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.

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References

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  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. 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]
  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]
  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]
  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]
  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]
  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]
  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]
  24. This peak power is slightly higher than the power Pcr = 2λ02/[4πn(ω0)n2] for spatial self-focusing.

2008 (4)

2006 (1)

2005 (3)

2004 (6)

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

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

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

2003 (1)

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]

2002 (1)

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

2001 (1)

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

1996 (1)

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]

1995 (2)

1993 (1)

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]

1990 (1)

Ackermann, T.

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]

Akhmediev, N.

Averchi, A.

Bergé, L.

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

Blair, S.

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

Bragheri, F.

Braun, A.

Chong, A.

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

Clerici, M.

Conti, C.

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]

Couairon, A.

Cristiani, I.

Di Trapani, P.

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]

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]

A. Dubietis, E. Gaizauskas, G. Tamosauskas, and P. Di Trapani, “Light Filaments without Self-Channeling,” Phys. Rev. Lett. 92, 253903 (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]

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]

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

Du, D.

Dubietis, A.

A. Dubietis, E. Gaizauskas, G. Tamosauskas, and P. Di Trapani, “Light Filaments without Self-Channeling,” Phys. Rev. Lett. 92, 253903 (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]

Eisenberg, H. S.

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

Faccio, D.

Fibich, G.

Firth, W. J.

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]

Gaeta, A. L.

Gaizauskas, E.

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

Grelu, P.

Henin, S.

Henz, S.

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]

Herrmann, J.

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]

Ilan, B.

Jäger, R.

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]

Jedrkiewicz, O.

Kolesik, M.

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

Korn, G.

Liu, X.

McLeod, R.

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

Moll, D.

Moloney, J. V.

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]

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

Morandotti, R.

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

Mourou, G.

Papazoglou, D.

Parola, A.

Piskarskas, A.

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]

Porras, M. A.

Renninger, W. H.

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

Silberberg, Y.

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

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

Skupin, S.

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

Soto Crespo, J. M.

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]

Soto-Crespo, J. M.

Squier, J.

Tamosauskas, G.

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

Tanguy, Y.

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]

Tartara, L.

Trillo, S.

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]

Trita, A.

Trull, J.

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]

Tzortzakis, S.

Valiulis, G.

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]

Wagner, K.

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

Wise, F.

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

Wise, F. W.

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

Wright, E. M.

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]

J. Opt. Soc. Am. B (1)

Opt. Express (3)

Opt. Lett. (6)

Opt. Photon. News (1)

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

Phys. Rev. A (3)

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

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

Phys. Rev. E (2)

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]

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

Phys. Rev. Lett. (6)

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. 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]

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]

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

Other (2)

This peak power is slightly higher than the power Pcr = 2λ02/[4πn(ω0)n2] for spatial self-focusing.

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

Supplementary Material (1)

» Media 1: MOV (1189 KB)     

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Figures (4)

Fig. 1.
Fig. 1. Peak intensity (solid curves) and FWHM width (dashed blue curves) along z for input Gaussian pulses of carrier frequency ω 0 = 1.21525 fs-1, Gaussian width σ = 0.00716 cm [duration σ(k 0∣k ″ 0∣)1/2 = 29 fs], and increasing power above the critical power P cr = 13 MW, calculated from (1) with the parameters of fused silica (k 0 = 5.854 × 104 cm-1, k ″ 0 = -279.4 cm-1fs2, n 2 = 2.2 × 10-16 cm2/W, β (K) = 5.11 × 10-116 cm17/W9, with K = 10[10]).

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Fig. 2.
Fig. 2. (a) Intensity profile a 2(r) (solid curves) and their asymptotic form (dotted blue curves), and (b) nonlinear losses profile (k 0∣k ″ 0∣)1/2 N(r) (equal to the inward energy flux profile -(k 0∣k ″ 0∣)1/2 F(r)) of DLBs in fused silica at λ0 = 1550 nm with increasing peak intensities I 0 = a 2(0). The intensity and loss profiles of the LB with maximum intensity I 0,m = 14.716 TW/cm2 [Eq. (4) with γK = 3.15294 for K = 10] are represented with dashed red curves. (c) Total losses (k 0∣k ″ 0∣)1/2 NT of DLBs in fused silica at 1550 nm as a function of their peak intensity (solid curves). The total losses of conical LBs with δ = 2 and 7 cm-1 are shown with dashed curves. (d) Shape of the DLBs of maximum intensity and losses (dashed red curves), and their asymptotic forms b 2/ρ 2/(2K-3) (dotted blue curves). The radial profiles are obtained by solving (2) and (3) in the dimensionless form f ″ + 2f ′/ρ + ϕ ′ 2 f + = 0, 8πγ∫0 ρ f 2K ρ 2 dρ = 4πρ 2 φ ′ f 2, with f = a/I 0 1/2, ρ = (2k 0 2 n 2 I 0/n 0)1/2 r, and γ = n 0 β (K) IK-2 /(4n 2 k 0), and using the maximum values of γ leading to localized solutions, which are γK ≃ 0.88283,1.43838,1.83801,2.16571,2.45008,2.70474,2.93740, 3.15294 for K = 3,4,… 10, respectively.

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Fig. 3.
Fig. 3. (a) Peak intensity (solid curve), energy (dashed blue curve) and losses (dotted green curve) along z in the same conditions as in Fig. 1 except the Gaussian width σ = 0.011 cm [Gaussian duration σ(k 0∣k ″ 0∣)1/2 =44.5 fs] and the peak power P = 50P cr. (b-d) Radial intensity profile of the pulse with increasing propagation distance z (solid curves), of the DLB of maximum losses (dashed red curve) and of the DLB fitting the pulse (open circles). A vertical off-set is introduced for clarity.

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Fig. 4.
Fig. 4. Left: For the same example as in Fig. 3, animation of the evolution along z of the radial intensity profile (solid curve) as it approaches and relaxes from the DLB of maximum losses (dashed red curve) (Media 1). Right: for the same example as in the left panel, visualization of the evolution as a trajectory of the pulse in the parameter space of DLBs and conical LBs in fused silica at 1550 nm. The inset shows a tiny region close to I 0,m .

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Equations (6)

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∂ z A = i 2 k 0 Δ ⊥ A − i k 0 ″ 2 ∂ t ′ 2 A + i k 0 n 2 n 0 ∣ A ∣ 2 A − β ( K ) 2 ∣ A ∣ 2 K − 2 A ,
a ″ + 2 a ′ r + 2 k 0 δa − φ ′ 2 a + 2 k 0 2 n 2 n 0 a 3 = 0 ,
β ( K ) 4 π ∫ 0 r drr 2 a 2 K = − 4 π r 2 k 0 a 2 φ ′ ,
− α ( 1 − α ) b r 2 + 2 k 0 δb − k 0 2 N T 2 16 π 2 b 3 r 4 ( α − 1 ) + 2 k 0 2 n 2 b 3 n 0 r − 2 α ~ 0 .
I 0 , m = ( 4 γ K n 2 k 0 n 0 β ( K ) ) 1 / ( K − 2 ) ,
− α ( 1 − α ) b r 2 − [ k 0 b 2 K − 2 β ( K ) 3 − 2 Kα ] 2 r 2 − 4 α ( K − 1 ) + 2 k 0 2 n 2 b 3 n 0 r − 2 α ~ 0 .

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