OSA's Digital Library

Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 26 — Dec. 22, 2008
  • pp: 21529–21543
« Show journal navigation

Self-compression of 2 µm laser filaments

Luc Bergé  »View Author Affiliations


Optics Express, Vol. 16, Issue 26, pp. 21529-21543 (2008)
http://dx.doi.org/10.1364/OE.16.021529


View Full Text Article

Acrobat PDF (384 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

We numerically study the filamentation of ultrashort laser pulses at 2 µm carrier wavelength in noble gases (argon, xenon) and in air. Compared with filamentation in the near-visible domain (800 nm), mid-infrared optical sources with durations close to a single cycle can be generically produced at various pressures and powers near the self-focusing threshold. The mechanism by which self-compression takes place mainly involves optical self-focusing, pulse steepening and plasma defocusing. On-axis spectra and spectral phases are discussed. Delivering single-cycled pulses at long wavelengths has important applications in the generation of high-order harmonics and isolated attosecond pulses.

© 2008 Optical Society of America

1. Introduction

Laser filaments in gases result from the competition between Kerr self-focusing and defocusing by a laser-induced electron plasma [1

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]

]. Besides their promising applications for remote sensing [2

2. 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, 863–905 (2005). [CrossRef]

, 3

3. A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Phys. Rep. 44, 47–189 (2007). [CrossRef]

, 4

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

, 5

5. J. Kasparian and J.-P. Wolf, “Physics and applications of atmospheric nonlinear optics and filamentation,” Opt. Express 16, 466–493 (2008). [CrossRef] [PubMed]

], ultrashort pulses with input powers close to the self-focusing threshold, P cr, produce a single filament whose spectrum broadens and duration can significantly shrink along the propagation axis. Many studies have been devoted to this remarkable property, in order to generate few-cycle optical sources [6

6. S. Champeaux and L. Bergé, “Femtosecond pulse compression in pressure-gas cells filled with argon,” Phys. Rev. E 68, 066603 (2003). [CrossRef]

, 7

7. 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, 673–677 (2004). [CrossRef]

, 8

8. 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]

, 9

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

, 10

10. A. Zaïr, 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]

]. Recently, a couple of papers [11

11. O. G. Kosareva, N. A. Panov, D. S. Uryupina, M. V. Kurilova, A. V. Mazhorova, A. B. Savel’ev, R. V. Volkov, V. P. Kandidov, and S. L. Chin, “Optimization of a femtosecond pulse self-compression region along a filament in air,” Appl. Phys. B 91, 35–43 (2008). [CrossRef]

, 12

12. L. Bergé and S. Skupin, “Few-cycle light bullets created by femtosecond filaments,” Phys. Rev. Lett. 100, 113902 (2008). [CrossRef] [PubMed]

] identified that, at 800 nm wavelength, maximum shortening in time happens when the filament diffracts beyond the self-focus point, after the shortest wavelength has been attained in the supercontinuum. Compression efficiency varies according to the propagation medium. From the experimental point of view, 800 nm pulses with about 5 fs full-width at half-maximum (FWHM) durations have been produced in argon at pressures close to atmospheric, with subsequent chirped mirror compression and spatial selection of the beam core [10

10. A. Zaïr, 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]

]. Without any post-compression stage, ouput filaments with 8 fs durations have been measured from a gas cell filled with krypton [8

8. 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]

]. From the numerical point of view, FWHM durations comprised between 5 and 7 fs have been obtained by propagating pulses in atom gases with moderate ionization potentials (Ui≤16 eV) at uniform pressure. As the compression rate increases with the saturation intensity fixed by the ionization threshold, filaments of 3–4 fs have been simulated in a neon gas, whose ionisation potential exceeds 21 eV [13

13. A. Couairon, H. S. Chakraborty, and M. B. Gaarde, “From single-cycle self-compressed filaments to isolated attosecond pulses in noble gases,” Phys. Rev. A 77, 053814 (2008). [CrossRef]

]. Sub-cycle (<2 fs) pulses were even expected by self-channeling 5 fs pulses centered at 750 nm in helium (Ui=24.6 eV) [14

14. E. Goulielmakis, S. Koehler, B. Reiter, M. Schultze, A. J. Verheof, E. E. Serebryannikov, A. M. Zheltikov, and F. Krausz, “Ultrabroadband, coherent light source based on self-channeling of few-cycle pulses in helium,” Opt. Lett. 33, 1407–1409 (2008). [CrossRef] [PubMed]

]. At smaller wavelengths, ultraviolet pulses (266 nm) compressed through the four-wave mixing of a 400 nm pump filament with a 800 nm idler beam have been shown to keep durations as short as 1.5 fs over 30 cm propagation ranges [15

15. L. Bergé and S. Skupin, “Sub-2 fs pulses generated by self-channeling in the deep ultraviolet,” Opt. Lett. 33, 750–752 (2008). [CrossRef] [PubMed]

].

The possibility to generate intense few-cycle pulses nowadays opens new avenues in strong field physics and its applications. Among those, high-order harmonic generation (HHG) rises as the key process giving access to the production of single attosecond pulses. HHG is usually achieved by focusing an intense driving pulse into a noble gas. Initiated by optical field ionization, HHG results from the coherent frequency conversion of the pump wave into numerous harmonics extending into the XUV spectrum. Whereas many-cycle pump pulses yield attosecond pulse trains [16

16. P. M. Paul, E. S. Toma, P. Breger, G. Mullot, F. Auge, Ph. Balcou, H. G. Muller, and P. Agostini, “Observation of a train of attosecond pulses from high harmonic generation,” Science 292, 1689–1692 (2001). [CrossRef] [PubMed]

], few-cycle pulses with stable carrier-envelope phase allow to sort out isolated attosecond wave-packets [17

17. G. Sansone, E. Benedetti, F. Galegari, C. Vozzi, L. Avaldi, R. Flammini, L. Poletto, P. Villoresi, C. Altucci, R. Velotta, S. Stagira, S. De Silvestri, and M. Nisoli, “Isolated single-cycle attosecond pulses,” Science 314, 443–446 (2008). [CrossRef]

]. Few-cycle pulses have often been produced by exploiting self-phase modulation in hollow fibers filled with noble gases [18

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

]. The alternative procedure employing filamentation, however, avoids the delicate coupling between an intense pulse and a long waveguide, and it currently preserves the carrier-envelope phase [19

19. D. Faccio, A. Lotti, M. Kolesik, J. V. Moloney, S. Tzortzakis, A. Couairon, and P. Di Trapani, “Spontaneous emergence of pulses with constant carrier-envelope phase in femtosecond filamentation,” Opt. Express 16, 11103–11114 (2008). [CrossRef] [PubMed]

].

So far, most of the HHG experiments have been performed using titanium:sapphire (Ti:Sa) sources operating at near-visible wavelengths (~800 nm). Nonetheless, scaling laser-atom interactions towards longer wavelengths should yield more energetic particles and shorter bursts of attosecond light [20

20. B. Sheehy, J. D. D. Martin, L. F. DiMauro, P. Agostini, K. J. Schafer, M. B. Gaarde, and K. C. Kulander, “High harmonic generation at long wavelengths,” Phys. Rev. Lett. 83, 5270–5273 (1999). [CrossRef]

, 21

21. B. Shan and Z. Chang, “Dramatic extension of the high-order harmonic cutoff by using a long-wavelength driving field,” Phys. Rev. A 65, 011804(R) (2001). [CrossRef]

, 22

22. K. D. Schultz, C. I. Blaga, R. Chirla, P. Colosimo, J. Cryan, A. M. March, C. Roedig, E. Sistrunk, J. tate, J. Wheeler, P. Agostini, and L. F. DiMauro, “Strong field physics with long wavelength lasers,” J. Mod. Opt. 54, 1075–1085 (2007). [CrossRef]

, 23

23. J. Tate, T. Auguste, H. G. Muller, P. Salières, P. Agostini, and L. F. DiMauro, “Scaling of wave-packet dynamics in an intense midinfrared field,” Phys. Rev. Lett. 98, 013901 (2007). [CrossRef] [PubMed]

]. First, the cutoff photon energy of HHG is described by E max=Ui+3.17Up, Up being the ponderomotive energy varying like I/ω 2 0 where I and ω0 are the laser intensity and central frequency, respectively. This cutoff is thus proportional to the square of the wavelength and bounds a wider harmonic spectrum at long wavelengths. Second, the maximum energy gained by electrons when they return to the parent ion is 10 Up. So, at constant intensity, longer wavelengths generate more energetic particles, opening the route to the production of multi-keV electrons. Third, depending on the electron trajectories, attosecond bursts carry a chirp (attochirp), which is inversely proportional to the laser wavelength. Longer wavelengths reduce this chirp, which increases coherence of the harmonics and the phase matching with the pump wave. Driving laser-atom interactions at mid-infrared wavelengths should thus help in generating energetic and extremely short x-ray pulses [23

23. J. Tate, T. Auguste, H. G. Muller, P. Salières, P. Agostini, and L. F. DiMauro, “Scaling of wave-packet dynamics in an intense midinfrared field,” Phys. Rev. Lett. 98, 013901 (2007). [CrossRef] [PubMed]

].

For this purpose, laser sources operating in the mid-IR are now available. Femtosecond optical parametric amplifiers (OPAs) pumped by multi-mJ Ti:Sa sources deliver multicycle pulses in the spectral range 1.3–2 µm with sufficient peak power to trigger the filamentation process. Hauri et al. [24

24. C. P. Hauri, R. B. Lopez-Martens, C. I. Blaga, K. D. Schultz, J. Cryan, R. Chirla, P. Colosimo, G. Doumy, A. M. March, C. Roedig, E. Sistrunk, J. Tate, J. Wheeler, L. F. DiMauro, and E. P. Power, “Intense self-compressed, self-phase-stabilized few-cycle pulses at 2µm from an optical filament,” Opt. Lett. 32, 868–870 (2007). [CrossRef] [PubMed]

] recently examined long-wavelength filamentation by using phase-stabilized 330 µJ, 55 fs pulses at 2 µm, produced by difference frequency generation in a Ti:Sa-pumped OPA. Pulses launched in loosely focused geometry into a xenon cell at 2 bar pressure under-went self-channeling and they shrank in time down to 3 optical cycles (17 fs) while keeping 270 µJ energy. Similar durations with higher pulse energy have also been reported at ~1.4 µm wavelengths by Vozzi et al. [25

25. C. Vozzi, C. Manzoni, F. Calegari, E. Benedetti, G. Sansone, G. Cerullo, M. Nisoli, S. De Silvestri, and S. Stagira, “Characterization of a high-energy self-phase-stabilized near-infrared parametric source,” J. Opt. Soc. Am. B 25, 112–117 (2008). [CrossRef]

], who applied difference frequency generation to a filament broad supercontinuum boosted by a two-stage OPA. Generation of 13 fs pulses centered at 3 µm wavelength was demonstrated by Fuji and Suzuki [26

26. T. Fuji and T. Suzuki, “Generation of sub-two-cycle mid-infrared pulses by four-wave mixing through filamentation in air,” Opt. Lett. 32, 3330–3332 (2007). [CrossRef] [PubMed]

], who exploited four-wave mixing through filamentation in air. By means of more classical compression techniques, sub-two-cycle (8.5 fs) pulses have been produced at 1.6 µm wavelength from a white-light seeded, 800 nm pumped degenerate OPA, using a deformable mirror compressor [27

27. D. Brida, G. Cirmi, C. Manzoni, S. Bonora, P. Villoresi, S. De Silvestri, and G. Cerullo, “Sub-two-cycle light pulses at 1.6 µm from an optical parametric amplifier,” Opt. Lett. 33, 741–743 (2008). [CrossRef] [PubMed]

].

In this paper, we numerically investigate the nonlinear dynamics of femtosecond filaments formed in different gases (argon, xenon and air) at the 2 µm laser wavelength. Emphasis will be given to filamentation regimes leading to robust self-compressed pulses. By “robust”, we signify that maximum compression rates will be preserved by singly-peaked filaments over distances exceeding far their Rayleigh length. Self-compression will be discussed only in terms of on-axis temporal distributions reaching the maximum, plasma-mediated intensity. Indeed, FWHM pulse durations averaged through a pinhole with diameter comparable with the filament core size routinely remain close to the duration of the on-axis pulse profile [9

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

, 11

11. O. G. Kosareva, N. A. Panov, D. S. Uryupina, M. V. Kurilova, A. V. Mazhorova, A. B. Savel’ev, R. V. Volkov, V. P. Kandidov, and S. L. Chin, “Optimization of a femtosecond pulse self-compression region along a filament in air,” Appl. Phys. B 91, 35–43 (2008). [CrossRef]

]. Self-compressed filaments created in argon, xenon and air at 800 nm are compared for different pressures with their counterparts at 2 µm. We show that, for interaction media with moderate ionization potentials, mid-IR filaments easily reach the single-cyle limit, unlike near-visible pulses whose durations approach 2–3 optical cycles. In connection, an impressive spectral broadening takes place owing to self-phase modulation and to a sustained action of pulse steepening. The major result is that pulse self-compression monitored by filamentation works at long wavelengths and can achieve pulse durations much less than those reported so far in the mid-infrared.

2. Model equations

The propagation model is elaborated on the nonlinear envelope equation earlier derived by Brabec and Krausz [28

28. T. Brabec and F. Krausz, “Nonlinear optical pulse propagation in the single-cycle regime,” Phys. Rev. Lett. 78, 3282–3285 (1997). [CrossRef]

]. This model governs the complex envelope of the laser electric field E=Ueik0ziω0t+c.c. around a central frequency ω 0=2πc/λ 0. A new time variable t→t-kz is utilized to replace the pulse into the frame moving with the group velocity k ′-1=(∂k/∂ω|ω=ω0)-1. The equation for the forward pump envelope U then expands as [4

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

]

zU=i2k0T12U+iDU+iω0cn2T×[(1xK)U2+xKth(tt)U(t)2dt]U
ik02n02ρcT1ρUσ2ρU(ρntρ)UiW(I)2IU,
(1)

h(t)=τ12+τ22τ1τ22etτ2sin(tτ1),
(2)

in the ratio xK=1/2. τ 1=62.5 fs is the inverse of the fundamental rotational frequency and τ 2=76.9 fs denotes the dipole dephasing time.

Besides, assuming electrons born at rest, the growth of the electron density, ρ, is only governed by external source terms,

tρ=W(I)(ρntρ)+σUiρI,
(3)

that include photo-ionization processes with rate W(I) and collisional ionization with cross-section σ=q2e/[meε0n0e(1+ω 2 0/ν2e)]. Here, ρnt is the density of neutral species; qe, me and νe are the electron charge, mass and collision time, respectively. Electron recombination in gases is efficient over ns time scales, and therefore neglected. In Eq. (3), the rate for photoionization W(I) follows from Perelomov, Popov and Terent’ev (PPT)’s theory [29

29. 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).

] (see also [30

30. R. Nuter and L. Bergé, “Pulse chirping and ionization of O2 molecules for the filamentation of femtosecond laser pulses in air,” J. Opt. Soc. Am. B 23, 874–884 (2006). [CrossRef]

]) yielding

W(Ep)=42πCn*,l*2(2E0Ep1+γ2)2n*32mf(l,m)m!e2ν[sinh1(γ)γ1+γ21+2γ2]
×Uiγ21+γ2κν0+eα(κν)Φm(β(κν)),
(4)

where, expressed in atomic units, Ep~√I denotes the peak optical amplitude, E0=(2Ui)3/2,γ=ω02Ui/Ep,ν=Ui/h¯ω0,β=2γ/1+γ2,α=2[sinh1(γ)γ/1+γ2],v0=<v+1> and Φm(x)=ex20x(x2y2)|m|ey2dy.n*=Z/2Ui is the effective quantum number, Z the residual ion charge, l*=n*-1 and n,l,m are the principal quantum number, the orbital momentum and the magnetic quantum number, respectively. The pre-exponential factors

Cn*,l*2=22n*n*Γ(n*+l*+1)Γ(n*l*),f(l,m)=(2l+1)(l+m)!2mm!(lm)!
(5)

are extracted from the tunneling theory derived by Ammosov, Delone and Krainov [31

31. M. V. Ammosov, N. B. Delone, and V. P. Krainov, “Tunnel ionization of complex atoms and of atomic ions in an alternating electromagnetic field,” Sov. Phys. JETP 64, 1191–1194 (1986).

]. This model reproduces with high fidelity (5% relative variations) the single ionization of Ar atoms by 800 nm driving pulses, compared with numerical solutions of the time-dependent Schrödinger equation [32

32. R. Wiehle, B. Witzel, H. Helm, and E. Cormier, “Dynamics of strong-field above-threshold ionization of argon: Comparison between experiment and theory,” Phys. Rev. A 67, 063405 (2003). [CrossRef]

, 33

33. E. Cormier, P.-A. Hervieux, R. Wiehle, B. Witzel, and H. Helm, “ATI of complex systems: Ar and C60,” Eur. Phys. J. D. 26, 83–90 (2003). [CrossRef]

]. The Keldysh parameter γ separates the tunneling (γ≪1) and multiphoton (γ≫1) regimes and it increases with ω0 at constant intensity. Longer wavelengths should thus drive the interaction preferably in the tunnel regime, closer to the semiclassical description of the three-step model [23

23. J. Tate, T. Auguste, H. G. Muller, P. Salières, P. Agostini, and L. F. DiMauro, “Scaling of wave-packet dynamics in an intense midinfrared field,” Phys. Rev. Lett. 98, 013901 (2007). [CrossRef] [PubMed]

].

Fig. 1. Ionization rates plotted from the PPT formula Eq. (4) with ion charge Z=1 and angular momentum equal to 1 for atom gases and 0 for air [30] at central wavelengths (a) 800 nm and (b) 2 µm. The blue curves refer to ionization in argon; the red curves correspond to xenon, and the green curves show the ionization rate in air for O2 molecules.

The propagation model is constituted by Eq. (1) and Eq. (3). It will be integrated numerically by using initially collimated Gaussian pulses,

U(r,z=0,t)=2Pinπw02er2w02t2tp2,
(6)

with input power P in, beam waist w 0 and 1/e2 pulse half-width tp. In linear regime, Gaussian pulses are expected to diffract over the Rayleigh distance z 0=π n0w20 0. The values of the physical parameters used in Eq. (1), including the GVD coefficient k″, are specified in Table 1 for the atmospheric pressure p=1 bar. Dispersion curves follow Dalgarno and Kingston’s refractive indices [34

34. A. Dalgarno and A. E. Kingston, “The refractive indices and Verdet constants of the inert gases,” Proc. Royal Soc. London A 259, 424–429 (1960). [CrossRef]

] for argon and xenon, and that of Peck and Reeder [35

35. E. R. Peck and K. Reeder, “Dispersion of Air,” J. Opt. Soc. Am. 62, 958–962 (1972). [CrossRef]

] for air. Nonlinear susceptibilities χω0(3) allowing to evaluate n 2 at different wavelengths are taken from Shelton’s Ref. 36. The selected Kerr index for air lies in orders of magnitude consistent with those of [37

37. W. Liu and S. L. Chin, “Direct measurement of the critical power of femtosecond Ti:sapphire laser pulse in air,” Opt. Express 13, 5750–5755 (2005). [CrossRef] [PubMed]

, 38

38. S. Skupin and L. Bergé, “Supercontinuum generation of ulrashort pulses in air at different central wavelengths,” Opt. Commun. 280, 173–182 (2007). [CrossRef]

, 39

39. S. Champeaux, L. Bergé, D. Gordon, A. Ting, J. Peñano, and P. Sprangle, “(3+1)-dimensional numerical simulations of femtosecond laser filaments in air: Toward a quantitative agreement with experiments,” Phys. Rev. E 77, 036406 (2008). [CrossRef]

]. Ar and Xe atoms have the ionization potentials Ui=16 eV and Ui=12.1 eV, respectively. Because the ionization potential of N 2 is higher than that of O2 molecules, we only consider the latter specy as generating an electron plasma with again Ui=12.1 eV. All ionization models used in the present work are illustrated in Fig. 1. Although different rates have been proposed for describing air ionization [40

40. A. Talebpour, J. Yang, and S. L. Chin, “Semi-empirical model for the rate of tunnel ionization of N2 and O2 molecule in an intense Ti:sapphire laser pulse,” Opt. Commun. 163, 29–32 (1999). [CrossRef]

], these currently differ from each other by less than 2 decades, which should not significantly modify the saturation intensity in the filaments at large photon numbers Kν 0=<Ui/h̄ω 0+1>. For the sake of clarity, all curves referring to argon are plotted in blue; those corresponding to xenon are in red, while curves describing air are plotted in green.

To understand the propagation dynamics at different carrier wavelengths, we find it convenient to fix the same ratio of input power over critical, whatever the local pressure may be. In

Table 1. Physical parameters for Ar, Xe and air at 800 nm and 2 µm for the pressure p=1 bar. ρnt is equal to 2.7×1019 cm-3 for the noble gases, and to 5.4×1018 cm-3 for the 20% dioxygen molecules of air. The critical plasma density takes the value ρcr≃1.11×1021/λ20[µm]. The last line contains estimations of Imax computed from Eq. (7).

table-icon
View This Table

all cases we choose input peak powers close to critical, i. e., P crP in≤3P cr for atom gases (P crλ 2 0/2πn0n 2). In air, increasing P in up to 5P cr will overcome the reduction of the Kerr nonlinearity due to Raman scattering. The input beam waist is w 0=500µm and the initial FWHM pulse extent is 2ln2tp=30. Numerical snapshots of temporal field distributions are collected every 10 cm. All numerical simulations have been performed in radial symmetry with 6144×6144 points in (r=x2+y2,t) with respective resolutions of 2.44 µm and 0.25 fs, for an adaptive step along z. We cross-checked that the dynamics remained similar when the pulses were simulated from a more complete model, such as the frequency-dependent unidirectional propagation equation describing the forward spectral amplitude of the total field (see [4

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

] and references therein). When neglecting third harmonic generation in noble gases [43

43. N. Aközbek, S. A. Trushin, A. Baltuška, W. Fuss, E. Goulielmakis, K. Kosma, F. Krausz, S. Panja, M. Uiberacker, W. E. Schmid, A. Becker, M. Scalora, and M. Bloemer, “Extending the supercontinuum spectrum down to 200 nm with few-cycle pulses,” New J. Phys. 8, 177 (2006). [CrossRef]

], the propagation pattern was found unchanged, yielding identical compression rates. When taking third-harmonic generation into account, as expected in air [4

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

], the coupling between the pump and the harmonic wave creates oscillations in the temporal structure. However, the maximum intensity remains of same magnitude and the supercontinuum extends over comparable band-with. Minimum durations are analogous to those given by the envelope model [Eq. (1)] and they are attained at comparable propagation distances.

From Table 1, it is clear that dispersion and collisional ionization are weak. Self-channeling then mainly relies on the dynamic balance between Kerr self-focusing and plasma defocusing, so that estimates for the filament intensities (I fil), electron densities (ρ fil) and mean filament waist (w fil) can be deduced from equating diffraction, Kerr and ionization responses in Eq. (1). This yields the simple relations [4

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

]

Ifilρfil2ρcn0n̅2,ρfilΔTρntW(Ifil),Wfil2π(2k02n̅2Ifiln0)12,
(7)

where

n̅2=n2(1xK)+n2xKmaxtth(tt)e2t2tp2dt
(8)

The magnitude of the saturation intensity directly impacts spectral broadening, which is initiated by self-phase modulation (SPM). SPM creates a strong supercontinuum as the intensity increases by self-focusing. In the limits T,T -1→1, frequency variations evolve like

Δωk0Δzt(n̅2Iρ2n0ρc),
(9)

over the longitudinal path Δz and they vary with the superimposed actions of the Kerr and plasma responses. When accounting for steepening terms (T,T -1≠1), shock edges occur in the back of the pulse [41

41. D. Anderson and M. Lisak, “Nonlinear asymmetric self-phase modulation and self-steepening of pulses in long optical waveguides,” Phys. Rev. A 27, 1393–1398 (1983). [CrossRef]

] and they usually blueshift the spectrum [4

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

, 42

42. N. Aközbek, M. Scalora, C. M. Bowden, and S. L. Chin, “White-light continuum generation and filamentation during the propagation of ultra-short laser pulses in air,” Opt. Commun. 191, 353–362 (2001). [CrossRef]

, 43

43. N. Aközbek, S. A. Trushin, A. Baltuška, W. Fuss, E. Goulielmakis, K. Kosma, F. Krausz, S. Panja, M. Uiberacker, W. E. Schmid, A. Becker, M. Scalora, and M. Bloemer, “Extending the supercontinuum spectrum down to 200 nm with few-cycle pulses,” New J. Phys. 8, 177 (2006). [CrossRef]

].

3. Filament compressors at 800 nm

Below we briefly recall scaling of the quantities in Eq. (7) when the pressure differs from 1 bar. In that case, the Kerr index n 2, the dispersion operator 𝓓, the neutral density ρ nt and the cross section for avalanche ionization σ varies linearly with the pressure p, whereas the ionization rate W(I) is pressure-independent [44

44. S. Champeaux and L. Bergé, “Long-range multifilamentation of femtosecond laser pulses versus air pressure,” Opt. Lett. 31, 1301–1303 (2006). [CrossRef] [PubMed]

]. Using the basic ordering Eq. (7), we deduce that I fil keeps a constant value, as confirmed by a recent experiment [45

45. J. Bernhardt, W. Liu, S. L. Chin, and R. Sauerbrey, “Pressure independence of intensity clamping during filamentation: theory and experiment,” Appl. Phys. B 91, 45–48 (2008). [CrossRef]

]. The filament waist w fil behaves like 1/√p while the peak electron density ρ fil evolves like p. If we conjecture that the self-channeling length, Δz fil, is driven by the Rayleigh length associated with the filament, zR filπn0w2fil 0, then the self-guiding range should decrease inversely proportional to the pressure.

Figure 2 summarizes macroscopic aspects of filamentation at 800 nm in Ar, Xe for the input power P in=3P cr, and in air for P in=5P cr [note that 5P cr(n 2)≃3.3P cr(n̄2) in the atmosphere]. The saturation intensity reached in argon at this wavelength lies in the range of 100 TW/cm2 for p=0.5,1 and 2 bar, in fair agreement with the value reported in Table 1. Of course, this evaluation does not fit the maximum peak intensity reached by the pulse, as Eq. (7) just provides a crude estimate of the mean intensity level from which plasma generation can arrest beam collapse. Since we are operating at constant ratio P in/P cr, the self-focus point is located at analogous distances, zc~0.4 m, whatever the pressure and propagation medium may be [Figs. 2(a)–2(b)]. Slight deviations originate from the action of GVD, which increases with the pressure and retards the beam collapse to some extent. The filamentation range, evaluated at half of the saturation intensity, increases by a factor of ~2 when the pressure is decreased by the same factor, as expected above. A longer self-guiding takes place in air, resulting from moderate intensity clamping (n̄2 I air fil~0.1n 2 I Xe fil), small dispersion compared with Xe, and refocusing of time slices in the back of the pulse owing to the non-local Raman response. Figure 2(c) shows peak electron densities in Ar and Xe. Their values do increase linearly with p. At decreasing pressures, the filament waist grows like 1/√p. Mean values of w fil [Eq. (7)] are ~100 µm for Ar, ~60 µm for Xe and ~150 µm for air at p=1 bar, which supports the comparison with Fig. 2(d). Note that n̄2 I fil reaches higher values in xenon, which explains the smaller filament sizes compared with, e. g., argon at the same pressure. In every case, the self-guiding range covers several times the Rayleigh distance of the filament, i. e., Δz fil~5×zR fil.

Figure 3 illustrates evolution patterns in the (t, z) plane. All pulses exhibit similar distortions. The pulse starts to self-focus then triggers a plasma sequence, which defocuses the back zone (z~0.4 m). At later distances, the trailing pulse refocuses, which forms a double-peaked distribution in the temporal profile (z~0.5 m in noble gases). The trailing edge self-steepens and develops a shock dynamics. Its intensity reaches the highest peak value, which triggers in turn the highest plasma density. Consequently, the trailing edge of the pulse is rapidly depleted by plasma generation, whereas the leading edge, coupled to weaker density levels, continues to propagate within a slow diffraction stage. Ultimately, the pulse shape relaxes to a very short peak in the front zone, where the smallest durations are attained (z≥0.6 m in Ar and Xe). These have been indicated in Fig. 3. In Ar at p=1 bar, the pulse time extent goes down to 6.5 fs from z=0.8 m, and it never exceeds 7 fs in the range 0.6≤z≤1.2 m. Identical durations have been measured over longer ranges at lower pressure (0.5 bar), and over shorter ranges at higher pressure (2 bar). The same property applies to xenon and air. In Xe at 1 bar pressure, the pulse does not shrink below 7.5 fs and maintains maximum compression upon 20 cm only. In contrast, diminishing the pressure down to 0.5 bar favors shorter durations over 40 cm. In air, the pulse profile develops several peaks in the range 0.5≤z<1 m, which we attribute to the asymmetry in time introduced by the Raman response. FWHM durations of 7.5 fs, reached from z=1 m, are preserved up to the distance z=1.4 m.

Fig. 2. Peak intensities at 800 nm (a) in Ar for p=1 bar (solid curve), p=0.5 bar (dashed curve) and p=2 bar (dash-dotted curve); (b) in Xe at 2 bar (dotted curve), 1 bar (dash-dotted curve), 0.5 bar (dashed curve), and in air at 1 bar pressure (solid curve). Power ratios are specified in the text. (c) Peak electron densities in the zoomed filamentation range 0.3≤z≤0.55 m for Ar at 1 bar (solid curve) and in Xe at 0.5 bar (dashed curve), 1 bar (dash-dotted curve) and 2 bar (dotted curve). (d) Filament waist for Ar at 1 bar (blue solid curve), Xe at 0.5 bar (dashed curve), 1 bar (dash-dotted curve) and 2 bar (dotted curve) in the same range. The green solid curve shows wfil for air at p=1 bar.

Figure 4(a) shows the shortest temporal profiles formed at maximum compression in Figs. 3(a) and 3(d). Pulses are singly-peaked. For comparison, the dash-dotted curve represents the latter pulse configuration obtained at the same propagation distance when the operator T is set equal to unity. Two quasi-symmetric peaks emerge, as predicted by early self-channeling scenarios in gases [46

46. M. Mlejnek, E. M. Wright, and J. V. Moloney, “Dynamic spatial replenishment of femtosecond pulses propagating in air,” Opt. Lett. 23, 382–384 (1998). [CrossRef]

]. These peaks have comparable intensity and they run over more than 20 cm along the filament path. Each of the spikes experiences an individual compression down to ~3 fs. However, both design a profile with overall FWHM duration of 36 fs. Pulse steepening, instead, enhances the revival of the Kerr response. It makes the trail pulse rapidly refocus and decay, so that a unique leading spike survives and slowly diffracts in the front zone. It is worth recalling that antecedent works reported shortening of the pulse in the trailing region [12

12. L. Bergé and S. Skupin, “Few-cycle light bullets created by femtosecond filaments,” Phys. Rev. Lett. 100, 113902 (2008). [CrossRef] [PubMed]

]. In collimated geometry this dynamics is, however, generic if the peak power is sufficiently close to critical and avoids pulse splitting in time, which is not the case here. When pulse splitting occurs, no general rule allows us to guess a priori that compression will systematically take place in the rear pulse. Let us remind in this regards that when the pulse splits up through, e. g., normal GVD, self-steepening amplifies the trailing pulse near the self-focus point at powers close to critical. However, the leading pulse may become the dominant component at higher powers beyond the self-focus point [47

47. J. K. Ranka and A. L. Gaeta, “Breakdown of the slowly varying envelope approximation in the self-focusing of ultrashort pulses,” Opt. Lett. 23, 534–536 (1998). [CrossRef]

].

Fig. 3. Temporal evolution of 800 nm Gaussian pulses in the (t,z) plane for (a) argon and (b) air at 1 bar pressure, and for xenon at (c) 1 bar pressure and (d) 0.5 bar pressure.

4. Filament compressors at 2 µm

Following Ref. [38

38. S. Skupin and L. Bergé, “Supercontinuum generation of ulrashort pulses in air at different central wavelengths,” Opt. Commun. 280, 173–182 (2007). [CrossRef]

], we briefly sketch the basic changes expected in the physics of femtosecond filaments at large wavelengths, namely,

• The supercontinuum increases with the carrier wavelength λ 0.

• At comparable ratios P in/P cr, infrared filaments are more energetic than near-visible ones. They survive along longer distances and have larger beam waist ~λ0.

• The influence of pulse steepening increases with λ 0.

Fig. 4. (a) On-axis temporal profiles and (b) on-axis spectra (left axis) with spectral phases (right axis) for argon at 1 bar pressure, z=0.9 m (solid curve, FWHM=6.5 fs) and xenon at 0.5 bar pressure, z=0.6 m (dashed curve, FWHM=7 fs). The dashed-dotted line in Fig. 4(a) illustrates the latter case when the self-steepening operators are artificially suppressed (T=T -1=1). The dotted curve of Fig. 4(b) refers to the spectrum in argon at z=0.

Figure 5 illustrates some examples of light self-guiding in argon, xenon and air at 2 µm wavelength. In Fig. 5(a), filamentation starts at 1 critical power in Ar for p=1 bar (dash-dotted curve). However, the pulse intensity reaches saturation thresholds in the magnitude of those theoretically expected (~100 TW/cm2) only at higher peak power (solid curve), or at lower pressure (dashed curve). At atmospheric pressure, power ratios strictly above unity are indeed required to realize a collapse dynamics triggering maximum electron densities. With P in=1P cr, the normalized GVD coefficient δ≡2z 0 k /t 2 p, although weak in argon, is still large enough to inhibit self-focusing at moderate pressures p≥1 bar. This limitation is overcome at 0.5 bar pressure [48

48. G. G. Luther, J. V. Moloney, A. C. Newell, and E. M. Wright, “Self-focusing threshold in normally dispersive media,” Opt. Lett. 19, 862–864 (1994). [CrossRef] [PubMed]

]. When the pulse self-focuses until triggering a plasma sequence, the scaling with respect to the pressure p still holds. The location of zc is practically unchanged at equal power ratios; the magnitude of I fil remains pressure-independent [Fig. 5(b)]; plasma densities follow pressure variations [Fig. 5(c)], and the filamentation ranges and waists increase when p decreases [Fig. 5(d)]. As expected in Sec. 2, filament intensities are comparable at near-visible and mid-IR wavelengths, although the peak intensity decreases to some extent at 2 µm in air. The most significant changes reported in Fig. 5 concern the filamentation range and waist. Engaging smaller power ratios than in Fig. 2, filament channels extend upon equal or longer distances. This can be explained by the dependency of Δz fil over λ 0 and the much larger values of the critical powers (see Table 1), rendering 2 µm pulses more energetic even at weaker ratios P in/P cr. On the other hand, the filament waist increases up to 290 µm in Ar, 180 µm in Xe, and 600 µm in air, as computed from Eq. (7) at atmospheric pressure. These estimates are again in rather good agreement with the filament sizes plotted in Fig. 5(d). Increase in the filament waist also follows from the direct proportionality between w fil and k -1 0. On the whole, the self-guiding ranges cover at least two filament Rayleigh lengths zR fil. The ratio between the input (500 µm) and filament waists differs in the computations done at 800 nm and 2 µm, which directly modifies the evolution pattern of the pulses.

Fig. 5. Peak intensities at 2 µm (a) in Ar for p=1 bar (dash-dotted curve), p=0.5 bar (dashed curve) with P in=1P cr, and for p=1 bar with P in=1.2P cr (solid curve); (b) in Xe at 1 bar (dash-dotted curve) and 0.5 bar (dashed curve) for P in=1.3P cr, and in air at 1 bar (solid curve) and 2 bar (dotted curve) for P in=2.2P cr. (c) Peak electron densities in Ar and air at different pressures with the same plotstyles as in Figs. 5(a)–5(b). (d) Filament waists for the previous cases. The red dash-dotted curve shows the filament waist in Xe at 1 bar.
Fig. 6. Temporal evolution of 2 µm Gaussian pulses in the (t,z) plane for (a) Xe at p=0.5 bar and P in=1.3P cr; (b) air at p=1 bar and P in=2.2P cr; (c) Ar at 1 bar pressure and P in=1.2P cr, and (d) Ar at 0.5 bar pressure and P in=1P cr.

The on-axis temporal profiles of the last two configurations of Fig. 6 have been detailed in Fig. 7(a). We can observe how steep the trailing edge of the pulses becomes in the vicinity of the self-focus point. Pulse shaping results from the self-steepening operator T=1+(i/ω0)∂t, whose impact increases at long wavelengths and sharpens temporal gradients in the filament [41

41. D. Anderson and M. Lisak, “Nonlinear asymmetric self-phase modulation and self-steepening of pulses in long optical waveguides,” Phys. Rev. A 27, 1393–1398 (1983). [CrossRef]

]. This statement can already be seen from the dash-dotted curve representing the intensity profile formed at the same distance in 1 bar argon without steepening terms. Steepening effects start along the early self-focusing stage. They push the pulse towards the rear zone and are responsible for shock formation on the trailing pulse. Their influence increases with λ0 and I fil, which contributes to shorten the field distribution in time. In this regards, Fig. 7(b) shows the on-axis spectra relative to argon and xenon. SPM broadens the initial spectrum and pulse steepening enhances asymmetry to large frequencies [4

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

]. Unlike the 800 nm pulses depicted in Fig. 4(b), the spectral phases associated to such extreme self-compressions are remarkably flat in the entire bandwidth around the central frequency.

Fig. 7. (a) On-axis temporal profiles for argon at 1 bar pressure, z=0.6 m with (solid curve, FWHM=6.5 fs) and without pulse steepening (dash-dotted curve, FWHM=7.5 fs), and at 0.5 bar pressure, z=0.8 m (dashed curve, FWHM=5.5 fs). (b) On-axis spectra (left axis) with spectral phases (right axis) in 1 bar argon and in the xenon case (dashed curves) shown in Fig. 6(a). The dotted curve represents the spectrum in argon at z=0.
Fig. 8. On-axis spectra of Gaussian pulses at distance of maximal compression (a) at 800 nm in 1 bar pressure Ar (blue solid curve), 0.5 bar Xe (red dashed curve) and in 1 bar air (green solid curve); (b) at 2 µm in 1 bar Ar with (blue solid curve) and without (blue dash-dotted curve) steepening terms [Fig. 7(a)], and in 0.5 bar Xe (red dashed curve).

To complete the comparison between 800 nm and 2 µm pulses, we find it instructive to display their respective spectral broadenings over several decades in logarithmic scales. In Fig. 8(a), on-axis spectra have been plotted for 800 nm pulses at distances of maximum compression in argon, xenon and air. SPM broadens the pulse spectrum preferentially to the left, i. e., to the red wavelengths at normalized spectral intensities >0.1. This is a signature of the first plasma response that limits the beam collapse [4

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

]. Blueshift owing to self-steepening occurs at lower spectral intensities. This blueshift is more pronounced in air, which we attribute to the combined action of plasma, self-steepening and Raman delay at higher power. In contrast, red-shifts appear limited in 2 µmfilamentation [Fig. 8(b)]. The reason is that the Kerr-induced SPM is early distorted asymmetrically by self-steepening prior to plasma generation, as confirmed by the dash-dotted curve showing the on-axis spectrum with no steepening term. The sharp shocks which mark the trailing edge of the pulse then cause an important blueshift. The resulting spectrum develops a bandwidth significantly broader than that characterizing near-visible pulses.

5. Conclusion

In summary, numerical simulations suggest that 2 µm filaments can be compressed down to durations very close to the single optical period, unlike 800 nm filaments whose best compression rates make them reach 23 optical cycles in gases with moderate ionization potentials <20 eV. At long wavelengths, pulse shrinking in time is justified by an amplified action of self-steepening for saturation intensities remaining comparable with those attained at 800 nm. Pulse compression is accompanied by a prominent spectral broadening, shifted towards the large frequencies. Further improvements of the theoretical model will be necessary to describe with more accuracy what happens when the pulses reach sub-cycle durations. However, all our numerical simulations emphasize the generic compression of 2 µm laser pulses to the single-cycle limit with no particular limitation. The remarkable compression rates achieved with long carrier wavelengths should offer novel perspectives for high-order harmonic generation and production of isolated attosecond pulses.

Acknowledgments

The author thanks Dr. Stefan Skupin for fruitful discussion and judicious pieces of advice. Numerical simulations have been performed on the computer cluster CCRT at CEA-France.

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.

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, 863–905 (2005). [CrossRef]

3.

A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Phys. Rep. 44, 47–189 (2007). [CrossRef]

4.

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

5.

J. Kasparian and J.-P. Wolf, “Physics and applications of atmospheric nonlinear optics and filamentation,” Opt. Express 16, 466–493 (2008). [CrossRef] [PubMed]

6.

S. Champeaux and L. Bergé, “Femtosecond pulse compression in pressure-gas cells filled with argon,” Phys. Rev. E 68, 066603 (2003). [CrossRef]

7.

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, 673–677 (2004). [CrossRef]

8.

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]

9.

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

10.

A. Zaïr, 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]

11.

O. G. Kosareva, N. A. Panov, D. S. Uryupina, M. V. Kurilova, A. V. Mazhorova, A. B. Savel’ev, R. V. Volkov, V. P. Kandidov, and S. L. Chin, “Optimization of a femtosecond pulse self-compression region along a filament in air,” Appl. Phys. B 91, 35–43 (2008). [CrossRef]

12.

L. Bergé and S. Skupin, “Few-cycle light bullets created by femtosecond filaments,” Phys. Rev. Lett. 100, 113902 (2008). [CrossRef] [PubMed]

13.

A. Couairon, H. S. Chakraborty, and M. B. Gaarde, “From single-cycle self-compressed filaments to isolated attosecond pulses in noble gases,” Phys. Rev. A 77, 053814 (2008). [CrossRef]

14.

E. Goulielmakis, S. Koehler, B. Reiter, M. Schultze, A. J. Verheof, E. E. Serebryannikov, A. M. Zheltikov, and F. Krausz, “Ultrabroadband, coherent light source based on self-channeling of few-cycle pulses in helium,” Opt. Lett. 33, 1407–1409 (2008). [CrossRef] [PubMed]

15.

L. Bergé and S. Skupin, “Sub-2 fs pulses generated by self-channeling in the deep ultraviolet,” Opt. Lett. 33, 750–752 (2008). [CrossRef] [PubMed]

16.

P. M. Paul, E. S. Toma, P. Breger, G. Mullot, F. Auge, Ph. Balcou, H. G. Muller, and P. Agostini, “Observation of a train of attosecond pulses from high harmonic generation,” Science 292, 1689–1692 (2001). [CrossRef] [PubMed]

17.

G. Sansone, E. Benedetti, F. Galegari, C. Vozzi, L. Avaldi, R. Flammini, L. Poletto, P. Villoresi, C. Altucci, R. Velotta, S. Stagira, S. De Silvestri, and M. Nisoli, “Isolated single-cycle attosecond pulses,” Science 314, 443–446 (2008). [CrossRef]

18.

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

19.

D. Faccio, A. Lotti, M. Kolesik, J. V. Moloney, S. Tzortzakis, A. Couairon, and P. Di Trapani, “Spontaneous emergence of pulses with constant carrier-envelope phase in femtosecond filamentation,” Opt. Express 16, 11103–11114 (2008). [CrossRef] [PubMed]

20.

B. Sheehy, J. D. D. Martin, L. F. DiMauro, P. Agostini, K. J. Schafer, M. B. Gaarde, and K. C. Kulander, “High harmonic generation at long wavelengths,” Phys. Rev. Lett. 83, 5270–5273 (1999). [CrossRef]

21.

B. Shan and Z. Chang, “Dramatic extension of the high-order harmonic cutoff by using a long-wavelength driving field,” Phys. Rev. A 65, 011804(R) (2001). [CrossRef]

22.

K. D. Schultz, C. I. Blaga, R. Chirla, P. Colosimo, J. Cryan, A. M. March, C. Roedig, E. Sistrunk, J. tate, J. Wheeler, P. Agostini, and L. F. DiMauro, “Strong field physics with long wavelength lasers,” J. Mod. Opt. 54, 1075–1085 (2007). [CrossRef]

23.

J. Tate, T. Auguste, H. G. Muller, P. Salières, P. Agostini, and L. F. DiMauro, “Scaling of wave-packet dynamics in an intense midinfrared field,” Phys. Rev. Lett. 98, 013901 (2007). [CrossRef] [PubMed]

24.

C. P. Hauri, R. B. Lopez-Martens, C. I. Blaga, K. D. Schultz, J. Cryan, R. Chirla, P. Colosimo, G. Doumy, A. M. March, C. Roedig, E. Sistrunk, J. Tate, J. Wheeler, L. F. DiMauro, and E. P. Power, “Intense self-compressed, self-phase-stabilized few-cycle pulses at 2µm from an optical filament,” Opt. Lett. 32, 868–870 (2007). [CrossRef] [PubMed]

25.

C. Vozzi, C. Manzoni, F. Calegari, E. Benedetti, G. Sansone, G. Cerullo, M. Nisoli, S. De Silvestri, and S. Stagira, “Characterization of a high-energy self-phase-stabilized near-infrared parametric source,” J. Opt. Soc. Am. B 25, 112–117 (2008). [CrossRef]

26.

T. Fuji and T. Suzuki, “Generation of sub-two-cycle mid-infrared pulses by four-wave mixing through filamentation in air,” Opt. Lett. 32, 3330–3332 (2007). [CrossRef] [PubMed]

27.

D. Brida, G. Cirmi, C. Manzoni, S. Bonora, P. Villoresi, S. De Silvestri, and G. Cerullo, “Sub-two-cycle light pulses at 1.6 µm from an optical parametric amplifier,” Opt. Lett. 33, 741–743 (2008). [CrossRef] [PubMed]

28.

T. Brabec and F. Krausz, “Nonlinear optical pulse propagation in the single-cycle regime,” Phys. Rev. Lett. 78, 3282–3285 (1997). [CrossRef]

29.

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).

30.

R. Nuter and L. Bergé, “Pulse chirping and ionization of O2 molecules for the filamentation of femtosecond laser pulses in air,” J. Opt. Soc. Am. B 23, 874–884 (2006). [CrossRef]

31.

M. V. Ammosov, N. B. Delone, and V. P. Krainov, “Tunnel ionization of complex atoms and of atomic ions in an alternating electromagnetic field,” Sov. Phys. JETP 64, 1191–1194 (1986).

32.

R. Wiehle, B. Witzel, H. Helm, and E. Cormier, “Dynamics of strong-field above-threshold ionization of argon: Comparison between experiment and theory,” Phys. Rev. A 67, 063405 (2003). [CrossRef]

33.

E. Cormier, P.-A. Hervieux, R. Wiehle, B. Witzel, and H. Helm, “ATI of complex systems: Ar and C60,” Eur. Phys. J. D. 26, 83–90 (2003). [CrossRef]

34.

A. Dalgarno and A. E. Kingston, “The refractive indices and Verdet constants of the inert gases,” Proc. Royal Soc. London A 259, 424–429 (1960). [CrossRef]

35.

E. R. Peck and K. Reeder, “Dispersion of Air,” J. Opt. Soc. Am. 62, 958–962 (1972). [CrossRef]

36.

D. P. Shelton, “Nonlinear-optical susceptibilities of gases measured at 1064 and 1319 nm,” Phys. Rev. A 42, 2578–2592 (1990). [CrossRef] [PubMed]

37.

W. Liu and S. L. Chin, “Direct measurement of the critical power of femtosecond Ti:sapphire laser pulse in air,” Opt. Express 13, 5750–5755 (2005). [CrossRef] [PubMed]

38.

S. Skupin and L. Bergé, “Supercontinuum generation of ulrashort pulses in air at different central wavelengths,” Opt. Commun. 280, 173–182 (2007). [CrossRef]

39.

S. Champeaux, L. Bergé, D. Gordon, A. Ting, J. Peñano, and P. Sprangle, “(3+1)-dimensional numerical simulations of femtosecond laser filaments in air: Toward a quantitative agreement with experiments,” Phys. Rev. E 77, 036406 (2008). [CrossRef]

40.

A. Talebpour, J. Yang, and S. L. Chin, “Semi-empirical model for the rate of tunnel ionization of N2 and O2 molecule in an intense Ti:sapphire laser pulse,” Opt. Commun. 163, 29–32 (1999). [CrossRef]

41.

D. Anderson and M. Lisak, “Nonlinear asymmetric self-phase modulation and self-steepening of pulses in long optical waveguides,” Phys. Rev. A 27, 1393–1398 (1983). [CrossRef]

42.

N. Aközbek, M. Scalora, C. M. Bowden, and S. L. Chin, “White-light continuum generation and filamentation during the propagation of ultra-short laser pulses in air,” Opt. Commun. 191, 353–362 (2001). [CrossRef]

43.

N. Aközbek, S. A. Trushin, A. Baltuška, W. Fuss, E. Goulielmakis, K. Kosma, F. Krausz, S. Panja, M. Uiberacker, W. E. Schmid, A. Becker, M. Scalora, and M. Bloemer, “Extending the supercontinuum spectrum down to 200 nm with few-cycle pulses,” New J. Phys. 8, 177 (2006). [CrossRef]

44.

S. Champeaux and L. Bergé, “Long-range multifilamentation of femtosecond laser pulses versus air pressure,” Opt. Lett. 31, 1301–1303 (2006). [CrossRef] [PubMed]

45.

J. Bernhardt, W. Liu, S. L. Chin, and R. Sauerbrey, “Pressure independence of intensity clamping during filamentation: theory and experiment,” Appl. Phys. B 91, 45–48 (2008). [CrossRef]

46.

M. Mlejnek, E. M. Wright, and J. V. Moloney, “Dynamic spatial replenishment of femtosecond pulses propagating in air,” Opt. Lett. 23, 382–384 (1998). [CrossRef]

47.

J. K. Ranka and A. L. Gaeta, “Breakdown of the slowly varying envelope approximation in the self-focusing of ultrashort pulses,” Opt. Lett. 23, 534–536 (1998). [CrossRef]

48.

G. G. Luther, J. V. Moloney, A. C. Newell, and E. M. Wright, “Self-focusing threshold in normally dispersive media,” Opt. Lett. 19, 862–864 (1994). [CrossRef] [PubMed]

OCIS Codes
(260.5950) Physical optics : Self-focusing
(320.5520) Ultrafast optics : Pulse compression
(320.7110) Ultrafast optics : Ultrafast nonlinear optics

ToC Category:
Ultrafast Optics

History
Original Manuscript: September 22, 2008
Revised Manuscript: November 7, 2008
Manuscript Accepted: November 7, 2008
Published: December 15, 2008

Citation
Luc Bergé, "Self-compression of 2 μm laser filaments," Opt. Express 16, 21529-21543 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-26-21529


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  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. 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. 83,863-905 (2005). [CrossRef]
  3. A. Couairon and A. Mysyrowicz, "Femtosecond filamentation in transparent media," Phys. Rep. 44,47-189 (2007). [CrossRef]
  4. L. Bergé, S. Skupin, R. Nuter, J. Kasparian, and J.-P. Wolf, "Ultrashort filaments of light in weakly ionized, optically transparent media," Rep. Prog. Phys. 70,1633-1713 (2007). [CrossRef]
  5. J. Kasparian and J.-P. Wolf, "Physics and applications of atmospheric nonlinear optics and filamentation," Opt. Express 16,466-493 (2008). [CrossRef] [PubMed]
  6. S. Champeaux and L. Bergé, "Femtosecond pulse compression in pressure-gas cells filled with argon," Phys. Rev. E 68,066603 (2003). [CrossRef]
  7. 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 79,673-677 (2004). [CrossRef]
  8. 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]
  9. S. Skupin, G. Stibenz, and L. Bergé, F. Lederer, T. Sokollik, M. Schnürer, N. Zhavoronkov, and G. Steinmeyer, "Self-compression by femtosecond pulse filamentation: Experiments versus numerical simulations," Phys. Rev. E 74,056604 (2006). [CrossRef]
  10. A. Za¨ır, 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]
  11. O. G. Kosareva, N. A. Panov, D. S. Uryupina, M. V. Kurilova, A. V. Mazhorova, A. B. Savel’ev, R. V. Volkov, V. P. Kandidov, and S. L. Chin, "Optimization of a femtosecond pulse self-compression region along a filament in air," Appl. Phys. B 91,35-43 (2008). [CrossRef]
  12. L. Bergé and S. Skupin, "Few-cycle light bullets created by femtosecond filaments," Phys. Rev. Lett. 100,113902 (2008). [CrossRef] [PubMed]
  13. A. Couairon and H. S. Chakraborty and M. B. Gaarde, "From single-cycle self-compressed filaments to isolated attosecond pulses in noble gases," Phys. Rev. A 77,053814 (2008). [CrossRef]
  14. E. Goulielmakis, S. Koehler, B. Reiter, M. Schultze, A. J. Verheof, E. E. Serebryannikov, A. M. Zheltikov, and F. Krausz, "Ultrabroadband, coherent light source based on self-channeling of few-cycle pulses in helium," Opt. Lett. 33,1407-1409 (2008). [CrossRef] [PubMed]
  15. L. Bergé and S. Skupin, "Sub-2 fs pulses generated by self-channeling in the deep ultraviolet," Opt. Lett. 33,750-752 (2008). [CrossRef] [PubMed]
  16. P. M. Paul, E. S. Toma, P. Breger, G. Mullot, F. Auge, Ph. Balcou, H. G. Muller, and P. Agostini, "Observation of a train of attosecond pulses from high harmonic generation," Science 292,1689-1692 (2001). [CrossRef] [PubMed]
  17. G. Sansone, E. Benedetti, F. Galegari, C. Vozzi, L. Avaldi, R. Flammini, L. Poletto, P. Villoresi, C. Altucci, R. Velotta, S. Stagira, S. De Silvestri, and M. Nisoli, "Isolated single-cycle attosecond pulses," Science 314,443-446 (2008). [CrossRef]
  18. M. Nisoli, S. de Silvestri, O. Svelto, R. Szip¨oecs, K. Ferencz, Ch. Spielmann, S. Sartania, and F. Krausz, "Compression of high-energy laser pulses below 5 fs," Opt. Lett. 22,522-524 (1997). [CrossRef] [PubMed]
  19. D. Faccio, A. Lotti, M. Kolesik, J. V. Moloney, S. Tzortzakis, A. Couairon, and P. Di Trapani, "Spontaneous emergence of pulses with constant carrier-envelope phase in femtosecond filamentation," Opt. Express 16,11103-11114 (2008). [CrossRef] [PubMed]
  20. B. Sheehy, J. D. D. Martin, L. F. DiMauro, P. Agostini, K. J. Schafer, M. B. Gaarde, and K. C. Kulander, "High harmonic generation at long wavelengths," Phys. Rev. Lett. 83,5270-5273 (1999). [CrossRef]
  21. B. Shan and Z. Chang, "Dramatic extension of the high-order harmonic cutoff by using a long-wavelength driving field," Phys. Rev. A 65,011804(R) (2001). [CrossRef]
  22. K. D. Schultz, C. I. Blaga, R. Chirla, P. Colosimo, J. Cryan, A. M. March, C. Roedig, E. Sistrunk, J. tate, J. Wheeler, P. Agostini, and L. F. DiMauro, "Strong field physics with long wavelength lasers," J. Mod. Opt. 54,1075-1085 (2007). [CrossRef]
  23. J. Tate, T. Auguste, H. G. Muller, P. Sali`eres, P. Agostini, and L. F. DiMauro, "Scaling of wave-packet dynamics in an intense midinfrared field," Phys. Rev. Lett. 98,013901 (2007). [CrossRef] [PubMed]
  24. C. P. Hauri, R. B. Lopez-Martens, C. I. Blaga, K. D. Schultz, J. Cryan, R. Chirla, P. Colosimo, G. Doumy, A. M. March, C. Roedig, E. Sistrunk, J. Tate, J. Wheeler, L. F. DiMauro, and E. P. Power, "Intense self-compressed, self-phase-stabilized few-cycle pulses at 2m from an optical filament," Opt. Lett. 32,868-870 (2007). [CrossRef] [PubMed]
  25. C. Vozzi, C. Manzoni, F. Calegari, E. Benedetti, G. Sansone, G. Cerullo, M. Nisoli, S. De Silvestri, and S. Stagira, "Characterization of a high-energy self-phase-stabilized near-infrared parametric source," J. Opt. Soc. Am. B 25,112-117 (2008). [CrossRef]
  26. T. Fuji and T. Suzuki, "Generation of sub-two-cycle mid-infrared pulses by four-wave mixing through filamentation in air," Opt. Lett. 32,3330-3332 (2007). [CrossRef] [PubMed]
  27. D. Brida, G. Cirmi, C. Manzoni, S. Bonora, P. Villoresi, S. De Silvestri, and G. Cerullo, "Sub-two-cycle light pulses at 1.6 m from an optical parametric amplifier," Opt. Lett. 33,741-743 (2008). [CrossRef] [PubMed]
  28. T. Brabec and F. Krausz, "Nonlinear optical pulse propagation in the single-cycle regime," Phys. Rev. Lett. 78,3282-3285 (1997). [CrossRef]
  29. 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).
  30. R. Nuter and L. Bergé, "Pulse chirping and ionization of O2 molecules for the filamentation of femtosecond laser pulses in air," J. Opt. Soc. Am. B 23,874-884 (2006). [CrossRef]
  31. M. V. Ammosov, N. B. Delone, and V. P. Krainov, "Tunnel ionization of complex atoms and of atomic ions in an alternating electromagnetic field," Sov. Phys. JETP 64,1191-1194 (1986).
  32. R. Wiehle, B. Witzel, H. Helm, and E. Cormier, "Dynamics of strong-field above-threshold ionization of argon: Comparison between experiment and theory," Phys. Rev. A 67,063405 (2003). [CrossRef]
  33. E. Cormier, P.-A. Hervieux, R. Wiehle, B. Witzel, and H. Helm, "ATI of complex systems: Ar and C60," Eur. Phys. J. D. 26,83-90 (2003). [CrossRef]
  34. A. Dalgarno and A. E. Kingston, "The refractive indices and Verdet constants of the inert gases," Proc. Royal Soc. London A 259,424-429 (1960). [CrossRef]
  35. E. R. Peck and K. Reeder, "Dispersion of Air," J. Opt. Soc. Am. 62,958-962 (1972). [CrossRef]
  36. D. P. Shelton, "Nonlinear-optical susceptibilities of gases measured at 1064 and 1319 nm," Phys. Rev. A 42,2578-2592 (1990). [CrossRef] [PubMed]
  37. W. Liu and S. L. Chin, "Direct measurement of the critical power of femtosecond Ti:sapphire laser pulse in air," Opt. Express 13,5750-5755 (2005). [CrossRef] [PubMed]
  38. S. Skupin and L. Bergé, "Supercontinuum generation of ulrashort pulses in air at different central wavelengths," Opt. Commun. 280,173-182 (2007). [CrossRef]
  39. S. Champeaux, L. Bergé, D. Gordon, A. Ting, J. Peñano, and P. Sprangle, "(3+1)-dimensional numerical simulations of femtosecond laser filaments in air: Toward a quantitative agreement with experiments," Phys. Rev. E 77,036406 (2008). [CrossRef]
  40. A. Talebpour, J. Yang, and S. L. Chin, "Semi-empirical model for the rate of tunnel ionization of N2 and O2 molecule in an intense Ti:sapphire laser pulse," Opt. Commun. 163,29-32 (1999). [CrossRef]
  41. D. Anderson and M. Lisak, "Nonlinear asymmetric self-phase modulation and self-steepening of pulses in long optical waveguides," Phys. Rev. A 27,1393-1398 (1983). [CrossRef]
  42. N. Ak¨ozbek, M. Scalora, C. M. Bowden, and S. L. Chin, "White-light continuum generation and filamentation during the propagation of ultra-short laser pulses in air," Opt. Commun. 191,353-362 (2001). [CrossRef]
  43. N. Ak¨ozbek, S. A. Trushin, A. Baltuška,W. Fuss, E. Goulielmakis, K. Kosma, F. Krausz, S. Panja, M. Uiberacker, W. E. Schmid, A. Becker, M. Scalora, and M. Bloemer, "Extending the supercontinuum spectrum down to 200 nm with few-cycle pulses," New J. Phys. 8,177 (2006). [CrossRef]
  44. S. Champeaux and L. Bergé, "Long-range multifilamentation of femtosecond laser pulses versus air pressure," Opt. Lett. 31,1301-1303 (2006). [CrossRef] [PubMed]
  45. J. Bernhardt, W. Liu, S. L. Chin, and R. Sauerbrey, "Pressure independence of intensity clamping during filamentation: theory and experiment," Appl. Phys. B 91,45-48 (2008). [CrossRef]
  46. M. Mlejnek, E. M. Wright, and J. V. Moloney, "Dynamic spatial replenishment of femtosecond pulses propagating in air," Opt. Lett. 23,382-384 (1998). [CrossRef]
  47. J. K. Ranka and A. L. Gaeta, "Breakdown of the slowly varying envelope approximation in the self-focusing of ultrashort pulses," Opt. Lett. 23,534-536 (1998). [CrossRef]
  48. G. G. Luther, J. V. Moloney, A. C. Newell, and E. M. Wright, "Self-focusing threshold in normally dispersive media," Opt. Lett. 19,862-864 (1994). [CrossRef] [PubMed]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.


« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited