3.4.1 Dependence of SGMF on the laser irradiance and probe to target distance
The calculated values of the SGMF with increasing the laser irradiances for the different probe to target distances of 1, 2, 3 and 4cm is shown in Fig. 11. The SGMF increases from the 235 G to 590 G at a distance of 1 cm, from 195 G to 415 G at a distance of 2 cm, from 120 G to 355 G at a distance of 3 cm and from 70 G to 260 G at a distance of 4 cm increasing the laser irradiances from 10.2 to 26 GWcm− 2. There is a linear increase in the SGMF with increasing the laser irradiance due to an increase in the density of emitted charged species after greater energy deposition. The hot electron gain more energy form the source [11]. The most of the laser pulse energy coverts to the kinetic energy of ions and electrons and remaining part of the energy goes to their magnetic field energy.
Due to the shielding effect, the laser absorption efficiency in plasma as compared to target increases as the laser irradiance increases. The component of electric field \({E}_{d}=2{E}_{L}\text{sin}\theta\) of incoming laser pulse is normal to the target surface. This electric field accelerates the charges species in the LIP. Therefore, an increase in the laser irradiance the electric field strength of the laser increases which in turns the generation and acceleration of ions / electrons and SGMF of LIP increases [35]. Figure 11 shows that the SGMF is reduced by increasing the distance of probe from the target. This is because the source of the SGMF is near the target and moving away from the magnetic field line, the magnetic field becomes weaker [19, 24]. In the Figs. 9 (a – h) the very noticeable feature is the peak arrival time which is reduced by reducing the probe to target distance. Since the propagation of magnetic field front and the laser produced expanding plasma front are moving with same velocity.
The SGMF signal is delayed because it moves too far from the target to magnetic probe [19]. Therefore both the amplitude of the signals and SGMF decrease at longer probe to target distance. This result shows that the SGMF expands radially according to the Biot-Savart relation [20].
The laser-induced plasma can produce self-generated magnetic and electric fields. The generation of these fields can be caused by a number of factors, including the abundance of charged particles, strong electric currents, the laser wake field, the ponder motive, and Coulomb forces.[36] Laser-induced self-generated magnetic fields have a significant impact on plasma conditions and dynamical evolution in terms of inertial confinement fusion, fast electron transport, and heat conduction in plasmas [37]. Among the various mechanisms proposed for their generation, among the various mechanisms proposed for their generation, four important physical mechanisms are (i) charge separation between fast moving electrons and slow moving ions. three important physical mechanisms are (ii) non-parallel temperature and density gradients (The Biermann battery effect) [36], (iii) the current of fast electrons produced by laser target interaction can lead to axial magnetic field generation [38], and (iv) the huge radiation pressure associated with laser pulse (Ponderomotive force) is a source of azimuthal magnetic fields in the critical density region [39]. The first theory of charge separation or charge imbalance is considered in the present investigations as a most favorable mechanism for generation of electric field in the plasma.
Figure 12 shows the separation of charge with double layer structure among fast and slow moving Zr-electrons, as well as fast and slow moving ions of Zr with ambipolar acceleration, which represents quadrupole distribution, during the LIP of Zr. In the LIP, during plume expansion, plasma species convert thermal energy into translational energy [40]. The kinetic energy and the fluence of ions and electrons are important parameter that describe the expansion of laser induced plasma dynamics [31]. The ejection process of ions and electrons occurs during the initial stage of laser matter interaction. As the plasma plume travels perpendicular to the target's surface, pressure gradients are formed. The plume expands according to the Maxwell distribution due to the large number of collision between the plasma species within the Knudsen layer [41]. The plasma plume expands because of the adiabatic process after laser pulse termination [42]. The hydrodynamic and electrostatic forces accelerate the ions and electrons in the forward direction. This forward peaking is responsible for higher the kinetic energy of charge species than the neutral species in the LIP. The Inverse Bremsstrahlung (IB) absorption heats the electrons, which then transfer their absorbed energy to the ions through electron-ions collisions in the electrostatic mode [33]. The thermalization time \({10}^{-10}-{10}^{-11}\text{s}\) of electrons-ions is smaller than the duration of laser pulse (\(6\times {10}^{-9}s\)) [5, 12]. Therefore during the early stage of laser matter interaction, the electrons and ions attain the same thermal temperature. Electrons achieve the highest thermal velocity, because of small size and lightest among the plasma species and they travel faster than ions. The ions lag behind which prevents the complete escaping of the electrons and the self-electrostatic field is developed which is called self-generated electric field. This SGEF is directly proportional to the K.E of the ions and the charge state. The LIP which is created by the nano second laser displays the thermal nature. The electrons obtain energy from the IB process, reheat the LIP at a fixed distance, which sustains the acceleration process [5]. The pressure gradients are caused by electrons density gradient being ejected from normal to the target. The ions gain kinetic energy by this process of acceleration caused by the separation of charges. The decrease in average kinetic energy of ions is responsible for the spatial distribution of acceleration potential [43]. Whereas, in the electrostatic model, the ions are accelerated, which is attributed to the ambipolar electric field (double layer) effect in the laser induced plasma. The double layer is formed by the ejection of energetic electrons, which is caused by the three body recombination and IB absorption process [44]. The charge imbalance generated in the LIP is responsible for the escaping process of energetic electrons, which also causes ions to accelerate. The acceleration of ejected species of plasma is maintained by the double layer potential difference [9]. The expansion of LIP is axisymmetric, so the \(\nabla {T}_{e}\) and \(\nabla {n}_{e}\) are non-parallel in the direction of expanding plasma. This combination of non-parallel \(\nabla {T}_{e}\) and \(\nabla {n}_{e}\) generates the SGMF in the LIP [13]. During the laser heating of the plasma, the \(\nabla {T}_{e}\) is directed along the axis of the target surface and \(\nabla {n}_{e}\) is directed along the laser axis [45]. These pressure gradients decrease as the probe moves away from the surface of target [18]. The overall mechanisms are explained in Fig. 12.
The SGEF can also be estimated by the following relation [46].
$${E}_{f}=\frac{\nabla {P}_{e}}{{n}_{e }e}=\frac{\nabla {T}_{e}}{e}+{T}_{e}\frac{{\nabla n}_{e}}{{n}_{e}}$$
4
where “e” is the charge on electron and “\(\nabla {P}_{e}={n}_{e}{T}_{e}\)”, is the pressure of plasma or electron pressure (\({T}_{e}\)in eV). At a laser irradiance of \(26 \text{G}\text{W}{\text{c}\text{m}}^{-2}\) and target-to-probe distance of 1 cm, the self-generated electric field obtained from Eq. (4) is, \({E}_{f}\tilde3.97\times {10}^{3} V/m\) with the temperature gradient of \(\nabla {T}_{e}\tilde3.9678 \text{K}\text{e}\text{V}\) and density gradient of \(\nabla {n}_{e}\tilde8.97\times {10}^{11}{cm}^{-2}\). The electrons and ions densities along the axially resolved Zr plasma provide information about temperature and density gradients of electrons and ions in axial direction and charge separation, which are supposed to be responsible for the generation of SGEMFs [47]. Fast electrons leave the plasma plume much earlier than ions due to the higher mobility of electrons than ions. Some electrons are prevented from escaping the plasma due to the space-charge separation between the fast electrons and the ions that are lagging behind, leading to the formation of a self-generated electric field in the expanding plasma-vacuum boundary [48]. Table 1 indicates the individual ion density gradients, electron density gradients, and ions-electrons density gradients of Zr plasma, with increasing laser irradiances at various distances of 1, 2, 3, and 4 cm of FC form the Zr target. The values of electron density gradients from 1 cm to 2 cm were used to estimate the electric field from Eq. (4).The SGEF depends upon the pressure gradients of plasma which in turn are dependent upon \(\nabla {T}_{e}\) and \(\nabla {n}_{e}\). The estimated values of electric field from Eq. (4) with directly measured values of electric field by electric probe are plotted in Fig. 13. These values are in good agreement with each other, which confirm that our experimental measured values of electric field by electric probe are accurate. The estimated electric field values from Eq. (4) and the directly measured electric field values by electric probe are presented in Fig. 13 and indicate significant differences. The density and temperature gradients are measured by the FC, whereas the self-generated electric field is measured by the electric probe. The difference in area between the FC (\(3.79 {\text{c}\text{m}}^{2}\)) and the electric probe (tip area \(0.95 {\text{m}\text{m}}^{2}\)) is responsible for the difference in estimated electric field values and directly measured electric field values by an electric probe. The adiabatic plume expansion, cooling, condensation, recombination, and collisional losses are another possible reason for the significantly higher analytically estimated electric field values than measured electric field [31].
Table 1
The ion density and electron density gradients of Zr plasma, with increasing laser irradiances at various distances of 1, 2, 3, and 4 cm of FC form Zr target.
Irradiances
(GWcm-2)
|
Ions density gradient
cm-2
(1 cm to 2 cm)
|
Ions density gradient
cm-2
(2 cm to 3 cm)
|
Ions density gradient
cm-2
(3 cm to 4 cm)
|
Electrons density gradient
cm-2
(1 cm to 2 cm)
|
Electrons density gradient
cm-2
(2 cm to 3 cm)
|
Electrons density gradient
cm-2
(3 cm to 4 cm)
|
Charge separation or Ions and electrons
density gradient
cm-2
(1 cm to 2 cm)
|
Charge separation or Ions and electrons
density gradient
cm-2
(2 cm to 3 cm)
|
Charge separation or Ions and electrons
density gradient
cm-2
(3 cm to 4 cm)
|
10.2
|
9.41E+11
|
1.01E+12
|
1.70E+12
|
6.97E+11
|
1.08E+12
|
2.38E+11
|
2.44E+11
|
7.00E+10
|
1.46E+12
|
12.4
|
9.40E+11
|
3.13E+11
|
1.69E+12
|
1.79E+11
|
2.03E+12
|
1.57E+11
|
7.61E+11
|
1.72E+12
|
1.53E+12
|
14.7
|
3.98E+11
|
2.28E+11
|
2.34E+12
|
2.53E+10
|
2.84E+11
|
7.70E+11
|
4.23E+11
|
5.12E+11
|
1.57E+12
|
17
|
2.05E+11
|
8.85E+11
|
2.48E+12
|
1.72E+11
|
1.43E+11
|
6.43E+11
|
3.30E+10
|
1.03E+12
|
1.84E+12
|
19.2
|
4.06E+10
|
8.96E+11
|
1.80E+12
|
7.16E+10
|
9.48E+10
|
1.14E+12
|
3.10E+10
|
8.01E+11
|
6.60E+11
|
21.5
|
9.33E+10
|
4.18E+11
|
2.23E+12
|
1.59E+10
|
4.27E+11
|
2.93E+12
|
7.74E+10
|
9.00E+09
|
7.00E+11
|
23.7
|
1.44E+11
|
7.79E+11
|
3.36E+12
|
5.26E+11
|
6.55E+11
|
2.22E+12
|
3.82E+11
|
1.24E+11
|
1.14E+12
|
26
|
3.70E+11
|
1.50E+11
|
9.05E+11
|
8.97E+11
|
3.63E+11
|
2.31E+12
|
5.27E+11
|
2.13E+11
|
1.41E+12
|
The double layer structure of the plasma is confirmed by both measurement i.e. time-of-flight measurement of ions by faraday cup and electric field by electric probe.