Abstract

We study the formation and interaction of spatial dark optical solitons in materials with a nonlocal nonlinear response. We show that unlike in local materials, where dark solitons typically repel, the nonlocal nonlinearity leads to a long-range attraction and formation of stable bound states of dark solitons.

© 2004 Optical Society of America

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  1. Y. S. Kivshar and G. P. Agrawal, Optical Solitons: from Fibers to Photonic Crystals (Academic, San Diego, Calif., 2003), p. 560.
  2. G. I. Stegeman and M. Segev, Science 286, 1518 (1999).
    [CrossRef] [PubMed]
  3. Y. S. Kivshar and B. Luther-Davies, Phys. Rep. 298, 81 (1998), and references therein.
    [CrossRef]
  4. K. J. Blow and N. Doran, Phys. Lett. A 107, 55 (1985).
    [CrossRef]
  5. W. Zhao and E. Bourkoff, Opt. Lett. 14, 1371 (1989).
    [CrossRef] [PubMed]
  6. D. Foursa and P. Emplit, Phys. Rev. Lett. 77, 4011 (1996).
    [CrossRef] [PubMed]
  7. G. Swartzlander, D. R. Andersen, J. J. Regan, H. Yin, and A. E. Kaplan, Phys. Rev. Lett. 66, 1583 (1991).
    [CrossRef] [PubMed]
  8. S. R. Skinner, G. R. Allan, D. R. Andersen, and A. L. Smirl, IEEE J. Quantum Electron. 27, 2211 (1991).
    [CrossRef]
  9. B. Luther-Davies and Y. Xiaoping, Opt. Lett. 17, 1755 (1992).
    [CrossRef] [PubMed]
  10. V. V. Afanasjev, P. L. Chu, and B. A. Malomed, Phys. Rev. E 57, 1088 (1998).
    [CrossRef]
  11. E. A. Ostrovskaya, Y. S. Kivshar, Z. Chen, and M. Segev, Opt. Lett. 24, 327 (1999).
    [CrossRef]
  12. D. Suter and T. Blasberg, Phys. Rev. A 48, 4583 (1993).
    [CrossRef] [PubMed]
  13. A. G. Litvak, JETP Lett. 4, 230 (1966).
  14. A. Parola, L. Salasnich, and L. Reatto, Phys. Rev. A 57, R3180 (1998).
    [CrossRef]
  15. O. Bang, W. Królikowski, J. Wyller, and J. J. Rasmussen, Phys. Rev. E 66, 046619 (2002).
    [CrossRef]
  16. A. W. Snyder and D. J. Mitchell, Science 276, 1538 (1997).
    [CrossRef]
  17. W. Królikowski and O. Bang, Phys. Rev. E 63, 016610 (2001).
    [CrossRef]
  18. J. Wyller, W. Królikowski, O. Bang, and J. J. Rasmussen, Phys. Rev. E 66, 066615 (2002).
    [CrossRef]
  19. W. Królikowski, O. Bang, J. J. Rasmussen, and J. Wyller, Phys. Rev. E 64, 016612 (2001).
    [CrossRef]
  20. I. V. Shadrivov and A. A. Zharov, J. Opt. Soc. Am. B 19, 596 (2002).
    [CrossRef]
  21. N. I. Nikolov, D. Neshev, O. Bang, and W. Krolikowski, Phys. Rev. E 68, 36614 (2003).
    [CrossRef]
  22. C. Conti, M. Peccianti, and G. Assanto, Phys. Rev. Lett. 91, 073901 (2003).
    [CrossRef]
  23. A. V. Buryak, P. Di. Trapani, D. V. Skyrabin, and S. Trillo, Phys. Rep. 370, 63 (2002).
    [CrossRef]
  24. V. E. Zakharov and A. B. Shabat, Sov. Phys. JETP 37, 823 (1973).

2003 (2)

N. I. Nikolov, D. Neshev, O. Bang, and W. Krolikowski, Phys. Rev. E 68, 36614 (2003).
[CrossRef]

C. Conti, M. Peccianti, and G. Assanto, Phys. Rev. Lett. 91, 073901 (2003).
[CrossRef]

2002 (4)

A. V. Buryak, P. Di. Trapani, D. V. Skyrabin, and S. Trillo, Phys. Rep. 370, 63 (2002).
[CrossRef]

O. Bang, W. Królikowski, J. Wyller, and J. J. Rasmussen, Phys. Rev. E 66, 046619 (2002).
[CrossRef]

J. Wyller, W. Królikowski, O. Bang, and J. J. Rasmussen, Phys. Rev. E 66, 066615 (2002).
[CrossRef]

I. V. Shadrivov and A. A. Zharov, J. Opt. Soc. Am. B 19, 596 (2002).
[CrossRef]

2001 (2)

W. Królikowski and O. Bang, Phys. Rev. E 63, 016610 (2001).
[CrossRef]

W. Królikowski, O. Bang, J. J. Rasmussen, and J. Wyller, Phys. Rev. E 64, 016612 (2001).
[CrossRef]

1999 (2)

1998 (3)

Y. S. Kivshar and B. Luther-Davies, Phys. Rep. 298, 81 (1998), and references therein.
[CrossRef]

V. V. Afanasjev, P. L. Chu, and B. A. Malomed, Phys. Rev. E 57, 1088 (1998).
[CrossRef]

A. Parola, L. Salasnich, and L. Reatto, Phys. Rev. A 57, R3180 (1998).
[CrossRef]

1997 (1)

A. W. Snyder and D. J. Mitchell, Science 276, 1538 (1997).
[CrossRef]

1996 (1)

D. Foursa and P. Emplit, Phys. Rev. Lett. 77, 4011 (1996).
[CrossRef] [PubMed]

1993 (1)

D. Suter and T. Blasberg, Phys. Rev. A 48, 4583 (1993).
[CrossRef] [PubMed]

1992 (1)

1991 (2)

G. Swartzlander, D. R. Andersen, J. J. Regan, H. Yin, and A. E. Kaplan, Phys. Rev. Lett. 66, 1583 (1991).
[CrossRef] [PubMed]

S. R. Skinner, G. R. Allan, D. R. Andersen, and A. L. Smirl, IEEE J. Quantum Electron. 27, 2211 (1991).
[CrossRef]

1989 (1)

1985 (1)

K. J. Blow and N. Doran, Phys. Lett. A 107, 55 (1985).
[CrossRef]

1973 (1)

V. E. Zakharov and A. B. Shabat, Sov. Phys. JETP 37, 823 (1973).

1966 (1)

A. G. Litvak, JETP Lett. 4, 230 (1966).

Afanasjev, V. V.

V. V. Afanasjev, P. L. Chu, and B. A. Malomed, Phys. Rev. E 57, 1088 (1998).
[CrossRef]

Agrawal, G. P.

Y. S. Kivshar and G. P. Agrawal, Optical Solitons: from Fibers to Photonic Crystals (Academic, San Diego, Calif., 2003), p. 560.

Allan, G. R.

S. R. Skinner, G. R. Allan, D. R. Andersen, and A. L. Smirl, IEEE J. Quantum Electron. 27, 2211 (1991).
[CrossRef]

Andersen, D. R.

S. R. Skinner, G. R. Allan, D. R. Andersen, and A. L. Smirl, IEEE J. Quantum Electron. 27, 2211 (1991).
[CrossRef]

G. Swartzlander, D. R. Andersen, J. J. Regan, H. Yin, and A. E. Kaplan, Phys. Rev. Lett. 66, 1583 (1991).
[CrossRef] [PubMed]

Assanto, G.

C. Conti, M. Peccianti, and G. Assanto, Phys. Rev. Lett. 91, 073901 (2003).
[CrossRef]

Bang, O.

N. I. Nikolov, D. Neshev, O. Bang, and W. Krolikowski, Phys. Rev. E 68, 36614 (2003).
[CrossRef]

O. Bang, W. Królikowski, J. Wyller, and J. J. Rasmussen, Phys. Rev. E 66, 046619 (2002).
[CrossRef]

J. Wyller, W. Królikowski, O. Bang, and J. J. Rasmussen, Phys. Rev. E 66, 066615 (2002).
[CrossRef]

W. Królikowski, O. Bang, J. J. Rasmussen, and J. Wyller, Phys. Rev. E 64, 016612 (2001).
[CrossRef]

W. Królikowski and O. Bang, Phys. Rev. E 63, 016610 (2001).
[CrossRef]

Blasberg, T.

D. Suter and T. Blasberg, Phys. Rev. A 48, 4583 (1993).
[CrossRef] [PubMed]

Blow, K. J.

K. J. Blow and N. Doran, Phys. Lett. A 107, 55 (1985).
[CrossRef]

Bourkoff, E.

Buryak, A. V.

A. V. Buryak, P. Di. Trapani, D. V. Skyrabin, and S. Trillo, Phys. Rep. 370, 63 (2002).
[CrossRef]

Chen, Z.

Chu, P. L.

V. V. Afanasjev, P. L. Chu, and B. A. Malomed, Phys. Rev. E 57, 1088 (1998).
[CrossRef]

Conti, C.

C. Conti, M. Peccianti, and G. Assanto, Phys. Rev. Lett. 91, 073901 (2003).
[CrossRef]

Doran, N.

K. J. Blow and N. Doran, Phys. Lett. A 107, 55 (1985).
[CrossRef]

Emplit, P.

D. Foursa and P. Emplit, Phys. Rev. Lett. 77, 4011 (1996).
[CrossRef] [PubMed]

Foursa, D.

D. Foursa and P. Emplit, Phys. Rev. Lett. 77, 4011 (1996).
[CrossRef] [PubMed]

Kaplan, A. E.

G. Swartzlander, D. R. Andersen, J. J. Regan, H. Yin, and A. E. Kaplan, Phys. Rev. Lett. 66, 1583 (1991).
[CrossRef] [PubMed]

Kivshar, Y. S.

E. A. Ostrovskaya, Y. S. Kivshar, Z. Chen, and M. Segev, Opt. Lett. 24, 327 (1999).
[CrossRef]

Y. S. Kivshar and B. Luther-Davies, Phys. Rep. 298, 81 (1998), and references therein.
[CrossRef]

Y. S. Kivshar and G. P. Agrawal, Optical Solitons: from Fibers to Photonic Crystals (Academic, San Diego, Calif., 2003), p. 560.

Krolikowski, W.

N. I. Nikolov, D. Neshev, O. Bang, and W. Krolikowski, Phys. Rev. E 68, 36614 (2003).
[CrossRef]

Królikowski, W.

O. Bang, W. Królikowski, J. Wyller, and J. J. Rasmussen, Phys. Rev. E 66, 046619 (2002).
[CrossRef]

J. Wyller, W. Królikowski, O. Bang, and J. J. Rasmussen, Phys. Rev. E 66, 066615 (2002).
[CrossRef]

W. Królikowski, O. Bang, J. J. Rasmussen, and J. Wyller, Phys. Rev. E 64, 016612 (2001).
[CrossRef]

W. Królikowski and O. Bang, Phys. Rev. E 63, 016610 (2001).
[CrossRef]

Litvak, A. G.

A. G. Litvak, JETP Lett. 4, 230 (1966).

Luther-Davies, B.

Y. S. Kivshar and B. Luther-Davies, Phys. Rep. 298, 81 (1998), and references therein.
[CrossRef]

B. Luther-Davies and Y. Xiaoping, Opt. Lett. 17, 1755 (1992).
[CrossRef] [PubMed]

Malomed, B. A.

V. V. Afanasjev, P. L. Chu, and B. A. Malomed, Phys. Rev. E 57, 1088 (1998).
[CrossRef]

Mitchell, D. J.

A. W. Snyder and D. J. Mitchell, Science 276, 1538 (1997).
[CrossRef]

Neshev, D.

N. I. Nikolov, D. Neshev, O. Bang, and W. Krolikowski, Phys. Rev. E 68, 36614 (2003).
[CrossRef]

Nikolov, N. I.

N. I. Nikolov, D. Neshev, O. Bang, and W. Krolikowski, Phys. Rev. E 68, 36614 (2003).
[CrossRef]

Ostrovskaya, E. A.

Parola, A.

A. Parola, L. Salasnich, and L. Reatto, Phys. Rev. A 57, R3180 (1998).
[CrossRef]

Peccianti, M.

C. Conti, M. Peccianti, and G. Assanto, Phys. Rev. Lett. 91, 073901 (2003).
[CrossRef]

Rasmussen, J. J.

O. Bang, W. Królikowski, J. Wyller, and J. J. Rasmussen, Phys. Rev. E 66, 046619 (2002).
[CrossRef]

J. Wyller, W. Królikowski, O. Bang, and J. J. Rasmussen, Phys. Rev. E 66, 066615 (2002).
[CrossRef]

W. Królikowski, O. Bang, J. J. Rasmussen, and J. Wyller, Phys. Rev. E 64, 016612 (2001).
[CrossRef]

Reatto, L.

A. Parola, L. Salasnich, and L. Reatto, Phys. Rev. A 57, R3180 (1998).
[CrossRef]

Regan, J. J.

G. Swartzlander, D. R. Andersen, J. J. Regan, H. Yin, and A. E. Kaplan, Phys. Rev. Lett. 66, 1583 (1991).
[CrossRef] [PubMed]

Salasnich, L.

A. Parola, L. Salasnich, and L. Reatto, Phys. Rev. A 57, R3180 (1998).
[CrossRef]

Segev, M.

Shabat, A. B.

V. E. Zakharov and A. B. Shabat, Sov. Phys. JETP 37, 823 (1973).

Shadrivov, I. V.

Skinner, S. R.

S. R. Skinner, G. R. Allan, D. R. Andersen, and A. L. Smirl, IEEE J. Quantum Electron. 27, 2211 (1991).
[CrossRef]

Skyrabin, D. V.

A. V. Buryak, P. Di. Trapani, D. V. Skyrabin, and S. Trillo, Phys. Rep. 370, 63 (2002).
[CrossRef]

Smirl, A. L.

S. R. Skinner, G. R. Allan, D. R. Andersen, and A. L. Smirl, IEEE J. Quantum Electron. 27, 2211 (1991).
[CrossRef]

Snyder, A. W.

A. W. Snyder and D. J. Mitchell, Science 276, 1538 (1997).
[CrossRef]

Stegeman, G. I.

G. I. Stegeman and M. Segev, Science 286, 1518 (1999).
[CrossRef] [PubMed]

Suter, D.

D. Suter and T. Blasberg, Phys. Rev. A 48, 4583 (1993).
[CrossRef] [PubMed]

Swartzlander, G.

G. Swartzlander, D. R. Andersen, J. J. Regan, H. Yin, and A. E. Kaplan, Phys. Rev. Lett. 66, 1583 (1991).
[CrossRef] [PubMed]

Trapani, P. Di.

A. V. Buryak, P. Di. Trapani, D. V. Skyrabin, and S. Trillo, Phys. Rep. 370, 63 (2002).
[CrossRef]

Trillo, S.

A. V. Buryak, P. Di. Trapani, D. V. Skyrabin, and S. Trillo, Phys. Rep. 370, 63 (2002).
[CrossRef]

Wyller, J.

O. Bang, W. Królikowski, J. Wyller, and J. J. Rasmussen, Phys. Rev. E 66, 046619 (2002).
[CrossRef]

J. Wyller, W. Królikowski, O. Bang, and J. J. Rasmussen, Phys. Rev. E 66, 066615 (2002).
[CrossRef]

W. Królikowski, O. Bang, J. J. Rasmussen, and J. Wyller, Phys. Rev. E 64, 016612 (2001).
[CrossRef]

Xiaoping, Y.

Yin, H.

G. Swartzlander, D. R. Andersen, J. J. Regan, H. Yin, and A. E. Kaplan, Phys. Rev. Lett. 66, 1583 (1991).
[CrossRef] [PubMed]

Zakharov, V. E.

V. E. Zakharov and A. B. Shabat, Sov. Phys. JETP 37, 823 (1973).

Zhao, W.

Zharov, A. A.

IEEE J. Quantum Electron. (1)

S. R. Skinner, G. R. Allan, D. R. Andersen, and A. L. Smirl, IEEE J. Quantum Electron. 27, 2211 (1991).
[CrossRef]

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

JETP Lett. (1)

A. G. Litvak, JETP Lett. 4, 230 (1966).

Opt. Lett. (3)

Phys. Lett. A (1)

K. J. Blow and N. Doran, Phys. Lett. A 107, 55 (1985).
[CrossRef]

Phys. Rep. (2)

Y. S. Kivshar and B. Luther-Davies, Phys. Rep. 298, 81 (1998), and references therein.
[CrossRef]

A. V. Buryak, P. Di. Trapani, D. V. Skyrabin, and S. Trillo, Phys. Rep. 370, 63 (2002).
[CrossRef]

Phys. Rev. A (2)

D. Suter and T. Blasberg, Phys. Rev. A 48, 4583 (1993).
[CrossRef] [PubMed]

A. Parola, L. Salasnich, and L. Reatto, Phys. Rev. A 57, R3180 (1998).
[CrossRef]

Phys. Rev. E (6)

O. Bang, W. Królikowski, J. Wyller, and J. J. Rasmussen, Phys. Rev. E 66, 046619 (2002).
[CrossRef]

W. Królikowski and O. Bang, Phys. Rev. E 63, 016610 (2001).
[CrossRef]

J. Wyller, W. Królikowski, O. Bang, and J. J. Rasmussen, Phys. Rev. E 66, 066615 (2002).
[CrossRef]

W. Królikowski, O. Bang, J. J. Rasmussen, and J. Wyller, Phys. Rev. E 64, 016612 (2001).
[CrossRef]

V. V. Afanasjev, P. L. Chu, and B. A. Malomed, Phys. Rev. E 57, 1088 (1998).
[CrossRef]

N. I. Nikolov, D. Neshev, O. Bang, and W. Krolikowski, Phys. Rev. E 68, 36614 (2003).
[CrossRef]

Phys. Rev. Lett. (3)

C. Conti, M. Peccianti, and G. Assanto, Phys. Rev. Lett. 91, 073901 (2003).
[CrossRef]

D. Foursa and P. Emplit, Phys. Rev. Lett. 77, 4011 (1996).
[CrossRef] [PubMed]

G. Swartzlander, D. R. Andersen, J. J. Regan, H. Yin, and A. E. Kaplan, Phys. Rev. Lett. 66, 1583 (1991).
[CrossRef] [PubMed]

Science (2)

G. I. Stegeman and M. Segev, Science 286, 1518 (1999).
[CrossRef] [PubMed]

A. W. Snyder and D. J. Mitchell, Science 276, 1538 (1997).
[CrossRef]

Sov. Phys. JETP (1)

V. E. Zakharov and A. B. Shabat, Sov. Phys. JETP 37, 823 (1973).

Other (1)

Y. S. Kivshar and G. P. Agrawal, Optical Solitons: from Fibers to Photonic Crystals (Academic, San Diego, Calif., 2003), p. 560.

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Figures (3)

Fig. 1
Fig. 1

Intensity profile ux2 (solid curve) and induced waveguide structure 1+Δnx (dashed curve) of (a) a single nonlocal dark soliton and (b) a bound state for λ=1. The degree of nonlocality is σ=4.

Fig. 2
Fig. 2

Evolution of the intensity profile of a bound state of two nonlocal dark solitons for σ=1 and λ=1 on (top) a cw background and (bottom) a Gaussian background.

Fig. 3
Fig. 3

Dark nonlocal solitons formed by (a), (b) phase or (c) intensity modulation of a cw background. In (a) the phase jump is π and the degree of nonlocality is σ=2, and the initial soliton separation is x0=5.5,4,2.5 from left to right. In (b) the phase jump is 0.95π and x0=2.5, and σ=0.1,1,2 from left to right. In (c) the width of the intensity gap is 7.5, and σ=0.1,3,6 from left to right.

Equations (3)

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

izu+x2u+ΔnIu=0, ΔnI=--Rx-τIτdτ,
izu+x2u+Δnu=0,
Δn-σ2x2Δn=-u2,

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