## 3 Electron Acceleration with Radially Polarized Laser Beams
### Direct Longitudinal Electron Acceleration with Radially Polarized Laser Beams  
By RPS and RPT, free electrons gain a substantial energy due to the combined action of the transverse and longitudinal electric field components. However, strong acceleration can take place at the center of RPLBs without the action of the transverse field components [16; 17; 63; 64]. Direct longitudinal acceleration with RPLBs allows for matching the electrons with the longitudinal field oscillations and effectively offers the possibility of producing well collimated quasi-monoenergetic relativistic attosecond electron pulses [16; 18; 19]. The fundamental issue with longitudinal acceleration with RPLBs is the superluminous axial phase velocity. In fact, RPLBs have an axial variation of the carrier phase of \((p+2)\pi\) from \(z=-\infty\) to \(\infty\), independently of the beam spot size and pulse duration [30; 39; 65]. This is (\(p+2\)) times that of the fundamental Gaussian beam, twice for the lowest order (\(p=0\)) RPLB. This higher value of the Gouy phase shift is due to the fact that RPLBs diffract more rapidly than the fundamental Gaussian mode [66]. When considering a point of constant phase along \(r=0\), the phase velocity is  
\[v_{phase}=c\left[1-\frac{(\lambda_{0}/w_{0})^{2}}{\pi^{2}(1+z^{2}/z_{R}^{2})} \right]^{-1} \tag{8}\]where it was assumed that \(w_{0}\gg\lambda_{0}\) and \(p=0\), for simplicity. Because of the velocity mismatch, a particle moving in the beam will inevitably drift with respect to the carrier oscillations. Within a given time interval \(\Delta t\), the net energy gain is optimal if that drift is less than half the laser wavelength, _i.e._, if \(\Delta t|v_{phase}-v_{z}|\lesssim\lambda_{0}/2\), where \(v_{z}\) is the particle's longitudinal velocity. During that same time interval, the particle has traveled over a distance  
\[\Delta z_{dph}=\frac{\lambda_{0}/2}{|v_{phase}/v_{z}-1|} \tag{9}\]  
which defines the dephasing length. An interaction with the laser field for exactly two dephasing lengths--corresponding to a complete cycle drift--results in no acceleration, on average. For the special case of an electron with a relativistic longitudinal velocity (\(v_{z}\simeq c\)), the dephasing length is:  
\[\Delta z_{dph}=\frac{\lambda_{0}}{2}\left[\left(\frac{\pi w_{0}}{\lambda_{0} }\right)^{2}+\left(\frac{z}{w_{0}}\right)^{2}-1\right] \tag{10}\]  
It is thus observed that around the beam waist (\(z\simeq 0\)), where the longitudinal field is the most intense, the distance over which a relativistic electron remains in phase with the laser field is only about the Rayleigh distance [\(\Delta z_{dph}\simeq(\pi z_{R})/2\)]. This sets a fundamental limit on the energy that can be transferred from the laser field to particles (see also [14]). Assuming a perfect synchronization between an electron moving along the \(z\) axis and the laser field, the maximum variation in the total electron energy from \(z_{i}\) to \(z_{f}\) is  
\[\Delta W|_{z_{i}}^{z_{f}} = \mathrm{Re}\left[-e\int_{z_{i}}^{z_{f}}\tilde{E}_{z}(0,z)\exp(-j \phi_{0})dz\right] \tag{11a}\] \[= -ez_{R}E_{z0}\bigg{\{}\left[A(z_{i})-A(z_{f})\right]\cos\phi_{0}+ \left[B(z_{i})-B(z_{f})\right]\sin\phi_{0}\bigg{\}} \tag{11b}\]  
where \(\tilde{E}_{z}\) is the complex envelope of the field component given at Equation (2b). Above, \(-e\) is the electron charge, \(E_{z0}=[2\sqrt{2}\exp(1/2)/(k_{0}w_{0})]E_{0}\) is the amplitude of the longitudinal electric field component, \(A(z)=[w_{0}/w(z)]^{2}\), and \(B(z)=z_{R}/R(z)\). Respectively, \(w(z)=w_{0}[1+(z/z_{R})^{2}]^{1/2}\) and \(R(z)=z+z_{R}^{2}/z\) are the beam waist size and radius of curvature of the wavefront at \(z\). According to Equation (11b), there are two situations where the energy gain is optimal. One is when the acceleration occurs between two axial positions where the wavefront curvature radius is minimum but of opposite sign. This happens when \([z_{i},z_{f}]=[-z_{R},z_{R}]\).