Abstract

We introduce a concept for stable spatial soliton formation, mediated by the competition between self-bending induced by a strongly asymmetric nonlocal nonlinearity and spatially localized gain superimposed on a wide pedestal with linear losses. When acting separately both effects seriously prevent stable localization of light, but under suitable conditions they counteract each other, forming robust soliton states that are attractors for a wide range of material and input light conditions.

© 2012 Optical Society of America

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  1. Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, 2003).
  2. N. N. Rosanov, Spatial Hysteresis and Optical Patterns (Springer, 2010).
  3. N. Akhmediev and A. Ankiewicz, eds. Dissipative Solitons: From Optics to Biology and Medicine (Springer, 2008).
  4. A. M. Sergeev and V. I. Petviashvili, Dokl. Akad. Nauk SSSR 276, 1380 (1984).
  5. B. A. Malomed, Phys. D 29155 (1987).
    [CrossRef]
  6. W. van Saarloos and P. C. Hohenberg, Phys. Rev. Lett. 64, 749 (1990).
    [CrossRef]
  7. N. N. Akhmediev and V. V. Afanasjev, Phys. Rev. Lett. 75, 2320 (1995).
    [CrossRef]
  8. J. M. Soto-Crespo, N. Akhmediev, and A. Ankiewicz, Phys. Rev. Lett. 85, 2937 (2000).
    [CrossRef]
  9. V. Skarka, N. B. Aleksic, H. Leblond, B. A. Malomed, and D. Mihalache, Phys. Rev. Lett. 105, 213901 (2010).
    [CrossRef]
  10. E. A. Ultanir, G. I. Stegeman, D. Michaelis, C. H. Lange, and F. Lederer, Phys. Rev. Lett. 90, 253903 (2003).
    [CrossRef]
  11. C.-K. Lam, B. A. Malomed, K. W. Chow, and P. K. A. Wai, Eur. Phys. J. Spec. Top. 173, 233 (2009).
    [CrossRef]
  12. Y. V. Kartashov, V. V. Konotop, V. A. Vysloukh, and L. Torner, Opt. Lett. 35, 1638 (2010).
    [CrossRef]
  13. M. O. Williams, C. W. McGrath, and J. N. Kutz, Opt. Express 18, 11671 (2010).
    [CrossRef]
  14. C. H. Tsang, B. A. Malomed, C. K. Lam, and K. W. Chow, Eur. Phys. J. D 59, 81 (2010).
    [CrossRef]
  15. Y. V. Kartashov, V. V. Konotop, and V. A. Vysloukh, Opt. Lett. 36, 82 (2011).
    [CrossRef]
  16. V. E. Lobanov, Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Opt. Lett. 36, 85 (2011).
    [CrossRef]
  17. V. E. Lobanov, O. V. Borovkova, Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Phys. Rev. A 85, 023804 (2012).
    [CrossRef]
  18. D. N. Christodoulides and T. H. Coskun, Opt. Lett. 21, 1220 (1996).
    [CrossRef]
  19. W. Krolikowski, N. Akhmediev, B. Luther-Davies, and M. Cronin-Golomb, Phys. Rev. E 54, 5761 (1996).
    [CrossRef]
  20. S. R. Singh, M. I. Carvalho, and D. N. Christodoulides, Opt. Commun. 130, 288 (1996).
    [CrossRef]
  21. J. Petter, C. Weilnau, C. Denz, A. Stepken, and F. Kaiser, Opt. Commun. 170, 291 (1999).
    [CrossRef]
  22. A. E. Siegman, J. Opt. Soc. Am. A 20, 1617 (2003).
    [CrossRef]

2012 (1)

V. E. Lobanov, O. V. Borovkova, Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Phys. Rev. A 85, 023804 (2012).
[CrossRef]

2011 (2)

2010 (4)

C. H. Tsang, B. A. Malomed, C. K. Lam, and K. W. Chow, Eur. Phys. J. D 59, 81 (2010).
[CrossRef]

Y. V. Kartashov, V. V. Konotop, V. A. Vysloukh, and L. Torner, Opt. Lett. 35, 1638 (2010).
[CrossRef]

M. O. Williams, C. W. McGrath, and J. N. Kutz, Opt. Express 18, 11671 (2010).
[CrossRef]

V. Skarka, N. B. Aleksic, H. Leblond, B. A. Malomed, and D. Mihalache, Phys. Rev. Lett. 105, 213901 (2010).
[CrossRef]

2009 (1)

C.-K. Lam, B. A. Malomed, K. W. Chow, and P. K. A. Wai, Eur. Phys. J. Spec. Top. 173, 233 (2009).
[CrossRef]

2003 (2)

E. A. Ultanir, G. I. Stegeman, D. Michaelis, C. H. Lange, and F. Lederer, Phys. Rev. Lett. 90, 253903 (2003).
[CrossRef]

A. E. Siegman, J. Opt. Soc. Am. A 20, 1617 (2003).
[CrossRef]

2000 (1)

J. M. Soto-Crespo, N. Akhmediev, and A. Ankiewicz, Phys. Rev. Lett. 85, 2937 (2000).
[CrossRef]

1999 (1)

J. Petter, C. Weilnau, C. Denz, A. Stepken, and F. Kaiser, Opt. Commun. 170, 291 (1999).
[CrossRef]

1996 (3)

D. N. Christodoulides and T. H. Coskun, Opt. Lett. 21, 1220 (1996).
[CrossRef]

W. Krolikowski, N. Akhmediev, B. Luther-Davies, and M. Cronin-Golomb, Phys. Rev. E 54, 5761 (1996).
[CrossRef]

S. R. Singh, M. I. Carvalho, and D. N. Christodoulides, Opt. Commun. 130, 288 (1996).
[CrossRef]

1995 (1)

N. N. Akhmediev and V. V. Afanasjev, Phys. Rev. Lett. 75, 2320 (1995).
[CrossRef]

1990 (1)

W. van Saarloos and P. C. Hohenberg, Phys. Rev. Lett. 64, 749 (1990).
[CrossRef]

1987 (1)

B. A. Malomed, Phys. D 29155 (1987).
[CrossRef]

1984 (1)

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

Afanasjev, V. V.

N. N. Akhmediev and V. V. Afanasjev, Phys. Rev. Lett. 75, 2320 (1995).
[CrossRef]

Agrawal, G. P.

Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, 2003).

Akhmediev, N.

J. M. Soto-Crespo, N. Akhmediev, and A. Ankiewicz, Phys. Rev. Lett. 85, 2937 (2000).
[CrossRef]

W. Krolikowski, N. Akhmediev, B. Luther-Davies, and M. Cronin-Golomb, Phys. Rev. E 54, 5761 (1996).
[CrossRef]

N. Akhmediev and A. Ankiewicz, eds. Dissipative Solitons: From Optics to Biology and Medicine (Springer, 2008).

Akhmediev, N. N.

N. N. Akhmediev and V. V. Afanasjev, Phys. Rev. Lett. 75, 2320 (1995).
[CrossRef]

Aleksic, N. B.

V. Skarka, N. B. Aleksic, H. Leblond, B. A. Malomed, and D. Mihalache, Phys. Rev. Lett. 105, 213901 (2010).
[CrossRef]

Ankiewicz, A.

J. M. Soto-Crespo, N. Akhmediev, and A. Ankiewicz, Phys. Rev. Lett. 85, 2937 (2000).
[CrossRef]

N. Akhmediev and A. Ankiewicz, eds. Dissipative Solitons: From Optics to Biology and Medicine (Springer, 2008).

Borovkova, O. V.

V. E. Lobanov, O. V. Borovkova, Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Phys. Rev. A 85, 023804 (2012).
[CrossRef]

Carvalho, M. I.

S. R. Singh, M. I. Carvalho, and D. N. Christodoulides, Opt. Commun. 130, 288 (1996).
[CrossRef]

Chow, K. W.

C. H. Tsang, B. A. Malomed, C. K. Lam, and K. W. Chow, Eur. Phys. J. D 59, 81 (2010).
[CrossRef]

C.-K. Lam, B. A. Malomed, K. W. Chow, and P. K. A. Wai, Eur. Phys. J. Spec. Top. 173, 233 (2009).
[CrossRef]

Christodoulides, D. N.

D. N. Christodoulides and T. H. Coskun, Opt. Lett. 21, 1220 (1996).
[CrossRef]

S. R. Singh, M. I. Carvalho, and D. N. Christodoulides, Opt. Commun. 130, 288 (1996).
[CrossRef]

Coskun, T. H.

Cronin-Golomb, M.

W. Krolikowski, N. Akhmediev, B. Luther-Davies, and M. Cronin-Golomb, Phys. Rev. E 54, 5761 (1996).
[CrossRef]

Denz, C.

J. Petter, C. Weilnau, C. Denz, A. Stepken, and F. Kaiser, Opt. Commun. 170, 291 (1999).
[CrossRef]

Hohenberg, P. C.

W. van Saarloos and P. C. Hohenberg, Phys. Rev. Lett. 64, 749 (1990).
[CrossRef]

Kaiser, F.

J. Petter, C. Weilnau, C. Denz, A. Stepken, and F. Kaiser, Opt. Commun. 170, 291 (1999).
[CrossRef]

Kartashov, Y. V.

Kivshar, Y. S.

Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, 2003).

Konotop, V. V.

Krolikowski, W.

W. Krolikowski, N. Akhmediev, B. Luther-Davies, and M. Cronin-Golomb, Phys. Rev. E 54, 5761 (1996).
[CrossRef]

Kutz, J. N.

Lam, C. K.

C. H. Tsang, B. A. Malomed, C. K. Lam, and K. W. Chow, Eur. Phys. J. D 59, 81 (2010).
[CrossRef]

Lam, C.-K.

C.-K. Lam, B. A. Malomed, K. W. Chow, and P. K. A. Wai, Eur. Phys. J. Spec. Top. 173, 233 (2009).
[CrossRef]

Lange, C. H.

E. A. Ultanir, G. I. Stegeman, D. Michaelis, C. H. Lange, and F. Lederer, Phys. Rev. Lett. 90, 253903 (2003).
[CrossRef]

Leblond, H.

V. Skarka, N. B. Aleksic, H. Leblond, B. A. Malomed, and D. Mihalache, Phys. Rev. Lett. 105, 213901 (2010).
[CrossRef]

Lederer, F.

E. A. Ultanir, G. I. Stegeman, D. Michaelis, C. H. Lange, and F. Lederer, Phys. Rev. Lett. 90, 253903 (2003).
[CrossRef]

Lobanov, V. E.

V. E. Lobanov, O. V. Borovkova, Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Phys. Rev. A 85, 023804 (2012).
[CrossRef]

V. E. Lobanov, Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Opt. Lett. 36, 85 (2011).
[CrossRef]

Luther-Davies, B.

W. Krolikowski, N. Akhmediev, B. Luther-Davies, and M. Cronin-Golomb, Phys. Rev. E 54, 5761 (1996).
[CrossRef]

Malomed, B. A.

C. H. Tsang, B. A. Malomed, C. K. Lam, and K. W. Chow, Eur. Phys. J. D 59, 81 (2010).
[CrossRef]

V. Skarka, N. B. Aleksic, H. Leblond, B. A. Malomed, and D. Mihalache, Phys. Rev. Lett. 105, 213901 (2010).
[CrossRef]

C.-K. Lam, B. A. Malomed, K. W. Chow, and P. K. A. Wai, Eur. Phys. J. Spec. Top. 173, 233 (2009).
[CrossRef]

B. A. Malomed, Phys. D 29155 (1987).
[CrossRef]

McGrath, C. W.

Michaelis, D.

E. A. Ultanir, G. I. Stegeman, D. Michaelis, C. H. Lange, and F. Lederer, Phys. Rev. Lett. 90, 253903 (2003).
[CrossRef]

Mihalache, D.

V. Skarka, N. B. Aleksic, H. Leblond, B. A. Malomed, and D. Mihalache, Phys. Rev. Lett. 105, 213901 (2010).
[CrossRef]

Petter, J.

J. Petter, C. Weilnau, C. Denz, A. Stepken, and F. Kaiser, Opt. Commun. 170, 291 (1999).
[CrossRef]

Petviashvili, V. I.

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

Rosanov, N. N.

N. N. Rosanov, Spatial Hysteresis and Optical Patterns (Springer, 2010).

Sergeev, A. M.

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

Siegman, A. E.

Singh, S. R.

S. R. Singh, M. I. Carvalho, and D. N. Christodoulides, Opt. Commun. 130, 288 (1996).
[CrossRef]

Skarka, V.

V. Skarka, N. B. Aleksic, H. Leblond, B. A. Malomed, and D. Mihalache, Phys. Rev. Lett. 105, 213901 (2010).
[CrossRef]

Soto-Crespo, J. M.

J. M. Soto-Crespo, N. Akhmediev, and A. Ankiewicz, Phys. Rev. Lett. 85, 2937 (2000).
[CrossRef]

Stegeman, G. I.

E. A. Ultanir, G. I. Stegeman, D. Michaelis, C. H. Lange, and F. Lederer, Phys. Rev. Lett. 90, 253903 (2003).
[CrossRef]

Stepken, A.

J. Petter, C. Weilnau, C. Denz, A. Stepken, and F. Kaiser, Opt. Commun. 170, 291 (1999).
[CrossRef]

Torner, L.

Tsang, C. H.

C. H. Tsang, B. A. Malomed, C. K. Lam, and K. W. Chow, Eur. Phys. J. D 59, 81 (2010).
[CrossRef]

Ultanir, E. A.

E. A. Ultanir, G. I. Stegeman, D. Michaelis, C. H. Lange, and F. Lederer, Phys. Rev. Lett. 90, 253903 (2003).
[CrossRef]

van Saarloos, W.

W. van Saarloos and P. C. Hohenberg, Phys. Rev. Lett. 64, 749 (1990).
[CrossRef]

Vysloukh, V. A.

Wai, P. K. A.

C.-K. Lam, B. A. Malomed, K. W. Chow, and P. K. A. Wai, Eur. Phys. J. Spec. Top. 173, 233 (2009).
[CrossRef]

Weilnau, C.

J. Petter, C. Weilnau, C. Denz, A. Stepken, and F. Kaiser, Opt. Commun. 170, 291 (1999).
[CrossRef]

Williams, M. O.

Dokl. Akad. Nauk SSSR (1)

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

Eur. Phys. J. D (1)

C. H. Tsang, B. A. Malomed, C. K. Lam, and K. W. Chow, Eur. Phys. J. D 59, 81 (2010).
[CrossRef]

Eur. Phys. J. Spec. Top. (1)

C.-K. Lam, B. A. Malomed, K. W. Chow, and P. K. A. Wai, Eur. Phys. J. Spec. Top. 173, 233 (2009).
[CrossRef]

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

Opt. Commun. (2)

S. R. Singh, M. I. Carvalho, and D. N. Christodoulides, Opt. Commun. 130, 288 (1996).
[CrossRef]

J. Petter, C. Weilnau, C. Denz, A. Stepken, and F. Kaiser, Opt. Commun. 170, 291 (1999).
[CrossRef]

Opt. Express (1)

Opt. Lett. (4)

Phys. D (1)

B. A. Malomed, Phys. D 29155 (1987).
[CrossRef]

Phys. Rev. A (1)

V. E. Lobanov, O. V. Borovkova, Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Phys. Rev. A 85, 023804 (2012).
[CrossRef]

Phys. Rev. E (1)

W. Krolikowski, N. Akhmediev, B. Luther-Davies, and M. Cronin-Golomb, Phys. Rev. E 54, 5761 (1996).
[CrossRef]

Phys. Rev. Lett. (5)

W. van Saarloos and P. C. Hohenberg, Phys. Rev. Lett. 64, 749 (1990).
[CrossRef]

N. N. Akhmediev and V. V. Afanasjev, Phys. Rev. Lett. 75, 2320 (1995).
[CrossRef]

J. M. Soto-Crespo, N. Akhmediev, and A. Ankiewicz, Phys. Rev. Lett. 85, 2937 (2000).
[CrossRef]

V. Skarka, N. B. Aleksic, H. Leblond, B. A. Malomed, and D. Mihalache, Phys. Rev. Lett. 105, 213901 (2010).
[CrossRef]

E. A. Ultanir, G. I. Stegeman, D. Michaelis, C. H. Lange, and F. Lederer, Phys. Rev. Lett. 90, 253903 (2003).
[CrossRef]

Other (3)

Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, 2003).

N. N. Rosanov, Spatial Hysteresis and Optical Patterns (Springer, 2010).

N. Akhmediev and A. Ankiewicz, eds. Dissipative Solitons: From Optics to Biology and Medicine (Springer, 2008).

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Figures (4)

Fig. 1.
Fig. 1.

Propagation of a Gaussian beam in a medium with localized gain for a = 2 at (a) μ = 0.2 and (b)  μ = 0.3 .

Fig. 2.
Fig. 2.

(a)  U versus μ at a = 1.57 (curve 1), a = 1.6 (curve 2), and a = 1.62 (curve 3). Circles correspond to the fields shown in Figs. 3(a) and 3(b). (b)  U versus μ at a = 1.64 (curve 1) and a = 2 (curve 2). In (a) and (b) stable branches are shown black. (c)  δ r versus μ for upper δ r upp and lower δ r low soliton branches at a = 1.57 . (d) Domains of soliton existence and soliton stability in the plane ( a , μ ) . Solitons are unstable in the shaded area for μ < μ cr and stable in the white area for μ cr < μ < μ upp .

Fig. 3.
Fig. 3.

Profiles of solutions at μ = 0.5 , a = 1.62 from the (a) lower and (b) upper branches. (c) Field modulus distributions at μ = 0.2 . Dashed lines indicate the center of the amplifying channel.

Fig. 4.
Fig. 4.

(a)  U versus a at μ = 0.6 . (b)  δ r for the upper soliton branch versus a at μ = 0.2 (curve 1) and μ = 0.23 (curve 2).

Equations (1)

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i q ξ = 1 2 2 q η 2 q | q | 2 + μ q η | q | 2 + i γ ( η ) q .

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