Fig. 5: Proton focusing in particle-in-cell simulations.

Series of snapshots of (a) the proton beam density n_b, b the water molecule ionisation state, c the azimuthal magnetic fields and d the proton beam transverse phase space as the proton beam propagates through the vapour. The progress of the proton beam in each snapshot is indicated by the z-axis (a–c) and the text labels (d) (note that the bunch stretches longitudinally due to velocity dispersion). e, f Proton beam angular distributions as functions of the proton beam centroid position z_b for propagation through vapour and vacuum respectively. Both plots are normalised to the same value to allow direct comparison. The red dashed vertical lines in (e) indicate the corresponding positions of the snapshots in (a–d) and the blue line indicates the normalised water vapour density profile (scaled to the plot window), which had a maximum molecular density of 6 × 10¹⁷ cm⁻³.