Abstract

We consider soliton formation in thermal nonlinear media bounded by rectangular cross sections and uncover what we believe to be a new class of nonlinear stationary topological state. Specifically, we find that stationary higher-order vortex states in standard shapes do not exist, but rather they take the form of multiple spatially separated single-charge singularities nested in an elliptical beam. Double-charge states are found to be remarkably robust despite their shape asymmetry and phase-singularity splitting. States with higher topological charges are found to be unstable.

© 2010 Optical Society of America

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  1. V. I. Kruglov and R. A. Vlasov, Phys. Lett. A 111, 401 (1985).
    [CrossRef]
  2. A. S. Desyatnikov, Y. S. Kivshar, and L. Torner, Prog. Opt. 47, 291 (2005).
    [CrossRef]
  3. V. I. Kruglov, Yu. A. Logvin, and V. M. Volkov, J. Mod. Opt. 39, 2277 (1992).
    [CrossRef]
  4. L. Torner and D. V. Petrov, Electron. Lett. 33, 608 (1997).
    [CrossRef]
  5. W. J. Firth and D. V. Skryabin, Phys. Rev. Lett. 79, 2450 (1997).
    [CrossRef]
  6. S. K. Turitsyn, Theor. Math. Phys. 64, 226 (1985).
    [CrossRef]
  7. A. M. Rubenchik and S. K. Turitsyn, Laser Part. Beams 5, 3 (1987).
    [CrossRef]
  8. D. Briedis, D. Petersen, D. Edmundson, W. Krolikowski, and O. Bang, Opt. Express 13, 435 (2005).
    [CrossRef] [PubMed]
  9. A. I. Yakimenko, Y. A. Zaliznyak, and Y. Kivshar, Phys. Rev. E 71, 065603 (2005).
    [CrossRef]
  10. S. Skupin, O. Bang, D. Edmundson, and W. Krolikowski, Phys. Rev. E 73, 066603 (2006).
    [CrossRef]
  11. A. I. Yakimenko, V. M. Lashkin, and O. O. Prikhodko, Phys. Rev. E 73, 066605 (2006).
    [CrossRef]
  12. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Opt. Express 15, 9378 (2007).
    [CrossRef] [PubMed]
  13. S. Skupin, M. Saffman, and W. Krolikowski, Phys. Rev. Lett. 98, 263902 (2007).
    [CrossRef] [PubMed]
  14. C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, Phys. Rev. Lett. 95, 213904 (2005).
    [CrossRef] [PubMed]
  15. A. Minovich, D. Neshev, A. Dreischuh, W. Krolikowski, and Y. Kivshar, Opt. Lett. 32, 1599 (2007).
    [CrossRef] [PubMed]
  16. B. Alfassi, C. Rotschild, O. Manela, M. Segev, and D. N. Christodoulides, Opt. Lett. 32, 154 (2007).
    [CrossRef]
  17. B. Alfassi, C. Rotschild, O. Manela, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 98, 213901 (2007).
    [CrossRef] [PubMed]
  18. Y. V. Kartashov, F. Ye, V. A. Vysloukh, and L. Torner, Opt. Lett. 32, 2260 (2007).
    [CrossRef] [PubMed]
  19. F. Ye, Y. V. Kartashov, and L. Torner, Phys. Rev. A 77, 043821 (2008).
    [CrossRef]
  20. D. Buccoliero, A. S. Desyatnikov, W. Krolikowski, and Y. Kivshar, J. Opt. A 11, 094014 (2009).
    [CrossRef]
  21. A. Alberucci, M. Peccianti, and G. Assanto, Opt. Lett. 32, 2795 (2007).
    [CrossRef] [PubMed]
  22. A. V. Mamaev, M. Saffman, and A. A. Zozulya, Phys. Rev. Lett. 78, 2108 (1997).
    [CrossRef]

2009 (1)

D. Buccoliero, A. S. Desyatnikov, W. Krolikowski, and Y. Kivshar, J. Opt. A 11, 094014 (2009).
[CrossRef]

2008 (1)

F. Ye, Y. V. Kartashov, and L. Torner, Phys. Rev. A 77, 043821 (2008).
[CrossRef]

2007 (7)

2006 (2)

S. Skupin, O. Bang, D. Edmundson, and W. Krolikowski, Phys. Rev. E 73, 066603 (2006).
[CrossRef]

A. I. Yakimenko, V. M. Lashkin, and O. O. Prikhodko, Phys. Rev. E 73, 066605 (2006).
[CrossRef]

2005 (4)

A. S. Desyatnikov, Y. S. Kivshar, and L. Torner, Prog. Opt. 47, 291 (2005).
[CrossRef]

A. I. Yakimenko, Y. A. Zaliznyak, and Y. Kivshar, Phys. Rev. E 71, 065603 (2005).
[CrossRef]

C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, Phys. Rev. Lett. 95, 213904 (2005).
[CrossRef] [PubMed]

D. Briedis, D. Petersen, D. Edmundson, W. Krolikowski, and O. Bang, Opt. Express 13, 435 (2005).
[CrossRef] [PubMed]

1997 (3)

A. V. Mamaev, M. Saffman, and A. A. Zozulya, Phys. Rev. Lett. 78, 2108 (1997).
[CrossRef]

L. Torner and D. V. Petrov, Electron. Lett. 33, 608 (1997).
[CrossRef]

W. J. Firth and D. V. Skryabin, Phys. Rev. Lett. 79, 2450 (1997).
[CrossRef]

1992 (1)

V. I. Kruglov, Yu. A. Logvin, and V. M. Volkov, J. Mod. Opt. 39, 2277 (1992).
[CrossRef]

1987 (1)

A. M. Rubenchik and S. K. Turitsyn, Laser Part. Beams 5, 3 (1987).
[CrossRef]

1985 (2)

V. I. Kruglov and R. A. Vlasov, Phys. Lett. A 111, 401 (1985).
[CrossRef]

S. K. Turitsyn, Theor. Math. Phys. 64, 226 (1985).
[CrossRef]

Alberucci, A.

Alfassi, B.

B. Alfassi, C. Rotschild, O. Manela, M. Segev, and D. N. Christodoulides, Opt. Lett. 32, 154 (2007).
[CrossRef]

B. Alfassi, C. Rotschild, O. Manela, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 98, 213901 (2007).
[CrossRef] [PubMed]

Assanto, G.

Bang, O.

S. Skupin, O. Bang, D. Edmundson, and W. Krolikowski, Phys. Rev. E 73, 066603 (2006).
[CrossRef]

D. Briedis, D. Petersen, D. Edmundson, W. Krolikowski, and O. Bang, Opt. Express 13, 435 (2005).
[CrossRef] [PubMed]

Briedis, D.

Buccoliero, D.

D. Buccoliero, A. S. Desyatnikov, W. Krolikowski, and Y. Kivshar, J. Opt. A 11, 094014 (2009).
[CrossRef]

Carmon, T.

C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, Phys. Rev. Lett. 95, 213904 (2005).
[CrossRef] [PubMed]

Christodoulides, D. N.

B. Alfassi, C. Rotschild, O. Manela, M. Segev, and D. N. Christodoulides, Opt. Lett. 32, 154 (2007).
[CrossRef]

B. Alfassi, C. Rotschild, O. Manela, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 98, 213901 (2007).
[CrossRef] [PubMed]

Cohen, O.

C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, Phys. Rev. Lett. 95, 213904 (2005).
[CrossRef] [PubMed]

Desyatnikov, A. S.

D. Buccoliero, A. S. Desyatnikov, W. Krolikowski, and Y. Kivshar, J. Opt. A 11, 094014 (2009).
[CrossRef]

A. S. Desyatnikov, Y. S. Kivshar, and L. Torner, Prog. Opt. 47, 291 (2005).
[CrossRef]

Dreischuh, A.

Edmundson, D.

S. Skupin, O. Bang, D. Edmundson, and W. Krolikowski, Phys. Rev. E 73, 066603 (2006).
[CrossRef]

D. Briedis, D. Petersen, D. Edmundson, W. Krolikowski, and O. Bang, Opt. Express 13, 435 (2005).
[CrossRef] [PubMed]

Firth, W. J.

W. J. Firth and D. V. Skryabin, Phys. Rev. Lett. 79, 2450 (1997).
[CrossRef]

Kartashov, Y. V.

Kivshar, Y.

D. Buccoliero, A. S. Desyatnikov, W. Krolikowski, and Y. Kivshar, J. Opt. A 11, 094014 (2009).
[CrossRef]

A. Minovich, D. Neshev, A. Dreischuh, W. Krolikowski, and Y. Kivshar, Opt. Lett. 32, 1599 (2007).
[CrossRef] [PubMed]

A. I. Yakimenko, Y. A. Zaliznyak, and Y. Kivshar, Phys. Rev. E 71, 065603 (2005).
[CrossRef]

Kivshar, Y. S.

A. S. Desyatnikov, Y. S. Kivshar, and L. Torner, Prog. Opt. 47, 291 (2005).
[CrossRef]

Krolikowski, W.

D. Buccoliero, A. S. Desyatnikov, W. Krolikowski, and Y. Kivshar, J. Opt. A 11, 094014 (2009).
[CrossRef]

A. Minovich, D. Neshev, A. Dreischuh, W. Krolikowski, and Y. Kivshar, Opt. Lett. 32, 1599 (2007).
[CrossRef] [PubMed]

S. Skupin, M. Saffman, and W. Krolikowski, Phys. Rev. Lett. 98, 263902 (2007).
[CrossRef] [PubMed]

S. Skupin, O. Bang, D. Edmundson, and W. Krolikowski, Phys. Rev. E 73, 066603 (2006).
[CrossRef]

D. Briedis, D. Petersen, D. Edmundson, W. Krolikowski, and O. Bang, Opt. Express 13, 435 (2005).
[CrossRef] [PubMed]

Kruglov, V. I.

V. I. Kruglov, Yu. A. Logvin, and V. M. Volkov, J. Mod. Opt. 39, 2277 (1992).
[CrossRef]

V. I. Kruglov and R. A. Vlasov, Phys. Lett. A 111, 401 (1985).
[CrossRef]

Lashkin, V. M.

A. I. Yakimenko, V. M. Lashkin, and O. O. Prikhodko, Phys. Rev. E 73, 066605 (2006).
[CrossRef]

Logvin, Yu. A.

V. I. Kruglov, Yu. A. Logvin, and V. M. Volkov, J. Mod. Opt. 39, 2277 (1992).
[CrossRef]

Mamaev, A. V.

A. V. Mamaev, M. Saffman, and A. A. Zozulya, Phys. Rev. Lett. 78, 2108 (1997).
[CrossRef]

Manela, O.

B. Alfassi, C. Rotschild, O. Manela, M. Segev, and D. N. Christodoulides, Opt. Lett. 32, 154 (2007).
[CrossRef]

B. Alfassi, C. Rotschild, O. Manela, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 98, 213901 (2007).
[CrossRef] [PubMed]

C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, Phys. Rev. Lett. 95, 213904 (2005).
[CrossRef] [PubMed]

Minovich, A.

Neshev, D.

Peccianti, M.

Petersen, D.

Petrov, D. V.

L. Torner and D. V. Petrov, Electron. Lett. 33, 608 (1997).
[CrossRef]

Prikhodko, O. O.

A. I. Yakimenko, V. M. Lashkin, and O. O. Prikhodko, Phys. Rev. E 73, 066605 (2006).
[CrossRef]

Rotschild, C.

B. Alfassi, C. Rotschild, O. Manela, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 98, 213901 (2007).
[CrossRef] [PubMed]

B. Alfassi, C. Rotschild, O. Manela, M. Segev, and D. N. Christodoulides, Opt. Lett. 32, 154 (2007).
[CrossRef]

C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, Phys. Rev. Lett. 95, 213904 (2005).
[CrossRef] [PubMed]

Rubenchik, A. M.

A. M. Rubenchik and S. K. Turitsyn, Laser Part. Beams 5, 3 (1987).
[CrossRef]

Saffman, M.

S. Skupin, M. Saffman, and W. Krolikowski, Phys. Rev. Lett. 98, 263902 (2007).
[CrossRef] [PubMed]

A. V. Mamaev, M. Saffman, and A. A. Zozulya, Phys. Rev. Lett. 78, 2108 (1997).
[CrossRef]

Segev, M.

B. Alfassi, C. Rotschild, O. Manela, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 98, 213901 (2007).
[CrossRef] [PubMed]

B. Alfassi, C. Rotschild, O. Manela, M. Segev, and D. N. Christodoulides, Opt. Lett. 32, 154 (2007).
[CrossRef]

C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, Phys. Rev. Lett. 95, 213904 (2005).
[CrossRef] [PubMed]

Skryabin, D. V.

W. J. Firth and D. V. Skryabin, Phys. Rev. Lett. 79, 2450 (1997).
[CrossRef]

Skupin, S.

S. Skupin, M. Saffman, and W. Krolikowski, Phys. Rev. Lett. 98, 263902 (2007).
[CrossRef] [PubMed]

S. Skupin, O. Bang, D. Edmundson, and W. Krolikowski, Phys. Rev. E 73, 066603 (2006).
[CrossRef]

Torner, L.

F. Ye, Y. V. Kartashov, and L. Torner, Phys. Rev. A 77, 043821 (2008).
[CrossRef]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Opt. Express 15, 9378 (2007).
[CrossRef] [PubMed]

Y. V. Kartashov, F. Ye, V. A. Vysloukh, and L. Torner, Opt. Lett. 32, 2260 (2007).
[CrossRef] [PubMed]

A. S. Desyatnikov, Y. S. Kivshar, and L. Torner, Prog. Opt. 47, 291 (2005).
[CrossRef]

L. Torner and D. V. Petrov, Electron. Lett. 33, 608 (1997).
[CrossRef]

Turitsyn, S. K.

A. M. Rubenchik and S. K. Turitsyn, Laser Part. Beams 5, 3 (1987).
[CrossRef]

S. K. Turitsyn, Theor. Math. Phys. 64, 226 (1985).
[CrossRef]

Vlasov, R. A.

V. I. Kruglov and R. A. Vlasov, Phys. Lett. A 111, 401 (1985).
[CrossRef]

Volkov, V. M.

V. I. Kruglov, Yu. A. Logvin, and V. M. Volkov, J. Mod. Opt. 39, 2277 (1992).
[CrossRef]

Vysloukh, V. A.

Yakimenko, A. I.

A. I. Yakimenko, V. M. Lashkin, and O. O. Prikhodko, Phys. Rev. E 73, 066605 (2006).
[CrossRef]

A. I. Yakimenko, Y. A. Zaliznyak, and Y. Kivshar, Phys. Rev. E 71, 065603 (2005).
[CrossRef]

Ye, F.

Zaliznyak, Y. A.

A. I. Yakimenko, Y. A. Zaliznyak, and Y. Kivshar, Phys. Rev. E 71, 065603 (2005).
[CrossRef]

Zozulya, A. A.

A. V. Mamaev, M. Saffman, and A. A. Zozulya, Phys. Rev. Lett. 78, 2108 (1997).
[CrossRef]

Electron. Lett. (1)

L. Torner and D. V. Petrov, Electron. Lett. 33, 608 (1997).
[CrossRef]

J. Mod. Opt. (1)

V. I. Kruglov, Yu. A. Logvin, and V. M. Volkov, J. Mod. Opt. 39, 2277 (1992).
[CrossRef]

J. Opt. A (1)

D. Buccoliero, A. S. Desyatnikov, W. Krolikowski, and Y. Kivshar, J. Opt. A 11, 094014 (2009).
[CrossRef]

Laser Part. Beams (1)

A. M. Rubenchik and S. K. Turitsyn, Laser Part. Beams 5, 3 (1987).
[CrossRef]

Opt. Express (2)

Opt. Lett. (4)

Phys. Lett. A (1)

V. I. Kruglov and R. A. Vlasov, Phys. Lett. A 111, 401 (1985).
[CrossRef]

Phys. Rev. A (1)

F. Ye, Y. V. Kartashov, and L. Torner, Phys. Rev. A 77, 043821 (2008).
[CrossRef]

Phys. Rev. E (3)

A. I. Yakimenko, Y. A. Zaliznyak, and Y. Kivshar, Phys. Rev. E 71, 065603 (2005).
[CrossRef]

S. Skupin, O. Bang, D. Edmundson, and W. Krolikowski, Phys. Rev. E 73, 066603 (2006).
[CrossRef]

A. I. Yakimenko, V. M. Lashkin, and O. O. Prikhodko, Phys. Rev. E 73, 066605 (2006).
[CrossRef]

Phys. Rev. Lett. (5)

S. Skupin, M. Saffman, and W. Krolikowski, Phys. Rev. Lett. 98, 263902 (2007).
[CrossRef] [PubMed]

C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, Phys. Rev. Lett. 95, 213904 (2005).
[CrossRef] [PubMed]

B. Alfassi, C. Rotschild, O. Manela, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 98, 213901 (2007).
[CrossRef] [PubMed]

W. J. Firth and D. V. Skryabin, Phys. Rev. Lett. 79, 2450 (1997).
[CrossRef]

A. V. Mamaev, M. Saffman, and A. A. Zozulya, Phys. Rev. Lett. 78, 2108 (1997).
[CrossRef]

Prog. Opt. (1)

A. S. Desyatnikov, Y. S. Kivshar, and L. Torner, Prog. Opt. 47, 291 (2005).
[CrossRef]

Theor. Math. Phys. (1)

S. K. Turitsyn, Theor. Math. Phys. 64, 226 (1985).
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

Field modulus (top) and phase (bottom) distributions for vortex solitons in thermal medium with d η = 20 , d ζ = 40 for (a) b = 2 , m = 1 ; (b) b = 2 , m = 2 ; (c) b = 4.64 , m = 3 ; (d) b = 5 , m = 4 ; (e) b = 5 , m = 5 ; and (f) b = 6 , m = 6 .

Fig. 2
Fig. 2

(a) U versus b for vortex with m = 2 . (b) W η , ζ versus d η for vortex with m = 2 , b = 2 . In all cases d ζ = 40 . Circles correspond to soliton in Fig. 1b.

Fig. 3
Fig. 3

Distance between singularities for m = 2 vortex (a) versus d η at b = 2 , and (b) versus b at d η = 20 . In all cases d ζ = 40 .

Fig. 4
Fig. 4

Propagation dynamics of vortex solitons perturbed by input noise with (a),(b) m = 1 , b = 2 ; (c),(d) m = 2 , b = 2 ; (e),(f) m = 3 , b = 6 in a sample with d η = 20 , d ζ = 40 . (g)–(i) Propagation of vortex with m = 2 , b = 2 that would be exact soliton in a sample with d η = 20 , d ζ = 40 in a sample with slightly changed dimensions d η = 21 , d ζ = 39 .

Equations (2)

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

i q ξ = 1 2 ( 2 q η 2 + 2 q ζ 2 ) q n ,
2 n η 2 + 2 n ζ 2 = | q | 2 .

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