## Azimuthons in nonlocal nonlinear media

Optics Express, Vol. 14, Issue 17, pp. 7903-7908 (2006)

http://dx.doi.org/10.1364/OE.14.007903

Acrobat PDF (369 KB)

### Abstract

We demonstrate that spatial nonlocal response provides an effective physical mechanism for stabilization of recently introduced azimuthally modulated self-trapped rotating singular optical beams or *azimuthons* [see A. S. Desyatnikov, A. A. Sukhorukov, and Yu. S. Kivshar, Phys. Rev. Lett. **95**, 203904 (2005)]. We find that stable azimuthons become possible when the nonlocality parameter exceeds a certain threshold value and, in a sharp contrast to local media, the azimuthons with *N* peaks can also exist for *N* < 2*m*, where *m* is the azimuthon topological charge.

© 2006 Optical Society of America

## 1. Introduction

*nonlocal optical solitons*is explained by a number of experimental observations of self-trapping effects and spatial solitons in different types of nonlocal nonlinear media [1]. It has already been shown that a diversity of nonlinear media such as atomic vapors [2

2. 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]

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

4. 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]

5. W. Krolikowski, 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 **6**, S288–S294 (2004). [CrossRef]

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

7. M. Peccianti, K. A. Brzdakiewicz, and G. Assanto, “Nonlocal spatial soliton interactions in nematic liquid crystals”, Opt. Lett. **27**, 1460–1462 (2002). [CrossRef]

8. N. I. Nikolov, D. Neshev, W. Krolikowski, O. Bang, J. J. Rasmussen, and P. L. Christiansen, “Attraction of nonlocal dark optical solitons,” Opt. Lett. **29**, 286–288 (2004). [CrossRef] [PubMed]

9. Z. Xu, Y. V. Kartashov, and L. Torner, “Upper threshold for stability of multipole-mode solitons in nonlocal nonlinear media,” Opt. Lett. **30**, 3171–3173 (2005). [CrossRef] [PubMed]

10. A. S. Desyatnikov, Yu. S. Kivshar, and L. Torner, “Optical vortices and vortex solitons,” in Prog. Opt.47, Ed. E. Wolf (North-Holland, Amsterdam, 2005), pp. 291–391. [CrossRef]

11. A. I. Yakimenko, Yu. A. Zaliznyak, and Yu. S. Kivshar, “Stable vortex solitons in nonlocal self-focusing nonlinear media,” Phys. Rev. E **71**, 065603 (2005). [CrossRef]

12. D. Briedis, D. E. Petersen, D. Edmundson, W. Krolikowski, and O. Bang, “Ring vortex solitons in nonlocal nonlinear media,” Opt. Express **13**, 435–443 (2005), http://www.opticsexpress.org/abstract.cfm?id=82380. [CrossRef] [PubMed]

*clusters*of many solitons [13

13. A. S. Desyatnikov and Yu. S. Kivshar, “Rotating optical soliton slusters,” Phys. Rev. Lett. **88**, 053901 (2002). [CrossRef] [PubMed]

*azimuthons*[14

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

15. 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**, 1100–1102 (2006). [CrossRef] [PubMed]

*azimuthon*[14

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

*N*peaks may exist when the nonlocality parameter exceeds a certain threshold value and, in a sharp contrast to local media [14

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

*N*≥ 2, becomes independent on the beam topological charge. The topological structure of azimuthons with

*N*< 2

*m*is presented by a circular array of

*N*single-charge vortices with one additional dislocation at the origin with the charge given by the rule

*m*-

*N*, so that the total topological charge of azimuthon

*m*is calculated as an algebraic sum of charges of all vortices.

## 2. Variational approach

*E*,

*z*and

*x*,

*y*) stand for the propagation and transverse coordinates, respectively, and ∆

_{⊥}is the transverse Laplacian The kernel

*K*of the nonlinear response is defined by a physical process responsible for the medium nonlinearity. Here we assume the Gaussian response

*z*=

*z*′σ

^{2}, and omit primes. Following Ref. [14

**95**, 203904 (2005). [CrossRef] [PubMed]

*r*,

*θ*=

*φ*-

*ωz*}, in the form

*E*(

*z*) = (√

*π*/σ)

*V*(

*r*,

*θ*)exp(

*ikz*), and employ variational method [16

16. B. A. Malomed, “Variational methods in nonlinear fiber optics and related fields,” Prog. Opt.43, Ed. E. Wolf (North-Holland, Amsterdam, 2002), p.71–191. [CrossRef]

**95**, 203904 (2005). [CrossRef] [PubMed]

*ω*

_{0},

*n*, and

*R*(

*r*) are the variational parameters. Solving variational equation for

*ω*

_{0}we find

*ω*

_{0}= 2

*R*

^{2}

*r*

^{-1}

*dr*/ (

*R*

^{2}

*rdr*), and we define the normalized rotational velocity Ω =

*ω*/

*ω*

_{0}. The parameter

*n*∊ [0,1] is the contrast of azimuthal modulation. The structure of azimuthon is defined by two integer indices, the number of peaks

*N*and topological charge

*m*.

16. B. A. Malomed, “Variational methods in nonlinear fiber optics and related fields,” Prog. Opt.43, Ed. E. Wolf (North-Holland, Amsterdam, 2002), p.71–191. [CrossRef]

*I*

_{N}is the modified Bessel function of the first kind of the order

*N*. The parameter

*k*is expressed as

*k*=

*k*

_{0}+

*ωS*, where

*S*=

*mδ*+ Ω(1 -

*δ*) and

*S*is a ratio of the orbital angular momentum,

*M*= Im ∫

*V*

^{*}

*V*

_{θ}

*d*

*P*= ∫ |

*V*|

^{2}

*d*

*S*=

*M*/

*P*. Parameter

*g*determines the radial profile of the azimuthon through the asymptotic relation

*R*→

*r*

^{g}at

*r*→ 0. It is given by

*g*

^{2}=

*δm*

^{2}+ (Ω

^{2}+

*N*

^{2}/4)(1 -

*δ*), the latter expression indicates that larger values of the topological charge

*m*, rotational velocity Ω, or the number of peaks

*N*correspond to a larger radius of the ring. Finally, from Eq. (4) and a variational equation for

*n*we obtain a condition that should be satisfied simultaneously with Eq. (4), and it defines the rotation frequency,

## 3. Existence conditions

*n*→ 0, the ansatz (3) generates a radially symmetric vortex soliton with

*θ*= 1,

*S*=

*g*=

*m*, and

*k*=

*k*

_{0}+

*mω*. Thus, solving Eq. (4) for the vortex solitons with

*n*= 0, from Eq. (5) we can obtain two cutoff branches

*ω*

_{0}

*N*. Cutoff frequencies determine the existence domains for azimuthon solutions, shown on the plane (

*ω*,

*k*) for azimuthons with topological charges

*m*= 1,2,3 [see Figs. 1(d–f)].

**95**, 203904 (2005). [CrossRef] [PubMed]

*N*< 2

*m*holds; these azimuthons always rotate with a positive angular velocity. In particular, for even

*N*> 2, e.g.,

*N*= 4 in Fig. 1(d–f), azimuthons with charges

*m*=

*N*/2 + 1 were found to exist with positive angular velocities only [see Fig. 1(f)], while for

*m*=

*N*/2 we obtain the azimuthons also with zero (or almost zero) angular velocity [see Fig. 1(e)], and for

*m*=

*N*/2 - 1 there exist azimuthons with either positive, zero, or negative angular velocity [see Fig. 1(d)].

*F*

_{N}in Eq. (5) vanishes as the beam power (and

*k*) grows, the azimuthon angular velocity approaches the value Ω

_{±}=

*m*±

*N*/2 for

*k*→ ∞. From this relation we find that, for

*N*= 2

*m*, one of the cutoff branches that bound the azimuthon existence domain always tends to zero,

*ω*_

^{c.o.}→ 0. In the case of dipole solitons [15

15. 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**, 1100–1102 (2006). [CrossRef] [PubMed]

*m*= 1, the angular velocity does not vanish but becomes small. For larger number of peaks

*N*, our method predicts the existence of stationary nonrotating azimuthons with four peaks and a double charge, six-peak azimuthons with a triple charge, etc. We note that the non-rotating azimuthons appear due to a balance between two contributions to nonzero angular momentum, the energy flow produced by a nontrivial singular phase, and the contribution due to the beam spiraling.

*m*= 2 and two peaks do not exist in local Kerr media [14

**95**, 203904 (2005). [CrossRef] [PubMed]

*k*→ 0 in our model. In contrast, here we find that the azimuthon with

*N*= 2 and

*m*= 2 is indeed possible for

*k*> 16.6, see Fig. 1(e). For the azimuthon with the indices

*N*= 3 and

*m*= 2, we also find a region

*k*>

*k*

_{offset}where it exists, this region is separated by a small gap

*k*

_{offset}from the local limit

*k*= 0. Similar situation occurs for the case

*m*= 3 [Fig. 1(f)]: as the number of peaks

*N*grows, the gap

*k*

_{offset}decreases as the general existence condition

*N*> 2

*n*→ 0 in Eq. (5).

*N*≥ 2, and it does not depend on the value of its charge

*m*; this is in a sharp contrast to the results obtained for the cases of local Kerr and saturable media [14

**95**, 203904 (2005). [CrossRef] [PubMed]

*N*,

*m*). However, for a given

*m*, the existence domain for azimuthons quickly shrinks with

*N*→ 2, as seen in Fig. 1.

## 4. Propagation dynamics and stability

*k*) is below the threshold, the azimuthons break up and split into the fundamental solitons which move away from each other; this is similar to what happens in local media [see Figs. 2(a,b) and 3(a)]. For larger degree of nonlocality the induced effective potential is strong enough to trap the beams even when the azimuthal instability develops. For example, the strong nonlocal potential can force the splinters generated by azimuthal instability to fuse into a single soliton [15

15. 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**, 1100–1102 (2006). [CrossRef] [PubMed]

*k*, azimuthons can remain localized as rings for very long distances, even when the number of peaks along the ring is not recognizable already after a short propagation, see Fig. 2(c). Nevertheless, similar to the dipole solitons [15

**31**, 1100–1102 (2006). [CrossRef] [PubMed]

*N*,

*m*) become stable for large enough

*k*, see Figs. 2(d) and 3(b).

*N*= 4,

*m*= 2 (c) and

*m*= 3 (d). Most interesting higher-order azimuthons are presented in Fig. 4. In (a) we show the rotating beam generated by the variational solution with

*N*= 2 and

*m*= 3. While initially it carries a single triple-charged vortex, the dislocation quickly splits to three elementary vortices, similar to the higher-charge dark vortex solitons in defocusing local media [10

10. A. S. Desyatnikov, Yu. S. Kivshar, and L. Torner, “Optical vortices and vortex solitons,” in Prog. Opt.47, Ed. E. Wolf (North-Holland, Amsterdam, 2005), pp. 291–391. [CrossRef]

*m*and

*N*leads to an interesting topological structures of azimuthon’s phase front for

*N*< 2

*m*. In Figs. 4(b–d) we show the dynamics of azimuthons with the charge

*m*= 6 for

*N*= 5, 6, and 7, correspondingly. We observe that an initial charge-six vortex evolves into topologically different states with different numbers of single-charge (“elementary”) vortices, depending on the symmetry order

*N*of intensity ring modulation. We conclude that the phase topology follows two constrains: first is the superimposed rotational symmetry of the order

*N*, surprisingly robust, and second is the total topological charge, conserved in all cases. Actually, the main configuration of the observed “vortex cluster” contain a ring of

*N*single-charged vortices (thus building a charge of

*m*

_{1}=

*N*), and an additional vortex of the charge

*m*

_{2}=

*m*-

*N*in the origin, keeping the total charge

*m*

_{1}+

*m*

_{2}=

*m*. It is clearly seen in Figs. 4(b–d) that the central dislocation is of the order

*m*

_{2}= 1 (b),

*m*

_{2}= 0 (c), and

*m*

_{2}= -1 (d).

## 5. Conclusions

*azimuthons*, characterized by two integer indices

*N*and

*m*. Some of those azimuthons can only exist in nonlocal media, and the stabilization is achieved when the nonlocality parameter exceeds a threshold value.

## References and links

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

2. | D. Suter and T. Blasberg, “Stabilization of transverse solitary waves by a nonlocal response of the nonlinear medium,” Phys. Rev. A |

3. | C. Conti, M. Peccianti, and G. Assanto, “Route to nonlocality and observation of accessible solitons,” Phys. Rev. Lett. |

4. | 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. |

5. | W. Krolikowski, 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 |

6. | O. Bang, W. Krolikowski, J. Wyller, and J. J. Rasmussen, “Collapse arrest and soliton stabilization in nonlocal nonlinear media,” Phys. Rev. E |

7. | M. Peccianti, K. A. Brzdakiewicz, and G. Assanto, “Nonlocal spatial soliton interactions in nematic liquid crystals”, Opt. Lett. |

8. | N. I. Nikolov, D. Neshev, W. Krolikowski, O. Bang, J. J. Rasmussen, and P. L. Christiansen, “Attraction of nonlocal dark optical solitons,” Opt. Lett. |

9. | Z. Xu, Y. V. Kartashov, and L. Torner, “Upper threshold for stability of multipole-mode solitons in nonlocal nonlinear media,” Opt. Lett. |

10. | A. S. Desyatnikov, Yu. S. Kivshar, and L. Torner, “Optical vortices and vortex solitons,” in Prog. Opt.47, Ed. E. Wolf (North-Holland, Amsterdam, 2005), pp. 291–391. [CrossRef] |

11. | A. I. Yakimenko, Yu. A. Zaliznyak, and Yu. S. Kivshar, “Stable vortex solitons in nonlocal self-focusing nonlinear media,” Phys. Rev. E |

12. | D. Briedis, D. E. Petersen, D. Edmundson, W. Krolikowski, and O. Bang, “Ring vortex solitons in nonlocal nonlinear media,” Opt. Express |

13. | A. S. Desyatnikov and Yu. S. Kivshar, “Rotating optical soliton slusters,” Phys. Rev. Lett. |

14. | A. S. Desyatnikov, A. A. Sukhorukov, and Yu. S. Kivshar, “Azimuthons: Spatially modulated vortex solitons,” Phys. Rev. Lett. |

15. | 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. |

16. | B. A. Malomed, “Variational methods in nonlinear fiber optics and related fields,” Prog. Opt.43, Ed. E. Wolf (North-Holland, Amsterdam, 2002), p.71–191. [CrossRef] |

**OCIS Codes**

(190.4420) Nonlinear optics : Nonlinear optics, transverse effects in

(190.5530) Nonlinear optics : Pulse propagation and temporal solitons

(190.5940) Nonlinear optics : Self-action effects

(350.5030) Other areas of optics : Phase

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: May 9, 2006

Revised Manuscript: July 26, 2006

Manuscript Accepted: July 26, 2006

Published: August 21, 2006

**Citation**

Servando Lopez-Aguayo, Anton S. Desyatnikov, and Yuri S. Kivshar, "Azimuthons in nonlocal nonlinear media," Opt. Express **14**, 7903-7908 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-17-7903

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### References

- Yu. S. Kivshar, and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, San Diego, 2003), p. 540.
- 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]
- C. Conti, M. Peccianti, and G. Assanto, "Route to nonlocality and observation of accessible solitons," Phys. Rev. Lett. 91, 073901 (2003). [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]
- W. Krolikowski, 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. Krolikowski, J. Wyller, and J. J. Rasmussen, "Collapse arrest and soliton stabilization in nonlocal nonlinear media," Phys. Rev. E 66, 046619 (2002). [CrossRef]
- M. Peccianti, K. A. Brzdakiewicz, and G. Assanto, "Nonlocal spatial soliton interactions in nematic liquid crystals", Opt. Lett. 27, 1460-1462 (2002). [CrossRef]
- N. I. Nikolov, D. Neshev, W. Krolikowski, O. Bang, J. J. Rasmussen, and P. L. Christiansen, "Attraction of nonlocal dark optical solitons," Opt. Lett. 29, 286-288 (2004). [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, 3171-3173 (2005). [CrossRef] [PubMed]
- A. S. Desyatnikov, Yu. S. Kivshar, and L. Torner, "Optical vortices and vortex solitons," in Prog. Opt. 47, Ed. E. Wolf (North-Holland, Amsterdam, 2005), pp. 291-391. [CrossRef]
- A. I. Yakimenko, Yu. A. Zaliznyak, and Yu. S. Kivshar, "Stable vortex solitons in nonlocal self-focusing nonlinear media," Phys. Rev. E 71, 065603 (2005). [CrossRef]
- D. Briedis, D. E. Petersen, D. Edmundson, W. Krolikowski, and O. Bang, "Ring vortex solitons in nonlocal nonlinear media," Opt. Express 13, 435-443 (2005). [CrossRef] [PubMed]
- A. S. Desyatnikov and Yu. S. Kivshar, "Rotating optical soliton slusters," Phys. Rev. Lett. 88, 053901 (2002). [CrossRef] [PubMed]
- A. S. Desyatnikov, A. A. Sukhorukov and Yu. S. Kivshar, "Azimuthons: Spatially modulated vortex solitons," Phys. Rev. Lett. 95, 203904 (2005). [CrossRef] [PubMed]
- 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, 1100-1102 (2006). [CrossRef] [PubMed]
- B. A. Malomed, "Variational methods in nonlinear fiber optics and related fields," Prog. Opt. 43, E. Wolf, ed., (North-Holland, Amsterdam, 2002), p.71-191. [CrossRef]

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