1. Introduction

It is well known that the propagation of intense laser pulses in any kinds of Kerr media (gases, liquids, and transparent solid materials) can result in the formation of filamentation under certain conditions. Recently various aspects of filamentation, such as long range propagation [1], light detection and ranging techniques (LIDAR) [2], virtual antenna [3], spectral broadening [4], pulse self-compression [5] and THz generation [6], have attracted lots of attention. Interestingly, recent experimental as well as theoretical studies have shown that the filamentation can be formed without self-guiding [7].

By recalling that the filamentation is an outcome of the balance of the diffraction, self-focusing due to the nonlinear index refraction and self-defocusing due to ionization, it is quite natural to come to the idea that the use of different transverse modes for incident laser pulses may result in different propagation dynamics, which may affect the filamentation formation, etc. Indeed, because the Bessel beam [8, 9] is known to be diffraction-free when it propagates in vacuum, the use of the Bessel beam draws lots of interests in recent years [10-12]. In a laboratory, however, it is not exactly an ideal Bessel beam but rather a quasi-Bessel beam that is produced. A quasi-Bessel beam is an apertured Bessel beam which carries large but still finite energy. Compared with a Gaussian beam with the same beam diameter, even a quasi-Bessel beam exhibits a remarkable resistant to diffraction during the propagation [13-15].

Comparisons between the Bessel and Gaussian beams for incident pulses have been reported both theoretically and experimentally from different aspects such as the beam divergence, power-transport efficiency [15], second harmonic generation efficiency [16], propagation diffracted by a circular aperture [17], and plasma channels [18, 19]. To our knowledge, however, a detailed study on the transverse-mode dependence of filamentation has not been reported in the literature.

The purpose of this paper is to theoretically investigate how the transverse-mode of the incident femtosecond pulse affects the propagation in the Kerr medium, in particular in terms of the formation of filament and plasma channel. We assume an Ar gas as a Kerr medium and numerically solve the extended nonlinear Schrödinger equation coupled with the electron density equation. Through the comparisons of the propagation dynamics for three different transverse-modes for incident pulses, i.e., Bessel, Gaussian, and Laguerre modes, we will show that the use of the Bessel mode significantly improves the propagation characteristics in the Kerr medium, for instance, in the formation of filament and plasma channel. Perhaps more importantly we will also show that the temporal distortion of the Bessel incident beam integrated over the beam radius is extremely small compared with the other modes. Since the pulse energy that the Bessel beam can carry is more than one order of magnitude larger than the other modes for the same peak intensity and pulse duration, these findings clearly indicate that the Bessel beam can be a very powerful tool in ultrafast nonlinear optics involving propagation in a Kerr medium.

2. Model

In this paper, we theoretically investigate the transverse-mode dependence of femtosecond filamentation in the Ar gas. Three different transverse modes are considered for incident pulses: Bessel, Gaussian, and Laguerre modes which have all axial symmetry with respect to the radial coordinate r. The extended nonlinear Schrödinger equation coupled with the electron density equation is used to describe the pulse propagation in the Ar gas.

2.1 Transverse modes of the incident beams

The transverse modes of incident pulses we employ in this paper are the Bessel

Gaussian

and Laguerre modes

where ε₀ is the peak amplitude of the incident electric field, t ₀ and w ₀ are the half temporal width and the spatial radius of the intensity (for 1/e²), respectively, and J ₀ is the zero order Bessel function. The parameters α and β are chosen in such a way that the full widths at half maximum of the beam diameter, $\left(w_{\mathrm{WFHM}}=\sqrt{2\ln 2}{w}_0\right)$, of different transverse modes becomes equal. The spatial profiles of the three transverse modes are shown in Fig. 1. In practice, a Bessel beam can be obtained from a Gaussian beam through the mode-conversion with an axicon lens [10, 18, 19], while a Laguerre beam can be obtained in a similar manner from a Gaussian beam with spiral phase plates [20, 21] or computer-generated holograms [22, 23].

 

Fig. 1. Spatial intensity distribution of the (a) Bessel, (b) Gaussian, and (c) Laguerre beams for the same beam diameter defined by the full width at the half maximum as shown by red arrows on the right panels.

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2.2 Equations

In order to describe the pulse propagation in the Ar gas, we employ the extended nonlinear Schrödinger equation coupled with the electron density equation. The extended nonlinear Schrödinger equation for the electric field envelop ε(r, t, z) in a reference frame (t=t_lab-z/v_g) moving at the group velocity v_g can be written as,

where the electron density ρ can be computed from the following equation,

In Eqs. (4) and (5), k ₀=2π/λ ₀ and k″=20 fs²/m are the wave number in vacuum and the second order dispersive coefficient, respectively. n ₀ and n ₂=5×10^-19 cm²/W are the linear refractive index and the nonlinear refractive index, respectively. ρ_c=1.73×10²¹ cm^-3 is the critical plasma density above which the plasma becomes opaque, while ρ _nt=2.7×10¹⁹ cm^-3 represents the neutral atom density. σ=1×10^-19 cm² stands for the cross-section for inverse bremsstrahlung, and U_i=15.76 eV is the ionization potential. I=εε* denotes the laser intensity and W(I) is the photoionization rate. Now Eqs. (4) and (5) are solved simultaneously by the split-step method [24, 25]; the diffraction part is calculated by the Crank-Nicholson method and the dispersion part is calculated in the frequency domain by Fourier transform, while the nonlinear part is solved in the time domain by the Runge-Kutta method.

2.3 Description of ionization

Filamentation takes place as a result of the balance between the diffraction, self-focusing due to the nonlinear index of refraction, and self-defocusing due to ionization. Clearly it is very important to choose the most appropriate model to describe the ionization processes. We note that different models for ionization may result in one or even a few orders of magnitude differences in the ionization rate at the same intensity, and naturally a filament with a wrong filament diameter and length will be numerically observed at a wrong intensity. This is so, since a filament originates from the dynamic balance between self-focusing and self-defocusing where the latter is directly related to the electron density in the medium. To describe the ionization process, it is often convenient to introduce the Keldysh parameter, γ, which is a measure of the adiabaticity [26]. For an atom with the ionization potential U_i exposed to the linearly polarized laser field with the field amplitude E and the frequency ω ₀, the Keldysh parameter, γ, is defined as,

where e is the electron charge and m the electron mass.

In this paper we employ the PPT (Perelemov-Popov-Terent’ev) model [27] instead of more commonly used tunneling or multiphoton ionization models, since the former is an improved version of the latters and valid for the wider intensity range.

2.3.1 PPT model

As mentioned above, the PPT model [27] is a kind of improved version of the Keldysh theory [26] where the ionization rate for any atom or ion with the orbital quantum number l and its projection m onto the quantization axis and the charge state Z is given by,

Naturally the PPT model can be reduced to the either tunneling or multiphoton ionization limits under certain conditions as we will briefly describe below. For detailed description, we refer to Refs. [26-28].

2.3.2 Tunneling and multiphoton ionization limits

Consider ionization of atoms in the initial state with l=m=0, n*=(U_i/U_H)^-1/2 and |C _n*,l*|²=2²ⁿ*/n*Γ(2n*). If the laser field is very strong, we have γ≪1 and tunneling ionization takes place. Under this condition we obtain A_m(ω ₀,γ)→1, and accordingly Eq. (7) can be reduced to

which is the ionization rate in the tunneling ionization regime (ADK model) [28]. From Eq. (8), we can see that the tunneling ionization rate is independent of the laser frequency. Similarly, if the laser field is sufficiently weak we have γ≫1 and multiphoton ionization takes place. Under this condition, Eq. (7) can be reduced to

where K=11 and σ_K=5.06×10^-140 s^-1 cm²²/W¹¹ for Ar at the central laser wavelength of 800 nm.

 

Fig. 2. Ionization probabilities of Ar calculated by the three different formulas, PPT model (solid), multiphoton ionization model (dashed), and ADK model (dotted), as a function of laser intensity. The duration and the central wavelength of the pulse are assumed to be 30 fs (FWHM) with a Gaussian temporal profile and 800 nm, respectively.

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In Fig. 2 we compare the ionization probabilities of Ar for the 30 fs pulse (FWHM with a Gaussian temporal profile) at the central wavelength of 800 nm calculated by the PPT model, and the tunneling (i.e., the ADK model) and multiphoton ionization limits, respectively. Figure 2 suggests that the PPT model is most appropriate to describe ionization for the intensity range of 10¹³ W/cm²~10¹⁴ W/cm² which is the intensity range we assume in this paper. Note that γ is about equal to or smaller than unity in this intensity range.

3. Results and discussions

To start with, we solve Eqs. (4) and (5) in vacuum to see the effect of diffraction. Figure 3 shows the variation of the beam diameter for the three different transverse modes of incident pulses as a function of propagation distance, z, which is normalized with respect to the Rayleigh length, z ₀, where z ₀=πw ² ₀/λ₀ with λ ₀ being the central laser wavelength in vacuum. We can see that the pulse with the Bessel incident mode is free of diffraction over a long distance while the pulse with the Gaussian incident mode has the largest diffraction compared with the other two modes. For the Gaussian beam with w _FWHM=100 µm and λ ₀=800 nm, the Rayleigh length is z ₀=2.83 cm.

 

Fig. 3. Variation of the beam diameters in vacuum for the Bessel (solid), Gaussian (dashed), and Laguerre (dotted) incident beams for the pulse with a 30 fs duration, which is nothing but the diffraction.

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Having checked the behavior of diffraction in vacuum, we now proceed to perform the propagation calculation in Ar at the atmospheric pressure with the incident pulse of 30 fs (FWHM) with a Gaussian temporal profile and a beam diameter of w _FWHM=100 µm for the three different transverse modes and the peak intensity of 3.2×10¹³ W/cm². Note that all the results shown below have been performed under these conditions except parameters mentioned specially, and the step sizes and the maximum beam radius we have employed to numerically solve Eqs. (4) and (5) are typically dt=0.5 fs, dr=5~10 µm, dz=0.01 cm and 2r _max=100w _FWHM.

It is very important to mention that each lobe of the Bessel beam (see Fig. 1(a)) contains approximately the same energy as the central part. This means that the pulse energy that the Bessel beam can carry is more than one order of magnitude larger than those of the Gaussian and Laguerre beams at the same peak intensity, pulse duration, and beam diameter. Indeed this is a very nice feature of the Bessel beam and we will make further remarks in terms of filamentation at the end of section 3.1.

3.1 Formation of filament

Now we perform propagation calculations in Ar gas at the atmospheric pressure and investigate the formation of filament for the three different transverse modes. Since the concept of filament is rather general as we have described in Sec. 1, we can always refer to the formation of filament for the incident pulse with any transverse mode. The length of the filament [29] can be defined as a distance of the propagation where the pulse spatially keeps a narrow and nearly constant diameter with a clamping intensity [30-32] which is sufficiently high to induce ionization in the medium. An alternative way to define the filament length is to refer to the length of the plasma channel. In this paper we have adopted the former definition for the filament length.

 

Fig. 4. Variation of the beam diameter as a function of propagation distance in the Ar gas for the (a) Bessel, (b) Gaussian, and (c) Laguerre incident beams.

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Fig. 5. Color-coded plot of the time-integrated propagating pulse as a function of propagation distance and beam radius for the (a) Bessel, (b) Gaussian, and (c) Laguerre incident beams.

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Figures 4(a)-4(c) show the variation of the beam diameter, w _FWHM, during the propagation for the Bessel, Gaussian, and Laguerre incident beams, respectively. Due to the self-focusing the beam diameter becomes smaller by ~20 % for all transverse modes during the first few cm of propagation in Figs. 4(a)-4(c). Accordingly the ionization probability at r=0 significantly increases from 10^-8 (at z=0 cm) to 10^-2~10^-3 (at z=3 cm) as shown in Figs. 10(a)-10(c), which now implies that self-defocusing starts into play. From the fact that both self-focusing and self-defocusing play important roles it is obvious that we are certainly not in the linear propagation regime but in the nonlinear propagation regime. Hence we can interpret the propagation of the pulse with a nearly constant beam diameter (which is smaller than the initial beam diameter as we have mentioned above) in Fig. 4 as a formation of filament. From Fig. 4 it is clear that the length of the filament is the longest for the Bessel incident beam, while that of the Gaussian incident beam is the shortest. We can also see that, for the same peak intensity (we have repeated calculations at intensities other than 3.2×10¹³ W/cm² which are not shown here), the diameters of the filaments are nearly the same for all three modes. It is interesting to point out that Fig. 4(c) for the Laguerre incident beam exhibits two separated filaments located at 0<z<10 cm and z>12 cm. This can be understood as a result of refocusing, as already discussed for the filament by the Gaussian incident beam [32-34]. Indeed refocusing takes place at different intensities for different transverse modes. For instance, similar to the case of Laguerre incident beam in Fig. 4(c), two separated filaments for the Bessel incident beam can be observed at a lower intensity (1.5×10¹³ W/cm² at which the Gaussian beam cannot form a filament).

Figures 5(a)-5(c) show the color-coded plot of the variation of the time-integrated pulse, i.e., I (r,z)=∫^{∞ -∞}|ε(r,t,z)|² dt for the Bessel, Gaussian, and Laguerre incident beams. We noted that the intensity distribution of the Bessel incident beam at small r is more uniform than the other two modes. It is also clear that the length of the filament for the Gaussian incident beam is the shortest among the three transverse modes. We would like to point out that the results shown in Fig. 4 are good measures about the detailed propagation characteristics as shown in Fig. 5. Clearly, provided the same peak intensity, pulse duration, and beam diameter, the Bessel incident beam has a remarkable superiority over the Gaussian and Laguerre incident beams in terms of the persistence of filamentation for a long propagation distance. This means that, even if the pulse energy is very large we can obtain a long filament by using the Bessel incident beam, since the Bessel beam can carry much more pulse energy than the Gaussian beam at the same peak intensity, pulse duration, and beam diameter as mentioned before. In contrast, the use of the Gaussian incident beam results in multi-filament [35-38] if the pulse energy is too large.

It is also interesting to compare the propagation of the Bessel and Gaussian incident beams with the same pulse energy and pulse duration. To investigate this, we perform similar calculations for the Bessel incident beam at the same peak intensity (3.2×10¹³ W/cm²) and pulse duration (30 fs) but with a much smaller beam diameter, w _FWHM=10 µm, which means that the pulse energy of the w _FWHM=10 µm Bessel incident beam is comparable to that of the w _FWHM=100 µm Gaussian incident beam. We find (results not shown here) that the beam diameter of the w _FWHM=10 µm Bessel incident beam is nearly unchanged during the propagation while the ionization probability is negligible, i.e., <10^-5 at r=0, which should be compared with Fig. 10(a) for the w _FWHM=100 µm Bessel incident beam where the ionization probability is much larger, i.e., ~10^-2. Such a small ionization probability for the w _FWHM=10 µm Bessel incident beam implies that the self-defocusing does not play any role during the propagation, indicating that the nearly unchanged beam diameter for the w _FWHM=10 µm Bessel incident beam during propagation has nothing to do with the filament. It simply comes from the non-diffractive nature of the Bessel beam in the linear propagation regime. As for the w _FWHM=10 µm Gaussian incident beam it suffers from huge diffraction. Thus, provided the comparable pulse energy and pulse duration (and naturally with different beam diameters), the use of the Bessel incident beam again leads to the significant improvement in propagation characteristics compared with the Gaussian incident beam, although it may not be called a filament if the beam diameter is chosen to be too small (such as 10 µm).

3.2 Change of the maximum intensity

In Fig. 6, we show the maximum intensity of the pulse (I_M(z)=max(|ε(r,t,z)|²)) as a function of propagation distance for the three transverse modes. We can see that, at the beginning of propagation, i.e., z<3 cm where self-focusing is the dominant process during propagation, the behavior of I_M(z) is almost the same for the three transverse modes. However, at z>3 cm where self-defocusing also set in due to significant ionization by the self-focused beam and hence self-focusing and self-defocusing compete to each other, quite different behaviors are seen for the three transverse modes: The Bessel incident beam has the best ability to maintain the beam diameter at the higher peak intensity during the propagation, which is followed by the Laguerre incident beam. The most commonly used Gaussian incident beam shows the worst feature.

 

Fig. 6. Maximum intensity of the pulse as a function of propagation distance for the Bessel (solid), Gaussian (dashed), and Laguerre (dotted) incident beams.

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3.3 Evolution of the temporal profile during the propagation

We further investigate the time-evolution of the three different modes during the propagation. As shown in Fig. 7(a) we have assumed that the temporal profile of the incident pulse is Gaussian and identical for the three transverse modes. In Figs. 7(b)-7(d) we show the temporal profile of the propagating pulse integrated over the beam radius, i.e., I(t, z)=∫^{∞ -∞}|ε(r,t,z)|² 2πrdr, at z=25 cm. We can see that the temporal profile, I(t, z=25 cm), for the Bessel incident beam is nearly undistorted, while those of the Gaussian and Laguerre incident beams exhibit significant distortions. The similar result at z=50 cm for the Bessel incident beam is shown in Fig. 7(e). Clearly the temporal profile still remains almost unchanged. Figures 8(a)-8(c) are the color-coded plot of the temporal profile, I(t, z), for the three different transverse modes as a function of propagation distance. Note that Figs. 7(a)-7(e) are nothing but the representative temporal profiles, I(t, z), at z=0, 25, and 50 cm in Fig. 8, respectively. As Fig. 8 clearly shows the Bessel incident beam is nearly undistorted during the propagation. This suggests that the use of the Bessel incident beam results in the significantly improved propagation characteristics in the Kerr medium.

 

Fig. 7. Temporal profile of the pulse integrated over the beam radius (a) at z=0 (note that this is exactly the same for the Bessel, Gaussian, and Laguerre incident beams), (b) at z=25 cm for the Bessel, (c) at z=25 cm for the Gaussian, (d) at z=25 cm for the Laguerre, and (e) at z=50 cm for the Bessel incident beams.

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Fig. 8. Color-coded plot of the temporal profiles integrated over the beam radius as a function of propagation distance, for the (a) Bessel, (b) Gaussian, and (c) Laguerre incident beams.

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Fig. 9. Temporal profiles of the pulse on axis (r=0) for the Bessel incident beam at different distances, (a) z=25 and (b) z=50 cm.

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In Figs. 9(a) and 9(b) we show the temporal profile of the pulse on axis, i.e., I(r=0, t, z)=|ε(r=0,t, z)|² for the Bessel incident beam at z=25 and 50 cm, respectively. It is interesting to point out that, although the temporal profile of the Bessel beam integrated over the beam radius remains nearly unchanged during the propagation (see Figs. 7(a), 7(b), and 7(e)), the on-axis temporal profile itself is distorted during the propagation. In this sense the behavior of the Bessel incident beam is still similar to those of the Gaussian and Laguerre incident beams. The results shown in Figs. 7-9 for the Bessel incident beam seem to suggest that the redistribution of pulse energy for the Bessel incident beam takes place in space only, while that for the Gaussian and Laguerre incident beams takes place in both space and time. To our knowledge, the peculiarity of the energy transfer for the Bessel incident beam has never been reported before.

3.4 Plasma channel

Having shown representative results for the pulse from the different viewpoints, we now present some results for the medium, i.e., Ar gas. Plasma channel is a spatial area with a nearly uniform plasma density along the propagation direction in the medium after significant ionization. In Fig. 10 we show the ionization probability at r=0 produced by the three different transverse modes for incident pulses. Clearly the plasma channel produced by the Bessel incident beam is much longer than that by the Gaussian incident beam. For the Gaussian incident beam, the plasma channel is usually limited to the short length due to the severe distortion of the pulse during the propagation. Increasing the peak intensity or beam diameter does not help to extend the length of the plasma channel for the Gaussian incident beam, since it simply results in the formation of multi-filaments [35-38], indicating that the plasma channel is lost. In contrast the Bessel incident beam is far superior in terms of the length of the plasma channel. The dips appearing in Figs. 10(a) and 10(c) are due to the reduction of the intensity at r=0 originating from the temporal distortion of the pulse. Figures 11(a)-11(c) are the color-coded plot of the ionization probability as a function of propagation distance and beam radius for the Bessel, Gaussian and Laguerre incident beams, respectively. We can see that the plasma channel formed by the Bessel incident beam is the longest and most uniform among the three transverse modes. A comparison between Fig. 5 and Fig. 11 shows that the diameter of the plasma channel is smaller than that of the pulse. This is simply because the formation of the plasma channel requires highly nonlinear processes in terms of ionization (see Eqs. (5), (7)–(9)). That is, a small change of intensity can result in a large change in the plasma density.

 

Fig. 10. Ionization probability at r=0 as a function of propagation distance for the (a) Bessel, (b) Gaussian, and (c) Laguerre incident beams.

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Fig. 11. Color-coded plot of the ionization probability as a function of propagation distance and beam radius for the (a) Bessel, (b) Gaussian, and (c) Laguerre incident beams.

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3.5 Robustness of the superiority of the Bessel incident beam

So far we have shown numerical results for the 100 µm beam diameter and 3.2×10¹³ W/cm² peak intensity for the incident beam. Since the formation of filament and plasma channel is based on the rather delicate dynamic balance of self-focusing and self-defocusing, it is very important to show that the superiority of the Bessel incident beam to the others is robust and not accidental. For that purpose we have undertaken further calculations for different beam diameters and peak intensities. Some representative results are presented in Fig. 12 for different beam diameters and peak intensities for the incident pulse. For the Gaussian incident beam, the length of the filament may become longer if we employ a larger beam diameter and higher intensity (compare Fig. 12 and Fig. 4(b)). But it is still shorter than that for the Bessel incident beam at the same beam diameter and peak intensity. We would like to point out that the filament lengths for the Bessel incident beams are beyond the distance we have calculated, 50 cm, for all five cases in Figs. 4(a), and Figs. 12(a), 12(b), 12(e) and 12(f). Moreover, increasing the beam diameter and peak intensity for the Gaussian incident beam to have a longer filament does not work forever, since multi-filament will take place at a certain intensity. Having the results presented in Fig. 12, we can now claim, with much more confidence, that the propagation characteristics of the Bessel incident beam is remarkably nicer compared with those of the Gaussian and Laguerre incident beams in terms of the lengths of filament and plasma channel and its potential to hold much more (possibly more than 1-2 orders of magnitude) energy in the pulse. Another results (not shown here) similar to Fig. 7 also show that the temporal profile of the pulse integrated over the beam radius is nearly undistorted for the Bessel incident beam. This also confirms the robustness of our findings.

 

Fig. 12. Variation of the beam diameter as a function of propagation distance in the Ar gas for the incident beams with (a) 150 µm, 3.2×10¹³ W/cm², (b) 200 µm, 3.2×10¹³ W/cm², (e) 100 µm, 2.0×10¹³ W/cm², and (f) 100 µm, 4.0×10¹³ W/cm². Ionization probability at r=0 corresponding to (a), (b), (e), and (f) are shown in (c), (d), (g), and (h), respectively.

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4. Conclusions

We have theoretically investigated the transverse-mode dependence of femtosecond pulse propagation in a Kerr medium (Ar gas) for the Bessel, Gaussian, and Laguerre incident beams by solving the extended nonlinear Schrodinger equation. Due to the different spatial intensity distributions and diffraction behaviors, we have found that the propagation characteristics represented by the filamentation and plasma channel are significantly different for the different transverse modes of the incident beams. Among others the Bessel incident beam turned out to have huge advantages over the others, since the length of the filament and also the plasma channel is much longer for the Bessel incident beam. Perhaps more importantly, we have also found that the temporal distortion of the pulse (integrated over the beam radius) is very small for the Bessel incident beam, while this is not the case for the Gaussian and Laguerre incident beams. A very similar trend has been found for different beam diameters and peak intensities, indicating that our findings are robust. This suggests that the femtosecond Bessel beam can be very useful for the high intensity signal transmission over the long distance. All these findings clearly indicate that the Bessel beam can be a very powerful tool in ultrafast nonlinear optics involving propagation in a Kerr medium.