## 3 Results and Discussion  
We first present the calculation results for a single test ion. Bare carbon nuclei (C\({}^{6+}\) ions) are chosen here because of their higher contrast of dose deposition in the tissue as compared to protons. In order to find the optimal parameters of the pulse and the optimal initial velocity of the ion, we performed numerical simulations by calculating the energy gain for a particle as a function of the chirp parameter \(b\) and the initial energy \(E_{0}=\gamma_{0}mc^{2}\) of the ion. The results are presented in Figure 2. The optimal chirping is in the range around \(b=\pm 0.1\), with low initial kinetic energies around \(\gamma_{0}\approx 1.05\). We note that for the circularly polarized pulses employed here, the dependence of the energy on the chirping parameter is a slowly-varying function. This is not the case for linearly polarized fields, where this function shows a strong oscillatory behavior [29]. Therefore, the appropriate chirping can be more practically implemented in experiments. The maximal energy gain that can be reached in this setting is around 100 MeV/u, reaching the range of interest in medical applications. As the previous figure shows, the acceleration by a chirped short pulse is optimal when the initial kinetic energy of particles is low. Motivated by this, we separately consider the experimentally advantageous situation when the particles are initially at rest, _i.e._, \(\gamma_{0}=1\). The energy gain is shown in Figure 3 with dependence of the chirp parameter \(b\). The maximal gain that can be achieved by such pulses when the particles are at rest initially is approximately 80 MeV/u. In the case of circularly polarized fields, similar gains can be reached when applying a negative frequency chirp (\(b<0\)), _i.e._, when the carrier frequency of the laser decreases in time. Additionally, circularly polarized laser beams have the advantage over linearly polarized beams that they are not sensitive to small variations of the chirp parameter \(b\)--_cf._ Figure 3 with Figure 2 of Ref. [29]. Figure 4 shows, for an optimal chirp parameter \(b=0.089\), the energy gain and the corresponding pulse as a function of the longitudinal displacement \(z\), confirming that the final kinetic energy around 100 MeV is reached on a sub-wavelength scale. The figure also illustrates--in accordance with earlier findings [29]--that it is the asymmetric part of the pulse, induced by the strong frequency modulation, that causes sudden acceleration of the ion. This is in contrast to an unchirped laser pulse, where a charged particle would gain kinetic energy during the first half of a laser cycle and subsequently loses this energy during the second half of the respective cycle due to the symmetry of the pulse. However, we want to emphasize that frequency modulation over a large frequency bandwidth is presently experimentally accessible only at lower field intensities [38,39]. In order to find the optimal pulse duration for acceleration, we performed single-particle simulations calculating the energy gain of a particle initially at rest, with dependence on this parameter, at a fixed total pulse energy. The results are displayed in Figure 5a, showing that maximal efficiency can be reached for pulses with a time duration of 3 to 4 cycles. Increasing the pulse duration can only decrease the final energy gain, since the peak electric field of the pulse decreases if the energy of the pulse is distributed over a longer time. Furthermore, in pulses with durations below 3-4 periods, the ion does not spend sufficient time interacting with the field to reach its maximal velocity. For each pulse duration, the chirp parameter \(b\) was optimized independently. The corresponding values of the chirp parameters are displayed in Figure 5b. Figure 2: Energy gain in units of MeV/u with dependence on the chirp parameter \(b\) and the initial dimensionless energy parameter \(\gamma_{0}\) (_i.e._, Lorentz factor, \(E_{0}=\gamma_{0}mc^{2}\)).