## 3 Results and Discussion
The wavelength of the laser is chosen to be 1 \(\upmu\)m and its power to 10 PW. The pulse duration is equal to 3 laser cycles, and the focal waist radius is equal to \(\lambda\). A circularly polarized laser pulse is employed. Figure 4: **(a)** Energy gain per nucleon for C\({}^{6+}\) nuclei and (**b**) the electric field of the pulse as a function of the phase \(\eta\), for a fixed optimal chirp parameter of \(b=0.089\). Other parameters are as described for Figure 2.  
Figure 5: **(a)** Energy gain per nucleon for C\({}^{6+}\) nuclei and (**b**) optimal chirp parameter \(b\) as a function of the pulse duration \(\tau\), given in units of the period \(\tau_{0}\). In these calculations, the pulse energy was kept fixed at a value of 100 J.  
Figure 3: Energy gain per nucleon for C\({}^{6+}\) nuclei as a function of the chirp parameter \(b\). The nuclei are assumed to be initially at rest, _i.e._, \(\gamma_{0}=1\). Other parameters are as described for Figure 2.  
The energy distribution of a beam of ions generated by laser acceleration was determined by many-particle calculations based on the coupled Newton-Lorentz equations (Equation (4)). We assume that the ions are randomly distributed in a small nano-scale cylindrical volume before interacting with the short focused laser pulse. The ions were assumed to be initially at rest, and have a solid-state density (2.4 \(\times\) 10\({}^{24}\) ions/cm\({}^{3}\)). Simulations were performed with and without the inclusion of the ion-ion interaction to assess the effect of Coulomb repulsion on the energy gain and its broadening. In both cases, the set of random initial coordinates utilized was kept constant. Results are shown in Figure 6. For this case, collective plasma effects are negligible since the induced plasma wavelength, \(\lambda_{p}=0.0088\)\(\upmu\)m, is much larger than the target thickness, \(L=0.0004\)\(\upmu\)m. Including the particle-particle repulsion yields an order-of-magnitude broader distribution than the simulation with artificially switching off the interaction. However, the average kinetic energy is not influenced by this: in the first, realistic simulation, the average gain is 82.557 MeV/u, with a standard deviation of 1.042 MeV/u (1.262%); in the latter case, the average gain is the same, while the standard deviation is 0.006 MeV/u (0.007%). Thus, we may conclude that for dense targets, it is indeed necessary to include inter-particle interactions in realistic simulations, even at the very high laser intensities considered. The beam energy spread is in the medically applicable range, _i.e._, approximately 1%, enabled by the small size of the target. The trajectories of the overdense interacting C\({}^{6+}\) ion beam are illustrated in Figure 7, and the emission angle is approximately 0.05 (radian), _i.e._, 2.87 (degree). Please note that for the angle we use the unit of radian in what follows. For the same conditions, except taking a larger chirp parameter, \(b=0.1\), the emission angle slightly increases from 0.05 to 0.051. Besides, for a longer laser pulse with the pulse duration \(\tau=4\tau_{0}\), the emission angle decreases to 0.048. In addition to a nano-scale target, simulations were made for a more extended underdense carbon plasma target, which may be realized as an expanding plasma created by a pre-pulse. The kinetic energy distribution for a target of ions with an initial density of 2.4 \(\times\) 10\({}^{20}\) ions/cm\({}^{3}\) is presented as a histogram in Figure 8. The induced plasma wavelength is \(\lambda_{p}=0.88\)\(\upmu\)m and the target thickness is \(L=0.04\)\(\upmu\)m. Therefore, collective plasma effects are ignored. In this case, particle-particle  
Figure 6: (**a**) Energy distribution histogram for an ensemble of 3000 interacting C\({}^{6+}\) ions, confined initially in a small cylinder (length 0.0004 \(\upmu\)m, diameter 0.002 \(\upmu\)m); (**b**) Energy distribution for the same set of ions, with the inter-particle Coulomb repulsion switched off. Note the different energy scales of the two sub-figures.