## 2 Materials and Methods  
Most high-intensity lasers generate beams with a TH, super-Gaussian energy distributor across the aperture. For the simulation of the LWFA of electrons using a TH laser beam, the quasi-3D Fourier-Bessel particle-in-cell (FBPIC) code version (0.25.0) was used [14]. The code employs the azimuthal modal decomposition method to describe the transverse profile of the laser beam, enabling the efficient simulation of the spatial-temporal evolution of the laser pulse in plasma with reduced computational resources. Three configuration cases were numerically investigated, using different approaches to focus the TH laser beam onto a gas target for laser wakefield acceleration:  
* TH beam focused using an OAP;
* TH beam focused using an AXP;
* TH beam focused using an AXP+STC, allowing control of the laser pulse's front curvature. In the simulations, a gas target consisting of two sections was used. The first section, which was L\({}_{\text{n}}\) = 200 \(\upmu\)m in length and with helium and a 1% nitrogen mixture, was implemented for the ionisation injection. The second section, which was 800 \(\upmu\)m in length and with pure helium, was only used to accelerate electrons. Such a gas target could be manufactured from a single block of fused silica using 3D laser machining technology developed by FTMC located in Vilnius, Lithuania [15]. The configuration used in the PIC simulation is presented in Figure 1. The plasma density corresponding to the highest energy of accelerated electrons was n\({}_{\text{e}}\) = 1.2 \(\times\) 10\({}^{19}\) cm\({}^{-3}\). In all the simulations, the total acceleration distance was 800 \(\upmu\)m. The beginning of the plasma zone was at a distance of 25 \(\upmu\)m, the ramp-up length of the plasma profile was 10 \(\upmu\)m, and the peak of the laser pulse at the start of the simulation was a distance of 15 \(\upmu\)m from z = 0 \(\upmu\)m, i.e., in the middle of the flying coordinate window. The key parameters of the laser beams and plasma are provided in Table 1. Furthermore, the simulation inputs and associated computational details are included in Table 2. The TH transverse distribution of the intensity of the laser pulse incident to the OAP/AXP mirror with a Gaussian temporal envelope of the pulse was applied in the simulation. The Fresnel approximation was used to calculate the electric field of the laser wave at the focal plane. Figure 1: The simulation setup: A TH laser beam is focused using an OAP or AXP onto a double gas target with a section of the He+1%N\({}_{2}\) mixture used for electron injection and a pure He section for electron acceleration. The length of the gas jet section of the He mixture with 1% N\({}_{2}\) was L\({}_{\text{n}}\) = 200 \(\upmu\)m.  
For the simulation, an incident TH laser pulse with an energy of 85 mJ and a diameter of \(d=50\) mm was chosen. The focal lengths of the OAP and AXP were \(f=400\) mm, corresponding to the _f/d_ = 8, the focal point \(z_{f}=430\) mm, and the focal depth of AXP \(\delta=800\) mm. The longitudinal distributions of the peak-normalised vector potential (laser strength parameter) \(a_{0}\) in the focal zone of the TH laser beam focused onto the vacuum using the OAP and the AXP used in the simulations are presented in Figure 2a. The simulations using an OAP were carried out for three cases: for the focal plane positions \(z_{f}=30\) mm, \(z_{f}=330\) mm, and \(z_{f}=430\) mm, starting the simulation at the plane \(z=0\) mm. This allowed us to evaluate the impact of initial starting values of laser strength parameter \(a_{0}\) on the electron injection and acceleration process. The vertical dotted line indicates the length of the injection section of the gas target. The focal depth of AXP \(\delta=800\) mm was chosen with the intention of maintaining the \(a_{0}\) in the range of 1.5-2.5 for a laser pulse with an energy of 85 mJ. It results in a similar longitudinal \(a_{0}\) profile of the beam, which was focused using both the OAP and AXP.