1. Introduction

Since the first observation of the spontaneous filament formation accompanying intense fs-pulse propagation in air in 1995 [1], much increasing attraction has been rapidly paid to the case of filament formation in air as documented in both experiment and theoretical studies [2–4], due to its potential values for applications. Recently, the filamentation of femtosecond pulses is used for the development of new methods of atmospheric monitoring and lighting control [5], even can be viewed as a new source of directed radiation to investigate the air pollution with a white light LIDAR and fluorescence spectroscopy [6].

In the view of practical applications, much attention has been focused on the long range propagation of filamentation in air for higher power laser beams in recent years [7, 8]. The usual mechanism responsible for the long range propagation of these filaments is identified as the dynamic balance between self-focusing due to the nonlinear refractive index of the air and self-defocusing due to the axial plasma generated by multiphoton ionization (MPI) of the air molecules and the diffraction of the laser beam, when the input power of the beam is higher than the critical power of self-focusing. Theoretically, several models based on self-channeling a moving focus [9] or dynamic spatial replenishment [10] addressed a dynamic equilibrium to comprehend the process of the filamentation. However, Méchain et al. [11] experimentally observed a new propagation regime with noionizated channels extending over the distance of 2 kilometers in air. They measured the maximum intensity that does not exceed a few${10}^{12}$ $W/c{m}^2$ in the light channels. Subsequently, Ruiz et al. [12] proposed the spontaneous generation of a filament without ionization in air due to soliton propagation for laser power lower than the collapse threshold. Based on numerical calculation, the observation of nonlinear saturation compressing the collapse of the self-focusing was reported in Ref [13].

In this work, we will investigate the filamentation process generated due to the higher-order terms of the nonlinear refractive index and Kerr self-focusing without the occurrence of the ionization effect. The interaction of self-focusing and the self-defocusing mainly related to the plasma generated by MPI, which is substituted by the higher-order Kerr terms should be re-recognized. This will inspire us to further consider whether the role of higher-order Kerr nonlinearity or the usual effect caused by plasma defocusing accurately predicts the deterministic filamentation patterns. It provides critical insight into the complex spatial-temporal dynamic process that collapse of each filament is not tuned by photoionization only. It shows that other potential optical mechanism should be further investigated.

2. Numerical simulation model

Recently, Loriot et al. [14] have measured the nonlinear refractive index that included the high-order indices terms, in$N_2$,$O_2$and$Ar$. They reported the nonlinear refractive index linearly increases, saturates, and then drops dramatically with the intensity of the laser beams. The refractive index variation along the polarization axis can be written as $\Delta n_{Kerr}=n_{2}I+n_4{I}^2+n_6{I}^3+n_8{I}^4+n_{10}{I}^5$, where I is the incident intensity. The higher order terms $\Delta n_{2*j}=2j/(2\ast j+1)n_{2j}{I}^j$ with$j\in N^{*}$, are related to the susceptibilities${\chi }^{(2*j+1)}$.

We consider a linearly polarized incident electric field at ${\lambda }_0$ = 800 nm, with cylindrical symmetry around the propagation axis z. The scalar envelope is assumed to be slowly varying in time and along z axis, it evolves according to the nonlinear propagation Schrödinger equation will be modified as follows:

3. Numerical simulations and discussions

Figure 1 shows the profiles for the case of filamentation propagation scenario. The transmitted beam firstly attempts to undergo self-focusing after a short distance until the intensity reaches a sufficiently high value to trigger the nonlinear singularity formation. Time dispersion is negligible and each cross section undergoes collapsing during the main stages of the focusing in the presence of higher-order Kerr effect arresting the collapse that would have occurred in the absence of any saturating mechanism. An input power 2$p_{cr}$of the Gaussian beam collapses towards the Townes profile (TP) similar to the observed behavior for a super-Gaussian (SG) beam, due to the nonlinearity effects domain over the diffraction. This is closely coincident with both the predictions of the experimental conformation and the numerical simulations analysis in the previous exploring works [15, 16].

 

Fig. 1 (Color online) Calculated transverse peak intensity profiles for the laser beam at the different z positions. (a-f) z = 9.57, 9.8, 10.8, 11.2, 11.35, 11.5 m, respectively. The narrow, weak, extended structures are due to self-focusing process in which the higher-order Kerr terms play a dominant role to be self-defocusing balance mechanism.

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However, the spatial solitonlike beam of self-channeling effect is not steady due to modulational instabilities in the transverse cross-section of the beam, which will exhibit a complex spatial feature. Figure 1 illustrates that the propagation dynamical processes will lead to the intensity spikes, and the spatial intensity profiles and their variation at different positions along the propagation axis are shown later for further details. At lower power, no ring appears during the self-focusing stage. The higher-order Kerr terms that occurs beyond the nonlinear focus does not even dig any hole in the transverse distribution. In contrast, at slightly higher power, inhomogeneities of the beam are amplified during the self-focusing stage and clearly engender a ring at z>9.8m [see Fig. 1(b)] as a result of modulational instability, and the higher-order Kerr effect is not significant until z~10.8m, as shown in Fig. 1(c). The tendency is even more enhanced at z~11.2m, where a deeper ring structure appears during the second self-focusing stage [Fig. 1(d)] and finally coalesces into a single filament, as can be seen in Figs. 1(e) and 1(f). Typically, the filament possessing a central core from tens to several hundreds of microns can propagate over tens of meters in air, much longer than the Rayleigh length [17].

At the low level of intensity, the energy losses due to MPI and dissipation caused by plasma absorption do not play dominant roles in the filamentation. When the central single spike is formed, the converged intensity increases quickly to the maximum. Then the higher-order terms of the Kerr effect will become to play a dominant role, and produce an additional nonlinear perturbation into the phase of the beam. Later on, modulational instability of the transverse cross-section gives rise to the gradient of the intensity flowing perpendicular to the propagation axis. A small portion of the energy of neighboring filaments will leak out and form a ring surrounding the main core that is shown in Fig. 2 for two different places. Subsequently the central intensity will undergo a decrease to form a hollow pipe and the higher-order effect also becomes weaker than the Kerr quadratic and the energy begin to flows from the peripheral region towards both the center filament and the near-filament region before the occurrence of another peak. Once more, the higher-order effect due to the increasing energy in the central region corresponding to the energy extraction from the external regions will cause defocusing. As a result, the combination of this focusing and spontaneous defocusing can maintain the long range propagation of the laser beam.

 

Fig. 2 (Color online) The intensity profile of the beam at (a) z = 12.7 m and (b) z = 14.1m, showing the presence of a narrow core surrounded by a weaker ring. It is shown that the robustness of the filament comes from the transverse low-intensity pedestal that controls the formation of the central hot spot.

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It should be noticed that the slow multiple refocusing of low-intensity background can continually replenish the energy loss in the core and supports the filamentation process over long distance. When the holes are dipped in the transverse beam section, they always appear near the nonlinear focus, as the intensity increase abruptly as shown in Figs. 3(a) and 3(b). Figure 3(c) shows the three-dimensional plot of the surface corresponding to equal values of the intensity along the propagating axis. They only result from the defocusing effect of the higher-order terms in the Eq. (1).

 

Fig. 3 (Color online) (a) The energy fluence distribution in the filament along the propagation axis z; (b) On-axis peak intensity vs propagation distance, and(c) Iso-intensity (iso = 0.5) distribution of the filament.

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4. Conclusion

We find it instructive to comment on the plasma channels excited by ionization of air molecules. It is to clear up the inner mechanisms that rebuild laser filaments after their formation. The analysis of the relevant nonlinear propagation equation reveals previously unrecognized behavior that the higher-order Kerr terms and the surrounding energy reservoir formed out side the filament core was proved to play a dominant role in this mechanism. Numerical simulation showed that the linear polarized cylindrically symmetric Gaussian beam had undergone an important transformation from a Gaussian to self-similar profile, featured by slowly decaying, oscillating tails. A self-similar ring-shaped profile which has been studied for a variety of conditions in nonlinear optics can be formed. The main result obtained from the simulation is that the general model is preserved but the first self-focusing stage is now significantly affected by higher-order Kerr terms and exhibits the formation of a ring-shaped pattern well before the excitation of electron plasma generated by MPI. In this case, it is not difficult to interpret the filamentation mechanism without ionization as reported in Ref [11]. Of course, the presence of the higher-order terms cannot prevent the generation of the plasma.