Discussion

Two pulsesE₁andE₂which co-propagates in aχ⁽³⁾medium are coupled by cross-phase mod-ulation (XPM). Hence the high intensity carried by a filament induces a transient refractive index change which is experienced by any electric field co-propagating with it. Moreover, the nonlinear modification of the medium induced by a filament, which is linearly polarized is a

Filament-probe delay (fs)

Wavelength (nm)

−150 −100 −50 0 50 100 150

395 400 405

Negative slope

Figure 3.5:Frequency resolved polarization of the probe beam gated by the interac-tion with the filament. The probe spectrum is recorded with the analyzer orthogonal to the initial probe polarization as a function of the relative delay between the fila-ment and the probe. A negative delay corresponds to the case where the filafila-ment is launched before the probe. The negative slope indicates a positive chirp of the gated slice of the probe. Because the probe is initially positively chirped, this constitutes an indirect piece of evidence of the temporal gating property of the filament.

priori non isotropic and then can break the symmetry in the optical response of the otherwise isotropic Argon gas. This anisotropy stems from the difference between the components of the χ⁽³⁾ tensor: bounded electrons oscillate along with the extremely strong filament field which stretches the formerly spherical electron cloud into an ellipsoid elongated along the polariza-tion direcpolariza-tion of the filament. This elongapolariza-tion typically increases the refractive index along the direction of the filament polarization and decreases it for the orthogonal direction.

In general, the third-order susceptibility is a fourth-rank tensor with 81 elements. In an isotropic medium, such as Argon, the symmetries imply that only three elements are indepen-dent of one another, and the third-order susceptibility can be written in terms of them as:

χ⁽³⁾_ijkl =χ³_xxyyδijδkl+χ³_xyxyδikδjl+χ³_xyyxδilδjk (3.1) whereδ_ij is the Kronecker delta function.

Hence if an intense pump pulse E_R polarized along −→e_R = −→e_x and a weak probe beam EB linearly polarized along−→eB co-propagate, only one term remains in the expression of the nonlinear polarization along the two main axis−→e_xand−→e_y. They are written respectively:

P_{XP M,x}^B = 3²₀

2 χ³_xxxx|ER|²EB,x (3.2)

P_{XP M,y}^B = 3²₀

2 χ³_yyxx|ER|²EB,y (3.3)

(3.4)

All the other non-linear Kerr polarization components are negligible here, as the ratio of the probe and driving beam energies is10⁻³. As the dominant contribution to theχ⁽³⁾ tensor is of electronic origin and considering that filament and probe frequencies are far from any resonant transition, one can consider thatχ⁽³⁾xxxx= 3χ⁽³⁾yyxx[137]. As a consequence, the filament induced birefringence is:

∆n = 1 2n₀(3

2Re(χ³_xxxx−χ³_yyxx))|E_x^R|² =n^{XP M}₂ If ilament (3.5) where n^{XP M}₂ = ⁴₃Re(χ³_xxxx)/(n²₀²₀c) = ⁴₃n^{SP M}₂ represents the XPM nonlinear refractive in-dex. As is usual in filamentation models, the third-order susceptibility of the plasma (ions and electrons) is not taken into account, because (i) electrons only have a significant contribution in the relativistic regime [138] and (ii) the small relative abundance of ions relative to neutral molecules (typically10⁻⁴ [49]), which is not reached in this experiment and the fact that their third-order susceptibility is smaller than that of neutrals and does not deviate by more than one order of magnitude [138] prevents them from having any measurable contribution.

Hence, the dephasing induced by the filament over the whole propagation distance is:

∆φ^probe_{XP M} = 2π The first approximation made here is to neglect group velocity dispersion between the probe and the filament in Argon. Secondly, we neglect the distortions of the spatial and temporal intensity distribution of the filament in considering only the maximal on-axis intensity for the fit. So the resulting inhomogeneities in the birefringence experienced by each spatio-temporal slice of the probe are not taken into account. These two approximations lead to underestimate n^{XP M}₂ .

A full development of the coupled polarization-dependent equations would be needed to obtain the full expression of the probe dephasing. From the experimental point of view, this corresponds to the pinhole inserted in the beam to select one small, homogeneous region only.

Moreover, because of thea priori time dependence of the dephasing, a temporal study of the probe polarization could give a better insight on the temporal dynamics of the filamentation.

Figure 3.6 displays the theoretical on-axis intensityI_{f ilament}(z)as a function of the propa-gation distance for several pressure as calculated with the model described in Chapter 1. The induced dephasing between the two polarization axes is then calculated as in Eq.3.6 and shows a linear dependence with the pressure. Indeed, our simulations show that higher pressure results in longer filaments with slightly lower intensity clamping. These effects roughly compensate each other when calculating R

I_{f ilament}dz, so that the dephasing ^2πn_λ ^{XP M2}

probe

R I_{f ilament}dz varies liken^{XP M}₂ , which is proportional to the pressure. The linear tendency is well reproduced by simulations. However, to fit the experimental results, we adoptedn^{XP M}₂ = 1.6 10⁻²⁰cm²W⁻¹, one order of magnitude below expected value of 10⁻¹⁹cm²W⁻¹. Further full simulations, in-cluding transverse intensity profiles and dispersion effect, are required to provide a quantitative transverse profile of filament-induced birefringence and achieve a better quantitative agree-ment. As the dephasing depends monotonically on the pressure, any value may be generated by

0.7 0.8 0.9 1 1.1 1.2 1.3

Figure 3.6: Theoretical on-axis intensity as a function of propagation distance for several Argon pressure. Inset: Dephasing induced on a probe co-propagating with the filament. Because both the intensity and the length of filaments depends on the pressure, the resulting dephasing shows a clear linear pressure dependence.

choosing an adequate Argon pressure. For example, an interpolation of the experimental data presented in Fig. 3.3 suggests that an equivalentλ/4plate can be generated for1.7±0.1bar.

In conclusion, we have demonstrated that laser-generated self-guided filaments can induce substantial birefringence in near-atmospheric pressure gases. An angle of 45^◦ between the filament and the probe polarizations allows the realization of Kerr-gates, with an unprecedented switching time ultimately limited by the duration of the filamenting pulse. An optical ultrafast switch could even be initiated remotely by self-guided filaments in the atmosphere [3], even in perturbed conditions [57, 58, 61], opening new perspectives for remote optical ultrafast data transmission and processing,e.g. remote ultrafast optical logical gates.