Lateral confinement of fast electrons and its impact on laser ion acceleration

In intense laser-plasma interactions, maximizing the density of fast electrons in the laser spot area is key to achieving plasma heating and particle acceleration. We find that when the laser spot size is large compared with the target foil thickness, fast electrons circulating in the foil show a “random walk” in the lateral direction due to the scattering by fluctuating fields at the plasma surface inside the spot area. We model the lateral motion as a diffusion, and find the resulting diffusion velocity is much slower than the speed of the ballistic transport. Hence, fast electrons accumulate in the spot region, and over time their density becomes typically 10 times greater than the laser-accelerated fast electron density. The enhancement of fast electron density in the target pushes the ion acceleration to more efficient regime.

Figure 1

2D PIC simulations with laser spot radii $1.4\mu \mathrm{m}$ (a), (b) and $35\mu \mathrm{m}$ (c)–(e). Panels (a) and (c) are trajectories of fast electrons in the simulations, (b) and (d) are snapshots of the electron energy densities at $t=1.2$ ps and $t=2.5$ ps, respectively, and (e) is the snapshot of the azimuthal magnetic field $B_z$ at $t=1.2$ ps.

Figure 2

Time evolutions of the lateral density profile of electrons having energies over 50 keV with $a_0=1.4$ for (a) small ($w=1.4\mu \mathrm{m}$) and (b) large ($w=35\mu \mathrm{m}$) spot lasers. The incident laser profiles are shown in the upper panels. (c) Electron energy distributions in the spot area $\left|y\right|<w$ for the small (blue) and large (red and orange) spots with $a_0=1.4$, and for the small spot with $a_0=7.0$ (blue dashed).

Figure 3

Lateral deviation length of electrons tracked in the 2D PIC simulation for $a_0=1.4$ with a uniform laser profile in the lateral direction. The average deviation for the tracked electrons is shown by the blue bold line. The diffusion scale length $L_{\mathrm{dif}}$ derived in Eq. (2) is presented by the black dashed line. A light speed trajectory is indicated by a black dash-dot line for reference.

Figure 4

Angular distribution of electrons whose energies are greater than about one fourth of the ponderomotive energy and less than 6 MeV in the velocity space inside the foil in the spot area. Panels (a) and (b) are the distributions in the large spot cases ($w=35\mu \mathrm{m}$) for $a_0=1.4$ and (b) $a_0=6.0$, respectively, and (c) and (d) are the distributions in the small spot cases ($w=1.4\mu \mathrm{m}$) for $a_0=1.4$ and $a_0=7.0$, respectively. The initial target thickness is $5\mu \mathrm{m}$ for all the cases. $\theta =0$ corresponds to the forward (positive $x$) direction.

Figure 5

(a) Fast electron densities at $y=0$ averaged inside the target in the PIC simulations (points) and that derived from the theory Eq. (10), $n_h(y=0)$ (background color). The black dashed lines are contour lines for the background color. The shaded areas are the regimes where the diffusion model is not applicable, i.e., the ballistic regime $\mathrm{\Delta }y\ge w$ where the solid vertical line corresponds to $\mathrm{\Delta }y=w$ (left), and the target disassembly regime ${\tau }_{\mathrm{conf}}\ge {\tau }_{\mathrm{dis}}$ for the initial target density $n_0=100n_c$ (right). The initial foil thickness $L_0$ in the simulations is $5\mu \mathrm{m}$ except for the squares. The white dotted line is a contour line for laser power for the same $L_0$. At $w/L_0=\infty$, simulation results for a quasi-1D condition with a periodic $y$-boundary are shown. The simulation values at $w/L_0=\infty$ can be reproduced well by Eq. (11). (b) The confinement time Eq. (6) for $L_0=5\mu \mathrm{m}$ and $10\mu \mathrm{m}$ with the relativistic approximation $\beta =1$.

Figure 6

2D snapshots of (a) electron energy density and (b) azimuthal magnetic field $B_z$ at $t=1.8$ ps in the PIC simulation with laser field amplitude $a_0=7.0$, spot radius $w=1.4\mu \mathrm{m}$, and initial foil thickness $L_0=5\mu \mathrm{m}$.

Figure 7

(a) Peak fast electron density, (b) maximum energy of deuteron ions from the target rear $E_{\mathrm{imax}}$, and (c) energy conversion efficiency $\eta$ from laser to ions with energies over 1 MeV/u and divergence angle $\pm 0.1$ rad from the target rear in the simulations. $E_{\mathrm{imax}}$ and $\eta$ are observed when the ion front expands $100\mu \mathrm{m}$ from the target rear. Spot radii $w$ divided by $L_0=5\mu \mathrm{m}$ are indicated in the plots. The black points are the cases where the diffusion model is not applicable ($w\le \mathrm{\Delta }y$). The triangles present simulations with the same laser power.