## Vortex solitons in photonic crystal fibers

Optics Express, Vol. 12, Issue 5, pp. 817-822 (2004)

http://dx.doi.org/10.1364/OPEX.12.000817

Acrobat PDF (711 KB)

### Abstract

We demonstrate the existence of vortex soliton solutions in photonic crystal fibers. We analyze the role played by the photonic crystal fiber defect in the generation of optical vortices. An analytical prediction for the angular dependence of the amplitude and phase of the vortex solution based on group theory is also provided. Furthermore, all the analysis is performed in the non-paraxial regime.

© 2004 Optical Society of America

## 1. Introduction

1. D. N. Christodoulides and R. I. Joseph, “Discrete self-focusing in nonlinear arrays of coupled waveguides,” Opt. Lett. **13**, 794 (1988). [CrossRef] [PubMed]

2. H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, “Discrete spatial optical solitons in waveguide arrays,” Phys. Rev. Lett. **81**, 3383 (1998). [CrossRef]

3. J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremedis, and D. N. Christodoulides, “Observation of discrete solitons in optically induced real time waveguide arrays,” Phys. Rev. Lett. **90**, 023902 (2003). [CrossRef] [PubMed]

4. D. Neshev, E. Ostrovskaya, Y. S. Kivshar, and W. Krolikowsky, “Spatial solitons in optically induced gratings,” Opt. Lett. **28**, 710 (2003). [CrossRef] [PubMed]

5. J. W. Fleischer, M. Segev, N. K. Efremedis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature **422**, 147 (2003). [CrossRef] [PubMed]

1. D. N. Christodoulides and R. I. Joseph, “Discrete self-focusing in nonlinear arrays of coupled waveguides,” Opt. Lett. **13**, 794 (1988). [CrossRef] [PubMed]

6. Y. S. Kivshar, “Self-localization in arrays of defocusing waveguides,” Opt. Lett. **18**, 1147 (1993). [CrossRef] [PubMed]

3. J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremedis, and D. N. Christodoulides, “Observation of discrete solitons in optically induced real time waveguide arrays,” Phys. Rev. Lett. **90**, 023902 (2003). [CrossRef] [PubMed]

5. J. W. Fleischer, M. Segev, N. K. Efremedis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature **422**, 147 (2003). [CrossRef] [PubMed]

7. B. A. Malomed and P. G. Keverkidis, “Discrete vortex solitons,” Phys. Rev. E **64**, 026601 (2001). [CrossRef]

8. J. Yang and Z. H. Musslimani, “Fundamental and vortex solitons in a two-dimensional optical lattice,” Opt. Lett. **28**, 2094 (2003). [CrossRef] [PubMed]

10. A. Ferrando, M. Zacarés, P. F. de Córdoba, D. Binosi, and J. A. Monsoriu, “Spatial soliton formation in photonic crystal fibers,” Opt. Express **11**, 452 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-5-452. [CrossRef] [PubMed]

5. J. W. Fleischer, M. Segev, N. K. Efremedis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature **422**, 147 (2003). [CrossRef] [PubMed]

8. J. Yang and Z. H. Musslimani, “Fundamental and vortex solitons in a two-dimensional optical lattice,” Opt. Lett. **28**, 2094 (2003). [CrossRef] [PubMed]

## 2. Description of the method

**E**≈0):

*k*

_{0}=

*ω*

_{0}/

*c*the vacuum wavenumber. The linear refractive index profile function

*n*

_{0}(

*x,y*) is 1 in the air-holes and equals nsilica in silica, whereas the nonlinear index profile function

*n*

_{2}(

*x,y*) is different from zero only in silica (

*χ*

^{(3)}(

_{silica)}/(2

*ε*

_{0}

*cn*

_{0(silica)})). We search for monochromatic (or quasi-monochromatic) stationary electric field solutions with well-defined constant polarization:

*L*

_{0}=

*x,y*) and

*L*

_{NL}(|

*ϕ*|)=

*x,y*)|

*ϕ*|

^{2}stand for the linear and nonlinear parts of the differential operator

*L*(|

*ϕ*|), respectively. Due to the geometry of the air-holes distribution, both

*n*

_{0}(

*x,y*) and

*n*

_{2}(

*x,y*) are invariant under the action of the

*𝓒*

_{6v}point-symmetry group. This group is constituted by discrete

*π*/3-rotations (

*R*

_{π/3}) plus specular reflections with respect to the

*x*and

*y*axes:

10. A. Ferrando, M. Zacarés, P. F. de Córdoba, D. Binosi, and J. A. Monsoriu, “Spatial soliton formation in photonic crystal fibers,” Opt. Express **11**, 452 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-5-452. [CrossRef] [PubMed]

11. M. J. Steel, T. P. White, C. M. de Sterke, R. C. McPhedran, and L. C. Botten, “Symmetry and degeneracy in microstructured optical fibers,” Opt. Lett. **26**, 488 (2001). [CrossRef]

*L*

_{0}is invariant under the

*𝓒*

_{6v}group, all its eigenfunctions have to lie on discrete representations of this group [12, 13

13. P. R. McIsaac, “Symmetry-induced modal characteristics of uniform waveguides,” IEEE Trans. Microw. Theory Tech. **23**, 421 (1975). [CrossRef]

*𝓒*

_{6v}. A pair of functions (

*ϕ*

_{l}

*,ϕ**

_{l}) (

*l*=1, 2) belonging to these representations have the following transformation properties:

*r,θ*+

*π*/3)=

*r,θ*) and

*r,-θ*)=

*r,π*-

*θ*)=

*r,θ*), whereas

*r,θ*+

*π*/3)=

*r,θ*) but

*r*,-

*θ*)=

*r,π*-

*θ*)=-

*r,θ*). Both

*θ*, so they can be expanded in Fourier series in cos(6

*nθ*) and sin(6

*nθ*) (

*n*∈

*N*). Reflection symmetry forces

*ϕ*

^{s}(

*ϕ*

^{p}) to depend on the cosine (sine) terms only. Thus,

*r,θ*)=∑

_{n}

*a*

_{ln}(

*r*)cos(6

*nθ*) and

*r,θ*)=∑

_{n}

*b*

_{ln}(

*r*) sin(6

*nθ*).

*ϕ*

_{l}of the form given by Eq. (3) can satisfy self-consistency, according to group theory. The full operator

*L*(|

*ϕ*

_{l}|)=

*L*

_{0}(

*n*

_{0})+

*L*

_{NL}(

*r*

^{l}

*;n*

_{2}) is invariant under the

*𝓒*

_{6v}group because

*L*

_{0},

*r*

^{l}

*n*

_{2}are all

*𝓒*

_{6v}invariants. Thus

*L*(|

*ϕ*

_{l}|), like in the linear case, provides the representation where

*ϕ*

_{l}lies on. However, group self-consistency is not sufficient to ensure that the resolution of the nonlinear equation (2) with the ansatz (3) provides a nontrivial solution in all cases, since Eq. (2) always admits the

*ϕ*=0 solution. We solve Eq. (2) by means of the Fourier iterative method previously used in Ref. [10

10. A. Ferrando, M. Zacarés, P. F. de Córdoba, D. Binosi, and J. A. Monsoriu, “Spatial soliton formation in photonic crystal fibers,” Opt. Express **11**, 452 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-5-452. [CrossRef] [PubMed]

*𝓒*

_{6v}. The important difference now is that we restrict ourselves to a different representation space; we search for nonlinear solutions in the

*l*=1,2 representations of

*𝓒*

_{6v}. The

*ϕ*

_{l}solution and its conjugate

*ϕ**

_{l}represent a vortex and an anti-vortex soliton of order

*l*, respectively.

## 3. Results

**11**, 452 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-5-452. [CrossRef] [PubMed]

*µ*m, and several air-hole radius at a fixed wavelength λ=1064nm (a convenient quasi-continuum source). We define a dimensionless nonlinear parameter as

*γ*=

*P/A*

_{0}(

*A*

_{0}is an area parameter:

*A*

_{0}=

*π*(Λ/2)

^{2};

*P*is the total power carried by the solution) and perform calculations for increasing values of γ. We discover that a family of optical vortex solutions of the type given by the ansatz (3) is found. The amplitudes of several solutions (with

*l*=1) for increasing values of

*γ*are represented in Fig. 1(b)–(d). According to group theory (Eq. (3)), these amplitudes have to be scalar functions, thus showing full invariance under

*𝓒*

_{6v}, and they have to vanish at the origin, as seen in Fig. 1(b)–(d). Besides, they become gradually narrower as the nonlinearity increases. It is interesting to plot the effective index of these solutions (

*n*

_{eff}=

*β/k*

_{0}) versus

*γ*, as shown in Fig. 2. The value of neff increases as

*γ*increases, accordingly to the narrowing of solutions depicted in Fig. 1(b)–(d). For comparison, we have also included the

*n*

_{eff}(

*γ*) curve of the fundamental soliton family for the same PCF structure, as in Ref. [10

**11**, 452 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-5-452. [CrossRef] [PubMed]

*γ*(power) to generate a nonlinear vortex soliton, i.e, there is a continuum of solutions in

*γ*starting from the linear (TE or TM) mode. Unlike in perfectly periodic structures, the defect can eliminate the presence of a threshold power [8

8. J. Yang and Z. H. Musslimani, “Fundamental and vortex solitons in a two-dimensional optical lattice,” Opt. Lett. **28**, 2094 (2003). [CrossRef] [PubMed]

14. A. Ferrando, E. Silvestre, J. J. Miret, P. Andrés, and M. V. Andrés, “Vector description of higher-order modes in photonic crystal fibers,” J. Opt. Soc. Am. A **17**, 1333 (2000). [CrossRef]

*γ*is necessary to generate the nonlinear vortex solution in the forbidden band. By playing with the geometric parameters of the PCF, it is possible to tune this threshold value. In all cases, this value is much lesser than in perfectly periodic structures, which can be important for experimental purposes.

*ϕ*

_{l})=

*lθ*+

*r,θ*)=

*lθ*+

*b*

_{ln}(

*r*) sin(6

*nθ*) and it points out that

*n*≈1. The phase of these vortex solutions differs from that of an ordinary vortex by the presence of the pseudo-scalar function extra term: the group phase. Besides the existence of the standard linear behavior in θ, the group phase provides an additional sinusoidal dependence on the angle (with period determined by the group order:

*π*/3 for order 6) not present in ordinary vortices.

*ϕ*

_{l}, we have to consider

*z*-dependent perturbations. This implies solving the non-paraxial equation (1) for the perturbed field

*ϕ*′=

*ϕ*

_{l}+

*δϕ*. In terms of group theory, (

*ϕ*

_{l}

*,ϕ**

_{l}) constitute an orthogonal basis of the

*l*representation of

*𝓒*

_{6v}, so we can consider two types of perturbations around a vortex solution

*ϕ*

_{l}: diagonal perturbations, in which

*ϕ*′ remains in the one-dimensional subspace spawned by

*ϕ*

_{l}, that is, 〈

*ϕ*′|

*ϕ**

_{l}〉=0, and non-diagonal, where the perturbation takes

*ϕ*′ out of this subspace, 〈

*ϕ*′|

*ϕ**

_{l}〉=0.

*ϕ*′=

*re*

^{iθ}exp[-(

*r/r*

_{0})

^{2}] is an example of diagonal perturbation for a vortex (

*l*=1). Another example is a perturbation of the form

*ϕ*′=(1+

*ε*)

*ϕ*

_{l},ε being a small real constant number. In both cases, 〈

*ϕ*′|

*ϕ**

_{l}〉=0. We have simulated the evolution of a perturbation of the latter type launched into the fiber core for different PCF configurations and values of

*ε*(

*ε*~0.025-0.1). In Fig. 4 we present an animation that displays the evolution of the transverse field amplitude along the fiber. The animation shows the transient from the initial amplitude towards an asymptotically stationary profile corresponding to a vortex. This transient is a few centimeters long and has the characteristic form of a damped oscillation. Non-paraxial evolution shows that a vortex is stable under diagonal perturbations. Diagonal perturbations generalize the concept of radial perturbations to the discrete symmetry case. We have also simulated non-diagonal perturbations by introducing random perturbations (|

*δϕ*(

*x,y*)|≤

*ε, ε*~0.05-0.1) that modify both the phase and amplitude of the vortex solution in such a way that 〈

*ϕ*′|

*ϕ**

_{l}〉≠0. Non-diagonal perturbations correspond to azimuthal ones in the radially symmetric case. In this case, an oscillatory instability occurs, as seen in the simulation presented in Fig. 5. This oscillatory instability initially appears as a rotating flux along the vortex ring. After a short evolution, the perturbed vortex breaks into a two strongly interacting single structures of zero vorticity with a cumbersome dynamics, as observed in our simulation. It is remarkable that, unlike vortices in an homogeneous medium [15

15. W. J. Firth and D. V. Skryabin, “Orbital solitons carrying orbital angular momentum,” Phys. Rev. Lett. **79**, 2450 (1997). [CrossRef]

16. N. Akhmediev, A. Ankiewicz, and J. M. Soto-Crespo, “Does the nonlinear Schrödinger equation correctly describe beam propagation?,” Opt. Lett. **18**, 411 (1993). [CrossRef] [PubMed]

## 4. Conclusions

## References and links

1. | D. N. Christodoulides and R. I. Joseph, “Discrete self-focusing in nonlinear arrays of coupled waveguides,” Opt. Lett. |

2. | H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, “Discrete spatial optical solitons in waveguide arrays,” Phys. Rev. Lett. |

3. | J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremedis, and D. N. Christodoulides, “Observation of discrete solitons in optically induced real time waveguide arrays,” Phys. Rev. Lett. |

4. | D. Neshev, E. Ostrovskaya, Y. S. Kivshar, and W. Krolikowsky, “Spatial solitons in optically induced gratings,” Opt. Lett. |

5. | J. W. Fleischer, M. Segev, N. K. Efremedis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature |

6. | Y. S. Kivshar, “Self-localization in arrays of defocusing waveguides,” Opt. Lett. |

7. | B. A. Malomed and P. G. Keverkidis, “Discrete vortex solitons,” Phys. Rev. E |

8. | J. Yang and Z. H. Musslimani, “Fundamental and vortex solitons in a two-dimensional optical lattice,” Opt. Lett. |

9. | D. N. Neshev, T. J. Alexander, E. A. Ostrovskaya, Y. Kivshar, H. Martin, and Z. Chen, “Observation of discrete vortex solitons in optically-induced photonic lattices,” arXiv:nlin/0309018 (2003). |

10. | A. Ferrando, M. Zacarés, P. F. de Córdoba, D. Binosi, and J. A. Monsoriu, “Spatial soliton formation in photonic crystal fibers,” Opt. Express |

11. | M. J. Steel, T. P. White, C. M. de Sterke, R. C. McPhedran, and L. C. Botten, “Symmetry and degeneracy in microstructured optical fibers,” Opt. Lett. |

12. | M. Hamermesh, |

13. | P. R. McIsaac, “Symmetry-induced modal characteristics of uniform waveguides,” IEEE Trans. Microw. Theory Tech. |

14. | A. Ferrando, E. Silvestre, J. J. Miret, P. Andrés, and M. V. Andrés, “Vector description of higher-order modes in photonic crystal fibers,” J. Opt. Soc. Am. A |

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

16. | N. Akhmediev, A. Ankiewicz, and J. M. Soto-Crespo, “Does the nonlinear Schrödinger equation correctly describe beam propagation?,” Opt. Lett. |

**OCIS Codes**

(060.4370) Fiber optics and optical communications : Nonlinear optics, fibers

(190.4370) Nonlinear optics : Nonlinear optics, fibers

**ToC Category:**

Research Papers

**History**

Original Manuscript: February 4, 2004

Revised Manuscript: February 23, 2004

Published: March 8, 2004

**Citation**

Albert Ferrando, Mario Zacarés, Pedro Fernández de Córdoba, Daniele Binosi, and Juan Monsoriu, "Vortex solitons in photonic crystal fibers," Opt. Express **12**, 817-822 (2004)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-5-817

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### References

- D. N. Christodoulides and R. I. Joseph, �??Discrete self-focusing in nonlinear arrays of coupled waveguides,�?? Opt. Lett. 13, 794 (1988). [CrossRef] [PubMed]
- H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, �??Discrete spatial optical solitons in waveguide arrays,�?? Phys. Rev. Lett. 81, 3383 (1998). [CrossRef]
- J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremedis, and D. N. Christodoulides, �??Observation of discrete solitons in optically induced real time waveguide arrays,�?? Phys. Rev. Lett. 90, 023902 (2003). [CrossRef] [PubMed]
- D. Neshev, E. Ostrovskaya, Y. S. Kivshar, and W. Krolikowsky, �??Spatial solitons in optically induced gratings,�?? Opt. Lett. 28, 710 (2003). [CrossRef] [PubMed]
- J.W. Fleischer, M. Segev, N. K. Efremedis, and D. N. Christodoulides, �??Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,�?? Nature 422, 147 (2003). [CrossRef] [PubMed]
- Y. S. Kivshar, �??Self-localization in arrays of defocusing waveguides,�?? Opt. Lett. 18, 1147 (1993). [CrossRef] [PubMed]
- B. A. Malomed and P. G. Keverkidis, �??Discrete vortex solitons,�?? Phys. Rev. E 64, 026601 (2001). [CrossRef]
- J. Yang and Z. H. Musslimani, �??Fundamental and vortex solitons in a two-dimensional optical lattice,�?? Opt. Lett. 28, 2094 (2003). [CrossRef] [PubMed]
- D. N. Neshev, T. J. Alexander, E. A. Ostrovskaya, Y. Kivshar, H. Martin, and Z. Chen, �??Observation of discrete vortex solitons in optically-induced photonic lattices,�?? arXiv:nlin/0309018 (2003).
- A. Ferrando, M. Zacarés, P. F. de Córdoba, D. Binosi, and J. A. Monsoriu, �??Spatial soliton formation in photonic crystal fibers,�?? Opt. Express 11, 452 (2003), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-5-452">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-5-452<a> [CrossRef] [PubMed]
- M. J. Steel, T. P. White, C. M. de Sterke, R. C. McPhedran, and L. C. Botten, �??Symmetry and degeneracy in microstructured optical fibers,�?? Opt. Lett. 26, 488 (2001). [CrossRef]
- M. Hamermesh, Group theory and its application to physical problems, (Addison-Wesley, Reading, Massachusetts, 1964).
- P. R. McIsaac, �??Symmetry-induced modal characteristics of uniform waveguides,�?? IEEE Trans. Microw. Theory Tech. 23, 421 (1975). [CrossRef]
- A. Ferrando, E. Silvestre, J. J. Miret, P. Andrés, and M. V. Andrés, �??Vector description of higher-order modes in photonic crystal fibers,�?? J. Opt. Soc. Am. A 17, 1333 (2000). [CrossRef]
- W. J. Firth and D. V. Skryabin, �??Orbital solitons carrying orbital angular momentum,�?? Phys. Rev. Lett. 79, 2450 (1997). [CrossRef]
- N. Akhmediev, A. Ankiewicz, and J. M. Soto-Crespo, �??Does the nonlinear Schrödinger equation correctly describe beam propagation?,�?? Opt. Lett. 18, 411 (1993). [CrossRef] [PubMed]

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