Abstract

We propose spatiotemporal solitons that consist of trains of short pulses. The pulses are collectively trapped in the transverse directions by a slow nonlinearity and each pulse is self-trapped in the longitudinal direction by a fast nonlinearity. We demonstrate numerically spatiotemporal bright pulse-train solitons (trains of light bullets) and temporally-dark spatiotemporal pulse-train solitons in an experimentally feasible scheme.

© 2009 Optical Society of America

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  1. F. Wise and P. Trapani, "The hunt for light bullets - spatiotemporal solitons," Opt. Photonics News 13, 28-32 (2002).
  2. B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, "Spatiotemporal optical solitons," J. Opt. B 7, R53-R72 (2005).
  3. A. Hasegawa, "Optical solitons in fibers," Opt. Photonics News 13, 33-37 (2002).
  4. G. Stegeman and M. Segev, "Optical spatial solitons and their interactions: universality and diversity," Science 286, 1518-1523 (1999).
    [PubMed]
  5. Y. Silberberg, "Collapse of optical pulses," Opt. Lett. 15, 1282-1284 (1990).
    [PubMed]
  6. X. Liu, L. J. Qian, and F. W. Wise, "Generation of optical spatiotemporal solitons," Phys. Rev. Lett. 82, 4631-4634 (1999).
  7. X. Liu, K. Beckwitt, andn F. Wise, "Two-dimensional optical spatiotemporal solitons in quadratic media," Phys. Rev. E 62, 1328-1340 (2000).
  8. D. Edmundson and R. H. Enns, "Robust bistable light bullets," Opt. Lett. 17, 586-588 (1992).
    [PubMed]
  9. N. Akhmediev and J. M. Soto-Crespo, "Generation of a train of three-dimensional optical solitons in self-focusing medium," Phys. Rev. A 47, 1358-1364 (1993).
    [PubMed]
  10. D. Mihalache, D. Mazilu, F. Lederer, B. A. Malomed, Y. V. Kartashov, L. C. Crasovan, and L. Torner, "Three-dimensional spatiotemporal optical solitons in nonlocal nonlinear media," Phys. Rev. E 73, 025601R (2006).
  11. L. Torner, S. Carrasco, J. P. Torres, L. C. Crasovan, and D. Mihalache, "Tandem light bullets," Opt. Commun. 199, 277-281 (2001).
  12. A. B. Aceves, C. De Angelis, A. M. Rubenchik, and S. K. Turitsyn, "Multidimensional solitons in fiber arrays," Opt. Lett. 19, 329-331 (1994).
    [PubMed]
  13. E. W. Laedke, K. H. Spatschek, and S. K. Turitsyn, "Stability of discrete solitons and quasicollapse to intrinsically localized modes," Phys. Rev. Lett. 73, 1055-1058 (1994).
    [PubMed]
  14. Z. Xu, Ya. V. Kartashov, L. C. Crasovan, D. Mihalache, and L. Torner, "Spatiotemporal discrete multicolor solitons," Phys. Rev. E 70, 066618 (2004).
  15. A. S. Sukhorukov and Y. S. Kivshar, "Slow-Light Optical Bullets in Arrays of Nonlinear Bragg-Grating Waveguides," Phys. Rev. Lett. 97, 233901 (2006).
  16. A.V.  Gorbach and D. V.  Skryabin, "Cascaded Generation of Multiply charged Optical Vortices and Spatiotemporal Helical Beams in a Raman Medium," Phys. Rev. Lett. 98, 243601 (2007).
    [PubMed]
  17. M. Segev, B. Crosignani, P. DiPorto, G. C. Valley, and A. Yariv, "Steady state spatial screening-solitons in photorefractive media with external applied field," Phys. Rev. Lett. 73, 3211-3214 (1994).
    [PubMed]
  18. M. Mitchell, Z. Chen, M. Shih, and M. Segev, "Self-trapping of partially spatially-incoherent light," Phys. Rev. Lett. 77, 490-493 (1996).
    [PubMed]
  19. M. Mitchell and M. Segev, "Self-trapping of incoherent white light," Nature 387, 880-883 (1997).
  20. M. Segev and D. N. Christodoulides, "Incoherent solitons: self-trapping of weakly-correlated wave-packets," Opt. Photonics News 13, 70-76 (2002).
  21. M. Peccianti, A. De Rossi, G. Assanto, A. De Luca, C. Umeton, and I. C. Khoo, "Electrically assisted self-confinement and waveguiding in planar nematic liquid crystal cells," Appl. Phys. Lett. 77, 7 (2000).
  22. M. Peccianti and G. Assanto, "Nematic liquid crystals: A suitable medium for self-confinement of coherent and incoherent light," Phys. Rev. E 65, R035603 (2002).
  23. M. D. Iturbe Castillo, J. J. Sánchez-Mondragón, and S. Stepanov, "Formation of steady-state cylindrical thermal lenses in dark stripes," Opt. Lett. 21, 1622-1624 (1996).
  24. C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, "Solitons in nonlinear media with an infinite range of nonlocality: first observation of coherent elliptic solitons and of vortex-ring solitons," Phys. Rev. Lett. 95, 213904 (2005).
    [PubMed]
  25. A. Barak, C. Rotschild, B. Alfassi, M. Segev, and D. N. Christodoulides, "Random-phase surface wave solitons in nonlocal nonlinear media," Opt. Lett. 32, 2450-2452 (2007).
    [PubMed]
  26. D. Kip, M. Soljacic, M. Segev, E. Eugenieva, and D. N. Christodoulides, "Modulation instability and pattern formation in spatially incoherent light beams," Science 290, 495-498 (2000)
    [PubMed]
  27. O. Cohen, G. Bartal, H. Buljan, T. Carmon, J. W. Fleischer, M. Segev, and D. N. Christodoulides, "Observation of random-phase lattice solitons," Nature 433, 500-503 (2005).
    [PubMed]
  28. A. W. Snyder and D. J. Mitchell, "Accessible solitons," Science 276, 1538-1541 (1997).
  29. A. W. Snyder, D. J. Mitchell, L. Poladian, and F. Ladouceur, "Self-induced optical fibers: spatial solitary waves," Opt. Lett. 16, 21-23 (1991).
    [PubMed]
  30. L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, "Experimental Observation of Picosecond Pulse Narrowing and Solitons in Optical Fibers," Phys. Rev. Lett. 45, 1095-1048 (1980).
  31. D. Krökel, N. J. Halas, G. Giuliani, and D. Grischkowsky, "Dark-Pulse Propagation in Optical Fibers," Phys. Rev. Lett. 60, 29-32 (1988).
    [PubMed]
  32. A. M. Weiner, J. P. Heritage, R. J. Hawkins, R. N. Thurston, E. M. Kirschner, D. E. Leaird, and W. J. Tomlinson, "Experimental Observation of the Fundamental Dark Soliton in Optical Fibers," Phys. Rev. Lett. 61, 2445-2448 (1988).
    [PubMed]
  33. S. K. Turitsyn, "Spatial dispersion of nonlinearity and stability of multidimensional solitons," Theor. Math. Phys. 64, 226-232 (1985).
  34. O. Bang, W. Krolikowski, J. Wyller, and J. Rasmussen, "Collapse arrest and soliton stabilization in nonlocal nonlinear media," Phys. Rev. E 66, 046619 (2002).
  35. S. Raghavan and G. P. Agrawal, "Spatiotemporal solitons in inhomogeneous nonlinear media," Opt. Commun. 180, 377-382 (2000).
  36. C. Rotschild, B. Alfassi, O. Cohen, and M. Segev, "Long-range interactions between optical solitons," Nature Phys. 2, 769-774 (2006).
  37. C. Rotschild, T. Schwartz, O. Cohen, and M. Segev, "Incoherent solitons in effectively-instantaneous nonlocal nonlinear media," Nat. Photonics 2, 371-376 (2008).
  38. O. Cohen, H. Buljan, T. Schwartz, J. W. Fleischer, and M. Segev, "Incoherent solitons in instantaneous nonlocal nonlinear media," Phys. Rev. E 73, 015601(R) (2006).

2008 (1)

C. Rotschild, T. Schwartz, O. Cohen, and M. Segev, "Incoherent solitons in effectively-instantaneous nonlocal nonlinear media," Nat. Photonics 2, 371-376 (2008).

2007 (2)

A.V.  Gorbach and D. V.  Skryabin, "Cascaded Generation of Multiply charged Optical Vortices and Spatiotemporal Helical Beams in a Raman Medium," Phys. Rev. Lett. 98, 243601 (2007).
[PubMed]

A. Barak, C. Rotschild, B. Alfassi, M. Segev, and D. N. Christodoulides, "Random-phase surface wave solitons in nonlocal nonlinear media," Opt. Lett. 32, 2450-2452 (2007).
[PubMed]

2006 (3)

A. S. Sukhorukov and Y. S. Kivshar, "Slow-Light Optical Bullets in Arrays of Nonlinear Bragg-Grating Waveguides," Phys. Rev. Lett. 97, 233901 (2006).

D. Mihalache, D. Mazilu, F. Lederer, B. A. Malomed, Y. V. Kartashov, L. C. Crasovan, and L. Torner, "Three-dimensional spatiotemporal optical solitons in nonlocal nonlinear media," Phys. Rev. E 73, 025601R (2006).

C. Rotschild, B. Alfassi, O. Cohen, and M. Segev, "Long-range interactions between optical solitons," Nature Phys. 2, 769-774 (2006).

2005 (3)

C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, "Solitons in nonlinear media with an infinite range of nonlocality: first observation of coherent elliptic solitons and of vortex-ring solitons," Phys. Rev. Lett. 95, 213904 (2005).
[PubMed]

O. Cohen, G. Bartal, H. Buljan, T. Carmon, J. W. Fleischer, M. Segev, and D. N. Christodoulides, "Observation of random-phase lattice solitons," Nature 433, 500-503 (2005).
[PubMed]

B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, "Spatiotemporal optical solitons," J. Opt. B 7, R53-R72 (2005).

2004 (1)

Z. Xu, Ya. V. Kartashov, L. C. Crasovan, D. Mihalache, and L. Torner, "Spatiotemporal discrete multicolor solitons," Phys. Rev. E 70, 066618 (2004).

2002 (5)

F. Wise and P. Trapani, "The hunt for light bullets - spatiotemporal solitons," Opt. Photonics News 13, 28-32 (2002).

A. Hasegawa, "Optical solitons in fibers," Opt. Photonics News 13, 33-37 (2002).

M. Segev and D. N. Christodoulides, "Incoherent solitons: self-trapping of weakly-correlated wave-packets," Opt. Photonics News 13, 70-76 (2002).

M. Peccianti and G. Assanto, "Nematic liquid crystals: A suitable medium for self-confinement of coherent and incoherent light," Phys. Rev. E 65, R035603 (2002).

O. Bang, W. Krolikowski, J. Wyller, and J. Rasmussen, "Collapse arrest and soliton stabilization in nonlocal nonlinear media," Phys. Rev. E 66, 046619 (2002).

2001 (1)

L. Torner, S. Carrasco, J. P. Torres, L. C. Crasovan, and D. Mihalache, "Tandem light bullets," Opt. Commun. 199, 277-281 (2001).

2000 (4)

M. Peccianti, A. De Rossi, G. Assanto, A. De Luca, C. Umeton, and I. C. Khoo, "Electrically assisted self-confinement and waveguiding in planar nematic liquid crystal cells," Appl. Phys. Lett. 77, 7 (2000).

S. Raghavan and G. P. Agrawal, "Spatiotemporal solitons in inhomogeneous nonlinear media," Opt. Commun. 180, 377-382 (2000).

D. Kip, M. Soljacic, M. Segev, E. Eugenieva, and D. N. Christodoulides, "Modulation instability and pattern formation in spatially incoherent light beams," Science 290, 495-498 (2000)
[PubMed]

X. Liu, K. Beckwitt, andn F. Wise, "Two-dimensional optical spatiotemporal solitons in quadratic media," Phys. Rev. E 62, 1328-1340 (2000).

1999 (2)

X. Liu, L. J. Qian, and F. W. Wise, "Generation of optical spatiotemporal solitons," Phys. Rev. Lett. 82, 4631-4634 (1999).

G. Stegeman and M. Segev, "Optical spatial solitons and their interactions: universality and diversity," Science 286, 1518-1523 (1999).
[PubMed]

1997 (2)

M. Mitchell and M. Segev, "Self-trapping of incoherent white light," Nature 387, 880-883 (1997).

A. W. Snyder and D. J. Mitchell, "Accessible solitons," Science 276, 1538-1541 (1997).

1996 (2)

M. Mitchell, Z. Chen, M. Shih, and M. Segev, "Self-trapping of partially spatially-incoherent light," Phys. Rev. Lett. 77, 490-493 (1996).
[PubMed]

M. D. Iturbe Castillo, J. J. Sánchez-Mondragón, and S. Stepanov, "Formation of steady-state cylindrical thermal lenses in dark stripes," Opt. Lett. 21, 1622-1624 (1996).

1994 (3)

M. Segev, B. Crosignani, P. DiPorto, G. C. Valley, and A. Yariv, "Steady state spatial screening-solitons in photorefractive media with external applied field," Phys. Rev. Lett. 73, 3211-3214 (1994).
[PubMed]

E. W. Laedke, K. H. Spatschek, and S. K. Turitsyn, "Stability of discrete solitons and quasicollapse to intrinsically localized modes," Phys. Rev. Lett. 73, 1055-1058 (1994).
[PubMed]

A. B. Aceves, C. De Angelis, A. M. Rubenchik, and S. K. Turitsyn, "Multidimensional solitons in fiber arrays," Opt. Lett. 19, 329-331 (1994).
[PubMed]

1993 (1)

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

1992 (1)

1991 (1)

1990 (1)

1988 (2)

D. Krökel, N. J. Halas, G. Giuliani, and D. Grischkowsky, "Dark-Pulse Propagation in Optical Fibers," Phys. Rev. Lett. 60, 29-32 (1988).
[PubMed]

A. M. Weiner, J. P. Heritage, R. J. Hawkins, R. N. Thurston, E. M. Kirschner, D. E. Leaird, and W. J. Tomlinson, "Experimental Observation of the Fundamental Dark Soliton in Optical Fibers," Phys. Rev. Lett. 61, 2445-2448 (1988).
[PubMed]

1985 (1)

S. K. Turitsyn, "Spatial dispersion of nonlinearity and stability of multidimensional solitons," Theor. Math. Phys. 64, 226-232 (1985).

1980 (1)

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, "Experimental Observation of Picosecond Pulse Narrowing and Solitons in Optical Fibers," Phys. Rev. Lett. 45, 1095-1048 (1980).

Aceves, A. B.

Agrawal, G. P.

S. Raghavan and G. P. Agrawal, "Spatiotemporal solitons in inhomogeneous nonlinear media," Opt. Commun. 180, 377-382 (2000).

Akhmediev, N.

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

Alfassi, B.

A. Barak, C. Rotschild, B. Alfassi, M. Segev, and D. N. Christodoulides, "Random-phase surface wave solitons in nonlocal nonlinear media," Opt. Lett. 32, 2450-2452 (2007).
[PubMed]

C. Rotschild, B. Alfassi, O. Cohen, and M. Segev, "Long-range interactions between optical solitons," Nature Phys. 2, 769-774 (2006).

Assanto, G.

M. Peccianti and G. Assanto, "Nematic liquid crystals: A suitable medium for self-confinement of coherent and incoherent light," Phys. Rev. E 65, R035603 (2002).

M. Peccianti, A. De Rossi, G. Assanto, A. De Luca, C. Umeton, and I. C. Khoo, "Electrically assisted self-confinement and waveguiding in planar nematic liquid crystal cells," Appl. Phys. Lett. 77, 7 (2000).

Bang, O.

O. Bang, W. Krolikowski, J. Wyller, and J. Rasmussen, "Collapse arrest and soliton stabilization in nonlocal nonlinear media," Phys. Rev. E 66, 046619 (2002).

Barak, A.

Bartal, G.

O. Cohen, G. Bartal, H. Buljan, T. Carmon, J. W. Fleischer, M. Segev, and D. N. Christodoulides, "Observation of random-phase lattice solitons," Nature 433, 500-503 (2005).
[PubMed]

Beckwitt, K.

X. Liu, K. Beckwitt, andn F. Wise, "Two-dimensional optical spatiotemporal solitons in quadratic media," Phys. Rev. E 62, 1328-1340 (2000).

Buljan, H.

O. Cohen, G. Bartal, H. Buljan, T. Carmon, J. W. Fleischer, M. Segev, and D. N. Christodoulides, "Observation of random-phase lattice solitons," Nature 433, 500-503 (2005).
[PubMed]

Carmon, T.

O. Cohen, G. Bartal, H. Buljan, T. Carmon, J. W. Fleischer, M. Segev, and D. N. Christodoulides, "Observation of random-phase lattice solitons," Nature 433, 500-503 (2005).
[PubMed]

C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, "Solitons in nonlinear media with an infinite range of nonlocality: first observation of coherent elliptic solitons and of vortex-ring solitons," Phys. Rev. Lett. 95, 213904 (2005).
[PubMed]

Carrasco, S.

L. Torner, S. Carrasco, J. P. Torres, L. C. Crasovan, and D. Mihalache, "Tandem light bullets," Opt. Commun. 199, 277-281 (2001).

Chen, Z.

M. Mitchell, Z. Chen, M. Shih, and M. Segev, "Self-trapping of partially spatially-incoherent light," Phys. Rev. Lett. 77, 490-493 (1996).
[PubMed]

Christodoulides, D. N.

A. Barak, C. Rotschild, B. Alfassi, M. Segev, and D. N. Christodoulides, "Random-phase surface wave solitons in nonlocal nonlinear media," Opt. Lett. 32, 2450-2452 (2007).
[PubMed]

O. Cohen, G. Bartal, H. Buljan, T. Carmon, J. W. Fleischer, M. Segev, and D. N. Christodoulides, "Observation of random-phase lattice solitons," Nature 433, 500-503 (2005).
[PubMed]

M. Segev and D. N. Christodoulides, "Incoherent solitons: self-trapping of weakly-correlated wave-packets," Opt. Photonics News 13, 70-76 (2002).

D. Kip, M. Soljacic, M. Segev, E. Eugenieva, and D. N. Christodoulides, "Modulation instability and pattern formation in spatially incoherent light beams," Science 290, 495-498 (2000)
[PubMed]

Cohen, O.

C. Rotschild, T. Schwartz, O. Cohen, and M. Segev, "Incoherent solitons in effectively-instantaneous nonlocal nonlinear media," Nat. Photonics 2, 371-376 (2008).

C. Rotschild, B. Alfassi, O. Cohen, and M. Segev, "Long-range interactions between optical solitons," Nature Phys. 2, 769-774 (2006).

O. Cohen, G. Bartal, H. Buljan, T. Carmon, J. W. Fleischer, M. Segev, and D. N. Christodoulides, "Observation of random-phase lattice solitons," Nature 433, 500-503 (2005).
[PubMed]

C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, "Solitons in nonlinear media with an infinite range of nonlocality: first observation of coherent elliptic solitons and of vortex-ring solitons," Phys. Rev. Lett. 95, 213904 (2005).
[PubMed]

Crasovan, L. C.

D. Mihalache, D. Mazilu, F. Lederer, B. A. Malomed, Y. V. Kartashov, L. C. Crasovan, and L. Torner, "Three-dimensional spatiotemporal optical solitons in nonlocal nonlinear media," Phys. Rev. E 73, 025601R (2006).

Z. Xu, Ya. V. Kartashov, L. C. Crasovan, D. Mihalache, and L. Torner, "Spatiotemporal discrete multicolor solitons," Phys. Rev. E 70, 066618 (2004).

L. Torner, S. Carrasco, J. P. Torres, L. C. Crasovan, and D. Mihalache, "Tandem light bullets," Opt. Commun. 199, 277-281 (2001).

Crosignani, B.

M. Segev, B. Crosignani, P. DiPorto, G. C. Valley, and A. Yariv, "Steady state spatial screening-solitons in photorefractive media with external applied field," Phys. Rev. Lett. 73, 3211-3214 (1994).
[PubMed]

De Angelis, C.

De Luca, A.

M. Peccianti, A. De Rossi, G. Assanto, A. De Luca, C. Umeton, and I. C. Khoo, "Electrically assisted self-confinement and waveguiding in planar nematic liquid crystal cells," Appl. Phys. Lett. 77, 7 (2000).

De Rossi, A.

M. Peccianti, A. De Rossi, G. Assanto, A. De Luca, C. Umeton, and I. C. Khoo, "Electrically assisted self-confinement and waveguiding in planar nematic liquid crystal cells," Appl. Phys. Lett. 77, 7 (2000).

DiPorto, P.

M. Segev, B. Crosignani, P. DiPorto, G. C. Valley, and A. Yariv, "Steady state spatial screening-solitons in photorefractive media with external applied field," Phys. Rev. Lett. 73, 3211-3214 (1994).
[PubMed]

Edmundson, D.

Enns, R. H.

Eugenieva, E.

D. Kip, M. Soljacic, M. Segev, E. Eugenieva, and D. N. Christodoulides, "Modulation instability and pattern formation in spatially incoherent light beams," Science 290, 495-498 (2000)
[PubMed]

Fleischer, J. W.

O. Cohen, G. Bartal, H. Buljan, T. Carmon, J. W. Fleischer, M. Segev, and D. N. Christodoulides, "Observation of random-phase lattice solitons," Nature 433, 500-503 (2005).
[PubMed]

Giuliani, G.

D. Krökel, N. J. Halas, G. Giuliani, and D. Grischkowsky, "Dark-Pulse Propagation in Optical Fibers," Phys. Rev. Lett. 60, 29-32 (1988).
[PubMed]

Gorbach, A. V.

A.V.  Gorbach and D. V.  Skryabin, "Cascaded Generation of Multiply charged Optical Vortices and Spatiotemporal Helical Beams in a Raman Medium," Phys. Rev. Lett. 98, 243601 (2007).
[PubMed]

Gordon, J. P.

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, "Experimental Observation of Picosecond Pulse Narrowing and Solitons in Optical Fibers," Phys. Rev. Lett. 45, 1095-1048 (1980).

Grischkowsky, D.

D. Krökel, N. J. Halas, G. Giuliani, and D. Grischkowsky, "Dark-Pulse Propagation in Optical Fibers," Phys. Rev. Lett. 60, 29-32 (1988).
[PubMed]

Halas, N. J.

D. Krökel, N. J. Halas, G. Giuliani, and D. Grischkowsky, "Dark-Pulse Propagation in Optical Fibers," Phys. Rev. Lett. 60, 29-32 (1988).
[PubMed]

Hasegawa, A.

A. Hasegawa, "Optical solitons in fibers," Opt. Photonics News 13, 33-37 (2002).

Hawkins, R. J.

A. M. Weiner, J. P. Heritage, R. J. Hawkins, R. N. Thurston, E. M. Kirschner, D. E. Leaird, and W. J. Tomlinson, "Experimental Observation of the Fundamental Dark Soliton in Optical Fibers," Phys. Rev. Lett. 61, 2445-2448 (1988).
[PubMed]

Heritage, J. P.

A. M. Weiner, J. P. Heritage, R. J. Hawkins, R. N. Thurston, E. M. Kirschner, D. E. Leaird, and W. J. Tomlinson, "Experimental Observation of the Fundamental Dark Soliton in Optical Fibers," Phys. Rev. Lett. 61, 2445-2448 (1988).
[PubMed]

Iturbe Castillo, M. D.

Kartashov, Y. V.

D. Mihalache, D. Mazilu, F. Lederer, B. A. Malomed, Y. V. Kartashov, L. C. Crasovan, and L. Torner, "Three-dimensional spatiotemporal optical solitons in nonlocal nonlinear media," Phys. Rev. E 73, 025601R (2006).

Kartashov, Ya. V.

Z. Xu, Ya. V. Kartashov, L. C. Crasovan, D. Mihalache, and L. Torner, "Spatiotemporal discrete multicolor solitons," Phys. Rev. E 70, 066618 (2004).

Khoo, I. C.

M. Peccianti, A. De Rossi, G. Assanto, A. De Luca, C. Umeton, and I. C. Khoo, "Electrically assisted self-confinement and waveguiding in planar nematic liquid crystal cells," Appl. Phys. Lett. 77, 7 (2000).

Kip, D.

D. Kip, M. Soljacic, M. Segev, E. Eugenieva, and D. N. Christodoulides, "Modulation instability and pattern formation in spatially incoherent light beams," Science 290, 495-498 (2000)
[PubMed]

Kirschner, E. M.

A. M. Weiner, J. P. Heritage, R. J. Hawkins, R. N. Thurston, E. M. Kirschner, D. E. Leaird, and W. J. Tomlinson, "Experimental Observation of the Fundamental Dark Soliton in Optical Fibers," Phys. Rev. Lett. 61, 2445-2448 (1988).
[PubMed]

Kivshar, Y. S.

A. S. Sukhorukov and Y. S. Kivshar, "Slow-Light Optical Bullets in Arrays of Nonlinear Bragg-Grating Waveguides," Phys. Rev. Lett. 97, 233901 (2006).

Krökel, D.

D. Krökel, N. J. Halas, G. Giuliani, and D. Grischkowsky, "Dark-Pulse Propagation in Optical Fibers," Phys. Rev. Lett. 60, 29-32 (1988).
[PubMed]

Krolikowski, W.

O. Bang, W. Krolikowski, J. Wyller, and J. Rasmussen, "Collapse arrest and soliton stabilization in nonlocal nonlinear media," Phys. Rev. E 66, 046619 (2002).

Ladouceur, F.

Laedke, E. W.

E. W. Laedke, K. H. Spatschek, and S. K. Turitsyn, "Stability of discrete solitons and quasicollapse to intrinsically localized modes," Phys. Rev. Lett. 73, 1055-1058 (1994).
[PubMed]

Leaird, D. E.

A. M. Weiner, J. P. Heritage, R. J. Hawkins, R. N. Thurston, E. M. Kirschner, D. E. Leaird, and W. J. Tomlinson, "Experimental Observation of the Fundamental Dark Soliton in Optical Fibers," Phys. Rev. Lett. 61, 2445-2448 (1988).
[PubMed]

Lederer, F.

D. Mihalache, D. Mazilu, F. Lederer, B. A. Malomed, Y. V. Kartashov, L. C. Crasovan, and L. Torner, "Three-dimensional spatiotemporal optical solitons in nonlocal nonlinear media," Phys. Rev. E 73, 025601R (2006).

Liu, X.

X. Liu, K. Beckwitt, andn F. Wise, "Two-dimensional optical spatiotemporal solitons in quadratic media," Phys. Rev. E 62, 1328-1340 (2000).

X. Liu, L. J. Qian, and F. W. Wise, "Generation of optical spatiotemporal solitons," Phys. Rev. Lett. 82, 4631-4634 (1999).

Malomed, B. A.

D. Mihalache, D. Mazilu, F. Lederer, B. A. Malomed, Y. V. Kartashov, L. C. Crasovan, and L. Torner, "Three-dimensional spatiotemporal optical solitons in nonlocal nonlinear media," Phys. Rev. E 73, 025601R (2006).

B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, "Spatiotemporal optical solitons," J. Opt. B 7, R53-R72 (2005).

Manela, O.

C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, "Solitons in nonlinear media with an infinite range of nonlocality: first observation of coherent elliptic solitons and of vortex-ring solitons," Phys. Rev. Lett. 95, 213904 (2005).
[PubMed]

Mazilu, D.

D. Mihalache, D. Mazilu, F. Lederer, B. A. Malomed, Y. V. Kartashov, L. C. Crasovan, and L. Torner, "Three-dimensional spatiotemporal optical solitons in nonlocal nonlinear media," Phys. Rev. E 73, 025601R (2006).

Mihalache, D.

D. Mihalache, D. Mazilu, F. Lederer, B. A. Malomed, Y. V. Kartashov, L. C. Crasovan, and L. Torner, "Three-dimensional spatiotemporal optical solitons in nonlocal nonlinear media," Phys. Rev. E 73, 025601R (2006).

B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, "Spatiotemporal optical solitons," J. Opt. B 7, R53-R72 (2005).

Z. Xu, Ya. V. Kartashov, L. C. Crasovan, D. Mihalache, and L. Torner, "Spatiotemporal discrete multicolor solitons," Phys. Rev. E 70, 066618 (2004).

L. Torner, S. Carrasco, J. P. Torres, L. C. Crasovan, and D. Mihalache, "Tandem light bullets," Opt. Commun. 199, 277-281 (2001).

Mitchell, D. J.

Mitchell, M.

M. Mitchell and M. Segev, "Self-trapping of incoherent white light," Nature 387, 880-883 (1997).

M. Mitchell, Z. Chen, M. Shih, and M. Segev, "Self-trapping of partially spatially-incoherent light," Phys. Rev. Lett. 77, 490-493 (1996).
[PubMed]

Mollenauer, L. F.

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, "Experimental Observation of Picosecond Pulse Narrowing and Solitons in Optical Fibers," Phys. Rev. Lett. 45, 1095-1048 (1980).

Peccianti, M.

M. Peccianti and G. Assanto, "Nematic liquid crystals: A suitable medium for self-confinement of coherent and incoherent light," Phys. Rev. E 65, R035603 (2002).

M. Peccianti, A. De Rossi, G. Assanto, A. De Luca, C. Umeton, and I. C. Khoo, "Electrically assisted self-confinement and waveguiding in planar nematic liquid crystal cells," Appl. Phys. Lett. 77, 7 (2000).

Poladian, L.

Qian, L. J.

X. Liu, L. J. Qian, and F. W. Wise, "Generation of optical spatiotemporal solitons," Phys. Rev. Lett. 82, 4631-4634 (1999).

Raghavan, S.

S. Raghavan and G. P. Agrawal, "Spatiotemporal solitons in inhomogeneous nonlinear media," Opt. Commun. 180, 377-382 (2000).

Rasmussen, J.

O. Bang, W. Krolikowski, J. Wyller, and J. Rasmussen, "Collapse arrest and soliton stabilization in nonlocal nonlinear media," Phys. Rev. E 66, 046619 (2002).

Rotschild, C.

C. Rotschild, T. Schwartz, O. Cohen, and M. Segev, "Incoherent solitons in effectively-instantaneous nonlocal nonlinear media," Nat. Photonics 2, 371-376 (2008).

A. Barak, C. Rotschild, B. Alfassi, M. Segev, and D. N. Christodoulides, "Random-phase surface wave solitons in nonlocal nonlinear media," Opt. Lett. 32, 2450-2452 (2007).
[PubMed]

C. Rotschild, B. Alfassi, O. Cohen, and M. Segev, "Long-range interactions between optical solitons," Nature Phys. 2, 769-774 (2006).

C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, "Solitons in nonlinear media with an infinite range of nonlocality: first observation of coherent elliptic solitons and of vortex-ring solitons," Phys. Rev. Lett. 95, 213904 (2005).
[PubMed]

Rubenchik, A. M.

Sánchez-Mondragón, J. J.

Schwartz, T.

C. Rotschild, T. Schwartz, O. Cohen, and M. Segev, "Incoherent solitons in effectively-instantaneous nonlocal nonlinear media," Nat. Photonics 2, 371-376 (2008).

Segev, M.

C. Rotschild, T. Schwartz, O. Cohen, and M. Segev, "Incoherent solitons in effectively-instantaneous nonlocal nonlinear media," Nat. Photonics 2, 371-376 (2008).

A. Barak, C. Rotschild, B. Alfassi, M. Segev, and D. N. Christodoulides, "Random-phase surface wave solitons in nonlocal nonlinear media," Opt. Lett. 32, 2450-2452 (2007).
[PubMed]

C. Rotschild, B. Alfassi, O. Cohen, and M. Segev, "Long-range interactions between optical solitons," Nature Phys. 2, 769-774 (2006).

C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, "Solitons in nonlinear media with an infinite range of nonlocality: first observation of coherent elliptic solitons and of vortex-ring solitons," Phys. Rev. Lett. 95, 213904 (2005).
[PubMed]

O. Cohen, G. Bartal, H. Buljan, T. Carmon, J. W. Fleischer, M. Segev, and D. N. Christodoulides, "Observation of random-phase lattice solitons," Nature 433, 500-503 (2005).
[PubMed]

M. Segev and D. N. Christodoulides, "Incoherent solitons: self-trapping of weakly-correlated wave-packets," Opt. Photonics News 13, 70-76 (2002).

D. Kip, M. Soljacic, M. Segev, E. Eugenieva, and D. N. Christodoulides, "Modulation instability and pattern formation in spatially incoherent light beams," Science 290, 495-498 (2000)
[PubMed]

G. Stegeman and M. Segev, "Optical spatial solitons and their interactions: universality and diversity," Science 286, 1518-1523 (1999).
[PubMed]

M. Mitchell and M. Segev, "Self-trapping of incoherent white light," Nature 387, 880-883 (1997).

M. Mitchell, Z. Chen, M. Shih, and M. Segev, "Self-trapping of partially spatially-incoherent light," Phys. Rev. Lett. 77, 490-493 (1996).
[PubMed]

M. Segev, B. Crosignani, P. DiPorto, G. C. Valley, and A. Yariv, "Steady state spatial screening-solitons in photorefractive media with external applied field," Phys. Rev. Lett. 73, 3211-3214 (1994).
[PubMed]

Shih, M.

M. Mitchell, Z. Chen, M. Shih, and M. Segev, "Self-trapping of partially spatially-incoherent light," Phys. Rev. Lett. 77, 490-493 (1996).
[PubMed]

Silberberg, Y.

Skryabin, D. V.

A.V.  Gorbach and D. V.  Skryabin, "Cascaded Generation of Multiply charged Optical Vortices and Spatiotemporal Helical Beams in a Raman Medium," Phys. Rev. Lett. 98, 243601 (2007).
[PubMed]

Snyder, A. W.

Soljacic, M.

D. Kip, M. Soljacic, M. Segev, E. Eugenieva, and D. N. Christodoulides, "Modulation instability and pattern formation in spatially incoherent light beams," Science 290, 495-498 (2000)
[PubMed]

Soto-Crespo, J. M.

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

Spatschek, K. H.

E. W. Laedke, K. H. Spatschek, and S. K. Turitsyn, "Stability of discrete solitons and quasicollapse to intrinsically localized modes," Phys. Rev. Lett. 73, 1055-1058 (1994).
[PubMed]

Stegeman, G.

G. Stegeman and M. Segev, "Optical spatial solitons and their interactions: universality and diversity," Science 286, 1518-1523 (1999).
[PubMed]

Stepanov, S.

Stolen, R. H.

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, "Experimental Observation of Picosecond Pulse Narrowing and Solitons in Optical Fibers," Phys. Rev. Lett. 45, 1095-1048 (1980).

Sukhorukov, A. S.

A. S. Sukhorukov and Y. S. Kivshar, "Slow-Light Optical Bullets in Arrays of Nonlinear Bragg-Grating Waveguides," Phys. Rev. Lett. 97, 233901 (2006).

Thurston, R. N.

A. M. Weiner, J. P. Heritage, R. J. Hawkins, R. N. Thurston, E. M. Kirschner, D. E. Leaird, and W. J. Tomlinson, "Experimental Observation of the Fundamental Dark Soliton in Optical Fibers," Phys. Rev. Lett. 61, 2445-2448 (1988).
[PubMed]

Tomlinson, W. J.

A. M. Weiner, J. P. Heritage, R. J. Hawkins, R. N. Thurston, E. M. Kirschner, D. E. Leaird, and W. J. Tomlinson, "Experimental Observation of the Fundamental Dark Soliton in Optical Fibers," Phys. Rev. Lett. 61, 2445-2448 (1988).
[PubMed]

Torner, L.

D. Mihalache, D. Mazilu, F. Lederer, B. A. Malomed, Y. V. Kartashov, L. C. Crasovan, and L. Torner, "Three-dimensional spatiotemporal optical solitons in nonlocal nonlinear media," Phys. Rev. E 73, 025601R (2006).

B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, "Spatiotemporal optical solitons," J. Opt. B 7, R53-R72 (2005).

Z. Xu, Ya. V. Kartashov, L. C. Crasovan, D. Mihalache, and L. Torner, "Spatiotemporal discrete multicolor solitons," Phys. Rev. E 70, 066618 (2004).

L. Torner, S. Carrasco, J. P. Torres, L. C. Crasovan, and D. Mihalache, "Tandem light bullets," Opt. Commun. 199, 277-281 (2001).

Torres, J. P.

L. Torner, S. Carrasco, J. P. Torres, L. C. Crasovan, and D. Mihalache, "Tandem light bullets," Opt. Commun. 199, 277-281 (2001).

Trapani, P.

F. Wise and P. Trapani, "The hunt for light bullets - spatiotemporal solitons," Opt. Photonics News 13, 28-32 (2002).

Turitsyn, S. K.

A. B. Aceves, C. De Angelis, A. M. Rubenchik, and S. K. Turitsyn, "Multidimensional solitons in fiber arrays," Opt. Lett. 19, 329-331 (1994).
[PubMed]

E. W. Laedke, K. H. Spatschek, and S. K. Turitsyn, "Stability of discrete solitons and quasicollapse to intrinsically localized modes," Phys. Rev. Lett. 73, 1055-1058 (1994).
[PubMed]

S. K. Turitsyn, "Spatial dispersion of nonlinearity and stability of multidimensional solitons," Theor. Math. Phys. 64, 226-232 (1985).

Umeton, C.

M. Peccianti, A. De Rossi, G. Assanto, A. De Luca, C. Umeton, and I. C. Khoo, "Electrically assisted self-confinement and waveguiding in planar nematic liquid crystal cells," Appl. Phys. Lett. 77, 7 (2000).

Valley, G. C.

M. Segev, B. Crosignani, P. DiPorto, G. C. Valley, and A. Yariv, "Steady state spatial screening-solitons in photorefractive media with external applied field," Phys. Rev. Lett. 73, 3211-3214 (1994).
[PubMed]

Weiner, A. M.

A. M. Weiner, J. P. Heritage, R. J. Hawkins, R. N. Thurston, E. M. Kirschner, D. E. Leaird, and W. J. Tomlinson, "Experimental Observation of the Fundamental Dark Soliton in Optical Fibers," Phys. Rev. Lett. 61, 2445-2448 (1988).
[PubMed]

Wise, F.

B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, "Spatiotemporal optical solitons," J. Opt. B 7, R53-R72 (2005).

F. Wise and P. Trapani, "The hunt for light bullets - spatiotemporal solitons," Opt. Photonics News 13, 28-32 (2002).

Wise, F. W.

X. Liu, L. J. Qian, and F. W. Wise, "Generation of optical spatiotemporal solitons," Phys. Rev. Lett. 82, 4631-4634 (1999).

Wyller, J.

O. Bang, W. Krolikowski, J. Wyller, and J. Rasmussen, "Collapse arrest and soliton stabilization in nonlocal nonlinear media," Phys. Rev. E 66, 046619 (2002).

Xu, Z.

Z. Xu, Ya. V. Kartashov, L. C. Crasovan, D. Mihalache, and L. Torner, "Spatiotemporal discrete multicolor solitons," Phys. Rev. E 70, 066618 (2004).

Yariv, A.

M. Segev, B. Crosignani, P. DiPorto, G. C. Valley, and A. Yariv, "Steady state spatial screening-solitons in photorefractive media with external applied field," Phys. Rev. Lett. 73, 3211-3214 (1994).
[PubMed]

Appl. Phys. Lett. (1)

M. Peccianti, A. De Rossi, G. Assanto, A. De Luca, C. Umeton, and I. C. Khoo, "Electrically assisted self-confinement and waveguiding in planar nematic liquid crystal cells," Appl. Phys. Lett. 77, 7 (2000).

J. Opt. B (1)

B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, "Spatiotemporal optical solitons," J. Opt. B 7, R53-R72 (2005).

Nat. Photonics (1)

C. Rotschild, T. Schwartz, O. Cohen, and M. Segev, "Incoherent solitons in effectively-instantaneous nonlocal nonlinear media," Nat. Photonics 2, 371-376 (2008).

Nature (2)

O. Cohen, G. Bartal, H. Buljan, T. Carmon, J. W. Fleischer, M. Segev, and D. N. Christodoulides, "Observation of random-phase lattice solitons," Nature 433, 500-503 (2005).
[PubMed]

M. Mitchell and M. Segev, "Self-trapping of incoherent white light," Nature 387, 880-883 (1997).

Nature Phys. (1)

C. Rotschild, B. Alfassi, O. Cohen, and M. Segev, "Long-range interactions between optical solitons," Nature Phys. 2, 769-774 (2006).

Opt. Commun. (2)

S. Raghavan and G. P. Agrawal, "Spatiotemporal solitons in inhomogeneous nonlinear media," Opt. Commun. 180, 377-382 (2000).

L. Torner, S. Carrasco, J. P. Torres, L. C. Crasovan, and D. Mihalache, "Tandem light bullets," Opt. Commun. 199, 277-281 (2001).

Opt. Lett. (6)

Opt. Photonics News (3)

F. Wise and P. Trapani, "The hunt for light bullets - spatiotemporal solitons," Opt. Photonics News 13, 28-32 (2002).

A. Hasegawa, "Optical solitons in fibers," Opt. Photonics News 13, 33-37 (2002).

M. Segev and D. N. Christodoulides, "Incoherent solitons: self-trapping of weakly-correlated wave-packets," Opt. Photonics News 13, 70-76 (2002).

Phys. Rev. A (1)

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

Phys. Rev. E (5)

D. Mihalache, D. Mazilu, F. Lederer, B. A. Malomed, Y. V. Kartashov, L. C. Crasovan, and L. Torner, "Three-dimensional spatiotemporal optical solitons in nonlocal nonlinear media," Phys. Rev. E 73, 025601R (2006).

Z. Xu, Ya. V. Kartashov, L. C. Crasovan, D. Mihalache, and L. Torner, "Spatiotemporal discrete multicolor solitons," Phys. Rev. E 70, 066618 (2004).

M. Peccianti and G. Assanto, "Nematic liquid crystals: A suitable medium for self-confinement of coherent and incoherent light," Phys. Rev. E 65, R035603 (2002).

X. Liu, K. Beckwitt, andn F. Wise, "Two-dimensional optical spatiotemporal solitons in quadratic media," Phys. Rev. E 62, 1328-1340 (2000).

O. Bang, W. Krolikowski, J. Wyller, and J. Rasmussen, "Collapse arrest and soliton stabilization in nonlocal nonlinear media," Phys. Rev. E 66, 046619 (2002).

Phys. Rev. Lett. (9)

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, "Experimental Observation of Picosecond Pulse Narrowing and Solitons in Optical Fibers," Phys. Rev. Lett. 45, 1095-1048 (1980).

D. Krökel, N. J. Halas, G. Giuliani, and D. Grischkowsky, "Dark-Pulse Propagation in Optical Fibers," Phys. Rev. Lett. 60, 29-32 (1988).
[PubMed]

A. M. Weiner, J. P. Heritage, R. J. Hawkins, R. N. Thurston, E. M. Kirschner, D. E. Leaird, and W. J. Tomlinson, "Experimental Observation of the Fundamental Dark Soliton in Optical Fibers," Phys. Rev. Lett. 61, 2445-2448 (1988).
[PubMed]

C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, "Solitons in nonlinear media with an infinite range of nonlocality: first observation of coherent elliptic solitons and of vortex-ring solitons," Phys. Rev. Lett. 95, 213904 (2005).
[PubMed]

X. Liu, L. J. Qian, and F. W. Wise, "Generation of optical spatiotemporal solitons," Phys. Rev. Lett. 82, 4631-4634 (1999).

M. Segev, B. Crosignani, P. DiPorto, G. C. Valley, and A. Yariv, "Steady state spatial screening-solitons in photorefractive media with external applied field," Phys. Rev. Lett. 73, 3211-3214 (1994).
[PubMed]

M. Mitchell, Z. Chen, M. Shih, and M. Segev, "Self-trapping of partially spatially-incoherent light," Phys. Rev. Lett. 77, 490-493 (1996).
[PubMed]

A. S. Sukhorukov and Y. S. Kivshar, "Slow-Light Optical Bullets in Arrays of Nonlinear Bragg-Grating Waveguides," Phys. Rev. Lett. 97, 233901 (2006).

E. W. Laedke, K. H. Spatschek, and S. K. Turitsyn, "Stability of discrete solitons and quasicollapse to intrinsically localized modes," Phys. Rev. Lett. 73, 1055-1058 (1994).
[PubMed]

Phys. Rev. Lett. (1)

A.V.  Gorbach and D. V.  Skryabin, "Cascaded Generation of Multiply charged Optical Vortices and Spatiotemporal Helical Beams in a Raman Medium," Phys. Rev. Lett. 98, 243601 (2007).
[PubMed]

Science (3)

G. Stegeman and M. Segev, "Optical spatial solitons and their interactions: universality and diversity," Science 286, 1518-1523 (1999).
[PubMed]

D. Kip, M. Soljacic, M. Segev, E. Eugenieva, and D. N. Christodoulides, "Modulation instability and pattern formation in spatially incoherent light beams," Science 290, 495-498 (2000)
[PubMed]

A. W. Snyder and D. J. Mitchell, "Accessible solitons," Science 276, 1538-1541 (1997).

Theor. Math. Phys. (1)

S. K. Turitsyn, "Spatial dispersion of nonlinearity and stability of multidimensional solitons," Theor. Math. Phys. 64, 226-232 (1985).

Other (1)

O. Cohen, H. Buljan, T. Schwartz, J. W. Fleischer, and M. Segev, "Incoherent solitons in instantaneous nonlocal nonlinear media," Phys. Rev. E 73, 015601(R) (2006).

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Figures (2)

Fig. 1.
Fig. 1.

Temporally-bright spatiotemporal pulse-train soliton. Intensity of the soliton in plane x-z (a) and t-z (b) demonstrating the stationary propagation. (c) Intensity in x-y-t at the input and output faces of the sample. Intensity profiles in x-direction (d) and t-direction (e) at the input and output faces of the sample. (f) Width (FWHM) of the intensity in the x-direction (f) and t-direction (g) versus propagation distance for linear and nonlinear propagations.

Fig. 2.
Fig. 2.

Temporally-dark spatiotemporal pulse-train soliton. Intensity of the soliton in plane x-z (a) and t-z (b) demonstrating the stationary propagation of the soliton. (c) Intensity in x-y-t at the input and output faces of the sample. Intensity profiles in x-direction (d) and t-direction (e) at the input and output faces of the sample. (f) Width (FWHM) of the intensity in the x-direction (f) and t-direction (g) versus propagation distance for linear and nonlinear propagations.

Equations (8)

Equations on this page are rendered with MathJax. Learn more.

i Ψ q z β 2 2 2 Ψ q t 2 + 1 2 k T 2 Ψ q + Δ n q k n 0 Ψ q = 0
Δ n q = n 2 Ψ q 2 + Δ n q slow
t NL Δ n q + 1 slow Δ n q slow T + Δ n q slow = T / 2 T / 2 f ( Ψ 2 ) dt
Δ n slow ( x , y ) = Δ n 0 exp [ ( x 2 + y 2 ) / σ 2 ] 1 T T / 2 T / 2 Ψ 2 dt
Ψ B = I 0 exp [ ( x 2 + y 2 ) / w 2 ] sech ( t / τ ) exp ( ( z ) )
Ψ D = I 0 exp [ ( x 2 + y 2 ) / w 2 ] exp [ ( t / Δ t ) 2 N ] tanh ( t / τ ) exp ( ( z ) )
P = 2 n 0 σ 2 Δ n 0 k 2 w 4
I 0 = 2 β 2 n 0 / n 2 k τ 2

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