## Observation of multi-component spatial vector solitons of four-wave mixing |

Optics Express, Vol. 20, Issue 13, pp. 14168-14182 (2012)

http://dx.doi.org/10.1364/OE.20.014168

Acrobat PDF (1401 KB)

### Abstract

We report the observation of multi-component dipole and vortex vector solitons composed of eight coexisting four-wave mixing (FWM) signals in two-level atomic system. The formation and stability of the multi-component dipole and vortex vector solitons are observed via changing the experiment parameters, including the frequency detuning, powers, and spatial configuration of the involved beams and the temperature of the medium. The transformation between modulated vortex solitons and rotating dipole solitons is observed at different frequency detunings. The interaction forces between different components of vector solitons are also investigated.

© 2012 OSA

## 1. Introduction

2. G. I. Stegeman and M. Segev, “Optical spatial solitons and their interactions: universality and diversity,” Science **286**(5444), 1518–1523 (1999). [CrossRef] [PubMed]

3. 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**(15), 2698–2701 (1996). [CrossRef] [PubMed]

4. A. S. Desyatnikov, Y. S. Kivshar, K. Motzek, F. Kaiser, C. Weilnau, and C. Denz, “Multicomponent dipole-mode spatial solitons,” Opt. Lett. **27**(8), 634–636 (2002). [CrossRef] [PubMed]

5. K. Motzek, F. Kaiser, C. Weilnau, C. Denz, G. McCarthy, W. Krolikowski, A. Desyatnikov, and Y. S. Kivshar, “Multi-component vector solitons in photorefractive crystals,” Opt. Commun. **209**(4-6), 501–506 (2002). [CrossRef]

6. L. J. Ge, Q. Wang, M. Shen, and J. L. Shi, “Dipole solitons in nonlocal nonlinear media with anisotropy,” Opt. Commun. **284**(9), 2351–2356 (2011). [CrossRef]

10. J. K. Yang, I. Makasyuk, A. Bezryadina, and Z. G. Chen, “Dipole solitons in optically induced two-dimensional photonic lattices,” Opt. Lett. **29**(14), 1662–1664 (2004). [CrossRef] [PubMed]

11. N. K. Efremidis, J. Hudock, D. N. Christodoulides, J. W. Fleischer, O. Cohen, and M. Segev, “Two-dimensional optical lattice solitons,” Phys. Rev. Lett. **91**(21), 213906 (2003). [CrossRef] [PubMed]

12. Y. P. Zhang, Z. G. Wang, Z. Q. Nie, C. B. Li, H. X. Chen, K. Q. Lu, and M. Xiao, “Four-wave mixing dipole soliton in laser-induced atomic gratings,” Phys. Rev. Lett. **106**(9), 093904 (2011). [CrossRef] [PubMed]

## 2. Basic theory and experimental scheme

^{−1}linewidth) with the frequency detuning

*x-o-z*plane. Other three laser beams

*y-o-z*plane. In this case, there will be eight FWM signals coexisting in the same atomic systems:

12. Y. P. Zhang, Z. G. Wang, Z. Q. Nie, C. B. Li, H. X. Chen, K. Q. Lu, and M. Xiao, “Four-wave mixing dipole soliton in laser-induced atomic gratings,” Phys. Rev. Lett. **106**(9), 093904 (2011). [CrossRef] [PubMed]

*i*= 1 for EIG1, and 2 for EIG2). The periodically modulated total linear and nonlinear refractive index is given by the expression

29. Y. Liu, M. Durand, S. Chen, A. Houard, B. Prade, B. Forestier, and A. Mysyrowicz, “Energy exchange between femtosecond laser filaments in air,” Phys. Rev. Lett. **105**(5), 055003 (2010). [CrossRef] [PubMed]

*N*-component vector soliton can be constructed from simple soliton components. For interactions between soliton components in medium with saturation nonlinearities, the critical angle

31. A. W. Snyder and A. P. Sheppard, “Collisions, steering, and guidance with spatial solitons,” Opt. Lett. **18**(7), 482–484 (1993). [CrossRef] [PubMed]

*p*= 1,2,5,6). Similarly, the superposition of four vertically-aligned dipole components

*q*= 3,4,7,8). Two nodeless probe beams

32. Y. P. Zhang, Z. Q. Nie, Y. Zhao, C. B. Li, R. M. Wang, J. H. Si, and M. Xiao, “Modulated vortex solitons of four-wave mixing,” Opt. Express **18**(11), 10963–10972 (2010). [CrossRef] [PubMed]

*m*and the number of intensity peaks

*M*. In our experiment, they are created jointly by the interference of multiple beams and the cross-phase modulation of the dressing and pump fields. The soliton solutions can be written as,

*m*of the vortex soliton is determined by

*n*being the number of laser beams which create the spiral phase plate and

## 3. Experimental observation of multi-component solitons

*x-z*plane, the grating EIG1 and EIG3 have horizontal orientation (Fig. 1(b)). Therefore, dipole components

*x*-axis. For the same reason, the dipole components

*y*-axis. The probe beam

*x*direction when

*y*direction when they are at the middle of the oven. Figure 3(c) shows the variation of the horizontal- and vertical-size of

*y*-axis. But in

*x*-direction [33

33. Y. P. Zhang, C. C. Zuo, H. B. Zheng, C. B. Li, Z. Q. Nie, J. P. Song, H. Chang, and M. Xiao, “Controlled spatial beamsplitter using four-wave mixing images,” Phys. Rev. A **80**(5), 055804 (2009). [CrossRef]

2. G. I. Stegeman and M. Segev, “Optical spatial solitons and their interactions: universality and diversity,” Science **286**(5444), 1518–1523 (1999). [CrossRef] [PubMed]

34. M. Marinescu and A. Dalgarno, “Dispersion forces and long-range electronic transition dipole moments of alkali-metal dimer excited states,” Phys. Rev. A **52**(1), 311–328 (1995). [CrossRef] [PubMed]

35. K. Singer, J. Stanojevic, M. Weidemüller, and R. Côté, “Long-range interactions between alkali Rydberg atoms pairs correlated to the *n*s-*n*s, *n*p-*n*p and *n*d-*n*d asymptotes,” J. Phys. At. Mol. Opt. Phys. **38**(2), S295–S307 (2005). [CrossRef]

*y*-axis. We denote this multi-component soliton structure as (0,1,1). The horizontal-splitting of the FWM signal in self-defocusing region disappears when

*x*-axis. We denote this multi-component soliton structure as (0,-1,-1).

**)**with circular cross section (as seen in Fig. 4(b)). The fundamental component

*z*for the involved beams. In the experiment, with the temperature increasing, the corresponding propagation distance increases about

*a*is the temperature increasing multiple. This propagation distance is 10.6 times longer than the diffraction length (

33. Y. P. Zhang, C. C. Zuo, H. B. Zheng, C. B. Li, Z. Q. Nie, J. P. Song, H. Chang, and M. Xiao, “Controlled spatial beamsplitter using four-wave mixing images,” Phys. Rev. A **80**(5), 055804 (2009). [CrossRef]

36. 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**(8), 1100–1102 (2006). [CrossRef]

37. Y. V. Izdebskaya, A. S. Desyatnikov, G. Assanto, and Y. S. Kivshar, “Dipole azimuthons and vortex charge flipping in nematic liquid crystals,” Opt. Express **19**(22), 21457–21466 (2011). [CrossRef] [PubMed]

20. J. R. Salgueiro and Y. S. Kivshar, “Single- and double-vortex vector solitons in self-focusing nonlinear media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **70**(5), 056613 (2004). [CrossRef] [PubMed]

38. A. H. Carlsson, J. N. Malmberg, D. Anderson, M. Lisak, E. A. Ostrovskaya, T. J. Alexander, and Y. S. Kivshar, “Linear and nonlinear waveguides induced by optical vortex solitons,” Opt. Lett. **25**(9), 660–662 (2000). [CrossRef] [PubMed]

39. A. P. Sheppard and M. Haelterman, “Polarization-domain solitary waves of circular symmetry in Kerr media,” Opt. Lett. **19**(12), 859–861 (1994). [CrossRef] [PubMed]

40. C. T. Law, X. Zhang, and G. A. Swartzlander Jr., “Waveguiding properties of optical vortex solitons,” Opt. Lett. **25**(1), 55–57 (2000). [CrossRef] [PubMed]

28. Y. V. Izdebskaya, J. Rebling, A. S. Desyatnikov, and Y. S. Kivshar, “Observation of vector solitons with hidden vorticity,” Opt. Lett. **37**(5), 767–769 (2012). [CrossRef] [PubMed]

## 4. Conclusion

## Acknowledgments

## References and links

1. | S. Trillo and W. Torruella, |

2. | G. I. Stegeman and M. Segev, “Optical spatial solitons and their interactions: universality and diversity,” Science |

3. | V. Tikhonenko, J. Christou, and B. Luther-Davies, “Three dimensional bright spatial soliton collision and fusion in a saturable nonlinear medium,” Phys. Rev. Lett. |

4. | A. S. Desyatnikov, Y. S. Kivshar, K. Motzek, F. Kaiser, C. Weilnau, and C. Denz, “Multicomponent dipole-mode spatial solitons,” Opt. Lett. |

5. | K. Motzek, F. Kaiser, C. Weilnau, C. Denz, G. McCarthy, W. Krolikowski, A. Desyatnikov, and Y. S. Kivshar, “Multi-component vector solitons in photorefractive crystals,” Opt. Commun. |

6. | L. J. Ge, Q. Wang, M. Shen, and J. L. Shi, “Dipole solitons in nonlocal nonlinear media with anisotropy,” Opt. Commun. |

7. | M. Shen, X. Chen, J. L. Shi, Q. Wang, and W. Krolikowski, “Incoherently coupled vector dipole soliton pairs in nonlocal media,” Opt. Commun. |

8. | F. W. Ye, Y. V. Kartashov, and L. Torner, “Stabilization of dipole solitons in nonlocal nonlinear media,” Phys. Rev. A |

9. | C. Rotschild, B. Alfassi, O. Cohen, and M. Segev, “Long-range interactions between optical solitons,” Nature |

10. | J. K. Yang, I. Makasyuk, A. Bezryadina, and Z. G. Chen, “Dipole solitons in optically induced two-dimensional photonic lattices,” Opt. Lett. |

11. | N. K. Efremidis, J. Hudock, D. N. Christodoulides, J. W. Fleischer, O. Cohen, and M. Segev, “Two-dimensional optical lattice solitons,” Phys. Rev. Lett. |

12. | Y. P. Zhang, Z. G. Wang, Z. Q. Nie, C. B. Li, H. X. Chen, K. Q. Lu, and M. Xiao, “Four-wave mixing dipole soliton in laser-induced atomic gratings,” Phys. Rev. Lett. |

13. | A. S. Desyatnikov, Y. S. Kivshar, and L. Torner, “Optical vortices and vortex solitons,” Prog. Opt. |

14. | A. S. Desyatnikov and Y. S. Kivshar, “Rotating optical soliton clusters,” Phys. Rev. Lett. |

15. | A. S. Desyatnikov, A. A. Sukhorukov, and Y. S. Kivshar, “Azimuthons: spatially modulated vortex solitons,” Phys. Rev. Lett. |

16. | A. Minovich, D. N. Neshev, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Observation of optical soliton azimuthons,” Opt. Express |

17. | W. J. Firth and D. V. Skryabin, “Optical solitons carrying Orbital angular momentum,” Phys. Rev. Lett. |

18. | J. K. Yang and D. E. Pelinovsky, “Stable vortex and dipole vector solitons in a saturable nonlinear medium,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

19. | A. S. Desyatnikov and Y. S. Kivshar, “Necklace-ring vector solitons,” Phys. Rev. Lett. |

20. | J. R. Salgueiro and Y. S. Kivshar, “Single- and double-vortex vector solitons in self-focusing nonlinear media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

21. | A. S. Desyatnikov, D. Neshev, E. A. Ostrovskaya, Y. S. Kivshar, W. Krolikowski, B. Luther-Davies, J. J. García-Ripoll, and V. M. Pérez-García, “Multipole spatial vector solitons,” Opt. Lett. |

22. | C. C. Jeng, M. F. Shih, K. Motzek, and Y. Kivshar, “Partially incoherent optical vortices in self-focusing nonlinear media,” Phys. Rev. Lett. |

23. | A. S. Desyatnikov, D. Mihalache, D. Mazilu, B. A. Malomed, and F. Lederer, “Stable counter-rotating vortex pairs in saturable media,” Phys. Lett. A |

24. | Z. H. Musslimani, M. Segev, and D. N. Christodoulides, “Multicomponent two-dimensional solitons carrying topological charges,” Opt. Lett. |

25. | C. Anastassiou, M. Soljačić, M. Segev, E. D. Eugenieva, D. N. Christodoulides, D. Kip, Z. H. Musslimani, and J. P. Torres, “Eliminating the transverse instabilities of Kerr solitons,” Phys. Rev. Lett. |

26. | Z. H. Musslimani, M. Soljacić, M. Segev, and D. N. Christodoulides, “Interactions between two-dimensional composite vector solitons carrying topological charges,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

27. | C. Pigier, R. Uzdin, T. Carmon, M. Segev, A. Nepomnyaschchy, and Z. H. Musslimani, “Collisions between (2+1)D rotating propeller solitons,” Opt. Lett. |

28. | Y. V. Izdebskaya, J. Rebling, A. S. Desyatnikov, and Y. S. Kivshar, “Observation of vector solitons with hidden vorticity,” Opt. Lett. |

29. | Y. Liu, M. Durand, S. Chen, A. Houard, B. Prade, B. Forestier, and A. Mysyrowicz, “Energy exchange between femtosecond laser filaments in air,” Phys. Rev. Lett. |

30. | H. Wang, D. Goorskey, and M. Xiao, “Enhanced Kerr nonlinearity via atomic coherence in a three-level atomic system,” Phys. Rev. Lett. |

31. | A. W. Snyder and A. P. Sheppard, “Collisions, steering, and guidance with spatial solitons,” Opt. Lett. |

32. | Y. P. Zhang, Z. Q. Nie, Y. Zhao, C. B. Li, R. M. Wang, J. H. Si, and M. Xiao, “Modulated vortex solitons of four-wave mixing,” Opt. Express |

33. | Y. P. Zhang, C. C. Zuo, H. B. Zheng, C. B. Li, Z. Q. Nie, J. P. Song, H. Chang, and M. Xiao, “Controlled spatial beamsplitter using four-wave mixing images,” Phys. Rev. A |

34. | M. Marinescu and A. Dalgarno, “Dispersion forces and long-range electronic transition dipole moments of alkali-metal dimer excited states,” Phys. Rev. A |

35. | K. Singer, J. Stanojevic, M. Weidemüller, and R. Côté, “Long-range interactions between alkali Rydberg atoms pairs correlated to the |

36. | 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. |

37. | Y. V. Izdebskaya, A. S. Desyatnikov, G. Assanto, and Y. S. Kivshar, “Dipole azimuthons and vortex charge flipping in nematic liquid crystals,” Opt. Express |

38. | A. H. Carlsson, J. N. Malmberg, D. Anderson, M. Lisak, E. A. Ostrovskaya, T. J. Alexander, and Y. S. Kivshar, “Linear and nonlinear waveguides induced by optical vortex solitons,” Opt. Lett. |

39. | A. P. Sheppard and M. Haelterman, “Polarization-domain solitary waves of circular symmetry in Kerr media,” Opt. Lett. |

40. | C. T. Law, X. Zhang, and G. A. Swartzlander Jr., “Waveguiding properties of optical vortex solitons,” Opt. Lett. |

**OCIS Codes**

(190.3270) Nonlinear optics : Kerr effect

(190.4180) Nonlinear optics : Multiphoton processes

(190.4380) Nonlinear optics : Nonlinear optics, four-wave mixing

(270.1670) Quantum optics : Coherent optical effects

(300.2570) Spectroscopy : Four-wave mixing

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: February 15, 2012

Revised Manuscript: May 4, 2012

Manuscript Accepted: May 21, 2012

Published: June 11, 2012

**Citation**

Ruimin Wang, Zhenkun Wu, Yiqi Zhang, Zhaoyang Zhang, Chenzhi Yuan, Huaibin Zheng, Yuanyuan Li, Jinhai Zhang, and Yanpeng Zhang, "Observation of multi-component spatial vector solitons of four-wave mixing," Opt. Express **20**, 14168-14182 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-13-14168

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### References

- S. Trillo and W. Torruella, Spatial Solitons (Springer-Verlag, Berlin, 2001).
- G. I. Stegeman and M. Segev, “Optical spatial solitons and their interactions: universality and diversity,” Science286(5444), 1518–1523 (1999). [CrossRef] [PubMed]
- 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(15), 2698–2701 (1996). [CrossRef] [PubMed]
- A. S. Desyatnikov, Y. S. Kivshar, K. Motzek, F. Kaiser, C. Weilnau, and C. Denz, “Multicomponent dipole-mode spatial solitons,” Opt. Lett.27(8), 634–636 (2002). [CrossRef] [PubMed]
- K. Motzek, F. Kaiser, C. Weilnau, C. Denz, G. McCarthy, W. Krolikowski, A. Desyatnikov, and Y. S. Kivshar, “Multi-component vector solitons in photorefractive crystals,” Opt. Commun.209(4-6), 501–506 (2002). [CrossRef]
- L. J. Ge, Q. Wang, M. Shen, and J. L. Shi, “Dipole solitons in nonlocal nonlinear media with anisotropy,” Opt. Commun.284(9), 2351–2356 (2011). [CrossRef]
- M. Shen, X. Chen, J. L. Shi, Q. Wang, and W. Krolikowski, “Incoherently coupled vector dipole soliton pairs in nonlocal media,” Opt. Commun.282(24), 4805–4809 (2009). [CrossRef]
- F. W. Ye, Y. V. Kartashov, and L. Torner, “Stabilization of dipole solitons in nonlocal nonlinear media,” Phys. Rev. A77(4), 043821 (2008). [CrossRef]
- C. Rotschild, B. Alfassi, O. Cohen, and M. Segev, “Long-range interactions between optical solitons,” Nature2, 769–774 (2006).
- J. K. Yang, I. Makasyuk, A. Bezryadina, and Z. G. Chen, “Dipole solitons in optically induced two-dimensional photonic lattices,” Opt. Lett.29(14), 1662–1664 (2004). [CrossRef] [PubMed]
- N. K. Efremidis, J. Hudock, D. N. Christodoulides, J. W. Fleischer, O. Cohen, and M. Segev, “Two-dimensional optical lattice solitons,” Phys. Rev. Lett.91(21), 213906 (2003). [CrossRef] [PubMed]
- Y. P. Zhang, Z. G. Wang, Z. Q. Nie, C. B. Li, H. X. Chen, K. Q. Lu, and M. Xiao, “Four-wave mixing dipole soliton in laser-induced atomic gratings,” Phys. Rev. Lett.106(9), 093904 (2011). [CrossRef] [PubMed]
- A. S. Desyatnikov, Y. S. Kivshar, and L. Torner, “Optical vortices and vortex solitons,” Prog. Opt.47, 291–391 (2005). [CrossRef]
- A. S. Desyatnikov and Y. S. Kivshar, “Rotating optical soliton clusters,” Phys. Rev. Lett.88(5), 053901 (2002). [CrossRef] [PubMed]
- A. S. Desyatnikov, A. A. Sukhorukov, and Y. S. Kivshar, “Azimuthons: spatially modulated vortex solitons,” Phys. Rev. Lett.95(20), 203904 (2005). [CrossRef] [PubMed]
- A. Minovich, D. N. Neshev, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Observation of optical soliton azimuthons,” Opt. Express17(26), 23610–23616 (2009). [CrossRef] [PubMed]
- W. J. Firth and D. V. Skryabin, “Optical solitons carrying Orbital angular momentum,” Phys. Rev. Lett.79(13), 2450–2453 (1997). [CrossRef]
- J. K. Yang and D. E. Pelinovsky, “Stable vortex and dipole vector solitons in a saturable nonlinear medium,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.67(1), 016608 (2003). [CrossRef] [PubMed]
- A. S. Desyatnikov and Y. S. Kivshar, “Necklace-ring vector solitons,” Phys. Rev. Lett.87(3), 033901 (2001). [CrossRef] [PubMed]
- J. R. Salgueiro and Y. S. Kivshar, “Single- and double-vortex vector solitons in self-focusing nonlinear media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.70(5), 056613 (2004). [CrossRef] [PubMed]
- A. S. Desyatnikov, D. Neshev, E. A. Ostrovskaya, Y. S. Kivshar, W. Krolikowski, B. Luther-Davies, J. J. García-Ripoll, and V. M. Pérez-García, “Multipole spatial vector solitons,” Opt. Lett.26(7), 435–437 (2001). [CrossRef] [PubMed]
- C. C. Jeng, M. F. Shih, K. Motzek, and Y. Kivshar, “Partially incoherent optical vortices in self-focusing nonlinear media,” Phys. Rev. Lett.92(4), 043904 (2004). [CrossRef] [PubMed]
- A. S. Desyatnikov, D. Mihalache, D. Mazilu, B. A. Malomed, and F. Lederer, “Stable counter-rotating vortex pairs in saturable media,” Phys. Lett. A364(3-4), 231–234 (2007). [CrossRef]
- Z. H. Musslimani, M. Segev, and D. N. Christodoulides, “Multicomponent two-dimensional solitons carrying topological charges,” Opt. Lett.25(1), 61–63 (2000). [CrossRef] [PubMed]
- C. Anastassiou, M. Soljačić, M. Segev, E. D. Eugenieva, D. N. Christodoulides, D. Kip, Z. H. Musslimani, and J. P. Torres, “Eliminating the transverse instabilities of Kerr solitons,” Phys. Rev. Lett.85(23), 4888–4891 (2000). [CrossRef] [PubMed]
- Z. H. Musslimani, M. Soljacić, M. Segev, and D. N. Christodoulides, “Interactions between two-dimensional composite vector solitons carrying topological charges,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.63(6), 066608 (2001). [CrossRef] [PubMed]
- C. Pigier, R. Uzdin, T. Carmon, M. Segev, A. Nepomnyaschchy, and Z. H. Musslimani, “Collisions between (2+1)D rotating propeller solitons,” Opt. Lett.26(20), 1577–1579 (2001). [CrossRef] [PubMed]
- Y. V. Izdebskaya, J. Rebling, A. S. Desyatnikov, and Y. S. Kivshar, “Observation of vector solitons with hidden vorticity,” Opt. Lett.37(5), 767–769 (2012). [CrossRef] [PubMed]
- Y. Liu, M. Durand, S. Chen, A. Houard, B. Prade, B. Forestier, and A. Mysyrowicz, “Energy exchange between femtosecond laser filaments in air,” Phys. Rev. Lett.105(5), 055003 (2010). [CrossRef] [PubMed]
- H. Wang, D. Goorskey, and M. Xiao, “Enhanced Kerr nonlinearity via atomic coherence in a three-level atomic system,” Phys. Rev. Lett.87(7), 073601 (2001). [CrossRef] [PubMed]
- A. W. Snyder and A. P. Sheppard, “Collisions, steering, and guidance with spatial solitons,” Opt. Lett.18(7), 482–484 (1993). [CrossRef] [PubMed]
- Y. P. Zhang, Z. Q. Nie, Y. Zhao, C. B. Li, R. M. Wang, J. H. Si, and M. Xiao, “Modulated vortex solitons of four-wave mixing,” Opt. Express18(11), 10963–10972 (2010). [CrossRef] [PubMed]
- Y. P. Zhang, C. C. Zuo, H. B. Zheng, C. B. Li, Z. Q. Nie, J. P. Song, H. Chang, and M. Xiao, “Controlled spatial beamsplitter using four-wave mixing images,” Phys. Rev. A80(5), 055804 (2009). [CrossRef]
- M. Marinescu and A. Dalgarno, “Dispersion forces and long-range electronic transition dipole moments of alkali-metal dimer excited states,” Phys. Rev. A52(1), 311–328 (1995). [CrossRef] [PubMed]
- K. Singer, J. Stanojevic, M. Weidemüller, and R. Côté, “Long-range interactions between alkali Rydberg atoms pairs correlated to the ns-ns, np-np and nd-nd asymptotes,” J. Phys. At. Mol. Opt. Phys.38(2), S295–S307 (2005). [CrossRef]
- 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(8), 1100–1102 (2006). [CrossRef]
- Y. V. Izdebskaya, A. S. Desyatnikov, G. Assanto, and Y. S. Kivshar, “Dipole azimuthons and vortex charge flipping in nematic liquid crystals,” Opt. Express19(22), 21457–21466 (2011). [CrossRef] [PubMed]
- A. H. Carlsson, J. N. Malmberg, D. Anderson, M. Lisak, E. A. Ostrovskaya, T. J. Alexander, and Y. S. Kivshar, “Linear and nonlinear waveguides induced by optical vortex solitons,” Opt. Lett.25(9), 660–662 (2000). [CrossRef] [PubMed]
- A. P. Sheppard and M. Haelterman, “Polarization-domain solitary waves of circular symmetry in Kerr media,” Opt. Lett.19(12), 859–861 (1994). [CrossRef] [PubMed]
- C. T. Law, X. Zhang, and G. A. Swartzlander., “Waveguiding properties of optical vortex solitons,” Opt. Lett.25(1), 55–57 (2000). [CrossRef] [PubMed]

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