Abstract

We show that weakly guiding nonlinear waveguides support stable propagation of rotating spatial solitons (azimuthons). We investigate the role of waveguide symmetry on the soliton rotation. We find that azimuthons in circular waveguides always rotate rigidly during propagation and the analytically predicted rotation frequency is in excellent agreement with numerical simulations. On the other hand, azimuthons in square waveguides may experience spatial deformation during propagation. Moreover, we show that there is a critical value for the modulation depth of azimuthons above which solitons just wobble back and forth, and below which they rotate continuously. We explain these dynamics using the concept of energy difference between different orientations of the azimuthon.

© 2010 Optical Society of America

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  5. D. Buccoliero, A. S. Desyatnikov, W. Królikowski, and Y. S. Kivshar, "Spiraling multivortex solitons in nonlocal nonlinear media," Opt. Lett. 33, 198-200 (2008).
    [CrossRef] [PubMed]
  6. S. Lopez-Aguayo, A. S. Desyatnikov, and Y. S. Kivshar, "Azimuthons in nonlocal nonlinear media," Opt. Express 14, 7903-7908 (2006).
    [CrossRef] [PubMed]
  7. D. Buccoliero, A. S. Desyatnikov, W. Królikowski, and Y. S. Kivshar, "Laguerre and Hermite soliton clusters in nonlocal nonlinear media," Phys. Rev. Lett. 98, 053901 (2007).
    [CrossRef] [PubMed]
  8. A. Minovich, D. N. Neshev, A. S. Desyatnikov, W. Królikowski, and Y. S. Kivshar, "Observation of optical azimuthons," Opt. Express 17, 23610-23616 (2009).
    [CrossRef]
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  11. D. Suter, and T. Blasberg, "Stabilization of transverse solitary waves by a nonlocal response of the nonlinear medium," Phys. Rev. A 48, 4583-4587 (1993).
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  12. O. Bang, W. Królikowski, J. Wyller, and J. J. Rasmussen, "Collapse arrest and soliton stabilization in nonlocal nonlinear media," Phys. Rev. E 66, 046619 (2002).
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  13. W. Królikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, "Modulational instability, solitons and beam propagation in spatially nonlocal nonlinear media," J. Opt. B Quantum Semiclassical Opt. 6, S288-S294 (2004).
    [CrossRef]
  14. D. Briedis, D. Petersen, D. Edmundson, W. Królikowski, and O. Bang, "Ring vortex solitons in nonlocal nonlinear media," Opt. Express 13, 435-443 (2005).
    [CrossRef] [PubMed]
  15. S. Lopez-Aguayo, A. S. Desyatnikov, Y. S. Kivshar, S. Skupin, W. Królikowski, and O. Bang, "Stable rotating dipole solitons in nonlocal optical media," Opt. Lett. 31, 1100-1102 (2006).
    [CrossRef] [PubMed]
  16. S. Skupin, M. Grech, and W. Królikowski, "Rotating soliton solutions in nonlocal nonlinear media," Opt. Express 16, 9118-9131 (2008).
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  17. F. Maucher, D. Buccoliero, S. Skupin, M. Grech, A. S. Desyatnikov, and W. Krolikowski, "Tracking azimuthons in nonlocal nonlinear media," Opt. Quantum Electron. 41, 337-348 (2009).
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    [CrossRef]
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    [CrossRef] [PubMed]
  20. C. Conti, M. Peccianti, and G. Assanto, "Observation of optical spatial solitons in a highly nonlocal medium," Phys. Rev. Lett. 92, 113902 (2004).
    [CrossRef] [PubMed]
  21. R. Nath, P. Pedri, and L. Santos, "Soliton-soliton scattering in dipolar Bose-Einstein condensates," Phys. Rev. A 76, 013606 (2007).
    [CrossRef]
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    [CrossRef]
  23. A. G. Litvak, V. Mironov, G. Fraiman, and A. Yunakovskii, "Thermal self-effect of wave beams in plasma with a nonlocal nonlinearity," Sov. J. Plasma Phys. 1, 31-37 (1975).
  24. 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]
  25. S. Skupin, U. Peschel, L. Bergé, and F. Lederer, "Stability of weakly nonlinear localized states in attractive potentials," Phys. Rev. E 70, 016614 (2004).
    [CrossRef]
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    [CrossRef]
  30. F. Maucher, S. Skupin, M. Shen, and W. Królikowski, "Rotating three-dimensional solitons in Bose-Einstein condensates with gravity like attractive nonlocal interaction," Phys. Rev. A 81, 063617 (2010).
    [CrossRef]
  31. Y. S. Kivshar, and D. K. Campbell, "Peierls-Nabarro potential barrier for highly localized nonlinear modes," Phys. Rev. E 48, 3077-3081 (1993).
    [CrossRef]
  32. In the linear system, any dipole orientation is possible, and can be constructed from orthogonal basis (D1,D2) by superposition.

2010

F. Maucher, S. Skupin, M. Shen, and W. Królikowski, "Rotating three-dimensional solitons in Bose-Einstein condensates with gravity like attractive nonlocal interaction," Phys. Rev. A 81, 063617 (2010).
[CrossRef]

2009

A. Minovich, D. N. Neshev, A. S. Desyatnikov, W. Królikowski, and Y. S. Kivshar, "Observation of optical azimuthons," Opt. Express 17, 23610-23616 (2009).
[CrossRef]

F. Maucher, D. Buccoliero, S. Skupin, M. Grech, A. S. Desyatnikov, and W. Krolikowski, "Tracking azimuthons in nonlocal nonlinear media," Opt. Quantum Electron. 41, 337-348 (2009).
[CrossRef]

V. M. Lashkin, E. A. Ostrovskaya, A. S. Desyatnikov, and Y. S. Kivshar, "Vector azimuthons in two-component Bose-Einstein condensates," Phys. Rev. A 80, 013615 (2009).
[CrossRef]

2008

2007

D. Buccoliero, A. S. Desyatnikov, W. Królikowski, and Y. S. Kivshar, "Laguerre and Hermite soliton clusters in nonlocal nonlinear media," Phys. Rev. Lett. 98, 053901 (2007).
[CrossRef] [PubMed]

R. Nath, P. Pedri, and L. Santos, "Soliton-soliton scattering in dipolar Bose-Einstein condensates," Phys. Rev. A 76, 013606 (2007).
[CrossRef]

2006

2005

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]

D. Briedis, D. Petersen, D. Edmundson, W. Królikowski, and O. Bang, "Ring vortex solitons in nonlocal nonlinear media," Opt. Express 13, 435-443 (2005).
[CrossRef] [PubMed]

A. S. Desyatnikov, A. A. Sukhorukov, and Y. S. Kivshar, "Azimuthons: Spatially modulated vortex solitons," Phys. Rev. Lett. 95, 203904 (2005).
[CrossRef] [PubMed]

2004

W. Królikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, "Modulational instability, solitons and beam propagation in spatially nonlocal nonlinear media," J. Opt. B Quantum Semiclassical Opt. 6, S288-S294 (2004).
[CrossRef]

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

S. Skupin, U. Peschel, L. Bergé, and F. Lederer, "Stability of weakly nonlinear localized states in attractive potentials," Phys. Rev. E 70, 016614 (2004).
[CrossRef]

2003

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

2002

O. Bang, W. Królikowski, J. Wyller, and J. J. Rasmussen, "Collapse arrest and soliton stabilization in nonlocal nonlinear media," Phys. Rev. E 66, 046619 (2002).
[CrossRef]

D. V. Skryabin, J. M. McSloy, and W. J. Firth, "Stability of spiraling solitary waves in Hamiltonian systems," Phys. Rev. E 66, 055602 (2002).
[CrossRef]

M. Peccianti, K. A. Brzdkiewicz, and G. Assanto, "Nonlocal spatial soliton interactions in nematic liquid crystals," Opt. Lett. 27, 1460-1462 (2002).
[CrossRef]

1999

1998

D. V. Skryabin, and W. J. Firth, "Dynamics of self-trapped beams with phase dislocation in saturable Kerr and quadratic nonlinear media," Phys. Rev. E 58, 3916-3930 (1998).
[CrossRef]

1993

D. Suter, and T. Blasberg, "Stabilization of transverse solitary waves by a nonlocal response of the nonlinear medium," Phys. Rev. A 48, 4583-4587 (1993).
[CrossRef] [PubMed]

Y. S. Kivshar, and D. K. Campbell, "Peierls-Nabarro potential barrier for highly localized nonlinear modes," Phys. Rev. E 48, 3077-3081 (1993).
[CrossRef]

1992

V. I. Kruglov, Y. A. Logvin, and V. M. Volkov, "The theory of spiral laser beams in nonlinear media," J. Mod. Opt. 39, 2277-2291 (1992).
[CrossRef]

1989

P. Coullet, L. Gil, and F. Rocca, "Optical vortices," Opt. Commun. 73, 403-408 (1989).
[CrossRef]

1975

A. G. Litvak, V. Mironov, G. Fraiman, and A. Yunakovskii, "Thermal self-effect of wave beams in plasma with a nonlocal nonlinearity," Sov. J. Plasma Phys. 1, 31-37 (1975).

Assanto, G.

C. Conti, M. Peccianti, and G. Assanto, "Observation of optical spatial solitons in a 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]

M. Peccianti, K. A. Brzdkiewicz, and G. Assanto, "Nonlocal spatial soliton interactions in nematic liquid crystals," Opt. Lett. 27, 1460-1462 (2002).
[CrossRef]

Bang, O.

S. Lopez-Aguayo, A. S. Desyatnikov, Y. S. Kivshar, S. Skupin, W. Królikowski, and O. Bang, "Stable rotating dipole solitons in nonlocal optical media," Opt. Lett. 31, 1100-1102 (2006).
[CrossRef] [PubMed]

D. Briedis, D. Petersen, D. Edmundson, W. Królikowski, and O. Bang, "Ring vortex solitons in nonlocal nonlinear media," Opt. Express 13, 435-443 (2005).
[CrossRef] [PubMed]

W. Królikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, "Modulational instability, solitons and beam propagation in spatially nonlocal nonlinear media," J. Opt. B Quantum Semiclassical Opt. 6, S288-S294 (2004).
[CrossRef]

O. Bang, W. Królikowski, J. Wyller, and J. J. Rasmussen, "Collapse arrest and soliton stabilization in nonlocal nonlinear media," Phys. Rev. E 66, 046619 (2002).
[CrossRef]

Bergé, L.

S. Skupin, U. Peschel, L. Bergé, and F. Lederer, "Stability of weakly nonlinear localized states in attractive potentials," Phys. Rev. E 70, 016614 (2004).
[CrossRef]

Blasberg, T.

D. Suter, and T. Blasberg, "Stabilization of transverse solitary waves by a nonlocal response of the nonlinear medium," Phys. Rev. A 48, 4583-4587 (1993).
[CrossRef] [PubMed]

Briedis, D.

Brzdkiewicz, K. A.

Buccoliero, D.

F. Maucher, D. Buccoliero, S. Skupin, M. Grech, A. S. Desyatnikov, and W. Krolikowski, "Tracking azimuthons in nonlocal nonlinear media," Opt. Quantum Electron. 41, 337-348 (2009).
[CrossRef]

D. Buccoliero, A. S. Desyatnikov, W. Królikowski, and Y. S. Kivshar, "Spiraling multivortex solitons in nonlocal nonlinear media," Opt. Lett. 33, 198-200 (2008).
[CrossRef] [PubMed]

D. Buccoliero, A. S. Desyatnikov, W. Królikowski, and Y. S. Kivshar, "Laguerre and Hermite soliton clusters in nonlocal nonlinear media," Phys. Rev. Lett. 98, 053901 (2007).
[CrossRef] [PubMed]

Campbell, D. K.

Y. S. Kivshar, and D. K. Campbell, "Peierls-Nabarro potential barrier for highly localized nonlinear modes," Phys. Rev. E 48, 3077-3081 (1993).
[CrossRef]

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]

Cohen, O.

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.

C. Conti, M. Peccianti, and G. Assanto, "Observation of optical spatial solitons in a 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]

Coullet, P.

P. Coullet, L. Gil, and F. Rocca, "Optical vortices," Opt. Commun. 73, 403-408 (1989).
[CrossRef]

Desyatnikov, A. S.

V. M. Lashkin, E. A. Ostrovskaya, A. S. Desyatnikov, and Y. S. Kivshar, "Vector azimuthons in two-component Bose-Einstein condensates," Phys. Rev. A 80, 013615 (2009).
[CrossRef]

A. Minovich, D. N. Neshev, A. S. Desyatnikov, W. Królikowski, and Y. S. Kivshar, "Observation of optical azimuthons," Opt. Express 17, 23610-23616 (2009).
[CrossRef]

F. Maucher, D. Buccoliero, S. Skupin, M. Grech, A. S. Desyatnikov, and W. Krolikowski, "Tracking azimuthons in nonlocal nonlinear media," Opt. Quantum Electron. 41, 337-348 (2009).
[CrossRef]

D. Buccoliero, A. S. Desyatnikov, W. Królikowski, and Y. S. Kivshar, "Spiraling multivortex solitons in nonlocal nonlinear media," Opt. Lett. 33, 198-200 (2008).
[CrossRef] [PubMed]

D. Buccoliero, A. S. Desyatnikov, W. Królikowski, and Y. S. Kivshar, "Laguerre and Hermite soliton clusters in nonlocal nonlinear media," Phys. Rev. Lett. 98, 053901 (2007).
[CrossRef] [PubMed]

S. Lopez-Aguayo, A. S. Desyatnikov, and Y. S. Kivshar, "Azimuthons in nonlocal nonlinear media," Opt. Express 14, 7903-7908 (2006).
[CrossRef] [PubMed]

S. Lopez-Aguayo, A. S. Desyatnikov, Y. S. Kivshar, S. Skupin, W. Królikowski, and O. Bang, "Stable rotating dipole solitons in nonlocal optical media," Opt. Lett. 31, 1100-1102 (2006).
[CrossRef] [PubMed]

A. S. Desyatnikov, A. A. Sukhorukov, and Y. S. Kivshar, "Azimuthons: Spatially modulated vortex solitons," Phys. Rev. Lett. 95, 203904 (2005).
[CrossRef] [PubMed]

Edmundson, D.

D. Briedis, D. Petersen, D. Edmundson, W. Królikowski, and O. Bang, "Ring vortex solitons in nonlocal nonlinear media," Opt. Express 13, 435-443 (2005).
[CrossRef] [PubMed]

W. Królikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, "Modulational instability, solitons and beam propagation in spatially nonlocal nonlinear media," J. Opt. B Quantum Semiclassical Opt. 6, S288-S294 (2004).
[CrossRef]

Firth, W. J.

D. V. Skryabin, J. M. McSloy, and W. J. Firth, "Stability of spiraling solitary waves in Hamiltonian systems," Phys. Rev. E 66, 055602 (2002).
[CrossRef]

D. V. Skryabin, and W. J. Firth, "Dynamics of self-trapped beams with phase dislocation in saturable Kerr and quadratic nonlinear media," Phys. Rev. E 58, 3916-3930 (1998).
[CrossRef]

Fraiman, G.

A. G. Litvak, V. Mironov, G. Fraiman, and A. Yunakovskii, "Thermal self-effect of wave beams in plasma with a nonlocal nonlinearity," Sov. J. Plasma Phys. 1, 31-37 (1975).

Gil, L.

P. Coullet, L. Gil, and F. Rocca, "Optical vortices," Opt. Commun. 73, 403-408 (1989).
[CrossRef]

Grech, M.

F. Maucher, D. Buccoliero, S. Skupin, M. Grech, A. S. Desyatnikov, and W. Krolikowski, "Tracking azimuthons in nonlocal nonlinear media," Opt. Quantum Electron. 41, 337-348 (2009).
[CrossRef]

S. Skupin, M. Grech, and W. Królikowski, "Rotating soliton solutions in nonlocal nonlinear media," Opt. Express 16, 9118-9131 (2008).
[CrossRef] [PubMed]

Kamimura, T.

Kamisugi, N.

Kivshar, Y. S.

A. Minovich, D. N. Neshev, A. S. Desyatnikov, W. Królikowski, and Y. S. Kivshar, "Observation of optical azimuthons," Opt. Express 17, 23610-23616 (2009).
[CrossRef]

V. M. Lashkin, E. A. Ostrovskaya, A. S. Desyatnikov, and Y. S. Kivshar, "Vector azimuthons in two-component Bose-Einstein condensates," Phys. Rev. A 80, 013615 (2009).
[CrossRef]

D. Buccoliero, A. S. Desyatnikov, W. Królikowski, and Y. S. Kivshar, "Spiraling multivortex solitons in nonlocal nonlinear media," Opt. Lett. 33, 198-200 (2008).
[CrossRef] [PubMed]

D. Buccoliero, A. S. Desyatnikov, W. Królikowski, and Y. S. Kivshar, "Laguerre and Hermite soliton clusters in nonlocal nonlinear media," Phys. Rev. Lett. 98, 053901 (2007).
[CrossRef] [PubMed]

S. Lopez-Aguayo, A. S. Desyatnikov, and Y. S. Kivshar, "Azimuthons in nonlocal nonlinear media," Opt. Express 14, 7903-7908 (2006).
[CrossRef] [PubMed]

S. Lopez-Aguayo, A. S. Desyatnikov, Y. S. Kivshar, S. Skupin, W. Królikowski, and O. Bang, "Stable rotating dipole solitons in nonlocal optical media," Opt. Lett. 31, 1100-1102 (2006).
[CrossRef] [PubMed]

A. S. Desyatnikov, A. A. Sukhorukov, and Y. S. Kivshar, "Azimuthons: Spatially modulated vortex solitons," Phys. Rev. Lett. 95, 203904 (2005).
[CrossRef] [PubMed]

Y. S. Kivshar, and D. K. Campbell, "Peierls-Nabarro potential barrier for highly localized nonlinear modes," Phys. Rev. E 48, 3077-3081 (1993).
[CrossRef]

Krolikowski, W.

F. Maucher, D. Buccoliero, S. Skupin, M. Grech, A. S. Desyatnikov, and W. Krolikowski, "Tracking azimuthons in nonlocal nonlinear media," Opt. Quantum Electron. 41, 337-348 (2009).
[CrossRef]

Królikowski, W.

F. Maucher, S. Skupin, M. Shen, and W. Królikowski, "Rotating three-dimensional solitons in Bose-Einstein condensates with gravity like attractive nonlocal interaction," Phys. Rev. A 81, 063617 (2010).
[CrossRef]

A. Minovich, D. N. Neshev, A. S. Desyatnikov, W. Królikowski, and Y. S. Kivshar, "Observation of optical azimuthons," Opt. Express 17, 23610-23616 (2009).
[CrossRef]

D. Buccoliero, A. S. Desyatnikov, W. Królikowski, and Y. S. Kivshar, "Spiraling multivortex solitons in nonlocal nonlinear media," Opt. Lett. 33, 198-200 (2008).
[CrossRef] [PubMed]

S. Skupin, M. Grech, and W. Królikowski, "Rotating soliton solutions in nonlocal nonlinear media," Opt. Express 16, 9118-9131 (2008).
[CrossRef] [PubMed]

D. Buccoliero, A. S. Desyatnikov, W. Królikowski, and Y. S. Kivshar, "Laguerre and Hermite soliton clusters in nonlocal nonlinear media," Phys. Rev. Lett. 98, 053901 (2007).
[CrossRef] [PubMed]

S. Lopez-Aguayo, A. S. Desyatnikov, Y. S. Kivshar, S. Skupin, W. Królikowski, and O. Bang, "Stable rotating dipole solitons in nonlocal optical media," Opt. Lett. 31, 1100-1102 (2006).
[CrossRef] [PubMed]

D. Briedis, D. Petersen, D. Edmundson, W. Królikowski, and O. Bang, "Ring vortex solitons in nonlocal nonlinear media," Opt. Express 13, 435-443 (2005).
[CrossRef] [PubMed]

W. Królikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, "Modulational instability, solitons and beam propagation in spatially nonlocal nonlinear media," J. Opt. B Quantum Semiclassical Opt. 6, S288-S294 (2004).
[CrossRef]

O. Bang, W. Królikowski, J. Wyller, and J. J. Rasmussen, "Collapse arrest and soliton stabilization in nonlocal nonlinear media," Phys. Rev. E 66, 046619 (2002).
[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, 2277-2291 (1992).
[CrossRef]

Kuzuu, N.

Lashkin, V. M.

V. M. Lashkin, E. A. Ostrovskaya, A. S. Desyatnikov, and Y. S. Kivshar, "Vector azimuthons in two-component Bose-Einstein condensates," Phys. Rev. A 80, 013615 (2009).
[CrossRef]

V. M. Lashkin, "Two-dimensional multisolitons and azimuthons in Bose-Einstein condensates," Phys. Rev. A 77, 025602 (2008).
[CrossRef]

Lederer, F.

S. Skupin, U. Peschel, L. Bergé, and F. Lederer, "Stability of weakly nonlinear localized states in attractive potentials," Phys. Rev. E 70, 016614 (2004).
[CrossRef]

Litvak, A. G.

A. G. Litvak, V. Mironov, G. Fraiman, and A. Yunakovskii, "Thermal self-effect of wave beams in plasma with a nonlocal nonlinearity," Sov. J. Plasma Phys. 1, 31-37 (1975).

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, 2277-2291 (1992).
[CrossRef]

Lopez-Aguayo, S.

Manela, O.

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]

Maucher, F.

F. Maucher, S. Skupin, M. Shen, and W. Królikowski, "Rotating three-dimensional solitons in Bose-Einstein condensates with gravity like attractive nonlocal interaction," Phys. Rev. A 81, 063617 (2010).
[CrossRef]

F. Maucher, D. Buccoliero, S. Skupin, M. Grech, A. S. Desyatnikov, and W. Krolikowski, "Tracking azimuthons in nonlocal nonlinear media," Opt. Quantum Electron. 41, 337-348 (2009).
[CrossRef]

McSloy, J. M.

D. V. Skryabin, J. M. McSloy, and W. J. Firth, "Stability of spiraling solitary waves in Hamiltonian systems," Phys. Rev. E 66, 055602 (2002).
[CrossRef]

Minovich, A.

Mironov, V.

A. G. Litvak, V. Mironov, G. Fraiman, and A. Yunakovskii, "Thermal self-effect of wave beams in plasma with a nonlocal nonlinearity," Sov. J. Plasma Phys. 1, 31-37 (1975).

Nath, R.

R. Nath, P. Pedri, and L. Santos, "Soliton-soliton scattering in dipolar Bose-Einstein condensates," Phys. Rev. A 76, 013606 (2007).
[CrossRef]

Neshev, D.

W. Królikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, "Modulational instability, solitons and beam propagation in spatially nonlocal nonlinear media," J. Opt. B Quantum Semiclassical Opt. 6, S288-S294 (2004).
[CrossRef]

Neshev, D. N.

Nikolov, N. I.

W. Królikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, "Modulational instability, solitons and beam propagation in spatially nonlocal nonlinear media," J. Opt. B Quantum Semiclassical Opt. 6, S288-S294 (2004).
[CrossRef]

Ostrovskaya, E. A.

V. M. Lashkin, E. A. Ostrovskaya, A. S. Desyatnikov, and Y. S. Kivshar, "Vector azimuthons in two-component Bose-Einstein condensates," Phys. Rev. A 80, 013615 (2009).
[CrossRef]

Peccianti, M.

C. Conti, M. Peccianti, and G. Assanto, "Observation of optical spatial solitons in a 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]

M. Peccianti, K. A. Brzdkiewicz, and G. Assanto, "Nonlocal spatial soliton interactions in nematic liquid crystals," Opt. Lett. 27, 1460-1462 (2002).
[CrossRef]

Pedri, P.

R. Nath, P. Pedri, and L. Santos, "Soliton-soliton scattering in dipolar Bose-Einstein condensates," Phys. Rev. A 76, 013606 (2007).
[CrossRef]

Peschel, U.

S. Skupin, U. Peschel, L. Bergé, and F. Lederer, "Stability of weakly nonlinear localized states in attractive potentials," Phys. Rev. E 70, 016614 (2004).
[CrossRef]

Petersen, D.

Rasmussen, J. J.

W. Królikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, "Modulational instability, solitons and beam propagation in spatially nonlocal nonlinear media," J. Opt. B Quantum Semiclassical Opt. 6, S288-S294 (2004).
[CrossRef]

O. Bang, W. Królikowski, J. Wyller, and J. J. Rasmussen, "Collapse arrest and soliton stabilization in nonlocal nonlinear media," Phys. Rev. E 66, 046619 (2002).
[CrossRef]

Rocca, F.

P. Coullet, L. Gil, and F. Rocca, "Optical vortices," Opt. Commun. 73, 403-408 (1989).
[CrossRef]

Rotschild, C.

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]

Santos, L.

R. Nath, P. Pedri, and L. Santos, "Soliton-soliton scattering in dipolar Bose-Einstein condensates," Phys. Rev. A 76, 013606 (2007).
[CrossRef]

Segev, M.

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]

G. I. Stegeman, and M. Segev, "Optical spatial solitons and their interactions: Universality and diversity," Science 286, 1518-1523 (1999).
[CrossRef] [PubMed]

Shen, M.

F. Maucher, S. Skupin, M. Shen, and W. Królikowski, "Rotating three-dimensional solitons in Bose-Einstein condensates with gravity like attractive nonlocal interaction," Phys. Rev. A 81, 063617 (2010).
[CrossRef]

Skryabin, D. V.

D. V. Skryabin, J. M. McSloy, and W. J. Firth, "Stability of spiraling solitary waves in Hamiltonian systems," Phys. Rev. E 66, 055602 (2002).
[CrossRef]

D. V. Skryabin, and W. J. Firth, "Dynamics of self-trapped beams with phase dislocation in saturable Kerr and quadratic nonlinear media," Phys. Rev. E 58, 3916-3930 (1998).
[CrossRef]

Skupin, S.

F. Maucher, S. Skupin, M. Shen, and W. Królikowski, "Rotating three-dimensional solitons in Bose-Einstein condensates with gravity like attractive nonlocal interaction," Phys. Rev. A 81, 063617 (2010).
[CrossRef]

F. Maucher, D. Buccoliero, S. Skupin, M. Grech, A. S. Desyatnikov, and W. Krolikowski, "Tracking azimuthons in nonlocal nonlinear media," Opt. Quantum Electron. 41, 337-348 (2009).
[CrossRef]

S. Skupin, M. Grech, and W. Królikowski, "Rotating soliton solutions in nonlocal nonlinear media," Opt. Express 16, 9118-9131 (2008).
[CrossRef] [PubMed]

S. Lopez-Aguayo, A. S. Desyatnikov, Y. S. Kivshar, S. Skupin, W. Królikowski, and O. Bang, "Stable rotating dipole solitons in nonlocal optical media," Opt. Lett. 31, 1100-1102 (2006).
[CrossRef] [PubMed]

S. Skupin, U. Peschel, L. Bergé, and F. Lederer, "Stability of weakly nonlinear localized states in attractive potentials," Phys. Rev. E 70, 016614 (2004).
[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]

Sukhorukov, A. A.

A. S. Desyatnikov, A. A. Sukhorukov, and Y. S. Kivshar, "Azimuthons: Spatially modulated vortex solitons," Phys. Rev. Lett. 95, 203904 (2005).
[CrossRef] [PubMed]

Suter, D.

D. Suter, and T. Blasberg, "Stabilization of transverse solitary waves by a nonlocal response of the nonlinear medium," Phys. Rev. A 48, 4583-4587 (1993).
[CrossRef] [PubMed]

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, 2277-2291 (1992).
[CrossRef]

Wyller, J.

W. Królikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, "Modulational instability, solitons and beam propagation in spatially nonlocal nonlinear media," J. Opt. B Quantum Semiclassical Opt. 6, S288-S294 (2004).
[CrossRef]

O. Bang, W. Królikowski, J. Wyller, and J. J. Rasmussen, "Collapse arrest and soliton stabilization in nonlocal nonlinear media," Phys. Rev. E 66, 046619 (2002).
[CrossRef]

Yoshida, H.

Yoshida, K.

Yunakovskii, A.

A. G. Litvak, V. Mironov, G. Fraiman, and A. Yunakovskii, "Thermal self-effect of wave beams in plasma with a nonlocal nonlinearity," Sov. J. Plasma Phys. 1, 31-37 (1975).

Appl. Opt.

J. Mod. Opt.

V. I. Kruglov, Y. A. Logvin, and V. M. Volkov, "The theory of spiral laser beams in nonlinear media," J. Mod. Opt. 39, 2277-2291 (1992).
[CrossRef]

J. Opt. B Quantum Semiclassical Opt.

W. Królikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, "Modulational instability, solitons and beam propagation in spatially nonlocal nonlinear media," J. Opt. B Quantum Semiclassical Opt. 6, S288-S294 (2004).
[CrossRef]

Opt. Commun.

P. Coullet, L. Gil, and F. Rocca, "Optical vortices," Opt. Commun. 73, 403-408 (1989).
[CrossRef]

Opt. Express

Opt. Lett.

Opt. Quantum Electron.

F. Maucher, D. Buccoliero, S. Skupin, M. Grech, A. S. Desyatnikov, and W. Krolikowski, "Tracking azimuthons in nonlocal nonlinear media," Opt. Quantum Electron. 41, 337-348 (2009).
[CrossRef]

Phys. Rev. A

R. Nath, P. Pedri, and L. Santos, "Soliton-soliton scattering in dipolar Bose-Einstein condensates," Phys. Rev. A 76, 013606 (2007).
[CrossRef]

V. M. Lashkin, E. A. Ostrovskaya, A. S. Desyatnikov, and Y. S. Kivshar, "Vector azimuthons in two-component Bose-Einstein condensates," Phys. Rev. A 80, 013615 (2009).
[CrossRef]

V. M. Lashkin, "Two-dimensional multisolitons and azimuthons in Bose-Einstein condensates," Phys. Rev. A 77, 025602 (2008).
[CrossRef]

D. Suter, and T. Blasberg, "Stabilization of transverse solitary waves by a nonlocal response of the nonlinear medium," Phys. Rev. A 48, 4583-4587 (1993).
[CrossRef] [PubMed]

F. Maucher, S. Skupin, M. Shen, and W. Królikowski, "Rotating three-dimensional solitons in Bose-Einstein condensates with gravity like attractive nonlocal interaction," Phys. Rev. A 81, 063617 (2010).
[CrossRef]

Phys. Rev. E

Y. S. Kivshar, and D. K. Campbell, "Peierls-Nabarro potential barrier for highly localized nonlinear modes," Phys. Rev. E 48, 3077-3081 (1993).
[CrossRef]

D. V. Skryabin, J. M. McSloy, and W. J. Firth, "Stability of spiraling solitary waves in Hamiltonian systems," Phys. Rev. E 66, 055602 (2002).
[CrossRef]

O. Bang, W. Królikowski, J. Wyller, and J. J. Rasmussen, "Collapse arrest and soliton stabilization in nonlocal nonlinear media," Phys. Rev. E 66, 046619 (2002).
[CrossRef]

D. V. Skryabin, and W. J. Firth, "Dynamics of self-trapped beams with phase dislocation in saturable Kerr and quadratic nonlinear media," Phys. Rev. E 58, 3916-3930 (1998).
[CrossRef]

S. Skupin, U. Peschel, L. Bergé, and F. Lederer, "Stability of weakly nonlinear localized states in attractive potentials," Phys. Rev. E 70, 016614 (2004).
[CrossRef]

Phys. Rev. Lett.

A. S. Desyatnikov, A. A. Sukhorukov, and Y. S. Kivshar, "Azimuthons: Spatially modulated vortex solitons," Phys. Rev. Lett. 95, 203904 (2005).
[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]

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 a highly nonlocal medium," Phys. Rev. Lett. 92, 113902 (2004).
[CrossRef] [PubMed]

D. Buccoliero, A. S. Desyatnikov, W. Królikowski, and Y. S. Kivshar, "Laguerre and Hermite soliton clusters in nonlocal nonlinear media," Phys. Rev. Lett. 98, 053901 (2007).
[CrossRef] [PubMed]

Science

G. I. Stegeman, and M. Segev, "Optical spatial solitons and their interactions: Universality and diversity," Science 286, 1518-1523 (1999).
[CrossRef] [PubMed]

Sov. J. Plasma Phys.

A. G. Litvak, V. Mironov, G. Fraiman, and A. Yunakovskii, "Thermal self-effect of wave beams in plasma with a nonlocal nonlinearity," Sov. J. Plasma Phys. 1, 31-37 (1975).

Other

R. W. Boyd, Nonlinear Optics (Academic Press, Amsterdam, 2008), 3rd ed.

G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, Singapore, 2009), 4th ed.

In the linear system, any dipole orientation is possible, and can be constructed from orthogonal basis (D1,D2) by superposition.

Supplementary Material (2)

» Media 1: AVI (4060 KB)     
» Media 2: AVI (4067 KB)     

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Figures (8)

Fig. 1
Fig. 1

Azimuthon rotation frequency ω versus amplitude factor A (left panel) and amplitude ratio B (right panel). Blue solid lines show analytical predictions from Eq. (9). Red dots denote results obtained from numerical simulations of Eq. (2). The values of A and B are shown next to the lines.

Fig. 2
Fig. 2

The propagation of a dipole azimuthon with A = 0.4, B = 0.5 in the circular waveguide. The iso-surface plot on the top clearly displays the spiraling of the azimuthon during propagation. Figures in the row below depict the azimuthon’s transverse intensity distribution at input, and after rotating π/2, respectively. The white line indicates the waveguide boundaries.

Fig. 3
Fig. 3

The propagation of the azimuthon-like dipole, that rotates in the square waveguide. Top row: intensity (left) and phase (right) distribution corresponding to the input and soliton rotation by π/4, respectively. The white line indicates the waveguide boundaries. The iso-surface plot at the bottom displays the rotation and deformation during propagation. The initial amplitude and modulation parameters are A = 0.4, B = 0.5.

Fig. 4
Fig. 4

The propagation of a dipole azimuthon that wobbles in the square waveguide. The first row depicts the dipole at input and maximum displacement, respectively. The white line indicates the waveguide boundaries. The iso-surface plot below displays the twist and deformation during propagation. The chosen amplitude factor and ratio are A = 0.4, B = 0.2.

Fig. 5
Fig. 5

Average rotation frequency ω̄ versus amplitude factor A (left panel) and amplitude ratio B (right panel) in a square waveguide. Red dots are numerical results obtained from Eq. (2), blue curves are the fitting results to the numerical simulations. The green line represents the superposed dipoles twist during propagation. The values of A and B are shown next to the curves. The black square represents the analytical estimate for Bcr.

Fig. 6
Fig. 6

Left panel: Hamiltonian of A(D1 + iBD2) (red curve) as a function of B and Hamiltonian of Dp (blue line) (for the same power P). The intersection of the two curves defines Bcr (depicted by the black square). Right panel: The angular momenta of the rotating super-posed dipoles (corresponding to Fig. 3, blue curve), and the twisting ones (corresponding to Fig. 4, red curve). The other curves depict the cases of B slightly above (black) and below (green) Bcr. All curves are obtained for A = 0.4.

Fig. 7
Fig. 7

The propagation of a hexapole azimuthon with A = 0.4, B = 0.5 in the circular waveguide. Left panel shows the original hexapole azimuthon and the corresponding phase; right panel shows the iso-surface plot of the propagation.

Fig. 8
Fig. 8

The propagation of higher order azimuthons in the square waveguide. The first row shows a rotating hexapole with A = 0.4, B = 0.5 > Bcr( Media 2), and the second row shows a twisting hexapole with A = 0.4, B = 0.2 < Bcr ( Media 1). The superposed dodecapoles (left two panels), and superposed icosapoles (right two panels) as well as their deformations are shown in the third row; in both cases we choose A = 0.4 and B = 0.5.

Equations (21)

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i ζ + 1 2 k 0 ( 2 ξ 2 + 2 η 2 ) + k 0 n 2 n b | | 2 + k 0 n ( ξ , η ) n b n b = 0 ,
i z ψ + ( 2 x 2 + 2 y 2 ) ψ + σ | ψ | 2 ψ + V ψ = 0 ,
V ( x , y ) = { V 0 where { x 2 + y 2 1 for circular waveguide | x | 1 & | y | 1 for square waveguide 0 elsewhere .
ψ ( r , ϕ , z ) = U ( r , ϕ ω z ) exp ( i κ z ) ,
U ( r , ϕ ω z ) = AF ( r ) [ cos ( ϕ ω z ) + iB sin ( ϕ ω z ) ] ,
κ P + ω L z + I + N = 0 ,
κ L z + ω P + I + N = 0 .
P = r | U ( r ) | 2 d r d ϕ ,
L z = i r U ( r ) ϕ U * ( r ) d r d ϕ ,
I = r U * ( r ) Δ U ( r ) d r d ϕ ,
N = r [ σ | U ( r ) | 2 + V ] | U ( r ) | 2 d r d ϕ ,
P = r | U ( r ) ϕ | 2 d r d ϕ ,
I = i r U * ( r ) ϕ Δ U ( r ) d r d ϕ ,
N = i r [ σ | U ( r ) | 2 + V ] U * ( r ) ϕ U ( r ) d r d ϕ .
ω = P ( I + N ) L z ( I + N ) L z 2 P P .
F ( r ) = C × { J 1 ( V 0 κ r ) for 0 r 1 J 1 ( V 0 κ ) K 1 ( κ ) K 1 ( κ r ) for r > 1 ,
ω = π σ 2 r | F ( r ) | 4 d r A 2 B .
= 8 k 0 r 0 2 A 2 B r | F ( r ) | 4 d r .
U ( x , y , z = 0 ) = A ( D 1 + iB D 2 ) .
= ( | ψ | 2 1 2 | ψ | 4 V | ψ | 2 ) d x d y ,
D p = D 1 + D 2 | D 1 + D 2 | 2 d x d y .

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