Abstract

We introduce the basic properties of solitons in elliptical photonic lattices induced optically by a superposition of Mathieu beams. Owing to the modulation of the intensity along its elliptical rings, these lattices allow novel dynamics of propagation, being possible, for the first time to our knowledge, to propagate solitons in an elliptic motion with varying rotation rate.

© 2008 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817 (2003).
    [CrossRef] [PubMed]
  2. J. W. Fleishcer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Nature 422, 147 (2003).
    [CrossRef]
  3. N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, Phys. Rev. E 66, 046602 (2002).
    [CrossRef]
  4. O. Bang and P. D. Miller, Opt. Lett. 21, 1105 (1996).
    [CrossRef] [PubMed]
  5. Y. V. Kartashov, A. S. Zelenina, L. Torner, and V. A. Vysloukh, Opt. Lett. 31, 238 (2006).
    [CrossRef] [PubMed]
  6. Z. Bouchal, Czech. J. Phys. 53, 537 (2003).
    [CrossRef]
  7. S. Chávez-Cerda, J. C. Gutiérrez-Vega, and G. H. C. New, Opt. Lett. 26, 1803 (2001).
    [CrossRef]
  8. M. A. Bandres, J. C. Gutiérrez-Vega, and S. Chávez-Cerda, Opt. Lett. 29, 44 (2004).
    [CrossRef] [PubMed]
  9. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Phys. Rev. Lett. 93, 093904 (2004).
    [CrossRef] [PubMed]
  10. R. Fischer, D. N. Neshev, S. Lopez-Aguayo, A. S. Desyatnikov, A. A. Sukhorukov, W. Krolikowski, and Y. S. Kivshar, Opt. Express 14, 2825 (2006).
    [CrossRef] [PubMed]
  11. Y. V. Kartashov, A. A. Egorov, V. A. Vysloukh, and L. Torner, Opt. Lett. 31, 238 (2006).
    [CrossRef] [PubMed]
  12. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Opt. Lett. 33, 141 (2006).
    [CrossRef]
  13. C. Lopez-Mariscal, J. C. Gutiérrez-Vega, G. Milne, and K. Dholakia, Opt. Express 14, 4182 (2006).
    [CrossRef] [PubMed]
  14. Y. Kartashov, V. Vysloukh, and L. Torner, Opt. Express 12, 2831 (2004).
    [CrossRef] [PubMed]
  15. D. E. Pelinovsky and Yu. A. Stepanyants, SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 42, 1110 (2004).
  16. Y. S. Kivshar and D. E. Pelinovsky, Phys. Rep. 331, 117 (2000).
    [CrossRef]
  17. A. V. Burvak, Y. S. Kivshar, and S. Trillo, Phys. Rev. Lett. 77, 5210 (1996).
    [CrossRef]
  18. X. Wang, Z. Chen, and P. G. Kevrekidis, Phys. Rev. Lett. 96, 083904 (2006).
    [CrossRef] [PubMed]

2006

2004

Y. Kartashov, V. Vysloukh, and L. Torner, Opt. Express 12, 2831 (2004).
[CrossRef] [PubMed]

D. E. Pelinovsky and Yu. A. Stepanyants, SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 42, 1110 (2004).

M. A. Bandres, J. C. Gutiérrez-Vega, and S. Chávez-Cerda, Opt. Lett. 29, 44 (2004).
[CrossRef] [PubMed]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Phys. Rev. Lett. 93, 093904 (2004).
[CrossRef] [PubMed]

2003

Z. Bouchal, Czech. J. Phys. 53, 537 (2003).
[CrossRef]

D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817 (2003).
[CrossRef] [PubMed]

J. W. Fleishcer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Nature 422, 147 (2003).
[CrossRef]

2002

N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, Phys. Rev. E 66, 046602 (2002).
[CrossRef]

2001

2000

Y. S. Kivshar and D. E. Pelinovsky, Phys. Rep. 331, 117 (2000).
[CrossRef]

1996

A. V. Burvak, Y. S. Kivshar, and S. Trillo, Phys. Rev. Lett. 77, 5210 (1996).
[CrossRef]

O. Bang and P. D. Miller, Opt. Lett. 21, 1105 (1996).
[CrossRef] [PubMed]

Bandres, M. A.

Bang, O.

Bouchal, Z.

Z. Bouchal, Czech. J. Phys. 53, 537 (2003).
[CrossRef]

Burvak, A. V.

A. V. Burvak, Y. S. Kivshar, and S. Trillo, Phys. Rev. Lett. 77, 5210 (1996).
[CrossRef]

Chávez-Cerda, S.

Chen, Z.

X. Wang, Z. Chen, and P. G. Kevrekidis, Phys. Rev. Lett. 96, 083904 (2006).
[CrossRef] [PubMed]

Christodoulides, D. N.

D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817 (2003).
[CrossRef] [PubMed]

J. W. Fleishcer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Nature 422, 147 (2003).
[CrossRef]

N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, Phys. Rev. E 66, 046602 (2002).
[CrossRef]

Desyatnikov, A. S.

Dholakia, K.

Efremidis, N. K.

J. W. Fleishcer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Nature 422, 147 (2003).
[CrossRef]

N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, Phys. Rev. E 66, 046602 (2002).
[CrossRef]

Egorov, A. A.

Fischer, R.

Fleischer, J. W.

N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, Phys. Rev. E 66, 046602 (2002).
[CrossRef]

Fleishcer, J. W.

J. W. Fleishcer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Nature 422, 147 (2003).
[CrossRef]

Gutiérrez-Vega, J. C.

Kartashov, Y.

Kartashov, Y. V.

Kevrekidis, P. G.

X. Wang, Z. Chen, and P. G. Kevrekidis, Phys. Rev. Lett. 96, 083904 (2006).
[CrossRef] [PubMed]

Kivshar, Y. S.

R. Fischer, D. N. Neshev, S. Lopez-Aguayo, A. S. Desyatnikov, A. A. Sukhorukov, W. Krolikowski, and Y. S. Kivshar, Opt. Express 14, 2825 (2006).
[CrossRef] [PubMed]

Y. S. Kivshar and D. E. Pelinovsky, Phys. Rep. 331, 117 (2000).
[CrossRef]

A. V. Burvak, Y. S. Kivshar, and S. Trillo, Phys. Rev. Lett. 77, 5210 (1996).
[CrossRef]

Krolikowski, W.

Lederer, F.

D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817 (2003).
[CrossRef] [PubMed]

Lopez-Aguayo, S.

Lopez-Mariscal, C.

Miller, P. D.

Milne, G.

Neshev, D. N.

New, G. H. C.

Pelinovsky, D. E.

D. E. Pelinovsky and Yu. A. Stepanyants, SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 42, 1110 (2004).

Y. S. Kivshar and D. E. Pelinovsky, Phys. Rep. 331, 117 (2000).
[CrossRef]

Sears, S.

N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, Phys. Rev. E 66, 046602 (2002).
[CrossRef]

Segev, M.

J. W. Fleishcer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Nature 422, 147 (2003).
[CrossRef]

N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, Phys. Rev. E 66, 046602 (2002).
[CrossRef]

Silberberg, Y.

D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817 (2003).
[CrossRef] [PubMed]

Stepanyants, Yu. A.

D. E. Pelinovsky and Yu. A. Stepanyants, SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 42, 1110 (2004).

Sukhorukov, A. A.

Torner, L.

Trillo, S.

A. V. Burvak, Y. S. Kivshar, and S. Trillo, Phys. Rev. Lett. 77, 5210 (1996).
[CrossRef]

Vysloukh, V.

Vysloukh, V. A.

Wang, X.

X. Wang, Z. Chen, and P. G. Kevrekidis, Phys. Rev. Lett. 96, 083904 (2006).
[CrossRef] [PubMed]

Zelenina, A. S.

Czech. J. Phys.

Z. Bouchal, Czech. J. Phys. 53, 537 (2003).
[CrossRef]

Nature

D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817 (2003).
[CrossRef] [PubMed]

J. W. Fleishcer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Nature 422, 147 (2003).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rep.

Y. S. Kivshar and D. E. Pelinovsky, Phys. Rep. 331, 117 (2000).
[CrossRef]

Phys. Rev. E

N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, Phys. Rev. E 66, 046602 (2002).
[CrossRef]

Phys. Rev. Lett.

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Phys. Rev. Lett. 93, 093904 (2004).
[CrossRef] [PubMed]

A. V. Burvak, Y. S. Kivshar, and S. Trillo, Phys. Rev. Lett. 77, 5210 (1996).
[CrossRef]

X. Wang, Z. Chen, and P. G. Kevrekidis, Phys. Rev. Lett. 96, 083904 (2006).
[CrossRef] [PubMed]

SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal.

D. E. Pelinovsky and Yu. A. Stepanyants, SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 42, 1110 (2004).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1
Fig. 1

Transverse profile of third-order EPLs with k t = 2 for (a) ϵ = 0 , (b) ϵ = 1 , and (c) ϵ = 2 . Profiles of centered solitons supported by a third-order EPL with s = 0.05 , ϵ = 1 , k t = 2 , and p = 2 for (d) β = 0.8 , (e) β = 15 , and (f) β = 20 .

Fig. 2
Fig. 2

Power of the soliton and their localization into the lattice (inset) versus the propagation constant for different values of lattice depth p for modes localized over (a) the semi-major axis and (b) the semi-minor axis of the second elliptic ring.

Fig. 3
Fig. 3

Dynamics of soliton propagation in the EPL with m = 3 and ϵ = 1 . In all cases s = 0.05 , p = 4 , and β = 10 . (a) Oscillatory motion. (b) Rotational motion. (c) Nontrapped motion. The loss of the power for the soliton propagation for each of these cases is shown in (d), (e), and (f), respectively. (g) Propagation trajectories of the soliton in the EPL for cases (a) and (b). In this example, g rot = 0.513 and g out = 1.53 .

Fig. 4
Fig. 4

(a) Transverse rotation rate [ S = Δ r Δ z ] of the solitons trapped in an EPL and a Bessel lattice. (b) Enclosed power for soliton trapped in the EPL and the Bessel lattice.

Equations (2)

Equations on this page are rendered with MathJax. Learn more.

i Ψ z = ( 2 x 2 + 2 y 2 ) Ψ Ψ 2 1 + s Ψ 2 Ψ p Γ ( r ) Ψ ,
τ ( r ) = 0 2 π [ C ce m ( ϕ ; ϵ ) + i S se m ( ϕ ; ϵ ) ] exp [ i k t ( x cos ϕ + y sin ϕ ) ] d ϕ ,

Metrics