## 4 Acceleration by Ultrashort and Tightly Focused Radially Polarized Laser Pulses  
Nowadays, state-of-the-art ultra-intense laser systems are able to generate terawatt laser pulses whose durations represent only a few optical cycles [76]. With high-aperture focusing systems, these ultrashort pulses can be focused on a spot of sub-wavelength dimension in the focal plane. While most theoretical studies on direct acceleration in RPLBs have limited themselves to the paraxial limit (\(k_{0}z_{0}\gg 1\)) [18, 67, 68], modeling the dynamics under nonparaxial and ultrashort pulse conditions is now necessary to bridge the gap between theory and current experiments (see Section 5). A pulsed beam consists of a sum of monochromatic beams with different frequencies. In the special case of an isodiffracting pulse, each frequency component has the same wavefront curvature. A rigorous solution for the lowest-order radially polarized isodiffracting pulsed beam may therefore be obtained by taking a coherent superposition of TM\({}_{01}\) beams [see Equations (1)] with different frequencies but identical confocal parameter \(z_{0}\)[77]. If we weight each frequency component by the spectral amplitude function \(F(\omega)\) in Fourier space and then calculate the inverse Fourier transform, we obtain the following field components in complex notation for a TM\({}_{01}\) pulsed beam [75, 77]:  
\[\tilde{E}_{r} =-\frac{3 jA_{0}\sin\tilde{2}\theta}{2k_{0}^{3}\tilde{R}}\bigg{(} \frac{G_{-}^{(0)}}{\tilde{R}^{2}}-\frac{G_{+}^{(1)}}{c\tilde{R}}+\frac{G_{-}^ {(2)}}{3c^{2}}\bigg{)}\exp(-j\omega_{0}t) \tag{16a}\] \[\tilde{E}_{z} =-\frac{jA_{0}}{k_{0}^{3}\tilde{R}}\bigg{[}\frac{(3\cos^{2} \tilde{\theta}-1)}{\tilde{R}}\bigg{(}\frac{G_{-}^{(0)}}{\tilde{R}}-\frac{G_{+ }^{(1)}}{c}\bigg{)}-\frac{\sin^{2}\tilde{\theta}}{c^{2}}G_{-}^{(2)}\bigg{]} \exp(-j\omega_{0}t)\] (16b) \[\tilde{B}_{\phi} =-\frac{jA_{0}\sin\tilde{\theta}}{k_{0}^{3}c^{2}\tilde{R}} \bigg{(}\frac{G_{-}^{(1)}}{\tilde{R}}-\frac{G_{+}^{(2)}}{c}\bigg{)}\exp(-j \omega_{0}t) \tag{16c}\]  
where \(G_{\pm}^{(n)}=\frac{\partial^{n}}{\partial t^{n}}[f(\tilde{t}_{+})\pm f( \tilde{t}_{-})]\), \(f(t)\) is the inverse Fourier transform of the pulse spectrum \(F(\omega)\) and \(\tilde{t}_{\pm}=t\pm\tilde{R}/c+jz_{0}/c\). It can be verified that in the limit of a delta frequency spectrum, \(F(\omega)=\delta(\omega-\omega_{0})\), the fields given in Equations (1) for a TM\({}_{01}\) beam of frequency \(\omega_{0}\) are recovered [77]. To model ultrashort pulses, a suitable frequency spectrum is the Poisson-like function [78, 79]:  
\[F(\omega)=2\pi\left(\frac{s}{\omega_{0}}\right)^{s+1}\frac{\omega^{s}e^{- \kappa\omega/\omega_{0}}}{\Gamma(s+1)}\,H(\omega) \tag{17}\]  
As previously defined, \(\omega_{0}\) is the frequency of maximum spectral amplitude. \(\Gamma(s+1)\) is the Gamma function, and \(H(\omega)\) is the Heaviside step function that ensures the absence of components with negative frequencies. The constant \(s\) is a real and positive parameter that controls the shape of the spectrum and the corresponding pulse. The inverse Fourier transform of Equation (17) is  
\[f(t)=\left(1-\frac{j\omega_{0}t}{s}\right)^{-(s+1)} \tag{18}\]  
which leads to exact closed-form expressions for the electromagnetic field components of the TM\({}_{01}\) pulse. Note that as \(s\) increases, the width of the spectrum \(F(\omega)\) decreases while the pulse duration increases. The spectrum \(F(\omega)\) and the corresponding temporal profile of the \(E_{z}\) field component are shown in Figure 5 for different values of \(s\). The direct acceleration of an on-axis electron by an ultrashort and nonparaxial TM\({}_{01}\) pulsed beam was investigated by some of us in a recent contribution [75]. More specifically, the laser power dependence of the maximum final kinetic energy that an electron initially at rest at \(r=0\) can acquire was studied for different pulse durations and degrees of focusing.