Ion wave breaking acceleration

Laser driven ion wave breaking acceleration (IWBA) in plasma wakefields is investigated by means of a one-dimensional (1D) model and 1D/3D particle-in-cell (PIC) simulations. IWBA operates in relativistic transparent plasma for laser intensities in the range of ${10}^{20}–{10}^{23}\mathrm{W}/{\mathrm{cm}}^2$. The threshold for IWBA is identified in the plane of plasma density and laser amplitude. In the region just beyond the threshold, self-injection takes place only for a fraction of ions and in a limited time period. This leads to well collimated ion pulses with peaked energy spectra, in particular for 3D geometry.

Figure 1

Results of 1D-PIC simulations. A laser pulse of amplitude $a_0=44$ (not shown) is incident from the left, driving a wakefield. Electron and ion densities as well as electric field are plotted for two different initial plasma densities: (a) $0.2n_c$, (b) and (c) $2.8n_c$; lower parts show ion phase space in blue. Results in (b) refer to an early time (93 fs), when the ion wave is just before breaking, while those in (c) refer to a later time (333 fs), when the ion wave is broken and accelerated ions have overtaken the electron layer at the laser front; (a) corresponds to a linear ion wave developing at low plasma density; (d) ion energy spectra corresponding to (c) and (a).

Figure 2

Schematic figure of ion dynamics, modeling the results of Fig. 1. Density distribution of ions ($n_i$), cold electrons ($n_e$), and heated electrons ($n_{eh}$) are denoted by red, blue, and gray areas, respectively. The charge separation electric field $E_z$ along the propagation direction is denoted by black solid lines. The lower panels show trajectories (green lines) in ion phase space $(z,v_{i,z})$ in a frame (marked by apostrophes) comoving with the wake at phase velocity $v_{\mathrm{ph}}$. Column (a) refers to ion waves with small amplitude, where ions just pass through, while (b) refers to a breaking ion wave where ion velocity just reaches $v_{\mathrm{ph}}$, and (c) refers to a stage after breaking when background ions have been partially injected and are accelerated to the right (indicated by a red spot). The wakefield is then more expanded and weaker such that further injection is suppressed. In cases (b) and (c), the downstream cold electron layer has disappeared, and a charge double layer wakefield has formed between the laser-driven front electrons and the trailing ions; it is modeled by width $d$ and constant field $E_{\max }$ in (b), and $d^{\mathrm{ab}}$ and $E_{\max }^{\mathrm{ab}}$ in (c).

Figure 3

1D model and simulation results. (a) Peak ion energy ${\mathcal{E}}_{i,\mathrm{peak}}$ (color scale) obtained from simulations plotted in plane of laser amplitude $a_0$ and density $n_0/n_c$. The black solid line shows the onset of IWBA derived from Eqs. (2)–(3), the black dashed line denotes the border to hole-boring (HB) (full ion trapping) as obtained from simulations. (b) Maximum of peak ion energy ${\mathcal{E}}_{i,\mathrm{peak}}$ versus $a_0$; red circles from simulations, black solid line from Eqs. (1)–(3), blue thin dashed line from HB model [21]. (c) Laser propagation velocity ${\beta }_{\mathrm{ph}}$ and (d) peak ion energy ${\mathcal{E}}_{i,\mathrm{peak}}$ as a function of density $n_0/n_c$ for $a_0=44$ (red thick solid line), $a_0=20$ (green thick broken line), and $a_0=8$ (blue thick dotted line) obtained from simulation. The corresponding thin lines show the HB model. The black crosses in (c) denote ${\beta }_{\mathrm{ph}}$ at IWBA threshold as obtained from simulations.

Figure 4

Results of a 3D-PIC simulation with initial plasma density $n_0=3n_c$ and laser amplitude $a_0=155$; results are shown as cuts in $(x,z)$ plane at $y=-0.1\mu \mathrm{m}$. From left to right, results at different times: (a) 84 fs, (b) 165 fs, (c) 246 fs. From top to bottom, distribution of electron density ($n_e/a_0{n}_c$), ion density ($n_i/a_0{n}_c$), electric field $E_z/E_L$, and ions in $(z,p_z)$ phase space (arbitrary units). The dashed black vertical lines mark the location of the laser-driven electron layer (right line) and the trailing ion layer (left line) with the positive electric field in between. Inserts ($1\mu \mathrm{m}\times 1\mu \mathrm{m}$) in the ion density panels show blow-ups of the ion blobs (marked by black arrows), captured and accelerated in this field [column (b)] and just emerging in front of the electron layer [column (c)].

Figure 5

3D-PIC simulation results. (a) Ion energy spectrum (black line) and angular-spectrum distribution (color scale) at time 246 fs for the simulation displayed in Fig. 4. (b) Peak ion energy ${\mathcal{E}}_{i,\mathrm{peak}}$ (blue squares) and numbers of accelerated ions (green asterisks) for different initial plasma densities and laser amplitude $a_0=155$, same as in Fig. 4. The black solid vertical line and black square mark the threshold density and the corresponding ion energy as obtained from Eqs. (1)–(3). For $1n_c$ and $1.5n_c$, the maximum energies are plotted since there are no spectral peaks.