## 3 Electron Acceleration with Radially Polarized Laser Beams
### Free-Space Electron Acceleration in Laser Beams  
The classical motion of a charged particle in a laser beam is given by the Lorentz force equation [35]:  
\[\frac{d\mathbf{p}}{dt}=q\left(\mathbf{E}+\mathbf{v}\times\mathbf{B}\right) \tag{6}\]  
where the particle's attributes are its mass, charge, velocity, kinetic momentum, and energy defined by the symbols \(m\), \(q\), \(\mathbf{v}\), \(\mathbf{p}=\gamma m\mathbf{v}\), and \(W=\gamma mc^{2}\), respectively, with \(\gamma=(1-\mathbf{v}^{2}/c^{2})^{-1/2}\). \(\mathbf{E}\) and \(\mathbf{B}\) are the electromagnetic field components of the laser beam [e.g., from Equations (1) or (2)]. For a non-relativistic motion in a weak harmonic electric field of angular frequency \(\omega_{0}\):  
\[\frac{|\tilde{\mathbf{v}}|}{c}\simeq\left(\frac{q}{mc\omega_{0}}\right)| \tilde{\mathbf{E}}| \tag{7}\]  
It is thus common to define a dimensionless parameter \(a_{0}^{2}=(q/mc\omega_{0})^{2}E_{0}^{2}\) whose value relative to 1 characterizes distinct regimes of particle dynamics (see Table 1). The non-relativistic regime is often referred to as the ponderomotive regime. Effectively, charged particles in a relatively weak laser beam (\(a_{0}^{2}\ll 1\)) drift away from the high-intensity regions with an average motion that is independent of the laser polarization [44]. The mathematical form of the ponderomotive force (PF)--the net force resulting from the temporal average of the quivering particle motion associated with the rapid field oscillations--was initially proposed in the 1960's [45,46]. Relativistic (\(a_{0}^{2}\sim 1\)) formulations came some 30 years later from independent authors [47, 48, 49]. In ponderomotive acceleration models, it is often assumed that the effect of the longitudinal electric field component cancels out when averaging over a full laser field cycle. For ultra-relativistic laser intensities (\(a_{0}^{2}\gg 1\)), this assumption fails [16]. To effectively reveal the phase-sensitive sub-cycle dynamics that characterizes longitudinal acceleration in RPLBs, it is necessary to take into account all the field components and their oscillations, as it is done when working directly with the Lorentz force equation. \begin{table}
\begin{tabular}{l c} \hline \hline
**Regime** & \(\mathbf{a_{0}^{2}}\) \\ \hline Non-relativistic & \(\ll 1\) \\ Relativistic & \(\sim 1\) \\ Ultra-relativistic & \(\gg 1\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The three dynamical regimes associated with the motion of a charged particle in a laser beam in terms of the normalized field parameter \(a_{0}=(q/mc\omega_{0})E_{0}\). Despite the controversy that has surrounded the experimental observation of ponderomotive electron acceleration in Gaussian laser beams [50; 51; 52; 53], the mechanism by which the particles acquire a significant (relativistic) momentum is now well understood and accepted [54]. The relativistic generalization of the original PF model [45; 46; 47; 48; 49] helped to explain and understand experimental observations in terms of relativistic ponderomotive scattering (RPS), where particles are expelled out of the beam focus within only a few laser cycles [55; 56]. Typically, RPS proceeds in three steps. First, electrons are released by ionization (predominantly tunnel and above-threshold ionization) and accelerated toward the low-intensity regions of the laser beam by the transverse electric field component. Second, the trajectories are bent in the \(\mathbf{k}\propto(\mathbf{E}\times\mathbf{B})\) direction by the magnetic field (_via_ the \(\mathbf{v}\times\mathbf{B}\) contribution of the Lorentz force). Third and finally, the longitudinal electric field component takes over and provides a final extra push. Although the amplitude of the longitudinal component is usually tiny (\(\propto E_{0}\lambda_{0}/w_{0}\)), including it into calculations changes the result from vanishingly small to considerable acceleration [54; 55; 57].