Self-Consistent Simulation of Transport and Energy Deposition of Intense Laser-Accelerated Proton Beams in Solid-Density Matter

The first self-consistent hybrid particle-in-cell (PIC) simulation of intense proton beam transport and energy deposition in solid-density matter is presented. Both the individual proton slowing-down and the collective beam-plasma interaction effects are taken into account with a new dynamic proton stopping power module that has been added to a hybrid PIC code. In this module, the target local stopping power can be updated at each time step based on its thermodynamic state. For intense proton beams, the reduction of target stopping power from the cold condition due to continuous proton heating eventually leads to broadening of the particle range and energy deposition far beyond the Bragg peak. For tightly focused beams, large magnetic field growth in collective interactions results in self-focusing of the beam and much stronger localized heating of the target.

Figure 1

(a) The calculated proton stopping power in lsp (with the new implementation) as a function of proton energy with target temperatures of $T_e=T_i=10$, 100, and 300 eV, (b) the corresponding Bragg curves of 2 MeV protons, (c) comparisons of the proton projected ranges in solid Al from lsp simulations with those from the scaalp theory [26, 27], and (d) comparison of the temporal evolution of the heated target temperature from lsp simulation and experimental measurement [11].

Figure 2

lsp simulation results of proton beam transport in a solid Al target, showing proton beam density (left) and target temperature distributions (right), at time $t=17\mathrm{ps}$. The beams have the same monoenergetic energies of 5 MeV, flattop distributed pulse durations 3 ps, and transversely Gaussian radii $14\mu \mathrm{m}$, but different current densities of, respectively, (a),(d) ${10}^9$, (b),(e) ${10}^{10}$, and (c),(f) ${10}^{11}\mathrm{A}/{\mathrm{cm}}^2$.

Figure 3

Results of lsp simulations similar to Fig. 2, with beam current density ${10}^{10}\mathrm{A}/{\mathrm{cm}}^2$ and a narrower beam radius $r_0=7\mu \mathrm{m}$, where the other parameters are the same. Panels (a)–(c) are the final heated target temperature maps at $t=17\mathrm{ps}$ for, respectively, the three categories (i)–(iii). Panel (d) corresponds to (c) but with the wide beam radius $14\mu \mathrm{m}$ case of Fig. 2. Panel (e) is the temporal evolutions of the total beam energy depositions, where dotted, dashed, and solid lines are for the cases (a), (b), and (c), respectively. Panel (f) is the corresponding evolution of the measured stopping power on axis and the evolution of the self-generated magnetic field at $r=r_0/4$ in (c). Panels (g) and (h) show the azimuthal magnetic fields generated in the simulations of (c) and (d), respectively.

Figure 4

Proton density (left) and heated target temperature (right) of Maxwellian proton beams in Al targets at $t=20\mathrm{ps}$. Panels (a) and (b) are for a broad beam, having current density $5\times {10}^9\mathrm{A}/{\mathrm{cm}}^2$ and a radius of $44\mu \mathrm{m}$; (c) and (d) are for a narrow beam of $5\times {10}^{10}\mathrm{A}/{\mathrm{cm}}^2$ and $12\mu \mathrm{m}$. Both beams have the same total energy of 6 J.