Open Access
Add to BibSonomyAbstract
Using computer simulations we explore the dynamics of nonlinear pulsed Bessel-like beams arising from axicon-focused Gaussian beams and circularly apodized beams propagating in air. These pulses exhibit similar self-action. We also note that noninear behavior can occur for these pulses in the absence of significant plasma density. This is especially interesting in light of recent experimental observations of self-guiding without plasma.
©2007 Optical Society of America
1. Introduction
Since the discovery of long distance self-guiding of optical pulses, a variety of models have been proposed to explain this phenomenon [1, 2, 3, 4]. Most of these models describe beams with bell-shaped initial profiles containing at least one critical power (Pcr) [5] undergoing self-induced collapse. Ultimately the collapse is arrested by a combination of diffraction, group velocity dispersion (GVD), plasma induced defocusing and/or absorption [6, 7, 8]. One such theory that has received general acceptance is that of dynamic spatial replenishment (DSR) [9]. In DSR, a combination of plasma- induced defocusing and GVD cause the pulse to split and reform temporally multiple times during its propagation. However, despite this very dynamic behavior, the average fluence of the pulse remains relatively constant giving the appearance of a single, continuous filament. So far, DSR has been very successful in describing the behavior of bell-shaped, super- critical pulses.
Another topic of vigorous study in the field of long distance propagation is that of “diffractionless” beams. Bessel beams first received wide attention when Durnin proposed them as nondiffracting solutions to the linear wave Eq. [10]. It was quickly recognized that Bessel beams contain infinite power and are therefore unphysical. However, envelope Bessel beams contain finite power and still exhibit quasi-diffractionless properties. Of these envelope Bessel beams, by far the most common is a Bessel beam with a Gaussian spatial envelope. The linear properties of Bessel-Gauss beams have been studied quite extensively [11, 12]. Among the most interesting of these linear properties is the quasi-diffractionless propagation of the small central spot of the beam and “self-healing” that is exhibited by the beam when it encounters obstructions [13]. This is reminiscent of the replenishment that occurs in the DSR model. While in the case of DSR, the replenishment occurs from energy that travels throughout the space-time envelope of the pulse due to a combination of plasma induced defocusing, Kerr nonlinearity and GVD, in linear Bessel beams, the replenishment is a result of the conical nature of the wavefront and is strictly a linear effect. In the nonlinear regime, Bessel-like beams, having an intense central core surrounded by much less intense rings, exhibit unique self-action. The intensity of the central core oscillates as the beam propagates in the nonlinear medium. In the last few years, a number of authors have studied the dynamics of these beams. The first explaination of the intensity oscillations of nonlinear Bessel beams was probably provided by Gadonas et. al [14]. They used phase matching arguments to argue that nonlinear action modifies the angular spectrum of the Bessel beam to produce two additional angular components, an on-axis component and a conical wave with a transverse wavenumber that is a factor of √2 times the original beam’s conical component. They further argue that the intensity oscillations result from interference between the original conical wave and the on-axis component. These results match well with experimentally obtained far-field angular spectra [14, 15]. Johannisson et al. performed a theoretical study of nonlinear Bessel and Bessel-Gauss beams [16]. This study included an analysis of the global behavior of the Bessel beams in terms of virial theory including a discussion of global and partial collapse within the beam. Also, Johannisson observed the characteristic intensity modulation in the numerical simulations. The affect of nonlinear losses on the dynamics of Bessel-like beams was addressed theoretically by Porras et al. [17] and experimentally by Polesana et al. [18, 19]. These treatments attributed the dynamic modulation of the Bessel beam’s intensity to the interaction between unbalanced incoming and outgoing Hankel beams. The unbalance arose because of nonlinear loss in the intense core of the beam. This theory was applied successfully to experiments in relatively dense materials (fused silica and Coumarine 120 in methanol) where nonlinear losses were significant and plasma densities remained small.
Recent experimental and numerical studies of nonlinear flat-top beams have yielded interesting results. Ruiz et al. recently reported on an experiment in which a circularly apodized, focused beam exhibited self-guiding over many Rayleigh lengths without generating significant amounts of plasma [20]. In this case, self-guiding was attributed to the beam acquiring a Townes-like profile during its propagation. Numerical simulations were also reported recently, that demonstrated that nonlinear, super-Gaussian beams carrying less than approximatly ten critical powers evolve towards plasmaless, self-guided profiles. Again, these profiles were described as Townes-like [21].
In this paper, we report on our numerical simulations of nonlinear pulsed Bessel-like beams in air. In all of our simulations the pulsed beams which we inject into the nonlinear medium do not initially have Bessel-like profiles, being axicon-focused, pulsed, Gaussian beams or focused, pulsed, super-Gaussian beams. The pulsed beams evolve Bessel-like character as they propagate. In fact, they would evolve into Bessel-like beams in the absence of the nonlinearity. The unique nonlinear self-action, which is characteristic of Bessel-like beams, is observed after power has been linearly partitioned within the pulsed beam. In this respect, to the best of our knowledge, our simulations differ from all of the work that has been conducted previously in which pulsed Bessel-like beams were directly injected into the nonlinear medium [14, 15, 16, 17, 18, 19]. Typically, nonlinear pulses are regularized by a combination of multiphoton absorption and plasma induced defocusing (PID). However, recent reports of plasmaless self-guiding [20] suggest that beams can be created that have distinctly nonlinear behavior without creating plasma. In order to accuratly judge the importance of the plasma density to Bessel-like pulse propagation in air, our simulations also include the effects of nonlinear losses and plasma induced defocusing, in addition to GVD. Other treatments have neglected PID and GVD. Also, our simulations demonstrate the quasi-linear development and subsequent nonlinear dynamics of pulsed Bessel-like beams from injection to termination over relatively long interaction distances compared with previous work.
In section 2 we discuss the model for nonlinear propagation in air. In sections 3 and 4 we present the results of our simulations of axicon- focused, pulsed Gaussian beams and circularly apodized, pulsed beams, respectively. We also argue in these sections that the two cases both result in Bessel-like profiles that exhibit nonlinear self-action without generating significant plasma density. We argue that this is a general feature of pulsed Bessel-like beams which carry more than one critical power but less than the power necessary to cause the central core to collapse. The total power necessary to cause the core collapse depends on the details of the Bessel-like profile, but it will be close to the total power that localizes one critical power or more into the core. This agrees well with previous work on linear power partitioning [22, 23, 24]. In section 5 we draw further comparisons between the results of sections 3 and 4 and the work in Ref. [14]. We argue that the Townes-like profiles reported in Refs. [20] and [21] are the end result of a cascaded self-action process of nonlinear Bessel-like beams that result from circularly apodized beams. Finally, section 6 conatains some concluding remarks about nonlinear Bessel-like beams.
2. Model
In this paper we are concerned with the paraxial, scalar wave Eq. within the slowly varying amplitude approximation [25]. In this approximation we assume that the electric field has the form
where kz is the magnitude of the longitudinal component of the wavevector, ω is the angular frequency of the field, and A(r, z) is a slowly varying, scalar amplitude such that ∂2|A|/∂z 2≪k 2 z|A|. In this case, A is scaled so that |A|2 has units of Wm-2. Under these assumptions, the extended nonlinear Schrödinger Eq. (NLS) reads
∇2 T is the radial Laplacian, . The first term on the right hand side of Eq. (2) is spatial diffraction, the second term is GVD, the third term causes plasma induced defocusing, the fourth term is the Kerr nonlinearity and the fifth term is multiphoton absorption. The plasma rate Eq. associated with Eq. (2) is
The first term on the right hand side of Eq. (3) is multiphoton ionization and the second term is electron-ion recombination. Unless stated otherwise, the parameters and varibles in Eqs. (2)–(3) are:
• k (wavenumber, nb ω/c): 7.854×106 m-1
• nb≈1.0 (linear index of refraction)
• ω (angular frequency of light)
• k” (coefficient of GVD): 0.3 fs2 cm-1
• σ (coefficient of inverse bremsstrahlung): 5.5×10-24 m2
• τ (mean time between electron-neutral collisions): 3.5×10-13 s
• ρ (plasma density)
• n 2 (nonlinear index of refraction)
• κ (multiphoton order): 7
• B (κ) (multiphoton ionization coefficient): 2.7×10-88 s-1m16W-8
• α (recombination coefficient): 5.0×10-13 m3s-1
Eq. (2) was solved with a split-step pseudo-spectral code [26].
Fig. 1. Maximum normalized intensity as a function of normalized propagation distance for axicon-focused Gaussian beams in linear (solid) and nonlinear (dashed) media. The propagation distance is normalized by the focal length of the axicon (zax=w 0 k/kr).
3. Simulations of axicon-focused, pulsed Gaussian beams
The initial condition for the axicon-focused, pulsed Gaussian beam reads
where w 0 is the 1/e radius of the initial Gaussian profile, tp is the pulse duration and kr is the radial wavenumber of the resulting Bessle-Gauss beam. The parameter kr is related to the angle (γ) of the axicon by
where n is the index of refraction of the axicon material and we are assuming that the axicon is in air. For the simulations presented here kr=3.4×104 m -1, w 0=1.0 mm and tp=120 fs. Also, n 2=5.57×10-23 W -1 m 2 resulting in Pcr≈1.7 GW. It can be shown [11, 14, 15] that after propagating a sufficient distance (>w 0 k/kr) the field amplitude, A, from Eq. 4 becomes
where J 0 is the zero order Bessel function of the first kind. Fig. 1 highlights the unique self-action of the nonlinear, pulsed, axicon-focused Gaussian beam. It shows the maximum normalized intensity for the pulses propagating in linear and nonlinear media. As mentioned in section 1, the intensity oscillations result from interference between on-axis and off-axis components of the angular spectrum. The on-axis component is the result of nonlinear self-action [14]. During each period, energy oscillates between adjacent rings in the resulting Bessel-Gauss beam. For reference, Fig. 2 shows the power oscillation in the rings of a continuous wave (CW), nonlinear, axicon-focused Gaussian beam. The intensity shown in the dashed curve in Fig. 1 corresponds to a 0.86 mJ, 120 fs pulse with an initial peak power of about 5.8 GW. Consequently, as the pulse propagates the beam power contained in the central core can be as high as Ppk=900 MW according to Fig. 2.
Fig. 2. Fraction of power contained in the central spot (solid) and the first ring (dashed) of a CW nonlinear axicon-focused Gaussian beam as a function of normalized propagation distance (z/zax,zax=w 0 k/kr).
An interesting property of the nonlinear beam shown in Fig. 1 is that the regularization of the self-focusing central core arises without significant plasma density. This is evident upon examination of Fig. 3. In this Fig., the integrated plasma generated by the axicon-focused, pulsed Gaussian beams of 0.86 mJ and 1.1 mJ are compared with that generated by an 85 fs, 1.0 mJ nonfocused, pulsed Gaussian beam. Here, the axicon focal length is 11 cm with the resulting Bessel-Gauss beams having a central core diameter of about 140 µm. The nonfocused, pulsed Gaussian had a 1/e radius of 0.7 mm and a Rayleigh range of 2.0 m. For comparison, a gaussian with a 1/e radius of 70 µm, similar to that of the Bessel-Gauss cores here, would have a Rayleigh range of only 5 mm. In the nonfocused, pulsed Gaussian beam, dynamic spatial replenishment is at work and plasma and GVD regularize the pulse. The higher energy pulses generate significantly more plasma than the 0.86 mJ pulse. Also, the GVD coefficient, k″, is about ten times larger in the case of the nonfocused pulsed Gaussian (2.0 fs 2 cm -1 as opposed to 0.3 fs 2 cm -1 for the axicon-focused Gaussians). This has the effect of decreasing the amount of integrated plasma generated by the pulse. So with weaker GVD, the nonfocused pulsed Gaussian would have generated even more plasma than it did in this case.
That the 0.86 mJ is in fact regularized can be seen by comparing the maximum intensity of the pulse with that of an equivalently intense CW beam. This comparision is made in Fig. 4. In this Fig., the solid curve shows the maximum intensity of a nonlinear CW axicon-focused Gaussian beam and the dashed curve shows the maximum intensity for a pulsed axicon-focused Gaussian beam(0.86mJ). The CW model does not include plasma effects or GVD, however, the two curves are quite similar. The similarity between these curves suggests that the regularizing mechanism is spatial diffraction and not plasma or GVD. This is in contrast to the work done by Polesana et al. in which it was observed that, in dense nonlinear media, multi-photon absorption regularized the nonlinear Bessel beam. In our air simulations, plasma and nonlinear losses play minimal roles. However, for more energetic pulsed Bessel beams, plasma can play a more significant role in air. The maximum intensities vs. propagation distance for a 1.1mJ pulsed axicon-focused Gaussian beam and its associated CW beam are shown in Fig. 5. This plot corresponds with the solid line in integrated plasma plot in Fig. 3. Here, the plasma arrests the self-focusing and clamps the maximum intensity below 5×10 17 Wm -2.
Fig. 3. Integrated plasma density from a 1.1 mJ axicon-focused, pulsed Gaussian beam (solid), a 0.86 mJ axicon-focused, pulsed Gaussian beam (dashed) and a 1.0 mJ non-focused, pulsed Gaussian beam (dot-dashed). The propagation distance is normalized by the axicon focal length, zax=w 0 k/kr, for each pulsed axicon-focused Gaussian beam and by the Rayleigh range for the nonfocused pulsed Gaussian beam. Here the nonfocused Gaussian has a Rayleigh range of 2.0 m and zax=22 cm.
Fig. 4. Maximum intensity as a function of propagation distance for nonlinear CW (solid, without plasma) and pulsed (dashed, with plasma) axicon-focused Gaussian beams. The pulse is a 0.86 mJ pulse with a duration of 120 fs. The peak power gets as high as Ppk=900 MW (Ppk/Pcr≈0.5). The propagation distance is normalized by the axicon focal length, zax=w 0 k/kr. Here, zax=22cm.
Fig. 5. Maximum intensity as a function of propagation distance for nonlinear CW (solid, without plasma) and pulsed (dashed, with plasma) axicon-focused Gaussian beams. The pulse is a 1.1mJ pulse with a duration of 120 fs. The peak power of the initial pulse is 7.4GW and the unclamped peak power contained in the central core could get as high as Ppk=1.1GW(Ppk/Pcr≈0.6). The propagation distance is normalized by the axicon focal length, zax=w 0 k/kr. Here, zax=22cm.
Finally, Fig. 6 shows the space-time field distributions for the 0.86 mJ pulse at several locations along its propagation path. The nonlinear self-action is clearly visible in the intense portion of the pulse at t=0. The nonlinear interaction causes what appears to be temporal splitting seen in (b) and (d) of Fig. 6. The splitting is the result of the intense central portion of the beam undergoing nonlinear self-action while the lower intensity leading and trailing edges propagate linearly. This is distinct from true splitting that occurs for nonlinear pulses in more strongly dispersive media [7].
In this section we have demonstrated that, for a range of input energies, high intensity, axicon-focused, pulsed Gaussian beams can undergo nonlinear self-action without significant plasma generation. In this regime the pulse is regularized by spatial diffraction alone and the nonlinear interaction is characterized by intensity oscillations of the central core. In the next section we demonstrate that circularly apodized beam exhibit qualitatively similar behavior.
Fig. 6. Field profile at several locations along the propagation axis for the 0.86mJ pulsed axicon-focused Gaussian beam. (a) z/zax=0.45, (b) z/zax=0.62, (c) z/zax=0.74, (d) z/zax=0.87, (e) z/zax=1.4, (f) z/zax=1.8.
4. Simulation of circularly apodized pulsed Gaussian beams
In this section we describe simulations of a focused pulsed super-Gaussian beam obtained from the initial condition
where w 0=2.5 mm is the radius of the super-Gaussian beam (circular aperture), tp=120 fs is the pulse duration, f=2.2 m is the focal length of the lens and m=8. Also, n 2=3.2×10-23 W -1 m 2 resulting in Pcr≈2.9 GW. These parameters were selected to approximate those in the experiment conducted by Ruiz et al. [20]. In our simulations here, the pulse is 0.56mJ with an initial peak power of 3.5 GW. Also GVD is neglected in this simulation (k″=0.0 fs 2 cm -1). Under nominally linear conditions, the field at the focal point of the lens would form an Airy pattern,
where f is the focal length of the lens, w 0 is the radius of the aperture, k is the wavenumber and J 1 is the first order Bessel function of the first kind. Fig. 7 shows the integrated plasma as a function of normalized propagation distance. We see that the integrated plasma is very low (compared to the nonfocused, pulsed Gaussian in Fig. 3), supporting the argument that plasma is not involvded in the regularization of the pulse. Fig. 8 shows the maximum intensity for the super-Gaussian pulse and a CW super-Gaussian beam of equal peak intensity. This is similar to Figs. 4 and 5 of section 3. As with the 0.86mJ axicon-focused, pulsed Gaussian beam, the peak intensity of the super-Gaussian pulse is very similar to that of the CW beam throughout the propagation, suggesting that the regularization is due to spatial diffraction and not plasma or GVD. The key indication of nonlinear action in Fig. 8 is the resurgence of the central core. This is indicated by the arrow in the Fig. Ruiz et al. noted a similar resurgence event in their experiment [20]. They attributed this to the formation of a Townes-like profile. We will argue in the next section that this Townes-like profile is the result of cascaded self-action of a nonlinear Bessel-like profile, given by Eq. (8), which evolves from the focused super-Gaussian beam. Finally, Fig. 9 shows the space-time field distribution of the pulsed super-Gaussian beam. Here too we observe the apparent temporal splitting that is characteristic of that induced by self-action in pulsed Bessel-like beams. Again, this is distinct from splitting that is induced by GVD.
The Bessel-like behavior of the nonlinear pulsed super-Gaussian beam makes sense when the influence of linear power partitioning is taken into account [24]. The pulse propagates linearly up until just before the focal point of the lens. At this point the power has been partitioned into an Airy pattern and the Kerr nonlinearity becomes significant. The Airy pattern is Bessel-like and is in many ways similar to the patterns that were generated by the axicon-focused Gaussian beams of section 3. Therefore propagation dynamics of the nonlinear pulsed Airy beam are similar to those of the nonlinear axicon-focused Gaussian beams. In this case, the central core of the Airy pattern contains about 86% of the input power, Ppk=3.0GW. In this case Ppk≈Pcr and if the initial condition contained more energy the central core would collapse around the focal point of the lens (i.e. the integrated plasma density would increase significantly resulting in plasma regularization of the pulse). Once the central spot has become sufficiently intense, nonlinear effects become significant and the pulse begins to undergo self-action. Therefore, the circularly apodized pulse exhibits similar behavior (oscillations and plasmaless self-guiding) to that of the axicon-focused pulsed Gaussian beam for the same reason; they both have far field distributions that are Bessel-like in character.
Fig. 7. Integrated plasma density for the 0.56mJ super-Gaussian pulse. The propagation distance is normalized by the focal length of the lens 2.2m.
Fig. 8. Maximum intensity as a function of normalized propagation distance for a CW super-Gaussian beam (solid, without plasma) and a 0.56mJ, 120 fs super-Gaussian pulse (dashed, with plasma). The propagation distance is normalized by the focal length of the lens 2.2m. The peak power of the pulse was Ppk≈3.0GW(Ppk/Pcr≈1)
Fig. 9. Field profile at several locations along the propagation axis for the 0.56mJ focused super-Gaussian pulse. (a) z/f=1.0, (b) z/f=1.1, (c) z/f=1.2, (d) z/f=1.4.
5. Discussion
In sections 3 and 4 we demonstrated two very different initial condition that evolve into Bessellike profiles and exhibit qualitatively similar nonlinear behavior. This section will compare the angular spectra of the two examples described earlier with the results in Ref. [14]. Fig. 10 shows the integrated angular spectra at z=0 m and after nonlinear propagation for the axicon-focused, pulsed Gaussian beam. The primary angular component, kr, is clearly visible in the initial spectrum. This component is also visible in the nonlinear modified spectrum, however, there is also a larger component at kr 0=0 m -1 (on-axis) and at kri≈√2kr (idler). The on-axis component is generated by the self-action of the primary angular component. The idler is generated due to parametric amplification of the on-axis component with the primary angular component as the pump [14]. The oscillation of the core intensity can be understood to first order as the interference between the on-axis component and the primary angular component. In this case, the interference would result in an oscillation period of L≈4πk/k 2 r or L=8.5 cm. This is longer by about a factor of 1.5 than the actual period in Fig. 4 (about 5.5 cm). This discrepancy is most likely the result of the fact that the estimation for the period used by Gadonas et al. [14] does not account for nonlinearity. The evolution of the angular spectrum of the focused, pulsed super-Gaussian beam is shown in Fig. 11. The initial portion of the propagation is linear. This is consistant with the concept of linear power partitioning which we have discussed in a previous paper [24]. The initial spectrum has a primary angular component kr≈8×103 m -1 and a relativly large on-axis component compared to the axiconfocused Gaussian. As the intensity of the beam increases, self-action produces the idler from parametric amplification (kri=√(2)kr≈11.3×104 m -1) of the on-axis component by the primary. This is manifested in Fig. 11 as broadening and shifting (dotted and dashed curves) of the primary component. Meanwhile, the intensity continues to increase. In the final spectrum, the new primary component, kr≈11.3×104 m -1, undergoes self action in a similar way to the axicon-focused Gaussian beam. This produces more on-axis radiation. The resurgence in Fig. 8 can be understood as the result of interference between the on-axis component and the other angular components in the spectrum.
6. Conclusion
In this article we have explored, with numerical simulation, nonlinear behavior of two very different initial conditions that evolve into Bessel-like beams, pulsed axicon-focused Gaussian beams and focused circularly apodized pulsed beams. In these beams linear power partitioning creates Bessel-like profiles. The angular spectrum of these resulting conical waves is modified by the self-action of the nonlinear beam. This produces an on-axis angular component which interferes with the original conical wave to produce the characteristic intensity oscillations of nonlinear Bessel beams. An idler is also produced at √2 times the transverse wavenumber of the original conical wave. The focused super-Gaussian beam differs from the axicon-focused Gaussian beam in that there is a second step in the evolution of the angular spectrum. The idler actually begins to dominate the spectrum, changing the primary wavenumber of the conical wave. This new conical wave then undergoes self-action and produces intensity oscillations due to the interference of the conical wave and the other components of the spectrum. To the best of our knowledge, this cascaded self-action of the nonlinear super-Gaussian beam has never been reported. We have also demonstrated that both the axicon-focused Gaussian beams and the focused super-Gaussian beams result in beams that propagate nonlinearly in qualitativly similar ways without significant plasma generation. In nonlinear pulses, plasma generation and its concomitant effects are complex and violent processes. Consequently they limit the degree to which pulse propagation can be controlled. Therefore, the potential for self-guiding with minimal plasma is interesting to those studying precision nonlinear beam control. Nonlinear, pulsed Bessel-like beams may provide a way, for a small range of input powers, to balance the nonlinear collapse that leads to self-guiding without the added high order nonlinearity associated with plasma. This is especially relevant to remote sensing applications. Because, circular apertures are common in optical system, it is not unreasonable for Bessel-like self-action to be common in nonlinear optical systems.
Fig. 10. Integrated angular spectrum for a nonlinear axicon-focused, pulsed Gaussian beam. The solid line is the initial spectrum and the dashed line is the spectrum after nonlinear propagation. On the abscissa, unity corresponds to a transverse wavenumber of 3.4×104 m -1.
Fig. 11. The evolution of the integrated spectrum of a focused, pulsed super-Gaussian beam in a nonlinear medium. Solid: z=0 m, Dotted: z=1.5 m, Dashed: z=2.1 m, Dotted-dashed: z=2.2 m. On the abscissa, unity corresponds to a transverse wavenumber of 8.0×103 m -1.
Acknowledgments
D.E. Roskey would like to thank the Directed Energy Professional Society for a graduate scholarship that supported his work.
This work was also supported by the Air Force Office for Scientific Research under grant FA9550-07-1-0010. Ewan Wright is supported in part by the Joint Services Optical Program (JSOP).
References and links
1. A. Braun, G. Korn, X. Liu, D. Du, J. Squier, and G. Mourou, “Self-channeling of high-peak-power femtosecond laser pulses in air,” Opt. Lett. 20, 73–75 (1995). [CrossRef] [PubMed]
2. E. Nibbering, P. Curley, G. Grillon, P. B.S., M. Franco, F. Salin, and A. Mysyrowicz, “Conical emission from self-guided femtosecond pulses in air,” Opt. Lett. 21, 62–64 (1996). [CrossRef] [PubMed]
3. A. Brodeur, C. Chien, F. Ilkov, S. Chin, O. Kosareva, and V. Kandidov, “Moving Focus in the propagation of ultrashort laser pulses in air,” Opt. Lett. 22, 304–306 (1997). [CrossRef] [PubMed]
4. H. Lange, G. Grillon, J.-F. Ripoche, M. Franco, B. Lamouroux, B. Prade, A. Mysyrowicz, E. Nibbering, and A. Chiron, “Anomalous long-range propagation of femtosecond laser pulses through air: moving focus or pulse self-guiding?” Opt. Lett. 23, 120–122 (1998). [CrossRef]
5. J. Marburger and Self-Focusing: Theory,” Prog. Quant. Electr. 4, 35–110 (1975). [CrossRef]
6. P. Monot, T. Auguste, L. Lompré, G. Mainfray, and C. Manus, “Focusing limits of a terawatt laser in an underdense plasma,” J. Opt. Soc. Am. B 9, 1579 (1992). [CrossRef]
7. P. Chernev and V. Petrov, “Self-focusing of light pulses in the presence of normal group-velocity dispersion,” Opt. Lett. 17, 172–174 (1992). [CrossRef] [PubMed]
8. J. Ranka, R. Schirmer, and A. Gaeta, “Observation of pulse splitting in nonlinear dispersive media,” Phys. Rev. Lett. 77, 3783–3786 (1996). [CrossRef] [PubMed]
9. M. Mlejnek, E. Wright, and J. Moloney, “Dynamic spatial replenishment of femtosecond pulses propagating in air,” Opt. Lett. 23, 382–384 (1998). [CrossRef]
10. J. Durnin, J. J. Miceli, and J. Eberly, “Diffraction-Free Beams,” Phys. Rev. Lett 58, 1499–1501 (1987). [CrossRef] [PubMed]
11. J. Arlt and K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. 177, 297–301 (2000). [CrossRef]
12. K. Dholakia and D. McGloin, “Bessel beams: Diffraction in a new light,” Contemporary Physics 46, 15–28 (2005). [CrossRef]
13. Z. Bouchal, J. Wagner, and M. Chlup, “Self-reconstruction of a distorted nondiffracting beam,” Opt. Commun. 151, 207–211 (1998). [CrossRef]
14. R. Gadonas, V. Jarutis, R. Paskauskas, V. Smilgevivcius, A. Stabinis, and V. Vaicaitis, “Self-action of Bessel beam in nonlinear medium,” Opt. Commun. 196, 309–316 (2001). [CrossRef]
15. V. Pyragaite, K. Regelskis, V. Smilgevicius, and A. Stabinis, “Self-action of Bessel light beams in medium with large nonlinearity,” Opt. Commun. 257, 139–145 (2005). [CrossRef]
16. P. Johannisson, D. Anderson, M. Lisak, and M. Marklund, “Nonlinear Bessel beams,” Opt. Commun. 222, 107–115 (2003). [CrossRef]
17. M. Porras, A. Parola, D. Faccio, A. Dubietis, and P. Di Trapani, “Nonlinear Unbalanced Bessel Beams: Stationary Conical Waves Supported by Nonlinear Losses,” Phys. Rev. Lett. 93(153902) (2004). [CrossRef] [PubMed]
18. P. Polesana, A. Dubietis, M. Porras, E. Kucinskas, D. Faccio, A. Couairon, and P. Di Trapani, “Near-field dynamics of ultrashort pulsed Bessel beams in media with Kerr nonlinearity,” Phys. Rev. E 73(056612) (2006). [CrossRef]
19. P. Polesana, P. Faccio, D. Di Trapani, A. Dubietis, A. Piskarskas, A. Couairon, and M. Porras, “High localization, focal depth and contrast by means of nonlinear Bessel Beams,” Opt. Express 13, 6160–6167 (2005). [CrossRef] [PubMed]
20. C. Ruiz, J. San Román, C. Méndez, V. Diáz, L. Plaja, I. Arias, and L. Roso, “Observation of Spontaneous Self-Channeling of Light in Air below the Collapse Threshold,” Phys. Rev. Lett. 95(053905) (2005). [CrossRef] [PubMed]
21. T. Grow, A. Ishaaya, L. Vuong, A. Gaeta, N. Gavish, and G. Fibich, “Collapse dynamics of super-Gaussian beams,” Opt. Express 14, 5468–5475 (2006). [CrossRef] [PubMed]
22. G. Mechain, A. Couairon, M. Franco, B. Prode, and A. Mysyrowicz, “Organizing multiple femtosecond filaments in air,” Phys. Rev. Lett. 93(035003) (2004). [CrossRef] [PubMed]
23. K. Cook, A. Kar, and R. Lamb, “White-light filaments induced by diffraction effects,” Optics Express 13, 2025–2031 (2005). [CrossRef] [PubMed]
24. D. Roskey, M. Kolesik, J. Moloney, and E. Wright, “The role of linear power partitioning in filamentation,” Appl. Phys. B 86, 249–258 (2007). [CrossRef]
25. R. Boyd, Nonlinear Optics (Academic Press,Inc., 1991).
26. P. DeVries, A First Course in Computational Physics (John Wiley and Sons, Inc., 1994).
