Abstract

We show that periodic optical lattices imprinted in cubic nonlinear media with strong two-photon absorption and localized linear gain landscapes support stable dissipative defect modes in both focusing and defocusing media. Their shapes and transverse extent are determined by the propagation constant that belongs to a gap of the lattice spectrum, which, in turn, is determined by the relation between gain and losses. One-hump and two-hump dissipative defect modes are obtained.

© 2010 Optical Society of America

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  1. F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008).
    [CrossRef]
  2. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Phys. Rep. 52, 63 (2009).
  3. F. Fedele, J. Yang, and Z. Chen, Opt. Lett. 30, 1506 (2005).
    [CrossRef] [PubMed]
  4. I. Makasyuk, Z. Chen, and J. Yang, Phys. Rev. Lett. 96, 223903 (2006).
    [CrossRef] [PubMed]
  5. N. K. Efremidis and K. Hizanidis, Opt. Express 13, 10571 (2005).
    [CrossRef] [PubMed]
  6. F. Ye, Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Opt. Lett. 33, 1288 (2008).
    [CrossRef] [PubMed]
  7. V. A. Brazhnyi, V. V. Konotop, and V. M. Pérez-García, Phys. Rev. Lett. 96, 060403 (2006).
    [CrossRef] [PubMed]
  8. V. A. Brazhnyi, V. V. Konotop, and V. M. Pérez-García, Phys. Rev. A 74, 023614 (2006).
    [CrossRef]
  9. W. C. Mak, B. A. Malomed, and P. L. Chu, Phys. Rev. E 67, 026608 (2003).
    [CrossRef]
  10. I. V. Mel’nikov and J. S. Aitchison, Appl. Phys. Lett. 87, 201111 (2005).
    [CrossRef]
  11. C.-K. Lam, B. A. Malomed, K. W. Chow, and P. K. A. Wai, Eur. J. Phys. 173, 233 (2009).
  12. K. Maruno, A. Ankiewicz, and N. Akhmediev, Opt. Commun. 221, 199 (2003).
    [CrossRef]
  13. N. K. Efremidis and D. N. Christodoulides, Phys. Rev. E 67, 026606 (2003).
    [CrossRef]
  14. N.Akhmediev and A.Ankiewicz, eds., Dissipative Solitons, Vol. 661 of Lecture Notes in Physics (Springer-Verlag, 2005).
    [CrossRef]

2009

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Phys. Rep. 52, 63 (2009).

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

2008

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008).
[CrossRef]

F. Ye, Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Opt. Lett. 33, 1288 (2008).
[CrossRef] [PubMed]

2006

I. Makasyuk, Z. Chen, and J. Yang, Phys. Rev. Lett. 96, 223903 (2006).
[CrossRef] [PubMed]

V. A. Brazhnyi, V. V. Konotop, and V. M. Pérez-García, Phys. Rev. Lett. 96, 060403 (2006).
[CrossRef] [PubMed]

V. A. Brazhnyi, V. V. Konotop, and V. M. Pérez-García, Phys. Rev. A 74, 023614 (2006).
[CrossRef]

2005

2003

K. Maruno, A. Ankiewicz, and N. Akhmediev, Opt. Commun. 221, 199 (2003).
[CrossRef]

N. K. Efremidis and D. N. Christodoulides, Phys. Rev. E 67, 026606 (2003).
[CrossRef]

W. C. Mak, B. A. Malomed, and P. L. Chu, Phys. Rev. E 67, 026608 (2003).
[CrossRef]

Aitchison, J. S.

I. V. Mel’nikov and J. S. Aitchison, Appl. Phys. Lett. 87, 201111 (2005).
[CrossRef]

Akhmediev, N.

K. Maruno, A. Ankiewicz, and N. Akhmediev, Opt. Commun. 221, 199 (2003).
[CrossRef]

Ankiewicz, A.

K. Maruno, A. Ankiewicz, and N. Akhmediev, Opt. Commun. 221, 199 (2003).
[CrossRef]

Assanto, G.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008).
[CrossRef]

Brazhnyi, V. A.

V. A. Brazhnyi, V. V. Konotop, and V. M. Pérez-García, Phys. Rev. A 74, 023614 (2006).
[CrossRef]

V. A. Brazhnyi, V. V. Konotop, and V. M. Pérez-García, Phys. Rev. Lett. 96, 060403 (2006).
[CrossRef] [PubMed]

Chen, Z.

I. Makasyuk, Z. Chen, and J. Yang, Phys. Rev. Lett. 96, 223903 (2006).
[CrossRef] [PubMed]

F. Fedele, J. Yang, and Z. Chen, Opt. Lett. 30, 1506 (2005).
[CrossRef] [PubMed]

Chow, K. W.

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

Christodoulides, D. N.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008).
[CrossRef]

N. K. Efremidis and D. N. Christodoulides, Phys. Rev. E 67, 026606 (2003).
[CrossRef]

Chu, P. L.

W. C. Mak, B. A. Malomed, and P. L. Chu, Phys. Rev. E 67, 026608 (2003).
[CrossRef]

Efremidis, N. K.

N. K. Efremidis and K. Hizanidis, Opt. Express 13, 10571 (2005).
[CrossRef] [PubMed]

N. K. Efremidis and D. N. Christodoulides, Phys. Rev. E 67, 026606 (2003).
[CrossRef]

Fedele, F.

Hizanidis, K.

Kartashov, Y. V.

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Phys. Rep. 52, 63 (2009).

F. Ye, Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Opt. Lett. 33, 1288 (2008).
[CrossRef] [PubMed]

Konotop, V. V.

V. A. Brazhnyi, V. V. Konotop, and V. M. Pérez-García, Phys. Rev. A 74, 023614 (2006).
[CrossRef]

V. A. Brazhnyi, V. V. Konotop, and V. M. Pérez-García, Phys. Rev. Lett. 96, 060403 (2006).
[CrossRef] [PubMed]

Lam, C.-K.

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

Lederer, F.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008).
[CrossRef]

Mak, W. C.

W. C. Mak, B. A. Malomed, and P. L. Chu, Phys. Rev. E 67, 026608 (2003).
[CrossRef]

Makasyuk, I.

I. Makasyuk, Z. Chen, and J. Yang, Phys. Rev. Lett. 96, 223903 (2006).
[CrossRef] [PubMed]

Malomed, B. A.

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

W. C. Mak, B. A. Malomed, and P. L. Chu, Phys. Rev. E 67, 026608 (2003).
[CrossRef]

Maruno, K.

K. Maruno, A. Ankiewicz, and N. Akhmediev, Opt. Commun. 221, 199 (2003).
[CrossRef]

Mel’nikov, I. V.

I. V. Mel’nikov and J. S. Aitchison, Appl. Phys. Lett. 87, 201111 (2005).
[CrossRef]

Pérez-García, V. M.

V. A. Brazhnyi, V. V. Konotop, and V. M. Pérez-García, Phys. Rev. A 74, 023614 (2006).
[CrossRef]

V. A. Brazhnyi, V. V. Konotop, and V. M. Pérez-García, Phys. Rev. Lett. 96, 060403 (2006).
[CrossRef] [PubMed]

Segev, M.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008).
[CrossRef]

Silberberg, Y.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008).
[CrossRef]

Stegeman, G. I.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008).
[CrossRef]

Torner, L.

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Phys. Rep. 52, 63 (2009).

F. Ye, Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Opt. Lett. 33, 1288 (2008).
[CrossRef] [PubMed]

Vysloukh, V. A.

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Phys. Rep. 52, 63 (2009).

F. Ye, Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Opt. Lett. 33, 1288 (2008).
[CrossRef] [PubMed]

Wai, P. K. A.

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

Yang, J.

I. Makasyuk, Z. Chen, and J. Yang, Phys. Rev. Lett. 96, 223903 (2006).
[CrossRef] [PubMed]

F. Fedele, J. Yang, and Z. Chen, Opt. Lett. 30, 1506 (2005).
[CrossRef] [PubMed]

Ye, F.

Appl. Phys. Lett.

I. V. Mel’nikov and J. S. Aitchison, Appl. Phys. Lett. 87, 201111 (2005).
[CrossRef]

Eur. J. Phys.

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

Opt. Commun.

K. Maruno, A. Ankiewicz, and N. Akhmediev, Opt. Commun. 221, 199 (2003).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rep.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008).
[CrossRef]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Phys. Rep. 52, 63 (2009).

Phys. Rev. A

V. A. Brazhnyi, V. V. Konotop, and V. M. Pérez-García, Phys. Rev. A 74, 023614 (2006).
[CrossRef]

Phys. Rev. E

W. C. Mak, B. A. Malomed, and P. L. Chu, Phys. Rev. E 67, 026608 (2003).
[CrossRef]

N. K. Efremidis and D. N. Christodoulides, Phys. Rev. E 67, 026606 (2003).
[CrossRef]

Phys. Rev. Lett.

I. Makasyuk, Z. Chen, and J. Yang, Phys. Rev. Lett. 96, 223903 (2006).
[CrossRef] [PubMed]

V. A. Brazhnyi, V. V. Konotop, and V. M. Pérez-García, Phys. Rev. Lett. 96, 060403 (2006).
[CrossRef] [PubMed]

Other

N.Akhmediev and A.Ankiewicz, eds., Dissipative Solitons, Vol. 661 of Lecture Notes in Physics (Springer-Verlag, 2005).
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

(a) Band spectrum of R ( η ) . For p r = 5 the lower edge of the semi-infinite gap is at b = 2.875 , while the first finite gap corresponds to the interval ( 0.645 , 1.840 ) . Shadowed (empty) regions indicate the allowed bands (gaps). Dependence U versus p i for the one-hump (curve 1) and two-hump (curve 2) modes is shown in (b) for the focusing and in (c) for defocusing media. (d) Propagation constant versus gain for one-hump modes in focusing (curve 1) and defocusing (curve 2) media. Circles in panels (b)–(d) correspond to the modes shown in Fig. 2. In all the cases α = 1 .

Fig. 2
Fig. 2

Profiles of (a), (b), (d), (e) one-hump (one-channel gain) and (c), (f) two-hump (two-channels gain) modes in focusing (left column) and defocusing (right column) media. The humps in two-hump modes in focusing and defocusing media are out-of-phase and in-phase, respectively.

Fig. 3
Fig. 3

Domains of existence for stable one-hump and two-hump modes (boundaries denoted by circles and pentagons, respectively) in (a), (b) defocusing and (c) focusing media. In the defocusing medium either (a) one-hump or (b) two-hump modes exist between the curves 1 and 2. In the focusing medium, one-hump and two-hump modes exist above the curves 1 and 2, respectively.

Equations (2)

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i q ξ = 1 2 q η η R ( η ) q + i γ ( η ) q σ | q | 2 q i α | q | 2 q .
b w = w η η 2 j 2 2 w 3 + σ w 3 + R w , j η = 2 γ w 2 2 α w 4 .

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