## 4 Acceleration by Ultrashort and Tightly Focused Radially Polarized Laser Pulses
The maximum energy gain, denoted \(\Delta W_{\text{max}}\), is calculated numerically by optimizing the electron's initial position on the optical axis and the laser pulse phase such that the final kinetic energy of the particle is maximal (see also [68]). Figure 6 shows the variation of \(\Delta W_{\text{max}}\) with the laser power for different combinations of \(k_{0}z_{0}\) and \(s\). Figure 6(a), in which \(\Delta W_{\text{max}}\) is expressed as a fraction of the theoretical energy gain limit \(\Delta W_{\text{lim}}\) [see Equation (12)] shows that for constant values of \(s\), the power above which significant acceleration occurs is greatly reduced as \(k_{0}z_{0}\) decreases, _i.e._, as the focusing is made tighter. According to Figure 6(b), MeV energy gains may be reached under tight focusing conditions (\(k_{0}z_{0}\sim 1\)) with laser peak powers as low as 15 GW. In contrast, previous works based on the paraxial approximation suggested that powers three orders of magnitude higher would be required to reach MeV kinetic energies [68]. At high peak power, Figure 6(a) shows that shorter pulses yield a more efficient acceleration, with a ratio \(\Delta W_{\text{max}}/\Delta W_{\text{lim}}\) reaching 80% for single-cycle (\(s=1\)) pulses. This is mainly a consequence of the fact that shorter pulses allow the electron to move close to the pulse peak; in longer pulses, the electron is trapped and accelerated by the front edge of the pulse. As explained in [75], where additional details about the results discussed in this paragraph may be found, the data shown in Figure 6 is completely independent of the dominant wavelength \(\lambda_{0}\) of the TM\({}_{01}\) laser pulse. Figure 5: **(a)** Poisson spectrum \(F(\omega)\) for different values of \(s\) with \(\phi_{0}=0\); **(b)** Temporal profile of the on-axis longitudinal electric field component at \(z=0\) of the corresponding TM\({}_{01}\) pulsed beam with \(\lambda_{0}=0.8\)\(\mu\)m and \(k_{0}z_{0}=10\). For moderately short pulses (\(s\gtrsim 50\)), Figure 6(b) shows well the gradual transition between the three acceleration regimes presented in Table 1. To illustrate this more clearly, the case \(k_{0}z_{0}=100\) and \(s=100\) is shown alone in Figure 6(c). At low laser power, \(\Delta W_{\text{max}}\) scales as \(P^{2}\). This is a consequence of the ponderomotive force, which is proportional to \(F_{p}\sim A_{0}^{2}\sim P\) and consequently leads to an energy gain that increases as \(P^{2}\). As the power is further increased, the dynamics undergoes a transition to the relativistic regime where sub-cycle acceleration begins. The maximum final kinetic energy of the electron increases rapidly since it is now allowed to copropagate with the laser pulse over longer distances. Eventually, the energy that the electron is able to extract from the pulse saturates toward a constant fraction of the theoretical energy gain limit \(\Delta W_{\text{lim}}\) [see Figure 6(a)]. This marks the onset of the ultrarelativistic regime, which is characterized by the scaling relation \(\Delta W_{\text{max}}\sim P^{1/2}\), in agreement with Equation (12). For ultrashort pulses (\(s\lesssim 50\)), \(\Delta W_{\text{max}}\) does not scale as \(P^{2}\) in the nonrelativistic regime. This shows that the ponderomotive force model is not appropriate to describe the interaction of electrons with pulses of a few-optical-cycle duration. Experimental Observation of Electron Acceleration with Tightly Focused Radially Polarized Laser Beams  
Recently, electron acceleration by tightly focused few-cycle RPLPs was demonstrated using an high-power infrared (IR) laser source available at the Advanced Laser Light Source (ALLS) facility (INRS, Varennes, Qc, Canada) [20]. In this section, we describe the method used to generate few-cycle RPLPs, specify the experimental conditions in which electron acceleration was observed, and present the characteristics of the accelerated electrons.