Abstract

We investigate 3D spatio-temporal focusing of elliptically-shaped beams in a bulk medium with Kerr nonlinearity and anomalous group-velocity dispersion (GVD). Strong space-time localization of the mode is observed through multi-filamentation with temporal compression by a factor of 3. This behavior is in contrast to the near-zero GVD regime in which minimal pulse temporal compression is observed. Our theoretical simulations qualitatively reproduce the experimental results showing the highly localized spatio-temporal profile in the anomalous-GVD regime, which contrasts to the weakly localized pulse in the normal-GVD regime.

© 2011 OSA

1. Introduction

High-power, ultrashort laser propagation in bulk media has drawn significant interest for more than a decade due to its potential applications in remote sensing, lightning guiding, and pulse compression [16]. When the laser power P is higher than the critical power P cr, the pulse undergoes self-focusing and as it collapses, other nonlinearities such as multi-photon absorption (MPA) and plasma formation can counterbalance self-focusing, resulting in long-distance confinement of the beam, which is known as filamentation. When the power is much higher than Pcr, the beam breaks up into smaller multiple beams [i.e., multiple-filamentation (MF)], which can be due to modulational instability [7] or as Fibich et al. [8] have shown, beams with moderate powers (P < 15P cr) collapse and form a single filament first and then MF occurs after experiencing multiple focusing-defocusing cycles due to nonlinear effects such as plasma de-focusing. These noise-generated MF patterns can be random and unpredictable. However, recent theoretical and experimental studies [920] show that MF patterns can be controlled using elliptically-shaped input beams and vortex beams [21].

Liu et al. [22] used elliptical beams to generate (1 + 1 + 1)-D (one transverse dimension + time + propagation direction) solitons in quadratic nonlinear media. Furthermore, they investigated [23] whether multi-filaments from the elliptical input beam may form a 3D solitary wave (i.e. light bullets) [24]. However, angular dispersion, which was used for controlling (1 + 1 + 1)-D soliton generation, prevented the formation of the light bullet. More recently, Eisenberg et al. [25] studied elliptical beams in a planar waveguide in the anomalous group-velocity dispersion (GVD) regime and achieved spatio-temporal focusing with Kerr nonlinearity. Their calculations showed nearly stable propagation of the focused beam for several diffraction/dispersion lengths as evidence for a (1 + 1 + 1)-D soliton. However they did not investigate the possibility of a full 3-dimensional spatio-temporal localization.

Here we investigate the collapse dynamics and 3D spatio-temporal focusing of a highly elliptical femtosecond laser pulse propagating in a Kerr medium (BK7 glass) in the anomalous-GVD regime. The high-power elliptical beam breaks up into a controllable MF pattern, and each filament undergoes strong spatial localization, showing a symmetric Townes-like beam profile [26,27]. We also observe simultaneous temporal focusing in which the pulse compresses by a factor of 3. For comparison, similar elliptical beams in the zero-GVD regime exhibit minimal pulse compression even though similar MF is observed. According to our simulations, after strong spatio-temporal collapse due to the Kerr nonlinearity and anomalous GVD, the beam can be localized in space and time mainly due to the balance between self-focusing and diffraction in space and between anomalous GVD and self-phase modulation (SPM) in time. Pulse compression and spatio-temporal localization in our case is different from that in the normal-GVD regime using elliptical beams by Majus et al. [20] since pulse compression and the formation of weakly-localized stationary waves, called the X-waves in the normal-GVD regime involves more complicated phenomena such as pulse splitting and plasma defocusing [4,5] than spectral broadening by SPM.

2. Experiment

In our experiments, pulses from an amplified Ti: Sapphire laser system operating at a 1-kHz repetition rate (805 nm, 100 fs FWHM) are passed through an optical parametric amplifier (OPA) to produce pulses in the anomalous- (1600 nm, 300 fs FWHM, 450 fs autocorrelation FWHM) and zero- (1320 nm, 110 fs FWHM, 160 fs autocorrelation FWHM) GVD regions of BK7 glass. To generate an elliptical beam, we first expand and collimate the beam horizontally using a cylindrical lens telescope and then focus the beam vertically using another cylindrical lens onto the front face of a 30-cm-long BK7 sample. The spot size is 2.5 mm by 0.1 mm in the horizontal and vertical directions, respectively. The idler signal of the OPA is blocked by a linear polarizer, and the laser energy is controlled by a combination of a half-wave plate and another linear polarizer. We record the mode profiles after 30-cm propagation at the output face of the sample by imaging onto an IR CCD cameras with a f/2 achromat lens. The pulse duration of the apertured central part of the output beam (about 1-mm diameter) is measured using a Si photodiode two-photon autocorrelator [28].

Since the eccentricity e of the beam is 25, self-focusing and beam collapse initially occur along the narrower dimension. In this case, as demonstrated by several groups [1216,18,19], the beam breaks up into a deterministic MF pattern along its wider dimension when the laser power is several times the critical power Pcr. We can control the number and the position of filaments using elliptical beams in contrast to the random, noise-generated filaments for a circular beam [7]. According to calculations [29,30], the critical power for elliptical beams is larger than that for circular beams (Pcr = αλ 0 2/4πn 0 n 2 ~12 MW at 1600 nm, where α = 1.8962 is a constant associated with the initial Gaussian beam [31], λ 0 is the central wavelength, n 0 is the refractive index of BK7, and n 2 is the nonlinear index coefficient.) by a factor (0.4e/2 + 0.6) for large values of e. Thus the critical power for the elliptical beam for our experiment is 70 MW and the beam with P = 67 MW does not collapse as is shown in Fig. 1 . For P = 117 MW, we observe the bright central portion due to self-focusing. For P = 143 MW, two filaments with Townes-like beam profiles [26,27] form near the central region where the intensity is highest. At higher powers, the number of filaments increases and filaments in the central region become less intense and larger than those in the outer region. This can be explained as follows: as the intense parts of the beam collapse and form filaments, those filaments diffract due to nonlinear losses such as multi-photon absorption (MPA) by the time they reach the output face of the sample. For the case of the zero-GVD regime, similar MF patterns also occur along the wider dimension (Fig. 2 ). Since the critical power is proportional to λ 0 2, beam collapse and filamentation for 1320 nm occurs at smaller powers.

 

Fig. 1 Mode profiles of elliptical beams at the output face of a BK7 sample in the anomalous-GVD regime for input pulse energies (powers) of 20 μJ (67 MW), 35 μJ (117 MW), 43 μJ (143 MW), 48 μJ (160 MW), and 50 μJ (167 MW).

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Fig. 2 Mode profiles of elliptical beams at the output face of a BK7 sample in the zero-GVD regime (1320 nm) for different pulse energies.

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For the anomalous-GVD regime (1600 nm), autocorrelation measurements show that there is simultaneous pulse compression during beam collapse and MF [Figs. 3(a) and 3(b)]. When the power is low (P < 100 MW) such that there is no collapse, pulse compression is negligible since the sample length (30 cm) is much shorter than the dispersion length (290 cm). There is slight pulse compression near 100 MW due to spectral broadening by SPM in the anomalous-GVD regime [37]. However as the power increases and SPM becomes stronger, temporal focusing is observed at the output face with a minimum duration of 150 fs FWHM autocorrelation, which corresponds to approximately a 100 fs-pulse duration and represents compression of the input pulse by a factor of 3 [Fig. 3(b)]. For comparison, the autocorrelation trace in the zero-GVD regime with P = 136 MW increases to more than 250 fs FWHM due to SPM as compared to 160 fs at P = 45 MW, which approximately corresponds to an autocorrelation of the 110-fs input pulse. However, there is little change in the pulse duration after the initial broadening even though the power increases further [Fig. 3(c)] [25]. The aperture in front of the autocorrelator transmits 3 or 4 filaments near the center of the beams.

 

Fig. 3 Measured (a) autocorrelation traces and (b) pulse durations (FWHM) in the anomalous-GVD regime (1600 nm). (c) Measured pulse durations in the zero-GVD regime (1320 nm) for different pulse powers.

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Although filamentation and pulse compression in the anomalous-GVD regime with circular beams were experimentally and theoretically investigated before [3237], the filamentation patterns using circular beams are random and thus cannot be easily controlled since they are generated by random noise compared with the elliptical beams. The pulse compression observed here is also different from that with elliptical beams in the normal-GVD regime since drastic pulse reshaping such as (multiple) pulse-splitting occurs in the normal-GVD regime [20].

3. Theoretical simulations

Modeling the spatio-temporal behaviors of elliptical beams by performing a full 3-D spatio-temporal simulation is highly numerically intensive. As a result, we perform numerical simulations ignoring diffraction in the minor (shorter) axis since the beam profile does not change much in the minor axis (see Figs. 1 and 2), and MF only occurs along the major axis. This approach can yield qualitative spatio-temporal behavior that is very similar to that observed in our experiments. The nonlinear propagation equation is given by,

uξ=i42uμ2iLdf2L22uτ2+Ldf6L33uτ3+iLdfLnl|u|2uLdf2Lmp|u|2(K1)uiLdfLpl(1iωτc)​ ​ ​  ηu   ,

where u is the electric field amplitude normalized by the initial electric field amplitude E 0, ξ = z/Ldf is the propagation distance normalized by the diffraction length Ldf = n 0 πw 0 2 0, w 0 = (wxwy)1/2is the effective spot size for the elliptical beam, wx = 2.5 mm is the spot size in the horizontal direction, wy = 0.1 mm is the spot size in the vertical direction, n 0 = 1.5 is the refractive index of BK7, λ 0 is the central wavelength, µ = x/w 0 is the transverse coordinate along the major axis, normalized by the spot size w 0, τ is the retarded time normalized by the 1/e 2 pulse duration τp, L 2 = τp 2 2 is the dispersion length, L 3 = τp 3/β 3 is the third-order dispersion length, β 2 = −3.09 x 10−28 s2/cm (1.906 x 10−30 s2/cm) is the GVD parameter at 1600 nm (1320 nm), and β 3 = 1.77 x 10−42 s3/cm (0.88 x 10−43 s3/cm) is the third-order dispersion at 1600 nm (1320 nm). Lnl = c/(ωn 2 I 0) is the nonlinear length, n 2 = 2.2 x 10−16 cm2/W (2.5 x 10−16 cm2/W) is the nonlinear index coefficient at 1600 nm (1320 nm) [38], I 0 = cn 0 |E 0 |2/2π is the peak input intensity of the laser, Lmp = 1/[β(K)I 0 (K- 1 )] is the K-photon absorption length, β(K) = 1.4 x 10−75 cm11/W6 (1.8 x 10−50 cm7/W4) is the 7 (5)-photon absorption coefficient at 1600 nm (1320 nm) [39], Lpl = 2/(σρ 0 ωτ c) is the plasma length, σ is the inverse bremsstrahlung cross section, ρ 0 = β(K)I 0 K τp/(Kℏω) is the total electron density that is produced by the input laser pulse via multi-photon ionization, τc ~2.33 x 10−14 s is the electron-ion collision time [40,41], and η = ρ/ρ 0 is the normalized electron density. Each term on the right side of Eq. (1) represents diffraction in the major axis, dispersion, third-order dispersion, Kerr-nonlinearity, MPA, and plasma defocusing and collisional absorption, respectively. Considering collisional ionization, multiphoton ionization, and recombination for the plasma, the normalized electron density satisfies the equation,

ητ=αη|u|2+|u|2Kτpτrη,

where α = 𝜎I 0 τp /(n 0 Eg) is the avalanche ionization coefficient, Eg = 4.7 eV is the band-gap energy for BK7 [41], and τr = 150 fs is the electron recombination time [42].

Since we ignore diffraction in the minor axis, we expect that there are discrepancies between experiments and simulations, particularly in the distance to collapse. Therefore, we choose a propagation distance z in the simulation to approximately match the measured MF patterns. We include spatial noise of 2% in the initial intensity profiles, which results in slight asymmetry similar to that observed in our experiments (see Figs. 1 and 2). Large noise (>10%) results significant asymmetry in the filament patterns.

Figure 4 shows examples of the calculated spatio-temporal profiles at z = 13 cm in the anomalous-GVD [Figs. 4(a)-4(d)] and the zero-GVD [Figs. 4(e)-4(h)] regimes. In each figure, time-integrated spatial profiles which represent beam profiles along the major axis are shown on the top. The calculations qualitatively reproduce the experiments showing MF patterns and diffraction of central filaments due to MPA. The most notable feature about the filament in the anomalous-GVD regime is formation of the highly-localized, spatio-temporal profile [Figs. 4(c) and 4(d)] which is not observed in the zero-GVD regime. This is also in contrast to the case of the normal-GVD regime in which multiple peaks in the temporal domain take place mainly due to pulse splitting [20]. Small oscillations of the trail edges in the temporal domain in Figs. 4(d) and 4(h) are due to third-order dispersion [43].

 

Fig. 4 Calculated spatio (major axis)-temporal intensity distributions of collapsing elliptical beams at z = 13 cm in [(a)-(d)] the anomalous-GVD regime (1600 nm) and [(e)-(h)] near zero-GVD regime (1320 nm) for various powers. Time-integrated spatial profiles along the major axis (x) representing MF patterns are shown on top of spatio-temporal profiles. Note that initial pulse durations for 1600 nm (300 fs FWHM) and 1320 nm (110 fs FWHM) are different.

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The calculated autocorrelation traces and pulse durations also agree qualitatively with those observed in the experiments (Fig. 5 ). For 1320-nm pulses with P >200 MW, the pulse duration increases. However, the calculation does not show the saturation of pulse broadening at higher powers which is observed in the experiment [see Fig. 3(c)].

 

Fig. 5 Calculated (a) autocorrelation traces and (b) pulse durations (FWHM) of the space-averaged fluence over 1 mm in the anomalous-GVD regime (1600 nm) at z = 13 cm ignoring diffraction along the minor axis. (c) Calculated pulse durations (FWHM) of the space-averaged fluence over 1 mm in the zero-GVD regime (1320 nm) versus input peak powers at z = 13 cm.

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For comparison, if we ignore diffraction in the major axis, the calculated autocorrelation traces and pulse durations (Fig. 6 ) at z = 30 cm agree better with those observed in the experiments. However fully 3-dimensional simulations are required for more quantitative comparisons.

 

Fig. 6 Calculated (a) autocorrelation traces and (b) pulse durations (FWHM) of the space-averaged fluence over 1 mm in the anomalous-GVD regime (1600 nm) at z = 30 cm ignoring diffraction along the major axis. (c) Calculated pulse durations (FWHM) of the space-averaged fluence over 1 mm in the zero-GVD regime (1320 nm) versus input peak powers at z = 30 cm.

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4. Conclusion

In summary, we observe spatio-temporal focusing of an elliptical laser beam in a bulk medium with Kerr nonlinearity and anomalous GVD in a controllable manner. The beam breaks into a deterministic MF pattern, with each filament showing a Townes-like symmetric profile [26,27]. Due to the combined effect of spectral broadening via SPM and anomalous GVD, pulse compression by a factor of 3 simultaneously occurs. The simulation results suggest that the highly localized spatio-temporal wave can form via multifilamentation, which is in contrast to the formation of multiple peaks via pulse splitting in the normal-GVD regime [20]. In the future, we will investigate whether each filament generated from the break-up of the elliptical beam can propagate as a 3-D optical bullet under appropriate conditions.

Acknowledgments

This work was supported by NSF under Grant No. PHY-0703870 and the Army Research Office under Grant No. 186695-PH. The authors gratefully acknowledge useful discussions with M. Foster and A. Chong.

References and links

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References

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  1. J. Kasparian, M. Rodriguez, G. Méjean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y. B. Andre, A. Mysyrowicz, R. Sauerbrey, J.-P. Wolf, and L. Wöste, “White-light filaments for atmospheric analysis,” Science 301(5629), 61–64 (2003).
    [CrossRef] [PubMed]
  2. A. L. Gaeta, “Optics. Collapsing light really shines,” Science 301(5629), 54–55 (2003).
    [CrossRef] [PubMed]
  3. S. L. Chin, S. A. Hosseini, W. Liu, Q. Luo, F. Théberge, N. Aközbek, A. Becker, V. P. Kandidov, O. G. Kosareva, and H. Schroeder, “The propagation of powerful femtosecond laser pulses in optical media: physics, applications and new challenges,” Can. J. Phys. 83(9), 863–905 (2005).
    [CrossRef]
  4. A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Phys. Rep. 441(2-4), 47–189 (2007).
    [CrossRef]
  5. L. BergéS. Skupin, R. Nuter, J. Kasparian, and J.-P. Wolf, “Ultrashort filaments of light in weakly ionized optically transparent media,” Rep. Prog. Phys. 70(10), 1633–1713 (2007).
    [CrossRef]
  6. C. P. Hauri, W. Kornelis, F. W. Helbing, A. Heinrich, A. Couairon, A. Mysyrowicz, J. Biegert, and U. Keller, “Generation of intense, carrier-envelope phase-locked few-cycle laser pulses through filamentation,” Appl. Phys. B 79(6), 673–677 (2004).
    [CrossRef]
  7. V. I. Bespalov and V. I. Talanov, “Filamentary structure of light beams in non-linear liquids,” JETP Lett. 3, 307–310 (1966).
  8. G. Fibich, S. Eisenmann, B. Ilan, Y. Erlich, M. Fraenkel, Z. Henis, A. L. Gaeta, and A. Zigler, “Self-focusing distance of very high power laser pulses,” Opt. Express 13(15), 5897–5903 (2005).
    [CrossRef] [PubMed]
  9. J. W. Grantham, H. M. Gibbs, G. Khitrova, J. F. Valley, and X. Jiajin, “Kaleidoscopic spatial instability: Bifurcations of optical transverse solitary waves,” Phys. Rev. Lett. 66(11), 1422–1425 (1991).
    [CrossRef] [PubMed]
  10. G. Fibich and B. Ilan, “Self-focusing of circularly polarized beams,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(3), 036622 (2003).
    [CrossRef] [PubMed]
  11. S. Carrasco, S. Polyakov, H. Kim, L. Jankovic, G. I. Stegeman, J. P. Torres, L. Torner, M. Katz, and D. Eger, “Observation of multiple soliton generation mediated by amplification of asymmetries,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(4), 046616 (2003).
    [CrossRef] [PubMed]
  12. A. Dubietis, G. Tamosauskas, G. Fibich, and B. Ilan, “Multiple filamentation induced by input-beam ellipticity,” Opt. Lett. 29(10), 1126–1128 (2004).
    [CrossRef] [PubMed]
  13. G. Fibich, S. Eisenmann, B. Ilan, and A. Zigler, “Control of multiple filamentation in air,” Opt. Lett. 29(15), 1772–1774 (2004).
    [CrossRef] [PubMed]
  14. T. D. Grow and A. L. Gaeta, “Dependence of multiple filamentation on beam ellipticity,” Opt. Express 13(12), 4594–4599 (2005).
    [CrossRef] [PubMed]
  15. M. Centurion, Y. Pu, and D. Psaltis, “Self-organization of spatial solitons,” Opt. Express 13(16), 6202–6211 (2005).
    [CrossRef] [PubMed]
  16. V. Kudriasov, E. Gaizauskas, and V. Sirutkaitis, “Beam transformation and permanent modification in fused silica induced by femtosecond filaments,” J. Opt. Soc. Am. B 22(12), 2619–2627 (2005).
    [CrossRef]
  17. V. Y. Fedorov, V. P. Kandidov, O. G. Kosareva, N. Akozbek, M. Scalora, and S. L. Chin, “Filamentation of a femtosecond laser pulse with the initial beam ellipticity,” Laser Phys. 16(8), 1227–1234 (2006).
    [CrossRef]
  18. D. Majus, V. Jukna, G. Valiulis, and A. Dubietis, “Generation of periodic filament arrays by self-focusing of highly elliptical ultrashort pulsed laser beams,” Phys. Rev. A 79(3), 033843 (2009).
    [CrossRef]
  19. J.-P. Bérubé, R. Vallée, M. Bernier, O. Kosareva, N. Panov, V. Kandidov, and S. L. Chin, “Self and forced periodic arrangement of multiple filaments in glass,” Opt. Express 18(3), 1801–1819 (2010).
    [CrossRef] [PubMed]
  20. D. Majus, V. Jukna, G. Tamošauskas, G. Valiulis, and A. Dubietis, “Three-dimensional mapping of multiple filament arrays,” Phys. Rev. A 81(4), 043811 (2010).
    [CrossRef]
  21. L. T. Vuong, T. D. Grow, A. Ishaaya, A. L. Gaeta, G. W. ’t Hooft, E. R. Eliel, and G. Fibich, “Collapse of optical vortices,” Phys. Rev. Lett. 96(13), 133901 (2006).
    [CrossRef] [PubMed]
  22. X. Liu, L. J. Qian, and F. W. Wise, “Generation of optical spatiotemporal solitons,” Phys. Rev. Lett. 82(23), 4631–4634 (1999).
    [CrossRef]
  23. X. Liu, K. Beckwitt, and F. W. Wise, “Transverse instability of optical spatiotemporal solitons in quadratic media,” Phys. Rev. Lett. 85(9), 1871–1874 (2000).
    [CrossRef] [PubMed]
  24. Y. Silberberg, “Collapse of optical pulses,” Opt. Lett. 15(22), 1282–1284 (1990).
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  25. H. S. Eisenberg, R. Morandotti, Y. Silberberg, S. Bar-Ad, D. Ross, and J. S. Aitchison, “Kerr spatiotemporal self-focusing in a planar glass waveguide,” Phys. Rev. Lett. 87(4), 043902 (2001).
    [CrossRef] [PubMed]
  26. K. D. Moll, A. L. Gaeta, and G. Fibich, “Self-similar optical wave collapse: observation of the Townes profile,” Phys. Rev. Lett. 90(20), 203902 (2003).
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  27. R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13(15), 479–482 (1964).
    [CrossRef]
  28. J. K. Ranka, A. L. Gaeta, A. Baltuska, M. S. Pshenichnikov, and D. A. Wiersma, “Autocorrelation measurement of 6-fs pulses based on the two-photon-induced photocurrent in a GaAsP photodiode,” Opt. Lett. 22(17), 1344–1346 (1997).
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  29. G. Fibich and B. Ilan, “Self-focusing of elliptical beams:an example of the failure of the aberrationless approximation,” J. Opt. Soc. Am. B 17(10), 1749–1758 (2000).
    [CrossRef]
  30. V. P. Kandidov and V. Y. Fedorov, “Properties of self-focusing of elliptic beams,” Quantum Electron. 34, 1163–1168 (2004).
    [CrossRef]
  31. G. Fibich and A. L. Gaeta, “Critical power for self-focusing in bulk media and in hollow waveguides,” Opt. Lett. 25(5), 335–337 (2000).
    [CrossRef]
  32. K. D. Moll and A. L. Gaeta, “Role of dispersion in multiple-collapse dynamics,” Opt. Lett. 29(9), 995–997 (2004).
    [CrossRef] [PubMed]
  33. A. Saliminia, S. L. Chin, and R. Vallée, “Ultra-broad and coherent white light generation in silica glass by focused femtosecond pulses at 1.5 um,” Opt. Express 13(15), 5731–5738 (2005).
    [CrossRef] [PubMed]
  34. M. A. Porras, A. Dubietis, A. Matijošius, R. Piskarskas, F. Bragheri, A. Averchi, and P. D. Trapani, “Characterization of conical emission of light filaments in media with anomalous dispersion,” J. Opt. Soc. Am. B 24(3), 581–584 (2007).
    [CrossRef]
  35. L. Bergé andS. Skupin, “Self-channeling of ultrashort laser pulses in materials with anomalous dispersion,” Phys. Rev. E 71(6), 065601(R) (2005).
  36. J. Liu, R. Li, and Z. Xu, “Few-cycle spatiotemporal soliton wave excited by filamentation of a femtosecond laser pulse in materials with anomalous dispersion,” Phys. Rev. A 74(4), 043801 (2006).
    [CrossRef]
  37. G. Tamošauskas, A. Dubietis, and G. Valiulis, “Soliton-effect optical pulse compression in bulk media with χ(3) nonlinerity,” Nonlinear Anal. Modelling Control 5, 99–105 (2000).
  38. S. Skupin and L. Berge, “Self-guiding of femtosecond light pulses in condensed media: Plasma generation versus chromatic dispersion,” Physica D 220(1), 14–30 (2006).
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  39. K. Kennedy, “A first-order model of computation of laser-induced breakdown thresholds in ocular and aqueous media: Part I – theory,” IEEE J. Quantum Electron. 31(12), 2241–2249 (1995).
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  40. L. Sudrie, A. Couairon, M. Franco, B. Lamouroux, B. Prade, S. Tzortzakis, and A. Mysyrowicz, “Femtosecond laser-induced damage and filamentary propagation in fused silica,” Phys. Rev. Lett. 89(18), 186601 (2002).
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  41. R. Gopal, V. Deepak, and S. Sivaramakrishnan, ““Sytematic study of spatiotemporal dynamics of intense femtosecond laser pulses in BK-7 glass,” Pramana J. Phys. 68(4), 547–569 (2007).
    [CrossRef]
  42. P. Audebert, Ph. Daguzan, A. Dos Santos, J. C. Gauthier, J. P. Geindre, S. Guizard, G. Hamoniaux, K. Krastev, P. Martin, G. Petite, and A. Antonetti, “Space-time observation of an electron gas in SiO2.,” Phys. Rev. Lett. 73(14), 1990–1993 (1994).
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2010 (2)

D. Majus, V. Jukna, G. Tamošauskas, G. Valiulis, and A. Dubietis, “Three-dimensional mapping of multiple filament arrays,” Phys. Rev. A 81(4), 043811 (2010).
[CrossRef]

J.-P. Bérubé, R. Vallée, M. Bernier, O. Kosareva, N. Panov, V. Kandidov, and S. L. Chin, “Self and forced periodic arrangement of multiple filaments in glass,” Opt. Express 18(3), 1801–1819 (2010).
[CrossRef] [PubMed]

2009 (1)

D. Majus, V. Jukna, G. Valiulis, and A. Dubietis, “Generation of periodic filament arrays by self-focusing of highly elliptical ultrashort pulsed laser beams,” Phys. Rev. A 79(3), 033843 (2009).
[CrossRef]

2007 (4)

R. Gopal, V. Deepak, and S. Sivaramakrishnan, ““Sytematic study of spatiotemporal dynamics of intense femtosecond laser pulses in BK-7 glass,” Pramana J. Phys. 68(4), 547–569 (2007).
[CrossRef]

A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Phys. Rep. 441(2-4), 47–189 (2007).
[CrossRef]

L. BergéS. Skupin, R. Nuter, J. Kasparian, and J.-P. Wolf, “Ultrashort filaments of light in weakly ionized optically transparent media,” Rep. Prog. Phys. 70(10), 1633–1713 (2007).
[CrossRef]

L. BergéS. Skupin, R. Nuter, J. Kasparian, and J.-P. Wolf, “Ultrashort filaments of light in weakly ionized optically transparent media,” Rep. Prog. Phys. 70(10), 1633–1713 (2007).
[CrossRef]

M. A. Porras, A. Dubietis, A. Matijošius, R. Piskarskas, F. Bragheri, A. Averchi, and P. D. Trapani, “Characterization of conical emission of light filaments in media with anomalous dispersion,” J. Opt. Soc. Am. B 24(3), 581–584 (2007).
[CrossRef]

2006 (4)

L. T. Vuong, T. D. Grow, A. Ishaaya, A. L. Gaeta, G. W. ’t Hooft, E. R. Eliel, and G. Fibich, “Collapse of optical vortices,” Phys. Rev. Lett. 96(13), 133901 (2006).
[CrossRef] [PubMed]

J. Liu, R. Li, and Z. Xu, “Few-cycle spatiotemporal soliton wave excited by filamentation of a femtosecond laser pulse in materials with anomalous dispersion,” Phys. Rev. A 74(4), 043801 (2006).
[CrossRef]

S. Skupin and L. Berge, “Self-guiding of femtosecond light pulses in condensed media: Plasma generation versus chromatic dispersion,” Physica D 220(1), 14–30 (2006).
[CrossRef]

V. Y. Fedorov, V. P. Kandidov, O. G. Kosareva, N. Akozbek, M. Scalora, and S. L. Chin, “Filamentation of a femtosecond laser pulse with the initial beam ellipticity,” Laser Phys. 16(8), 1227–1234 (2006).
[CrossRef]

2005 (7)

L. Bergé andS. Skupin, “Self-channeling of ultrashort laser pulses in materials with anomalous dispersion,” Phys. Rev. E 71(6), 065601(R) (2005).

L. Bergé andS. Skupin, “Self-channeling of ultrashort laser pulses in materials with anomalous dispersion,” Phys. Rev. E 71(6), 065601(R) (2005).

S. L. Chin, S. A. Hosseini, W. Liu, Q. Luo, F. Théberge, N. Aközbek, A. Becker, V. P. Kandidov, O. G. Kosareva, and H. Schroeder, “The propagation of powerful femtosecond laser pulses in optical media: physics, applications and new challenges,” Can. J. Phys. 83(9), 863–905 (2005).
[CrossRef]

T. D. Grow and A. L. Gaeta, “Dependence of multiple filamentation on beam ellipticity,” Opt. Express 13(12), 4594–4599 (2005).
[CrossRef] [PubMed]

A. Saliminia, S. L. Chin, and R. Vallée, “Ultra-broad and coherent white light generation in silica glass by focused femtosecond pulses at 1.5 um,” Opt. Express 13(15), 5731–5738 (2005).
[CrossRef] [PubMed]

G. Fibich, S. Eisenmann, B. Ilan, Y. Erlich, M. Fraenkel, Z. Henis, A. L. Gaeta, and A. Zigler, “Self-focusing distance of very high power laser pulses,” Opt. Express 13(15), 5897–5903 (2005).
[CrossRef] [PubMed]

M. Centurion, Y. Pu, and D. Psaltis, “Self-organization of spatial solitons,” Opt. Express 13(16), 6202–6211 (2005).
[CrossRef] [PubMed]

V. Kudriasov, E. Gaizauskas, and V. Sirutkaitis, “Beam transformation and permanent modification in fused silica induced by femtosecond filaments,” J. Opt. Soc. Am. B 22(12), 2619–2627 (2005).
[CrossRef]

2004 (5)

C. P. Hauri, W. Kornelis, F. W. Helbing, A. Heinrich, A. Couairon, A. Mysyrowicz, J. Biegert, and U. Keller, “Generation of intense, carrier-envelope phase-locked few-cycle laser pulses through filamentation,” Appl. Phys. B 79(6), 673–677 (2004).
[CrossRef]

V. P. Kandidov and V. Y. Fedorov, “Properties of self-focusing of elliptic beams,” Quantum Electron. 34, 1163–1168 (2004).
[CrossRef]

K. D. Moll and A. L. Gaeta, “Role of dispersion in multiple-collapse dynamics,” Opt. Lett. 29(9), 995–997 (2004).
[CrossRef] [PubMed]

A. Dubietis, G. Tamosauskas, G. Fibich, and B. Ilan, “Multiple filamentation induced by input-beam ellipticity,” Opt. Lett. 29(10), 1126–1128 (2004).
[CrossRef] [PubMed]

G. Fibich, S. Eisenmann, B. Ilan, and A. Zigler, “Control of multiple filamentation in air,” Opt. Lett. 29(15), 1772–1774 (2004).
[CrossRef] [PubMed]

2003 (5)

J. Kasparian, M. Rodriguez, G. Méjean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y. B. Andre, A. Mysyrowicz, R. Sauerbrey, J.-P. Wolf, and L. Wöste, “White-light filaments for atmospheric analysis,” Science 301(5629), 61–64 (2003).
[CrossRef] [PubMed]

A. L. Gaeta, “Optics. Collapsing light really shines,” Science 301(5629), 54–55 (2003).
[CrossRef] [PubMed]

G. Fibich and B. Ilan, “Self-focusing of circularly polarized beams,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(3), 036622 (2003).
[CrossRef] [PubMed]

S. Carrasco, S. Polyakov, H. Kim, L. Jankovic, G. I. Stegeman, J. P. Torres, L. Torner, M. Katz, and D. Eger, “Observation of multiple soliton generation mediated by amplification of asymmetries,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(4), 046616 (2003).
[CrossRef] [PubMed]

K. D. Moll, A. L. Gaeta, and G. Fibich, “Self-similar optical wave collapse: observation of the Townes profile,” Phys. Rev. Lett. 90(20), 203902 (2003).
[CrossRef] [PubMed]

2002 (1)

L. Sudrie, A. Couairon, M. Franco, B. Lamouroux, B. Prade, S. Tzortzakis, and A. Mysyrowicz, “Femtosecond laser-induced damage and filamentary propagation in fused silica,” Phys. Rev. Lett. 89(18), 186601 (2002).
[CrossRef] [PubMed]

2001 (1)

H. S. Eisenberg, R. Morandotti, Y. Silberberg, S. Bar-Ad, D. Ross, and J. S. Aitchison, “Kerr spatiotemporal self-focusing in a planar glass waveguide,” Phys. Rev. Lett. 87(4), 043902 (2001).
[CrossRef] [PubMed]

2000 (4)

X. Liu, K. Beckwitt, and F. W. Wise, “Transverse instability of optical spatiotemporal solitons in quadratic media,” Phys. Rev. Lett. 85(9), 1871–1874 (2000).
[CrossRef] [PubMed]

G. Tamošauskas, A. Dubietis, and G. Valiulis, “Soliton-effect optical pulse compression in bulk media with χ(3) nonlinerity,” Nonlinear Anal. Modelling Control 5, 99–105 (2000).

G. Fibich and B. Ilan, “Self-focusing of elliptical beams:an example of the failure of the aberrationless approximation,” J. Opt. Soc. Am. B 17(10), 1749–1758 (2000).
[CrossRef]

G. Fibich and A. L. Gaeta, “Critical power for self-focusing in bulk media and in hollow waveguides,” Opt. Lett. 25(5), 335–337 (2000).
[CrossRef]

1999 (1)

X. Liu, L. J. Qian, and F. W. Wise, “Generation of optical spatiotemporal solitons,” Phys. Rev. Lett. 82(23), 4631–4634 (1999).
[CrossRef]

1997 (1)

1995 (1)

K. Kennedy, “A first-order model of computation of laser-induced breakdown thresholds in ocular and aqueous media: Part I – theory,” IEEE J. Quantum Electron. 31(12), 2241–2249 (1995).
[CrossRef]

1994 (1)

P. Audebert, Ph. Daguzan, A. Dos Santos, J. C. Gauthier, J. P. Geindre, S. Guizard, G. Hamoniaux, K. Krastev, P. Martin, G. Petite, and A. Antonetti, “Space-time observation of an electron gas in SiO2.,” Phys. Rev. Lett. 73(14), 1990–1993 (1994).
[CrossRef] [PubMed]

1991 (1)

J. W. Grantham, H. M. Gibbs, G. Khitrova, J. F. Valley, and X. Jiajin, “Kaleidoscopic spatial instability: Bifurcations of optical transverse solitary waves,” Phys. Rev. Lett. 66(11), 1422–1425 (1991).
[CrossRef] [PubMed]

1990 (1)

1966 (1)

V. I. Bespalov and V. I. Talanov, “Filamentary structure of light beams in non-linear liquids,” JETP Lett. 3, 307–310 (1966).

1964 (1)

R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13(15), 479–482 (1964).
[CrossRef]

’t Hooft, G. W.

L. T. Vuong, T. D. Grow, A. Ishaaya, A. L. Gaeta, G. W. ’t Hooft, E. R. Eliel, and G. Fibich, “Collapse of optical vortices,” Phys. Rev. Lett. 96(13), 133901 (2006).
[CrossRef] [PubMed]

Aitchison, J. S.

H. S. Eisenberg, R. Morandotti, Y. Silberberg, S. Bar-Ad, D. Ross, and J. S. Aitchison, “Kerr spatiotemporal self-focusing in a planar glass waveguide,” Phys. Rev. Lett. 87(4), 043902 (2001).
[CrossRef] [PubMed]

Akozbek, N.

V. Y. Fedorov, V. P. Kandidov, O. G. Kosareva, N. Akozbek, M. Scalora, and S. L. Chin, “Filamentation of a femtosecond laser pulse with the initial beam ellipticity,” Laser Phys. 16(8), 1227–1234 (2006).
[CrossRef]

Aközbek, N.

S. L. Chin, S. A. Hosseini, W. Liu, Q. Luo, F. Théberge, N. Aközbek, A. Becker, V. P. Kandidov, O. G. Kosareva, and H. Schroeder, “The propagation of powerful femtosecond laser pulses in optical media: physics, applications and new challenges,” Can. J. Phys. 83(9), 863–905 (2005).
[CrossRef]

Andre, Y. B.

J. Kasparian, M. Rodriguez, G. Méjean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y. B. Andre, A. Mysyrowicz, R. Sauerbrey, J.-P. Wolf, and L. Wöste, “White-light filaments for atmospheric analysis,” Science 301(5629), 61–64 (2003).
[CrossRef] [PubMed]

Antonetti, A.

P. Audebert, Ph. Daguzan, A. Dos Santos, J. C. Gauthier, J. P. Geindre, S. Guizard, G. Hamoniaux, K. Krastev, P. Martin, G. Petite, and A. Antonetti, “Space-time observation of an electron gas in SiO2.,” Phys. Rev. Lett. 73(14), 1990–1993 (1994).
[CrossRef] [PubMed]

Audebert, P.

P. Audebert, Ph. Daguzan, A. Dos Santos, J. C. Gauthier, J. P. Geindre, S. Guizard, G. Hamoniaux, K. Krastev, P. Martin, G. Petite, and A. Antonetti, “Space-time observation of an electron gas in SiO2.,” Phys. Rev. Lett. 73(14), 1990–1993 (1994).
[CrossRef] [PubMed]

Averchi, A.

Baltuska, A.

Bar-Ad, S.

H. S. Eisenberg, R. Morandotti, Y. Silberberg, S. Bar-Ad, D. Ross, and J. S. Aitchison, “Kerr spatiotemporal self-focusing in a planar glass waveguide,” Phys. Rev. Lett. 87(4), 043902 (2001).
[CrossRef] [PubMed]

Becker, A.

S. L. Chin, S. A. Hosseini, W. Liu, Q. Luo, F. Théberge, N. Aközbek, A. Becker, V. P. Kandidov, O. G. Kosareva, and H. Schroeder, “The propagation of powerful femtosecond laser pulses in optical media: physics, applications and new challenges,” Can. J. Phys. 83(9), 863–905 (2005).
[CrossRef]

Beckwitt, K.

X. Liu, K. Beckwitt, and F. W. Wise, “Transverse instability of optical spatiotemporal solitons in quadratic media,” Phys. Rev. Lett. 85(9), 1871–1874 (2000).
[CrossRef] [PubMed]

Berg, L.

L. BergéS. Skupin, R. Nuter, J. Kasparian, and J.-P. Wolf, “Ultrashort filaments of light in weakly ionized optically transparent media,” Rep. Prog. Phys. 70(10), 1633–1713 (2007).
[CrossRef]

L. Bergé andS. Skupin, “Self-channeling of ultrashort laser pulses in materials with anomalous dispersion,” Phys. Rev. E 71(6), 065601(R) (2005).

Berge, L.

S. Skupin and L. Berge, “Self-guiding of femtosecond light pulses in condensed media: Plasma generation versus chromatic dispersion,” Physica D 220(1), 14–30 (2006).
[CrossRef]

Bernier, M.

Bérubé, J.-P.

Bespalov, V. I.

V. I. Bespalov and V. I. Talanov, “Filamentary structure of light beams in non-linear liquids,” JETP Lett. 3, 307–310 (1966).

Biegert, J.

C. P. Hauri, W. Kornelis, F. W. Helbing, A. Heinrich, A. Couairon, A. Mysyrowicz, J. Biegert, and U. Keller, “Generation of intense, carrier-envelope phase-locked few-cycle laser pulses through filamentation,” Appl. Phys. B 79(6), 673–677 (2004).
[CrossRef]

Bourayou, R.

J. Kasparian, M. Rodriguez, G. Méjean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y. B. Andre, A. Mysyrowicz, R. Sauerbrey, J.-P. Wolf, and L. Wöste, “White-light filaments for atmospheric analysis,” Science 301(5629), 61–64 (2003).
[CrossRef] [PubMed]

Bragheri, F.

Carrasco, S.

S. Carrasco, S. Polyakov, H. Kim, L. Jankovic, G. I. Stegeman, J. P. Torres, L. Torner, M. Katz, and D. Eger, “Observation of multiple soliton generation mediated by amplification of asymmetries,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(4), 046616 (2003).
[CrossRef] [PubMed]

Centurion, M.

Chiao, R. Y.

R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13(15), 479–482 (1964).
[CrossRef]

Chin, S. L.

J.-P. Bérubé, R. Vallée, M. Bernier, O. Kosareva, N. Panov, V. Kandidov, and S. L. Chin, “Self and forced periodic arrangement of multiple filaments in glass,” Opt. Express 18(3), 1801–1819 (2010).
[CrossRef] [PubMed]

V. Y. Fedorov, V. P. Kandidov, O. G. Kosareva, N. Akozbek, M. Scalora, and S. L. Chin, “Filamentation of a femtosecond laser pulse with the initial beam ellipticity,” Laser Phys. 16(8), 1227–1234 (2006).
[CrossRef]

S. L. Chin, S. A. Hosseini, W. Liu, Q. Luo, F. Théberge, N. Aközbek, A. Becker, V. P. Kandidov, O. G. Kosareva, and H. Schroeder, “The propagation of powerful femtosecond laser pulses in optical media: physics, applications and new challenges,” Can. J. Phys. 83(9), 863–905 (2005).
[CrossRef]

A. Saliminia, S. L. Chin, and R. Vallée, “Ultra-broad and coherent white light generation in silica glass by focused femtosecond pulses at 1.5 um,” Opt. Express 13(15), 5731–5738 (2005).
[CrossRef] [PubMed]

Couairon, A.

A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Phys. Rep. 441(2-4), 47–189 (2007).
[CrossRef]

C. P. Hauri, W. Kornelis, F. W. Helbing, A. Heinrich, A. Couairon, A. Mysyrowicz, J. Biegert, and U. Keller, “Generation of intense, carrier-envelope phase-locked few-cycle laser pulses through filamentation,” Appl. Phys. B 79(6), 673–677 (2004).
[CrossRef]

L. Sudrie, A. Couairon, M. Franco, B. Lamouroux, B. Prade, S. Tzortzakis, and A. Mysyrowicz, “Femtosecond laser-induced damage and filamentary propagation in fused silica,” Phys. Rev. Lett. 89(18), 186601 (2002).
[CrossRef] [PubMed]

Daguzan, Ph.

P. Audebert, Ph. Daguzan, A. Dos Santos, J. C. Gauthier, J. P. Geindre, S. Guizard, G. Hamoniaux, K. Krastev, P. Martin, G. Petite, and A. Antonetti, “Space-time observation of an electron gas in SiO2.,” Phys. Rev. Lett. 73(14), 1990–1993 (1994).
[CrossRef] [PubMed]

Deepak, V.

R. Gopal, V. Deepak, and S. Sivaramakrishnan, ““Sytematic study of spatiotemporal dynamics of intense femtosecond laser pulses in BK-7 glass,” Pramana J. Phys. 68(4), 547–569 (2007).
[CrossRef]

Dos Santos, A.

P. Audebert, Ph. Daguzan, A. Dos Santos, J. C. Gauthier, J. P. Geindre, S. Guizard, G. Hamoniaux, K. Krastev, P. Martin, G. Petite, and A. Antonetti, “Space-time observation of an electron gas in SiO2.,” Phys. Rev. Lett. 73(14), 1990–1993 (1994).
[CrossRef] [PubMed]

Dubietis, A.

D. Majus, V. Jukna, G. Tamošauskas, G. Valiulis, and A. Dubietis, “Three-dimensional mapping of multiple filament arrays,” Phys. Rev. A 81(4), 043811 (2010).
[CrossRef]

D. Majus, V. Jukna, G. Valiulis, and A. Dubietis, “Generation of periodic filament arrays by self-focusing of highly elliptical ultrashort pulsed laser beams,” Phys. Rev. A 79(3), 033843 (2009).
[CrossRef]

M. A. Porras, A. Dubietis, A. Matijošius, R. Piskarskas, F. Bragheri, A. Averchi, and P. D. Trapani, “Characterization of conical emission of light filaments in media with anomalous dispersion,” J. Opt. Soc. Am. B 24(3), 581–584 (2007).
[CrossRef]

A. Dubietis, G. Tamosauskas, G. Fibich, and B. Ilan, “Multiple filamentation induced by input-beam ellipticity,” Opt. Lett. 29(10), 1126–1128 (2004).
[CrossRef] [PubMed]

G. Tamošauskas, A. Dubietis, and G. Valiulis, “Soliton-effect optical pulse compression in bulk media with χ(3) nonlinerity,” Nonlinear Anal. Modelling Control 5, 99–105 (2000).

Eger, D.

S. Carrasco, S. Polyakov, H. Kim, L. Jankovic, G. I. Stegeman, J. P. Torres, L. Torner, M. Katz, and D. Eger, “Observation of multiple soliton generation mediated by amplification of asymmetries,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(4), 046616 (2003).
[CrossRef] [PubMed]

Eisenberg, H. S.

H. S. Eisenberg, R. Morandotti, Y. Silberberg, S. Bar-Ad, D. Ross, and J. S. Aitchison, “Kerr spatiotemporal self-focusing in a planar glass waveguide,” Phys. Rev. Lett. 87(4), 043902 (2001).
[CrossRef] [PubMed]

Eisenmann, S.

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Fedorov, V. Y.

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Gaizauskas, E.

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R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13(15), 479–482 (1964).
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P. Audebert, Ph. Daguzan, A. Dos Santos, J. C. Gauthier, J. P. Geindre, S. Guizard, G. Hamoniaux, K. Krastev, P. Martin, G. Petite, and A. Antonetti, “Space-time observation of an electron gas in SiO2.,” Phys. Rev. Lett. 73(14), 1990–1993 (1994).
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P. Audebert, Ph. Daguzan, A. Dos Santos, J. C. Gauthier, J. P. Geindre, S. Guizard, G. Hamoniaux, K. Krastev, P. Martin, G. Petite, and A. Antonetti, “Space-time observation of an electron gas in SiO2.,” Phys. Rev. Lett. 73(14), 1990–1993 (1994).
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J. W. Grantham, H. M. Gibbs, G. Khitrova, J. F. Valley, and X. Jiajin, “Kaleidoscopic spatial instability: Bifurcations of optical transverse solitary waves,” Phys. Rev. Lett. 66(11), 1422–1425 (1991).
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J. W. Grantham, H. M. Gibbs, G. Khitrova, J. F. Valley, and X. Jiajin, “Kaleidoscopic spatial instability: Bifurcations of optical transverse solitary waves,” Phys. Rev. Lett. 66(11), 1422–1425 (1991).
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L. T. Vuong, T. D. Grow, A. Ishaaya, A. L. Gaeta, G. W. ’t Hooft, E. R. Eliel, and G. Fibich, “Collapse of optical vortices,” Phys. Rev. Lett. 96(13), 133901 (2006).
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P. Audebert, Ph. Daguzan, A. Dos Santos, J. C. Gauthier, J. P. Geindre, S. Guizard, G. Hamoniaux, K. Krastev, P. Martin, G. Petite, and A. Antonetti, “Space-time observation of an electron gas in SiO2.,” Phys. Rev. Lett. 73(14), 1990–1993 (1994).
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C. P. Hauri, W. Kornelis, F. W. Helbing, A. Heinrich, A. Couairon, A. Mysyrowicz, J. Biegert, and U. Keller, “Generation of intense, carrier-envelope phase-locked few-cycle laser pulses through filamentation,” Appl. Phys. B 79(6), 673–677 (2004).
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C. P. Hauri, W. Kornelis, F. W. Helbing, A. Heinrich, A. Couairon, A. Mysyrowicz, J. Biegert, and U. Keller, “Generation of intense, carrier-envelope phase-locked few-cycle laser pulses through filamentation,” Appl. Phys. B 79(6), 673–677 (2004).
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Hosseini, S. A.

S. L. Chin, S. A. Hosseini, W. Liu, Q. Luo, F. Théberge, N. Aközbek, A. Becker, V. P. Kandidov, O. G. Kosareva, and H. Schroeder, “The propagation of powerful femtosecond laser pulses in optical media: physics, applications and new challenges,” Can. J. Phys. 83(9), 863–905 (2005).
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Ishaaya, A.

L. T. Vuong, T. D. Grow, A. Ishaaya, A. L. Gaeta, G. W. ’t Hooft, E. R. Eliel, and G. Fibich, “Collapse of optical vortices,” Phys. Rev. Lett. 96(13), 133901 (2006).
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J. W. Grantham, H. M. Gibbs, G. Khitrova, J. F. Valley, and X. Jiajin, “Kaleidoscopic spatial instability: Bifurcations of optical transverse solitary waves,” Phys. Rev. Lett. 66(11), 1422–1425 (1991).
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D. Majus, V. Jukna, G. Tamošauskas, G. Valiulis, and A. Dubietis, “Three-dimensional mapping of multiple filament arrays,” Phys. Rev. A 81(4), 043811 (2010).
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D. Majus, V. Jukna, G. Valiulis, and A. Dubietis, “Generation of periodic filament arrays by self-focusing of highly elliptical ultrashort pulsed laser beams,” Phys. Rev. A 79(3), 033843 (2009).
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Kandidov, V. P.

V. Y. Fedorov, V. P. Kandidov, O. G. Kosareva, N. Akozbek, M. Scalora, and S. L. Chin, “Filamentation of a femtosecond laser pulse with the initial beam ellipticity,” Laser Phys. 16(8), 1227–1234 (2006).
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S. L. Chin, S. A. Hosseini, W. Liu, Q. Luo, F. Théberge, N. Aközbek, A. Becker, V. P. Kandidov, O. G. Kosareva, and H. Schroeder, “The propagation of powerful femtosecond laser pulses in optical media: physics, applications and new challenges,” Can. J. Phys. 83(9), 863–905 (2005).
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V. P. Kandidov and V. Y. Fedorov, “Properties of self-focusing of elliptic beams,” Quantum Electron. 34, 1163–1168 (2004).
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L. BergéS. Skupin, R. Nuter, J. Kasparian, and J.-P. Wolf, “Ultrashort filaments of light in weakly ionized optically transparent media,” Rep. Prog. Phys. 70(10), 1633–1713 (2007).
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J. Kasparian, M. Rodriguez, G. Méjean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y. B. Andre, A. Mysyrowicz, R. Sauerbrey, J.-P. Wolf, and L. Wöste, “White-light filaments for atmospheric analysis,” Science 301(5629), 61–64 (2003).
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C. P. Hauri, W. Kornelis, F. W. Helbing, A. Heinrich, A. Couairon, A. Mysyrowicz, J. Biegert, and U. Keller, “Generation of intense, carrier-envelope phase-locked few-cycle laser pulses through filamentation,” Appl. Phys. B 79(6), 673–677 (2004).
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S. Carrasco, S. Polyakov, H. Kim, L. Jankovic, G. I. Stegeman, J. P. Torres, L. Torner, M. Katz, and D. Eger, “Observation of multiple soliton generation mediated by amplification of asymmetries,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(4), 046616 (2003).
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C. P. Hauri, W. Kornelis, F. W. Helbing, A. Heinrich, A. Couairon, A. Mysyrowicz, J. Biegert, and U. Keller, “Generation of intense, carrier-envelope phase-locked few-cycle laser pulses through filamentation,” Appl. Phys. B 79(6), 673–677 (2004).
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Kosareva, O. G.

V. Y. Fedorov, V. P. Kandidov, O. G. Kosareva, N. Akozbek, M. Scalora, and S. L. Chin, “Filamentation of a femtosecond laser pulse with the initial beam ellipticity,” Laser Phys. 16(8), 1227–1234 (2006).
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S. L. Chin, S. A. Hosseini, W. Liu, Q. Luo, F. Théberge, N. Aközbek, A. Becker, V. P. Kandidov, O. G. Kosareva, and H. Schroeder, “The propagation of powerful femtosecond laser pulses in optical media: physics, applications and new challenges,” Can. J. Phys. 83(9), 863–905 (2005).
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P. Audebert, Ph. Daguzan, A. Dos Santos, J. C. Gauthier, J. P. Geindre, S. Guizard, G. Hamoniaux, K. Krastev, P. Martin, G. Petite, and A. Antonetti, “Space-time observation of an electron gas in SiO2.,” Phys. Rev. Lett. 73(14), 1990–1993 (1994).
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Lamouroux, B.

L. Sudrie, A. Couairon, M. Franco, B. Lamouroux, B. Prade, S. Tzortzakis, and A. Mysyrowicz, “Femtosecond laser-induced damage and filamentary propagation in fused silica,” Phys. Rev. Lett. 89(18), 186601 (2002).
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S. L. Chin, S. A. Hosseini, W. Liu, Q. Luo, F. Théberge, N. Aközbek, A. Becker, V. P. Kandidov, O. G. Kosareva, and H. Schroeder, “The propagation of powerful femtosecond laser pulses in optical media: physics, applications and new challenges,” Can. J. Phys. 83(9), 863–905 (2005).
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S. L. Chin, S. A. Hosseini, W. Liu, Q. Luo, F. Théberge, N. Aközbek, A. Becker, V. P. Kandidov, O. G. Kosareva, and H. Schroeder, “The propagation of powerful femtosecond laser pulses in optical media: physics, applications and new challenges,” Can. J. Phys. 83(9), 863–905 (2005).
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D. Majus, V. Jukna, G. Tamošauskas, G. Valiulis, and A. Dubietis, “Three-dimensional mapping of multiple filament arrays,” Phys. Rev. A 81(4), 043811 (2010).
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D. Majus, V. Jukna, G. Valiulis, and A. Dubietis, “Generation of periodic filament arrays by self-focusing of highly elliptical ultrashort pulsed laser beams,” Phys. Rev. A 79(3), 033843 (2009).
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P. Audebert, Ph. Daguzan, A. Dos Santos, J. C. Gauthier, J. P. Geindre, S. Guizard, G. Hamoniaux, K. Krastev, P. Martin, G. Petite, and A. Antonetti, “Space-time observation of an electron gas in SiO2.,” Phys. Rev. Lett. 73(14), 1990–1993 (1994).
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J. Kasparian, M. Rodriguez, G. Méjean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y. B. Andre, A. Mysyrowicz, R. Sauerbrey, J.-P. Wolf, and L. Wöste, “White-light filaments for atmospheric analysis,” Science 301(5629), 61–64 (2003).
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H. S. Eisenberg, R. Morandotti, Y. Silberberg, S. Bar-Ad, D. Ross, and J. S. Aitchison, “Kerr spatiotemporal self-focusing in a planar glass waveguide,” Phys. Rev. Lett. 87(4), 043902 (2001).
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C. P. Hauri, W. Kornelis, F. W. Helbing, A. Heinrich, A. Couairon, A. Mysyrowicz, J. Biegert, and U. Keller, “Generation of intense, carrier-envelope phase-locked few-cycle laser pulses through filamentation,” Appl. Phys. B 79(6), 673–677 (2004).
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J. Kasparian, M. Rodriguez, G. Méjean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y. B. Andre, A. Mysyrowicz, R. Sauerbrey, J.-P. Wolf, and L. Wöste, “White-light filaments for atmospheric analysis,” Science 301(5629), 61–64 (2003).
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L. Sudrie, A. Couairon, M. Franco, B. Lamouroux, B. Prade, S. Tzortzakis, and A. Mysyrowicz, “Femtosecond laser-induced damage and filamentary propagation in fused silica,” Phys. Rev. Lett. 89(18), 186601 (2002).
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L. BergéS. Skupin, R. Nuter, J. Kasparian, and J.-P. Wolf, “Ultrashort filaments of light in weakly ionized optically transparent media,” Rep. Prog. Phys. 70(10), 1633–1713 (2007).
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Petite, G.

P. Audebert, Ph. Daguzan, A. Dos Santos, J. C. Gauthier, J. P. Geindre, S. Guizard, G. Hamoniaux, K. Krastev, P. Martin, G. Petite, and A. Antonetti, “Space-time observation of an electron gas in SiO2.,” Phys. Rev. Lett. 73(14), 1990–1993 (1994).
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Polyakov, S.

S. Carrasco, S. Polyakov, H. Kim, L. Jankovic, G. I. Stegeman, J. P. Torres, L. Torner, M. Katz, and D. Eger, “Observation of multiple soliton generation mediated by amplification of asymmetries,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(4), 046616 (2003).
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Prade, B.

L. Sudrie, A. Couairon, M. Franco, B. Lamouroux, B. Prade, S. Tzortzakis, and A. Mysyrowicz, “Femtosecond laser-induced damage and filamentary propagation in fused silica,” Phys. Rev. Lett. 89(18), 186601 (2002).
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X. Liu, L. J. Qian, and F. W. Wise, “Generation of optical spatiotemporal solitons,” Phys. Rev. Lett. 82(23), 4631–4634 (1999).
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Rodriguez, M.

J. Kasparian, M. Rodriguez, G. Méjean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y. B. Andre, A. Mysyrowicz, R. Sauerbrey, J.-P. Wolf, and L. Wöste, “White-light filaments for atmospheric analysis,” Science 301(5629), 61–64 (2003).
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H. S. Eisenberg, R. Morandotti, Y. Silberberg, S. Bar-Ad, D. Ross, and J. S. Aitchison, “Kerr spatiotemporal self-focusing in a planar glass waveguide,” Phys. Rev. Lett. 87(4), 043902 (2001).
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Salmon, E.

J. Kasparian, M. Rodriguez, G. Méjean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y. B. Andre, A. Mysyrowicz, R. Sauerbrey, J.-P. Wolf, and L. Wöste, “White-light filaments for atmospheric analysis,” Science 301(5629), 61–64 (2003).
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J. Kasparian, M. Rodriguez, G. Méjean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y. B. Andre, A. Mysyrowicz, R. Sauerbrey, J.-P. Wolf, and L. Wöste, “White-light filaments for atmospheric analysis,” Science 301(5629), 61–64 (2003).
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V. Y. Fedorov, V. P. Kandidov, O. G. Kosareva, N. Akozbek, M. Scalora, and S. L. Chin, “Filamentation of a femtosecond laser pulse with the initial beam ellipticity,” Laser Phys. 16(8), 1227–1234 (2006).
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S. L. Chin, S. A. Hosseini, W. Liu, Q. Luo, F. Théberge, N. Aközbek, A. Becker, V. P. Kandidov, O. G. Kosareva, and H. Schroeder, “The propagation of powerful femtosecond laser pulses in optical media: physics, applications and new challenges,” Can. J. Phys. 83(9), 863–905 (2005).
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H. S. Eisenberg, R. Morandotti, Y. Silberberg, S. Bar-Ad, D. Ross, and J. S. Aitchison, “Kerr spatiotemporal self-focusing in a planar glass waveguide,” Phys. Rev. Lett. 87(4), 043902 (2001).
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L. BergéS. Skupin, R. Nuter, J. Kasparian, and J.-P. Wolf, “Ultrashort filaments of light in weakly ionized optically transparent media,” Rep. Prog. Phys. 70(10), 1633–1713 (2007).
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L. Sudrie, A. Couairon, M. Franco, B. Lamouroux, B. Prade, S. Tzortzakis, and A. Mysyrowicz, “Femtosecond laser-induced damage and filamentary propagation in fused silica,” Phys. Rev. Lett. 89(18), 186601 (2002).
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S. L. Chin, S. A. Hosseini, W. Liu, Q. Luo, F. Théberge, N. Aközbek, A. Becker, V. P. Kandidov, O. G. Kosareva, and H. Schroeder, “The propagation of powerful femtosecond laser pulses in optical media: physics, applications and new challenges,” Can. J. Phys. 83(9), 863–905 (2005).
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S. Carrasco, S. Polyakov, H. Kim, L. Jankovic, G. I. Stegeman, J. P. Torres, L. Torner, M. Katz, and D. Eger, “Observation of multiple soliton generation mediated by amplification of asymmetries,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(4), 046616 (2003).
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S. Carrasco, S. Polyakov, H. Kim, L. Jankovic, G. I. Stegeman, J. P. Torres, L. Torner, M. Katz, and D. Eger, “Observation of multiple soliton generation mediated by amplification of asymmetries,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(4), 046616 (2003).
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Tzortzakis, S.

L. Sudrie, A. Couairon, M. Franco, B. Lamouroux, B. Prade, S. Tzortzakis, and A. Mysyrowicz, “Femtosecond laser-induced damage and filamentary propagation in fused silica,” Phys. Rev. Lett. 89(18), 186601 (2002).
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D. Majus, V. Jukna, G. Tamošauskas, G. Valiulis, and A. Dubietis, “Three-dimensional mapping of multiple filament arrays,” Phys. Rev. A 81(4), 043811 (2010).
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D. Majus, V. Jukna, G. Valiulis, and A. Dubietis, “Generation of periodic filament arrays by self-focusing of highly elliptical ultrashort pulsed laser beams,” Phys. Rev. A 79(3), 033843 (2009).
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J. W. Grantham, H. M. Gibbs, G. Khitrova, J. F. Valley, and X. Jiajin, “Kaleidoscopic spatial instability: Bifurcations of optical transverse solitary waves,” Phys. Rev. Lett. 66(11), 1422–1425 (1991).
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L. T. Vuong, T. D. Grow, A. Ishaaya, A. L. Gaeta, G. W. ’t Hooft, E. R. Eliel, and G. Fibich, “Collapse of optical vortices,” Phys. Rev. Lett. 96(13), 133901 (2006).
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Wille, H.

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L. BergéS. Skupin, R. Nuter, J. Kasparian, and J.-P. Wolf, “Ultrashort filaments of light in weakly ionized optically transparent media,” Rep. Prog. Phys. 70(10), 1633–1713 (2007).
[CrossRef]

J. Kasparian, M. Rodriguez, G. Méjean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y. B. Andre, A. Mysyrowicz, R. Sauerbrey, J.-P. Wolf, and L. Wöste, “White-light filaments for atmospheric analysis,” Science 301(5629), 61–64 (2003).
[CrossRef] [PubMed]

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J. Kasparian, M. Rodriguez, G. Méjean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y. B. Andre, A. Mysyrowicz, R. Sauerbrey, J.-P. Wolf, and L. Wöste, “White-light filaments for atmospheric analysis,” Science 301(5629), 61–64 (2003).
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Yu, J.

J. Kasparian, M. Rodriguez, G. Méjean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y. B. Andre, A. Mysyrowicz, R. Sauerbrey, J.-P. Wolf, and L. Wöste, “White-light filaments for atmospheric analysis,” Science 301(5629), 61–64 (2003).
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J. Liu, R. Li, and Z. Xu, “Few-cycle spatiotemporal soliton wave excited by filamentation of a femtosecond laser pulse in materials with anomalous dispersion,” Phys. Rev. A 74(4), 043801 (2006).
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[CrossRef]

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L. BergéS. Skupin, R. Nuter, J. Kasparian, and J.-P. Wolf, “Ultrashort filaments of light in weakly ionized optically transparent media,” Rep. Prog. Phys. 70(10), 1633–1713 (2007).
[CrossRef]

Science (2)

J. Kasparian, M. Rodriguez, G. Méjean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y. B. Andre, A. Mysyrowicz, R. Sauerbrey, J.-P. Wolf, and L. Wöste, “White-light filaments for atmospheric analysis,” Science 301(5629), 61–64 (2003).
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Figures (6)

Fig. 1
Fig. 1

Mode profiles of elliptical beams at the output face of a BK7 sample in the anomalous-GVD regime for input pulse energies (powers) of 20 μJ (67 MW), 35 μJ (117 MW), 43 μJ (143 MW), 48 μJ (160 MW), and 50 μJ (167 MW).

Fig. 2
Fig. 2

Mode profiles of elliptical beams at the output face of a BK7 sample in the zero-GVD regime (1320 nm) for different pulse energies.

Fig. 3
Fig. 3

Measured (a) autocorrelation traces and (b) pulse durations (FWHM) in the anomalous-GVD regime (1600 nm). (c) Measured pulse durations in the zero-GVD regime (1320 nm) for different pulse powers.

Fig. 4
Fig. 4

Calculated spatio (major axis)-temporal intensity distributions of collapsing elliptical beams at z = 13 cm in [(a)-(d)] the anomalous-GVD regime (1600 nm) and [(e)-(h)] near zero-GVD regime (1320 nm) for various powers. Time-integrated spatial profiles along the major axis (x) representing MF patterns are shown on top of spatio-temporal profiles. Note that initial pulse durations for 1600 nm (300 fs FWHM) and 1320 nm (110 fs FWHM) are different.

Fig. 5
Fig. 5

Calculated (a) autocorrelation traces and (b) pulse durations (FWHM) of the space-averaged fluence over 1 mm in the anomalous-GVD regime (1600 nm) at z = 13 cm ignoring diffraction along the minor axis. (c) Calculated pulse durations (FWHM) of the space-averaged fluence over 1 mm in the zero-GVD regime (1320 nm) versus input peak powers at z = 13 cm.

Fig. 6
Fig. 6

Calculated (a) autocorrelation traces and (b) pulse durations (FWHM) of the space-averaged fluence over 1 mm in the anomalous-GVD regime (1600 nm) at z = 30 cm ignoring diffraction along the major axis. (c) Calculated pulse durations (FWHM) of the space-averaged fluence over 1 mm in the zero-GVD regime (1320 nm) versus input peak powers at z = 30 cm.

Equations (2)

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u ξ = i 4 2 u μ 2 i L d f 2 L 2 2 u τ 2 + L d f 6 L 3 3 u τ 3 + i L d f L n l | u | 2 u L d f 2 L m p | u | 2 ( K 1 ) u i L d f L p l ( 1 i ω τ c ) ​ ​ ​   η u     ,
η τ = α η | u | 2 + | u | 2 K τ p τ r η ,

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