## 2 Materials and Methods
In Figure 2b, the longitudinal distributions of the \(a_{0}\) of the TH laser beam, which was focused on the vacuum for different values of the focal depth of the AXP \(\delta=0.8\) mm, \(\delta=2.5\) mm, and \(\delta=5\) mm, are shown. The implementation of advantages of the long plateau of \(a_{0}\), which is typical for the electron acceleration using an AXP to focus the pulse of the Joule class laser [9; 10; 11], is not relevant in the case of the limited pulse energy of kHz lasers The a\({}_{0}\) values drop below one for \(\delta>2.5\) mm. For a small length of focal zone, the laser intensity is not constant along the focal zone of the AXP. It reaches its maximum value in the centre of the focal zone and drops down at the \begin{table}
\begin{tabular}{c c} \hline
**Physical Parameters** & **Values** \\ \hline Pulse duration (FWHM), \(\tau_{0}\) (fs) & 10 \\ Pulse centre start position, \(z_{0}\) (\(\mu\)m) & 20 \\ Pulse length (FWHM), \(L_{0}=\tau C_{0}\) (\(\mu\)m) & 3 \\ Laser strength parameter, \(a_{0}\) & 1.5–2.5 \\ Laser wavelength, \(\lambda\) (\(\mu\)m) & 0.8 \\ Laser pulse energy, (mJ) & 85 \\ Focus position, \(z_{\mathrm{gc}}\) (\(\mu\)m) & 430 \\ Beam waist, \(w_{0}\) (\(\mu\)m) & 5 \\ Rayleigh range for Gaussian beam, \(z_{R}\) (\(\mu\)m) & 98 \\ Rayleigh range for Bessel–Gauss beam, \(z_{R}\) (\(\mu\)m) & 800 \\ Hydrogen concentration, \(n_{atoms}\) (\(10^{19}\) cm\({}^{-3}\)) & 1.2 \\ Start position in \(z\) (\(\mu\)m) & 0 \\ Total z-extension, (\(\mu\)m) & 800 \\ Ramp length of the first nozzle (\(\mu\)m) & 10 \\ Ionisation injection zone length (\(\mu\)m) & 200 \\ Plasma wavelength, \(\lambda\)p (\(\mu\)m) & 9.5 \\ \hline \end{tabular}
\end{table} 
Table 1: Simulation parameters for laser and plasma. \begin{table}
\begin{tabular}{c c} \hline
**Simulation Parameters** & **Values** \\ \hline Number of grid points along \(z\) & 600 \\ Number of grid points for Gaussian beam along \(r\) & 300 \\ Number of grid points for Bessel–Gauss beam along \(r\) & 600 \\ Number of particles per cell along \(z\) & 2 \\ Number of particles per cell along \(r\) & 2 \\ Number of particles per cell along \(\theta\) & 3 \\ Number of azimuthal modes, \(n_{m}\) & 3 \\ Simulation box size along, \(z\) (\(\mu\)m) & 30 \\ Simulation box size for Gaussian beam along, \(r\) (\(\mu\)m) & 60 \\ Simulation box size for Bessel-Gaussian beam along, \(r\) (\(\mu\)m) & 200 \\ Speed of the moving window, \(v_{w}(c)\) & 1 \\ Simulation time step, \(\Delta t\) (fs) & 1/3 \\ \hline \end{tabular}
\end{table} 
Table 2: Computing parameters used in the FBPIC simulations. edges. By reflecting the beam with the radial echelon, introducing a delay within the laser pulse depending on the radial coordinate, and focusing the laser beam with the AXP, it is possible to control the group velocity of the laser pulse in the focal zone of an AXP [11]. In this way, the spatial-temporal structuring of laser radiation allows us to compensate for the dephasing of the laser pulse in the plasma. When the pulse velocity in the plasma is equal to the speed of light in the vacuum, we have a dephasingless laser wakefield accelerator (DLWFA) mode when dephasing has a negligible effect on electron acceleration [16,17]. To model the DLWFA mode, the temporal envelope of the laser pulse with the radial dependence of the pulse peak delay \(E(r,t)=\exp\Bigl{(}-\bigl{(}\bigl{(}t-\beta r^{2}\bigr{)}/\tau\bigr{)}^{2} \Bigr{)}\) in the Fresnel integral was used to calculate the electric field at the focal plane, where the delay parameter \(\beta=(\delta/c)\alpha\), and \(\alpha=(v-c)/c\) defines the relative change in group velocity due to STC. The LWFA simulation of electrons using the AXP+STC was carried out for three cases of the STC parameter \(\alpha=0.005\), \(\alpha=0.007\), and \(\alpha=0.009\).