(1)
where λ is the central wavelength, n0 the linear and n2 the nonlinear index of refraction. Inserting into Eq. (1) yields a nonlinear refractive index of . This value is essentially equal to the one cited above [11].
To verify the reliability of the value obtained from the moving focus method, we have measured the electron densities of helium plasma as a function of energy and pressure. The electron densities have been derived from Stark broadening of the atomic line He I 587.56 nm (1s2p3P°-1s3d3D) [18] and [19]. Note that the Stark broadening approach [20] has been applied to plasma filaments in gaseous media before ([21] and references therein).
Fig. 2 shows the electron density as a function of energy in the range of 1–63 mJ. The pressure is fixed at 320 Torr. It can be seen in Fig. 2 that the electron density increases until the change of slope tends towards a constant above a critical energy of around 25 mJ (Fig. 2, linear fit). This behaviour is characteristic for intensity clamping, which has been discussed in detail in [22], [23], [24] and [21]. The nonlinear index of refraction is linearly proportional to the pressure. Thus, the critical power is inversely proportional to the pressure
(2)
Using Eq. (2), the critical point ( 25 mJ, 320 Torr) corresponds to a critical power of 251 GW (pulse duration: 42 fs), which is in agreement with the measurement obtained in Fig. 1.
Fig. 2. Intensity clamping: electron density versus energy. The data (black squares) shows the electron density as a function of energy in the range of 1–63 mJ. The pressure is fixed at 320 Torr. The electron density increases until the change of slope tends towards a constant above a critical energy of around 25 mJ (linear fit). This behaviour is characteristic for intensity clamping (see text). The critical point ( 25 mJ, 320 Torr) corresponds to a critical power of (pulse duration: 42 fs). There is no saturation by depletion (ionization degree: 10−2).
Fig. 3 shows the electron density as a function of pressure in the range of 50–760 Torr. The energy is fixed at 35 mJ. The linear dependence on pressure above a critical pressure of around 250 Torr (Fig. 3, linear fit) can again be explained by intensity clamping (to be published [25]). In short, the electron density (Ne) is a function of the intensity (I) according to the formula
(3)
where NHe is the neutral density and RHe the (tunnel) ionization rate. The critical (clamped) intensity is independent of pressure Icrit≠Icrit(p) [26], [27], [28] and [25]. Thus, the electron density is linearly proportional to pressure above the critical pressure (Eq. (3)). The critical point ( 35 mJ, 250 Torr) corresponds to a critical power of 274 GW (pulse duration: 42 fs). Note that the electron densities are in the order of 1017 cm−3 (Fig. 2 and Fig. 3). This corresponds to an ionization degree of 10−2, which is well below the depletion limit [29].
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