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

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References

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  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]
  6. Y. S. Kivshar, �??Self-localization in arrays of defocusing waveguides,�?? Opt. Lett. 18, 1147 (1993).
    [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]
  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, 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]
  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]
  12. M. Hamermesh, Group theory and its application to physical problems, (Addison-Wesley, Reading, Massachusetts, 1964).
  13. P. R. McIsaac, �??Symmetry-induced modal characteristics of uniform waveguides,�?? IEEE Trans. Microw. Theory Tech. 23, 421 (1975).
    [CrossRef]
  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]
  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]

IEEE Trans. Microw. Theory Tech. (1)

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

J. Opt. Soc. Am. A (1)

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]

Nature (1)

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]

Opt. Express (1)

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]

Opt. Lett. (6)

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]

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

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

D. N. Christodoulides and R. I. Joseph, �??Discrete self-focusing in nonlinear arrays of coupled waveguides,�?? Opt. Lett. 13, 794 (1988).
[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]

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]

Phys. Rev. E (1)

B. A. Malomed and P. G. Keverkidis, �??Discrete vortex solitons,�?? Phys. Rev. E 64, 026601 (2001).
[CrossRef]

Phys. Rev. Lett. (3)

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]

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

Other (2)

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).

M. Hamermesh, Group theory and its application to physical problems, (Addison-Wesley, Reading, Massachusetts, 1964).

Supplementary Material (2)

» Media 1: GIF (997 KB)     
» Media 2: GIF (965 KB)     

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Figures (5)

Fig. 1.
Fig. 1.

(a) Schematic representation of a PCF. (b)–(d) Amplitudes of several vortices for increasing values of γ in a PCF with Λ=31µm and a=6µm (λ=1064nm): (b) γ=0,45×10-3; (c) γ=0,95×10-3; (d) γ=1,75×10-3.

Fig. 2.
Fig. 2.

Effective index of a family of vortex solutions as a function of γ (solid line). Same for a family of fundamental solitons (dashed line), as in Ref. [10]. The shadow region corresponds to the conduction band, constituted by Bloch modes, of the 2D photonic cladding with Λ=31µm and a=10µm (λ=1064nm).

Fig. 3.
Fig. 3.

Phase of a vortex with l=1 at r=21µm. We represent both the total phase arg(ϕl ) (solid line) and the group phase arg(ϕl )- (dashed line).

Fig. 4.
Fig. 4.

(997 KB) Evolution of the field amplitude in z under a diagonal perturbation (ε=0.095) for a large-scale PCF with Λ=31µm, a=10µm and λ=1064nm. We show the transient from an initial profile towards an asymptotic vortex solution.

Fig. 5.
Fig. 5.

(965 KB) Evolution of the field amplitude in z under a non-diagonal perturbation (ε=0.05) for a large-scale PCF with Λ=31µm, a=10µm and λ=1064nm.

Equations (3)

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[ t 2 + k 0 2 ( n 0 2 ( x ) + n 2 2 ( x ) E 2 ) ] E = 2 E z 2 ,
L ( ϕ ) ϕ = β 2 ϕ L ( ϕ ) = L 0 + L NL ( ϕ ) ,
ϕ l = r l e i l θ ϕ l s ( r , θ ) exp [ i ϕ l p ( r , θ ) ] l = 1 , 2 ,

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