Abstract

We report dynamic regimes supported by a sharp quasi-one-dimensional (1D) (“razor”), pyramid-shaped (“dagger”), and conical (“needle”) potentials in the 2D complex Ginzburg–Landau (CGL) equation with cubic-quintic non linearity. This is a model of an active optical medium with respective expanding antiwaveguiding structures. If the potentials are strong enough, they give rise to continuous generation of expanding soliton patterns by a 2D soliton initially placed at the center. In the case of the pyramidal potential with M edges, the generated patterns are sets of M jets for M5, or expanding polygonal chains of solitons for M6. In the conical geometry, these are concentric waves expanding in the radial direction.

© 2010 Optical Society of America

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  1. I. S. Aranson and L. Kramer, Rev. Mod. Phys. 74, 99(2002).
    [CrossRef]
  2. N. Rosanov, in Dissipative Solitons, N.Akhmediev and A.Ankievicz, eds. (Springer-Verlag, 2005), pp. 1–17.
  3. B. A. Malomed, in Encyclopedia of Nonlinear Science, A.Scott, ed. (Routledge, 2005), p. 157–160.
  4. V. I. Petviashvili and A. M. Sergeev, Dokl. Akad. Nauk SSSR 276, 1380 (1984).
  5. N. N. Rosanov, Spatial Hysteresis and Optical Patterns (Springer, 2002).
  6. P. Mandel and M. Tlidi, J. Opt. B 6, R60 (2004).
    [CrossRef]
  7. N. N. Rosanov, S. V. Fedorov, and A. N. Shatsev, Appl. Phys. B 81, 937 (2005).
    [CrossRef]
  8. C. O. Weiss and Ye. Larionova, Rep. Prog. Phys. 70, 255 (2007).
    [CrossRef]
  9. A. Ankiewicz, N. Devine, N. Akhmediev, and J. M. Soto-Crespo, Phys. Rev. A 77, 033840 (2008).
    [CrossRef]
  10. L.-C. Crasovan, B. A. Malomed, and D. Mihalache, Phys. Rev. E 63, 016605 (2000).
    [CrossRef]
  11. A. Desyatnikov, A. Maimistov, and B. Malomed, Phys. Rev. E 61, 3107 (2000).
    [CrossRef]
  12. L.-C. Crasovan, B. A. Malomed, and D. Mihalache, Phys. Lett. A 289, 59 (2001).
    [CrossRef]
  13. J. M. Soto-Crespo, N. Akhmediev, C. Mejia-Cortes, and N. Devine, Opt. Express 17, 4236 (2009).
    [CrossRef] [PubMed]
  14. D. V. Skryabin and A. G. Vladimirov, Phys. Rev. Lett. 89, 044101 (2002).
    [CrossRef] [PubMed]
  15. J. M. Soto-Crespo, P. Grelu, and N. Akhmediev, Opt. Express 14, 4013 (2006).
    [CrossRef] [PubMed]
  16. D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, Phys. Rev. A 75, 033811 (2007).
    [CrossRef]
  17. A. Kamagate, P. Grelu, and P. Tchofo-Dinda, Phys. Rev. E 79, 026609 (2009).
    [CrossRef]
  18. D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, Phys. Rev. E 78, 056601 (2008).
    [CrossRef]
  19. D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, Phys. Rev. A 77, 033817 (2008).
    [CrossRef]
  20. Y. J. He, B. A. Malomed, and H. Z. Wang, Opt. Express 15, 17502 (2007).
    [CrossRef] [PubMed]
  21. B. Liu, Y. J. He, Z. R. Qiu, and H. Z. Wang, Opt. Express 17, 12203 (2009).
    [CrossRef] [PubMed]
  22. H. Leblond, B. A. Malomed, and D. Mihalache, Phys. Rev. A 80, 033835 (2009).
    [CrossRef]
  23. H. Sakaguchi and B. A. Malomed, Phys. Rev. E 80, 026606 (2009).
    [CrossRef]
  24. Y. J. He, B. A. Malomed, D. Mihalache, B. Liu, H. C. Huang, H. Yang, and H. Z. Wang, Opt. Lett. 34, 2976(2009).
    [CrossRef] [PubMed]
  25. C. Cleff, B. Gütlich, and C. Denz, Phys. Rev. Lett. 100, 233902 (2008).
    [CrossRef] [PubMed]
  26. A. Szameit, J. Burghoff, T. Pertsch, S. Nolte, A. Tünnermann, and F. Lederer, Opt. Express 14, 6055 (2006).
    [CrossRef] [PubMed]

2009

2008

C. Cleff, B. Gütlich, and C. Denz, Phys. Rev. Lett. 100, 233902 (2008).
[CrossRef] [PubMed]

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, Phys. Rev. E 78, 056601 (2008).
[CrossRef]

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, Phys. Rev. A 77, 033817 (2008).
[CrossRef]

A. Ankiewicz, N. Devine, N. Akhmediev, and J. M. Soto-Crespo, Phys. Rev. A 77, 033840 (2008).
[CrossRef]

2007

C. O. Weiss and Ye. Larionova, Rep. Prog. Phys. 70, 255 (2007).
[CrossRef]

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, Phys. Rev. A 75, 033811 (2007).
[CrossRef]

Y. J. He, B. A. Malomed, and H. Z. Wang, Opt. Express 15, 17502 (2007).
[CrossRef] [PubMed]

2006

2005

N. N. Rosanov, S. V. Fedorov, and A. N. Shatsev, Appl. Phys. B 81, 937 (2005).
[CrossRef]

2004

P. Mandel and M. Tlidi, J. Opt. B 6, R60 (2004).
[CrossRef]

2002

I. S. Aranson and L. Kramer, Rev. Mod. Phys. 74, 99(2002).
[CrossRef]

D. V. Skryabin and A. G. Vladimirov, Phys. Rev. Lett. 89, 044101 (2002).
[CrossRef] [PubMed]

2001

L.-C. Crasovan, B. A. Malomed, and D. Mihalache, Phys. Lett. A 289, 59 (2001).
[CrossRef]

2000

L.-C. Crasovan, B. A. Malomed, and D. Mihalache, Phys. Rev. E 63, 016605 (2000).
[CrossRef]

A. Desyatnikov, A. Maimistov, and B. Malomed, Phys. Rev. E 61, 3107 (2000).
[CrossRef]

1984

V. I. Petviashvili and A. M. Sergeev, Dokl. Akad. Nauk SSSR 276, 1380 (1984).

Akhmediev, N.

Ankiewicz, A.

A. Ankiewicz, N. Devine, N. Akhmediev, and J. M. Soto-Crespo, Phys. Rev. A 77, 033840 (2008).
[CrossRef]

Aranson, I. S.

I. S. Aranson and L. Kramer, Rev. Mod. Phys. 74, 99(2002).
[CrossRef]

Burghoff, J.

Cleff, C.

C. Cleff, B. Gütlich, and C. Denz, Phys. Rev. Lett. 100, 233902 (2008).
[CrossRef] [PubMed]

Crasovan, L.-C.

L.-C. Crasovan, B. A. Malomed, and D. Mihalache, Phys. Lett. A 289, 59 (2001).
[CrossRef]

L.-C. Crasovan, B. A. Malomed, and D. Mihalache, Phys. Rev. E 63, 016605 (2000).
[CrossRef]

Denz, C.

C. Cleff, B. Gütlich, and C. Denz, Phys. Rev. Lett. 100, 233902 (2008).
[CrossRef] [PubMed]

Desyatnikov, A.

A. Desyatnikov, A. Maimistov, and B. Malomed, Phys. Rev. E 61, 3107 (2000).
[CrossRef]

Devine, N.

J. M. Soto-Crespo, N. Akhmediev, C. Mejia-Cortes, and N. Devine, Opt. Express 17, 4236 (2009).
[CrossRef] [PubMed]

A. Ankiewicz, N. Devine, N. Akhmediev, and J. M. Soto-Crespo, Phys. Rev. A 77, 033840 (2008).
[CrossRef]

Fedorov, S. V.

N. N. Rosanov, S. V. Fedorov, and A. N. Shatsev, Appl. Phys. B 81, 937 (2005).
[CrossRef]

Grelu, P.

A. Kamagate, P. Grelu, and P. Tchofo-Dinda, Phys. Rev. E 79, 026609 (2009).
[CrossRef]

J. M. Soto-Crespo, P. Grelu, and N. Akhmediev, Opt. Express 14, 4013 (2006).
[CrossRef] [PubMed]

Gütlich, B.

C. Cleff, B. Gütlich, and C. Denz, Phys. Rev. Lett. 100, 233902 (2008).
[CrossRef] [PubMed]

He, Y. J.

Huang, H. C.

Kamagate, A.

A. Kamagate, P. Grelu, and P. Tchofo-Dinda, Phys. Rev. E 79, 026609 (2009).
[CrossRef]

Kramer, L.

I. S. Aranson and L. Kramer, Rev. Mod. Phys. 74, 99(2002).
[CrossRef]

Larionova, Ye.

C. O. Weiss and Ye. Larionova, Rep. Prog. Phys. 70, 255 (2007).
[CrossRef]

Leblond, H.

H. Leblond, B. A. Malomed, and D. Mihalache, Phys. Rev. A 80, 033835 (2009).
[CrossRef]

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, Phys. Rev. E 78, 056601 (2008).
[CrossRef]

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, Phys. Rev. A 77, 033817 (2008).
[CrossRef]

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, Phys. Rev. A 75, 033811 (2007).
[CrossRef]

Lederer, F.

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, Phys. Rev. E 78, 056601 (2008).
[CrossRef]

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, Phys. Rev. A 77, 033817 (2008).
[CrossRef]

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, Phys. Rev. A 75, 033811 (2007).
[CrossRef]

A. Szameit, J. Burghoff, T. Pertsch, S. Nolte, A. Tünnermann, and F. Lederer, Opt. Express 14, 6055 (2006).
[CrossRef] [PubMed]

Liu, B.

Maimistov, A.

A. Desyatnikov, A. Maimistov, and B. Malomed, Phys. Rev. E 61, 3107 (2000).
[CrossRef]

Malomed, B.

A. Desyatnikov, A. Maimistov, and B. Malomed, Phys. Rev. E 61, 3107 (2000).
[CrossRef]

Malomed, B. A.

H. Leblond, B. A. Malomed, and D. Mihalache, Phys. Rev. A 80, 033835 (2009).
[CrossRef]

H. Sakaguchi and B. A. Malomed, Phys. Rev. E 80, 026606 (2009).
[CrossRef]

Y. J. He, B. A. Malomed, D. Mihalache, B. Liu, H. C. Huang, H. Yang, and H. Z. Wang, Opt. Lett. 34, 2976(2009).
[CrossRef] [PubMed]

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, Phys. Rev. A 77, 033817 (2008).
[CrossRef]

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, Phys. Rev. E 78, 056601 (2008).
[CrossRef]

Y. J. He, B. A. Malomed, and H. Z. Wang, Opt. Express 15, 17502 (2007).
[CrossRef] [PubMed]

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, Phys. Rev. A 75, 033811 (2007).
[CrossRef]

L.-C. Crasovan, B. A. Malomed, and D. Mihalache, Phys. Lett. A 289, 59 (2001).
[CrossRef]

L.-C. Crasovan, B. A. Malomed, and D. Mihalache, Phys. Rev. E 63, 016605 (2000).
[CrossRef]

B. A. Malomed, in Encyclopedia of Nonlinear Science, A.Scott, ed. (Routledge, 2005), p. 157–160.

Mandel, P.

P. Mandel and M. Tlidi, J. Opt. B 6, R60 (2004).
[CrossRef]

Mazilu, D.

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, Phys. Rev. E 78, 056601 (2008).
[CrossRef]

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, Phys. Rev. A 77, 033817 (2008).
[CrossRef]

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, Phys. Rev. A 75, 033811 (2007).
[CrossRef]

Mejia-Cortes, C.

Mihalache, D.

H. Leblond, B. A. Malomed, and D. Mihalache, Phys. Rev. A 80, 033835 (2009).
[CrossRef]

Y. J. He, B. A. Malomed, D. Mihalache, B. Liu, H. C. Huang, H. Yang, and H. Z. Wang, Opt. Lett. 34, 2976(2009).
[CrossRef] [PubMed]

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, Phys. Rev. A 77, 033817 (2008).
[CrossRef]

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, Phys. Rev. E 78, 056601 (2008).
[CrossRef]

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, Phys. Rev. A 75, 033811 (2007).
[CrossRef]

L.-C. Crasovan, B. A. Malomed, and D. Mihalache, Phys. Lett. A 289, 59 (2001).
[CrossRef]

L.-C. Crasovan, B. A. Malomed, and D. Mihalache, Phys. Rev. E 63, 016605 (2000).
[CrossRef]

Nolte, S.

Pertsch, T.

Petviashvili, V. I.

V. I. Petviashvili and A. M. Sergeev, Dokl. Akad. Nauk SSSR 276, 1380 (1984).

Qiu, Z. R.

Rosanov, N.

N. Rosanov, in Dissipative Solitons, N.Akhmediev and A.Ankievicz, eds. (Springer-Verlag, 2005), pp. 1–17.

Rosanov, N. N.

N. N. Rosanov, S. V. Fedorov, and A. N. Shatsev, Appl. Phys. B 81, 937 (2005).
[CrossRef]

N. N. Rosanov, Spatial Hysteresis and Optical Patterns (Springer, 2002).

Sakaguchi, H.

H. Sakaguchi and B. A. Malomed, Phys. Rev. E 80, 026606 (2009).
[CrossRef]

Sergeev, A. M.

V. I. Petviashvili and A. M. Sergeev, Dokl. Akad. Nauk SSSR 276, 1380 (1984).

Shatsev, A. N.

N. N. Rosanov, S. V. Fedorov, and A. N. Shatsev, Appl. Phys. B 81, 937 (2005).
[CrossRef]

Skryabin, D. V.

D. V. Skryabin and A. G. Vladimirov, Phys. Rev. Lett. 89, 044101 (2002).
[CrossRef] [PubMed]

Soto-Crespo, J. M.

Szameit, A.

Tchofo-Dinda, P.

A. Kamagate, P. Grelu, and P. Tchofo-Dinda, Phys. Rev. E 79, 026609 (2009).
[CrossRef]

Tlidi, M.

P. Mandel and M. Tlidi, J. Opt. B 6, R60 (2004).
[CrossRef]

Tünnermann, A.

Vladimirov, A. G.

D. V. Skryabin and A. G. Vladimirov, Phys. Rev. Lett. 89, 044101 (2002).
[CrossRef] [PubMed]

Wang, H. Z.

Weiss, C. O.

C. O. Weiss and Ye. Larionova, Rep. Prog. Phys. 70, 255 (2007).
[CrossRef]

Yang, H.

Appl. Phys. B

N. N. Rosanov, S. V. Fedorov, and A. N. Shatsev, Appl. Phys. B 81, 937 (2005).
[CrossRef]

Dokl. Akad. Nauk SSSR

V. I. Petviashvili and A. M. Sergeev, Dokl. Akad. Nauk SSSR 276, 1380 (1984).

J. Opt. B

P. Mandel and M. Tlidi, J. Opt. B 6, R60 (2004).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Lett. A

L.-C. Crasovan, B. A. Malomed, and D. Mihalache, Phys. Lett. A 289, 59 (2001).
[CrossRef]

Phys. Rev. A

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, Phys. Rev. A 75, 033811 (2007).
[CrossRef]

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, Phys. Rev. A 77, 033817 (2008).
[CrossRef]

H. Leblond, B. A. Malomed, and D. Mihalache, Phys. Rev. A 80, 033835 (2009).
[CrossRef]

A. Ankiewicz, N. Devine, N. Akhmediev, and J. M. Soto-Crespo, Phys. Rev. A 77, 033840 (2008).
[CrossRef]

Phys. Rev. E

L.-C. Crasovan, B. A. Malomed, and D. Mihalache, Phys. Rev. E 63, 016605 (2000).
[CrossRef]

A. Desyatnikov, A. Maimistov, and B. Malomed, Phys. Rev. E 61, 3107 (2000).
[CrossRef]

H. Sakaguchi and B. A. Malomed, Phys. Rev. E 80, 026606 (2009).
[CrossRef]

A. Kamagate, P. Grelu, and P. Tchofo-Dinda, Phys. Rev. E 79, 026609 (2009).
[CrossRef]

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, Phys. Rev. E 78, 056601 (2008).
[CrossRef]

Phys. Rev. Lett.

C. Cleff, B. Gütlich, and C. Denz, Phys. Rev. Lett. 100, 233902 (2008).
[CrossRef] [PubMed]

D. V. Skryabin and A. G. Vladimirov, Phys. Rev. Lett. 89, 044101 (2002).
[CrossRef] [PubMed]

Rep. Prog. Phys.

C. O. Weiss and Ye. Larionova, Rep. Prog. Phys. 70, 255 (2007).
[CrossRef]

Rev. Mod. Phys.

I. S. Aranson and L. Kramer, Rev. Mod. Phys. 74, 99(2002).
[CrossRef]

Other

N. Rosanov, in Dissipative Solitons, N.Akhmediev and A.Ankievicz, eds. (Springer-Verlag, 2005), pp. 1–17.

B. A. Malomed, in Encyclopedia of Nonlinear Science, A.Scott, ed. (Routledge, 2005), p. 157–160.

N. N. Rosanov, Spatial Hysteresis and Optical Patterns (Springer, 2002).

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Figures (3)

Fig. 1
Fig. 1

(a) 2D profile of one of the solitons (marked by the arrow) forming the generated pattern in panel (c1). (b) Regions of different dynamic regimes in the plane of n and a. Other panels display examples of the evolution: (c1), (c2) in region c, for n = 1 , a = 0.2 and n = 1 , a = 0.4 ; (d) in region d, for n = 0.5 , a = 2 ; (e) in region e, for n = 1 , a = 0.85 ; (f) in region f, for n = 1 , a = 2 ; (g) in region g for n = 1 , a = 0.1 .

Fig. 2
Fig. 2

Dynamic regimes in the model including the pyramid-shaped potentials whose cross sections are displayed in panels at the right edge. (a), (b) Continuous generation of three and four soliton jets, respectively. The regime of the four-jet generation is observed at values of the potential’s strength 0.13 < a < 0.8 . (c) Gradual stretching of the central soliton, without the generation of secondary solitons, in the potential with the square-shaped cross section, at smaller values of the potential strength than in (b), 0 < a < 0.13 . (d) Generation of stretching and eventually breaking ring-shaped hexagonal patterns z = 0 , 20, 40, 60, 80, and 100.

Fig. 3
Fig. 3

Evolution initiated by the central soliton in the case of the conical (needle-shaped) potential; see Eq. (3): (a) continuous generation of expanding annular solitons, at a = 0.25 in Eq. (3), (b) stretch of the central soliton, for a = 0.08 , (c) transformation of the initial central soliton into a stretching ring, at a = 0.13 .

Equations (4)

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

i u z + i δ · u + ( 1 / 2 i β ) ( u x x + u y y ) + ( 1 i ε ) | u | 2 u ( ν i μ ) | u | 4 u = F ( x , y ) u ,
F ( x , y ) = a | x | n ,
F ( x , y ) = a | sin [ ( 1 2 1 M + 2 ( n 1 ) M ) π x ] + cos [ ( 1 2 1 M + 2 ( n 1 ) M ) π y ] | , 1 + 2 ( n 1 ) M < θ π 1 + 2 π n M , n = 1 , 2 M ,
F ( x , y ) = a r , r x 2 + y 2 .

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