Abstract

We analyze the salient features of vortex-ring solitons supported by cylindrically symmetric media with nonlocal thermal nonlinearity. We discover the existence of a maximum allowed topological charge for such vortex solitons to be stable on propagation: Only vortex-ring solitons with topological charge m≤2 are found to be stable. This remarkable result holds independently of the radius of the sample.

© 2007 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. W. Krolikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, "Modula-tional instability, solitons and beam propagation in spatially nonlocal nonlinear media," J. Opt. B: Quantum and Semiclass. Opt. 6, S288 (2004).
    [CrossRef]
  2. C. Conti, M. Peccianti, and G. Assanto, "Route to nonlocality and observation of accessible solitons," Phys. Rev. Lett. 91, 073901 (2003).
    [CrossRef] [PubMed]
  3. C. Conti, M. Peccianti, and G. Assanto, "Observation of optical spatial solitons in highly nonlocal medium," Phys. Rev. Lett. 92, 113902 (2004).
    [CrossRef] [PubMed]
  4. M. Peccianti, C. Conti, and G. Assanto, "Interplay between nonlocality and nonlinearity in nematic liquid crystals," Opt. Lett. 30, 415 (2005).
    [CrossRef] [PubMed]
  5. V. A. Mironov, A. M. Sergeev, and E. M. Sher, "Non-unidimensional coupled solitons in field non-linear equations," Dokl. Akad. Nauk SSSR 260, 325 (1981).
  6. S. Lopez-Aguayo, A. S. Desyatnikov, Y. S. Kivshar, S. Skupin, W. Krolikowski, and O. Bang, "Stable rotating dipole solitons in nonlocal optical media," Opt. Lett. 31, 1100 (2006).
    [CrossRef] [PubMed]
  7. Y. V. Kartashov, L. Torner, V. A. Vysloukh, and D. Mihalache, "Multipole vector solitons in nonlocal nonlinear media," Opt. Lett. 31, 1483 (2006).
    [CrossRef] [PubMed]
  8. C. Rotschild, M. Segev, Z. Xu, Y. V. Kartashov, L. Torner, and O. Cohen, "Two-dimensional multipole solitons in nonlocal nonlinear media," Opt. Lett. 31, 3312 (2006).
    [CrossRef] [PubMed]
  9. S. Lopez-Aguayo, A. S. Desyatnikov, and Y. S. Kivshar, "Azimuthons in nonlocal nonlinear media," Opt. Express 14, 7903 (2006).
    [CrossRef] [PubMed]
  10. V. I. Kruglov, Y. A. Logvin, and V. M. Volkov, "The theory of spiral laser beams in nonlinear media," J. Mod. Opt. 39, 2277 (1992).
    [CrossRef]
  11. D. Briedis, D. Petersen, D. Edmundson, W. Krolikowski, and O. Bang, "Ring vortex solitons in nonlocal non-linear media," Opt. Express 13, 435 (2005).
    [CrossRef] [PubMed]
  12. A. I. Yakimenko, Y. A. Zaliznyak, and Y. Kivshar, "Stable vortex solitons in nonlocal self-focusing nonlinear media," Phys. Rev. E 71, 065603(R).
  13. A. M. Deykoon and G. A. Swartzlander, "Pinched optical-vortex soliton," J. Opt. Soc. Am. B 18, 804 (2001).
    [CrossRef]
  14. S. Skupin, O. Bang, D. Edmundson, and W. Krolikowski, "Stability of two-dimensional spatial solitons in non-local nonlinear media," Phys. Rev. E 73, 066603 (2006).
    [CrossRef]
  15. A. I. Yakimenko, V. M. Lashkin, and O. O. Prikhodko, "Dynamics of two-dimensional coherent structures in nonlocal nonlinear media," Phys. Rev. E 73, 066605 (2006).
    [CrossRef]
  16. A. S. Desyatnikov, Y. S. Kivshar, L. Torner, "Optical vortices and vortex solitons," Prog. Opt. 47, 291 (2005).
    [CrossRef]
  17. 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).
    [CrossRef] [PubMed]
  18. C. Rotschild, B. Alfassi, O. Cohen, and M. Segev, "Long-range interactions between optical solitons," Nat. Phys. 2, 769 (2006).
    [CrossRef]
  19. B. Alfassi, C. Rotschild, O. Manela, M. Segev, and D. N. Christodoulides, "Boundary force effects exerted on solitons in highly nonlocal nonlinear media," Opt. Lett. 32, 154 (2007).
    [CrossRef]
  20. A. Minovich, D. Neshev, A. Dreischuh, W. Krolikowski, and Y. Kivshar, "Experimental reconstruction of non-local response of thermal nonlinear optical media," Opt. Lett., Doc. ID 78152 (in press).
  21. Z. Xu, Y. V. Kartashov, and L. Torner, "Upper threshold for stability of multipole-mode solitons in nonlocal nonlinear media," Opt. Lett. 30, 3171 (2005).
    [CrossRef] [PubMed]

2007 (1)

2006 (7)

2005 (5)

D. Briedis, D. Petersen, D. Edmundson, W. Krolikowski, and O. Bang, "Ring vortex solitons in nonlocal non-linear media," Opt. Express 13, 435 (2005).
[CrossRef] [PubMed]

M. Peccianti, C. Conti, and G. Assanto, "Interplay between nonlocality and nonlinearity in nematic liquid crystals," Opt. Lett. 30, 415 (2005).
[CrossRef] [PubMed]

Z. Xu, Y. V. Kartashov, and L. Torner, "Upper threshold for stability of multipole-mode solitons in nonlocal nonlinear media," Opt. Lett. 30, 3171 (2005).
[CrossRef] [PubMed]

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

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).
[CrossRef] [PubMed]

2004 (2)

W. Krolikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, "Modula-tional instability, solitons and beam propagation in spatially nonlocal nonlinear media," J. Opt. B: Quantum and Semiclass. Opt. 6, S288 (2004).
[CrossRef]

C. Conti, M. Peccianti, and G. Assanto, "Observation of optical spatial solitons in highly nonlocal medium," Phys. Rev. Lett. 92, 113902 (2004).
[CrossRef] [PubMed]

2003 (1)

C. Conti, M. Peccianti, and G. Assanto, "Route to nonlocality and observation of accessible solitons," Phys. Rev. Lett. 91, 073901 (2003).
[CrossRef] [PubMed]

2001 (1)

1992 (1)

V. I. Kruglov, Y. A. Logvin, and V. M. Volkov, "The theory of spiral laser beams in nonlinear media," J. Mod. Opt. 39, 2277 (1992).
[CrossRef]

1981 (1)

V. A. Mironov, A. M. Sergeev, and E. M. Sher, "Non-unidimensional coupled solitons in field non-linear equations," Dokl. Akad. Nauk SSSR 260, 325 (1981).

Alfassi, B.

Assanto, G.

M. Peccianti, C. Conti, and G. Assanto, "Interplay between nonlocality and nonlinearity in nematic liquid crystals," Opt. Lett. 30, 415 (2005).
[CrossRef] [PubMed]

C. Conti, M. Peccianti, and G. Assanto, "Observation of optical spatial solitons in highly nonlocal medium," Phys. Rev. Lett. 92, 113902 (2004).
[CrossRef] [PubMed]

C. Conti, M. Peccianti, and G. Assanto, "Route to nonlocality and observation of accessible solitons," Phys. Rev. Lett. 91, 073901 (2003).
[CrossRef] [PubMed]

Bang, O.

S. Lopez-Aguayo, A. S. Desyatnikov, Y. S. Kivshar, S. Skupin, W. Krolikowski, and O. Bang, "Stable rotating dipole solitons in nonlocal optical media," Opt. Lett. 31, 1100 (2006).
[CrossRef] [PubMed]

S. Skupin, O. Bang, D. Edmundson, and W. Krolikowski, "Stability of two-dimensional spatial solitons in non-local nonlinear media," Phys. Rev. E 73, 066603 (2006).
[CrossRef]

D. Briedis, D. Petersen, D. Edmundson, W. Krolikowski, and O. Bang, "Ring vortex solitons in nonlocal non-linear media," Opt. Express 13, 435 (2005).
[CrossRef] [PubMed]

W. Krolikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, "Modula-tional instability, solitons and beam propagation in spatially nonlocal nonlinear media," J. Opt. B: Quantum and Semiclass. Opt. 6, S288 (2004).
[CrossRef]

Briedis, D.

Carmon, T.

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).
[CrossRef] [PubMed]

Christodoulides, D. N.

Cohen, O.

C. Rotschild, M. Segev, Z. Xu, Y. V. Kartashov, L. Torner, and O. Cohen, "Two-dimensional multipole solitons in nonlocal nonlinear media," Opt. Lett. 31, 3312 (2006).
[CrossRef] [PubMed]

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

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).
[CrossRef] [PubMed]

Conti, C.

M. Peccianti, C. Conti, and G. Assanto, "Interplay between nonlocality and nonlinearity in nematic liquid crystals," Opt. Lett. 30, 415 (2005).
[CrossRef] [PubMed]

C. Conti, M. Peccianti, and G. Assanto, "Observation of optical spatial solitons in highly nonlocal medium," Phys. Rev. Lett. 92, 113902 (2004).
[CrossRef] [PubMed]

C. Conti, M. Peccianti, and G. Assanto, "Route to nonlocality and observation of accessible solitons," Phys. Rev. Lett. 91, 073901 (2003).
[CrossRef] [PubMed]

Desyatnikov, A. S.

Deykoon, A. M.

Edmundson, D.

S. Skupin, O. Bang, D. Edmundson, and W. Krolikowski, "Stability of two-dimensional spatial solitons in non-local nonlinear media," Phys. Rev. E 73, 066603 (2006).
[CrossRef]

D. Briedis, D. Petersen, D. Edmundson, W. Krolikowski, and O. Bang, "Ring vortex solitons in nonlocal non-linear media," Opt. Express 13, 435 (2005).
[CrossRef] [PubMed]

W. Krolikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, "Modula-tional instability, solitons and beam propagation in spatially nonlocal nonlinear media," J. Opt. B: Quantum and Semiclass. Opt. 6, S288 (2004).
[CrossRef]

Kartashov, Y. V.

Kivshar, Y.

A. I. Yakimenko, Y. A. Zaliznyak, and Y. Kivshar, "Stable vortex solitons in nonlocal self-focusing nonlinear media," Phys. Rev. E 71, 065603(R).

Kivshar, Y. S.

Krolikowski, W.

S. Lopez-Aguayo, A. S. Desyatnikov, Y. S. Kivshar, S. Skupin, W. Krolikowski, and O. Bang, "Stable rotating dipole solitons in nonlocal optical media," Opt. Lett. 31, 1100 (2006).
[CrossRef] [PubMed]

S. Skupin, O. Bang, D. Edmundson, and W. Krolikowski, "Stability of two-dimensional spatial solitons in non-local nonlinear media," Phys. Rev. E 73, 066603 (2006).
[CrossRef]

D. Briedis, D. Petersen, D. Edmundson, W. Krolikowski, and O. Bang, "Ring vortex solitons in nonlocal non-linear media," Opt. Express 13, 435 (2005).
[CrossRef] [PubMed]

W. Krolikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, "Modula-tional instability, solitons and beam propagation in spatially nonlocal nonlinear media," J. Opt. B: Quantum and Semiclass. Opt. 6, S288 (2004).
[CrossRef]

Kruglov, V. I.

V. I. Kruglov, Y. A. Logvin, and V. M. Volkov, "The theory of spiral laser beams in nonlinear media," J. Mod. Opt. 39, 2277 (1992).
[CrossRef]

Lashkin, V. M.

A. I. Yakimenko, V. M. Lashkin, and O. O. Prikhodko, "Dynamics of two-dimensional coherent structures in nonlocal nonlinear media," Phys. Rev. E 73, 066605 (2006).
[CrossRef]

Logvin, Y. A.

V. I. Kruglov, Y. A. Logvin, and V. M. Volkov, "The theory of spiral laser beams in nonlinear media," J. Mod. Opt. 39, 2277 (1992).
[CrossRef]

Lopez-Aguayo, S.

Manela, O.

B. Alfassi, C. Rotschild, O. Manela, M. Segev, and D. N. Christodoulides, "Boundary force effects exerted on solitons in highly nonlocal nonlinear media," Opt. Lett. 32, 154 (2007).
[CrossRef]

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).
[CrossRef] [PubMed]

Mihalache, D.

Mironov, V. A.

V. A. Mironov, A. M. Sergeev, and E. M. Sher, "Non-unidimensional coupled solitons in field non-linear equations," Dokl. Akad. Nauk SSSR 260, 325 (1981).

Neshev, D.

W. Krolikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, "Modula-tional instability, solitons and beam propagation in spatially nonlocal nonlinear media," J. Opt. B: Quantum and Semiclass. Opt. 6, S288 (2004).
[CrossRef]

Nikolov, N. I.

W. Krolikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, "Modula-tional instability, solitons and beam propagation in spatially nonlocal nonlinear media," J. Opt. B: Quantum and Semiclass. Opt. 6, S288 (2004).
[CrossRef]

Peccianti, M.

M. Peccianti, C. Conti, and G. Assanto, "Interplay between nonlocality and nonlinearity in nematic liquid crystals," Opt. Lett. 30, 415 (2005).
[CrossRef] [PubMed]

C. Conti, M. Peccianti, and G. Assanto, "Observation of optical spatial solitons in highly nonlocal medium," Phys. Rev. Lett. 92, 113902 (2004).
[CrossRef] [PubMed]

C. Conti, M. Peccianti, and G. Assanto, "Route to nonlocality and observation of accessible solitons," Phys. Rev. Lett. 91, 073901 (2003).
[CrossRef] [PubMed]

Petersen, D.

Prikhodko, O. O.

A. I. Yakimenko, V. M. Lashkin, and O. O. Prikhodko, "Dynamics of two-dimensional coherent structures in nonlocal nonlinear media," Phys. Rev. E 73, 066605 (2006).
[CrossRef]

Rasmussen, J. J.

W. Krolikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, "Modula-tional instability, solitons and beam propagation in spatially nonlocal nonlinear media," J. Opt. B: Quantum and Semiclass. Opt. 6, S288 (2004).
[CrossRef]

Rotschild, C.

B. Alfassi, C. Rotschild, O. Manela, M. Segev, and D. N. Christodoulides, "Boundary force effects exerted on solitons in highly nonlocal nonlinear media," Opt. Lett. 32, 154 (2007).
[CrossRef]

C. Rotschild, M. Segev, Z. Xu, Y. V. Kartashov, L. Torner, and O. Cohen, "Two-dimensional multipole solitons in nonlocal nonlinear media," Opt. Lett. 31, 3312 (2006).
[CrossRef] [PubMed]

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

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).
[CrossRef] [PubMed]

Segev, M.

B. Alfassi, C. Rotschild, O. Manela, M. Segev, and D. N. Christodoulides, "Boundary force effects exerted on solitons in highly nonlocal nonlinear media," Opt. Lett. 32, 154 (2007).
[CrossRef]

C. Rotschild, M. Segev, Z. Xu, Y. V. Kartashov, L. Torner, and O. Cohen, "Two-dimensional multipole solitons in nonlocal nonlinear media," Opt. Lett. 31, 3312 (2006).
[CrossRef] [PubMed]

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

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).
[CrossRef] [PubMed]

Sergeev, A. M.

V. A. Mironov, A. M. Sergeev, and E. M. Sher, "Non-unidimensional coupled solitons in field non-linear equations," Dokl. Akad. Nauk SSSR 260, 325 (1981).

Sher, E. M.

V. A. Mironov, A. M. Sergeev, and E. M. Sher, "Non-unidimensional coupled solitons in field non-linear equations," Dokl. Akad. Nauk SSSR 260, 325 (1981).

Skupin, S.

S. Lopez-Aguayo, A. S. Desyatnikov, Y. S. Kivshar, S. Skupin, W. Krolikowski, and O. Bang, "Stable rotating dipole solitons in nonlocal optical media," Opt. Lett. 31, 1100 (2006).
[CrossRef] [PubMed]

S. Skupin, O. Bang, D. Edmundson, and W. Krolikowski, "Stability of two-dimensional spatial solitons in non-local nonlinear media," Phys. Rev. E 73, 066603 (2006).
[CrossRef]

Swartzlander, G. A.

Torner, L.

Volkov, V. M.

V. I. Kruglov, Y. A. Logvin, and V. M. Volkov, "The theory of spiral laser beams in nonlinear media," J. Mod. Opt. 39, 2277 (1992).
[CrossRef]

Vysloukh, V. A.

Wyller, J.

W. Krolikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, "Modula-tional instability, solitons and beam propagation in spatially nonlocal nonlinear media," J. Opt. B: Quantum and Semiclass. Opt. 6, S288 (2004).
[CrossRef]

Xu, Z.

Yakimenko, A. I.

A. I. Yakimenko, V. M. Lashkin, and O. O. Prikhodko, "Dynamics of two-dimensional coherent structures in nonlocal nonlinear media," Phys. Rev. E 73, 066605 (2006).
[CrossRef]

A. I. Yakimenko, Y. A. Zaliznyak, and Y. Kivshar, "Stable vortex solitons in nonlocal self-focusing nonlinear media," Phys. Rev. E 71, 065603(R).

Zaliznyak, Y. A.

A. I. Yakimenko, Y. A. Zaliznyak, and Y. Kivshar, "Stable vortex solitons in nonlocal self-focusing nonlinear media," Phys. Rev. E 71, 065603(R).

Dokl. Akad. Nauk SSSR (1)

V. A. Mironov, A. M. Sergeev, and E. M. Sher, "Non-unidimensional coupled solitons in field non-linear equations," Dokl. Akad. Nauk SSSR 260, 325 (1981).

J. Mod. Opt. (1)

V. I. Kruglov, Y. A. Logvin, and V. M. Volkov, "The theory of spiral laser beams in nonlinear media," J. Mod. Opt. 39, 2277 (1992).
[CrossRef]

J. Opt. B: Quantum and Semiclass. Opt. (1)

W. Krolikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, "Modula-tional instability, solitons and beam propagation in spatially nonlocal nonlinear media," J. Opt. B: Quantum and Semiclass. Opt. 6, S288 (2004).
[CrossRef]

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

Nat. Phys. (1)

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

Opt. Express (2)

Opt. Lett. (6)

Phys. Rev. E (3)

A. I. Yakimenko, Y. A. Zaliznyak, and Y. Kivshar, "Stable vortex solitons in nonlocal self-focusing nonlinear media," Phys. Rev. E 71, 065603(R).

S. Skupin, O. Bang, D. Edmundson, and W. Krolikowski, "Stability of two-dimensional spatial solitons in non-local nonlinear media," Phys. Rev. E 73, 066603 (2006).
[CrossRef]

A. I. Yakimenko, V. M. Lashkin, and O. O. Prikhodko, "Dynamics of two-dimensional coherent structures in nonlocal nonlinear media," Phys. Rev. E 73, 066605 (2006).
[CrossRef]

Phys. Rev. Lett. (3)

C. Conti, M. Peccianti, and G. Assanto, "Route to nonlocality and observation of accessible solitons," Phys. Rev. Lett. 91, 073901 (2003).
[CrossRef] [PubMed]

C. Conti, M. Peccianti, and G. Assanto, "Observation of optical spatial solitons in highly nonlocal medium," Phys. Rev. Lett. 92, 113902 (2004).
[CrossRef] [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).
[CrossRef] [PubMed]

Prog. Opt. (1)

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

Other (1)

A. Minovich, D. Neshev, A. Dreischuh, W. Krolikowski, and Y. Kivshar, "Experimental reconstruction of non-local response of thermal nonlinear optical media," Opt. Lett., Doc. ID 78152 (in press).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1.
Fig. 1.

(a). Profiles of solitons with topological charges m=0, 1, and 2 at b=10. (b) Refractive index profile for solitons with m=0 and 2 at b=10. (c) Profiles of vortex solitons with m=3 and 5 at b=10.

Fig. 2.
Fig. 2.

Energy flow (a) and integral radius (b) versus propagation constant for solitons with different topological charges.

Fig. 3.
Fig. 3.

Real part of the perturbation growth rate versus the propagation constant for vortex solitons with m=3 (a) and m=4 (b) for different azimuthal indices k. The point labeled by the circle in (a) corresponds to the perturbation profile shown in (c).

Fig. 4.
Fig. 4.

Evolution of perturbed vortex solitons. Field modulus distributions are shown at different propagation distances for (a) m=1, (b) m=2, (c) and (d) m=3, (e) and (f) m=4.

Equations (4)

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

i q ξ = 1 2 ( 2 q r 2 + 1 r q r + 1 r 2 2 q ϕ 2 ) q n ,
2 n r 2 + 1 r n r + 1 r 2 2 n ϕ 2 = q 2 .
i δ u = 1 2 ( d 2 u d r 2 + 1 r du dr ( m + k ) 2 r 2 u ) w Δ n k un + bu ,
i δ v = 1 2 ( d 2 v d r 2 + 1 r dv dr ( m k ) 2 r 2 v ) + w Δ n k + vn bv ,

Metrics