# Laser-Accelerated Plasma-Propulsion System
## 6 LAPPS Analytical Model
### TNSA Acceleration  
In this section, we describe the TNSA scheme and its relevant parameters. We consider the case, investigated in a large number of experiments, where a few-\(\upmu\)m thick target is irradiated by a relativistic (\(I\lambda_{L}^{2}\gtrsim 10^{18}\) W \(\upmu\)m\({}^{2}\)/cm\({}^{2}\)) laser pulse of a few tens of femtoseconds, which is nowadays available by using Chirped Pulse Amplification laser systems. In these conditions and for a Gaussian laser beam, the density of the electrons which are accelerated in the forward direction during laser-plasma interaction can be expressed by the relation [22]:  
\[n_{0}(I_{0})\simeq\frac{3I_{0}\Omega(I_{0})}{2 ck_{b}T_{hot}(I_{0})}, \tag{30}\]  
where \(I_{0}\) is the laser intensity, \(\Omega(I_{0})=1.2\times 10^{-15}I_{0}^{0.74}\left[\text{W/cm${}^{2}$}\right]\) is the conversion efficiency in fast electrons with a maximum value \(\Omega=0.5\) for \(I_{0}>5.7\times 10^{19}\) W/cm\({}^{2}\), and \(T_{hot}(I_{0})\) is the fast electrons temperature, which can be determined by the ponderomotive scaling [23]  
\[\frac{k_{b}T_{hot}}{m_{e}c^{2}}=\sqrt{1+\frac{I_{0}[\text{W/cm${}^{2}$}]\lambda _{L}^{2}[\upmu\text{m${}^{2}$}]}{1.37\times 10^{18}}}-1, \tag{31}\]  
where \(m_{e}\) is the electron rest mass and \(\lambda_{L}\) is the laser wavelength. As shown in Figure 4a, the fast electron density is a monotonically increasing function of \(I_{0}\). Moreover, as evident from Equation (30), \(n_{0}\) does not depend, in this approximation, on the specific target properties. The thickness and the atomic properties of the target, however, affect the  
Figure 3: Basic scheme of the Laser-Accelerated Plasma-Propulsion System (LAPPS). The laser system is represented by the red box. LAPPS is a closed cycle engine. propagation of the fast electrons into the target bulk, and therefore determine their density when they reach the rear target surface, which can be calculated by:  
\[n_{0e}(I_{0},d,\phi)=\frac{n_{0}(I_{0})}{\left(1+\frac{d\tan(\phi/2)}{w_{0}} \right)^{2}}, \tag{32}\]  
where \(w_{0}\) is the laser waist, \(d\) is the target thickness, and \(\tan(\phi/2)\) expresses the fast electron divergence. The laser waist defines the size of the focal spot and for a Gaussian beam with peak intensity \(I_{0}\) is equal to the radius where laser intensity \(I=I_{0}e^{-2}\). The detailed dependence of the broadening angle \(\phi\) is not trivial [24], since it is affected by many factors, including, among the others, the laser intensity, the mechanism of fast electron generation, the ionization \(Z\) of the target (determining multiple Coulomb scattering), and the self-generated magnetic fields induced into the target; for the moment, we only observe that \(n_{0e}\) can be remarkably lower than \(n_{0}\) for thick targets or high-\(Z\) ions. Ion acceleration is driven by the electrostatic sheath field generated by the electrons that arrive at the rear side of the target and try to escape into the vacuum. Assuming that the acceleration of ions with density \(n_{el}=n_{0I}/Z\) can be described by using a self-similar, isothermal, fluid model [22], their energy distribution can be expressed by:  
\[\frac{dN}{dE_{k}}=\frac{n_{0e}C_{s}t}{\sqrt{2 Zk_{b}T_{hot}E_{k}}}\exp\left(- \sqrt{\frac{2E_{k}}{Zk_{b}T_{hot}}}\right), \tag{33}\]  
where \(C_{s}=\sqrt{Zk_{b}T_{hot}/m_{I}}\) is the ion-acoustic velocity, \(m_{I}\) and \(Z\) are the ion rest mass and charge, and \(N\) is the number of ions accelerated per unit of surface. The kinetic energy \(E_{k}=m_{I}v_{t}^{2}/2\) can be considered classical. According to the above model, described by [PERSON] [22], the number of ions accelerated per unit surface \(N\) grows with time, suggesting that the calculation must be stopped at an opportune time \(t=\tau_{acc}\). Their energy distribution is here constant with time; however, the model predicts that the accelerated protons exhibit a high energy cutoff increasing with the duration of the laser pulse, and therefore with the acceleration time [22].