## Supercontinuum generation and pulse compression from gas filamentation of femtosecond laser pulses with different durations

Optics Express, Vol. 17, Issue 16, pp. 13841-13850 (2009)

http://dx.doi.org/10.1364/OE.17.013841

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### Abstract

Self-compression and spectral supercontinuum (SC) generated by filamentation of femtosecond laser pulses with duration from 45 fs down to 6 fs in argon gas have been numerically investigated. A 45-fs pulse can be self-compressed into a few-cycle pulse with duration of 12 fs at the post-filamentation region. By properly employing a high-pass filter to select the broadening high-frequency spectra which are almost in phase, the pulse can be further shortened to about 7 fs. By contrast, a 6-fs pulse cannot be further self-compressed into a shorter pulse by filamentation although it can generate much broader SC extending from 200 nm to 1300 nm. It is also found that a separate and strong SC in the ultraviolet (UV) region extending from 220 nm to 300 nm and peaked at about 255nm can be generated at proper propagation distances, which corresponds to a pulse with duration of about 5 fs.

© 2009 OSA

## 1. Introduction

30. S. Trushin, S. Panja, K. Kosma, W. E. Schmid, and W. Fuß, “Supercontinuum extending from >1000 to 250 nm, generated by focusing ten-fs laser pulses at 805 nm into Ar,” Appl. Phys. B **80**(4–5), 399–403 (2005). [CrossRef]

33. K. Kosma, S. A. Trushin, W. Fuß, and W. E. Schmid, “Characterization of supercontinuum radiation generated by self-focusing of few-cycle 800-nm pulses in Argon,” J. Mod. Opt. **55**(13), 2141–2177 (2008). [CrossRef]

30. S. Trushin, S. Panja, K. Kosma, W. E. Schmid, and W. Fuß, “Supercontinuum extending from >1000 to 250 nm, generated by focusing ten-fs laser pulses at 805 nm into Ar,” Appl. Phys. B **80**(4–5), 399–403 (2005). [CrossRef]

32. S. A. Trushin, K. Kosma, W. Fuss, and W. E. Schmid, “Sub-10-fs supercontinuum radiation generated by filamentation of few-cycle 800 nm pulses in argon,” Opt. Lett. **32**(16), 2432–2434 (2007). [CrossRef] [PubMed]

31. 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. Bloeme, “Extending the supercontinuum spectrum down to 200nm with few-cycle pulses,” N. J. Phys. **8**(9), 177 (2006). [CrossRef]

*E*

_{in}=1.3 mJ, the pulse will undergo self-compression and form a temporally compressed “light bullet” with a duration of ~12 fs at the post-filamentation domain as it is well-known [5

5. M. Mlejnek, E. M. Wright, and J. V. Moloney, “Femtosecond pulse propagation in argon: A pressure dependence study,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics **58**(4), 4903–4910 (1998). [CrossRef]

6. M. Nurhuada, A. Suda, M. Hatayama, K. Nagasaka, and K. Midorikawa, “propagation dynamics of femtosecond laser pulses in argon,” Phys. Rev. A **66**(2), 023811 (2002). [CrossRef]

28. L. Bergé and S. Skupin, “Few-cycle light bullets created by femtosecond filaments,” Phys. Rev. Lett. **100**(11), 113902 (2008). [CrossRef] [PubMed]

## 2. Simulations

*et*al. [34

34. T. Brabec and F. Krausz, “Intense few-cycle laser fields: Frontiers of nonlinear optics,” Rev. Mod. Phys. **72**(2), 545–591 (2000). [CrossRef]

*k*(

*ω*)=n(

*ω*)

*ω*/

*c*, where

*ω*is the optical frequency, and

*n*(

*ω*) is the linear refractive index of the material, the equation of propagation of electric field has the following form [35

35. M. Kolesik and J. V. Moloney, “Nonlinear optical pulse propagation simulation: From Maxwell’s to unidirectional equations,” Phys. Rev. Lett. **92**, 253901 (2004). [CrossRef] [PubMed]

38. J. Liu, H. Schroeder, S. L. Chin, W. Yu, R. Li, and Z. Xu, “Space-frequency coupling, conical waves, and small-scale filamentation in water,” Phys. Rev. A **72**(5), 053817 (2005). [CrossRef]

*E*̂ is the Fourier transform of the forward electric field component,

*P*̂

*is the nonlinear polarization in frequency domain of*

_{NL}*P*(

_{NL}*r*,

*z*,

*t*) given by

*P*(

_{NL}*r*,

*z*,

*t*)=2

*ε*

_{0}

*n*

_{b}

*n*

_{2}

*I*(

*r*,

*z*,

*t*)

*E*(

*r*,

*z*,

*t*), where

*n*

_{b}and

*n*

_{2}are linear refractive index and nonlinear coefficient at center wavelength

*λ*

_{0}, respectively. The current density

*J*caused by free electrons can be expressed in frequency domain as [36

_{f}36. J. Liu, X. Chen, J. Liu, Y. Zhu, Y. Leng, J. Dai, R. Li, and Z. Xu, “Spectrum reshaping and pulse self-compression in normally dispersive media with negatively chirped femtosecond pulses,” Opt. Express **14**(2), 979–987 (2006). [CrossRef] [PubMed]

38. J. Liu, H. Schroeder, S. L. Chin, W. Yu, R. Li, and Z. Xu, “Space-frequency coupling, conical waves, and small-scale filamentation in water,” Phys. Rev. A **72**(5), 053817 (2005). [CrossRef]

*k*

_{0}=0.5

*n*

_{b}cε_{0}. The evolution of free electron density can be described as ∂

_{t}

*ρ*=

*β*

^{(K)}

*I*

*K*+

*η*-

_{cas}ρ*η*

_{rec}ρ^{2}. The initial value of electron density is assumed as

*ρ*

_{0}≡10

^{9}cm

^{-3}[39

39. L. Bergé and A. Couairon, “Nonlinear propagation of self-guided ultra-short pulses in ionized gases,” Phys. Plasmas **7**(1), 210–231 (2000). [CrossRef]

*P*,

_{in}*w*

_{0},

*T*

_{0}, and

*f*denote input peak power, beam waist, pulse duration, and effective focal length, respectively.

*k*

_{(0)}=

*n*

_{b}

*ω*

_{0}/c is the wave number at the laser frequency. Throughout the paper, we will use full-width at half maximum (FWHM) values with respect to the intensity and beam energy to characterize the pulses, namely,

14. A. Mysyrowicz, A. Couairon, and U. Keller, “Self-compression of optical laser pulses by filamentation,” N. J. Phys. **10**(2), 025023 (2008). [CrossRef]

*I*=15.76 eV and the number of photons required for ionization is

_{p}*K*=11. The multiphoton ionization (MPI) coefficient can be calculated as

*β*

^{(11)}=1.34×10

^{-114}

*S*

^{-1}·

*m*

^{19}·

*W*-

^{11}. The collision time can be taken as

*τ*=190 fs. The linear refractive index

_{c}*n*(

*ω*) of argon is expressed as a formula given in Ref [40

40. A. Dalgarno and A. E. Kingston, “The Refractive Indices and Verdet Constants of the Inert Gases,” Proc. R. Soc. Lond. A Math. Phys. Sci. **259**(1298), 424–434 (1960). [CrossRef]

*Pσ*=

*λ*

^{2}

_{0}/2

*π*

*n*b

*n*2 for a nonlinear coefficient

*n*

_{2}=3×10

^{-19}

*cm*2/

*W*.

*η*=3.1×10

_{cas}^{-6}

*s*

^{-1}·

*m*

^{2}·

*W*

^{-1}and

*η*=7×10

_{rec}^{-13}

*m*

^{3}·

*s*

^{-1}represent cascade ionization and electron-ion recombination rate, respectively. The Eq. (1) was solved by a fourth-order Runge-Kutta method with adaptive stepsize control. The transversal Laplace operator ∇

^{2}

_{⊥}is calculated here by a finite difference method [41] instead of Fourier transformation method used in Ref [38

38. J. Liu, H. Schroeder, S. L. Chin, W. Yu, R. Li, and Z. Xu, “Space-frequency coupling, conical waves, and small-scale filamentation in water,” Phys. Rev. A **72**(5), 053817 (2005). [CrossRef]

## 3. Results and Discussions

*f*=60 cm. The beam waist of the pulse is assumed as

*w*

_{0}=2.5 mm (which corresponds to

*w*

*=2.94 mm). When the input peak power is higher than the critical power for self-focusing, the beam collapse can be expected to occur near the so-called nonlinear focus z*

_{FWHM}_{c}, which can be expressed by the well-known Marburger formula

*z*

_{0}=

*πn*

_{0}

*w*

^{2}

_{0}/

*λ*

_{0}is the Rayleigh length of the beam. In the case of a convergent beam, the position of the collapse

*z*

*c*,

*f*moves to

*z*

*,*

_{c}*=*

_{f}*z*

_{c}*/(*

_{f}*z*

*+*

_{c}*f*).

^{2}and about 10

^{17}cm

^{-3}, respectively, which almost do not change with the input power. This phenomenon has been called intensity clamping. When the input pulse power increases from 5 P

_{cr}to 12 P

_{cr}, the filament and the plasma channel are elongated from 9 cm to 18 cm. As a comparison and shown in Fig. 1(d), we simulated the propagation of 6-fs pulse at 5 P

_{cr}(E

_{in}=0.11 mJ). The length of the filament is about 19 cm, which is much longer than the corresponding 45-fs pulse with the same input power. The peak intensity in the filament will alter from about 62 to 70 TW/cm

^{2}along the propagation axis z, which is a little higher than that for the corresponding 45-fs pulse. As analyzed in ref [38

**72**(5), 053817 (2005). [CrossRef]

*T*, the clamped intensity will be higher than that for a long pulse.

_{0}_{cr}(E

_{in}=0.8 mJ), 8.2 P

_{cr}(E

_{in}=1.3 mJ), 12 P

_{cr}(E

_{in}=2.0 mJ), respectively. As described extensively [5

5. M. Mlejnek, E. M. Wright, and J. V. Moloney, “Femtosecond pulse propagation in argon: A pressure dependence study,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics **58**(4), 4903–4910 (1998). [CrossRef]

6. M. Nurhuada, A. Suda, M. Hatayama, K. Nagasaka, and K. Midorikawa, “propagation dynamics of femtosecond laser pulses in argon,” Phys. Rev. A **66**(2), 023811 (2002). [CrossRef]

13. 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 Stat. Nonlin. Soft Matter Phys. **74**(5 Pt 2), 056604 (2006). [CrossRef]

_{in}=5 P

_{cr}, this single-sided depletion will eventually give rise to a few-cycle pulse with a duration of ~12 fs after z~63 cm, but the obtained few-cycle pulse is very weak (Fig. 2(a)). When the input power increases to P

_{in}=8.2 P

_{cr}, an intense temporal-compressed “light bullet” with duration about 12 fs can be obtained from z~63 cm and maintain over 30 cm as shown in Fig. 2(b). The pulse self-compression seems to be less effective with increasing input pulse power. When the input peak power is further increased to P

_{in}=12 P

_{cr}, a pulse with a duration of ~12 fs can still be obtained at z~65 cm. However, it is very interesting that the resulted few-cycle pulse can filament immediately because its peak power exceeds corresponding critical threshold for self-focusing. The self-compressed pulse will experience another splitting, which also can be seen from Fig. 1(c) and Fig. 2(c).

_{in}=8.2 P

_{cr}and analyze the propagation dynamics in detail. Figures 3 shows the typical intensity distribution in time and space domain at z=53.6, 62.6, 67.0, 86.5 cm, respectively. Initially, the pulse is focused by a lens and self-focused due to Kerr effect. The peak intensity will gradually increase along with the way of propagation. The trailing edge of the pulse suffers trivial defocusing due to very weak ionization, which induces the slight asymmetry of the spatial-temporal profile. When the pulse reaches to the nonlinear focus (

*z*,

_{c}_{f}~53 cm), the pulse peak intensity quickly increases and reaches to about 57 TW/cm

^{2}on axis, which causes intense ionization and plasma is generated at the trailing edge of the pulse which defocuses the light on-axis. The pulse forms a ‘light cone’ as shown in Fig. 3(a). Figure 3(b,c) show that the rear pulse may refocus, which will generate a typical double-peaked temporal distribution. Eventually, the energy of the leading edge of the pulse will deplete and cause strong pulse shortening as shown in Figs. 3(d), where a pulse with duration of ~12 fs is generated.

*ω*>2.4 PHz) is very small while it is large in low frequency region. This kind of asymmetric spectral phase aberration is hard to compensate. However, when we properly cut out low-frequency spectra by using a high-pass filter, we can obtain a broad high-frequency spectrum whose phase aberration is small and can be compensated more easily because its spectral phase is almost quadratic. By using different spectral filters with different cut-off frequency to reshape the temporal profiles of the self-compressed pulses, we calculate the profiles of obtained pulses. As shown in Fig. 4(e), for the self-compressed pulse with a duration of ~12 fs at z=79.9 cm, shorter pulses with durations of about 9 fs, 7 fs, and 7 fs can be obtained by using high-pass filters with cut-off frequency of 2.4, 2.5, 2.6 PHz, respectively. However, when the cut-off frequency increases, the peak intensity of the obtained pulse will become lower.

_{cr}. For lower input power such as 2 P

_{cr}, the propagation dynamics is similar (not shown here). The on-axis temporal dynamics of 6-fs pulse was shown in Fig. 2(d). Figure 5 shows spatial-temporal intensity profile of 6-fs pulse at 800 nm propagating in argon at z=(a) 40.2, (b) 52.88, (c) 55.5, (d) 61.6 cm, respectively. The propagation dynamics is very different from that of 45-fs pulse as shown in Fig. 3. Initially, the input pulse is assumed as Gaussian both in space domain and time domain. The corresponding profile of spatial-temporal intensity looks like an ‘ellipse’. When the pulse propagates, the profile of the ‘ellipse’ becomes asymmetric (shown in Fig. 5(a)) along time axis mainly due to the external focusing that is numerically applied. When arriving at nonlinear focus (z

_{c},

_{f}~52.9 cm) the pulse begins to filament (Fig. 5(b)). The trailing edge of the pulse suffers strong self-steepening which can bring about some trivial self-compression effect. When further propagating, the trailing part of the pulse will split and form many short temporal peaks, which will increase in number and peak intensity along with the increase of propagation distance (Fig. 5(c,d)). The short temporal splitting can be explained by the shock profile induced by self-steepening.

^{2}can be obtained at propagation distances from 63.96 to 65.48 cm just by spectral filtering method.

_{0}=2.36 PHz) pump pulse with a 400-nm (ω

_{s}=4.72 PHz) seed generated by intense self-phase modulation and self-steepening during filamentation following the scheme 2ωs-ω

_{0}=ω

_{4WM}. This is reasonable because the phase match can be satisfied as discussed by Bergé

*et*al. [27

27. L. Bergé and S. Skupin, “Sub-2 fs pulses generated by self-channeling in the deep ultraviolet,” Opt. Lett. **33**(7), 750–752 (2008). [CrossRef] [PubMed]

*et al*. have described the experimental observation of the spectral SC generated by different pulse duration in Ref [31

31. 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. Bloeme, “Extending the supercontinuum spectrum down to 200nm with few-cycle pulses,” N. J. Phys. **8**(9), 177 (2006). [CrossRef]

31. 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. Bloeme, “Extending the supercontinuum spectrum down to 200nm with few-cycle pulses,” N. J. Phys. **8**(9), 177 (2006). [CrossRef]

*ik*

^{(0)}

*r*

^{2}/2

*f*in the wavefront of the initial laser pulse has significant effect on the filamentation dynamics of femtosecond pulse, especially for few-cycle pulse. We also investigated the propagation dynamics of 45-fs and 6-fs pulses with different quadratic phase such as -

*ik*(

*ω*)

*r*

^{2}/2

*f*� which in fact can result in much different femtosecond dynamics. Therefore, by adjusting the initial laser parameters, the filamentation process can be controlled. The control of filamentation process by changing the initial laser parameters will be discussed in detail in another paper.

## 4. Conclusion

_{in}=8.2 P

_{cr}and last over 30 cm. By properly employing high-pass filter, the pulse can be further shortened to about 7 fs which approaches to the Fourier transform limit of the corresponding spectral SC. As a comparison, the propagation dynamics of a 6-fs pulse is very different from that of a 45-fs pulse, which cannot be further self-compressed into a shorter pulse by filamentation although it can generate much broader SC extending from 200 nm to 1300 nm. It is also found that a separate and strong SC in the ultraviolet region extending from 220 nm to 300 nm and peaked at about 255nm can be generated, which can be explained as 4WM of a 800-nm pump pulse (i.e. input pulse) and a 400-nm seed pulse mainly generated by intense SPM and self-steepening during filamentation. The UV SC generated at proper propagation distance corresponds to a few-cycle pulse with duration of about 5 fs in time domain without spectral phase compensation.

## Acknowledgements

## References and links

1. | M. Hentschel, R. Kienberger, Ch. Spielmann, G. A. Reider, N. Milosevic, T. Brabec, P. Corkum, U. Heinzmann, M. Drescher, and F. Krausz, “Attosecond metrology,” Nature |

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3. | G. A. Mourou, T. Tajima, and S. V. Bulanov, “Optics in the relativistic regime,” Rev. Mod. Phys. |

4. | 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 |

5. | M. Mlejnek, E. M. Wright, and J. V. Moloney, “Femtosecond pulse propagation in argon: A pressure dependence study,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics |

6. | M. Nurhuada, A. Suda, M. Hatayama, K. Nagasaka, and K. Midorikawa, “propagation dynamics of femtosecond laser pulses in argon,” Phys. Rev. A |

7. | S. Champeaux and L. Bergé, “Femtosecond pulse compression in pressure-gas cells filled with argon,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

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9. | A. Couairon, M. Franco, A. Mysyrowicz, J. Biegert, and U. Keller, “Pulse self-compression to the single-cycle limit by filamentation in a gas with a pressure gradient,” Opt. Lett. |

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12. | A. Couairon, J. Biegert, C. P. Hauri, W. Kornelis, F. W. Helbing, U. Keller, and A. Mysyrowicz, “Self-compression of ultra-short laser pulses down to one optical cycle by filamentation,” J. Mod. Opt. |

13. | 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 Stat. Nonlin. Soft Matter Phys. |

14. | A. Mysyrowicz, A. Couairon, and U. Keller, “Self-compression of optical laser pulses by filamentation,” N. J. Phys. |

15. | X. Chen, Y. Leng, J. Liu, Y. Zhu, R. Li, and Z. Xu, “Pulse self-compression in normally dispersive bulk media,” Opt. Commun. |

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27. | L. Bergé and S. Skupin, “Sub-2 fs pulses generated by self-channeling in the deep ultraviolet,” Opt. Lett. |

28. | L. Bergé and S. Skupin, “Few-cycle light bullets created by femtosecond filaments,” Phys. Rev. Lett. |

29. | Z. Wang, J. Liu, R. Li, and Z. Xu, “Spectral analysis and control to obtain sub-5 fs pulses by femtosecond filamentation,” Opt. Lett. |

30. | S. Trushin, S. Panja, K. Kosma, W. E. Schmid, and W. Fuß, “Supercontinuum extending from >1000 to 250 nm, generated by focusing ten-fs laser pulses at 805 nm into Ar,” Appl. Phys. B |

31. | 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. Bloeme, “Extending the supercontinuum spectrum down to 200nm with few-cycle pulses,” N. J. Phys. |

32. | S. A. Trushin, K. Kosma, W. Fuss, and W. E. Schmid, “Sub-10-fs supercontinuum radiation generated by filamentation of few-cycle 800 nm pulses in argon,” Opt. Lett. |

33. | K. Kosma, S. A. Trushin, W. Fuß, and W. E. Schmid, “Characterization of supercontinuum radiation generated by self-focusing of few-cycle 800-nm pulses in Argon,” J. Mod. Opt. |

34. | T. Brabec and F. Krausz, “Intense few-cycle laser fields: Frontiers of nonlinear optics,” Rev. Mod. Phys. |

35. | M. Kolesik and J. V. Moloney, “Nonlinear optical pulse propagation simulation: From Maxwell’s to unidirectional equations,” Phys. Rev. Lett. |

36. | J. Liu, X. Chen, J. Liu, Y. Zhu, Y. Leng, J. Dai, R. Li, and Z. Xu, “Spectrum reshaping and pulse self-compression in normally dispersive media with negatively chirped femtosecond pulses,” Opt. Express |

37. | 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 |

38. | J. Liu, H. Schroeder, S. L. Chin, W. Yu, R. Li, and Z. Xu, “Space-frequency coupling, conical waves, and small-scale filamentation in water,” Phys. Rev. A |

39. | L. Bergé and A. Couairon, “Nonlinear propagation of self-guided ultra-short pulses in ionized gases,” Phys. Plasmas |

40. | A. Dalgarno and A. E. Kingston, “The Refractive Indices and Verdet Constants of the Inert Gases,” Proc. R. Soc. Lond. A Math. Phys. Sci. |

41. | J. W. Thomas, |

**OCIS Codes**

(320.5520) Ultrafast optics : Pulse compression

(320.6629) Ultrafast optics : Supercontinuum generation

**ToC Category:**

Ultrafast Optics

**History**

Original Manuscript: June 10, 2009

Revised Manuscript: July 16, 2009

Manuscript Accepted: July 16, 2009

Published: July 24, 2009

**Citation**

Zhanxin Wang, Jiansheng Liu, Ruxin Li, and Zhizhan Xu, "Supercontinuum generation and pulse compression from gas filamentation of femtosecond laser pulses with different durations," Opt. Express **17**, 13841-13850 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-16-13841

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### References

- M. Hentschel, R. Kienberger, Ch. Spielmann, G. A. Reider, N. Milosevic, T. Brabec, P. Corkum, U. Heinzmann, M. Drescher, and F. Krausz, “Attosecond metrology,” Nature 414(6863), 509–513 (2001). [CrossRef] [PubMed]
- 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(5798), 443–446 (2008). [CrossRef]
- G. A. Mourou, T. Tajima, and S. V. Bulanov, “Optics in the relativistic regime,” Rev. Mod. Phys. 78(2), 309–371 (2006). [CrossRef]
- 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(1), 35–43 (2008). [CrossRef]
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