## 3 Discussions  
The results of the LWFA simulation of electrons accelerated in plasma using the TH beam, focused using the OAP and the AXP+STC, are presented in Figures 3-5. Figure 4: The maximal energy \(E_{max}\) of accelerated electrons using the TH laser pulse with an energy of 85 mJ, which was focused using the OAP towards the focal plane \(z_{f}=30\)\(\mu\)m, \(z_{f}=330\)\(\mu\)m and \(z_{f}=430\)\(\mu\)m, using the AXP, and using the AXP with STC \(\alpha=0.007\). The plasma electron concentration was 1.2 \(\times\)\(10^{19}\) cm\({}^{-3}\), and the length of the ionisation injection zone in all cases was \(L_{n}=200\)\(\mu\)m.  
Figure 3: The dependence of the laser strength parameter \(a_{0}\) of the TH laser pulse with an energy of 85 mJ, which was focused using an OAP towards the focal plane at \(z_{f}=30\)\(\mu\)m, \(z_{f}=330\)\(\mu\)m, and \(z_{f}=430\)\(\mu\)m, using an AXP with a focal zone length of 800 \(\mu\)m, and using an AXP with STC \(\alpha=0.007\). The plasma electron concentration was \(1.2\times 10^{19}\) cm\({}^{-3}\), and the length of the ionisation injection zone in all cases was \(L_{n}=200\)\(\upmu\)m. The colour-marked areas represent the ionisation injection zone; area size correlates with the amount of accelerated charge and spread of the energy spectrum. In Figure 3, the dependence of the laser strength parameter \(a_{0}\) of the TH laser pulse, which was focused using an OAP towards the focal plane at \(z_{f}=30\)\(\upmu\)m, \(z_{f}=330\)\(\upmu\)m, and \(z_{f}=430\)\(\upmu\)m, using the AXP, and using the AXP+STC with \(\alpha=0.007\), propagating in a plasma medium with the electron concentration \(n_{e}=1.2\times 10^{19}\) cm\({}^{-3}\), are presented. In all cases, the laser pulse undergoes self-focusing, and the peak of the laser strength parameter \(a_{0}\) reaches a maximum value in the range of 3.3-4.4 in the focal zone. Despite the quite short Rayleigh length of \(z_{R}=98\)\(\upmu\)m of the OAP, the distance where \(a_{0}>2\) is approximately 500\(\upmu\)m long. In the case of the AXP, with a focal zone length of 800\(\upmu\)m, the distance where \(a_{0}>2\) is slightly higher and extends over 700\(\upmu\)m. The initial starting values of the laser strength parameter \(a_{0}\) affect the injection mechanism of electrons. The ionisation injection starts when \(a_{0}>2\), and with a limited length of the ionisation injection zone \(L_{n}=200\)\(\upmu\)m, the actual injection continues for as long as 200\(\upmu\)m for the OAP when focused onto the focal plane \(z_{f}=30\)\(\upmu\)m, 100\(\upmu\)m when focused onto the focal plane \(z_{f}=330\)\(\upmu\)m, and \(\sim\)50\(\upmu\)m when focused onto the focal plane \(z_{f}=430\)\(\upmu\)m, and when focused using the AXP. A shorter injection zone results in the formation of quasi-monoenergetic electron bunches, preferable for VHEE applications. However, the total charge of accelerated electrons becomes smaller. In such a way, adjusting the focal position inside the gas jet could help to control the ionisation injection length of electrons. The longitudinal distribution of the laser strength parameter \(a_{0}\) in the focal zone of the focused TH laser beam for the AXP+STC case presented in Figure 3 is similar to the longitudinal distribution of the a\({}_{0}\) of the TH laser beam which was focused using the AXP. However, because of the longer dephasing length, the electrons reach higher energies (Figures 4 and 5). In Figure 4, the dependence of the maximal energy  
Figure 5: The energy spectra of accelerated electrons using the TH laser pulse with an energy of 85 mJ, which was focused using an OAP towards the focal planes \(z_{f}=30\)\(\upmu\)m (**a**) and \(z_{f}=330\)\(\upmu\)m. (**b**), using the AXP (**c**), and using the AXP with STC parameter \(\alpha=0.007\) (**d**).