Fig. 1. Schematic of simulation setup. (a) Position of proton source; (b) initial state of proton beam in the simulation;(c) left-side position of solenoid; (d) right-side position of solenoid; (e) focus spot position of 250 MeV proton

Fig. 2. Proton y-py phase space diagram at the initial time

Fig. 3. (a) Magnetic field distribution on the axis; (b) magnetic field axial distribution at different r in a solenoid

Fig. 4. Controlled proton beam spatial divergence angle by using a solenoid. (a) Proton divergence angle distribution at initial time; (b) spatial divergence angle and proton beam central position changes with time

Fig. 5. x-y and y-z views of proton beam density distribution. (a)(e) Initial moment, t=160 ps; (b)(f) after protons pass completely through the right end of the solenoid, t=6500 ps; (c)(g) low-energy protons start to focus, t=9750 ps; (d)(h) protons around 250 MeV start to focus, t=13000 ps

Fig. 6. Proton parameters at focal spot. (a) x-y view of proton beam density distribution near focal spot; (b) mesh plot of proton density distribution at focal spot; (c) proton energy spectrum near focus position; (d) spatial distribution of all protons passing through the focal spot transverse profile