Fig. 1. (a) Schematic of a single helical coil. (b) A helical coil with a tube. In the model, the helix is replaced by a thin cylinder that is symmetric under rotation around the z axis, as in Fig. 2. Reproduced with permission from Hirsch-Passicos et al., “Helical coil design with controlled dispersion for bunching enhancement of protons generated by the target normal sheath acceleration,” Phys. Rev. E 109, 025211 (2024). Copyright 2024 American Physical Society.¹²

Fig. 2. (a) View of TWT along the symmetry axis. (b) Schematic of loaded TWT.¹³ The helical coil (orange) is confined in the tube (gray) and supported by nonconductive ties (green). Reproduced with permission from Jiao et al., “A key design and experiment of a broadband high-power pulsed helix TWT,” in 2019 International Vacuum Electronics Conference (IVEC) (IEEE, 2019), pp. 1–2. Copyright 2019 IEEE.

Fig. 3. Examples of helical coil geometries with (a) varying radius and (b) varying pitch along the z axis. The tube is symmetric under rotation around the symmetry axis as in Fig. 2.

Fig. 4. Schematic of model construction.

Fig. 5. Diagram showing the direction of current propagation at the surface of a cylinder at an angle Ψ to the azimuthal direction.

Fig. 6. Dependence of frequency ω on wavenumber k for an HC of radius a = 0.5 mm and linearly varying pitch (on a length of l_c = 40 mm) of h = 0.35–0.7 mm, without tube (red), with tubes of radii b/a = 1.4 (blue) and 1.8 (green), and in a perfect case without any dispersion (dashed black).

Fig. 7. Diagram of DoPPLIGHT code. The coil parameters correspond to the radius a, the pitch h and the length L. The source term provides the distribution of particles in energy, angle, and time. The electric and magnetic fields in the helix are calculated according to Eqs. (28) and (29). The space-charge fields account for Coulomb repulsion. Particle displacement is calculated with the Boris pusher.

Fig. 8. Coordinate dependence of the axial electric field E_z at r = 0.2 mm for coils (a) without and (b) with tube. Dashed lines show the result of the PIC simulation (SOPHIE), and solid lines are obtained with the DoPPLIGHT code. The helix radius is 0.5 mm, the pitch 0.35 mm, and the tube radius 0.7 mm.

Fig. 9. Coordinate dependence of the axial electric field E_z at r = 0.2 mm for coils (a) without and (b) with tube and pitch varying linearly with the coordinate [Eq. (30)]. Dashed lines show the results of PIC simulations (SOPHIE), and solid lines are obtained with the DoPPLIGHT code. Other parameters are given in the text.

Fig. 10. Energy distribution of the protons accelerated in the helical coil without tube (a) and with tube (b) calculated with SOPHIE code (red line) and DoPPLIGHT code (blue line) with a linearly varying pitch. Panel (c) compare the case of a linearly varying pitch and the constant pitch. Black line shows the input proton spectrum. The parameters of the coil and the current are given in the text.

Fig. 11. Dependence of WKB validity conditions (A2) on wavenumber k for the simulation parameters used in this work.

Fig. 12. Coordinate dependence of the axial electric field E_r at r = 0.2 mm for coils (a) without and (b) with tube, and for a pitch varying linearly for coils (c) without and (d) with tube. Dashed lines show the results of the PIC simulation (SOPHIE) and solid lines those obtained with the DoPPLIGHT code. The helix radius is 0.5 mm, the pitch 0.35 mm, and the tube radius 0.7 mm.

Fig. 13. Coordinate dependence of the current propagating in the helix j₀/sin Ψ for coils (a) without and (b) with tube, and for a pitch varying linearly for coils (c) without and (d) with tube. Dashed lines show the results of the PIC simulation (SOPHIE) and solid lines those obtained with the DoPPLIGHT code. The helix radius is 0.5 mm, the pitch 0.35 mm, and the tube radius 0.7 mm.

Fig. 14. Energy distribution of protons accelerated in helical coils (a) without and (b) with tube calculated with the SOPHIE code (red lines) and DoPPLIGHT code (blue lines). The black line shows the input proton spectrum. The parameters of the coil and the current are given in the text.