Fig. 1

Laser-driven magnetic-field generation and resulting particle dynamics. a, d, e, h 2D particle-in-cell (PIC) simulation results obtained using the (a, e) PICLS²⁶ and (d, h) PICADOR⁵⁹ codes (see Methods). The laser pulse impinges from the left onto a 2-μm thick Al foil, coated on its rear side (located at x = 12 μm) with a 20-nm thick proton layer. The laser FWHM spot size ϕ_L, duration τ_L, wavelength λ_L and intensity \(I_{\mathrm{L}}\lambda _{\mathrm{L}}^2\) are, respectively, (a, d) 1 μm, 400 fs, 0.5 μm, 6.5 × 10¹⁹ W μm² cm⁻² and (e, h) 1.6 μm, 700 fs, 1 μm and 2 × 10²¹ W μm² cm⁻². a, e Magnetostatic field B _z (in MG) developing inside and outside the target at 100 fs after the laser peak. d, h Sample electron trajectories from the PIC simulations, exiting the target at the laser peak. In d, the electron energies lie in the same range: 13.8 MeV (green), 14.3 MeV (red), 12.3 MeV (blue) and 18.7 MeV (cyan), yet the green electron proves more strongly magnetized because it is ejected into the vacuum about 50 fs later, and thus experiences a higher B-field. In h, the electron energies are 25.2 MeV (blue), 27.7 MeV (green), 90.9 MeV (red) and 162.7 MeV (cyan). b, c, f, g Results from the 1D expansion model (see Methods) at a laser intensity of (b, c) \(I_{\mathrm{L}}\lambda _{\mathrm{L}}^2 = 6.5 \times 10^{19}\,{\mathrm{W}}\,\mu {\mathrm{m}}^2{\mathrm{cm}}^{ - 2}\) and (f, g) \(I_{\mathrm{L}}\lambda _{\mathrm{L}}^2 = 2 \times 10^{21}\,{\mathrm{W}}\,\mu {\mathrm{m}}^2{\mathrm{cm}}^{ - 2}\) using the corresponding PIC simulation parameters. b, f Time evolutions of the proton velocity, ν_p(t), normalized to its final value (dashed green line), of the electron temperature, T_e(t), normalized to its initial value (T₀ = 1.1 MeV in b and T₀= 5.6 MeV in f, see Methods) (dashed-dotted red line), and of the inductive B-field, normalized to its predicted saturation value \(B_{{\mathrm{max}}} \equiv \left( {2\mu _0n_{{\mathrm{h}},{\mathrm{rear}}}k_{\mathrm{B}}T_0} \right)^{1/2}\) (B_max =144 MG in b and B_max =537 MG in f) (solid blue line). c, g Time evolutions of the electron (\(R_{\mathrm{L}}^{\mathrm{e}}\), blue) and proton (\(R_{\mathrm{L}}^{\mathrm{p}}\), red) radii, normalized to the instantaneous longitudinal extent of the proton plasma, x_front(t). The horizontal dashed black line delimitates the boundary between the regimes of strong (\(R_{\mathrm{L}}^{{\mathrm{e}},{\mathrm{p}}}/x_{{\mathrm{front}}} < 1\)) and weak (\(R_{\mathrm{L}}^{{\mathrm{e}},{\mathrm{p}}}/x_{{\mathrm{front}}} > 1\)) magnetization. The vertical dashed-dotted line indicates the time when the ion front has moved a distance larger than the local Debye length