Proton sheet crossing in thin relativistic plasma irradiated by a femtosecond petawatt laser pulse

Leveraging on analyses of Hamiltonian dynamics to examine the ion motion, we explicitly demonstrate that the proton sheet crossing and plateau-type energy spectrum are two intrinsic features of the effectively accelerated proton beams driven by a drift quasistatic longitudinal electric field. Via two-dimensional particle-in-cell simulations, we show the emergence of proton sheet crossing in a relativistically transparent plasma foil irradiated by a linearly polarized short pulse with the power of one petawatt. Instead of successively blowing the whole foil forward, the incident laser pulse readily penetrates through the plasma bulk, where the proton sheet crossing takes place and the merged self-generated longitudinal electric field traps and reflects the protons to yield a group of protons with plateau-type energy spectrum.

Figure 1

The black dashed line represents the electric potential $\phi \left(\xi \right)$ while the blue solid one illustrates the longitudinal electric field $E\left(\xi \right)$. (b) The background contour with brown color map renders the Hamiltonian value $\mathcal{H}(\xi ,p)$ in the unit of proton rest energy $m_p{c}^2$ and the blue dashed lines denote the boundary of the electric field at $\xi =-1\mu \text{m}$ and $2\mu \text{m}$. The lines with the rainbow colorcode exhibit the evolution of protons (originally locating at $-1⩽\xi \left[\mu \text{m}\right]⩽2$) in $\xi -p$ phase space. Panels (a) and (b) share the same horizontal axis.

Figure 2

The scattering distribution of protons in $x-p$ phase space, where the protons initially locating at $0⩽\xi \left[\mu \mathrm{m}\right]⩽2$ is coded by rainbow color map. The inset illustrates the proton energy spectrum at time $t=100$ fs. (b) The trajectories of proton motion, where the background cool-warm color map renders the strength of drifting QSELF. Panels (a) and (b) share the same horizontal axis and rainbow color scale.

Figure 3

The trajectories of proton in $(\xi ,p)$ space. All the parameters of the simulations are same as those in Fig. 1 except for the drift velocity being ${\beta }_d=0.45$ in panel (a) and ${\beta }_d=0.43$ in panel (b).

Figure 4

The spatial distributions of the transverse electric field $E_y$ and plasma electron density $n_e$ are presented in panels (a, b, c) for different time snapshots. The gray arrow in panel (a) marks the propagation direction of the incident laser pulse. The black lines in panels (b, c) outline the interface of plasma critical density ${\overline{n}}_e=n_c$, where the overline denotes the value which is temporally averaged over four laser period. The spatial distributions of the time averaged longitudinal field ${\overline{E}}_x$ and proton density $n_p$ are exhibited in panels (d, e, f), which corresponds to the same time as that in panels (a, b, c), respectively. The scattering plot of proton particles is illustrated in panel (g) where the colors represent the different time.

Figure 5

Panels (a, b, c) show the distribution of protons with the color rendering their original position $x_0$, while panels (d, e, f) exhibit the proton distribution in $p_x-x$ phase space with the same color scheme. Panel (g) outlines the profile of longitudinal electric field $E_x$ for four simulation time, where $E_x$ is transversely averaged over $-2.5⩽y\left[\mu \text{m}\right]⩽2.5$ and the dotted black lines mark the target initial position. Panel (h) shows the evolution of typical electron trajectories (rendered in winter color map) during $107<t<120$ fs, where the distribution of proton particles and transverse electric field are presented as well. Panel (i) exhibits the electron energy contribution from transverse work ${\mathrm{W}}_y$ and longitudinal work ${\mathrm{W}}_x$.

Figure 6

Panel (a) exhibits electron energy spectral for different laser field amplitude $a_0$ at time $t=67$ fs, where the reciprocal slope of dashed lines indicate the electron temperature. Panel (b) is same as panel (a) but for $t=93$ fs. Panel (c) presents the dependence of electron temperature on laser intensity $a_0$.

Figure 7

Panel (a) plots the proton energy spectra for three cases. Panels (b–d) show the ratio of “left,” “mid,” and “right” proton occupation over the proton energy spectra, respectively.

Figure 8

(a) The electric potential $\phi$ extracted from 2D PIC simulations and the dotted black line refers to the fitting $\phi \left(a_0\right)$. (b) Drifting velocity ${\beta }_d$ extracted from simulations and the dotted black line marks the fitting ${\beta }_d$ while the blue solid line ${\beta }^{*}\left(\phi \right)$ is based on the fitting $\phi \left(a_0\right)$. Panel (c) illustrates dependence of the proton cutoff energy on the laser intensity $a_0$, where the dashed red line corresponds to the theoretical prediction of Eq. (10) while the dashed blue (green) line refers to the scaling of conventional TNSA (LS) model. The blue shadow area represents the model of thermal acceleration with plasma-bound electrons and the gray dashed line denotes the model of Coulomb explosion with mass-limited target.