Abstract

We introduce a new species of gap solitons (GSs) supported by an azimuthally modulated guiding ring in defocusing cubic media. The periodicity in the azimuthal direction strongly modifies properties and existence domains of GSs. In addition to the fundamental solitons, we report even and twisted bound states. The former type is found to be stable, while the twisted states are always unstable in defocusing media.

© 2008 Optical Society of America

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  1. D. N. Christodoulides and R. I. Joseph, Phys. Rev. Lett. 62, 1746 (1989).
    [CrossRef] [PubMed]
  2. A. B. Aceves and S. Wabnitz, Phys. Lett. A 141, 37 (1989).
    [CrossRef]
  3. B. J. Eggleton, R. E. Slusher, C. M. de Sterke, and J. E. Sipe, Phys. Rev. Lett. 76, 1627 (1996).
    [CrossRef] [PubMed]
  4. J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Phys. Rev. Lett. 90, 023902 (2003).
    [CrossRef] [PubMed]
  5. J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Nature 422, 147 (2003).
    [CrossRef] [PubMed]
  6. D. Neshev, A. A. Sukhorukov, B. Hanna, W. Krolikowski, and Y. S. Kivshar, Phys. Rev. Lett. 93, 083905 (2004).
    [CrossRef] [PubMed]
  7. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Opt. Express 12, 2831 (2004).
    [CrossRef] [PubMed]
  8. X. Wang, Z. Chen, J. Wang, and J. Yang, Phys. Rev. Lett. 99, 243901 (2007).
    [CrossRef]
  9. D. Mandelik, R. Morandotti, J. S. Aitchison, and Y. Silberberg, Phys. Rev. Lett. 92, 093904 (2004).
    [CrossRef] [PubMed]
  10. F. Chen, M. Stepic, C. E. Rüter, D. Runde, D. Kip, V. Shandarov, O. Manela, and M. Segev, Opt. Express 13, 4314 (2005).
    [CrossRef] [PubMed]
  11. E. Smirnov, C. E. Rüter, D. Kip, Y. V. Kartashov, and L. Torner, Opt. Lett. 32, 1950 (2007).
    [CrossRef] [PubMed]
  12. Y. Kominis and K. Hizanidis, Opt. Lett. 31, 2888 (2006).
    [CrossRef] [PubMed]
  13. B. B. Baizakov, V. V. Konotop, and M. Salerno, J. Phys. B 35, 5105 (2002).
    [CrossRef]
  14. P. J. Y. Louis, E. A. Ostrovskaya, C. M. Savage, and Y. S. Kivshar, Phys. Rev. A 67, 013602 (2003).
    [CrossRef]
  15. N. K. Efremidis and D. N. Christodoulides, Phys. Rev. A 67, 063608 (2003).
    [CrossRef]
  16. O. Morsch and M. Oberthaler, Rev. Mod. Phys. 78, 179 (2006).
    [CrossRef]
  17. Y. V. Kartashov, A. A. Egorov, V. A. Vysloukh, and L. Torner, Phys. Rev. E 70, 065602(R) (2004).
    [CrossRef]
  18. J. Scheuer and B. Malomed, Phys. Rev. A 75, 063805 (2007).
    [CrossRef]
  19. A. V. Carpentier and H. Michinel, EPL 78, 10002 (2007).
    [CrossRef]

2007 (4)

X. Wang, Z. Chen, J. Wang, and J. Yang, Phys. Rev. Lett. 99, 243901 (2007).
[CrossRef]

J. Scheuer and B. Malomed, Phys. Rev. A 75, 063805 (2007).
[CrossRef]

A. V. Carpentier and H. Michinel, EPL 78, 10002 (2007).
[CrossRef]

E. Smirnov, C. E. Rüter, D. Kip, Y. V. Kartashov, and L. Torner, Opt. Lett. 32, 1950 (2007).
[CrossRef] [PubMed]

2006 (2)

Y. Kominis and K. Hizanidis, Opt. Lett. 31, 2888 (2006).
[CrossRef] [PubMed]

O. Morsch and M. Oberthaler, Rev. Mod. Phys. 78, 179 (2006).
[CrossRef]

2005 (1)

2004 (4)

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

Y. V. Kartashov, A. A. Egorov, V. A. Vysloukh, and L. Torner, Phys. Rev. E 70, 065602(R) (2004).
[CrossRef]

D. Mandelik, R. Morandotti, J. S. Aitchison, and Y. Silberberg, Phys. Rev. Lett. 92, 093904 (2004).
[CrossRef] [PubMed]

D. Neshev, A. A. Sukhorukov, B. Hanna, W. Krolikowski, and Y. S. Kivshar, Phys. Rev. Lett. 93, 083905 (2004).
[CrossRef] [PubMed]

2003 (4)

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Phys. Rev. Lett. 90, 023902 (2003).
[CrossRef] [PubMed]

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

P. J. Y. Louis, E. A. Ostrovskaya, C. M. Savage, and Y. S. Kivshar, Phys. Rev. A 67, 013602 (2003).
[CrossRef]

N. K. Efremidis and D. N. Christodoulides, Phys. Rev. A 67, 063608 (2003).
[CrossRef]

2002 (1)

B. B. Baizakov, V. V. Konotop, and M. Salerno, J. Phys. B 35, 5105 (2002).
[CrossRef]

1996 (1)

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, and J. E. Sipe, Phys. Rev. Lett. 76, 1627 (1996).
[CrossRef] [PubMed]

1989 (2)

D. N. Christodoulides and R. I. Joseph, Phys. Rev. Lett. 62, 1746 (1989).
[CrossRef] [PubMed]

A. B. Aceves and S. Wabnitz, Phys. Lett. A 141, 37 (1989).
[CrossRef]

Aceves, A. B.

A. B. Aceves and S. Wabnitz, Phys. Lett. A 141, 37 (1989).
[CrossRef]

Aitchison, J. S.

D. Mandelik, R. Morandotti, J. S. Aitchison, and Y. Silberberg, Phys. Rev. Lett. 92, 093904 (2004).
[CrossRef] [PubMed]

Baizakov, B. B.

B. B. Baizakov, V. V. Konotop, and M. Salerno, J. Phys. B 35, 5105 (2002).
[CrossRef]

Carmon, T.

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Phys. Rev. Lett. 90, 023902 (2003).
[CrossRef] [PubMed]

Carpentier, A. V.

A. V. Carpentier and H. Michinel, EPL 78, 10002 (2007).
[CrossRef]

Chen, F.

Chen, Z.

X. Wang, Z. Chen, J. Wang, and J. Yang, Phys. Rev. Lett. 99, 243901 (2007).
[CrossRef]

Christodoulides, D. N.

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

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Phys. Rev. Lett. 90, 023902 (2003).
[CrossRef] [PubMed]

N. K. Efremidis and D. N. Christodoulides, Phys. Rev. A 67, 063608 (2003).
[CrossRef]

D. N. Christodoulides and R. I. Joseph, Phys. Rev. Lett. 62, 1746 (1989).
[CrossRef] [PubMed]

de Sterke, C. M.

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, and J. E. Sipe, Phys. Rev. Lett. 76, 1627 (1996).
[CrossRef] [PubMed]

Efremidis, N. K.

N. K. Efremidis and D. N. Christodoulides, Phys. Rev. A 67, 063608 (2003).
[CrossRef]

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Phys. Rev. Lett. 90, 023902 (2003).
[CrossRef] [PubMed]

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

Eggleton, B. J.

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, and J. E. Sipe, Phys. Rev. Lett. 76, 1627 (1996).
[CrossRef] [PubMed]

Egorov, A. A.

Y. V. Kartashov, A. A. Egorov, V. A. Vysloukh, and L. Torner, Phys. Rev. E 70, 065602(R) (2004).
[CrossRef]

Fleischer, J. W.

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Phys. Rev. Lett. 90, 023902 (2003).
[CrossRef] [PubMed]

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

Hanna, B.

D. Neshev, A. A. Sukhorukov, B. Hanna, W. Krolikowski, and Y. S. Kivshar, Phys. Rev. Lett. 93, 083905 (2004).
[CrossRef] [PubMed]

Hizanidis, K.

Joseph, R. I.

D. N. Christodoulides and R. I. Joseph, Phys. Rev. Lett. 62, 1746 (1989).
[CrossRef] [PubMed]

Kartashov, Y. V.

Kip, D.

Kivshar, Y. S.

D. Neshev, A. A. Sukhorukov, B. Hanna, W. Krolikowski, and Y. S. Kivshar, Phys. Rev. Lett. 93, 083905 (2004).
[CrossRef] [PubMed]

P. J. Y. Louis, E. A. Ostrovskaya, C. M. Savage, and Y. S. Kivshar, Phys. Rev. A 67, 013602 (2003).
[CrossRef]

Kominis, Y.

Konotop, V. V.

B. B. Baizakov, V. V. Konotop, and M. Salerno, J. Phys. B 35, 5105 (2002).
[CrossRef]

Krolikowski, W.

D. Neshev, A. A. Sukhorukov, B. Hanna, W. Krolikowski, and Y. S. Kivshar, Phys. Rev. Lett. 93, 083905 (2004).
[CrossRef] [PubMed]

Louis, P. J. Y.

P. J. Y. Louis, E. A. Ostrovskaya, C. M. Savage, and Y. S. Kivshar, Phys. Rev. A 67, 013602 (2003).
[CrossRef]

Malomed, B.

J. Scheuer and B. Malomed, Phys. Rev. A 75, 063805 (2007).
[CrossRef]

Mandelik, D.

D. Mandelik, R. Morandotti, J. S. Aitchison, and Y. Silberberg, Phys. Rev. Lett. 92, 093904 (2004).
[CrossRef] [PubMed]

Manela, O.

Michinel, H.

A. V. Carpentier and H. Michinel, EPL 78, 10002 (2007).
[CrossRef]

Morandotti, R.

D. Mandelik, R. Morandotti, J. S. Aitchison, and Y. Silberberg, Phys. Rev. Lett. 92, 093904 (2004).
[CrossRef] [PubMed]

Morsch, O.

O. Morsch and M. Oberthaler, Rev. Mod. Phys. 78, 179 (2006).
[CrossRef]

Neshev, D.

D. Neshev, A. A. Sukhorukov, B. Hanna, W. Krolikowski, and Y. S. Kivshar, Phys. Rev. Lett. 93, 083905 (2004).
[CrossRef] [PubMed]

Oberthaler, M.

O. Morsch and M. Oberthaler, Rev. Mod. Phys. 78, 179 (2006).
[CrossRef]

Ostrovskaya, E. A.

P. J. Y. Louis, E. A. Ostrovskaya, C. M. Savage, and Y. S. Kivshar, Phys. Rev. A 67, 013602 (2003).
[CrossRef]

Runde, D.

Rüter, C. E.

Salerno, M.

B. B. Baizakov, V. V. Konotop, and M. Salerno, J. Phys. B 35, 5105 (2002).
[CrossRef]

Savage, C. M.

P. J. Y. Louis, E. A. Ostrovskaya, C. M. Savage, and Y. S. Kivshar, Phys. Rev. A 67, 013602 (2003).
[CrossRef]

Scheuer, J.

J. Scheuer and B. Malomed, Phys. Rev. A 75, 063805 (2007).
[CrossRef]

Segev, M.

F. Chen, M. Stepic, C. E. Rüter, D. Runde, D. Kip, V. Shandarov, O. Manela, and M. Segev, Opt. Express 13, 4314 (2005).
[CrossRef] [PubMed]

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Phys. Rev. Lett. 90, 023902 (2003).
[CrossRef] [PubMed]

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

Shandarov, V.

Silberberg, Y.

D. Mandelik, R. Morandotti, J. S. Aitchison, and Y. Silberberg, Phys. Rev. Lett. 92, 093904 (2004).
[CrossRef] [PubMed]

Sipe, J. E.

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, and J. E. Sipe, Phys. Rev. Lett. 76, 1627 (1996).
[CrossRef] [PubMed]

Slusher, R. E.

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, and J. E. Sipe, Phys. Rev. Lett. 76, 1627 (1996).
[CrossRef] [PubMed]

Smirnov, E.

Stepic, M.

Sukhorukov, A. A.

D. Neshev, A. A. Sukhorukov, B. Hanna, W. Krolikowski, and Y. S. Kivshar, Phys. Rev. Lett. 93, 083905 (2004).
[CrossRef] [PubMed]

Torner, L.

Vysloukh, V. A.

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

Y. V. Kartashov, A. A. Egorov, V. A. Vysloukh, and L. Torner, Phys. Rev. E 70, 065602(R) (2004).
[CrossRef]

Wabnitz, S.

A. B. Aceves and S. Wabnitz, Phys. Lett. A 141, 37 (1989).
[CrossRef]

Wang, J.

X. Wang, Z. Chen, J. Wang, and J. Yang, Phys. Rev. Lett. 99, 243901 (2007).
[CrossRef]

Wang, X.

X. Wang, Z. Chen, J. Wang, and J. Yang, Phys. Rev. Lett. 99, 243901 (2007).
[CrossRef]

Yang, J.

X. Wang, Z. Chen, J. Wang, and J. Yang, Phys. Rev. Lett. 99, 243901 (2007).
[CrossRef]

EPL (1)

A. V. Carpentier and H. Michinel, EPL 78, 10002 (2007).
[CrossRef]

J. Phys. B (1)

B. B. Baizakov, V. V. Konotop, and M. Salerno, J. Phys. B 35, 5105 (2002).
[CrossRef]

Nature (1)

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

Opt. Express (2)

Opt. Lett. (2)

Phys. Lett. A (1)

A. B. Aceves and S. Wabnitz, Phys. Lett. A 141, 37 (1989).
[CrossRef]

Phys. Rev. A (3)

P. J. Y. Louis, E. A. Ostrovskaya, C. M. Savage, and Y. S. Kivshar, Phys. Rev. A 67, 013602 (2003).
[CrossRef]

N. K. Efremidis and D. N. Christodoulides, Phys. Rev. A 67, 063608 (2003).
[CrossRef]

J. Scheuer and B. Malomed, Phys. Rev. A 75, 063805 (2007).
[CrossRef]

Phys. Rev. E (1)

Y. V. Kartashov, A. A. Egorov, V. A. Vysloukh, and L. Torner, Phys. Rev. E 70, 065602(R) (2004).
[CrossRef]

Phys. Rev. Lett. (6)

D. N. Christodoulides and R. I. Joseph, Phys. Rev. Lett. 62, 1746 (1989).
[CrossRef] [PubMed]

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, and J. E. Sipe, Phys. Rev. Lett. 76, 1627 (1996).
[CrossRef] [PubMed]

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Phys. Rev. Lett. 90, 023902 (2003).
[CrossRef] [PubMed]

D. Neshev, A. A. Sukhorukov, B. Hanna, W. Krolikowski, and Y. S. Kivshar, Phys. Rev. Lett. 93, 083905 (2004).
[CrossRef] [PubMed]

X. Wang, Z. Chen, J. Wang, and J. Yang, Phys. Rev. Lett. 99, 243901 (2007).
[CrossRef]

D. Mandelik, R. Morandotti, J. S. Aitchison, and Y. Silberberg, Phys. Rev. Lett. 92, 093904 (2004).
[CrossRef] [PubMed]

Rev. Mod. Phys. (1)

O. Morsch and M. Oberthaler, Rev. Mod. Phys. 78, 179 (2006).
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

Absolute value of the field in fundamental GSs on the ring for (a) b = 0.75 , (b) b = 3.50 , and (c) b = 6.39 . In all cases, p = 16 , n ϕ = 10 , and d = 0.5 . The continuous circle designates the azimuthally modulated guiding ring.

Fig. 2
Fig. 2

(a) Total power versus the propagation constant of fundamental GSs and even bound states for p = 16 , n ϕ = 10 , and d = 0.5 . Circles correspond to profiles shown in Fig. 1. Existence domains of the fundamental solitons (b) in the ( p , b ) plane for n ϕ = 10 , d = 0.5 , (c) in the ( n ϕ , b ) plane for p = 16 , d = 0.5 , and (d) in the ( d , b ) plane for p = 16 , n ϕ = 10 . The real part of the perturbation growth rate versus the propagation constant for (e) twisted and (f) even bound states for p = 16 , n ϕ = 10 , and d = 0.5 .

Fig. 3
Fig. 3

Absolute value of the field in a stable even bound state with n ϕ = 10 , d = 0.5 , p = 16 , and b = 5.6 at (a) ξ = 0 and (b) ξ = 500 . A broadband noise with variance σ noise 2 = 0.01 was added to the input configuration.

Fig. 4
Fig. 4

Same as in Fig. 3, but for an unstable twisted bound state with b = 6.1 . Configuration (b) is displayed at ξ = 42 .

Equations (3)

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i q ξ = 1 2 ( 2 q η 2 + 2 q ζ 2 ) + q q 2 p R ( η , ζ ) q ,
δ u = 1 2 ( 2 v η 2 + 2 v ζ 2 ) + b v + v w 2 p R v ,
δ v = 1 2 ( 2 u η 2 + 2 u ζ 2 ) b u 3 u w 2 + p R u .

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