Fig. 4

Magnetic deflections of protons at high laser intensity. a, b Maps of the proton energy density (in units of 100 n_c keV, where n_c = 1.1 × 10²¹ cm⁻³ is the critical density at 1 μm wavelength) as recorded, at the time of the laser peak, from the 2D PICLS simulations illustrated in Fig. 1a and e, respectively. b clearly reveals that at high laser intensity, the protons at the acceleration front are deflected by the magnetic field, forming a ring-like pattern. c–h Experimental 2D proton dose distributions (in Gy) measured using stacks of calibrated radiochromic films. In c–e, the laser parameters are \(I_{\mathrm{L}}\lambda _{\mathrm{L}}^2 = 9 \times 10^{19}\,{\mathrm{W}}\,\mu {\mathrm{m}}^2\,{\mathrm{cm}}^{ - 2}\), λ_L = 0.5 μm, ϕ_L = 0.9 μm, τ_L = 400 fs, and the target consists of a 0.5 μm thick Al foil. In f–h, the laser parameters are \(I_{\mathrm{L}}\lambda _{\mathrm{L}}^2 = 1.3 \times 10^{21}\,{\mathrm{W}}\,\mu {\mathrm{m}}^2\,{\mathrm{cm}}^{ - 2}\), λ_L = 1 μm, ϕ_L = 1.5 μm, τ_L = 800 fs, and the target is a 1.1 μm thick Au foil, yielding a ring-like pattern on the proton dose distribution, consistent with the simulation shown in b. For the > 28 MeV protons, this ring pattern encircles what looks like a central jet, which may result from some high-energy protons emitted on axis and having experienced relatively weak deflections. The white bars in c–h indicate an angular spread of 20°. i–k Proton dose vs. angle with respect to the target-rear normal, as extracted from the proton distribution displayed in f–h, respectively. Arrows indicate the angular peaks