## 3 Results
Within the discussed range of parameters, however, such a problem does not arise, although, as can be seen from Figure 3, the region in which \(\delta<1\) (depicted in blue at Figure 3c,f,i) turns out to be remarkably smaller than the region in which \(\sigma>1\) (depicted in red at Figure 3b,e,h) and \(\beta<1\) (depicted in blue at Figure 3a,d,g). Despite this, it can be seen that at the beginning of the interaction, the plasma is non-magnetized (Figure 3b). However, by the arrival of the radiation maximum, a noticeable region of cold magnetized plasma appears (Figure 3d,e), which further begins to expand (Figure 3g,h). The achieved values of \(\sigma\) are relatively small and do not exceed 10. Achievement of larger values can be expected at higher intensities; therefore, we performed calculations in the range of intensities \(3\times 10^{19}\)-\(10^{21}\) W/cm\({}^{2}\). The results are shown in Figure 4. It can be seen that at low intensities (Figure 4a-c) no magnetized plasma is formed. Although there is a noticeable region of low electron temperature in the sheath, the magnetic field is too weak to exceed the rest energy density of the electrons. With an increase in intensity, the plasma expands with increasingly high speeds, which is in accordance with the theory of collisionless plasma expansion into vacuum [27,28]. The magnitude of the generated magnetic fields also increases, and a significant region of cold magnetized plasma is formed. In this case, the degree of magnetization also increases and reaches \(\sigma\sim 10\) for \(10^{21}\) W/cm\({}^{2}\). Simultaneously, the value of \(\delta\) remains small. Figure 2: Time evolution of the peak magnetic field generated in the expanding plasma sheath for different laser intensities. Figure 3: Distributions of \(\beta\) parameter (left column), \(\sigma\) parameter (central column) and \(\delta\) parameter (right column) at (**a-c**) \(t=1000\) fs, (**d-f**) \(t=1500\) fs, and (**g-i**) \(t=2000\) fs after start of the simulation. The parameters of the simulation are the same as at Figure 1.  
Furthermore, note that efficient magnetization of the expanding sheath was observed only for relatively long pulses with the duration exceeding 400 fs. The case of a shorter 100 fs pulse is shown at Figure 5. The shown instant of time is corresponding to a moment of a peak intensity arrival. For such a short laser pulse, even at the highest investigated intensity of \(10^{21}\) W/cm\({}^{2}\), there is not enough time for the sheath to expand significantly (Figure 5a) and the electrons are still relativistically hot all over the sheath (Figure 5b), leading to high \(\beta\) parameter (Figure 5d). Moreover, despite the fact that the magnetic field is relatively high (\(B_{z}\sim 500\) MG at the peak, which should be compared to 800 MG for the same intensity for 700 fs pulse as shown at Figure 2), the weakly expanded sheath is rather dense so that the magnetization parameter \(\sigma<1\) (Figure 5e). Finally, due to a small dimension of the sheath the parameter \(\delta<1\) as well (Figure 5f). An exact pulse duration sufficient to obtain the magnetized plasmas depends on the pulse intensity and likely on the details of interaction but it seems that in a wide set of parameters it lays somewhere between 100 and 500 fs.