## 4 Analytical Model of EMP Emission and Theoretical Results  
The experimental results presented in the previous section led us to a new and simple theoretical model for EMP emission. In our model, the electrons generated in front of the target due to absorption of a fixed laser energy are pushed into the target by the ponderomotive force, and the electric charge leaving the rear side of the target depends on d, \(\rho\), and \(Z\) as follows:  
\[Q_{e}\ltimes-\frac{1}{d}\left(\frac{\rho}{Z}\right)^{0.46} \tag{1}\]  
The influence of \(d\), \(\rho\), and \(Z\) on EMP spectrum was previously analyzed in several papers [4,37,38]. The specific dependence indicated in Equation (1) ensures correlation between our experimental and theoretical data.The hypotheses given by Equation (1) are consequent with the experimental data presented in Figure 5a-c, which enable us to evaluate the total number of electrons generated in the forward direction at the rear side of the target by the under the curve area. Thus, in plot a) from Figure 5, the magenta curve (corresponding to \(d=10\) um Ag target) has an area under the curve approximately 4 times smaller as compared to the yellow curve (corresponding to 5 um thicknesses), while the area under the curve of the yellow curve is approximately 1.5 times smaller as compared to the red curve (corresponding to 4 um thickness). Similarly, the (b) and (c) plots of Figure 5 indicate that the Ag target has an intermediate area under the curve between Al and Au due to its intermediate ratio \(\rho/Z\). In Equation (1), the inverse dependence of \(Q\) on \(d\) and \(Z\) accounts for the spatial spreading, energy loss, and recombination of the electrons due to their collisions with the target atoms/ions along their path through the target, while direct dependence on \(\rho\) accounts for the larger skin-depth (i.e., light penetration depth within the target) and stronger reduction in the target neutralization currents during the initial stage of charge building on the target. More specifically, the inverse dependence of \(Q\) on \(d\) is related to the larger number of collisions with target atoms/ions and higher energy transfer along the path of the electrons through the target as d increases. The dependence of \(Q\) on the ratio \(\rho/Z\) is related to two interrelated phenomena. First, a higher electrical resistivity material has a larger optical penetration depth of the laser radiation at the front surface, enabling laser pulses to excite more electrons and, consequently, to generate a larger bunch of electrons at the rear surface of the target. Also, higher electrical resistivity leads to a stronger reduction in the target neutralization currents during the initial stages of charge building on the target. Second, a larger mass number of the material leads to strong spatial spreading and energy loss and, to recombination of the electrons propagating through the target due to collisions with massive atoms/ions, so that fewer and less energetic electrons escape at the rear surface. The laser electrons accelerated at the back side of the target leave behind the positive charge on the target \(Q=|Q_{e}|\), which, in the experimental conditions simulated here, is of the order of nC in agreement with previous experimental and theoretical studies [33,36,39]. Due to this positive charge of the target, a neutralization current occurs within the target stalk (made of a conductive material) [40]. During neutralization, the total positive charge accumulated on the target \(Q(t)\) decreases to zero as a lognormal function of time [41] with the neutralization time of \(\approx\)1 ns (Figure 7a (a--top plot/right)). The magnitude of the neutralization time is related to the length of the target stalk, which is approximately 15 cm. The target stalk of length \(L\) with the positive charge \(Q\) of the target is equivalent to a time-dependent electric dipole:  
\[P(t)=2 LQ(t) \tag{2}\]  
and generates electromagnetic radiation like a \(\lambda/4\) linear antenna [33; 40].