## A dissipative attractor in the spatiotemporal collapse of ultrashort light pulses

Optics Express, Vol. 18, Issue 7, pp. 7376-7383 (2010)

http://dx.doi.org/10.1364/OE.18.007376

Acrobat PDF (660 KB)

### Abstract

Spatiotemporal self-focusing in nonlinear lossy media pushes ultrashort pulses towards a universal, non-solitary and non-conical light-bullet wave state defined by the medium solely, and characterized by maximum energy losses. Its stationary propagation relies on a balance between nonlinear losses and the refuelling effect of self-focusing. No balancing gain is required for stationarity. These purely lossy dissipative light-bullets can explain many aspects of the filamentary dynamics in nonlinear media with anomalous dispersion.

© 2010 Optical Society of America

2. H. S. Eisenberg, R. Morandotti, and Y. Silberberg, “Kerr spatiotemporal self-Focusing in a planar glass waveguide,” Phys. Rev. Lett. **82**, 043902 (2001). [CrossRef]

3. R. McLeod, K. Wagner, and S. Blair, “(3+1)-dimensional soliton dragging logic,” Phys. Rev. A **52**, 3254–3278 (1995). [CrossRef] [PubMed]

4. Y. Silberberg, “Collapse of optical pulses,” Opt. Lett. **15**, 1282–1284 (1990). [CrossRef] [PubMed]

5. N. Akhmediev and J. M. Soto Crespo, “Generation of a train of three-dimensional optical solitons in a self-focusing medium,” Phys. Rev. A **47**, 1358–1364 (1993). [CrossRef] [PubMed]

6. G. Fibich and B. Ilan, “Optical light bullets in a pure Kerr medium,” Opt. Lett. **29**, 887–889 (2004). [CrossRef] [PubMed]

*solitary*type, or spatiotemporal solitons. In filamentation experiments, the phase mechanism is widely accepted to be the defocusing effect of the lowering of the refractive index due to the light-induced plasma [7

7. 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]

8. S. Henz and J. Herrmann, “Two-dimensional spatial optical solitons in bulk Kerr media stabilized by self-induced multiphoton ionization: Variational approach,” Phys. Rev. E **53**, 4092–4097 (1996). [CrossRef]

2. H. S. Eisenberg, R. Morandotti, and Y. Silberberg, “Kerr spatiotemporal self-Focusing in a planar glass waveguide,” Phys. Rev. Lett. **82**, 043902 (2001). [CrossRef]

7. 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]

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

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

13. A. Dubietis, E. Gaizauskas, G. Tamosauskas, and P. Di Trapani, “Light Filaments without Self-Channeling,” Phys. Rev. Lett. **92**, 253903 (2004). [CrossRef] [PubMed]

14. M. Kolesik and J. V. Moloney, “Self-healing femtosecond light filaments,” Opt. Lett. **29**, 590–592 (2004). [CrossRef] [PubMed]

*conical*LBs that are resistant to nonlinear losses [15

15. M. A. Porras, A. Parola, D. Faccio, A. Dubietis, and P. Di Trapani, “Nonlinear unbalanced Bessel beams: Stationary conical waves supported by nonlinear losses,” Phys. Rev. Lett. **93**, 153902 (2004). [CrossRef] [PubMed]

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

15. M. A. Porras, A. Parola, D. Faccio, A. Dubietis, and P. Di Trapani, “Nonlinear unbalanced Bessel beams: Stationary conical waves supported by nonlinear losses,” Phys. Rev. Lett. **93**, 153902 (2004). [CrossRef] [PubMed]

17. M. A. Porras and A. Parola, “Nonlinear unbalanced Bessel beams in the collapse of Gaussian beams arrested by nonlinear losses,” Opt. Lett. **33**, 1738–1740 (2008). [CrossRef] [PubMed]

18. D. Faccio, M. Clerici, A. Averchi, O. Jedrkiewicz, S. Tzortzakis, D. Papazoglou, F. Bragheri, L. Tartara, A. Trita, S. Henin, I. Cristiani, A. Couairon, and P. Di Trapani, “Kerr-induced spontaneous Bessel beam formation in the regime of strong two-photon absorption,” Opt. Express **16**, 8213–8218 (2008). [CrossRef] [PubMed]

*dissipative*type, or dissipative LBs (DLBs), with the peculiarity of featuring only losses (no gain). Stationarity in purely absorbing media is possible because these DLBs carry, as conical LBs, infinite energy. Further, stationarity does not require the conical inward energy flux characteristic of conical LBs, but nonlinear compression alone provides the needed inward flux to sustain the stationarity.

17. M. A. Porras and A. Parola, “Nonlinear unbalanced Bessel beams in the collapse of Gaussian beams arrested by nonlinear losses,” Opt. Lett. **33**, 1738–1740 (2008). [CrossRef] [PubMed]

18. D. Faccio, M. Clerici, A. Averchi, O. Jedrkiewicz, S. Tzortzakis, D. Papazoglou, F. Bragheri, L. Tartara, A. Trita, S. Henin, I. Cristiani, A. Couairon, and P. Di Trapani, “Kerr-induced spontaneous Bessel beam formation in the regime of strong two-photon absorption,” Opt. Express **16**, 8213–8218 (2008). [CrossRef] [PubMed]

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

10. L. Bergé and S. Skupin, “Self-channeling of ultrashort laser pulses in materials with anomalous dispersion,” Phys. Rev. E **71**, 065601 (2005). [CrossRef]

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

10. L. Bergé and S. Skupin, “Self-channeling of ultrashort laser pulses in materials with anomalous dispersion,” Phys. Rev. E **71**, 065601 (2005). [CrossRef]

19. N. Akhmediev and A. Ankiewicz (Eds.), *Dissipative solitons*, Lecture Notes in Physics 661 (Springer, 2005). [CrossRef]

20. P. Grelu, J. M. Soto-Crespo, and N. Akhmediev, “Light bullets and dynamic pattern formation in nonlinear dissipative systems,” Opt. Express **13**, 9352–9360 (2005). [CrossRef] [PubMed]

21. J. M. Soto-Crespo, P. Grelu, and N. Akhmediev, “Optical bullets and “rockets” in nonlinear dissipative systems and their transformations and interactions,” Opt. Express **14**, 4013–4025 (2006). [CrossRef] [PubMed]

22. W. H. Renninger, A. Chong, and F. W. Wise, “Dissipative solitons in normal-dispersion fiber lasers,” Phys. Rev. A **77**, 023814 (2008). [CrossRef]

23. Y. Tanguy, T. Ackermann, W. J. Firth, and R. Jäger, “Realization of a semiconductor-based cavity soliton laser,” Phys. Rev. Lett. **100**, 013907 (2008). [CrossRef] [PubMed]

*A*of a wave packet

*E*=

*A*exp(-

*iω*

_{0}

*t*+

*ik*

_{0}

*z*) of carrier frequency

*ω*

_{0}, or wave length λ

_{0}= 2

*πc*/

*ω*

_{0}, propagating along

*z*. In Eq. (1), ∆

_{⊥}=

*∂*

_{x}

^{2}+

*∂*

_{y}

^{2}

*t*

^{′}=

*t*-

*k*

^{′}

_{0}

*z*is the local time,

*k*=

*n*(

*ω*)

*ω*/

*c*is the propagation constant in the medium,

*n*(

*ω*) the refractive index,

*c*the speed of light in vacuum,

*k*

^{(n)}

_{0}=

*d*/

^{n}k*dω*∣

^{n}_{ω0}), and

*n*

_{2}> 0 the nonlinear refractive index. In media with anomalous dispersion (

*k*

^{″}

_{0}< 0), the diffraction and dispersion terms have equal signs, and a pulse whose envelope A depends on the transversal and temporal coordinates through

*r*= (

*x*

^{2}+

*y*

^{2}+

*t*

^{′}

^{2}/

*k*

_{0}∣

*k*

^{″}

_{0}∣)

^{1/2}, will retain this property on propagation, and can self-focus symmetrically in space and time [4

4. Y. Silberberg, “Collapse of optical pulses,” Opt. Lett. **15**, 1282–1284 (1990). [CrossRef] [PubMed]

*β*

^{(K)}> 0 in Eq. (1) describes nonlinear losses due

*K*-photon absorption.

*λ*

_{0}= 1550 nm, Fig. 1 shows the change along

*z*of the peak intensity and width of the pulse, calculated from Eq. (1) for spatiotemporal symmetric Gaussian pulses

*A*= (2

*P*/

*πσ*

^{2})

^{1/2}exp(-

*r*

^{2}/

*σ*

^{2}) launched in the medium with different peak powers

*P*above the critical peak power

*P*

_{cr}= 2.157

*λ*

_{0}

^{2}/4

*πn*(

*ω*

_{0})

*n*

_{2}for spatiotemporal self-focusing [24]. Increasing the energy of the input pulse, the light “segments” of nearly constant high intensity and narrow width become longer, and a number of light “bursts” beyond the segment are formed. These segments may extend beyond the Rayleigh distance

*z*=

_{R}*πσ*

^{2}/λ

_{0}of the input pulse, and survive by several hundred times the Rayleigh distance expected from their width. These facts have been observed in self-channelling experiments and numerical simulations [9

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

10. L. Bergé and S. Skupin, “Self-channeling of ultrashort laser pulses in materials with anomalous dispersion,” Phys. Rev. E **71**, 065601 (2005). [CrossRef]

**71**, 065601 (2005). [CrossRef]

*A*(

*r,z*) =

*a*(

*r*)exp[

*iφ*(

*r*)]exp(-

*iδz*), where the real amplitude

*a*(

*r*) > 0 and phase

*φ*(

*r*) must satisfy

*d*/

*dr*. The second equation, written as

*N*(

*r*) = -

*F*(

*r*) for short, establishes that in stationary solutions the energy losses

*N*(

*r*) within any sphere or radius

*r*must be refuelled by an inward radial energy flux -

*F*(

*r*) from outside [the actual energy losses and flux per unit propagation length in, e.g., J/cm, are (

*k*

_{0}∣

*k*

^{″}

_{0}∣)

^{1/2}times

*N*(

*r*) and

*F*(

*r*)]. Localized solutions (i.e., approaching zero at large

*r*) with

*δ*< 0 are solitary LBs. They exist only in transparent media (

*β*

^{(K)}= 0), require a stabilizing phase mechanism other than the Kerr nonlinearity [an additional term in (2)], and a precise relation between the peak intensity

*I*

_{0}=

*a*

^{2}(0) and the wave vector shift

*δ*. Solutions with

*δ*> 0 are conical LBs, or nonlinear O waves [16

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

*r*, and then they carry infinite energy. In their two-dimensional versions (nonlinear Bessel beams [15

15. M. A. Porras, A. Parola, D. Faccio, A. Dubietis, and P. Di Trapani, “Nonlinear unbalanced Bessel beams: Stationary conical waves supported by nonlinear losses,” Phys. Rev. Lett. **93**, 153902 (2004). [CrossRef] [PubMed]

*r*during propagation. In three dimensions, the energy flows along hyper-cones of angle

*r*. Given

*δ*, or a cone angle, the conical flux can sustain the stationarity for any

*I*

_{0}up to a maximum value that depends on the cone angle and the medium properties [15

**93**, 153902 (2004). [CrossRef] [PubMed]

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

*δ*= 0 whose stationarity with losses does not rely on a specific wave geometry, but only on the nonlinear properties of the medium. Seeking for a balance between the Kerr nonlinearity and nonlinear losses, we try solutions of the form

*a*(

*r*) ~

*b*/

*r*at large

^{α}*r*, where

*b*> 0 is a constant, and localization requires

*α*> 0. Assuming for the moment that losses occur only in a region about the pulse center, and that the total losses are finite, i.e., that

*N*(

*r*) grows up to a maximum value

*N*< ∞ with increasing

_{T}*r*, Eq. (3) yields

*ϕ*

^{′}~ -

*k*

_{0}

*N*

_{T}*r*

^{2}

^{(α-1)}/4

*πb*

^{2}at large

*r*, and Eq. (2) yields

*α*must be consistent with this equation. For instance, for

*δ*> 0, the first and the Kerr terms are subleading with respect to the conical term 2

*k*

_{0}

*δb*, which is constant, and must be balanced by the loss term. This implies the decay with

*α*= 1 of conical LBs. In absence of a cone angle (

*δ*= 0), the decay with

*α*= 1 is no longer consistent. Instead, a balance between the loss and Kerr terms is possible for 4(

*α*- 1) = -2

*α*, which yields

*α*= 2/3, the first term being then subleading. Equating the coefficients of the loss and Kerr terms, we also obtain

*b*

^{6}= (

*N*

_{T}

^{2}

*n*

_{0}/32

*π*

^{2}

*n*

_{2}).

*δ*= 0 [Fig. 2(a), solid curves], and to present the decay

*b*/

*r*

^{2/3}for nonlinear losses compensation by Kerr nonlinearity [Fig. 2(a), dotted blue curves]. Losses occur only in the pulse center since the

*N*(

*r*) stabilizes in a constant equal to the total losses

*N*[Fig. 2(b), solid curves]. The amplitude and phase of these LBs are such that an inward energy flux -

_{T}*F*(

*r*) equals to the losses

*N*(

*r*). In absence of a cone angle, this flux is an effect of Kerr compression only. As their peak intensity

*I*

_{0}=

*a*

^{2}(0) is higher, they become wider and dissipate more total energy

*N*. The total losses

_{T}*N*with increasing peak intensity are compiled in Fig. 2(c, solid curve). For comparison, the total losses of conical LBs (

_{T}*δ*> 0) with increasing peak intensity are also shown in Fig. 2(c, dashed curves), and are seen to be lower.

*α*= 1/(2

*K*- 3) and

*b*

^{4K-6}= 2(3 - 2

*Kα*)

^{2}

*n*

_{2}/[

*β*

^{(K)2}

*n*

_{0}]. These formulas give indeed the behavior of the DLB of maximum losses [Fig. 2(d), dotted blue curves], which is numerically calculated for different values of

*K*[Fig.2(d), dashed red curves]. Differently from DLBs of lower intensity, this LB dissipates energy not only in the center, but everywhere along its radial profile, and is unique given a material and a carrier wave length.

*I*

_{0,m}, of the DLB with maximum losses. This coincidence holds irrespective of the self-focusing pulse, e. g., of its initial power [Figs. 1 from (a) to (c)] and of its width (Figs. 1 and 3). Also, the intensity remains nearly constant in spite that the energy decreases drastically along the segment [Fig. 3(a), dashed blue curve] due to an energy loss per unit length (dotted green curve) comparable to the total energy. The formation of these quasi-LBs upon spatiotemporal self-focusing can be explained from a spontaneous balance of Kerr compression and nonlinear losses to form a DLB. As suggested by previous works, [15

**93**, 153902 (2004). [CrossRef] [PubMed]

*I*

_{0,m}[Fig. 3(c)], the inner part of the radial profile (solid curves) matches that of the attractive DLB (dashed red curve). However, the finite energy of the pulse prevents the pulse from reaching completely the DLB attractor with infinite loses. As

*z*increases, the slowly evolving radial profile of the pulse fits up to a larger radial distance

*r*to the profiles of DLBs with slightly changing intensities extremely close to

*I*

_{0,m}[Fig. 3(c), open circles] and finite losses. At longer propagation distances, the increasingly lack of energy forces the pulse to decay into less lossy conical LBs. The pulse radial profiles [Fig. 3(d) solid curves] at different values of

*z*are seen to match now the radial profiles of conical LBs having same peak intensity and total losses as the pulse (open circles). Contrary to what is stated in Ref. [17

17. M. A. Porras and A. Parola, “Nonlinear unbalanced Bessel beams in the collapse of Gaussian beams arrested by nonlinear losses,” Opt. Lett. **33**, 1738–1740 (2008). [CrossRef] [PubMed]

*I*

_{0},

*δ*) and of DLBs (

*I*

_{0},

*δ*= 0). The dashed area indicates the region of parameters where conical and DLBs do not exist. Self-focusing carries the pulse directly to the point (

*I*

_{0,m},0) representing the DLB with maximum losses in the medium, or to points so close to it that they cannot be discerned at the scale of the figure or of the inset, remaining in this vicinity for about the first half of the collapse segment. Relaxation follows the indicated trajectory, where it is seen that the cone angle grows initially, but tends to zero at the end of the segment. The same dynamics explains also the “bursts” after the segments, if any, but since the remaining energy is considerably smaller, the DLB attractor is less approached and relaxation is faster.

## References and links

1. | F. Wise and P. Di Trapani, “The hunt for light bullets—Spatiotemporal solitons,” Opt. Photon. News , 29–32 (2002). |

2. | H. S. Eisenberg, R. Morandotti, and Y. Silberberg, “Kerr spatiotemporal self-Focusing in a planar glass waveguide,” Phys. Rev. Lett. |

3. | R. McLeod, K. Wagner, and S. Blair, “(3+1)-dimensional soliton dragging logic,” Phys. Rev. A |

4. | Y. Silberberg, “Collapse of optical pulses,” Opt. Lett. |

5. | N. Akhmediev and J. M. Soto Crespo, “Generation of a train of three-dimensional optical solitons in a self-focusing medium,” Phys. Rev. A |

6. | G. Fibich and B. Ilan, “Optical light bullets in a pure Kerr medium,” Opt. Lett. |

7. | 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. |

8. | S. Henz and J. Herrmann, “Two-dimensional spatial optical solitons in bulk Kerr media stabilized by self-induced multiphoton ionization: Variational approach,” Phys. Rev. E |

9. | D. Moll and A. L. Gaeta, “Role of dispersion in multiple-collapse dynamics,” Opt. Lett. |

10. | L. Bergé and S. Skupin, “Self-channeling of ultrashort laser pulses in materials with anomalous dispersion,” Phys. Rev. E |

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

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

13. | A. Dubietis, E. Gaizauskas, G. Tamosauskas, and P. Di Trapani, “Light Filaments without Self-Channeling,” Phys. Rev. Lett. |

14. | M. Kolesik and J. V. Moloney, “Self-healing femtosecond light filaments,” Opt. Lett. |

15. | M. A. Porras, A. Parola, D. Faccio, A. Dubietis, and P. Di Trapani, “Nonlinear unbalanced Bessel beams: Stationary conical waves supported by nonlinear losses,” Phys. Rev. Lett. |

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

17. | M. A. Porras and A. Parola, “Nonlinear unbalanced Bessel beams in the collapse of Gaussian beams arrested by nonlinear losses,” Opt. Lett. |

18. | D. Faccio, M. Clerici, A. Averchi, O. Jedrkiewicz, S. Tzortzakis, D. Papazoglou, F. Bragheri, L. Tartara, A. Trita, S. Henin, I. Cristiani, A. Couairon, and P. Di Trapani, “Kerr-induced spontaneous Bessel beam formation in the regime of strong two-photon absorption,” Opt. Express |

19. | N. Akhmediev and A. Ankiewicz (Eds.), |

20. | P. Grelu, J. M. Soto-Crespo, and N. Akhmediev, “Light bullets and dynamic pattern formation in nonlinear dissipative systems,” Opt. Express |

21. | J. M. Soto-Crespo, P. Grelu, and N. Akhmediev, “Optical bullets and “rockets” in nonlinear dissipative systems and their transformations and interactions,” Opt. Express |

22. | W. H. Renninger, A. Chong, and F. W. Wise, “Dissipative solitons in normal-dispersion fiber lasers,” Phys. Rev. A |

23. | Y. Tanguy, T. Ackermann, W. J. Firth, and R. Jäger, “Realization of a semiconductor-based cavity soliton laser,” Phys. Rev. Lett. |

24. |
This peak power is slightly higher than the power |

**OCIS Codes**

(190.4180) Nonlinear optics : Multiphoton processes

(190.5940) Nonlinear optics : Self-action effects

(190.7110) Nonlinear optics : Ultrafast nonlinear optics

(260.5950) Physical optics : Self-focusing

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: February 11, 2010

Revised Manuscript: March 18, 2010

Manuscript Accepted: March 18, 2010

Published: March 24, 2010

**Citation**

Miguel A. Porras, "A dissipative attractor in the spatiotemporal collapse of ultrashort light pulses," Opt. Express **18**, 7376-7383 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-7-7376

Sort: Year | Journal | Reset

### References

- F. Wise and P. Di Trapani, "The hunt for light bullets—Spatiotemporal solitons," Opt. Photon. News, 29-32 (2002).
- H. S. Eisenberg, R. Morandotti, and Y. Silberberg, "Kerr spatiotemporal self-Focusing in a planar glass waveguide," Phys. Rev. Lett. 82,043902 (2001). [CrossRef]
- R. McLeod, K. Wagner, and S. Blair, "(3+1)-dimensional soliton dragging logic," Phys. Rev. A 52,3254-3278 (1995). [CrossRef] [PubMed]
- Y. Silberberg, "Collapse of optical pulses," Opt. Lett. 15,1282-1284 (1990). [CrossRef] [PubMed]
- N. Akhmediev and J. M. Soto Crespo, "Generation of a train of three-dimensional optical solitons in a self focusing medium," Phys. Rev. A 47,1358-1364 (1993). [CrossRef] [PubMed]
- G. Fibich and B. Ilan, "Optical light bullets in a pure Kerr medium," Opt. Lett. 29,887-889 (2004). [CrossRef] [PubMed]
- 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]
- S. Henz and J. Herrmann, "Two-dimensional spatial optical solitons in bulk Kerr media stabilized by self-induced multiphoton ionization: Variational approach," Phys. Rev. E 53,4092-4097 (1996). [CrossRef]
- D. Moll and A. L. Gaeta, "Role of dispersion in multiple-collapse dynamics," Opt. Lett. 29, 995-997 (2004). [CrossRef] [PubMed]
- L. Bergé and S. Skupin, "Self-channeling of ultrashort laser pulses in materials with anomalous dispersion," Phys. Rev. E 71,065601 (2005). [CrossRef]
- P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, "Spontaneously generated X-shaped light bullets," Phys. Rev. Lett. 91,093904 (2003). [CrossRef] [PubMed]
- M. Kolesik, E. M. Wright, and J. V. Moloney, "Dynamic nonlinear X waves for femtosecond pulse propagation in water," Phys. Rev. Lett. 92,253901 (2004). [CrossRef] [PubMed]
- A. Dubietis, E. Gaizauskas, G. Tamosauskas, and P. Di Trapani, "Light Filaments without Self-Channeling," Phys. Rev. Lett. 92,253903 (2004). [CrossRef] [PubMed]
- M. Kolesik and J. V. Moloney, "Self-healing femtosecond light filaments," Opt. Lett. 29,590-592 (2004). [CrossRef] [PubMed]
- M. A. Porras, A. Parola, D. Faccio, A. Dubietis, and P. Di Trapani, "Nonlinear unbalanced Bessel beams: Stationary conical waves supported by nonlinear losses," Phys. Rev. Lett. 93,153902 (2004). [CrossRef] [PubMed]
- M. A. Porras, A. Parola, and P. Di Trapani, "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 and A. Parola, "Nonlinear unbalanced Bessel beams in the collapse of Gaussian beams arrested by nonlinear losses," Opt. Lett. 33,1738-1740 (2008). [CrossRef] [PubMed]
- D. Faccio, M. Clerici, A. Averchi, O. Jedrkiewicz, S. Tzortzakis, D. Papazoglou, F. Bragheri, L. Tartara, A. Trita, S. Henin, I. Cristiani, A. Couairon, and P. Di Trapani, "Kerr-induced spontaneous Bessel beam formation in the regime of strong two-photon absorption," Opt. Express 16,8213-8218 (2008). [CrossRef] [PubMed]
- N. Akhmediev and A. Ankiewicz (Eds.), Dissipative solitons, Lecture Notes in Physics 661 (Springer, 2005). [CrossRef]
- P. Grelu, J. M. Soto-Crespo, and N. Akhmediev, "Light bullets and dynamic pattern formation in nonlinear dissipative systems," Opt. Express 13,9352-9360 (2005). [CrossRef] [PubMed]
- J. M. Soto-Crespo, P. Grelu, and N. Akhmediev, "Optical bullets and "rockets" in nonlinear dissipative systems and their transformations and interactions," Opt. Express 14,4013-4025 (2006). [CrossRef] [PubMed]
- W. H. Renninger, A. Chong, and F. W. Wise, "Dissipative solitons in normal-dispersion fiber lasers," Phys. Rev. A 77,023814 (2008). [CrossRef]
- Y. Tanguy, T. Ackermann, W. J. Firth, and R. Jäger, "Realization of a semiconductor-based cavity soliton laser," Phys. Rev. Lett. 100,013907 (2008). [CrossRef] [PubMed]
- This peak power is slightly higher than the power Pcr = 2λ20/[4πn(ω0)n2] for spatial self-focusing.

## 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.