Figure 4.

(a) Energy gain as a function of length of the HC target obtained from experiment (red) and simulations (Blue). The HC targets ( $0.7\pm 0.015\text{mm}$ internal diameter and $0.35\pm 0.015\text{mm}$ pitch) used in this case were similar to that used for the data shown in Fig. 3, except the length of HC which was varied from 2 to 10 mm. The simulations were carried out using the particle tracing code, as in Fig. 2, reconstructing the physical dimensions of the HCs used in the experiment and scanning for the HC length beyond the range studied in the experiment. The black circles show the simulated energy gain from a suitably designed varying pitch HC as calculated from the analytical model discussed in the text. (b) and (c) show dynamics of transiting protons (in a similar fashion as Fig. 2) through a constant and a variable pitched HCs, respectively. The HCs in both cases extend from z=1.5 to 16.5 mm with internal diameter 0.5 mm and initial pitch 0.5 mm. While the HC in (b) maintains a constant pitch over its full length, the pitch in (c) was varied according to Eq. (1), which gives a pitch increase of $\sim 13\mu \text{m}$ per turn. The beam divergence shown in (b,c) were taken at $z=100\text{mm}$ and the energy spectra were plotted for a half-angle divergence $<0.4^{\circ }$.