## 4 Acceleration by Ultrashort and Tightly Focused Radially Polarized Laser Pulses
### Electron Acceleration Measurements  
For the electron acceleration experiment, the few-cycle RPLP presented in the previous section was focused into a low-density oxygen gas with a controlled pressure of 800 mTorr. According to the barrier suppression ionization model [81], up to 6 electrons per oxygen atom can be released during the interaction with the intensity mentioned above. This suggests a maximum photo-electron density of \(\sim 3\times 10^{17}\) cm\({}^{-3}\) in the focal region at the peak of the laser pulse. The experimental configuration for electron acceleration and measurement consisted of a tilted mirror, a high NA (0.7) on-axis parabola, and the electron deflector and detectors. All components were located inside a low-pressure (800 mTorr) gas chamber filled with oxygen and aligned with the focusing axis (see Figure 8). A small aperture hole was drilled in the reflective mirror to let the electron beam propagate freely to the detectors. To detect the electrons, a particle-sensitive plastic scintillator with a thin aluminum overcoat was used to convert the incoming electron beam into photons in the visible region (500 nm). These photons were then collected by a photo-mutiplier tube (PMT) to produce the correlated signal. To measure the electron distribution, the same scintillator-overcoat combination was coupled to a high dynamic range (16 bits) CCD camera. It is worth mentioning that the carrier-envelope phase of the IR pulse was not stabilized. Therefore, the carrier phase \(\phi_{0}\) was fluctuating randomly between \(-\pi\) and \(\pi\) from shot to shot. The average transverse profile of the electron beam that propagates through the hole of the reflective mirror is shown in Figure 9(a). The measured divergence of the electron beam, from the focal plane to the detector, is 37 mrad (half-angle at 1/e). This includes probable space-charge effects and scattering. Figure 8: RPLP electron acceleration and measurement set-up. The 25-mm diameter RPLP is reflected off a mirror placed at \(45^{\circ}\) and sent toward the focusing on-axis parabola. The resulting longitudinal field at focus is strong enough to ionize the oxygen molecules from the ambient gas (800 mTorr) and accelerate the photo-electrons along the propagation axis. The electron beam created passes through a small aperture (4 mm) in the reflecting mirror and is collected by different detectors positioned 10 cm away from the focal plane. By rotating the linear polarization before the PSC with the help of a half-wave plate, the intensity of the longitudinal field at focus can be controlled. With a vertical input polarization (\(90^{\circ}\)), the output mode of the PSC has a radial polarization (TM\({}_{01}\) mode) that generates a maximum longitudinal field in the focal plane. With a horizontal input polarization (0 and \(180^{\circ}\)), the output beam has an azimuthal polarization (TE\({}_{01}\) mode), without a longitudinal field in the focal plane. Figure 9(b) shows the electron signal measured by the PMT while shifting gradually from a TE\({}_{01}\) mode with a purely transverse electric field to a TM\({}_{01}\) mode with a dominant longitudinal electric field. It is observed that the signal peaks when the effective mode approaches the ideal TM\({}_{01}\) mode, where the longitudinal electric field is the strongest. On the other hand, no signal is observed with the TE\({}_{01}\) or TEM\({}_{00}\) mode (when removing the PSC). This is consistent with the RPLB acceleration scheme described in Section 3 where the electrons are accelerated along the propagation axis by the longitudinal electric field. Figure 9: Experimental observation of direct longitudinal electron acceleration in tightly-focused RPLPs.