Fig. 1. Geometry of the radiography of E and B fields with protons of multiple energies. The field region contains isosurfaces of the E and B fields at E_s = 10⁸ V/m and B_s = −1 T.

Fig. 2. (a) and (b) Spatial distributions of flux density perturbations (δn/n₀)₁ and (δn/n₀)₂ in the detection plane during radiography of the fields shown in Fig. 1 with protons in parallel, corresponding to protons of ε_k1 = 20 MeV and ε_k2 = 40 MeV, respectively. (c) and (d) Reconstructed spatial distributions of the deflection velocities u_d1 and u_d2 corresponding to protons of ε_k1 = 20 MeV and ε_k2 = 40 MeV, respectively.

Fig. 3. (a) and (b) 2D distributions of the reconstructed and pre-set ∫Edx, respectively; (c) 1D comparison of these distributions along y = 0. (d) and (e) 2D distributions of the reconstructed and pre-set ∫Bdx, respectively; (f) 1D comparison of these distributions along y = 0.

Fig. 4. If the field imaged with lower-energy probe protons is changed by a given factor of f_E,B = 0.2 compared with that imaged with higher-energy probes, the reconstructed (a) ∫Edx and (b) ∫ex×Bdx will be closer to the pre-set distributions when the energy gap ε_k2 − ε_k1 is larger.

Fig. 5. (a) and (b) There will be differences between the reconstructed and preset ∫Edx and ∫ex×Bdx if the field imaged with lower-energy probes is changed by a factor of f_E,B compared with that imaged with higher energy probes. (c) and (d) Difference between the reconstructed and pre-set fields at z = 50 µm for different values of the field growth rate η: (c) ∫(ER−EP)dx; (d) ∫(BR−BP)dx. This indicates that for larger ε_k1 − ε_k2, the reconstructed fields will differ more from the pre-set fields.

Fig. 6. When f_E,B are estimated as f_E0,B0, the difference between the pre-set distributions and the reconstructed (a) ∫Edx and (b) ∫ex×Bdx can be largely eliminated.

Fig. 7. δn/n₀ in the detection plane in the proton radiography of a current filamentary instability for (a) ε_k1 = 20 MeV and (b) ε_k2 = 40 MeV.

Fig. 8. Deflection velocities for proton beams of kinetic energy ε_k1 = 20 MeV. (a) u_dy1 directly obtained from radiography. (b) u_dy1 reconstructed from (δn/n₀)₁ in Fig. 7(a). (c) 1D comparison along z = 80 µm. (d) Directly obtained u_dz1. (e) Reconstructed u_dz1. (f) 1D comparison along y = 160 µm.

Fig. 9. Deflection velocities for proton beams of kinetic energy ε_k2 = 40 MeV. (a) u_dy2 directly obtained from radiography. (b) u_dy2 reconstructed from (δn/n₀) ₂ in Fig. 7(b). (c) 1D comparison along z = 80 µm. (d) Directly obtained u_dz2. (e) Reconstructed u_dz2. (f) 1D comparison along y = 160 µm.

Fig. 10. Comparison of ∫Eydx: (a) reconstructed from proton radiography; (b) directly obtained from PIC simulation; (c) 1D comparison along z = 44 µm. Comparison of ∫Ezdx: (d) reconstructed from proton radiography; (e) directly obtained from PIC simulation; (f) 1D comparison along y = 88 µm.

Fig. 11. Comparison of ∫Bzdx: (a) reconstructed from proton radiography; (b) directly obtained from PIC simulation; (c) 1D comparison along z = 44 µm. Comparison of ∫Bydx: (d) reconstructed from proton radiography; (e) directly obtained from PIC simulation; (f) 1D comparison along y = 88 µm.

Fig. 12. ∫Edx and ∫u0×Bdx directly obtained from PIC simulation in (a) the y and (b) the z direction along z = 80 µm, in which the amplitudes of ∫u0×Bdx have been amplified by a factor of 10. These results indicate that the E field greatly dominates the B field in deflecting the probe protons.