## Modulated vortex solitons of four-wave mixing

Optics Express, Vol. 18, Issue 11, pp. 10963-10972 (2010)

http://dx.doi.org/10.1364/OE.18.010963

Acrobat PDF (1434 KB)

### Abstract

We experimentally demonstrate the vortex solitons of four-wave mixing (FWM) in multi-level atomic media created by the interference patterns with superposing three or more waves. The modulation effect of the vortex solitons is induced by the cross-Kerr nonlinear dispersion due to atomic coherence in the multi-level atomic system. These FWM vortex patterns are explained via the three-, four- and five-wave interference topologies.

© 2010 OSA

## 1. Introduction

2. G. A. Swartzlander Jr and C. T. Law, “Optical vortex solitons observed in Kerr nonlinear media,” Phys. Rev. Lett. **69**(17), 2503–2506 (1992). [CrossRef] [PubMed]

3. B. P. Anderson, P. C. Haljan, C. A. Regal, D. L. Feder, L. A. Collins, C. W. Clark, and E. A. Cornell, “Watching dark solitons decay into vortex rings in a Bose-Einstein condensate,” Phys. Rev. Lett. **86**(14), 2926–2929 (2001). [CrossRef] [PubMed]

5. M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall, C. E. Wieman, and E. A. Cornell, “Vortices in a Bose-Einstein Condensate,” Phys. Rev. Lett. **83**(13), 2498–2501 (1999). [CrossRef]

4. M. J. Holland and J. E. Williams, “Preparing topological states of a Bose-Einstein condensate,” Nature **401**(6753), 568–572 (1999). [CrossRef]

6. A. V. Gorbach, D. V. Skryabin, and C. N. Harvey, “Vortex solitons in an off-resonant Raman medium,” Phys. Rev. A **77**(6), 063810 (2008). [CrossRef]

7. A. V. Gorbach and D. V. Skryabin, “Cascaded generation of multiply charged optical vortices and spatiotemporal helical beams in a Raman medium,” Phys. Rev. Lett. **98**(24), 243601 (2007). [CrossRef] [PubMed]

8. A. S. Desyatnikov, A. A. Sukhorukov, and Y. S. Kivshar, “Azimuthons: spatially modulated vortex solitons,” Phys. Rev. Lett. **95**(20), 203904 (2005). [CrossRef] [PubMed]

*φ*. The necklace-ring solitons can merge into vortex and fundamental solitons in dissipative media [9

9. Y. J. He, H. Z. Wang, and B. A. Malomed, “Fusion of necklace-ring patterns into vortex and fundamental solitons in dissipative media,” Opt. Express **15**(26), 17502–17508 (2007). [CrossRef] [PubMed]

10. G. P. Agrawal, “Induced focusing of optical beams in self-defocusing nonlinear media,” Phys. Rev. Lett. **64**(21), 2487–2490 (1990). [CrossRef] [PubMed]

11. D. Bortman-Arbiv, A. D. Wilson-Gordon, and H. Friedmann, “Induced optical spatial solitons,” Phys. Rev. A **58**(5), R3403–R3406 (1998). [CrossRef]

12. M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. Malos, and N. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A **56**(5), 4064–4075 (1997). [CrossRef]

15. W. Jiang, Q. F. Chen, Y. S. Zhang, and G.-C. Guo, “Computation of topological charges of optical vortices via nondegenerate four-wave mixing,” Phys. Rev. A **74**(4), 043811 (2006). [CrossRef]

16. 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]

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

## 2. Theoretical model and experimental scheme

*I*is the beam intensity) are involved in the experimental schemes. In Fig. 1(b), energy levels

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

18. Y. P. Zhang, U. Khadka, B. Anderson, and M. Xiao, “Temporal and Spatial Interference between Four-Wave Mixing and Six-Wave Mixing Channels,” Phys. Rev. Lett. **102**(1), 013601 (2009). [CrossRef] [PubMed]

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

18. Y. P. Zhang, U. Khadka, B. Anderson, and M. Xiao, “Temporal and Spatial Interference between Four-Wave Mixing and Six-Wave Mixing Channels,” Phys. Rev. Lett. **102**(1), 013601 (2009). [CrossRef] [PubMed]

16. 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]

**80**(5), 055804 (2009). [CrossRef]

8. A. S. Desyatnikov, A. A. Sukhorukov, and Y. S. Kivshar, “Azimuthons: spatially modulated vortex solitons,” Phys. Rev. Lett. **95**(20), 203904 (2005). [CrossRef] [PubMed]

*w*. We can obtain the stationary transverse solution of the modulated vortex soliton as [8

8. A. S. Desyatnikov, A. A. Sukhorukov, and Y. S. Kivshar, “Azimuthons: spatially modulated vortex solitons,” Phys. Rev. Lett. **95**(20), 203904 (2005). [CrossRef] [PubMed]

9. Y. J. He, H. Z. Wang, and B. A. Malomed, “Fusion of necklace-ring patterns into vortex and fundamental solitons in dissipative media,” Opt. Express **15**(26), 17502–17508 (2007). [CrossRef] [PubMed]

13. J. Leach, M. R. Dennis, J. Courtial, and M. J. Padgett, “Laser beams: knotted threads of darkness,” Nature **432**(7014), 165 (2004). [CrossRef] [PubMed]

14. K. O’Holleran, M. J. Padgett, and M. R. Dennis, “Topology of optical vortex lines formed by the interference of three, four, and five plane waves,” Opt. Express **14**(7), 3039–3044 (2006). [CrossRef] [PubMed]

## 3. Modulated vortex solitons

*z*) rises, which leads to several splitting parts with weak absorption. As the temperature gets higher with an increased absorption, the beam intensity decreases.

*α*approaches to zero and

*z*increases with the temperature. According to the solution of Eq. (1)

*z*values.

13. J. Leach, M. R. Dennis, J. Courtial, and M. J. Padgett, “Laser beams: knotted threads of darkness,” Nature **432**(7014), 165 (2004). [CrossRef] [PubMed]

14. K. O’Holleran, M. J. Padgett, and M. R. Dennis, “Topology of optical vortex lines formed by the interference of three, four, and five plane waves,” Opt. Express **14**(7), 3039–3044 (2006). [CrossRef] [PubMed]

11. D. Bortman-Arbiv, A. D. Wilson-Gordon, and H. Friedmann, “Induced optical spatial solitons,” Phys. Rev. A **58**(5), R3403–R3406 (1998). [CrossRef]

## 4. Conclusion

## Acknowledgments

## References and links

1. | Y. S. Kivshar, and G. P. Agrawal, |

2. | G. A. Swartzlander Jr and C. T. Law, “Optical vortex solitons observed in Kerr nonlinear media,” Phys. Rev. Lett. |

3. | B. P. Anderson, P. C. Haljan, C. A. Regal, D. L. Feder, L. A. Collins, C. W. Clark, and E. A. Cornell, “Watching dark solitons decay into vortex rings in a Bose-Einstein condensate,” Phys. Rev. Lett. |

4. | M. J. Holland and J. E. Williams, “Preparing topological states of a Bose-Einstein condensate,” Nature |

5. | M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall, C. E. Wieman, and E. A. Cornell, “Vortices in a Bose-Einstein Condensate,” Phys. Rev. Lett. |

6. | A. V. Gorbach, D. V. Skryabin, and C. N. Harvey, “Vortex solitons in an off-resonant Raman medium,” Phys. Rev. A |

7. | A. V. Gorbach and D. V. Skryabin, “Cascaded generation of multiply charged optical vortices and spatiotemporal helical beams in a Raman medium,” Phys. Rev. Lett. |

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

9. | Y. J. He, H. Z. Wang, and B. A. Malomed, “Fusion of necklace-ring patterns into vortex and fundamental solitons in dissipative media,” Opt. Express |

10. | G. P. Agrawal, “Induced focusing of optical beams in self-defocusing nonlinear media,” Phys. Rev. Lett. |

11. | D. Bortman-Arbiv, A. D. Wilson-Gordon, and H. Friedmann, “Induced optical spatial solitons,” Phys. Rev. A |

12. | M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. Malos, and N. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A |

13. | J. Leach, M. R. Dennis, J. Courtial, and M. J. Padgett, “Laser beams: knotted threads of darkness,” Nature |

14. | K. O’Holleran, M. J. Padgett, and M. R. Dennis, “Topology of optical vortex lines formed by the interference of three, four, and five plane waves,” Opt. Express |

15. | W. Jiang, Q. F. Chen, Y. S. Zhang, and G.-C. Guo, “Computation of topological charges of optical vortices via nondegenerate four-wave mixing,” Phys. Rev. A |

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

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

18. | Y. P. Zhang, U. Khadka, B. Anderson, and M. Xiao, “Temporal and Spatial Interference between Four-Wave Mixing and Six-Wave Mixing Channels,” Phys. Rev. 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

(080.4865) Geometric optics : Optical vortices

(190.6135) Nonlinear optics : Spatial solitons

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: February 25, 2010

Revised Manuscript: April 9, 2010

Manuscript Accepted: April 13, 2010

Published: May 10, 2010

**Citation**

Yanpeng Zhang, Zhiqiang Nie, Yan Zhao, Changbiao Li, Ruimin Wang, Jinhai Si, and Min Xiao, "Modulated vortex solitons of four-wave mixing," Opt. Express **18**, 10963-10972 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-11-10963

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### References

- Y. S. Kivshar, and G. P. Agrawal, Optical solitons: From Fibers to Photonic Crystals (Academic, San Diego, 2003).
- G. A. Swartzlander and C. T. Law, “Optical vortex solitons observed in Kerr nonlinear media,” Phys. Rev. Lett. 69(17), 2503–2506 (1992). [CrossRef] [PubMed]
- B. P. Anderson, P. C. Haljan, C. A. Regal, D. L. Feder, L. A. Collins, C. W. Clark, and E. A. Cornell, “Watching dark solitons decay into vortex rings in a Bose-Einstein condensate,” Phys. Rev. Lett. 86(14), 2926–2929 (2001). [CrossRef] [PubMed]
- M. J. Holland and J. E. Williams, “Preparing topological states of a Bose-Einstein condensate,” Nature 401(6753), 568–572 (1999). [CrossRef]
- M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall, C. E. Wieman, and E. A. Cornell, “Vortices in a Bose-Einstein Condensate,” Phys. Rev. Lett. 83(13), 2498–2501 (1999). [CrossRef]
- A. V. Gorbach, D. V. Skryabin, and C. N. Harvey, “Vortex solitons in an off-resonant Raman medium,” Phys. Rev. A 77(6), 063810 (2008). [CrossRef]
- A. V. Gorbach and D. V. Skryabin, “Cascaded generation of multiply charged optical vortices and spatiotemporal helical beams in a Raman medium,” Phys. Rev. Lett. 98(24), 243601 (2007). [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]
- Y. J. He, H. Z. Wang, and B. A. Malomed, “Fusion of necklace-ring patterns into vortex and fundamental solitons in dissipative media,” Opt. Express 15(26), 17502–17508 (2007). [CrossRef] [PubMed]
- G. P. Agrawal, “Induced focusing of optical beams in self-defocusing nonlinear media,” Phys. Rev. Lett. 64(21), 2487–2490 (1990). [CrossRef] [PubMed]
- D. Bortman-Arbiv, A. D. Wilson-Gordon, and H. Friedmann, “Induced optical spatial solitons,” Phys. Rev. A 58(5), R3403–R3406 (1998). [CrossRef]
- M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. Malos, and N. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56(5), 4064–4075 (1997). [CrossRef]
- J. Leach, M. R. Dennis, J. Courtial, and M. J. Padgett, “Laser beams: knotted threads of darkness,” Nature 432(7014), 165 (2004). [CrossRef] [PubMed]
- K. O’Holleran, M. J. Padgett, and M. R. Dennis, “Topology of optical vortex lines formed by the interference of three, four, and five plane waves,” Opt. Express 14(7), 3039–3044 (2006). [CrossRef] [PubMed]
- W. Jiang, Q. F. Chen, Y. S. Zhang, and G.-C. Guo, “Computation of topological charges of optical vortices via nondegenerate four-wave mixing,” Phys. Rev. A 74(4), 043811 (2006). [CrossRef]
- 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]
- Y. P. Zhang, C. C. Zuo, H. B. Zheng, C. Li, Z. Nie, J. Song, H. Chang, and M. Xiao, “Controlled spatial beam splitter using four-wave-mixing images,” Phys. Rev. A 80(5), 055804 (2009). [CrossRef]
- Y. P. Zhang, U. Khadka, B. Anderson, and M. Xiao, “Temporal and Spatial Interference between Four-Wave Mixing and Six-Wave Mixing Channels,” Phys. Rev. Lett. 102(1), 013601 (2009). [CrossRef] [PubMed]

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