## 4 Acceleration by Ultrashort and Tightly Focused Radially Polarized Laser Pulses
### Method for Generating Tightly Focused Ultrashort RPLPs  
The ALLS few-cycle IR beam line can deliver 1-mJ 15-fs pulses in the fundamental Gaussian (TEM\({}_{00}\)) transverse mode at a repetition rate of 100 Hz [80]. The central wavelength is 1.8 \(\mu\)m and the laser pulse energy stability is of the order of 2.5% rms. The \(s\) parameter associated with these IR pulses is approximately \(s=125\).  
To create the RPLB profile, we used a combination of wave plates specifically oriented in such a way that the phase of the electric field taken at any position on the beam plane is shifted by \(\pi\) compared to its antipodal counterpart (see Figure 7). The polarization state converter (PSC) used in the experiments is composed of 4 sections of achromatic IR half-wave plates, each having a different fast axis orientation, as shown in Figure 7(b). The created RPLB far-field mode profile, when focused by low numerical aperture (NA) optics, is shown in Figure 7(d). It is comparable to the theoretical RPLB shown in Figure 1(a), with an annular intensity profile and a minimum at the center. Weak focusing was used to image the beam with sufficient resolution. In this limit, the longitudinal field is negligible (\(|E_{z}|^{2}\ll|E_{r}|^{2}\)). To reach the longitudinal acceleration threshold (\(a_{z}^{2}\sim 1\)), the few-cycle RPLP was focused to a sub-wavelength focal spot with an on-axis high NA (0.7) parabolic mirror.  
In the experimental configuration, characterization of the focal spot at high intensity is not possible by conventional methods due to the high numerical aperture and the on-axis geometry of the focusing parabola. Local surface quality of the parabola was tested and found to be better than \(\lambda_{0}/4\) at \(\lambda_{0}=675\,\mathrm{nm}\) with a diode laser source. Performances were extrapolated to 1.8 \(\mu\)m. The beam waist is estimated to \(w_{0}\approx 0.6\lambda_{0}\), corresponding to \(k_{0}z_{0}\approx 8\) and a transverse beam profile somewhere in between those shown in Figure 1(b) and 1(c) (\(|E_{z}|^{2}\gtrsim|E_{r}|^{2}\)). The energy measured after the parabola is 550 \(\mu\)J per pulse. This results in a peak power of 36 GW and an intensity of \(7.2\times 10^{17}\) W/cm\({}^{2}\), approximately. According to Figure 2, this corresponds to a normalized parameter \(a_{z}^{2}\simeq 1.7\). However, a comparison with Figure 6 suggests that this value of \(a_{z}^{2}\) overestimates the real strength of the interaction. In the light of the predictions made in Section 4 and the experimental results presented below, it is most likely  
Figure 7: Method to generate the lowest-order RPLB. (**a**) Transverse profile of the incident linearly polarized Gaussian (TEM\({}_{00}\)) beam; (**b**) The polarization state converter (PSC) composed of 4 sections of IR achromatic half-wave plates with the fast axis orientation of \(0^{\circ}\) (top), \(+45^{\circ}\) (right), \(+90^{\circ}\) (bottom), \(-45^{\circ}\) (left); (**c**) The near field beam profile and relative electric field direction at the output of the PSC; (**d**) The far field image of the weakly focused RPLB. (**a**) to (**c**) are theoretical representations while (**d**) is an experimental measurement.  
that electron acceleration proceeded in the early phase of the relativistic regime where the longitudinal electric field is dominant but too weak to create attosecond electron pulses.