## Perturbative and non-perturbative aspects of optical filamentation in bulk dielectric media

Optics Express, Vol. 16, Issue 5, pp. 2971-2988 (2008)

http://dx.doi.org/10.1364/OE.16.002971

Acrobat PDF (815 KB)

### Abstract

The field of optical filament formation from initial ultrashort laser pulses in bulk dielectric media has now reached a high state of maturity, and has been studied in all three phases of matter, including long distance propagation in air, also termed light string propagation, water, and glass. From the earliest studies of light string propagation in air it was observed that conical emission, namely colored light emission off-axis from the filament, was a byproduct that accompanied the filamentation process.Since then several other byproducts accompanying optical filamentation have been studied, namely, white light or supercontinuum (SC) generation, third-harmonic (TH) generation, and X- and O-waves. Our goal in this paper is to review the theory and simulation of the byproducts accompanying optical filamentation, and to show that a unified approach is possible. Employing the angularly resolved spectrum, or *K* - *Ω* spectrum, a notion that has been used to great effect in the area of nonlinear conical waves, we demonstrate that a unified approach to the byproducts accompanying optical filamentation can be achieved using the twin notions of the Effective Three-Wave-Mixing (ETWM) picture of wave-mixing in the presence of filaments, which determines the locus of phase-matched wave generation in the angularly resolved spectrum, and the first-Born approximation to determine the profile of the angularly resolved spectrum. We summarize results of previous works and show that unlike the essentially non-perturbative core of the filament, several byproducts of filamentation can be treated as perturbative effects that have negligible feed-back effects on the filament itself. This should be of great utility for future studies of optimization of the yield of a given byproduct.

© 2008 Optical Society of America

## 1. Introduction

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

2. E. Garmire, R. Y. Chiao, and C. H. Townes, “Dynamics and characteristics of the self-trapping of intense light beams,” Phys. Rev. Lett. **16**, 347–349 (1966). [CrossRef]

3. P. L. Kelley, “Self-focusing of optical beams,” Phys. Rev. Lett. **15**, 1005–1008 (1965). [CrossRef]

4. P. Lallemand and N. Bloembergen, “Self-focusing of laser beams and stimulated Raman gain in liquids,” Phys. Rev. Lett. **15**, 1010–1012 (1965). [CrossRef]

5. Y. R. Shen and Y. J. Shaham, “Beam deterioration and stimulated Raman effect,” Phys. Rev. Lett. **15**, 1008–1010 (1965). [CrossRef]

6. 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**, 73–5 (1995). [CrossRef] [PubMed]

8. 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]

9. E. T. J. Nibbering, P. F. Curley, G. Grillon, B. S. Prade, M. A. Franco, F. Salin, and A. Mysyrowicz, “Conical emission from self-guided femtosecond pulses in air,” Opt.Lett. **21**, 62–64 (1996). [CrossRef] [PubMed]

10. J. Kasparian, R. Sauerbrey, D. Mondelain, S. Niedermeier, J. Yu, J. P. Wolf, Y. B. Andre, M. A. Franco, B. S. Prade, S. Tzortzakis, A. Mysyrowicz, M. Rodriguez, H Wille, and L Wöste, “Infrared extension of the supercontinuum generated by femtosecond terrawatt laser pulses propagating in the atmosphere,” Opt. Lett. **25**, 1397–1399 (2000). [CrossRef]

11. A. L. Gaeta, “Catastrophic Collapse of Ultrashort Pulses,” Phys. Rev. Lett. **84**, 3582–3585 (2000). [CrossRef] [PubMed]

12. N. Akozbek, A. Iwasaki, A. Becker, M. Scalora, S. L. Chin, and C. M. Bowden, “Third-harmonic generation and self-channeling in air using high-power femtosecond laser pulses,” Phys. Rev. Lett. **89**, 143901–143904 (2002). [CrossRef] [PubMed]

13. S. Tzortzakis, G. Mechain, G. Patalano, Y. B. Andre, B. Prade, M. Franco, A. Mysyrowicz, J. M. Munier, M. Gheudin, G. Beaudin, and P. Encrenaz, “Coherent subterahertz radiation from femtosecond infrared filaments in air,” Opt. Lett. **27**, 1944–1946 (2002). [CrossRef]

14. C. D’Amico, A. Houard, M. Franco, B. Prade, A. Mysyrowicz, A. Couairon, and V. T. Tikhonchuk, “Conical forward THz emission from femtosecond-laser-beam filamentation in air,” Phys. Rev. Lett. **98**, 235002–235004 (2007). [CrossRef] [PubMed]

15. P. DiTrapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, “Spontaneously generated X-shaped light bullets,” Phys. Rev. Lett. **91**, 093904–093907 (2003). [CrossRef]

16. M. Kolesik, E. M. Wright, and J. V. Moloney, “Dynamic nonlinear X-Waves for femtosecond pulse propagation in water,” Phys. Rev. Lett. **92**, 253901–253904 (2004). [CrossRef] [PubMed]

17. D. Faccio, M. A. Porras, A. Dubietis, F. Bragheri, A. Couairon, and P. DiTrapani, “Conical emission, pulse splitting, and X-Wave parametric amplification in nonlinear dynamics of ultrashort light pulses,” Phys. Rev. Lett. **96**, 193901–193904 (2006). [CrossRef] [PubMed]

18. M. A. Porras, A. Parola, and P. DiTrapani, “Nonlinear unbalanced O-waves: nonsolitary, conical light bullets in nonlinear dissipative media,” J. Opt. Soc. Am. B **22**, 1406–1413 (2005). [CrossRef]

19. M. A. Porras, A. Dubietis, E. Kucinskas, F. Bragheri, V. Degiorgio, A. Couairon, D. Faccio, and P. DiTrapani, “From X- to O-shaped spatiotemporal spectra of light filaments in water,” Opt. Lett. **30**, 3398–3400 (2005). [CrossRef]

20. A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Physics Reports **441**, 47–189 (2007). [CrossRef]

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

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

24. C. Sulem and P.-L. Sulem, “Focusing nonlinear Schrodinger equation and wave-packet collapse,” Nonlinear Anal.-Theory Methods Appl. **30**(2), 833–844 (1997). [CrossRef]

*R*is the finite-norm solution to

26. K. D. Moll, A. L. Gaeta, and G. Fibich, “Self-similar optical wave collapse: Observation of the Townes profile,” Phys. Rev. Lett. **90**, 203902–203905 (2003). [CrossRef] [PubMed]

27. G. G. Luther, A. C. Newell, and J. V. Moloney, “The effects of normal dispersion on collapse events,” Physica D **74**(1–2), 59–74 (1994). [CrossRef]

*O*(

*ε*) perturbation, the results extend to strong GVD (

*O*(1)) as shown in [27

27. G. G. Luther, A. C. Newell, and J. V. Moloney, “The effects of normal dispersion on collapse events,” Physica D **74**(1–2), 59–74 (1994). [CrossRef]

28. M. A. Porras, A. Parola, D. Faccio, A. Couairon, and P. Di Trapani, “Light-filament dynamics and the spatiotemporal instability of the Townes profile,” Phys. Rev. A **76**, 011803–011804 (2007). [CrossRef]

16. M. Kolesik, E. M. Wright, and J. V. Moloney, “Dynamic nonlinear X-Waves for femtosecond pulse propagation in water,” Phys. Rev. Lett. **92**, 253901–253904 (2004). [CrossRef] [PubMed]

16. M. Kolesik, E. M. Wright, and J. V. Moloney, “Dynamic nonlinear X-Waves for femtosecond pulse propagation in water,” Phys. Rev. Lett. **92**, 253901–253904 (2004). [CrossRef] [PubMed]

*K*- Ω spectrum, for the propagating pulse, a notion that has been used to great effect in the area of conical waves [29

29. D. Faccio, A. Matijosius, A. Dubietis, R. Piskarskas, A. Varanavicius, E. Gaizauskas, A. Piskarskas, A. Couairon, and P. DiTrapani, “Near- and far-field evolution of laser pulse filaments in Kerr media,” Phys. Rev. E **72**, 037601–037604 (2005). [CrossRef]

**92**, 253901–253904 (2004). [CrossRef] [PubMed]

30. M. Kolesik, E. M. Wright, and J. V. Moloney, “Interpretation of the spectrally resolved far field of femtosecond pulses propagating in bulk nonlinear dispersive media,” Opt. Express **13**, 10729–10741 (2005). [CrossRef] [PubMed]

*K*- Ω plane where phase-matched nonlinear mixing can occur and generate the range of byproducts. Second, we introduce an approximation to the spectrum that is formally akin to the first-Born approximation [31

31. M. Kolesik, E. M. Wright, and J. V. Moloney, “Supercontinuum and third-harmonic generation accompanying optical filamentation as first-order scattering processes,” Opt. Lett. **32**, 2816–2818 (2007). [CrossRef] [PubMed]

*K*- Ω spectrum and use it to explain specifically what we mean by byproducts accompanying optical filaments. Section 3 will introduce the ideas underlying the ETWM picture and the first-Born approximation, and we apply them to the examples of X-and O-waves in water, and conical emission and third-harmonic radiation in air and its relation to SC generation. Section 4 will give our summary and conclusions.

## 2. Optical filament propagation

### 2.1. Basic propagation and medium response equations

*r⃗*={

*x, y*} is the transverse position vector. The effects of linear medium dispersion on the propagating pulse are fully taken into account here through the dispersion relation

*z*-component of the wave-vector for an angular frequency

*ω*, transverse wavevector

*k⃗*, and for a medium of linear permittivity

*ε*(

*ω*). The spectral amplitudes

*A*(

*z*,

*k⃗, ω*) of the field evolve as functions of the propagation distance

*z*solely due to the medium nonlinearities, and obey the

*z*-propagated unidirectional pulse propagating equation (UPPE) equation [32

32. M. Kolesik, J. V. Moloney, and M. Mlejnek, “Unidirectional optical pulse propagation equation,” Phys. Rev. Lett. **89**, 28392–28395 (2002). [CrossRef]

33. M. Kolesik and J. V. Moloney, “Nonlinear optical pulse propagation simulation: From Maxwell’s to unidirectional equations,” Phys. Rev. E **70**, 036604–036614 (2004). [CrossRef]

*P*(

*r⃗, z, t*) implicitly depends on the propagating optical field through the medium nonlinearity.We write it in the form

### 2.2. (K - Ω) spectrum & byproducts

29. D. Faccio, A. Matijosius, A. Dubietis, R. Piskarskas, A. Varanavicius, E. Gaizauskas, A. Piskarskas, A. Couairon, and P. DiTrapani, “Near- and far-field evolution of laser pulse filaments in Kerr media,” Phys. Rev. E **72**, 037601–037604 (2005). [CrossRef]

*A*(

*z, k⃗, ω*) is actually the native representation of the ultrashort pulse (see Eqn.(3,4)). But even if using other implementations of the femtosecond pulse propagation model, the far field spectrum is straightforward to obtain since it is a mere Fourier transform of the real-space representation of the pulse at any given propagation distance

*z*:

*S*(

*ω*)|

^{2}, or just on-axis spectrum |

*A*(

*z, k⃗*=0,

*ω*)|

^{2}that is also frequently measured in experiments contain much less information about the optical waveform than the angularly resolved spectrum. However, it is instructive to illustrate this difference on the following example to emphasize the utility of far-field spectra.

36. F. Theberge, N. Akozbek, W. Liu, J. F. Gravel, and S. L. Chin, “Third harmonic beam profile generated in atmospheric air using femtosecond laser pulses,” Opt. Commun. **245**, 399–405 (2005). [CrossRef]

37. L. Berge, S. Skupin, G. Mejean, J. Kasparian, J. Yu, S. Frey, E. Salmon, and J. P Wolf, “Supercontinuum emission and enhanced self-guiding of infrared femtosecond filaments sustained by third-harmonic generation in air,” Phys. Rev. E **71**, 016602–016614 (2005). [CrossRef]

38. M. Kolesik, E. M. Wright, A. Becker, and J. V. Moloney, “Simulation of third-harmonic and supercontinuum generation for femtosecond pulses in air,” Appl. Phys. B **85**, 531–538 (2006). [CrossRef]

## 3. Effective Three-Wave Mixing & First Born Approximation

39. M. Kolesik, G. Katona, J. V. Moloney, and E. M. Wright, “Theory and simulation of supercontinuum generation in transparent bulk media,” Appl. Phys. B **77**, 185–195 (2003). [CrossRef]

40. M. Kolesik, G. Katona, J. V. Moloney, and E. M. Wright, “Physical factors limiting the spectral extent and band gap dependence of supercontinuum generation,” Phys. Rev. Lett. **91**, 043905–4 (2003). [CrossRef] [PubMed]

**92**, 253901–253904 (2004). [CrossRef] [PubMed]

30. M. Kolesik, E. M. Wright, and J. V. Moloney, “Interpretation of the spectrally resolved far field of femtosecond pulses propagating in bulk nonlinear dispersive media,” Opt. Express **13**, 10729–10741 (2005). [CrossRef] [PubMed]

31. M. Kolesik, E. M. Wright, and J. V. Moloney, “Supercontinuum and third-harmonic generation accompanying optical filamentation as first-order scattering processes,” Opt. Lett. **32**, 2816–2818 (2007). [CrossRef] [PubMed]

*n¯*

_{2}

*E*

^{2}(

*z, r⃗, t*). Next we will see how the non-perturbative Δχ leaves its perturbative signatures in the far field spectrum. Understanding the signatures will give us a tool for interpreting measured K-Ω spectra and thus infer Δχ with all the information about the pulse-medium interaction it carries.

### 3.1. Effective Three-Wave Mixing

*v*the velocity of the

_{p}*p*-th peak in the nonlinear response, and approximate the total nonlinear change in susceptibility Δχ as a sum of contributions from the main peaks:

^{χp}(

*z, r⃗, t*-

*z*/

*v*) becomes a slowly evolving function of its first argument

_{p}*z*. That is the only property we use, while the concrete shape of these response peaks is unimportant on the ETWM level. When (9,10) are inserted into (4), we change the time integration variable for each of the peak contributions to τ=→

*t*-

*z*/

*v*to get

_{p}*v*is chosen appropriately, then the function Δ

_{p}_{χp}(

*z*,

*r⃗*, τ) changes slower than the

*z*-dependent phase factor in the integrand. Consequently, the |

*A*(

*k⃗, ω*)|

^{2}will be highest for those

*k⃗, ω*which ensure that the latter vanishes:

*ω*-

*ω*

_{0}and the transverse wavenumber

*k*scatters an incident optical wave with frequency

*ω*

_{0}and the wave-vector {

*k⃗*=0

*, k*=

_{z}*K*(

*ω*

_{0}

*,*0)} to produce a scattered or output wave of frequency

*ω*, transverse wavenumber

*k*, and the

*z*component of the wavevector

*K*(

*ω, k*). We term this Effective Three-Wave Mixing, since the material wave is a “composite excitation” that arises in the core of the filament as a result of complex nonlinear processes. As such, it doesn’t obey any photon-like dispersion relation, and its velocity

*v*can in principle be “arbitrary.”

_{p}### 3.2. ETWM picture of X- & O-waves in water

41. D. Faccio, A. Averchi, A. Dubietis, P. Polesana, A. Piskarskas, P. DiTrapani, and A. Couairon, “Stimulated Raman X-waves in ultrashort optical pulse filamentation,” Opt. Lett. **32**, 184–186 (2007). [CrossRef]

42. D. Faccio, A. Averchi, A. Couairon, M. Kolesik, J. V. Moloney, A. Dubietis, G. Tamosauskas, P. Polesana, A. Piskarskas, and P. DiTrapani, “Spatio-temporal reshaping and X-Wave dynamics in optical filaments,” Opt. Express **15**, 13077–13095 (2007). [CrossRef] [PubMed]

*µ*J and 20 fs, while the other pulse is weaker, 0.2

*µ*J, and longer, 100 fs, with the initial intensity ten times less. The reason for having a week seed pulse is to use the strong pulse as as reshaping tool to act on the weaker pulse that will be transformed into an X-wave such that the group velocities of the ensuing waveforms will be equal.

*v*is a fitting parameter that can be determined by fitting the resulting curves to the spectrum. In the left panel of Fig. 5 the upper and lower curve (the latter including two disjoint components) correspond to Eq.(14) and Eq.(13), respectively, for the response peak velocity

_{p}*v*=

_{p}*c*/1.3423 (Note: the group index at pump wavelength is 1.3416). This value is in agreement with the velocity obtained directly from the simulated response temporal and spatial profiles. Thus, the three-wave mixing phase matching relation allowed us to measure the group velocity of the response peak and thus also the group velocity of the optical pulse inside the filament, and identified the long “arms” in the spectrum as due to a scattering process in which the original carrier wave scatters off the response peak potential. Also, it shows that the seed pulse is transformed due to its scattering on the potential created by the pump pulse. The intensity of the seed pulse can be arbitrary small, and this is an example of a purely “perturbative” X-wave.

*v*. Thus, the effective three-wave mixing picture gives us actually some quantitative information about the filament, namely the pulse group velocity inside the filament core.

_{p}*v*is the velocity of the response peak that generated the X-wave from the seed radiation. After elimination of

_{p}*K*(

*ω*

_{X},

*k*) and

_{X}*ω*

_{X}, we obtain this phase-matching relation:

### 3.3. First Born approximation

*z*to obtain an expression for the far-field spectrum

*A*(

*k⃗, ω*)≡

*A*(

*z*→∞,

*k⃗, ω*)

30. M. Kolesik, E. M. Wright, and J. V. Moloney, “Interpretation of the spectrally resolved far field of femtosecond pulses propagating in bulk nonlinear dispersive media,” Opt. Express **13**, 10729–10741 (2005). [CrossRef] [PubMed]

31. M. Kolesik, E. M. Wright, and J. V. Moloney, “Supercontinuum and third-harmonic generation accompanying optical filamentation as first-order scattering processes,” Opt. Lett. **32**, 2816–2818 (2007). [CrossRef] [PubMed]

**32**, 2816–2818 (2007). [CrossRef] [PubMed]

*E*

^{2}profile of a pulse that has just gone through splitting. This represents the shape of the first-Born scattering potential at a fixed propagation distance. It can be decomposed into the “envelope,” or DC part (shown in Fig. 8 in the middle panel) and into a component oscillating at about second-harmonic frequency (shown in Fig. 8 in the right panel). The former is responsible for the SC production. The high-frequency of the latter causes the scattering waves to acquire additional frequency shift roughly twice the fundamental and thus produces the third-harmonic radiation. Thus, SC and TH are generated through the same scattering process, but they each respond to a different aspect of the nonlinear response.

*A*(

*k⃗, ω*)| to the “scattering potential” Δχ through a simple linear formula. This make it possible, in principle, to obtain knowledge about the pulse evolution inside the filament from a relatively simple measurement of the angularly resolved spectrum.

## 4. Conclusions

## 5. Acknowledgments

## References and links

1. | R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. |

2. | E. Garmire, R. Y. Chiao, and C. H. Townes, “Dynamics and characteristics of the self-trapping of intense light beams,” Phys. Rev. Lett. |

3. | P. L. Kelley, “Self-focusing of optical beams,” Phys. Rev. Lett. |

4. | P. Lallemand and N. Bloembergen, “Self-focusing of laser beams and stimulated Raman gain in liquids,” Phys. Rev. Lett. |

5. | Y. R. Shen and Y. J. Shaham, “Beam deterioration and stimulated Raman effect,” Phys. Rev. Lett. |

6. | 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. |

7. | L. Wöste, C. Wedekind, H. Wille, P. Rairoux, B. Stein, S. Nikolov, C. Werner, S. Niedermeier, F. Ronneberger, H. Schillinger, and R. Sauerbrey, “Femtosecond atmospheric lamp.” AT-Fachverlag, Stuttgart, Laser und Opto-electronik |

8. | M. Mlejnek, E. M. Wright, and J. V. Moloney, “Dynamic spatial replenishment of femtosecond pulses propagating in air,” Opt. Lett. |

9. | E. T. J. Nibbering, P. F. Curley, G. Grillon, B. S. Prade, M. A. Franco, F. Salin, and A. Mysyrowicz, “Conical emission from self-guided femtosecond pulses in air,” Opt.Lett. |

10. | J. Kasparian, R. Sauerbrey, D. Mondelain, S. Niedermeier, J. Yu, J. P. Wolf, Y. B. Andre, M. A. Franco, B. S. Prade, S. Tzortzakis, A. Mysyrowicz, M. Rodriguez, H Wille, and L Wöste, “Infrared extension of the supercontinuum generated by femtosecond terrawatt laser pulses propagating in the atmosphere,” Opt. Lett. |

11. | A. L. Gaeta, “Catastrophic Collapse of Ultrashort Pulses,” Phys. Rev. Lett. |

12. | N. Akozbek, A. Iwasaki, A. Becker, M. Scalora, S. L. Chin, and C. M. Bowden, “Third-harmonic generation and self-channeling in air using high-power femtosecond laser pulses,” Phys. Rev. Lett. |

13. | S. Tzortzakis, G. Mechain, G. Patalano, Y. B. Andre, B. Prade, M. Franco, A. Mysyrowicz, J. M. Munier, M. Gheudin, G. Beaudin, and P. Encrenaz, “Coherent subterahertz radiation from femtosecond infrared filaments in air,” Opt. Lett. |

14. | C. D’Amico, A. Houard, M. Franco, B. Prade, A. Mysyrowicz, A. Couairon, and V. T. Tikhonchuk, “Conical forward THz emission from femtosecond-laser-beam filamentation in air,” Phys. Rev. Lett. |

15. | P. DiTrapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, “Spontaneously generated X-shaped light bullets,” Phys. Rev. Lett. |

16. | M. Kolesik, E. M. Wright, and J. V. Moloney, “Dynamic nonlinear X-Waves for femtosecond pulse propagation in water,” Phys. Rev. Lett. |

17. | D. Faccio, M. A. Porras, A. Dubietis, F. Bragheri, A. Couairon, and P. DiTrapani, “Conical emission, pulse splitting, and X-Wave parametric amplification in nonlinear dynamics of ultrashort light pulses,” Phys. Rev. Lett. |

18. | M. A. Porras, A. Parola, and P. DiTrapani, “Nonlinear unbalanced O-waves: nonsolitary, conical light bullets in nonlinear dissipative media,” J. Opt. Soc. Am. B |

19. | M. A. Porras, A. Dubietis, E. Kucinskas, F. Bragheri, V. Degiorgio, A. Couairon, D. Faccio, and P. DiTrapani, “From X- to O-shaped spatiotemporal spectra of light filaments in water,” Opt. Lett. |

20. | A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Physics Reports |

21. | J. Kasparian and J.-P. Wolf, “Physics and applications of atmospheric nonlinear optics and filamentation,” Opt. Express |

22. | V. E. Zakharov and A. B. Shabat, “Exact theory of 2-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP |

23. | S. N. Vlasov, V. A. Petrishschev, and V. I. Talanov, “Averaged description of wave beams in linear and nonlinear media (the method of moments),” Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofyzika |

24. | C. Sulem and P.-L. Sulem, “Focusing nonlinear Schrodinger equation and wave-packet collapse,” Nonlinear Anal.-Theory Methods Appl. |

25. | C. Sulem and P.-L. Sulem, |

26. | K. D. Moll, A. L. Gaeta, and G. Fibich, “Self-similar optical wave collapse: Observation of the Townes profile,” Phys. Rev. Lett. |

27. | G. G. Luther, A. C. Newell, and J. V. Moloney, “The effects of normal dispersion on collapse events,” Physica D |

28. | M. A. Porras, A. Parola, D. Faccio, A. Couairon, and P. Di Trapani, “Light-filament dynamics and the spatiotemporal instability of the Townes profile,” Phys. Rev. A |

29. | D. Faccio, A. Matijosius, A. Dubietis, R. Piskarskas, A. Varanavicius, E. Gaizauskas, A. Piskarskas, A. Couairon, and P. DiTrapani, “Near- and far-field evolution of laser pulse filaments in Kerr media,” Phys. Rev. E |

30. | M. Kolesik, E. M. Wright, and J. V. Moloney, “Interpretation of the spectrally resolved far field of femtosecond pulses propagating in bulk nonlinear dispersive media,” Opt. Express |

31. | M. Kolesik, E. M. Wright, and J. V. Moloney, “Supercontinuum and third-harmonic generation accompanying optical filamentation as first-order scattering processes,” Opt. Lett. |

32. | M. Kolesik, J. V. Moloney, and M. Mlejnek, “Unidirectional optical pulse propagation equation,” Phys. Rev. Lett. |

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

34. | A. Couairon and L. Berge, “Modeling the filamentation of the ultra-short pulses in ionizing media,” Phys. Plasmas |

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

36. | F. Theberge, N. Akozbek, W. Liu, J. F. Gravel, and S. L. Chin, “Third harmonic beam profile generated in atmospheric air using femtosecond laser pulses,” Opt. Commun. |

37. | L. Berge, S. Skupin, G. Mejean, J. Kasparian, J. Yu, S. Frey, E. Salmon, and J. P Wolf, “Supercontinuum emission and enhanced self-guiding of infrared femtosecond filaments sustained by third-harmonic generation in air,” Phys. Rev. E |

38. | M. Kolesik, E. M. Wright, A. Becker, and J. V. Moloney, “Simulation of third-harmonic and supercontinuum generation for femtosecond pulses in air,” Appl. Phys. B |

39. | M. Kolesik, G. Katona, J. V. Moloney, and E. M. Wright, “Theory and simulation of supercontinuum generation in transparent bulk media,” Appl. Phys. B |

40. | M. Kolesik, G. Katona, J. V. Moloney, and E. M. Wright, “Physical factors limiting the spectral extent and band gap dependence of supercontinuum generation,” Phys. Rev. Lett. |

41. | D. Faccio, A. Averchi, A. Dubietis, P. Polesana, A. Piskarskas, P. DiTrapani, and A. Couairon, “Stimulated Raman X-waves in ultrashort optical pulse filamentation,” Opt. Lett. |

42. | D. Faccio, A. Averchi, A. Couairon, M. Kolesik, J. V. Moloney, A. Dubietis, G. Tamosauskas, P. Polesana, A. Piskarskas, and P. DiTrapani, “Spatio-temporal reshaping and X-Wave dynamics in optical filaments,” Opt. Express |

**OCIS Codes**

(190.5530) Nonlinear optics : Pulse propagation and temporal solitons

(320.2250) Ultrafast optics : Femtosecond phenomena

**ToC Category:**

Nonlinear Localization and Filamentation Effects

**History**

Original Manuscript: January 11, 2008

Revised Manuscript: February 12, 2008

Manuscript Accepted: February 12, 2008

Published: February 19, 2008

**Virtual Issues**

Focus Serial: Frontiers of Nonlinear Optics (2007) *Optics Express*

**Citation**

M. Kolesik and J. V. Moloney, "Perturbative and non-perturbative
aspects of optical filamentation in bulk
dielectric media," Opt. Express **16**, 2971-2988 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-5-2971

Sort: Year | Journal | Reset

### References

- R. Y. Chiao, E. Garmire, and C. H. Townes, "Self-trapping of optical beams," Phys. Rev. Lett. 13, 479-482 (1964). [CrossRef]
- E. Garmire, R. Y. Chiao, and C. H. Townes, "Dynamics and characteristics of the self-trapping of intense light beams," Phys. Rev. Lett. 16, 347-349 (1966). [CrossRef]
- P. L. Kelley, "Self-focusing of optical beams," Phys. Rev. Lett. 15, 1005-1008 (1965). [CrossRef]
- P. Lallemand and N. Bloembergen, "Self-focusing of laser beams and stimulated Raman gain in liquids," Phys. Rev. Lett. 15, 1010-1012 (1965). [CrossRef]
- Y. R. Shen and Y. J. Shaham, "Beam deterioration and stimulated Raman effect," Phys. Rev. Lett. 15, 1008-1010 (1965). [CrossRef]
- 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]
- L. Woste, C. Wedekind, H. Wille, P. Rairoux, B. Stein, S. Nikolov, C. Werner, S. Niedermeier, F. Ronneberger, H. Schillinger, and R. Sauerbrey, "Femtosecond atmospheric lamp." AT-Fachverlag, Stuttgart, Laser und Optoelectronik 29, 51-53 (1997).
- 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]
- E. T. J. Nibbering, P. F. Curley, G. Grillon, B. S. Prade, M. A. Franco, F. Salin, and A. Mysyrowicz, "Conical emission from self-guided femtosecond pulses in air," Opt. Lett. 21, 62-64 (1996). [CrossRef] [PubMed]
- J. Kasparian, R. Sauerbrey, D. Mondelain, S. Niedermeier, J. Yu, J. P. Wolf, Y. B. Andre, M. A. Franco, B. S. Prade, S. Tzortzakis, A. Mysyrowicz, M. Rodriguez, H. Wille, and L. Woste, "Infrared extension of the supercontinuum generated by femtosecond terrawatt laser pulses propagating in the atmosphere," Opt. Lett. 25, 1397-99 (2000). [CrossRef]
- A. L. Gaeta, "Catastrophic collapse of ultrashort pulses," Phys. Rev. Lett. 84, 3582-3585 (2000). [CrossRef] [PubMed]
- N. Akozbek, A. Iwasaki, A. Becker, M. Scalora, S. L. Chin, and C. M. Bowden, "Third-harmonic generation and self-channeling in air using high-power femtosecond laser pulses," Phys. Rev. Lett. 89, 143901-4 (2002). [CrossRef] [PubMed]
- S. Tzortzakis, G. Mechain, G. Patalano, Y. B. Andre, B. Prade, M. Franco, A. Mysyrowicz, J. M. Munier, M. Gheudin, G. Beaudin, and P. Encrenaz, "Coherent subterahertz radiation from femtosecond infrared filaments in air," Opt. Lett. 27, 1944-1946 (2002). [CrossRef]
- C. D’Amico, A. Houard, M. Franco, B. Prade, A. Mysyrowicz, A. Couairon. and V. T. Tikhonchuk, " Conical forward THz emission from femtosecond-laser-beam filamentation in air, " Phys. Rev. Lett. 98, 235002-4 (2007). [CrossRef] [PubMed]
- P. DiTrapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, "Spontaneously generated X-shaped light bullets," Phys. Rev. Lett. 91, 093904-7 (2003). [CrossRef]
- M. Kolesik, E. M. Wright, and J. V. Moloney, "Dynamic nonlinear X-Waves for femtosecond pulse propagation in water," Phys. Rev. Lett. 92, 253901-4 (2004). [CrossRef] [PubMed]
- D. Faccio, M. A. Porras, A. Dubietis, F. Bragheri, A. Couairon, and P. DiTrapani, "Conical emission, pulse splitting, and X-Wave parametric amplification in nonlinear dynamics of ultrashort light pulses," Phys. Rev. Lett. 96, 193901-4 (2006). [CrossRef] [PubMed]
- M. A. Porras, A. Parola, and P. DiTrapani, "Nonlinear unbalanced O-waves: nonsolitary, conical light bullets in nonlinear dissipative media," J. Opt. Soc. Am. B 22, 1406-1413 (2005). [CrossRef]
- M. A. Porras, A. Dubietis, E. Kucinskas, F. Bragheri, V. Degiorgio, A. Couairon, D. Faccio, and P. DiTrapani, "From X- to O-shaped spatiotemporal spectra of light filaments in water," Opt. Lett. 30, 3398-3400 (2005). [CrossRef]
- A. Couairon and A. Mysyrowicz, "Femtosecond filamentation in transparent media," Physics Reports 441, 47-189 (2007). [CrossRef]
- J. Kasparian, and J.-P. Wolf, "Physics and applications of atmospheric nonlinear optics and filamentation," Opt. Express 16, 466-493 (2008). [CrossRef] [PubMed]
- V. E. Zakharov and A. B. Shabat, "Exact theory of 2-dimensional self-focusing and one-dimensional selfmodulation of waves in nonlinear media," Sov. Phys. JETP 34, 62-69 (1972).
- S. N. Vlasov, V. A. Petrishschev, and V. I. Talanov, "Averaged description of wave beams in linear and nonlinear media (the method of moments)," Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofyzika 14, 1353-1363 (1971) (in Russian).
- C. Sulem and P.-L. Sulem, "Focusing nonlinear Schrodinger equation and wave-packet collapse," Nonlinear Anal.-Theory Methods Appl. 30, 833-844 (1997). [CrossRef]
- C. Sulem and P.-L. Sulem, The Nonlinear Schr¨odinger Equation: Self-focusing and wave collapse, (Springer Series on Applied Mathematical Sciences, 1999) Vol. 139.
- K. D. Moll, A. L. Gaeta, and G. Fibich, "Self-similar optical wave collapse: Observation of the Townes profile," Phys. Rev. Lett. 90, 203902-5 (2003). [CrossRef] [PubMed]
- G. G. Luther, A. C. Newell, and J. V. Moloney, "The effects of normal dispersion on collapse events," Physica D 74, 59-74 (1994). [CrossRef]
- M. A. Porras, A. Parola, D. Faccio, A. Couairon, and P. Di Trapani, " Light-filament dynamics and the spatiotemporal instability of the Townes profile, " Phys. Rev. A 76, 011803-4 (2007). [CrossRef]
- D. Faccio, A. Matijosius, A. Dubietis, R. Piskarskas, A. Varanavicius, E. Gaizauskas, A. Piskarskas, A. Couairon, and P. DiTrapani, "Near- and far-field evolution of laser pulse filaments in Kerr media," Phys. Rev. E 72, 037601-4 (2005). [CrossRef]
- M. Kolesik, E. M. Wright, and J. V. Moloney, "Interpretation of the spectrally resolved far field of femtosecond pulses propagating in bulk nonlinear dispersive media," Opt. Express 13, 10729-10741 (2005). [CrossRef] [PubMed]
- M. Kolesik, E. M. Wright, and J. V. Moloney, "Supercontinuum and third-harmonic generation accompanying optical filamentation as first-order scattering processes," Opt. Lett. 32, 2816-2818 (2007). [CrossRef] [PubMed]
- M. Kolesik, J. V. Moloney, and M. Mlejnek, "Unidirectional optical pulse propagation equation," Phys. Rev. Lett. 89, 28392-5 (2002). [CrossRef]
- M. Kolesik and J. V. Moloney, "Nonlinear optical pulse propagation simulation: From Maxwell’s to unidirectional equations," Phys. Rev. E 70, 036604-14 (2004). [CrossRef]
- A. Couairon and L. Berge, "Modeling the filamentation of the ultra-short pulses in ionizing media," Phys. Plasmas 7, 193-209 (2000). [CrossRef]
- L. Berge and A. Couairon, "Nonlinear propagation of self-guided ultra-short pulses in ionized gases," Phys. Plasmas 7, 210-230 (2000). [CrossRef]
- F. Theberge, N. Akozbek, W. Liu, J. F. Gravel, and S. L. Chin, "Third harmonic beam profile generated in atmospheric air using femtosecond laser pulses," Opt. Commun. 245, 399-405 (2005). [CrossRef]
- L. Berge, S. Skupin, G. Mejean, J. Kasparian, J. Yu, S. Frey, E. Salmon, and J. P. Wolf, "Supercontinuum emission and enhanced self-guiding of infrared femtosecond filaments sustained by third-harmonic generation in air," Phys. Rev. E 71, 016602-14 (2005). [CrossRef]
- M. Kolesik, E. M. Wright, A. Becker, and J. V. Moloney, "Simulation of third-harmonic and supercontinuum generation for femtosecond pulses in air," Appl. Phys. B 85, 531-538 (2006). [CrossRef]
- M. Kolesik, G. Katona, J. V. Moloney, and E. M. Wright, "Theory and simulation of supercontinuum generation in transparent bulk media," Appl. Phys. B 77, 185-195 (2003). [CrossRef]
- M. Kolesik, G. Katona, J. V. Moloney, and E. M. Wright, "Physical factors limiting the spectral extent and band gap dependence of supercontinuum generation," Phys. Rev. Lett. 91, 043905-4 (2003). [CrossRef] [PubMed]
- D. Faccio, A. Averchi, A. Dubietis, P. Polesana, A. Piskarskas, P. DiTrapani, and A. Couairon, "Stimulated Raman X-waves in ultrashort optical pulse filamentation," Opt. Lett. 32, 184-186 (2007). [CrossRef]
- D. Faccio, A. Averchi, A. Couairon, M. Kolesik, J. V. Moloney, A. Dubietis, G. Tamosauskas, P. Polesana, A. Piskarskas, and P. DiTrapani, "Spatio-temporal reshaping and X-Wave dynamics in optical filaments," Opt. Express 15, 13077-95 (2007). [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.

### Multimedia

Multimedia Files | Recommended Software |

» Media 1: MPG (1530 KB) |

« Previous Article | Next Article »

OSA is a member of CrossRef.