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We theoretically investigate the formation of filament and plasma channel in Ar gas by intense femtosecond pulses in the Bessel, truncated Bessel, and combination of two Gaussian modes. Through the numerical results obtained by solving the generalized nonlinear Schrödinger equation coupled with the electron density evolution equation, we find that there is a radial energy flow during the propagation, which implies that the outer part of the Bessel beam serves as an energy reservoir for the filament formed around the central peak. The results we obtain for the Bessel and truncated Bessel incident beams are consistent in that we can obtain a longer filament and plasma channel if more energy is reserved in the outer part of the Bessel incident beam. More interestingly we show that the combined use of two Gaussian beams with different beam diameters increases the energy stored in the outer part of the beam, and as a result the lengths of the filament and plasma channel become remarkably longer. This can be a practical choice to improve the propagation properties.
©2010 Optical Society of America
1. Introduction
Since the Bessel beam has been introduced to the beam profile family by Durnin et al. [1,2], the non-diffraction property [3–6] of the Bessel beam has drawn more and more attention in recent years from both scientific and technological points of view such as harmonic generation [7–12], microfabrication, microlithography, micromanipulation [13–18], and filamentation [19–24]. In our previous work [25], we have theoretically studied the filamentation in Ar gas by the intense femtosecond beams in three different transverse modes (Bessel, Laguerre, and Gaussian), and found that the Bessel incident beam has a huge advantages over the others. The lengths of the filament and the plasma channel induced by the Bessel incident beam are much longer than those by the other transverse modes under the same peak intensity, pulse duration, and beam diameter. 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.
In this paper, we focus on the Bessel incident beam and undertake more detailed study to better understand the peculiar propagation dynamics we have found in our previous work [25]. For that purpose we numerically solve the generalized nonlinear Schrödinger equation coupled with the electron density evolution equation, and study the propagation dynamics of the Bessel incident beams. To start with, we investigate the temporal and spectral evolutions at different segments (lobes) of the Bessel incident beam and the electron density of the medium during the propagation. Change of the field current at different segments of the beam during the propagation provides us with more information. All those results consistently suggest that the outer part of the beam serves as an energy reservoir to maintain the filamentation at the inner part of the beam. After studying the propagation dynamics of the Bessel incident beam, we show that the truncated Bessel incident beam still possesses an advantage over the Gaussian incident beam in terms of the formation of filament and plasma channel if the truncated radius is not too small. Finally, we propose an idea of combining two Gaussian beams with different beam diameters and show that the combined Gaussian beams result in better propagation properties in terms of the lengths of the filament and plasma channel.
Before going into the detail, we should note that higher order nonlinear terms such as self-steepening and space-time focusing are not included in our calculations. Strictly speaking such terms are important for short pulses as we have assumed in this paper. However, as we will show later on, the results for the 30 fs pulses we present in this work and also in our previous work [25] without such terms are very similar with those for 50 fs pulses or longer where higher order nonlinear terms do not play any important role. If someone intends to compare their results with ours this point has to be kept in mind.
2. Model
The model we employed in this paper is the generalized nonlinear Schrödinger equation coupled with the electron density evolution equation. The Bessel incident beam with a Gaussian temporal profile can be expressed as:
where ε 0 is the peak field amplitude of the incident laser pulse, kr is the radial wave number, t 0 is the half temporal width for 1/e 2 of the intensity, and J 0 is the zero order Bessel function. The wavelength λ, wave number k, radial wave number kr and longitude wave number kz are connected through the relation of:In Fig. 1 , we plot a Bessel beam with a 100 µm FWHM (full width at half maximum) diameter. We can see that the Bessel beam has a maximum at the center of the beam and a lot of sub maxima (lobes).
Fig. 1 The transverse profiles of a 100 µm diameter (FWHM as shown by the red arrows) Bessel beam as a function of (a) x and y (b) r.
The generalized nonlinear Schrödinger equation for the electric field envelope, ε(r, t, z), in the reference frame (t = t lab-z/v g) moving at the group velocity, v g, can be written as:
where the evolution equation of the electron density, ρ, can be written as:In the above equations, k 0 = 2π/λ 0 with λ 0 being the wavelength in vacuum, k׳׳ the second order dispersive coefficient, n 0 and n 2 the linear and nonlinear refractive indices, respectively, ρ c the critical plasma density above which the plasma becomes opaque, ρ nt the neutral atom density, σ the cross-section for inverse bremsstrahlung, U i the ionization potential, I = εε* the laser intensity, and finally W(I) the photoionization rate. Equations (3) and (4) under the incident pulse described by Eq. (1) can be solved by the split-step method [26,27]. For more detail about the meaning and the value of the parameters and the method of solving the equations, we refer to Refs [19,25–30].
3. Filament and plasma channel by the Bessel beam
Following our previous work [25] we assume that the incident pulse is a 30 fs, 100 µm (for the FWHM of the intensity) Bessel pulse with a 3.2 × 1013 W/cm2 peak intensity unless otherwise mentioned, and calculate the pulse propagation in Ar gas at the atmospheric pressure. The step sizes and the maximum beam radius we have employed to numerically solve Eqs. (3) and (4) are typically dt = 0.5 fs, dr = 5 ~10 µm, dz = 0.01 cm and 2r max = 120 w FWHM. In the following, it should be understood that whenever we refer to the pulse duration, beam diameter and beam radius, they are defined for the FWHM.
3.1 Evolution of the temporal and spectral profiles of the beam
To start with, we look into the temporal and spectral profiles of the pulse. For that purpose, we numerically solve Eqs. (3) and (4) under the conditions mentioned above, and perform the spatial integration over different beam radii. Note that this spatial integration does not affect the propagation dynamics, since it is carried out after the propagation calculation has been completed. First, we focus on the evolution of the temporal profile. The evolutions of the partial powers (integration of the pulse intensity over a finite domain of r) at each instant t as a function of propagation z are shown in Fig. 2 . When the maximum r for the spatial integration is small (100 µm) (Fig. 2(a)), the temporal distortion is serious. However, as the maximum r for the spatial integration is increased from 100 to 500 µm (Figs. 2(b), 2(c)), the distortion becomes smaller, and finally when the maximum r is set to r max (Fig. 2(d)), the distortion is negligible. This clearly implies that the central part of the beam where the intensity is very high suffers from a serious temporal distortion during the propagation, while the outer part of the beam where the intensity is much lower is nearly undistorted. To understand this trend, recall that each lobe of the Bessel beam contains nearly the same energy [5]. It is then clear that including the outer part for the spatial integration means that the undistorted parts of the pulse makes more contribution to the spatially integrated temporal profile. Another fact we can find from Fig. 2 is that, along the filament, the trailing edge of the pulse experiences more nonlinear effects and hence more distortion than the leading edge. In fact, the plasma defocusing will result in refocusing the trailing edge of the pulse at moderate z [31,34]. This is because the filament is a dynamic balance between focusing and defocusing effect and the ionization effect in the leading part will be accumulated to the trailing edge of the pulse.
Fig. 2 Evolution of the partial powers integrated over the different beam radii within (a) 100 µm, (b) 200 µm, (c) 500 µm, and (d) rmax as a function of time t and propagation distance z.
Having shown the evolution of the temporal profile of the pulse, we now show the evolution of the spectral profile of the pulse, which can be obtained by taking the Fourier transform of the temporal profile. The results are shown in Fig. 3 . As we can see from Fig. 3(a), the central part of the beam integrated over the maximum r of 100 µm has a large spectral broadening. When the maximum r for the spatial integration increases, the spectral profiles converge to its original profile as we can see from Fig. 3(a) to 3(d). The reason for this tendency is exactly the same with that we have seen in Fig. 2.
Fig. 3 Evolution of the spectral profiles integrated over the beam radius within (a) 100 µm, (b) 200 µm, (c) 500 µm, and (d) rmax as a function of propagation distance.
More information can be obtained if we look at the local change of the temporal and spectral profiles of the beam. In Fig. 4 , we plot the temporal profiles for the different lobes of the beam, which are obtained by integrating over the central peak only, and the first eight lobes only. Note again that this integration does not affect the propagation dynamics, since it is carried out after the propagation calculation has been completed. As we go from the central peak to the outer lobes of the beam, we can see that the outer lobes propagate without temporal distortion for a longer propagation distance. This is consistent with the trend we have shown in Fig. 2. It is interesting to point out that the distorted parts of the lobes have a similar characteristic structure and its position moves along the propagation direction as the order of the lobes increases. See the distorted structures in the circles appearing in Figs. 4(c)-4(i).
Fig. 4 Evolution of the temporal profiles at the (a) central peak and (b)-(i) first eight lobes as a function of propagation distance.
Since we have seen the progressive changes in the temporal profiles for the different lobes in Fig. 4, we expect to see similar results in terms of the spectral profile. Indeed Fig. 5 shows the changes of the spectral profiles for the different lobes as a function of propagation distance. Note that the structures in the spectral profile are more obvious than those in the temporal profile. From Figs. 4 and 5, we can see that, when the high intensity Bessel beam propagates in the Kerr medium, the outer lobes will start to change after the longer propagation distance since the “local” peak intensity is lower, and maintain the original (undistorted) temporal as well as spectral profiles for the longer propagation distance compared with the inner lobes. The ideal (untruncated) Bessel beam has a lot of lobes and the temporal and spectral profiles will start to change (or start to suffer from the nonlinear effects) from the inner lobes first and then the outer lobes as the pulse propagates in the Kerr medium.
Fig. 5 Evolution of the spectral profiles at the (a) central peak and (b)-(i) first eight lobes as a function of propagation distance.
3.2 Energy flow
From the results presented in the previous subsection, one may naively think that the outer part of the beam does not make any important contribution to the Bessel beam propagation. That is not true. Although the inner part of the beam contributes much more to the spectral broadening and temporal distortion of the pulse, the outer part also plays an important role in the propagation. Actually the outer part of the beam serves as an energy reservoir and provides energy to the inner part to maintain the filamentation, which is consistent with the previous findings for the Gaussian beam [31,34] and the Bessel beam [19]. In this subsection, we do some analysis for the energy flow of the Bessel beam during propagation.
In Fig. 6 we show the change of the percentage of pulse energy contained in the different beam radii as a function of propagation distance. Note the decrease of energy for the larger radius (in particular 500 µm) and nearly unchanged energy for the small radius (100 µm). This seems to suggest that there is an energy flow from the outer part to the inner part.
Fig. 6 Change of the percentage of pulse energy contained in the different beam radii as a function of propagation distance.
In order to confirm the above interpretation, we have calculated the transverse field current j during the propagation. We have also calculated the current j for the Gaussian beam with the same peak intensity, beam diameter and pulse duration for comparison. The current j is defined as [19]:
With this definition, when the current takes a negative value the energy flow is understood to be inward to the cylindrical surface of radius r.In Fig. 7 , we plot the maximum and minimum values of the field current, i.e., max{j(r,t,z)} and min{j(r,t,z)}, at a given z, for the Bessel and Gaussian beams with the same peak intensity, beam radius, and pulse duration. Note that positive (negative) field current stands for the outflow (inflow) of the current. At the very beginning of the propagation, the outflow current is very small while the inflow current becomes more significant within a short propagation distance. This is associated with the self-focusing of the beam. After that, both Bessel and Gaussian beams can take positive as well as negative field currents depending on the value of r. However, after about 25 cm of propagation, the field current of the Gaussian beam becomes negligible because of the annihilation of the filament at around 25 cm [25].
Fig. 7 Change of the maximum and minimum values of the field current, i.e., max{j(r,t,z)} and min{j(r,t,z)} for a given z, for the Bessel and Gaussian incident beams during the propagation. Positive (negative) field current stands for the outflow (inflow) of the energy.
Figure 8 shows the field current integrated over time as a function of radius after the different propagation distances. At the beginning (i.e. z = 1 cm, see Fig. 8(a)), the field current is mainly inflow (negative) for both Bessel and Gaussian beam at small r (r < 100 μm). This is simply because of the self-focusing. After propagating some distance (see Figs. 8(b)-8(d)), we can see that both inflow and outflow are present, at different radial positions, i.e., at some radius there is an inflow while at another radius there is an outflow. We can see the energy flow not only at a small radius (r < 100 μm), but also at a large radius (r > 100 μm). In the 3D simulation for the Gaussian incident beam [35] Skupin et al. have shown that, beyond a circular zone of 300 μm diameter, the energy exchange becomes small and of weak influence, which was experimentally confirmed by Liu et al. [34]. In our work, we also find similar results for the Bessel incident beam in Fig. 8, but the zone of the Bessel beam seems larger than that of the Gaussian beam [34,35]. By comparing Figs. 8(b)-8(d) with Fig. 8(a), we notice that the field current at the distance where the filament is formed is not so large compared with the field current originating solely from the self-focusing at the very beginning of the propagation. Consistently with Fig. 7, we find that the field current disappears at z > 25 cm for the Gaussian beam where the filament disappears.
Fig. 8 Normalized field current integrated over time as a function of radius at (a) z = 1 cm, (b) 3 cm, (c) 25 cm, and (d) 50 cm for the Bessel and Gaussian incident beams.
In order to obtain more comprehensive information on the field current, we make color-coded plots as functions of time and beam radius. The results are shown in Figs. 9 and 10 for the Bessel and Gaussian incident beams, respectively. Note that Figs. 9 and 10 have been respectively normalized but with different scalings. Clearly time integration on Figs. 9 and 10 would result in Fig. 8. We can see that, at the beginning, the field current only flows towards the center for both Bessel and Gaussian incident beams. After some propagation distance, i.e., z = 3 cm, however, the current is inflow or outflow depending on the time and radial segments. Note the energy flow near the center of the beam. When the radius r is large, the energy flow becomes very small. It is clear that the presence of the localized field current is strongly associated with the formation and maintain of the filament. The energy flow from the outer part to the center must mean that the outer part of the beam serves as an energy reservoir for the filament.
Fig. 9 Normalized field current as functions of time and radius at (a) z = 1 cm, (b) 3 cm, (c) 15 cm, and (d) 50 cm for the Bessel incident beam.
4. Filament and plasma channel by the truncated Bessel beam
In our previous work [25], we have studied the dynamics in terms of the formation of the filament and plasma channel by the Bessel beam and Gaussian beam. In this section, we investigate the influence of the truncation of the transverse profile on the formation of filament and plasma channel, which is more realistic from the experimental point of view.
For that purpose we employ the Bessel incident beams with diameters of 100 µm and 150 µm truncated at 250 µm and 500 µm. All other parameters have been kept to be exactly the same with those in Section 3 and also in our previous work [25].
In Fig. 11(a) , we plot the variation of the beam diameter of the Bessel beam for the 100 µm incident beam diameter truncated at the radius of 250 µm and 500 µm. We can see the two filaments due to the defocusing-refocusing cycles. This is similar to what we have found for the Laguerre incident mode in our previous work [25]. Similar results for the 150 µm incident beam diameter are shown in Fig. 11(b). In this case, the filament turns out to be longer than that for the 100 µm incident beam diameter. From Figs. 11(a) and 11(b), we notice that the truncation at a larger beam diameter results in the better quality (longer and more continuous) of the filament, as we expect. Figure 12 presents the transverse profiles of the pulse as functions of propagation distance and radius for the four cases corresponding to Fig. 11. From Figs. 11 and 12, we find that the transverse profile of the pulse is not very sensitive to the truncation radius at the beginning of the propagation, since it is the intense central part of the beam, after all, that induces the self-focusing. However, as the pulse propagates for a longer distance, the outer part of the pulse, which did not participate the self-focusing during the first few cm of propagation, plays more role and the transverse mode becomes more sensitive to the truncation radius.
Fig. 11 Variation of the beam diameter of the truncated Bessel beam with (a) 100 µm and (b) 150 µm incident beam diameters as a function of propagation distance. Incident beams are truncated at the radii of 250 µm and 500 µm for both (a) and (b).
Fig. 12 Transverse profiles of the pulse as functions of propagation distance and beam radius for the 100 µm incident beam diameter truncated at the radii of (a) 250 µm and (b) 500 µm. Similar results are shown for the 150 µm incident beam diameter truncated at the radii of (c) 250 µm and (d) 500 µm.
Having studied the filament by the truncated Bessel beam, we now turn to the ionization dynamics in the medium. When the intense fs pulse passes through the Ar gas, ionization takes place in the gas. The term “ionization probability” defines the ratio between the numbers of ions and original neutral atoms. Note that the ionization probability can be obtained from the time integration of ionization rate. In Fig. 13 we show the ionization probability at r = 0 as a function of propagation distance. Note that Figs. 13(a) and 13(b) correspond to Figs. 11(a) and 11(b), respectively. Similar to the filament, the quality of the plasma channel also improves by using the larger beam diameter and/or larger truncation radius. We also notice that the ionization probabilities at r = 0 are nearly the same at the beginning of the propagation. Finally, Fig. 14 shows the ionization probability as functions of propagation distance and radius. It turns out that the plasma channel induced by the 100 µm diameter beam is not continuous, (Fig. 14(b)) while it is more continuous for the 150 µm diameter beam (Fig. 14(d)). Another remark is that, for the 100 µm case, the second plasma channel is much farther away in z for the truncation at the radius of 250 µm (Fig. 14 (a)) than that for the truncation at the radius of 500 µm (Fig. 14 (b)). If we choose more appropriate incident beam diameter and the truncation radius, we can achieve a long and continuous plasma channel (Figs. 14 (c) and 14(d)).
Fig. 13 Variation of the ionization probability at r = 0 for the truncated Bessel beam with (a) 100 and (b) 150 µm incident diameter (FWHM) as a function of propagation distance. In both cases the beam is truncated at the radius of 250 and 500 µm.
Fig. 14 Ionization probability as functions of propagation distance and beam radius for the 100 µm incident beam diameter truncated at the radii of (a) 250 µm and (b) 500 µm. Similar results for the 150 µm incident beam diameter are shown for the truncation radii of (c) 250 µm and (d) 500 µm.
In our previous work [25], we have shown the superiority of the Bessel beam to the Gaussian beam where we have assumed that the Bessel beam is nearly ideal, i.e., it spatially extends to 2r max = 120 w FWHM for the Bessel beam, However, the truncated Bessel beam is more realistic. In this section we have compared the results by the truncated Bessel beam with those by the Gaussian beam to find that we can still see the advantage of the truncated Bessel beam over the Gaussian beam as long as the truncation radius is not very small. To confirm the above statement, we recall the results in our previous work [25] for the Bessel beam without a truncation and Gaussian beam where we have employed the identical beam diameter, peak intensity, and the Ar gas pressure: For the incident beam diameters of 100 µm and 150 µm, the lengths of the filaments were 15 cm and 20 cm, respectively, for the Gaussian beam, while those for the ideal Bessel beam were more than 50 cm. What we find from Figs. 11 and 12 is that the length of the filament by the truncated Bessel beam is already a bit longer than that by the Gaussian beam if the truncation radius is 500 µm. They are about 18 cm and 28 cm for the 100 µm and 150 µm diameter beams. Similar argument holds for the plasma channel. These findings confirm the superiority of the Bessel beam to the Gaussian beam under real experimental conditions. If the Bessel beam is truncated at a larger beam radius, the truncated Bessel beam shows even better performance. In other words, if the beam truncation is done at a small radius, there is not sufficient energy reservoir in the outer part of the beam to provide energy to the inner part to sustain a long filament.
5. Combination of two Gaussian beams
Based on what we have learned in Sec. 3 and 4 for the Bessel incident beam, we ask if there is any way to improve the propagation properties for the Gaussian incident mode. We come to an idea of combining two Gaussian beams with different beam diameters.
5.1 Concept of combining two Gaussian beams
First, we clarify the idea of combining two Gaussian beams. A Gaussian beam has a well known form:
where the FWHM of the beam diameter, w FWHM, equals to . Now we introduce two Gaussian beams with the same temporal shape, amplitude, but different beam diameters, and, in which the temporal part has been dropped off for simplicity and the beam diameter of the latter is four times larger than the former. The idea of combining two Gaussian beams is to increase the energy stored in the outer part of the beam through the introduction of the second broader Gaussian beam. This way the filamentation induced by the first narrower Gaussian beam could persist for a longer distance. Specifically we consider the following four cases: [case A] ε g = ε g1, [case B] ε g = ε g2, [case C] ε g = 0.9ε g1 + 0.1ε g2 and [case D] ε g = 0.8ε g1 + 0.2ε g2. Note that all four cases have the same peak intensity at r = 0, but different total (spatially integrated) pulse energies. If they are normalized by the total energy for case A, they are 1, 16, 1.309, and 1.883, respectively. Figure 15 shows the spatial profiles for case A, C, and D. From Fig. 15, we can see that combining the two Gaussian beams with narrower and broader beam diameters results in practically the very similar (but not exactly identical) beam diameters around r = 0 but different spatial distributions at the tail (dashed circle in Fig. 15), which would serve as an energy reservoir.5.2 Superioty of the combined Gaussian beams as an energy reservoir
Now we focus on the filament and plasma channel by the combined Gaussian beams described above. To make meaningful comparisons, we consider the following eight cases, A-H:
A: single beam with a 100 µm beam diameter and 3.2 × 1013 W/cm2 peak intensity.
B: single beam with a 400 µm beam diameter and 3.2 × 1013 W/cm2 peak intensity.
C: combined (A and B) beams with the field amplitude ratio of 0.9A + 0.1B.
D: combined (A and B) beams with the field amplitude ratio of 0.8A + 0.2B.
E: single beam with the same total energy and peak intensity with C (accordingly the beam diameter is a little larger than 100 µm).
F: single beam with the same total energy and peak intensity with D (accordingly the beam diameter is a little larger than that for case E).
G: single beam with a 100 µm beam diameter and the same total energy with C (accordingly the peak intensity is larger than that for case C).
H: single beam with a 100 µm beam diameter and the same total energy with D (accordingly the peak intensity is larger than that for case D).
The total energy, beam diameter, and peak intensity for the above eight cases are summarized in Table 1 .
Table 1. Comparison of Total energy, Beam Diameter, and Peak Intensity for Cases A–H Where Values are Normalized with Respect to Those for Case A
Now we perform the propagation calculations for case A-H with the identical pulse duration of 30 fs. For the systematic understanding, we make comparisons by grouping the numerical results for case A-H into three: Group 1 includes case A, B, C, and D, Group 2 includes case C, D, E, and F, Group 3 includes case C, D, G, and H. In Figs. 16(a) , 16(b), and 16(c) we compare the variation of beam diameter as a function of propagation distance for Group 1-3, respectively. Similarly in Figs. 17(a) , 17(b), and 17(c) we compare the variation of ionization probability at r = 0 as a function of propagation distance for Group 1-3, respectively.
Fig. 16 Variation of beam diameter as a function of propagation distance for (a) Group 1 (case A, B, C, and D), (b) Group 2 (case C, D, E, and F), and (c) Group 3 (case C, D, G, and H).
Fig. 17 Variation of ionization probability at r = 0 as a function of propagation distance for (a) Group 1 (case A, B, C, and D), (b) Group 2 (case C, D, E, and F), and (c) Group 3 (case C, D, G, and H).
From the comparison in Group 1 (see Figs. 16(a) and 17 (a)), we can see that, when the broadening of the outer part of the incident beam is small (case C), the improvement in terms of the filament length seems negligible. However, if more energy is stored in the outer part (case D), we can see the clear improvement in both filament and plasma channel. Because the pulse for case B has an order of magnitude larger total energy than the other cases (see Table 1), it has the longest lengths for the filament and plasma channel. The comparison in Group 2 (see Figs. 16(b) and 17 (b)) further verifies the importance of the energy reservoir in the outer part of the beam. From comparison between C and E, and D and F, the importance of the reservoir for the persistence of filament and plasma channel is clear: from Figs. 16(b) and 17(b), we can see that the filament and plasma channel produced by the combined Gaussian beams which have a larger reservoir is much longer than that by the single Gaussian beam with the same total energy and peak intensity but a larger diameter. The comparison in Group 3 (see Figs. 16(c) and 17(c)) confirms the effect of the energy reservoir from another respect: If the total energy is chosen to be the same, we can obtain longer filament and plasma channel by the beam with a larger beam diameter and accordingly larger energy reservoir (case D) compared with the beam at a higher peak intensity (case H).
As a last remark in this section, we would like to note that the results for case D are quite remarkable. We have shown that combining two Gaussian beams (case D) results in better propagation properties compared with the cases of a single Gaussian beam with the same total energy (case F and H).
6. Validity for longer pulses
Generally speaking, for pulses as short as 30 fs, higher order nonlinear effects such as self-steepening and space-time focusing should be taken into account [32,33]. In this paper, however, we have employed 30 fs for the incident pulse duration so that we can make a detailed study under the same conditions as in our previous work [25]. In fact, we have numerically ensured that the results for the 30 fs pulses calculated without the higher order nonlinear terms are very similar to those for the 50 fs or longer pulses where the higher order nonlinear terms play very minor roles. This means that most of the features we have found in this paper for the 30 fs pulses without taken into account the higher order nonlinear effects can be found for longer pulses where such effects do not play any important roles, anyway. As an example, we now make a comparison between the Bessel and Gaussian incident beams with a 50 fs duration, 100 µm beam diameter, and 3.2 × 1013 W/cm2 peak intensity propagating in Ar gas. The results for the beam diameter and ionization probability at r = 0 are shown in Fig. 18 . By comparing Figs. 18(a) and 18(b) with Figs. 4 and 10 in our previous work [25], respectively, we can say that the Bessel incident beam provides a much longer filament and plasma channel than the Gaussian beam for the 50 fs pulses as well. Having done these comparisons, we can surely say that very similar results should be obtained for longer pulses.
Fig. 18 Variation of the (a) beam diameter and (b) ionization probability at r = 0 for the 50 fs Bessel and Gaussian beam as a function of propagation distance.
7. Conclusions
In this paper, we have carried out the detailed study on the performance of the Bessel, truncated Bessel, and the combination of two Gaussian incident beams by solving the generalized nonlinear Schrödinger equation coupled with the electron density evolution equation. For the Bessel beam, comparisons of the temporal and spectral profiles at the different segments (lobes) of the beam have led us to the conclusion that many things are happening at the inner part of the beam. Moreover, the results on the field current at the different transverse segments of the Bessel beam indicate that the outer part of the Bessel beam serves as an energy reservoir so that the filament formed at the inner part persists for a long propagation distance. For the truncated Bessel beam, the lengths of the filament and plasma channel strongly depends on the incident beam diameter and the truncation radius, and they are naturally shorter than those by the ideal (untruncated) Bessel beam. Nevertheless, the lengths of the filament and plasma channel by the truncated Bessel beam can be still better than those by the Gaussian beam if the truncation radius is not very small.
Perhaps more interestingly, we have shown that the combined use of two Gaussian beams with different beam diameters can remarkably improve the quality of the filament and plasma channel, since more energy is stored in the outer part of the combined Gaussian beams. This can be a practical choice to improve the quality of the filament. Although we have employed 30 fs pulses for the calculation without taking into account the higher order nonlinear effects so that the direct comparison can be made with our previous work [25], we have numerically ensured that very similar results can be obtained for longer pulses where such effects do not play any important role. If someone intends to compare their results with ours for the 30 fs pulses this point has to be kept in mind.
Acknowledgments
This work was supported by a Grant-in-Aid for Scientific Research from the Ministry of Education and Science of Japan.
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