Abstract

The effects caused by nonresonant nonlinear interaction between noncollinear self-focusing beams are considered in two-dimensional optical samples by use of multiscale analysis. An analytical expression for beam trajectory shifts that are due to mutual interaction is derived, and the range of parameters is given, beyond which the mentioned consideration fails. We compare our results with the naive geometrical-optics model. It is shown that these two approaches give the same results. This justifies use of the geometrical-optics approach to describe elastic and almost-elastic collision processes both in Kerr and saturable nonlinear media. The results we obtained could be useful for the design of phase independent nonlinear photonic switches and all-optical logic elements.

© 2004 Optical Society of America

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References

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  1. G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, San Diego, Calif., 1995).
  2. R. K. Dodd, J. C. Eilbeck, J. D. Gibbon, and H. C. Morris, Solitons and Nonlinear Wave Equations (Academic, London, 1982).
  3. V. E. Zakharov and E. A. Kuznetsov, “Hamiltonian formalism for nonlinear waves,” Phys. Usp. 40, 1087–1116 (1997).
    [CrossRef]
  4. S. O. Demokritov, B. Hillebrands, and A. N. Slavin, “Brillouin light scattering studies of confined spin waves: linear and nonlinear confinement,” Phys. Rep. 348, 441–489 (2001).
    [CrossRef]
  5. G. I. Stegeman and M. Segev, “Optical spatial solitons and their interactions: universality and diversity,” Science 286, 1518–1523 (1999).
    [CrossRef] [PubMed]
  6. N. Giorgadze and R. Khomeriki, “Nonresonant interaction of noncollinear weakly nonlinear modulated waves of magnetization,” J. Magn. Magn. Mater. 186, 239–247 (1998).
    [CrossRef]
  7. N. Giorgadze and R. Khomeriki, “Interaction of envelope solitons in yttrium iron garnet films,” Phys. Rev. B 60, 1247–1251 (1999).
    [CrossRef]
  8. R. Khomeriki and L. Tkeshelashvili, “A generalized approach for the description of magnetostatic soliton interaction in yttrium-iron-garnet films,” J. Phys.: Condens. Matter 12, 8875–8882 (2000).
  9. A. W. Snyder, D. J. Mitchell, and A. V. Buryak, “Qualitative theory of bright solitons: the soliton sketch,” J. Opt. Soc. Am. B 13, 1146–1150 (1996).
    [CrossRef]
  10. V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).
  11. V. I. Karpman and V. V. Solov’ev, “A perturbational approach to the two-soliton systems,” Physica D 3, 487–502 (1981).
    [CrossRef]
  12. K. A. Gorshkov and L. A. Ostrovsky, “Interactions of solitons in nonintegrable systems: direct perturbation method and applications,” Physica D 3, 428–438 (1981).
    [CrossRef]
  13. D. Anderson and M. Lisak, “Bandwidth limits due to incoherent soliton interaction in optical-fiber communication systems,” Phys. Rev. A 32, 2270–2274 (1985).
    [CrossRef] [PubMed]
  14. F. Reynaud and A. Barthelemy, “Optically controlled interaction between two fundamental soliton beams,” Europhys. Lett. 12, 401–405 (1990).
    [CrossRef]
  15. J. S. Aitchison, A. M. Weiner, Y. Silberberg, D. E. Leaird, M. K. Oliver, J. L. Jackel, and P. W. E. Smith, “Experimental observation of spatial soliton interactions,” Opt. Lett. 16, 15–17 (1991).
    [CrossRef] [PubMed]
  16. J. S. Aitchison, Y. Silberberg, A. M. Weiner, D. E. Leaird, M. K. Oliver, J. L. Jackel, E. M. Vogel, and P. W. E. Smith, “Spatial optical solitons in planar glass waveguides,” J. Opt. Soc. Am. B 8, 1290–1297 (1991).
    [CrossRef]
  17. M. Shih and M. Segev, “Incoherent collisions between two-dimensional bright steady-state photorefractive spatial screening solitons,” Opt. Lett. 21, 1538–1540 (1996).
    [CrossRef] [PubMed]
  18. V. Tikhonenko, J. Christou, and B. Luther-Davies, “Three dimensional bright spatial soliton collision and fusion in a saturable nonlinear medium,” Phys. Rev. Lett. 76, 2698–2701 (1996).
    [CrossRef] [PubMed]
  19. H. Meng, G. Salamo, M. Shih, and M. Segev, “Coherent collisions of photorefractive solitons,” Opt. Lett. 22, 448–450 (1997).
    [CrossRef] [PubMed]
  20. W. Krolikowski and S. A. Holmstrom, “Fusion and birth of spatial solitons upon collision,” Opt. Lett. 22, 369–371 (1997).
    [CrossRef] [PubMed]
  21. M. Shih, M. Segev, and G. Salamo, “Three-dimensional spiraling of interacting spatial solitons,” Phys. Rev. Lett. 78, 2551–2554 (1997).
    [CrossRef]
  22. A. V. Buryak, Y. S. Kivshar, M. Shih, and M. Segev, “Induced coherence and stable soliton spiraling,” Phys. Rev. Lett. 82, 81–84 (1999).
    [CrossRef]
  23. T. Taniuti, “Reductive perturbation method and far fields of wave equations,” Suppl. Prog. Theor. Phys. 55, 1–35 (1975).
    [CrossRef]
  24. M. Oikawa and N. Yajima, “Perturbation approach to nonlinear systems. II. Interaction of nonlinear modulated waves,” J. Phys. Soc. Jpn. 37, 486–496 (1974).
    [CrossRef]
  25. T.-T. Shi and S. Chi, “Nonlinear photonic switching by using the spatial soliton collision,” Opt. Lett. 15, 1123–1125 (1990).
    [CrossRef] [PubMed]
  26. O. V. Kolokoltsev, R. Salas, and V. Vountesmeri, “All-optical phase-independent logic elements based on phase shift induced by coherent soliton collisions,” J. Lightwave Technol. 20, 1048–1053 (2002).
    [CrossRef]
  27. S. Blair, K. H. Wagner, and R. McLeod, “Material figures of merit for spatial soliton interactions in the presence of absorption,” J. Opt. Soc. Am. B 13, 2141–2153 (1996).
    [CrossRef]
  28. O. Cohen, R. Uzdin, T. Carmon, J. W. Fleischer, M. Segev, and S. Odoulov, “Collisions between optical spatial solitons propagating in opposite directions,” Phys. Rev. Lett. 89, 133901 (2002).
    [CrossRef] [PubMed]
  29. A. W. Snyder and D. J. Mitchell, “Accessible solitons,” Science 276, 1538–1541 (1997).
    [CrossRef]

2002 (2)

O. Cohen, R. Uzdin, T. Carmon, J. W. Fleischer, M. Segev, and S. Odoulov, “Collisions between optical spatial solitons propagating in opposite directions,” Phys. Rev. Lett. 89, 133901 (2002).
[CrossRef] [PubMed]

O. V. Kolokoltsev, R. Salas, and V. Vountesmeri, “All-optical phase-independent logic elements based on phase shift induced by coherent soliton collisions,” J. Lightwave Technol. 20, 1048–1053 (2002).
[CrossRef]

2001 (1)

S. O. Demokritov, B. Hillebrands, and A. N. Slavin, “Brillouin light scattering studies of confined spin waves: linear and nonlinear confinement,” Phys. Rep. 348, 441–489 (2001).
[CrossRef]

2000 (1)

R. Khomeriki and L. Tkeshelashvili, “A generalized approach for the description of magnetostatic soliton interaction in yttrium-iron-garnet films,” J. Phys.: Condens. Matter 12, 8875–8882 (2000).

1999 (3)

G. I. Stegeman and M. Segev, “Optical spatial solitons and their interactions: universality and diversity,” Science 286, 1518–1523 (1999).
[CrossRef] [PubMed]

N. Giorgadze and R. Khomeriki, “Interaction of envelope solitons in yttrium iron garnet films,” Phys. Rev. B 60, 1247–1251 (1999).
[CrossRef]

A. V. Buryak, Y. S. Kivshar, M. Shih, and M. Segev, “Induced coherence and stable soliton spiraling,” Phys. Rev. Lett. 82, 81–84 (1999).
[CrossRef]

1998 (1)

N. Giorgadze and R. Khomeriki, “Nonresonant interaction of noncollinear weakly nonlinear modulated waves of magnetization,” J. Magn. Magn. Mater. 186, 239–247 (1998).
[CrossRef]

1997 (5)

V. E. Zakharov and E. A. Kuznetsov, “Hamiltonian formalism for nonlinear waves,” Phys. Usp. 40, 1087–1116 (1997).
[CrossRef]

A. W. Snyder and D. J. Mitchell, “Accessible solitons,” Science 276, 1538–1541 (1997).
[CrossRef]

M. Shih, M. Segev, and G. Salamo, “Three-dimensional spiraling of interacting spatial solitons,” Phys. Rev. Lett. 78, 2551–2554 (1997).
[CrossRef]

W. Krolikowski and S. A. Holmstrom, “Fusion and birth of spatial solitons upon collision,” Opt. Lett. 22, 369–371 (1997).
[CrossRef] [PubMed]

H. Meng, G. Salamo, M. Shih, and M. Segev, “Coherent collisions of photorefractive solitons,” Opt. Lett. 22, 448–450 (1997).
[CrossRef] [PubMed]

1996 (4)

1991 (2)

1990 (2)

T.-T. Shi and S. Chi, “Nonlinear photonic switching by using the spatial soliton collision,” Opt. Lett. 15, 1123–1125 (1990).
[CrossRef] [PubMed]

F. Reynaud and A. Barthelemy, “Optically controlled interaction between two fundamental soliton beams,” Europhys. Lett. 12, 401–405 (1990).
[CrossRef]

1985 (1)

D. Anderson and M. Lisak, “Bandwidth limits due to incoherent soliton interaction in optical-fiber communication systems,” Phys. Rev. A 32, 2270–2274 (1985).
[CrossRef] [PubMed]

1981 (2)

V. I. Karpman and V. V. Solov’ev, “A perturbational approach to the two-soliton systems,” Physica D 3, 487–502 (1981).
[CrossRef]

K. A. Gorshkov and L. A. Ostrovsky, “Interactions of solitons in nonintegrable systems: direct perturbation method and applications,” Physica D 3, 428–438 (1981).
[CrossRef]

1975 (1)

T. Taniuti, “Reductive perturbation method and far fields of wave equations,” Suppl. Prog. Theor. Phys. 55, 1–35 (1975).
[CrossRef]

1974 (1)

M. Oikawa and N. Yajima, “Perturbation approach to nonlinear systems. II. Interaction of nonlinear modulated waves,” J. Phys. Soc. Jpn. 37, 486–496 (1974).
[CrossRef]

1972 (1)

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).

Aitchison, J. S.

Anderson, D.

D. Anderson and M. Lisak, “Bandwidth limits due to incoherent soliton interaction in optical-fiber communication systems,” Phys. Rev. A 32, 2270–2274 (1985).
[CrossRef] [PubMed]

Barthelemy, A.

F. Reynaud and A. Barthelemy, “Optically controlled interaction between two fundamental soliton beams,” Europhys. Lett. 12, 401–405 (1990).
[CrossRef]

Blair, S.

Buryak, A. V.

A. V. Buryak, Y. S. Kivshar, M. Shih, and M. Segev, “Induced coherence and stable soliton spiraling,” Phys. Rev. Lett. 82, 81–84 (1999).
[CrossRef]

A. W. Snyder, D. J. Mitchell, and A. V. Buryak, “Qualitative theory of bright solitons: the soliton sketch,” J. Opt. Soc. Am. B 13, 1146–1150 (1996).
[CrossRef]

Carmon, T.

O. Cohen, R. Uzdin, T. Carmon, J. W. Fleischer, M. Segev, and S. Odoulov, “Collisions between optical spatial solitons propagating in opposite directions,” Phys. Rev. Lett. 89, 133901 (2002).
[CrossRef] [PubMed]

Chi, S.

Christou, J.

V. Tikhonenko, J. Christou, and B. Luther-Davies, “Three dimensional bright spatial soliton collision and fusion in a saturable nonlinear medium,” Phys. Rev. Lett. 76, 2698–2701 (1996).
[CrossRef] [PubMed]

Cohen, O.

O. Cohen, R. Uzdin, T. Carmon, J. W. Fleischer, M. Segev, and S. Odoulov, “Collisions between optical spatial solitons propagating in opposite directions,” Phys. Rev. Lett. 89, 133901 (2002).
[CrossRef] [PubMed]

Demokritov, S. O.

S. O. Demokritov, B. Hillebrands, and A. N. Slavin, “Brillouin light scattering studies of confined spin waves: linear and nonlinear confinement,” Phys. Rep. 348, 441–489 (2001).
[CrossRef]

Fleischer, J. W.

O. Cohen, R. Uzdin, T. Carmon, J. W. Fleischer, M. Segev, and S. Odoulov, “Collisions between optical spatial solitons propagating in opposite directions,” Phys. Rev. Lett. 89, 133901 (2002).
[CrossRef] [PubMed]

Giorgadze, N.

N. Giorgadze and R. Khomeriki, “Interaction of envelope solitons in yttrium iron garnet films,” Phys. Rev. B 60, 1247–1251 (1999).
[CrossRef]

N. Giorgadze and R. Khomeriki, “Nonresonant interaction of noncollinear weakly nonlinear modulated waves of magnetization,” J. Magn. Magn. Mater. 186, 239–247 (1998).
[CrossRef]

Gorshkov, K. A.

K. A. Gorshkov and L. A. Ostrovsky, “Interactions of solitons in nonintegrable systems: direct perturbation method and applications,” Physica D 3, 428–438 (1981).
[CrossRef]

Hillebrands, B.

S. O. Demokritov, B. Hillebrands, and A. N. Slavin, “Brillouin light scattering studies of confined spin waves: linear and nonlinear confinement,” Phys. Rep. 348, 441–489 (2001).
[CrossRef]

Holmstrom, S. A.

Jackel, J. L.

Karpman, V. I.

V. I. Karpman and V. V. Solov’ev, “A perturbational approach to the two-soliton systems,” Physica D 3, 487–502 (1981).
[CrossRef]

Khomeriki, R.

R. Khomeriki and L. Tkeshelashvili, “A generalized approach for the description of magnetostatic soliton interaction in yttrium-iron-garnet films,” J. Phys.: Condens. Matter 12, 8875–8882 (2000).

N. Giorgadze and R. Khomeriki, “Interaction of envelope solitons in yttrium iron garnet films,” Phys. Rev. B 60, 1247–1251 (1999).
[CrossRef]

N. Giorgadze and R. Khomeriki, “Nonresonant interaction of noncollinear weakly nonlinear modulated waves of magnetization,” J. Magn. Magn. Mater. 186, 239–247 (1998).
[CrossRef]

Kivshar, Y. S.

A. V. Buryak, Y. S. Kivshar, M. Shih, and M. Segev, “Induced coherence and stable soliton spiraling,” Phys. Rev. Lett. 82, 81–84 (1999).
[CrossRef]

Kolokoltsev, O. V.

Krolikowski, W.

Kuznetsov, E. A.

V. E. Zakharov and E. A. Kuznetsov, “Hamiltonian formalism for nonlinear waves,” Phys. Usp. 40, 1087–1116 (1997).
[CrossRef]

Leaird, D. E.

Lisak, M.

D. Anderson and M. Lisak, “Bandwidth limits due to incoherent soliton interaction in optical-fiber communication systems,” Phys. Rev. A 32, 2270–2274 (1985).
[CrossRef] [PubMed]

Luther-Davies, B.

V. Tikhonenko, J. Christou, and B. Luther-Davies, “Three dimensional bright spatial soliton collision and fusion in a saturable nonlinear medium,” Phys. Rev. Lett. 76, 2698–2701 (1996).
[CrossRef] [PubMed]

McLeod, R.

Meng, H.

Mitchell, D. J.

Odoulov, S.

O. Cohen, R. Uzdin, T. Carmon, J. W. Fleischer, M. Segev, and S. Odoulov, “Collisions between optical spatial solitons propagating in opposite directions,” Phys. Rev. Lett. 89, 133901 (2002).
[CrossRef] [PubMed]

Oikawa, M.

M. Oikawa and N. Yajima, “Perturbation approach to nonlinear systems. II. Interaction of nonlinear modulated waves,” J. Phys. Soc. Jpn. 37, 486–496 (1974).
[CrossRef]

Oliver, M. K.

Ostrovsky, L. A.

K. A. Gorshkov and L. A. Ostrovsky, “Interactions of solitons in nonintegrable systems: direct perturbation method and applications,” Physica D 3, 428–438 (1981).
[CrossRef]

Reynaud, F.

F. Reynaud and A. Barthelemy, “Optically controlled interaction between two fundamental soliton beams,” Europhys. Lett. 12, 401–405 (1990).
[CrossRef]

Salamo, G.

M. Shih, M. Segev, and G. Salamo, “Three-dimensional spiraling of interacting spatial solitons,” Phys. Rev. Lett. 78, 2551–2554 (1997).
[CrossRef]

H. Meng, G. Salamo, M. Shih, and M. Segev, “Coherent collisions of photorefractive solitons,” Opt. Lett. 22, 448–450 (1997).
[CrossRef] [PubMed]

Salas, R.

Segev, M.

O. Cohen, R. Uzdin, T. Carmon, J. W. Fleischer, M. Segev, and S. Odoulov, “Collisions between optical spatial solitons propagating in opposite directions,” Phys. Rev. Lett. 89, 133901 (2002).
[CrossRef] [PubMed]

A. V. Buryak, Y. S. Kivshar, M. Shih, and M. Segev, “Induced coherence and stable soliton spiraling,” Phys. Rev. Lett. 82, 81–84 (1999).
[CrossRef]

G. I. Stegeman and M. Segev, “Optical spatial solitons and their interactions: universality and diversity,” Science 286, 1518–1523 (1999).
[CrossRef] [PubMed]

M. Shih, M. Segev, and G. Salamo, “Three-dimensional spiraling of interacting spatial solitons,” Phys. Rev. Lett. 78, 2551–2554 (1997).
[CrossRef]

H. Meng, G. Salamo, M. Shih, and M. Segev, “Coherent collisions of photorefractive solitons,” Opt. Lett. 22, 448–450 (1997).
[CrossRef] [PubMed]

M. Shih and M. Segev, “Incoherent collisions between two-dimensional bright steady-state photorefractive spatial screening solitons,” Opt. Lett. 21, 1538–1540 (1996).
[CrossRef] [PubMed]

Shabat, A. B.

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).

Shi, T.-T.

Shih, M.

A. V. Buryak, Y. S. Kivshar, M. Shih, and M. Segev, “Induced coherence and stable soliton spiraling,” Phys. Rev. Lett. 82, 81–84 (1999).
[CrossRef]

M. Shih, M. Segev, and G. Salamo, “Three-dimensional spiraling of interacting spatial solitons,” Phys. Rev. Lett. 78, 2551–2554 (1997).
[CrossRef]

H. Meng, G. Salamo, M. Shih, and M. Segev, “Coherent collisions of photorefractive solitons,” Opt. Lett. 22, 448–450 (1997).
[CrossRef] [PubMed]

M. Shih and M. Segev, “Incoherent collisions between two-dimensional bright steady-state photorefractive spatial screening solitons,” Opt. Lett. 21, 1538–1540 (1996).
[CrossRef] [PubMed]

Silberberg, Y.

Slavin, A. N.

S. O. Demokritov, B. Hillebrands, and A. N. Slavin, “Brillouin light scattering studies of confined spin waves: linear and nonlinear confinement,” Phys. Rep. 348, 441–489 (2001).
[CrossRef]

Smith, P. W. E.

Snyder, A. W.

Solov’ev, V. V.

V. I. Karpman and V. V. Solov’ev, “A perturbational approach to the two-soliton systems,” Physica D 3, 487–502 (1981).
[CrossRef]

Stegeman, G. I.

G. I. Stegeman and M. Segev, “Optical spatial solitons and their interactions: universality and diversity,” Science 286, 1518–1523 (1999).
[CrossRef] [PubMed]

Taniuti, T.

T. Taniuti, “Reductive perturbation method and far fields of wave equations,” Suppl. Prog. Theor. Phys. 55, 1–35 (1975).
[CrossRef]

Tikhonenko, V.

V. Tikhonenko, J. Christou, and B. Luther-Davies, “Three dimensional bright spatial soliton collision and fusion in a saturable nonlinear medium,” Phys. Rev. Lett. 76, 2698–2701 (1996).
[CrossRef] [PubMed]

Tkeshelashvili, L.

R. Khomeriki and L. Tkeshelashvili, “A generalized approach for the description of magnetostatic soliton interaction in yttrium-iron-garnet films,” J. Phys.: Condens. Matter 12, 8875–8882 (2000).

Uzdin, R.

O. Cohen, R. Uzdin, T. Carmon, J. W. Fleischer, M. Segev, and S. Odoulov, “Collisions between optical spatial solitons propagating in opposite directions,” Phys. Rev. Lett. 89, 133901 (2002).
[CrossRef] [PubMed]

Vogel, E. M.

Vountesmeri, V.

Wagner, K. H.

Weiner, A. M.

Yajima, N.

M. Oikawa and N. Yajima, “Perturbation approach to nonlinear systems. II. Interaction of nonlinear modulated waves,” J. Phys. Soc. Jpn. 37, 486–496 (1974).
[CrossRef]

Zakharov, V. E.

V. E. Zakharov and E. A. Kuznetsov, “Hamiltonian formalism for nonlinear waves,” Phys. Usp. 40, 1087–1116 (1997).
[CrossRef]

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).

Europhys. Lett. (1)

F. Reynaud and A. Barthelemy, “Optically controlled interaction between two fundamental soliton beams,” Europhys. Lett. 12, 401–405 (1990).
[CrossRef]

J. Lightwave Technol. (1)

J. Magn. Magn. Mater. (1)

N. Giorgadze and R. Khomeriki, “Nonresonant interaction of noncollinear weakly nonlinear modulated waves of magnetization,” J. Magn. Magn. Mater. 186, 239–247 (1998).
[CrossRef]

J. Opt. Soc. Am. B (3)

J. Phys. Soc. Jpn. (1)

M. Oikawa and N. Yajima, “Perturbation approach to nonlinear systems. II. Interaction of nonlinear modulated waves,” J. Phys. Soc. Jpn. 37, 486–496 (1974).
[CrossRef]

J. Phys.: Condens. Matter (1)

R. Khomeriki and L. Tkeshelashvili, “A generalized approach for the description of magnetostatic soliton interaction in yttrium-iron-garnet films,” J. Phys.: Condens. Matter 12, 8875–8882 (2000).

Opt. Lett. (5)

Phys. Rep. (1)

S. O. Demokritov, B. Hillebrands, and A. N. Slavin, “Brillouin light scattering studies of confined spin waves: linear and nonlinear confinement,” Phys. Rep. 348, 441–489 (2001).
[CrossRef]

Phys. Rev. A (1)

D. Anderson and M. Lisak, “Bandwidth limits due to incoherent soliton interaction in optical-fiber communication systems,” Phys. Rev. A 32, 2270–2274 (1985).
[CrossRef] [PubMed]

Phys. Rev. B (1)

N. Giorgadze and R. Khomeriki, “Interaction of envelope solitons in yttrium iron garnet films,” Phys. Rev. B 60, 1247–1251 (1999).
[CrossRef]

Phys. Rev. Lett. (4)

V. Tikhonenko, J. Christou, and B. Luther-Davies, “Three dimensional bright spatial soliton collision and fusion in a saturable nonlinear medium,” Phys. Rev. Lett. 76, 2698–2701 (1996).
[CrossRef] [PubMed]

M. Shih, M. Segev, and G. Salamo, “Three-dimensional spiraling of interacting spatial solitons,” Phys. Rev. Lett. 78, 2551–2554 (1997).
[CrossRef]

A. V. Buryak, Y. S. Kivshar, M. Shih, and M. Segev, “Induced coherence and stable soliton spiraling,” Phys. Rev. Lett. 82, 81–84 (1999).
[CrossRef]

O. Cohen, R. Uzdin, T. Carmon, J. W. Fleischer, M. Segev, and S. Odoulov, “Collisions between optical spatial solitons propagating in opposite directions,” Phys. Rev. Lett. 89, 133901 (2002).
[CrossRef] [PubMed]

Phys. Usp. (1)

V. E. Zakharov and E. A. Kuznetsov, “Hamiltonian formalism for nonlinear waves,” Phys. Usp. 40, 1087–1116 (1997).
[CrossRef]

Physica D (2)

V. I. Karpman and V. V. Solov’ev, “A perturbational approach to the two-soliton systems,” Physica D 3, 487–502 (1981).
[CrossRef]

K. A. Gorshkov and L. A. Ostrovsky, “Interactions of solitons in nonintegrable systems: direct perturbation method and applications,” Physica D 3, 428–438 (1981).
[CrossRef]

Science (2)

G. I. Stegeman and M. Segev, “Optical spatial solitons and their interactions: universality and diversity,” Science 286, 1518–1523 (1999).
[CrossRef] [PubMed]

A. W. Snyder and D. J. Mitchell, “Accessible solitons,” Science 276, 1538–1541 (1997).
[CrossRef]

Sov. Phys. JETP (1)

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).

Suppl. Prog. Theor. Phys. (1)

T. Taniuti, “Reductive perturbation method and far fields of wave equations,” Suppl. Prog. Theor. Phys. 55, 1–35 (1975).
[CrossRef]

Other (2)

G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, San Diego, Calif., 1995).

R. K. Dodd, J. C. Eilbeck, J. D. Gibbon, and H. C. Morris, Solitons and Nonlinear Wave Equations (Academic, London, 1982).

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Figures (2)

Fig. 1
Fig. 1

Schematic of the interaction process between self-focusing beams in an off-resonant optical medium. Solid curves borders of the beams; α0, the angle between the first (narrow) beam and the normal vector n2 of the second (wide) beam; k1 and k2, carrier wave vectors, respectively; δl1, a trajectory shift of the first beam, which was caused by nonlinear interaction effects. Note that the second beam trajectory is also slightly shifted.

Fig. 2
Fig. 2

Optical ray refraction through the self-focusing beam area with borders denoted by horizontal solid lines.

Equations (13)

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

P=χ(1)E+χ(3)E3,
2E-βEtt=γ(E3)tt,
E=α=1αl=-φl(α)(ξ, τ)exp[il(kr-ωt)],
iφ1(1)τ+2φ1(1)ξ2+3γω2φ1(1)|φ1(1)|2=0,
E=α=1αl1l2=-φl1l2(α)exp[i(kl1l2r-ωl1l2t+Ωl1l2)],
ξp=[(npr)-ψp(ξ1, ξ2, τ1, τ2)],τp=2 (kpr)2kp2,
E=φ10(1)exp{i[k1r-ω1t+Ω10(1)]}+φ01(1)×exp{i[k2r-ω2t+Ω01(1)]}+c.c.,
|φ10(1)|=|A1|sech{|A1|6γω12[(n1r)-ψ1(1)]},
|φ01(1)|=|A2|sech{|A2|6γω22[(n2r)-ψ2(1)]},
ψ1(1)ξ2=(n1n2)(k1n2)Ω10(1)ξ2=3γω12|φ01(1)|2 (n1n2)(k1n2)2.
|A2|3γβ|sin α0|cos2 α01,
δl1=ψ1(1)()-ψ1(1)(-)=3γβsin α0cos2 α0-|φ01(1)(ξ2)|2dξ2.
δl1=|A2|6γβω2sin α0cos2 α0.

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