## Waveguide properties of the asymmetric collision between two bright spatial solitons in Kerr media |

Optics Express, Vol. 20, Issue 24, pp. 27411-27418 (2012)

http://dx.doi.org/10.1364/OE.20.027411

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### Abstract

In this work, we numerically characterize the waveguide properties of the asymmetric collision between two bright spatial solitons in a nonlinear Kerr media. The results demonstrate that the energy carried by a probe beam guided by one soliton can be transferred after the collision to the waveguide created by the other soliton depending on the initial separation between the solitons, the angle of their collision, and in some cases, the particular soliton that initially guides the probe beam. The observed behavior is equivalent to that obtained for the symmetrical collision when there is an initial relative phase between the solitons.

© 2012 OSA

## 1. Introduction

1. A. W. Synder and D. J. Mitchell, “Accessible solitons,” Science **276**(5318), 1538–1541 (1997). [CrossRef]

1. A. W. Synder and D. J. Mitchell, “Accessible solitons,” Science **276**(5318), 1538–1541 (1997). [CrossRef]

4. R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. **13**(15), 479–482 (1964). [CrossRef]

5. B. Luther-Davies and Y. Xiaoping, “Waveguides and Y junctions formed in bulk media by using dark spatial solitons,” Opt. Lett. **17**(7), 496–498 (1992). [CrossRef] [PubMed]

6. G. E. Torres-Cisneros, J. J. Sánchez-Mondragon, and V. A. Vysloukh, “Asymmetric optical Y junctions and switching of weak beams using bright spatial-soliton collisions,” Opt. Lett. **18**(16), 1299–1301 (1993). [CrossRef] [PubMed]

7. B. L. Davies and X. Yang, “Steerable optical waveguides formed in self-defocusing media by using dark spatial solitons,” Opt. Lett. **17**, 496–498 (1992). [CrossRef] [PubMed]

9. P. D. Miller and N. N. Akhmediev, “Transfer matrices for multiport devices made from solitons,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics **53**(4), 4098–4106 (1996). [CrossRef] [PubMed]

10. Y. Kodama and A. Hasegawa, “Effects of initial overlap on the propagation of optical solitons at different wavelengths,” Opt. Lett. **16**(4), 208–210 (1991). [CrossRef] [PubMed]

11. M. Shalaby and A. Barthelemy, “Ultrafast photonic switching and splitting through cross-phase modulation with a spatial solitons couple,” Opt. Commun. **94**(5), 341–345 (1992). [CrossRef]

*et al.*[12

12. J. S. Aitchison, A. M. Weiner, Y. Silberberg, D. E. Leaird, M. K. Oliver, J. L. Jackel, and P. W. Smith, “Experimental observation of spatial soliton interactions,” Opt. Lett. **16**(1), 15–17 (1991). [CrossRef] [PubMed]

13. J. S. Aitchison, A. M. Weiner, Y. Silberberg, D. E. Leaird, M. K. Oliver, J. L. Jackel, and P. W. Smith, “Spatial optical solitons in planar glass waveguides,” J. Opt. Soc. Am. B **8**(6), 1290–1297 (1991). [CrossRef]

*et al.*[14

14. M. Shalaby, F. Reynaud, and A. Barthelemy, “Experimental observation of spatial soliton interactions with a *π*/2 relative phase difference,” Opt. Lett. **17**(11), 778–780 (1992). [CrossRef] [PubMed]

*π*/2 phase difference, one soliton gained energy at the expense of the other soliton. Furthermore, the energy exchange direction is switched when the phase is increased to 3

*π*/2. These effects can be viewed as the consequence of the well-known four-wave mixing term in nonlinear optics. These experiments demonstrated the basic properties of the coherent Kerr soliton interactions. In [15

15. P. Chamorro-Posada and G. S. McDonald, “Spatial Kerr soliton collisions at arbitrary angles,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **74**(3), 036609 (2006). [CrossRef] [PubMed]

16. K. Steiglitz and D. Rand, “Photon trapping and transfer with solitons,” Phys. Rev. A **79**(2), 021802 (2009). [CrossRef]

6. G. E. Torres-Cisneros, J. J. Sánchez-Mondragon, and V. A. Vysloukh, “Asymmetric optical Y junctions and switching of weak beams using bright spatial-soliton collisions,” Opt. Lett. **18**(16), 1299–1301 (1993). [CrossRef] [PubMed]

16. K. Steiglitz and D. Rand, “Photon trapping and transfer with solitons,” Phys. Rev. A **79**(2), 021802 (2009). [CrossRef]

18. K. Steiglitz, “Making beam splitters with dark soliton collisions,” Phys. Rev. A **82**(4), 043831 (2010). [CrossRef]

16. K. Steiglitz and D. Rand, “Photon trapping and transfer with solitons,” Phys. Rev. A **79**(2), 021802 (2009). [CrossRef]

18. K. Steiglitz, “Making beam splitters with dark soliton collisions,” Phys. Rev. A **82**(4), 043831 (2010). [CrossRef]

## 2. Theoretical model

6. G. E. Torres-Cisneros, J. J. Sánchez-Mondragon, and V. A. Vysloukh, “Asymmetric optical Y junctions and switching of weak beams using bright spatial-soliton collisions,” Opt. Lett. **18**(16), 1299–1301 (1993). [CrossRef] [PubMed]

*q*

_{1}) and a weak, probe, (

*q*

_{2}) beam in a Kerr media were described by the following coupled equations: where

*q*

_{j}is the amplitude of the optical field normalized to the maximum intensity

*I*,

_{mj}*L*

_{D}= n_{01}

*k*

_{01}

*x*

_{01}

^{2}/2 is the diffraction length with

*n*

_{01}the linear refractive index,

*k*

_{01}the wave number and

*x*

_{01}the initial beam width for beam

*q*

_{1},

*L*

_{NL}= (|

*n*

_{2}|

*k*

_{0}

*I*)

_{m}^{−1}is the nonlinear length with

*n*

_{2}is the nonlinear refractive index,

*r*=

_{n}*n*

_{01}/

*n*

_{02}with

*n*

_{02}the linear refractive index for beam

*q*

_{2}, and

*r*

_{k}= k_{01}/

*k*

_{02}with

*k*

_{02}the wave number for beam

*q*

_{2}. In this work, by simplicity we used

*r*=

_{n}*r*= 1, this means that the soliton and probe beam have the same wavelength and direction (experimentally, different polarizations can be used to differentiate both beams), the results considering other values of these parameters give similar behavior.

_{k}*Z*axis and other soliton (S2) is initially separated by a distance

*c*from first soliton and makes an angle

*θ*with respect to the

*Z*axis. The following equation was used as initial condition:where

*tanθ*=

*V/2.*Note that this initial condition is equivalent to the symmetric collision but with a relative phase difference between the solitons equal to

*Vc/2.*As a probe beam we considered a Gaussian beam, with unitary amplitude, as initial condition for Eq. (2), given bywhere

*w*is a factor to adjust the width of the beam.

## 3. Numerical results

*θ*with

*Z*axis, for different separation distances

*c*with

*V =*1. We can observe that the amount of light in each waveguide depended on the separation distance. For example in Figs. 2(a) and 2(d) for a distance of

*c*= 10.5 and

*c*= 16, respectively, the energy of the probe beam is almost entirely transferred to other waveguide; for a initial separation of

*c*= 11.5, the amount of energy in each waveguide is almost the same (Fig. 2(b)); and for

*c*= 12.5, Fig. 2(c), the probe beam energy was mainly retained in the initial waveguide.

**79**(2), 021802 (2009). [CrossRef]

18. K. Steiglitz, “Making beam splitters with dark soliton collisions,” Phys. Rev. A **82**(4), 043831 (2010). [CrossRef]

*V = 1*and

*c =*8. We can observe that after the soliton collision the modes are separated, the fundamental mode is deflected while the higher mode is not. For

*V =*1 and initial separations between the beams of 10

*<c<*15 the splitter was not very efficient. This behavior was independent of which solitons guided the probe beam.

*V*= 2 and

*c*= 8 between the solitons: in (a) when the probe beam was initially guided by S1 and in (b) when it was initially guided by S2. We observed that at the end of the propagation path in both cases the major amount of energy was confined in the other waveguide, however in Fig. 5(a) this energy is less than in Fig. 5(b). It is important to note that the light confined in the case of Fig. 5(a), after the collision, is not the fundamental mode, in support of the idea that the asymmetric collision can be used as a mode separator [17

17. K. Steiglitz, “Soliton-guided phase shifter and beam splitter,” Phys. Rev. A **81**(3), 033835 (2010). [CrossRef]

*V*= 2 as a function of the separation is shown in Fig. 7 , where the probe beam was initially guided by: a) soliton S1 and b) soliton S2. The general behavior of both cases is similar; a difference was obtained in the maximum amount of energy confined in each waveguide. However, when the final confined energy, of the waveguide where the probe beam was initially guided, is plotted as a function of the separation

*c*differences in magnitude and position are very clear, see Fig. 7(c). Larger values of

*V*produced similar results with differences in: the separation between the solitons, the amplitude and mode of the confined field.

17. K. Steiglitz, “Soliton-guided phase shifter and beam splitter,” Phys. Rev. A **81**(3), 033835 (2010). [CrossRef]

*V*= 1 and the behavior of a probe beam initially guided by the central soliton. In our case the initial relative phase between the solitons was zero and the separation was chosen to give the best energy transference (

*c*= 8).

## 5. Conclusions

*V*less than 1, the probe beam energy confined by each waveguide after the collision, is independent of which soliton initially guided the probe beam. However, for

*V*larger than 1, the amount of energy confined by each waveguide depended on which soliton initially guided the probe beam. It is demonstrated that as in the temporal case the collision of spatial solitons can be used as a mode separator when the probe beam is a combination of fundamental and high order modes. The results can be used as reference for other types of nonlinearities and to design more complicated soliton collision configurations in order to predict the waveguide properties of such elements.

## Acknowledgments

## References and links

1. | A. W. Synder and D. J. Mitchell, “Accessible solitons,” Science |

2. | G. P. Agrawal, |

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

4. | R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. |

5. | B. Luther-Davies and Y. Xiaoping, “Waveguides and Y junctions formed in bulk media by using dark spatial solitons,” Opt. Lett. |

6. | G. E. Torres-Cisneros, J. J. Sánchez-Mondragon, and V. A. Vysloukh, “Asymmetric optical Y junctions and switching of weak beams using bright spatial-soliton collisions,” Opt. Lett. |

7. | B. L. Davies and X. Yang, “Steerable optical waveguides formed in self-defocusing media by using dark spatial solitons,” Opt. Lett. |

8. | N. Akhmediev and A. Ankiewicz, “Spatial soliton X-junctions and couplers,” Opt. Commun. |

9. | P. D. Miller and N. N. Akhmediev, “Transfer matrices for multiport devices made from solitons,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics |

10. | Y. Kodama and A. Hasegawa, “Effects of initial overlap on the propagation of optical solitons at different wavelengths,” Opt. Lett. |

11. | M. Shalaby and A. Barthelemy, “Ultrafast photonic switching and splitting through cross-phase modulation with a spatial solitons couple,” Opt. Commun. |

12. | J. S. Aitchison, A. M. Weiner, Y. Silberberg, D. E. Leaird, M. K. Oliver, J. L. Jackel, and P. W. Smith, “Experimental observation of spatial soliton interactions,” Opt. Lett. |

13. | J. S. Aitchison, A. M. Weiner, Y. Silberberg, D. E. Leaird, M. K. Oliver, J. L. Jackel, and P. W. Smith, “Spatial optical solitons in planar glass waveguides,” J. Opt. Soc. Am. B |

14. | M. Shalaby, F. Reynaud, and A. Barthelemy, “Experimental observation of spatial soliton interactions with a |

15. | P. Chamorro-Posada and G. S. McDonald, “Spatial Kerr soliton collisions at arbitrary angles,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

16. | K. Steiglitz and D. Rand, “Photon trapping and transfer with solitons,” Phys. Rev. A |

17. | K. Steiglitz, “Soliton-guided phase shifter and beam splitter,” Phys. Rev. A |

18. | K. Steiglitz, “Making beam splitters with dark soliton collisions,” Phys. Rev. A |

19. | D. Ramirez Martinez, M. M. Mendez Otero, M. L. Arroyo Carrasco, and M. D. Iturbe Castillo, “Alternative (1+1)-D dark spatial soliton-like distributions in Kerr media,” J. Phys. Sci. Appl. |

**OCIS Codes**

(190.0190) Nonlinear optics : Nonlinear optics

(190.3270) Nonlinear optics : Kerr effect

(190.6135) Nonlinear optics : Spatial solitons

**ToC Category:**

Spatial Solitons in Nonlinear Materials

**History**

Original Manuscript: September 5, 2012

Revised Manuscript: October 18, 2012

Manuscript Accepted: October 26, 2012

Published: November 19, 2012

**Virtual Issues**

Nonlinear Photonics (2012) *Optics Express*

**Citation**

D. Ramírez Martínez, M. M. Méndez Otero, M. L. Arroyo Carrasco, and M. D. Iturbe Castillo, "Waveguide properties of the asymmetric collision between two bright spatial solitons in Kerr media," Opt. Express **20**, 27411-27418 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-24-27411

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### References

- A. W. Synder and D. J. Mitchell, “Accessible solitons,” Science276(5318), 1538–1541 (1997). [CrossRef]
- G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 1989).
- Y. S. Kivshar and G. P. Agrawal, Optical Solitons (Academic Press, 2003).
- R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett.13(15), 479–482 (1964). [CrossRef]
- B. Luther-Davies and Y. Xiaoping, “Waveguides and Y junctions formed in bulk media by using dark spatial solitons,” Opt. Lett.17(7), 496–498 (1992). [CrossRef] [PubMed]
- G. E. Torres-Cisneros, J. J. Sánchez-Mondragon, and V. A. Vysloukh, “Asymmetric optical Y junctions and switching of weak beams using bright spatial-soliton collisions,” Opt. Lett.18(16), 1299–1301 (1993). [CrossRef] [PubMed]
- B. L. Davies and X. Yang, “Steerable optical waveguides formed in self-defocusing media by using dark spatial solitons,” Opt. Lett.17, 496–498 (1992). [CrossRef] [PubMed]
- N. Akhmediev and A. Ankiewicz, “Spatial soliton X-junctions and couplers,” Opt. Commun.100(1-4), 186–192 (1993). [CrossRef]
- P. D. Miller and N. N. Akhmediev, “Transfer matrices for multiport devices made from solitons,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics53(4), 4098–4106 (1996). [CrossRef] [PubMed]
- Y. Kodama and A. Hasegawa, “Effects of initial overlap on the propagation of optical solitons at different wavelengths,” Opt. Lett.16(4), 208–210 (1991). [CrossRef] [PubMed]
- M. Shalaby and A. Barthelemy, “Ultrafast photonic switching and splitting through cross-phase modulation with a spatial solitons couple,” Opt. Commun.94(5), 341–345 (1992). [CrossRef]
- J. S. Aitchison, A. M. Weiner, Y. Silberberg, D. E. Leaird, M. K. Oliver, J. L. Jackel, and P. W. Smith, “Experimental observation of spatial soliton interactions,” Opt. Lett.16(1), 15–17 (1991). [CrossRef] [PubMed]
- J. S. Aitchison, A. M. Weiner, Y. Silberberg, D. E. Leaird, M. K. Oliver, J. L. Jackel, and P. W. Smith, “Spatial optical solitons in planar glass waveguides,” J. Opt. Soc. Am. B8(6), 1290–1297 (1991). [CrossRef]
- M. Shalaby, F. Reynaud, and A. Barthelemy, “Experimental observation of spatial soliton interactions with a π/2 relative phase difference,” Opt. Lett.17(11), 778–780 (1992). [CrossRef] [PubMed]
- P. Chamorro-Posada and G. S. McDonald, “Spatial Kerr soliton collisions at arbitrary angles,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.74(3), 036609 (2006). [CrossRef] [PubMed]
- K. Steiglitz and D. Rand, “Photon trapping and transfer with solitons,” Phys. Rev. A79(2), 021802 (2009). [CrossRef]
- K. Steiglitz, “Soliton-guided phase shifter and beam splitter,” Phys. Rev. A81(3), 033835 (2010). [CrossRef]
- K. Steiglitz, “Making beam splitters with dark soliton collisions,” Phys. Rev. A82(4), 043831 (2010). [CrossRef]
- D. Ramirez Martinez, M. M. Mendez Otero, M. L. Arroyo Carrasco, and M. D. Iturbe Castillo, “Alternative (1+1)-D dark spatial soliton-like distributions in Kerr media,” J. Phys. Sci. Appl.1, 196–203 (2011).

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