## 3 Electron Acceleration with Radially Polarized Laser Beams
### Threshold for Sub-Cycle Acceleration and Attosecond Bunching  
We now proceed with the evaluation of the threshold for sub-cycle acceleration. To account for the contribution of the longitudinal electric field component, the normalized field parameter \(a_{0}\) is split into radial (\(a_{r}\)) and axial (\(a_{z}\)) components. These two new parameters are defined as follows:  
\[a_{r} =\left(\frac{q}{mc\omega_{0}}\right)|\tilde{E}_{r}|_{peak} \tag{13a}\] \[a_{z} =\left(\frac{q}{mc\omega_{0}}\right)|\tilde{E}_{z}|_{peak} \tag{13b}\] \begin{table}
\begin{tabular}{l c l} \hline \([z_{i},z_{f}]\) & \(\Delta W|_{z_{i}}^{z_{f}}\) & **Scenario** \\ \hline \([-\infty,\infty]\) & 0 & Lawson-Woodward [14] \\ \([-z_{R},z_{R}]\) & \(\Delta W_{\mathrm{lim}}\) & Limited interaction [14; 15; 69] \\ \([0,\infty]\) & \(\Delta W_{\mathrm{lim}}\) & Single pulse [19; 64; 72] \\ \([z_{R},\infty]\) & \(\Delta W_{\mathrm{lim}}/2\) & Pump-probe [16; 18; 67] \\ \hline \end{tabular}
\end{table} 
Table 2: Comparison between different electron acceleration scenarios in RPLBs. In the first column is the interaction range, followed by the corresponding change in energy calculated with Equation (11b), and finally a short description (scenario) with references. where \(|\tilde{E}_{r}|_{peak}\) and \(|\tilde{E}_{z}|_{peak}\) represent the peak values of the field envelope of the radial and longitudinal components, respectively. With Equations (1b) and (4), the threshold power corresponding to \(a_{z}^{2}=1\) is found to be  
\[P^{*}=\frac{9}{32}\frac{\pi}{\eta_{0}(k_{0}z_{0})^{3}}\left(\frac{mc^{2}}{q} \right)^{2}\frac{1-2(k_{0}z_{0})^{2}+2k_{0}z_{0}\sinh(2k_{0}z_{0})-\cosh(2k_{0 }z_{0})}{|j_{0}(jk_{0}z_{0})+j_{2}(jk_{0}z_{0})|^{2}} \tag{14}\]  
In the paraxial limit [18, 67]:  
\[\lim_{k_{0}z_{0}\gg 1}P^{*}\simeq\frac{\pi}{8\eta_{0}}\left(\frac{mc^{2}}{q} \right)^{2}(k_{0}z_{0})^{2}\simeq\frac{\pi^{5}}{2\eta_{0}}\left(\frac{mc^{2}} {q}\right)^{2}\left(\frac{w_{0}}{\lambda_{0}}\right)^{4} \tag{15}\]  
On the other hand, the analytical evaluation of \(a_{r}^{2}=1\), and of the corresponding threshold power, is not straightforward and best obtained numerically. Results are shown in Figure 2, where the different acceleration regimes in RPLBs are summarized. Figure 2: Overview of electron acceleration in RPLBs. The far right of the graph corresponds to the paraxial limit, where the dynamics is dominated by the transverse electromagnetic field. In this limit, the sub-cycle longitudinal acceleration regime is reached only if the average beam power is in the PW range (see also [18, 67]). Alternatively, the beam can be tightly focused to increase the peak intensity and lower the threshold down to the TWs and GWs [68, 75]. The sub-cycle regime characterized by \(a_{z}^{2}>1\) can itself be split into two sub-regimes corresponding to cases where the strength of the radial component is above threshold [Sub-cycle (1)] or not [Sub-cycle (2)]. Figure 4: In the sub-cycle regime, a collection of electrons initially at rest at the waist of an ultra-intense 12-fs (FWHM) RPLP (\(\lambda_{0}=800\) nm) is bunched to form an attosecond pulse. The initial electron positions (\(\bullet\)) followed a 100-nm spherical Gaussian distribution centered at \((r,z)=(0,0)\). Here, a magnified view reveals the extreme longitudinal compression experienced by the electrons during sub-cycle acceleration. The position of the leading edge of the accelerated electron distributions is about 6 mm away from beam waist. It was translated along \(z\) for a direct comparison of the relative durations. The size of the characters used to represent the particles gives the impression that the electron pulses are longer than they really are. For (+), we estimate that the duration along the longitudinal axis is as short as 170 zs. Simulation parameters were the following: (\(\times\)) \(P=100\) TW (\(k_{0}z_{0}\simeq 280\), \(a_{z}^{2}\simeq 4.7\)), (+) \(P=1\) PW (\(k_{0}z_{0}\simeq 850\), \(a_{z}^{2}\simeq 5.1\)). The carrier phase \(\phi_{0}\) was effectively optimized for the shortest durations.