1. Introduction

Typically a nanosecond, hundred picosecond laser pulse or laser beam with power little above the critical for self-focusing can propagate from one to few diffraction lengths in materials before collapsing. The femtosecond pulse propagation in air and other gases demonstrate significant increasing of this collapse/filamentation distance from a few meters up to several kilometers, and this effect is named arrest of the collapse. Various physical effects were suggested to prevent from early collapse, and to increase the distance of propagation before significant self-focusing and fragmentation of the pulse. The most popular theories include plasma-induced defocusing. However, when the intensity of the pulses is below the threshold for plasma generation [1] and in absence of ionization, collapse arrest is also observed. Other mechanisms for increasing the self-focusing distance such as non-paraxiality [2, 3], higher order of nonlinearity and dispersion were also investigated. The group velocity dispersion (GVD) can play some role in solids, where the dispersion length can be made of the order of the diffraction for femtosecond pulses, while for experiments in air, other gases and liquids the GVD is negligible and can not add significant effects on measurable distances. In [4] it is found that in fs domain the non-paraxial terms in the normalized envelope equation are not small and in some case bigger than transverse Laplacian. As a result, diffraction of fs optical pulse significantly smaller that beam diffraction is obtained. The purpose of this work is to investigate the nonlinear propagation dynamics of femtosecond laser pulses in air, gases and liquids, governed by the non-paraxial nonlinear evolution equation presented in [4].

2. Nonlinear envelope equation in dimensionless form

Starting from the Maxwell’s equations for an isotropic, dispersive, nonlinear Kerr-type media and using a linearly polarized one directional amplitude envelope E⃗=x⃗A(x,y, z, t)exp(i(k ₀ z-ωt)) the following scalar nonlinear envelope equation (NEE) can be obtained [5, 6, 7, 4]:

where k ₀ is the carrying wave number, v is the group velocity, k ^” is the dispersion of the group velocity and n ₂ is the nonlinear refractive index. We change to a nondimensional Galilean coordinate system with A = A₀ψ; x=r⊥x ^”; y=r⊥y ^”; z ^′=z₀z ^”; t ^′=t₀t ^”; z′=z₀z ^”; t=t₀t ^”. t ^′=t; z ^′=z-vt. We denote the initial transverse dimension by r _⊥, while z ₀ denotes the initial longitudinal dimension which is simply spatial analog of the initial time duration t ₀, determined by the relation z ₀=v_grt ₀. After dropping the second on dimensionless variables the equation for the nondimensional envelope ψ becomes

where ${\Delta }_{\perp }=\frac{{\partial }^2}{\partial x^2}+\frac{{\partial }^2}{\partial y^2}$ is the transverse Laplacian and

The number β = k₀r² _⊥/z_dis determines the ratio between diffraction and dispersion length and is very small in the transparency region of air and gases. The provided analysis of dimensionless parameters allows neglecting the dispersion terms with β coefficient and the self-steepening term with 2αδ ² γ ₁ coefficient (when γ~1) as small ones in equation (2). In this case the equation which governs pulse evolution in media with weak dispersion is

The NEE (5) is normalized to diffraction length z_diff=1 when 2αδ ²≅=1. This gives the possibility to calculate evolution of optical pulses in standard diffraction length, and to compare the results with evolution in paraxial approximation. The role of the mixed second derivative term was studied in [8] in the framework of ordinary one dimensional model. It is interesting that for fs laser pulses this term arrests the pulse self-compression in the nonlinear focus. The role of this term becomes more crucial in the frame of 3D+1 dimensional model.

 

Fig. 1. Nonlinear dynamics of the intensity profile of a Gaussian beam governed by 2D paraxial Eq. (6) under initial conditions ψ(x, y, t^′=0)=exp(-(x ²+y ²)/2), and γ=2.25. The beam is focused to a minimal diameter with large pedestal on distance 2–3z_diff.

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3. Non-paraxial dynamics of long optical pulses, light bullets and light disks

We consider optical pulses from 40–50 fs up to ps region. The optical period of center frequency has order 2 fs and this is the reason to fix the first constant α≫1; (here and below we have α=40, where a is the number of harmonics under pulse with precise 2π). We investigate the case when the power is slightly above the critical for self-focusing and use in our calculations the nonlinear constant γ≥1; γ=2.25. Therefore from three parameters in Eq.(5) we fix two:a and γ. The third parameter δ ² can be changed considerably in the ps and fs regions. For example,

the typical transverse size of laser pulses varies from r _⊥≅ 1-5 mm up to r⊥≅100µm, while for the longitudinal size of pulses there are three possibilities: pulses from nanosecond up to 30–40 ps, pulses from few ps up to 300–400 fs, and pulses from 250–300 fs up to 40–50 fs. Using the relation z ₀=v_grt ₀ one can obtain that in the former case z ₀≫r _⊥, δ²≪1 and the shape is closer to optical filaments while in the second case one can arrange z ₀≃r _⊥, δ²≅1 and the pulses look like light bullets. In the later case z ₀≪r _⊥, δ²≫1 and the form is closer to the light disks. All next numerical computations are performed with initial Gaussian pulses which satisfy the boundary conditions lim_x,y,z↦ ψ(x,y, z)=0 and also lim_{kx,ky,kz↦ψ} ̂(kx,ky,kz)=0.

 

Fig. 2. View of the spot of long Gaussian pulse (surfaces | ψ(x, y, z^′=0, t ^′=0; π/15; 2π/15; 3π/15) |² (x, y plane, y-vertical, x-horizontal)) governed by nonparaxial Eq. (5) under initial conditions ψ(x, y, z^′, t^′=0)=exp(-(x ²+y ²+δ ² z ^{′ 2})/2), α=40; δ ²=1/81; γ=2.25. The self-focusing regime is similar to self-focusing of a laser beam. Compare with Fig. 1.

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Fig. 3. View of surfaces | ψ(x, y=0, z ^′, t ^′=0; π/15; 2π/15; 3π/15) |² (x, z ^′-plane, x-vertical, z ^′-horizontal) at the same long Gaussian pulse governed by Eq. (5) and under the same initial conditions as on Fig. 2. Large spatial longitudinal modulation is observed.

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3.1. Laser beam

The standard model for beam propagation in optical media, incorporating the diffraction and Kerr effect is the two-dimensional NLS equation

 

Fig. 4. Nonlinear evolution of a Gaussian light bullet governed by Eq. (5) under initial conditions ψ(x, y, z^′, t^′=0)=exp(-(x ²+y ²+δ ² z ^′2)/2), α=40; δ2=1; γ=2.25. The surfaces | ψ(x, y=0, z^′, t^′=0; 2π; 4π; 6π) |2 are plotted. Significant increase at the self-focusing distance z^′≃18z_diff and a collapse without pedestal (Townes profile) are observed.

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Fig. 5. Nonlinear evolution of a light disk governed by Eq. (5) under initial conditions ψ(x, y, z^′, t^′=0)=exp(-(x ²+y ²+2z ^′2)/2),α=40; δ²=81; γ=2.25. The surfaces | ψ(x, y=0, z ^′, t ^′=0; 3π; 6π; 9π) |² are presented (x-vertical, z ^′-horizontal). The optical disk preserves its shape in the numerical experiment over a distance 28z_diff.

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Figure 1 presents intensity profile of the numerical solution to Eq. (6) for initial Gaussian beam in nonlinear regime with beam power P ₀ higher than the critical P ₀=2.25>P_cr. The numerical experiment demonstrates a typical picture, that is the profile is focused on very small area with large pedestal. Other well known result represented on Fig. 1 is that the wave collapse appear at normalized distance of two - three diffraction length z^′ _selffoc~2–3z_diff.

3.2. Long optical pulses

The nonlinear regime of a long optical pulse is performed by Eq. (5) and the following choice of dimensionless constants: γ=2.25; δ²=1/81; α=40. Figure 2 shows the spot, of a long optical pulse (| ψ(x,y, z ^′=0, t ^′=0; π/15; 2π/15; 3π/15) |²- surfaces), while Fig. 3 shows the xz ^′ plane of the same pulse (|ψ(x,y=0, z ^′, t ^′=0; π/15; 2π/15; 3π/15) |²- surfaces). An important difference is observed when we compare the propagation of a laser beam and a long pulse. While the transverse dynamics is approximately equal to the dynamics of a light beam (compare Fig. 1 with Fig. 2), the longitudinal size is influenced significantly by the nonlinearity, since the coefficient of the longitudinal temporal second derivative and the mixed term δ²=1/81 in Eq. (5) is negligible with respect to the nonlinear parameter γ=2.25. Thus, the nonlinearity becomes significant in longitudinal spatial and time projection and leads to self-phase modulation, spatial modulation in z direction (see Fig. 3) and enlargement of the spectra. The numerical analysis of non-paraxial Eq. (5) for long pulses demonstrates new challenges: a possible longitudinal spatial modulation, spectral enlarging of the pulse and obtaining a pulse with large spectral bandwidth.

3.3. Light bullets

The intensity profiles of a light bullet in nonlinear regime, which exceed the critical for selffocusing are displayed in Fig. 4. We solve numerically Eq. (5) for Gaussian bullet under initial condition ψ(x,y, z^′, t^′=0)=exp(-(x ²+y ²+² z ^′2)/2) and the following choice of dimensionless constants: γ=2.25; δ ²=1; α=40. The nonlinear dynamics demonstrates a new regime of week self-focusing without pedestal (scaled Townsian profile [9]) and a significant increase of the distance from where the self-focusing starts. Comparing the collapse of light beam (Fig. 1), long optical pulse (Fig. 2) and the bullet’s collapse (Fig. 4) it can be seen immediately the collapse arrest. The bullet’s self-focusing practically starts weakly on distance z≃18z_diff, while the collapses of light beam and long pulse appear at shorter distance of z≃(0.5-3)z_diff. The observation that the universal collapse profile is a Townes profile rather than blowup one, was demonstrated experimentally in [9]. We should note here that our nonlinear solution and the linear one in [4] are different from bullet’s solution obtained in the framework of model [10], where linear paraxial spatiotemporal equation has been used.

3.4. Light disks

We solve Eq. (5) for Gaussian light disk under initial conditions ψ(x,y, z^′, t^′=0)=exp(-(x ²+y ²+δ ² z ^′2)/2) and the following choice of dimensionless constants: γ=2.25; δ2=81; α=40. Figure 5 plotted the surfaces |ψ(x,y=0, z^′, t^′=0; 3π; 6π; 9π)|² where the x-direction is choused to be vertical and z ^′- direction, horizontal. It can be seen that the disk propagates without diffraction and self-focusing at a distance z≅28z_diff. From Eq. (5) and the numerical experiment we observe that the non-paraxial terms (δ ² terms) and first order term (2αδ ²) dominate over the nonlinear and diffraction terms (2αδ ²>δ ²≫γ>1) and practically blocked the self-focusing. Therefore, at a long distance, we observe stable propagation of light disk without change of its spatial, temporal and spectral form.

4. Conclusion

The numerical solutions of non-paraxial Eq. (5) demonstrate that the collapse process is highly sensitive to the initial 3D shape of the pulses. Moreover, it is shown that the non-paraxial model can explain such effects as longitudinal spatial and spectral broadening, collapse arrest and nonlinear wave guide behavior of ultrashort optical pulses in nonlinear regime near the critical power for self-focusing. This work is partially supported by the Bulgarian Science Foundation under grant F 1515/2005.