## 8 Interaction with cm Long Gas-Jet Target  
The gas-jet nozzle (Figure 9) is characterised by a 10 mm long, 4 mm wide slit and is mounted on a micrometric motorised support that allows the interaction point to be moved along the three Cartesian axes (position scan). During the pulsed operation, the gas flows out of the slit at supersonic speed to produce steep interfaces between gas and vacuum. The vacuum in the chamber before the shot is maintained at a pressure below \(\sim 10^{-5}\) Torr by a turbo-molecular pump connected to the chamber by a gate-valve. We perform numerical simulation to have quantitative information about electron density during propagation of the laser pulse in the gas-jet. The simulation is based on the tunnel ionisation model [44] and calculates the rate of tunnel ionisation in a non-adiabatic regime, _i.e._, as a function of the instantaneous laser phase, and assumes purely geometrical propagation of the beam in the plasma defined according to the real input laser parameters. According to this model, the rate of ionisation is given by  
Figure 8: Cross-correlation curve (left) showing the detailed temporal structure of the laser pulse in the 500 ps window before the main pulse. The plot on the left shows that intensity of the amplified spontaneous emission is just above \(10^{-9}\) of the peak intensity. The plot on the right shows the detail of the curve in the ps domain, just before the peak intensity. Figure 9: Schematic view of the target and interaction geometry, showing the gas-jet nozzle, the slab of gas, the main laser pulse and the probe pulse for plasma interferometry. \(\Gamma=N(t)e^{-\left(\frac{E^{2}f^{2}(\mu)}{\omega_{L}^{2}}\phi(\gamma,\theta) \right)}\) where \(E\) is the laser amplitude, \(f(t)\) is the pulse envelope and \(\omega_{L}\) is the laser angular frequency. In the above equation the term \(\phi(\gamma,\theta)\) includes the dependence upon the Keldysh parameter \(\gamma\), the ionisation potential \(I_{p}\), the average energy of electron oscillations in the laser field \(U_{p}\) and the phase of the laser field \(\theta\). The ionisation rate has a non-linear dependence on the instantaneous electric field. The exponential factor gives the strongest dependence on the electric field, while the pre-exponential factor includes dependence on the initial state that are defined by its quantum numbers. Figure 10 shows the ionisation profile in case of He gas, for three different positions of the laser pulse relative to the waist position. Figure 10 (left) shows the ionisation profile when the pulse is located at two Rayleigh ranges before the best focus. Figure 10 (centre) shows the ionisation profile when the pulse is located at one Rayleigh range before the best focus. Figure 10 (right) shows the ionisation profile when the pulse is located at the best focus. Ionisation maps have also been calculated for nitrogen. Figure 11 (left) shows the ionisation profiles when the pulse enters two Rayleigh ranges before the best focus. Figure 11 (left) shows the ionisation profiles for the case when the pulse enters one Rayleigh range before the best focus, and finally, Figure 11 (left) shows the ionisation profiles for the case when the pulse enters the best focus position. These calculations show that ionisation of nitrogen is very much different from that occurring in helium, where full ionisation is easily achieved even before the best focus position. In the case of nitrogen, ionisation up to He-like (two electrons left in the K-shell) is relatively easy to achieve before the pulse enters the best focus position. In fact, according to Figure 11 (left and centre), the ionisation  
Figure 11: Ionisation map of N with the laser pulse at two Rayleigh ranges before the best focus (**left**), at one Rayleigh range (**centre**) and at the best focus position. The graph shows that full ionisation in N occurs well before the pulse enters the focal region.