1. Introduction

One of the major goals in the study of nonlinear wave dynamics in optics is the generation of waves that propagate in the form of localized wavepackets in all the transverse dimensions of space, as well as in time i.e, nondiffracting and nondispersing [1]. Localization is typically achieved by nonlinearity that competes with the natural spreading of the wave in space and time. In a seminal paper, Silberberg proposed for the first time that Kerr nonlinearity can compensate for anomalous dispersion as well as diffraction leading to the formation of (3+1)D spatiotemporal solitons or light bullets [2]. Although the Silberberg light bullets are intrinsically unstable [3,4], the possibility to stabilize the soliton dynamics by means of different effects attracted a lot of attention. Among the candidates, higher order dispersion [5], competing higher order [6] or saturating [7] nonlinearities, materials with nonlocal nonlinear response [8] as well as filamentation [9] were investigated. A different approach was suggested to stabilize discrete light bullets by means of periodic waveguide configurations [10]. Similar approaches have also been carried out in the continuous limit both in periodic two-dimensional lattices [11,12] as well as in radially symmetric lattices [13]. Arrays of waveguides with Bragg gratings [14] have also been considered. Experimentally, 2D spatiotemporal solitons have been demonstrated in quadratic media [15]. Different types of spatiotemporal localized structures such as X-waves [16], Bessel-Airy linear bullets [17] and Airy-Airy-Airy nonlinear bullets [18] have been experimentally observed, whereas three-dimensional vortices [19] have been predicted. The main limiting factors in the experimental realization of light bullets are the anomalous dispersion and the slow time response of nonlinear media that mathematically support stable bullets. The availability of intense ultra-short light bullets in normally dispersive media would however offer ultimate control over a high power wavepacket, which could be used in various materials and have applications in fields like THz generation [20], telecommunications, optical metrology and sensing, long-range femtosecond filamentation [21] and attosecond pulse generation [22, 23]. In a recent work, we showed that filamentation of intense femtosecond laser pulses can be controlled with respect to its spatial properties by using optical lattices [24].

In this work, we exploit the filament dynamics in the presence of a lattice in both the spatial and the temporal domain. We find from numerical simulations that it is possible to generate spatiotemporal structures supporting intense laser propagation in normally dispersive air at a wavelength of 800 nm and remaining almost invariant (in space, time and maximum intensity) for over 90 cm in a lattice. We call these structures intense dynamic bullets (IDB) to distinguish them from perfectly stationary light bullets or solitons. The lattice is essential in the formation of the IDB. The role of each physical mechanism in the formation of the IDB is also analyzed.

2. Method

The lattice consists of concentric rings that describe a perturbation of the refractive index. These perturbations can be realized using different approaches for various transparent media. For example plasma can be used in the case of gases. One dimensional plasma lattices have recently been generated in air by interference of intense IR light beams for periodicities ranging from ∼500 nm [25] to 100 μm [26], and electron densities higher than 10¹⁸ cm ⁻³. The generation of cylindrically symmetric lattices is demanding but feasible since their Fourier transform is simply a sum of zero order Bessel functions. Note that the effect of such lattices on intense pulse propagation has not been explored yet. Thus, in principle, illumination of an amplitude mask allows the generation of intense concentric rings in the Rayleigh range of a long Fourier transforming lens. Furthermore, one could consider the use of positive or negative Δn lattices exploiting the molecular alignment of air molecules (or other gases). In the latter case the laser intensities needed are even lower than the ones needed for ionization and one can use pulse trains to further enhance the alignment [27] and consequently the strength of the lattice. Finally, in the case of filamentation in transparent solids, like glasses, one can use permanently written lattices in the bulk of the glass [28].

In our case, the cylindrically symmetric refractive index modulation is given by n _cyl(r) = n ₀ + Δn, for Δn = Δn ₀ Σ_{m =0} f(r – r_m). f(r) = exp[–(r/w)^{2 p}] is the function describing the refractive index distribution of each ring, in our case a super-Gaussian of order p = 8 with width w = 100 μm. The position of each ring is defined by r_m ≡ [m + 1/2]Λ. The period of the lattice is Λ = 350 μm and the refractive index modulation is Δn ₀ = −3 × 10⁻⁷. Assuming that plasma is used to generate such a modulation, this corresponds to a moderate plasma density of 10¹⁵ cm⁻³. The laser pulse used in the simulations has a Gaussian spatiotemporal profile with a duration of 35 fs at FWHM and 500 μm beam width (1/e² radius). The 150 μJ pulse is lauched in the center cylinder of the refractive index modulation as it is shown in Fig. 1.

 

Fig. 1 (Color Online) Input radial intensity distribution superimposed on the lattice potential.

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3. Numerical model

Our model relies on a nonlinear propagation equation along the z direction for the frequency components Ê (r,z,ω) of the envelope E(r,z,t) of the laser pulse coupled with an evolution equation for the electron density ρ(r,z,t) generated by the intense pulse [22]:

The first term on the right hand side (rhs) of Eq. (1a) describes diffraction and space-time focusing through the operator 𝒦_ω ≡ k ₀ + k′₀ δω, δω ≡ (ω – ω ₀), k ₀ ≡ k(ω ₀). $k(\omega )\equiv {\sum }_m{k}_0^{(m)}{(\delta \omega )}^m/m!$ denotes the dispersion relation in air and $k_0^{(n)}={\frac{{\partial }^{n}k}{\partial {\omega }^n}|}_{{\omega }_0}$ denote the dispersive coefficients corresponding to the frequency ω ₀ of the carrier wave. Dispersion and the effect of the lattice are described by D(ω) ≡ k(ω) – 𝒦_ω and Δn, respectively. The nonlinear polarization P̂ _NL(r, z,ω) [Eq. (1b)] accounts for the optical Kerr effect, plasma defocusing, and multiphoton absorption (MPA), first calculated in the temporal domain and transformed into spectral components of P̂ _NL. Pulse self-steepening is given by the explicit frequency dependence ω ≡ ω ₀ + δω in Eq. (1a). In some simulations discussed below and referred to as “Shock term off”, its effect was switched off by setting ω = ω ₀ in the explicit frequency dependence of the nonlinear polarization in Eqs. (1a) and (1b). The gas filling the gap of the lattice is air with nonlinear index n ₂ = 3.2 × 10⁻¹⁹ cm²/W, multiphoton ionization and absorption cross sections σ_K = 3.4 × 10⁻⁹⁶ s⁻¹cm¹⁶W⁻⁸ and β_K = Kh̄ω ₀ ρ _nt σ_K for K = 8 photons. The density of neutral oxygen molecules is ρ _nt = 0.5 × 10¹⁹ cm⁻³ and the critical plasma density is ρ_c ≃ 1.7 × 10²¹ cm⁻³.

From Marburger’s formula $P_{cr}=\frac{3.77{\lambda }^2}{8\pi n_0{n}_2}$, the critical power for collapse of a Gaussian beam is P_cr ∼ 3 GW. In the following we consider pulses with peak power of 1.25 P_cr and we stress that the critical power is only used as a reference indicating that a beam with the same input power would collapse in a pure Kerr medium. Due to the weak dispersion in air (k″ ∼ 0.2 fs²/cm), self-focusing prevails for input powers only a few percent above P_cr [29].

4. Simulation results and discussion

Figure 2 (Media 1) depicts the spatiotemporal reshaping of the laser pulse as it propagates in air without (top row) or with (bottom row) the lattice. In the case of standard filamentation in air, the intense pulse initially shrinks both in space and time due to the Kerr nonlinearity. At the nonlinear focus (z ∼ 130 cm), the pulse becomes sufficiently intense to generate an underdense plasma which defocuses the pulse trailing part while nonlinear effects including the Kerr effect and multiphoton absorption compete to sustain propagation of the leading part in the form of a filament (130 cm < z < 150 cm). Beyond the filamentation stage z > 150 cm, the pulse finally widens due to diffraction and dispersion. Such behavior is typical for pulses undergoing filamentation and is accompanied by high nonlinear losses [22].

 

Fig. 2 (Color Online) 3D iso-surface plots of the intensity distributions for the cases of (a) standard filamentation (red iso-surfaces, first row) and (b) IDB formation inside the periodic lattice (blue iso-surfaces, second row), for various propagation distances (Media 1). Isovalue is set to half of the peak intensity at each z position.

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Propagation of the same intense pulse inside the lattice is significantly different: during the initial self-focusing stage, the beam does not shrink as much as it does without the lattice. After the initial self-focusing stage, the spatiotemporal dynamics reaches a quasi-equilibrium. The intensity profile of the resulting IDB remains almost stationary in both space and time for about 1 m, i.e a factor of 5 larger than the filamentation distance in air for the same input pulse. The IDB finally starts to spread out slowly at z ∼ 240 cm. The peak intensity for the two regimes as a function of the propagation distance is shown in Fig. 3(a).

 

Fig. 3 (Color Online) (a) Peak intensity vs z for a standard filament in air (continuous line) and the IDB (dashed line). (b) Pulse duration (Radially averaged over 100 μm) of the IDB vs propagation distance z when all effects are accounted for (continuous curve) or when a specific effect is switched off at z = 150 cm: dotted curve: the lattice is removed; dashed curve: self-steepening is switched off; dash-doted curve: GVD is switched off.

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In order to investigate the interplay between different physical mechanisms responsible for the IDB formation, we performed numerical experiments by switching off each of the relevant effects, at z = 150 cm, where the IDB is already formed. Namely, we considered the effects of dispersion, self-steepening (shock-terms) and the periodic potential. Figure 3(b) shows the radially averaged pulse duration over a 100 μm radius which corresponds to the central cylinder of the lattice. The duration of the IDB remains nearly constant to about 15 fs over 90 cm (continuous curve). When group velocity dispersion (GVD) is switched off at z = 150 cm, the IDB with peak power above P_cr undergoes self-compression due to the competition between self-steepening and nonlinearity (Kerr focusing and plasma defocusing). This leads to a continuous decrease of the pulse duration [dash-doted curve in Fig. 3(b)] that ultimately becomes shorter than the single cycle limit (not shown here) and results from a well identified singularity of the standard filamentation model in the absence of GVD [30]. The periodic lattice modifies the effective diffraction of the beam and plays a crucial role in its stabilization since the pulse duration increases [dotted curve in Fig. 3(b)] if the lattice is replaced by air. In this case, the beam survives until z = 170 cm but then rapidly disperses. Finally the dashed curve depicts the pulse duration if the shock-term is switched off at z = 150 cm. The comparison of the dashed and continuous curves shows that the shock-term is essentially limiting dispersion since the pulse broadens in its absence.

From these numerical experiments, we identified the physical effects that dynamically balance each other to generate an IDB via nonlinear propagation in the periodic potential. Figure 4 shows the evolution of typical lengths characterizing these effects, namely the Kerr effect L _Kerr = (k ₀ n ₂ I)⁻¹, diffraction L _Diff = k ₀ R²/2, Dispersion L _GVD = T ²/2k″₀, MPA L _MPA = 1/(2β_KI^{K −1}), self-steepening L _Shock = cT/n ₂ I and the effect of the lattice L _Lat = (k ₀Δn)⁻¹. In these expressions, the peak intensity of the pulse I varies as a function of propagation distance, as well as the filament width R and the shortest pulse duration T supported by the pulse spectrum. For standard filamentation [Fig. 4(a)], the main prevailing effect is MPA as indicated by the shortest lengths of all effects over the whole filamentation distance 130 < z < 150 cm. The Kerr and self steepening effects need centimetric lengths to play a role while diffraction and dispersion need about 10 cm. Thus the filament lives for about 20 L _MPA or equivalently one L _Diff or L _Disp. For the IDB [Fig. 4(b)], the intensity is stabilized by the lattice at a lower level than that reached in the filament, thus the Kerr effect prevails with a typical length in the range 10–15 cm over the whole propagation distance 150 < z < 240 cm. Two independent equilibria take place in the spatial and temporal dimensions: For the spatial equilibrium, diffraction and the lattice effect with lengths in the range 30–40 cm both compete with the Kerr effect as 1/L_K ∼ 1/L _Diff + 1/L _Lat. For the equilibrium of the pulse profile, self-steepening and dispersion compete with self-phase modulation induced by the Kerr effect as 1/L_K ∼ 1/L _Shock + 1/L _Disp. These two separate equilibria maintain the peak intensity below ionization threshold for a propagation distance of about 10 L_K or 2 L _Disp, which is a factor of at least 5 larger than the filamentation length for the same input pulse.

 

Fig. 4 (Color Online) The smallest lengths characterize the most important physical effects in competition along the propagation axis. (a) Case of a standard filament. (b) Case of an IDB in a periodic lattice.

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5. Conclusion

In conclusion, we have shown numerically that filamentation of intense pulses in a radial symmetric periodic potential leads to the generation of intense dynamic bullets. Their intensity is above 10¹² W/cm² and their power is above P _cr while they exhibit a quasi-stationary spatiotemporal profile for long propagation distances as a result of the competition of linear and nonlinear effects. In the spatial domain the Kerr self-focusing is mainly balanced by the combined action of the effective lattice diffraction whereas in the temporal domain self-steepening and normal dispersion compete with self-phase modulation. Since the lattice parameters are in a feasible regime for experimental realization (refractive index changes in the order of 10⁻⁷ and periodicity of 350 μm), these intense dynamic bullets can be realized in transparent media using permanent or transient refractive index modification induced by high power commercial fs laser sources.