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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 19 — Sep. 14, 2009
  • pp: 16429–16435
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Self-pinching of pulsed laser beams during filamentary propagation

Carsten Brée, Ayhan Demircan, Stefan Skupin, Luc Bergé, and Günter Steinmeyer  »View Author Affiliations


Optics Express, Vol. 17, Issue 19, pp. 16429-16435 (2009)
http://dx.doi.org/10.1364/OE.17.016429


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Abstract

Competing nonlinear optical effects that act on femtosecond laser pulses propagating in a self-generated light filament may give rise to a pronounced radial beam deformation, similar to the z-pinch contraction of pulsed high-current discharges. This self-generated spatial beam contraction is accompanied by a pulse break-up that can be beneficially exploited for on-axis temporal compression of the pulse. The pinching mechanism therefore explains the recently observed self-compression and the complicated spatio-temporal shapes typical for filament propagation experiments.

© 2009 Optical Society of America

z𝓔=i2k0rrrr𝓔+iω0cn2𝓔2𝓔iω02n0cρcρ(I)𝓔,
(1)
ρ(I)=ρnt(1exp(tdtW[I(t)])).
(2)

Here, z is the propagation variable, t the retarded time, and ω0 is the central laser frequency at λ 0=2πn 0/k 0=800 nm. n 2 is the nonlinear refraction index. Photon densities are described via the complex optical field envelope 𝓔, with I=|𝓔|2. The wavelength-dependent critical plasma density is calculated from the Drude model according to ρcω 2 0 me ε 0/q 2 e, where qe and me are electron charge and mass, respectively, ε 0 is the dielectric constant, c the speed of light, and ρ nt denotes the neutral density. Plasma generation is driven by the ionization rate W[I], which is suitably described by Perelomov-Popov-Terent’ev (PPT) theory [14

14. A. M. Perelomov, V. S. Popov, and M. V. Terent’ev, “Ionization of atoms in an alternating electric field,” Sov. Phys. JETP 23, 924–934 (1966).

]. For our investigations, we use data for argon [7

7. S. Skupin, G. Stibenz, L. Bergé, F. Lederer, T. Sokollik, M. Schnuerer, N. Zhavoronkov, and G. Steinmeyer, “Self-compression by femtosecond pulse filamentation: Experiments versus numerical simulations,” Phys. Rev. E 74, 056604 (2006). [CrossRef]

] at atmospheric pressure.

𝓔=PπR2exp[r22R2+ik0r2zR2R].
(3)

The quadratic phase guarantees preservation of continuity equations through self-similar substitutions, and the pulse radius RR(z, t) depends on both the longitudinal and temporal variables. For conservative systems preserving the power P(t)≡2π∫∞0I(t)rdr along z, straightforward algebra provides the virial-type identity [15

15. L. Bergé and A. Couairon, “Nonlinear propagation of self-guided ultra-short pulses in ionized gases,” Phys. Plasmas 7, 210–230 (2000). [CrossRef]

]

z20r3𝓔2dr=2k020rr𝓔2dr
2n2n00r𝓔4dr1n02ρc0𝓔2r2rρdr.
(4)

0=1P(t)Pcr+μP2(t)tdtIN*+1(t)P(t)1(I(t)+N*I(t)P(t)P(t))2,
(5)
Fig. 1. (a) Spectrum of solutions I(t) of Eq. (5). (b) Spatio-temporal representation of the solution of Eq. (5) marked in red [see (a)], obtained by rotating the line segment generated by R(t)=P(t)/πI(t) around the t-axis. Color corresponds to on-axis intensity.

with P cr=λ 2 0/(2πn 0 n 2) and μ=k02N*σN*ρnt/πρc.

Equation (5) is basically a generalization of a Volterra-Urysohn integral equation [16

16. M. A. Darwish, “On integral equations of Urysohn-Volterra type,” Appl. Math. Comput. 136, 93–98 (2003). [CrossRef]

], with a kernel depending not only on I(t′) but also on I(t). Using additional simplifying assumptions, steady-state solutions with soliton-like qualities have been previously discussed [13

13. L. Bergé and A. Couairon, “Gas-induced solitons,” Phys. Rev. Lett. 86, 1003–1006 (2001). [CrossRef] [PubMed]

]. Here we solve Eq. (5) without the approximations made in Ref. [13

13. L. Bergé and A. Couairon, “Gas-induced solitons,” Phys. Rev. Lett. 86, 1003–1006 (2001). [CrossRef] [PubMed]

]. Taking into account that the integral term of Eq. (5) is strictly positive, it immediately follows that nontrivial solutions only exist on the temporal interval -t *<t<t * where P(t)>P cr, with t*=(lnPin/Pcr)1/2tp. From a physical point of view, Kerr self-focusing can compensate for diffraction only on this interval, enabling the existence of a stationary state. For computing a stationary solution I(t) of the integral equation, we use the method presented in [17

17. E. Babolian, F. Fattahzadeh, and E. Golpar Raboky, “A Chebyshev approximation for solving nonlinear integral equations of Hammerstein type,” Appl. Math. Comput. 189, 641–646 (2007). [CrossRef]

] which combines a Chebyshev approximation of the unknown I(t) with a Clenshaw-Curtis quadrature formula [18

18. C. W. Clenshaw and A. R. Curtis, “A method for numerical integration on an automatic computer,” Numer. Math. 2, 197–205 (1960). [CrossRef]

] for the integral term. As the laser beam parameters, we choose a ratio P in/P cr=2 and a pulse duration tFWHM=2ln2tp100 fs,, leading to t *≈50 fs. The spectrum of solutions thus obtained is depicted in Fig. 1(a). As 1-P(t)/Pcr vanishes at the boundaries, there exists a continuum of multiple roots. All solutions show a strongly asymmetric temporal shape, with an intense leading subpulse localized at t=-t * [13

13. L. Bergé and A. Couairon, “Gas-induced solitons,” Phys. Rev. Lett. 86, 1003–1006 (2001). [CrossRef] [PubMed]

] and a minimum [dashed line in Fig. 1(a)] localized near zero delay, followed by a region of rapid intensity increase, suggesting singular behavior of the solutions. Filamentation is known to proceed from a dynamical balance between the Kerr and plasma responses, and a steady-state solution cannot strictly be reached by the physical system. Nevertheless, Eq. (5) provides deep insight into the configuration that the pulse profile tends to achieve in the filamentary regime. The structure of the emerging solutions [Fig. 1(b)] indeed indicates the formation of two areas of high on-axis intensity being separated by an approximately 20 fs wide defocused zone of strongly reduced intensity. While similar double-peaked on-axis intensities have already been observed in numerical simulations and experiments [6

6. G. Stibenz, N. Zhavoronkov, and G. Steinmeyer, “Self-compression of millijoule pulses to 7.8 fs duration in a white-light filament,” Opt. Lett. 31, 274–276 (2006). [CrossRef] [PubMed]

, 21

21. M. Mlejnek, E. M. Wright, and J. V. Moloney, “Dynamic spatial replenishment of femtosecond pulses propagating in air,” Opt. Lett. 23, 382–384 (1997). [CrossRef]

24

24. S. L. Chin, Y. Chen, O. Kosareva, V. P. Kandidov, and F. Théberge, “What is a filament ?” Laser Phys. 18, 962–964 (2008). [CrossRef]

] many authors considered a parasitic dispersive break-up in bulk media or optical fibers. Despite its superficial similarity, however, such a break-up cannot be exploited for the compression of isolated femtosecond pulses as will be done below. Interestingly, we observe a comparable dynamical behavior as reported for condensed media, where temporal break-up around zero-delay and the subsequent emergence of nonlinear X-waves occurs. These X-waves were recently proposed to constitute attractors of the filament dynamics [19

19. C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, “Nonlinear electromagnetic X waves,” Phys. Rev. Lett. 90, 170406 (2003). [CrossRef] [PubMed]

, 20

20. A. Couairon, E. Gaižauskas, D. Faccio, A. Dubietis, and P. Di Trapani, “Nonlinear X-wave formation by femtosecond filamentation in Kerr media,” Phys. Rev. E 73, 016608 (2006). [CrossRef]

].

For a deeper substantiation of our analytical model, we perform direct numerical simulations using the reduced radially symmetric evolution model Eq. (1) for the envelope of the optical field. The incident field is modeled as a Gaussian in space and time with w 0 :=w(z=0, t)=2.5mm and identical peak input power and pulse duration as used for the solutions of Eq. (5). The field is focused into the medium with an f=1.5m lens. The result of these simulations can be considered as prototypical for the pulse shaping effect inside filaments. These simulations also demonstrate that spatial effects alone already suffice for filamentary self-compression. As the evolution of the on-axis temporal intensity profile in Fig. 2(a) reveals, filamentary compression always undergoes two distinct phases. Initially, while z approaches the nonlinear focus (z=1.4-1.5 m), a dominant leading peak is observed. When the trailing part of the pulse refocuses in the efficiently ionized zone (ρ max≈5×1016cm-3) a double-spiked structure emerges. This transient double pulse structure confirms the pulse break-up predicted from the analysis of Eq. (5), see Figs. 3(a) and (c), and is compatible with the stationary shapes detailed in Fig. 1(a). Subsequently, only one of the emerging peaks survives and experiences further pulse shaping in the filamentary channel.

Fig. 2. (a) Evolution of the on-axis temporal intensity profile along z for the reduced numerical model governed by Eq. (1). (b) Same for the simulation of the full model equations [7].

For an analytic description of temporal compression during further filamentary propagation, we use the dynamical equation for the time dependent beam radius derived from Eqs. (3) and (4) [15

15. L. Bergé and A. Couairon, “Nonlinear propagation of self-guided ultra-short pulses in ionized gases,” Phys. Plasmas 7, 210–230 (2000). [CrossRef]

], yet neglecting the plasma term. With the initial conditions R(z=z 0, t)=R 0 and zR(z=z 0, t)≡0 the resulting problem is analytically solvable, and we find R(z,t)=R01+[(zz0)k0R02]2(1P(t)Pcr). This equation models the evolution of the plasma-free filamentary channel from z>1.6m, assuming P(t)≤Pcr. The profile P(t) represents the power contained in the filament core region only. For simplicity, we assume here P(t)=Pcr exp(-2t 2/t 2 p), R 0=100 µm and tp=23 fs. This corresponds to the duration of the pulse at z 0=1.7m shown in Fig. 3(a). Resulting characteristic spatio-temporal shapes are shown in Fig. 4, clearly revealing the presence of self-pinching in this Kerr-dominated stage of propagation and the dominant role it plays for on-axis temporal compression.

Fig. 3. (a) Pulse sequence illustrating the two-stage self-compression mechanism. Shown are the on-axis intensity profiles for z=1.5m (solid line), z=1.55m (dashed line) and z=1.7m (dashed-dotted line). (b) Self-compressed few-cycle pulse at z=2.5m. (c) Spatiotemporal characteristics of the double-spiked structure at z=1.55 m. (d) Same for the fewcycle pulse at z=2.5m.

So far our analysis has completely neglected dispersion, self-steepening and losses. To convince ourselves that dissipation and temporal coupling between time slices have only a modifying effect on the discussed self-compression scenario, we pursued full simulations of the filament propagation, including few-cycle corrections and space-time focusing [7

7. S. Skupin, G. Stibenz, L. Bergé, F. Lederer, T. Sokollik, M. Schnuerer, N. Zhavoronkov, and G. Steinmeyer, “Self-compression by femtosecond pulse filamentation: Experiments versus numerical simulations,” Phys. Rev. E 74, 056604 (2006). [CrossRef]

]. As shown in Fig. 2(b), minor parameter adjustment, setting w0=3.5mm and leaving all other laser beam parameters the same value, suffices to see pulse self-compression within the full model equations. Now self-steepening provides a much more effective compression mechanism in the trailing part. However, the comparison of Fig. 2(a) with (b) also reveals that the dynamical behavior changes only slightly upon inclusion of temporal effects. Clearly, the same two-stage compression mechanism is observed as in the reduced model. We therefore conclude that the pulse break-up dynamics in the efficiently ionized zone is already inherent to the reduced dynamical system governed by Eq. (1). Rather than relying on the interplay of self-phase modulation and dispersion as in traditional laser pulse compression, filament self-compression is essentially a spatial effect, conveyed by the interplay of Kerr self-focusing and plasma self-defocusing. This dominance of spatial effects favorably agrees with the spatial replenishment model of Mlejnek et al. [21

21. M. Mlejnek, E. M. Wright, and J. V. Moloney, “Dynamic spatial replenishment of femtosecond pulses propagating in air,” Opt. Lett. 23, 382–384 (1997). [CrossRef]

]. However, our model indicates previously undiscussed consequences on the temporal pulse structure on axis of the filament, leading to the emergence of the pinch-like structure [Fig. 3(d)] that restricts effective self-compression to the spatial center of the filament [7

7. S. Skupin, G. Stibenz, L. Bergé, F. Lederer, T. Sokollik, M. Schnuerer, N. Zhavoronkov, and G. Steinmeyer, “Self-compression by femtosecond pulse filamentation: Experiments versus numerical simulations,” Phys. Rev. E 74, 056604 (2006). [CrossRef]

, 25

25. A. Guandalini, F. Schapper, M. Holler, J. Biegert, L. Gallmann, A. Couairon, M. Franco, A. Mysyrowicz, and U. Keller, “Spatio-temporal characterization of few-cycle pulses obtained by filamentation,” Opt. Express 15, 5394–5405 (2007). [CrossRef] [PubMed]

]. Our analysis confirms the existence of a leading subpulse, in the wake of which the short self-compressed pulse is actually shaped during the first stage of filamentary propagation. This leading structure gives rise to a pronounced temporal asymmetry of self-compressed pulses, which is confirmed in experiments [7

7. S. Skupin, G. Stibenz, L. Bergé, F. Lederer, T. Sokollik, M. Schnuerer, N. Zhavoronkov, and G. Steinmeyer, “Self-compression by femtosecond pulse filamentation: Experiments versus numerical simulations,” Phys. Rev. E 74, 056604 (2006). [CrossRef]

].

While qualitatively similar break-up processes have frequently been observed in laser filaments, our analysis identifies the spatially induced temporal break-up as a first step for efficient on-axis compression of an isolated pulse. In our case, the leading break-up portion is eventually observed to diffract out and to reduce its intensity, while the trailing pulse can maintain its peak intensity. A subsequent stage, dominated by diffraction and Kerr nonlinearity, serves to further compress the emerging isolated pulse, and may give rise to almost tenfold on-axis pulse compression. The main driver behind this complex scenario is a dynamic interplay between radial effects, namely diffraction, Kerr-type self-focusing, and, exclusively close to the geometric focus, plasma defocusing. The dominance of spatial effects clearly indicates the unavoidability of a pronounced spatio-temporal pinch structure of self-compressed pulses. The frequently observed pedestals in this method are identified as remainders of the suppressed leading pulse from the original split-up. Our analysis also indicates that lower pulse energies <1mJ requiring more nonlinear gases or higher pressures will see an increased influence of dispersive coupling, which can eventually render pulse self-compression difficult to achieve. Higher energies, however, may not see such limitation, opening a perspective for future improvement of few-cycle pulse self-compression schemes.

Fig. 4. Sequence of pulses illustrating temporal self-compression due to Kerr-induced spatial self-pinching in the variational model corresponding to (a) z = 1.7m, (b) z = 1.9m, (c) z = 2.1m and (d) z = 2.3m.

Financial support by the Deutsche Forschungsgemeinschaft, grants DE 1209/1-1 and STE 762/7-1, is gratefully acknowledged. We acknowledge support by the GENCI project No x2009106003.

References and links

1.

V. F. D’Yachenko and V. S. Imshenik, “Magnetohydrodynamic Theory of the Pinch Effect in a Dense High-Temperature Plasma (dense Plasma Focus),” Rev. Plasma. Phys. 5, 447–495 (1970).

2.

J. B. Taylor, “Relaxation of Toroidal Plasma and Generation of Reverse Magnetic Fields,” Phys. Rev. Lett. 33, 1139–1141 (1974). [CrossRef]

3.

E. B. Treacy, “Optical pulse compression with diffraction gratings,” IEEE J. Quantum Electron. 5, 454–458 (1969). [CrossRef]

4.

C. V. Shank, R. L. Fork, R. Yen, R. H. Stolen, and W. J. Tomlinson, “Compression of femtosecond optical pulses,” Appl. Phys. Lett. 40, 761–763(1982). [CrossRef]

5.

M. Nisoli, S. DeSilvestri, O. Svelto, R. Szipocs, K. Ferencz, C. Spielmann, S. Sartania, and F. Krausz, “Compression of high-energy laser pulses below 5 fs,” Opt. Lett. 22, 522–524 (1997). [CrossRef] [PubMed]

6.

G. Stibenz, N. Zhavoronkov, and G. Steinmeyer, “Self-compression of millijoule pulses to 7.8 fs duration in a white-light filament,” Opt. Lett. 31, 274–276 (2006). [CrossRef] [PubMed]

7.

S. Skupin, G. Stibenz, L. Bergé, F. Lederer, T. Sokollik, M. Schnuerer, N. Zhavoronkov, and G. Steinmeyer, “Self-compression by femtosecond pulse filamentation: Experiments versus numerical simulations,” Phys. Rev. E 74, 056604 (2006). [CrossRef]

8.

A. Couairon, M. Franco, A. Mysyrowicz, J. Biegert, and U. Keller, “Pulse self-compression to the single-cycle limit by filamentation in a gas with a pressure gradient,” Opt. Lett. 30, 2657–2659 (2005). [CrossRef] [PubMed]

9.

L. T. Vuong, R. B. Lopez-Martens, C. P. Hauri, and A. L. Gaeta, “Spectral reshaping and pulse compression via sequential filamentation in gases,” Opt. Express 16, 390–401 (2008). [CrossRef] [PubMed]

10.

R. Fedele and P. K. Shukla, “Self-consistent interaction between the plasma wake field and the driving relativistic electron-beam,” Phys. Rev. A 45, 4045–4049 (1992). [CrossRef] [PubMed]

11.

T. Passot and P. L. Sulem, “Alfven wave filamentation: from Hall-MHD to kinetic theory,” Physica Scripta T113, 89–91 (2004).

12.

N. L. Wagner, E. A. Gibson, T. Popmintchev, I. P. Christov, M. M. Murnane, and H. C. Kapteyn, “Selfcompression of ultrashort pulses through ionization-induced spatiotemporal reshaping,” Phys. Rev. Lett. 93, 173902 (2004). [CrossRef] [PubMed]

13.

L. Bergé and A. Couairon, “Gas-induced solitons,” Phys. Rev. Lett. 86, 1003–1006 (2001). [CrossRef] [PubMed]

14.

A. M. Perelomov, V. S. Popov, and M. V. Terent’ev, “Ionization of atoms in an alternating electric field,” Sov. Phys. JETP 23, 924–934 (1966).

15.

L. Bergé and A. Couairon, “Nonlinear propagation of self-guided ultra-short pulses in ionized gases,” Phys. Plasmas 7, 210–230 (2000). [CrossRef]

16.

M. A. Darwish, “On integral equations of Urysohn-Volterra type,” Appl. Math. Comput. 136, 93–98 (2003). [CrossRef]

17.

E. Babolian, F. Fattahzadeh, and E. Golpar Raboky, “A Chebyshev approximation for solving nonlinear integral equations of Hammerstein type,” Appl. Math. Comput. 189, 641–646 (2007). [CrossRef]

18.

C. W. Clenshaw and A. R. Curtis, “A method for numerical integration on an automatic computer,” Numer. Math. 2, 197–205 (1960). [CrossRef]

19.

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, “Nonlinear electromagnetic X waves,” Phys. Rev. Lett. 90, 170406 (2003). [CrossRef] [PubMed]

20.

A. Couairon, E. Gaižauskas, D. Faccio, A. Dubietis, and P. Di Trapani, “Nonlinear X-wave formation by femtosecond filamentation in Kerr media,” Phys. Rev. E 73, 016608 (2006). [CrossRef]

21.

M. Mlejnek, E. M. Wright, and J. V. Moloney, “Dynamic spatial replenishment of femtosecond pulses propagating in air,” Opt. Lett. 23, 382–384 (1997). [CrossRef]

22.

S. Akturk, A. Couairon, M. Franco, and A. Mysyrowicz, “Spectrogram representation of pulse self compression by filamentation,” Opt. Express 16, 17626–17636 (2008) [CrossRef] [PubMed]

23.

S. Akturk, C. D’Amico, M. Franco, A. Couairon, and A. Mysyrowicz, “Pulse shortening, spatial mode cleaning, and intense terahertz generation by filamentation in xenon,” Phys. Rev. A 76, 063819 (2007). [CrossRef]

24.

S. L. Chin, Y. Chen, O. Kosareva, V. P. Kandidov, and F. Théberge, “What is a filament ?” Laser Phys. 18, 962–964 (2008). [CrossRef]

25.

A. Guandalini, F. Schapper, M. Holler, J. Biegert, L. Gallmann, A. Couairon, M. Franco, A. Mysyrowicz, and U. Keller, “Spatio-temporal characterization of few-cycle pulses obtained by filamentation,” Opt. Express 15, 5394–5405 (2007). [CrossRef] [PubMed]

OCIS Codes
(320.5520) Ultrafast optics : Pulse compression
(320.7110) Ultrafast optics : Ultrafast nonlinear optics

ToC Category:
Ultrafast Optics

History
Original Manuscript: June 25, 2009
Revised Manuscript: August 5, 2009
Manuscript Accepted: August 6, 2009
Published: August 31, 2009

Citation
Carsten Brée, Ayhan Demircan, Stefan Skupin, Luc Bergé, and Günter Steinmeyer, "Self-pinching of pulsed laser beams during filamentary propagation," Opt. Express 17, 16429-16435 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-19-16429


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References

  1. V. F. D’Yachenko and V. S. Imshenik, "Magnetohydrodynamic Theory of the Pinch Effect in a Dense High-Temperature Plasma (dense Plasma Focus)," Rev. Plasma. Phys. 5, 447-495 (1970).
  2. J. B. Taylor, "Relaxation of Toroidal Plasma and Generation of Reverse Magnetic Fields," Phys. Rev. Lett. 33, 1139-1141 (1974). [CrossRef]
  3. E. B. Treacy, "Optical pulse compression with diffraction gratings," IEEE J. Quantum Electron. 5, 454-458 (1969). [CrossRef]
  4. C. V. Shank, R. L. Fork, R. Yen, R. H. Stolen, and W. J. Tomlinson, "Compression of femtosecond optical pulses," Appl. Phys. Lett. 40, 761-763 (1982). [CrossRef]
  5. M. Nisoli, S. DeSilvestri, O. Svelto, R. Szipocs, K. Ferencz, C. Spielmann, S. Sartania, and F. Krausz, "Compression of high-energy laser pulses below 5 fs," Opt. Lett. 22, 522-524 (1997). [CrossRef] [PubMed]
  6. G. Stibenz, N. Zhavoronkov, and G. Steinmeyer, "Self-compression of millijoule pulses to 7.8 fs duration in a white-light filament," Opt. Lett. 31, 274-276 (2006). [CrossRef] [PubMed]
  7. S. Skupin, G. Stibenz, L. Bergé, F. Lederer, T. Sokollik, M. Schnuerer, N. Zhavoronkov, and G. Steinmeyer, "Selfcompression by femtosecond pulse filamentation: Experiments versus numerical simulations," Phys. Rev. E 74, 056604 (2006). [CrossRef]
  8. A. Couairon, M. Franco, A. Mysyrowicz, J. Biegert, and U. Keller, "Pulse self-compression to the single-cycle limit by filamentation in a gas with a pressure gradient," Opt. Lett. 30, 2657-2659 (2005). [CrossRef] [PubMed]
  9. L. T. Vuong, R. B. Lopez-Martens, C. P. Hauri, and A. L. Gaeta, "Spectral reshaping and pulse compression via sequential filamentation in gases," Opt. Express 16, 390-401 (2008). [CrossRef] [PubMed]
  10. R. Fedele and P. K. Shukla, "Self-consistent interaction between the plasma wake field and the driving relativistic electron-beam," Phys. Rev. A 45, 4045-4049 (1992). [CrossRef] [PubMed]
  11. T. Passot and P. L. Sulem, "Alfven wave filamentation: from Hall-MHD to kinetic theory," Physica Scripta T113, 89-91 (2004).
  12. N. L. Wagner, E. A. Gibson, T. Popmintchev, I. P. Christov, M. M. Murnane, and H. C. Kapteyn, "Selfcompression of ultrashort pulses through ionization-induced spatiotemporal reshaping," Phys. Rev. Lett. 93, 173902 (2004). [CrossRef] [PubMed]
  13. L. Bergé and A. Couairon, "Gas-induced solitons," Phys. Rev. Lett. 86, 1003-1006 (2001). [CrossRef] [PubMed]
  14. A. M. Perelomov, V. S. Popov, and M. V. Terent’ev, "Ionization of atoms in an alternating electric field," Sov. Phys. JETP 23, 924-934 (1966).
  15. L. Bergé and A. Couairon, "Nonlinear propagation of self-guided ultra-short pulses in ionized gases," Phys. Plasmas 7, 210-230 (2000). [CrossRef]
  16. M. A. Darwish, "On integral equations of Urysohn-Volterra type," Appl. Math. Comput. 136, 93-98 (2003). [CrossRef]
  17. E. Babolian, F. Fattahzadeh, and E. Golpar Raboky, "A Chebyshev approximation for solving nonlinear integral equations of Hammerstein type," Appl. Math. Comput. 189, 641-646 (2007). [CrossRef]
  18. C. W. Clenshaw and A. R. Curtis, "A method for numerical integration on an automatic computer," Numer. Math. 2, 197-205 (1960). [CrossRef]
  19. C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, "Nonlinear electromagnetic X waves," Phys. Rev. Lett. 90, 170406 (2003). [CrossRef] [PubMed]
  20. A. Couairon, E. Gaižauskas, D. Faccio, A. Dubietis, and P. Di Trapani, "Nonlinear X-wave formation by femtosecond filamentation in Kerr media," Phys. Rev. E 73, 016608 (2006). [CrossRef]
  21. M. Mlejnek, E. M. Wright, and J. V. Moloney, "Dynamic spatial replenishment of femtosecond pulses propagating in air," Opt. Lett. 23, 382-384 (1997). [CrossRef]
  22. S. Akturk, A. Couairon, M. Franco, and A. Mysyrowicz, "Spectrogram representation of pulse self compression by filamentation," Opt. Express 16, 17626-17636 (2008) [CrossRef] [PubMed]
  23. S. Akturk, C. D’Amico, M. Franco, A. Couairon, and A. Mysyrowicz, "Pulse shortening, spatial mode cleaning, and intense terahertz generation by filamentation in xenon," Phys. Rev. A 76, 063819 (2007). [CrossRef]
  24. S. L. Chin, Y. Chen, O. Kosareva, V. P. Kandidov, and F. Théberge, "What is a filament?" Laser Phys. 18, 962-964 (2008). [CrossRef]
  25. A. Guandalini, F. Schapper, M. Holler, J. Biegert, L. Gallmann, A. Couairon, M. Franco, A. Mysyrowicz, and U. Keller, "Spatio-temporal characterization of few-cycle pulses obtained by filamentation," Opt. Express 15, 5394-5405 (2007). [CrossRef] [PubMed]

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