Abstract

We report experimental studies of spatial optical solitons in a nonlinear glass waveguide. Spatial solitons are self-trapped beams that propagate without diffraction. We demonstrate the formation of spatial solitons, and we investigate interactions between them. At high intensities two-photon absorption is observed to lead to the breakup of solitons. A beam propagation method is used to simulate the experimental conditions numerically, and the results agree well with experiment.

© 1991 Optical Society of America

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  1. R. Y. Chiao, E. Garmire, and C. H. Townes, Phys. Rev. Lett. 13, 479 (1964).
    [CrossRef]
  2. P. L. Kelley, Phys. Rev. Lett. 15, 1005 (1965).
    [CrossRef]
  3. V. E. Zakharov and A. B. Shabat, Sov. Phys. JETP 34, 62 (1972).
  4. A. Hasegawa and F. Tappert, Appl. Phys. Lett. 23, 142, 171 (1973).
    [CrossRef]
  5. L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, Phys. Rev. Lett. 45, 1095 (1980).
    [CrossRef]
  6. L. F. Mollenauer and K. Smith, Opt. Lett. 13, 675 (1988).
    [CrossRef]
  7. A. Barthelemy, S. Maneuf, and C. Froehly, Opt. Commun. 55, 201 (1985); S. Maneuf, R. Desailly, and C. Froehly, Opt. Commun. 65, 193 (1988); S. Maneuf and F. Reynaud, Opt. Commun. 66, 325 (1988).
    [CrossRef]
  8. J. S. Aitchison, A. M. Weiner, Y. Silberberg, M. K. Oliver, J. L. Jackel, D. E. Leaird, E. M. Vogel, and P. W. E. Smith, Opt. Lett. 15, 471 (1990).
    [CrossRef] [PubMed]
  9. A. M. Weiner, J. P. Heritage, R. J. Hawkins, R. N. Thurston, E. M. Kirchner, D. E. Leaird, and W. J. Tomlinson, Phys. Rev. Lett. 61, 2445 (1990).
    [CrossRef]
  10. G. A. Swartzlander, D. R. Andersen, and A. E. Kaplan, in Conference on Lasers and Electro-Optics, Vol. 7 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), p. 336.
  11. D. Milam and M. J. Weber, J. Appl. Phys. 47, 2497 (1976).
    [CrossRef]
  12. S. R. Friberg and P. W. Smith, IEEE J. Quantum Electron. QE-23, 2089 (1987).
    [CrossRef]
  13. T. Findakly, Opt. Eng. 24, 244 (1985).
    [CrossRef]
  14. W. H. Knox, M. C. Downer, R. L. Fork, and C. V. Shank, Opt. Lett. 9, 552 (1984).
    [CrossRef] [PubMed]
  15. R. L. Fork, O. E. Martinez, and J. P. Gordon, Opt. Lett. 9, 150 (1984).
    [CrossRef] [PubMed]
  16. Y. Silberberg, Opt. Lett. 15, 1005 (1990).
    [CrossRef] [PubMed]
  17. V. Mizrahi, K. W. DeLong, G. I. Stegeman, M. A. Saifi, and M. J. Andrejco, Opt. Lett. 14, 1140 (1989).
    [CrossRef] [PubMed]
  18. M. D. Feit and J. A. Fleck, Appl. Opt. 17, 3990 (1978).
    [CrossRef] [PubMed]
  19. Y. Silberberg, Opt. Lett. 15, 1282 (1990).
    [CrossRef] [PubMed]
  20. J. P. Gordon, Opt. Lett. 11, 596 (1983).
    [CrossRef]
  21. C. Desem and P. L. Chu, Proc. Inst. Electr. Eng. Part J 134, 145 (1987).
  22. F. M. Mitschke and L. F. Mollenauer, Opt. Lett. 12, 355 (1987).
    [CrossRef] [PubMed]
  23. A. Barthelemy, F. Reynaud, and B. Colombeau, SPIE Proc. 862, 27 (1987); F. Reynaud and A. Barthelemy, Europhys. Lett. 12, 401 (1990).
    [CrossRef]
  24. J. S. Aitchison, A. M. Weiner, Y. Silberberg, D. E. Leaird, M. K. Oliver, J. L. Jackel, and P. W. E. Smith, Opt. Lett 16, 15 (1991).
    [CrossRef] [PubMed]
  25. E. M. Vogel, D. M. Krol, J. L. Jackel, and J. S. Aitchison, MRS Symp. Proc. 152, 83 (1989).
    [CrossRef]
  26. J. L. Jackel, E. M. Vogel, and J. S. Aitchison, Appl. Opt. 29, 3126 (1990).
    [CrossRef] [PubMed]
  27. J. S. Aitchison, M. K. Oliver, E. Kapon, E. Colas, and P. W. E. Smith, Appl. Phys. Lett. 56, 1305 (1990).
    [CrossRef]

1991 (1)

J. S. Aitchison, A. M. Weiner, Y. Silberberg, D. E. Leaird, M. K. Oliver, J. L. Jackel, and P. W. E. Smith, Opt. Lett 16, 15 (1991).
[CrossRef] [PubMed]

1990 (6)

1989 (2)

V. Mizrahi, K. W. DeLong, G. I. Stegeman, M. A. Saifi, and M. J. Andrejco, Opt. Lett. 14, 1140 (1989).
[CrossRef] [PubMed]

E. M. Vogel, D. M. Krol, J. L. Jackel, and J. S. Aitchison, MRS Symp. Proc. 152, 83 (1989).
[CrossRef]

1988 (1)

1987 (4)

S. R. Friberg and P. W. Smith, IEEE J. Quantum Electron. QE-23, 2089 (1987).
[CrossRef]

C. Desem and P. L. Chu, Proc. Inst. Electr. Eng. Part J 134, 145 (1987).

F. M. Mitschke and L. F. Mollenauer, Opt. Lett. 12, 355 (1987).
[CrossRef] [PubMed]

A. Barthelemy, F. Reynaud, and B. Colombeau, SPIE Proc. 862, 27 (1987); F. Reynaud and A. Barthelemy, Europhys. Lett. 12, 401 (1990).
[CrossRef]

1985 (2)

T. Findakly, Opt. Eng. 24, 244 (1985).
[CrossRef]

A. Barthelemy, S. Maneuf, and C. Froehly, Opt. Commun. 55, 201 (1985); S. Maneuf, R. Desailly, and C. Froehly, Opt. Commun. 65, 193 (1988); S. Maneuf and F. Reynaud, Opt. Commun. 66, 325 (1988).
[CrossRef]

1984 (2)

1983 (1)

J. P. Gordon, Opt. Lett. 11, 596 (1983).
[CrossRef]

1980 (1)

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, Phys. Rev. Lett. 45, 1095 (1980).
[CrossRef]

1978 (1)

1976 (1)

D. Milam and M. J. Weber, J. Appl. Phys. 47, 2497 (1976).
[CrossRef]

1973 (1)

A. Hasegawa and F. Tappert, Appl. Phys. Lett. 23, 142, 171 (1973).
[CrossRef]

1972 (1)

V. E. Zakharov and A. B. Shabat, Sov. Phys. JETP 34, 62 (1972).

1965 (1)

P. L. Kelley, Phys. Rev. Lett. 15, 1005 (1965).
[CrossRef]

1964 (1)

R. Y. Chiao, E. Garmire, and C. H. Townes, Phys. Rev. Lett. 13, 479 (1964).
[CrossRef]

Aitchison, J. S.

J. S. Aitchison, A. M. Weiner, Y. Silberberg, D. E. Leaird, M. K. Oliver, J. L. Jackel, and P. W. E. Smith, Opt. Lett 16, 15 (1991).
[CrossRef] [PubMed]

J. S. Aitchison, A. M. Weiner, Y. Silberberg, M. K. Oliver, J. L. Jackel, D. E. Leaird, E. M. Vogel, and P. W. E. Smith, Opt. Lett. 15, 471 (1990).
[CrossRef] [PubMed]

J. L. Jackel, E. M. Vogel, and J. S. Aitchison, Appl. Opt. 29, 3126 (1990).
[CrossRef] [PubMed]

J. S. Aitchison, M. K. Oliver, E. Kapon, E. Colas, and P. W. E. Smith, Appl. Phys. Lett. 56, 1305 (1990).
[CrossRef]

E. M. Vogel, D. M. Krol, J. L. Jackel, and J. S. Aitchison, MRS Symp. Proc. 152, 83 (1989).
[CrossRef]

Andersen, D. R.

G. A. Swartzlander, D. R. Andersen, and A. E. Kaplan, in Conference on Lasers and Electro-Optics, Vol. 7 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), p. 336.

Andrejco, M. J.

Barthelemy, A.

A. Barthelemy, F. Reynaud, and B. Colombeau, SPIE Proc. 862, 27 (1987); F. Reynaud and A. Barthelemy, Europhys. Lett. 12, 401 (1990).
[CrossRef]

A. Barthelemy, S. Maneuf, and C. Froehly, Opt. Commun. 55, 201 (1985); S. Maneuf, R. Desailly, and C. Froehly, Opt. Commun. 65, 193 (1988); S. Maneuf and F. Reynaud, Opt. Commun. 66, 325 (1988).
[CrossRef]

Chiao, R. Y.

R. Y. Chiao, E. Garmire, and C. H. Townes, Phys. Rev. Lett. 13, 479 (1964).
[CrossRef]

Chu, P. L.

C. Desem and P. L. Chu, Proc. Inst. Electr. Eng. Part J 134, 145 (1987).

Colas, E.

J. S. Aitchison, M. K. Oliver, E. Kapon, E. Colas, and P. W. E. Smith, Appl. Phys. Lett. 56, 1305 (1990).
[CrossRef]

Colombeau, B.

A. Barthelemy, F. Reynaud, and B. Colombeau, SPIE Proc. 862, 27 (1987); F. Reynaud and A. Barthelemy, Europhys. Lett. 12, 401 (1990).
[CrossRef]

DeLong, K. W.

Desem, C.

C. Desem and P. L. Chu, Proc. Inst. Electr. Eng. Part J 134, 145 (1987).

Downer, M. C.

Feit, M. D.

Findakly, T.

T. Findakly, Opt. Eng. 24, 244 (1985).
[CrossRef]

Fleck, J. A.

Fork, R. L.

Friberg, S. R.

S. R. Friberg and P. W. Smith, IEEE J. Quantum Electron. QE-23, 2089 (1987).
[CrossRef]

Froehly, C.

A. Barthelemy, S. Maneuf, and C. Froehly, Opt. Commun. 55, 201 (1985); S. Maneuf, R. Desailly, and C. Froehly, Opt. Commun. 65, 193 (1988); S. Maneuf and F. Reynaud, Opt. Commun. 66, 325 (1988).
[CrossRef]

Garmire, E.

R. Y. Chiao, E. Garmire, and C. H. Townes, Phys. Rev. Lett. 13, 479 (1964).
[CrossRef]

Gordon, J. P.

R. L. Fork, O. E. Martinez, and J. P. Gordon, Opt. Lett. 9, 150 (1984).
[CrossRef] [PubMed]

J. P. Gordon, Opt. Lett. 11, 596 (1983).
[CrossRef]

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, Phys. Rev. Lett. 45, 1095 (1980).
[CrossRef]

Hasegawa, A.

A. Hasegawa and F. Tappert, Appl. Phys. Lett. 23, 142, 171 (1973).
[CrossRef]

Hawkins, R. J.

A. M. Weiner, J. P. Heritage, R. J. Hawkins, R. N. Thurston, E. M. Kirchner, D. E. Leaird, and W. J. Tomlinson, Phys. Rev. Lett. 61, 2445 (1990).
[CrossRef]

Heritage, J. P.

A. M. Weiner, J. P. Heritage, R. J. Hawkins, R. N. Thurston, E. M. Kirchner, D. E. Leaird, and W. J. Tomlinson, Phys. Rev. Lett. 61, 2445 (1990).
[CrossRef]

Jackel, J. L.

J. S. Aitchison, A. M. Weiner, Y. Silberberg, D. E. Leaird, M. K. Oliver, J. L. Jackel, and P. W. E. Smith, Opt. Lett 16, 15 (1991).
[CrossRef] [PubMed]

J. S. Aitchison, A. M. Weiner, Y. Silberberg, M. K. Oliver, J. L. Jackel, D. E. Leaird, E. M. Vogel, and P. W. E. Smith, Opt. Lett. 15, 471 (1990).
[CrossRef] [PubMed]

J. L. Jackel, E. M. Vogel, and J. S. Aitchison, Appl. Opt. 29, 3126 (1990).
[CrossRef] [PubMed]

E. M. Vogel, D. M. Krol, J. L. Jackel, and J. S. Aitchison, MRS Symp. Proc. 152, 83 (1989).
[CrossRef]

Kaplan, A. E.

G. A. Swartzlander, D. R. Andersen, and A. E. Kaplan, in Conference on Lasers and Electro-Optics, Vol. 7 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), p. 336.

Kapon, E.

J. S. Aitchison, M. K. Oliver, E. Kapon, E. Colas, and P. W. E. Smith, Appl. Phys. Lett. 56, 1305 (1990).
[CrossRef]

Kelley, P. L.

P. L. Kelley, Phys. Rev. Lett. 15, 1005 (1965).
[CrossRef]

Kirchner, E. M.

A. M. Weiner, J. P. Heritage, R. J. Hawkins, R. N. Thurston, E. M. Kirchner, D. E. Leaird, and W. J. Tomlinson, Phys. Rev. Lett. 61, 2445 (1990).
[CrossRef]

Knox, W. H.

Krol, D. M.

E. M. Vogel, D. M. Krol, J. L. Jackel, and J. S. Aitchison, MRS Symp. Proc. 152, 83 (1989).
[CrossRef]

Leaird, D. E.

J. S. Aitchison, A. M. Weiner, Y. Silberberg, D. E. Leaird, M. K. Oliver, J. L. Jackel, and P. W. E. Smith, Opt. Lett 16, 15 (1991).
[CrossRef] [PubMed]

A. M. Weiner, J. P. Heritage, R. J. Hawkins, R. N. Thurston, E. M. Kirchner, D. E. Leaird, and W. J. Tomlinson, Phys. Rev. Lett. 61, 2445 (1990).
[CrossRef]

J. S. Aitchison, A. M. Weiner, Y. Silberberg, M. K. Oliver, J. L. Jackel, D. E. Leaird, E. M. Vogel, and P. W. E. Smith, Opt. Lett. 15, 471 (1990).
[CrossRef] [PubMed]

Maneuf, S.

A. Barthelemy, S. Maneuf, and C. Froehly, Opt. Commun. 55, 201 (1985); S. Maneuf, R. Desailly, and C. Froehly, Opt. Commun. 65, 193 (1988); S. Maneuf and F. Reynaud, Opt. Commun. 66, 325 (1988).
[CrossRef]

Martinez, O. E.

Milam, D.

D. Milam and M. J. Weber, J. Appl. Phys. 47, 2497 (1976).
[CrossRef]

Mitschke, F. M.

Mizrahi, V.

Mollenauer, L. F.

Oliver, M. K.

J. S. Aitchison, A. M. Weiner, Y. Silberberg, D. E. Leaird, M. K. Oliver, J. L. Jackel, and P. W. E. Smith, Opt. Lett 16, 15 (1991).
[CrossRef] [PubMed]

J. S. Aitchison, A. M. Weiner, Y. Silberberg, M. K. Oliver, J. L. Jackel, D. E. Leaird, E. M. Vogel, and P. W. E. Smith, Opt. Lett. 15, 471 (1990).
[CrossRef] [PubMed]

J. S. Aitchison, M. K. Oliver, E. Kapon, E. Colas, and P. W. E. Smith, Appl. Phys. Lett. 56, 1305 (1990).
[CrossRef]

Reynaud, F.

A. Barthelemy, F. Reynaud, and B. Colombeau, SPIE Proc. 862, 27 (1987); F. Reynaud and A. Barthelemy, Europhys. Lett. 12, 401 (1990).
[CrossRef]

Saifi, M. A.

Shabat, A. B.

V. E. Zakharov and A. B. Shabat, Sov. Phys. JETP 34, 62 (1972).

Shank, C. V.

Silberberg, Y.

Smith, K.

Smith, P. W.

S. R. Friberg and P. W. Smith, IEEE J. Quantum Electron. QE-23, 2089 (1987).
[CrossRef]

Smith, P. W. E.

J. S. Aitchison, A. M. Weiner, Y. Silberberg, D. E. Leaird, M. K. Oliver, J. L. Jackel, and P. W. E. Smith, Opt. Lett 16, 15 (1991).
[CrossRef] [PubMed]

J. S. Aitchison, A. M. Weiner, Y. Silberberg, M. K. Oliver, J. L. Jackel, D. E. Leaird, E. M. Vogel, and P. W. E. Smith, Opt. Lett. 15, 471 (1990).
[CrossRef] [PubMed]

J. S. Aitchison, M. K. Oliver, E. Kapon, E. Colas, and P. W. E. Smith, Appl. Phys. Lett. 56, 1305 (1990).
[CrossRef]

Stegeman, G. I.

Stolen, R. H.

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, Phys. Rev. Lett. 45, 1095 (1980).
[CrossRef]

Swartzlander, G. A.

G. A. Swartzlander, D. R. Andersen, and A. E. Kaplan, in Conference on Lasers and Electro-Optics, Vol. 7 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), p. 336.

Tappert, F.

A. Hasegawa and F. Tappert, Appl. Phys. Lett. 23, 142, 171 (1973).
[CrossRef]

Thurston, R. N.

A. M. Weiner, J. P. Heritage, R. J. Hawkins, R. N. Thurston, E. M. Kirchner, D. E. Leaird, and W. J. Tomlinson, Phys. Rev. Lett. 61, 2445 (1990).
[CrossRef]

Tomlinson, W. J.

A. M. Weiner, J. P. Heritage, R. J. Hawkins, R. N. Thurston, E. M. Kirchner, D. E. Leaird, and W. J. Tomlinson, Phys. Rev. Lett. 61, 2445 (1990).
[CrossRef]

Townes, C. H.

R. Y. Chiao, E. Garmire, and C. H. Townes, Phys. Rev. Lett. 13, 479 (1964).
[CrossRef]

Vogel, E. M.

Weber, M. J.

D. Milam and M. J. Weber, J. Appl. Phys. 47, 2497 (1976).
[CrossRef]

Weiner, A. M.

J. S. Aitchison, A. M. Weiner, Y. Silberberg, D. E. Leaird, M. K. Oliver, J. L. Jackel, and P. W. E. Smith, Opt. Lett 16, 15 (1991).
[CrossRef] [PubMed]

A. M. Weiner, J. P. Heritage, R. J. Hawkins, R. N. Thurston, E. M. Kirchner, D. E. Leaird, and W. J. Tomlinson, Phys. Rev. Lett. 61, 2445 (1990).
[CrossRef]

J. S. Aitchison, A. M. Weiner, Y. Silberberg, M. K. Oliver, J. L. Jackel, D. E. Leaird, E. M. Vogel, and P. W. E. Smith, Opt. Lett. 15, 471 (1990).
[CrossRef] [PubMed]

Zakharov, V. E.

V. E. Zakharov and A. B. Shabat, Sov. Phys. JETP 34, 62 (1972).

Appl. Opt. (2)

Appl. Phys. Lett. (2)

J. S. Aitchison, M. K. Oliver, E. Kapon, E. Colas, and P. W. E. Smith, Appl. Phys. Lett. 56, 1305 (1990).
[CrossRef]

A. Hasegawa and F. Tappert, Appl. Phys. Lett. 23, 142, 171 (1973).
[CrossRef]

IEEE J. Quantum Electron. (1)

S. R. Friberg and P. W. Smith, IEEE J. Quantum Electron. QE-23, 2089 (1987).
[CrossRef]

J. Appl. Phys. (1)

D. Milam and M. J. Weber, J. Appl. Phys. 47, 2497 (1976).
[CrossRef]

MRS Symp. Proc. (1)

E. M. Vogel, D. M. Krol, J. L. Jackel, and J. S. Aitchison, MRS Symp. Proc. 152, 83 (1989).
[CrossRef]

Opt. Commun. (1)

A. Barthelemy, S. Maneuf, and C. Froehly, Opt. Commun. 55, 201 (1985); S. Maneuf, R. Desailly, and C. Froehly, Opt. Commun. 65, 193 (1988); S. Maneuf and F. Reynaud, Opt. Commun. 66, 325 (1988).
[CrossRef]

Opt. Eng. (1)

T. Findakly, Opt. Eng. 24, 244 (1985).
[CrossRef]

Opt. Lett (1)

J. S. Aitchison, A. M. Weiner, Y. Silberberg, D. E. Leaird, M. K. Oliver, J. L. Jackel, and P. W. E. Smith, Opt. Lett 16, 15 (1991).
[CrossRef] [PubMed]

Opt. Lett. (9)

Phys. Rev. Lett. (4)

A. M. Weiner, J. P. Heritage, R. J. Hawkins, R. N. Thurston, E. M. Kirchner, D. E. Leaird, and W. J. Tomlinson, Phys. Rev. Lett. 61, 2445 (1990).
[CrossRef]

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, Phys. Rev. Lett. 45, 1095 (1980).
[CrossRef]

R. Y. Chiao, E. Garmire, and C. H. Townes, Phys. Rev. Lett. 13, 479 (1964).
[CrossRef]

P. L. Kelley, Phys. Rev. Lett. 15, 1005 (1965).
[CrossRef]

Proc. Inst. Electr. Eng. Part J (1)

C. Desem and P. L. Chu, Proc. Inst. Electr. Eng. Part J 134, 145 (1987).

Sov. Phys. JETP (1)

V. E. Zakharov and A. B. Shabat, Sov. Phys. JETP 34, 62 (1972).

SPIE Proc. (1)

A. Barthelemy, F. Reynaud, and B. Colombeau, SPIE Proc. 862, 27 (1987); F. Reynaud and A. Barthelemy, Europhys. Lett. 12, 401 (1990).
[CrossRef]

Other (1)

G. A. Swartzlander, D. R. Andersen, and A. E. Kaplan, in Conference on Lasers and Electro-Optics, Vol. 7 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), p. 336.

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Figures (11)

Fig. 1
Fig. 1

Schematic view of the apparatus used in these experiments: BS’s, beam splitters; M’s, mirrors; Det’s, detectors; BS’s, beam splitters.

Fig. 2
Fig. 2

Experimentally measured time-averaged output-mode profiles as a function of increasing input power: a, 15-μm input spot; b, output beam profile, P = 24 kW; c, output beam profile, P = 400 kW; d, output beam profile P = 1.25 MW.

Fig. 3
Fig. 3

Experimental measurements of aperture transmission versus input power: a, aperture positioned at the center of the output mode; b, aperture positioned on one wing of the output; c, total transmission of the waveguide.

Fig. 4
Fig. 4

Simulation of propagation of a beam with a Gaussian initial profile, 15 μm FWHM, and power of 5Pt in a waveguide with a normalized TPA coefficient of K = 0.033. The total propagation distance is 5 mm.

Fig. 5
Fig. 5

Calculated intensity profile at the output of the 5-mm waveguide for a 15-μm FWHM Gaussian input beam and for various input powers.

Fig. 6
Fig. 6

Spatial–temporal profile of the a, input pulse and b, out put pulse for a peak power of 4.5Pt after propagating of the beam 5 mm in the waveguide.

Fig. 7
Fig. 7

Calculated time-averaged profile for the output pulse of Fig. 6.

Fig. 8
Fig. 8

Simulation of the transmission through an aperture centered on the optical axis versus the input peak power. The aperture width is equal to the input pulse width (FWHM).

Fig. 9
Fig. 9

Simulations of soliton interactions. Two fundamental solitons are launched in parallel. The relative phase is a, 0, resulting in an attractive force, and b, π, resulting in a repulsive force.

Fig. 10
Fig. 10

Experimental results of soliton interactions: (a) light distribution at the input of the waveguide, (b) output light distribution when the two beams do not overlap in time, (c) interacting solitons with a π phase shift, (d) interacting solitons with 0 phase shift.

Fig. 11
Fig. 11

Experimentally measured time-averaged output mode profiles for the high-nonlinearity glass: a, input spot; b, low-power output mode; c, Pin = 41 kW; d, Pin = 110 kW.

Equations (8)

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

2 i k E z = - 2 E x 2 - 2 k 2 n 2 n 0 E 2 E ,
E ( x , z ) = 1 k a 0 ( n 0 n 2 ) 1 / 2 exp ( i z 2 k a 0 2 ) sech ( x - x 0 a 0 ) ,
U s = E 2 d x = n 0 n 2 2 a 0 k 2 .
P t = 2 n 0 w n 2 a 0 k 2 .
2 i k E z = - 2 E x 2 - 2 k 2 n 2 n 0 ( 1 + i K ) E 2 E ,
K = β n 0 2 k n 2 .
I ( z ) = I 0 α 0 exp ( - α 0 z ) α 0 + β I 0 [ 1 - exp ( - α 0 z ) ] .
T = C ( 1 - R ) 2 ( 1 - α 0 L ) [ 1 - C ( 1 - R ) β L P inc 2 3 / 2 π A eff ] ,

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