electron wake in a proton plasma. (c) Transverse ion-density profile at $z=15c/{\omega }_{\mathrm{pe}}$. Notice that the ion density perturbation in this excitation phase is still building up and is a fraction of the background ion density, $\frac{\delta n_i}{n_0}<1$.

Figure 2

Electron beam-driven nonlinear ion wake at late time ($t=460{\omega }_{\mathrm{pe}}^{-1}=1.7f_{\mathrm{pi}}^{-1}$) in $m_i=m_p=1836m_e$ plasma. (a) Beam-driven ion-wake electron density in cylindrical coordinates (fixed box). The beam parameters are $n_b=5n_0$, ${\sigma }_r=0.5c/{\omega }_{\mathrm{pe}}$, ${\sigma }_z=1.5c/{\omega }_{\mathrm{pe}}$, ${\gamma }_b=38,000$, these beam-plasma parameters are quite similar to [5]. (b) Corresponding ion density in cylindrical coordinates (fixed box). Note the N-soliton formation in the ion density, $50c/{\omega }_{\mathrm{pe}}\le z\le 100c/{\omega }_{\mathrm{pe}}$. The later times in the time evolution of the ion wake are also inferred from density snapshots farther behind the beam. (c) Radial electron and ion density profile at $z=150c/{\omega }_{\mathrm{pe}}$. A full movie of radial electron and ion density dynamics is presented in Supplemental Material [20].

Figure 3

Bubble-wake train behind an ultrarelativistic electron beam with bubble: ${\beta }_g\ll {\beta }_{\varphi }\simeq {\beta }_{\mathrm{beam}}$. (a) Electron density in 2D cylindrical real space, (b) corresponding longitudinal electric field profile and (c) corresponding radial-field profile. Here the beam is located between 170 and $180\frac{c}{{\omega }_{\mathrm{pe}}}$. The bubbles just behind the driver in (a) undergo phase mixing over several cycles. The intermediate stages of the extent of phase mixing can be inferred from the bubbles that are closer to the beam. The beam-plasma parameters are the same as in Fig. 2 but the electron wake is shown at an earlier time $t=150{\omega }_{\mathrm{pe}}^{-1}$.

Figure 4

Ion dynamics in longitudinally asymmetric phases of the radial forces in an electron bubble. (a) Electron density of a bubble in 2D cylindrical real space. (b) Longitudinal on-axis profile of the electron density (black), longitudinal field (blue), focusing field (red). (c) Radial-field profile close to the back of the bubble. This is the focusing suck-in phase for the ions. (d) The fields at the center of the ion cavity of the bubble. This is the defocusing “push-out” phase for the ions.

Figure 5

Radial profile of the root-mean-square radial electron momentum (proportional to the square root of the electron temperature, $\sqrt{T_e}$) at 460 ${\omega }_{\mathrm{pe}}^{-1}$ for the beam-driven ion wake in Fig. 2. The blue curve shows the root-mean-squared radial electron momentum, $p_{\mathrm{th}}^e(r)=\sqrt{[{\mathrm{\Sigma }}_k{p}_r^2(k,r)2\pi r{n}_e(k,r)]/{\mathrm{\Sigma }}_{k}2\pi r{n}_e(k,r)}$ profile of the wake electrons corresponding to the time in Fig. 2 at 460 ${\omega }_{\mathrm{pe}}^{-1}$. This represents the square root of the electron temperature, $p_{\mathrm{th}}^e\propto \sqrt{T_e}$. The radial gradient of the temperature, $\frac{\partial }{\partial r}{T}_e$ is thus computed at the peak of the soliton (red) and in its vicinity (green). It is interesting to note that $\frac{\partial }{\partial r}{T}_e^{(1)}{|}_{\mathrm{peak}}=0$.

Figure 6

Time evolution of the cylindrical ion soliton. (a) Electron (black) and ion (red) spike radial positions (in terms of $c{\omega }_{\mathrm{pe}}^{-1}$) with time and a third-order fit (green) for the position of the ion density-spike of the soliton. (b) Radial fields of the electron bubble oscillations (in terms of $m_{e}c{\omega }_{\mathrm{pe}}{e}^{-1}$) at the electron density spike (magenta) and at the ion density spike (blue). (c) Radial velocity of the ion density spike of the soliton calculated from the third-order fit curve (red). An estimate of the sound speed (green) using the mean temperature, between the axis and the soliton location (green) and in the vicinity of the soliton peak (blue), in the expression $c_s=\sqrt{k_B⟨T_e⟩/m_i}$. Since the plasma is not isothermal the mean temperature is calculated by averaging the temperature of electrons over the indicated spatial region. (d) Gradient of the electron temperature at the soliton ion density peak (blue) and in the vicinity of the peak (red). The vicinity of the ion density peak of the soliton is defined as shown in Fig. 5.

Figure 7

Radial phase-space snapshots of the electron and ion density in Fig. 2. (a) Electron $p_r-r$ radial momentum phase space showing the accumulation of thermalized electrons within the ion soliton. (b) Ion $p_r-r$ radial momentum phase space showing the on-axis and ion-wake edge ion accumulations.

Figure 8

2D-PIC simulations: (a) electron density in real space showing electron density excitation in the crunch-in regime; (b) corresponding radial profile of fields excited by a positron beam in an ideal-channel versus an ion-wake channel. The radial profile of the normalized electron density (black) in an ion-wake channel (normalized to the maximum electron compression) at longitudinal location of the peak accelerating field ($r_{\mathrm{pb}}=2.3c/{\omega }_{\mathrm{pe}}$, ${\gamma }_{\mathrm{pb}}=38000$, $n_{\mathrm{pb}}=1.3n_0$). The radial profile of the accelerating field and the normalized potential of the focusing field ($E_r\text{−}{B}_{\theta }$) (radial field integrated from the edge of the box to a radius on x-axis).