Abstract

We show that the balance between localized gain and nonlinear cubic dissipation in the two-dimensional nonlinear Schrödinger equation allows for the existence of stable localized modes that we identify as solitons. Such modes exist only when the gain is strong enough and the energy flow exceeds certain threshold value. Above the critical value of the gain, symmetry breaking occurs and asymmetric dissipative solitons emerge.

© 2011 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2001).
  2. N. N. Rosanov, Spatial Hysteresis and Optical Patterns (Springer, 2002).
  3. M. Segev, Opt. Quantum Electron. 30, 503 (1998).
    [Crossref]
  4. B. A. Malomed, M. Gölles, I. M. Uzunov, and F. Lederer, Phys. Scr. 55, 73 (1997).
    [Crossref]
  5. C.-K. Lam, B. A. Malomed, K. W. Chow, and P. K. A. Wai, Eur. Phys. J. Spec. Top. 173, 233 (2009).
    [Crossref]
  6. M. O. Williams, C. W. McGrath, and J. N. Kutz, Opt. Express 18, 11671 (2010).
    [Crossref] [PubMed]
  7. Y. V. Kartashov, V. V. Konotop, V. A. Vysloukh, and L. Torner, Opt. Lett. 35, 1638 (2010).
    [Crossref] [PubMed]
  8. M. N. Ouarzazi, P. A. Bois, and M. Taki, Phys. Rev. A 53, 4408 (1996).
    [Crossref] [PubMed]
  9. A. Couairon and J.-M. Chomaz, Physica D 158, 129 (2001).
    [Crossref]
  10. H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, Phys. Rev. Lett. 81, 3383 (1998).
    [Crossref]
  11. M. J. Connelly, Semiconductor Optical Amplifiers (Springer, 2002).

2010 (2)

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]

2001 (1)

A. Couairon and J.-M. Chomaz, Physica D 158, 129 (2001).
[Crossref]

1998 (2)

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, Phys. Rev. Lett. 81, 3383 (1998).
[Crossref]

M. Segev, Opt. Quantum Electron. 30, 503 (1998).
[Crossref]

1997 (1)

B. A. Malomed, M. Gölles, I. M. Uzunov, and F. Lederer, Phys. Scr. 55, 73 (1997).
[Crossref]

1996 (1)

M. N. Ouarzazi, P. A. Bois, and M. Taki, Phys. Rev. A 53, 4408 (1996).
[Crossref] [PubMed]

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2001).

Aitchison, J. S.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, Phys. Rev. Lett. 81, 3383 (1998).
[Crossref]

Bois, P. A.

M. N. Ouarzazi, P. A. Bois, and M. Taki, Phys. Rev. A 53, 4408 (1996).
[Crossref] [PubMed]

Boyd, A. R.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, Phys. Rev. Lett. 81, 3383 (1998).
[Crossref]

Chomaz, J.-M.

A. Couairon and J.-M. Chomaz, Physica D 158, 129 (2001).
[Crossref]

Chow, K. W.

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

Connelly, M. J.

M. J. Connelly, Semiconductor Optical Amplifiers (Springer, 2002).

Couairon, A.

A. Couairon and J.-M. Chomaz, Physica D 158, 129 (2001).
[Crossref]

Eisenberg, H. S.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, Phys. Rev. Lett. 81, 3383 (1998).
[Crossref]

Gölles, M.

B. A. Malomed, M. Gölles, I. M. Uzunov, and F. Lederer, Phys. Scr. 55, 73 (1997).
[Crossref]

Kartashov, Y. V.

Konotop, V. V.

Kutz, J. N.

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]

Lederer, F.

B. A. Malomed, M. Gölles, I. M. Uzunov, and F. Lederer, Phys. Scr. 55, 73 (1997).
[Crossref]

Malomed, B. A.

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, M. Gölles, I. M. Uzunov, and F. Lederer, Phys. Scr. 55, 73 (1997).
[Crossref]

McGrath, C. W.

Morandotti, R.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, Phys. Rev. Lett. 81, 3383 (1998).
[Crossref]

Ouarzazi, M. N.

M. N. Ouarzazi, P. A. Bois, and M. Taki, Phys. Rev. A 53, 4408 (1996).
[Crossref] [PubMed]

Rosanov, N. N.

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

Segev, M.

M. Segev, Opt. Quantum Electron. 30, 503 (1998).
[Crossref]

Silberberg, Y.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, Phys. Rev. Lett. 81, 3383 (1998).
[Crossref]

Taki, M.

M. N. Ouarzazi, P. A. Bois, and M. Taki, Phys. Rev. A 53, 4408 (1996).
[Crossref] [PubMed]

Torner, L.

Uzunov, I. M.

B. A. Malomed, M. Gölles, I. M. Uzunov, and F. Lederer, Phys. Scr. 55, 73 (1997).
[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]

Williams, M. O.

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]

Opt. Express (1)

Opt. Lett. (1)

Opt. Quantum Electron. (1)

M. Segev, Opt. Quantum Electron. 30, 503 (1998).
[Crossref]

Phys. Rev. A (1)

M. N. Ouarzazi, P. A. Bois, and M. Taki, Phys. Rev. A 53, 4408 (1996).
[Crossref] [PubMed]

Phys. Rev. Lett. (1)

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, Phys. Rev. Lett. 81, 3383 (1998).
[Crossref]

Phys. Scr. (1)

B. A. Malomed, M. Gölles, I. M. Uzunov, and F. Lederer, Phys. Scr. 55, 73 (1997).
[Crossref]

Physica D (1)

A. Couairon and J.-M. Chomaz, Physica D 158, 129 (2001).
[Crossref]

Other (3)

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2001).

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

M. J. Connelly, Semiconductor Optical Amplifiers (Springer, 2002).

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 (5)

Fig. 1
Fig. 1

Profiles of radially symmetric dissipative solitons at (a) p i = 0.642 , α = 1.2 and (b) p i = 4 , α = 1.9 . Real and imaginary parts of the field and the field modulus are shown with curves labeled w r , w i , and u, respectively. In the both cases, d = 1.5 .

Fig. 2
Fig. 2

Energy flow of radially symmetric dissipative soliton (a), (b), its propagation constant (c), and the real part of perturbation growth rate (d) versus p i for different α and d = 1.5 . Cutoff and critical values of gain coefficient (e) versus α at d = 1.5 and (f) versus d at α = 1.5 .

Fig. 3
Fig. 3

Stable propagation of radially symmetric solitons at (a) p i = 0.8 , α = 1.2 and (b) p i = 4.2 , α = 1.9 and (c) transformation of unstable radially symmetric soliton into asymmetric stable state at p i = 2.0 , α = 1.2 . Field modulus distributions are shown at different distances. In all cases d = 1.5 .

Fig. 4
Fig. 4

(a), (b) Field modulus (top) and phase (bottom) distributions in the asymmetric solitons for different p i at α = 1.9 , d = 1.5 . (c) Initial and final distribution of the modulus for stable propagation of a perturbed asymmetric soliton at p i = 2.0 , α = 1.2 , d = 1.5 .

Fig. 5
Fig. 5

(a) U versus p i for asymmetric (as) and for radially symmetric (rs) solitons at α = 1.9 , d = 1.5 . (b) d η ζ versus p i for the family of asymmetric solitons illustrated in Figs. 4a, 4b.

Equations (3)

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

i q ξ = 1 2 2 q + i γ ( r ) q | q | 2 q i α | q | 2 q ,
2 b u + 2 u u v 2 + 2 u 3 = 0 ,
· ( u 2 v ) + 2 γ ( r ) u 2 2 α u 4 = 0.

Metrics