Abstract

We investigate families of optical solitons supported by two-dimensional disordered lattices. The linear modes of the system are identified, and their eigenvalues are arranged in a band structure. Introducing Kerr nonlinearity, we find that families of solitons originate from linear modes. Such solutions, depending on the eigenvalue of the supporting linear mode, as the power increases may become less localized/delocalized via resonant interactions with the modes of the linearized lattice. In addition, families of highly confined solitons exist in every waveguide of the lattice and for both signs of the nonlinearity.

© 2009 Optical Society of America

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  1. D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817 (2003).
    [CrossRef] [PubMed]
  2. F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008).
    [CrossRef]
  3. C. Kittel, Introduction to Solid State Physics (Wiley, 1986).
  4. P. Anderson, Phys. Rev. 109, 1492 (1958).
    [CrossRef]
  5. P. A. Lee and T. V. Ramakrishnan, Rev. Mod. Phys. 57, 287 (1985).
    [CrossRef]
  6. D. Belitz and T. R. Kirkpatrick, Rev. Mod. Phys. 66, 261 (1994).
    [CrossRef]
  7. J. D. Maynard, Rev. Mod. Phys. 73, 401 (2001).
    [CrossRef]
  8. P. Sheng, Introduction to Wave Scattering, Localization and Mesoscopic Phenomena (Springer, 2006).
  9. T. Pertsch, U. Peschel, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tünnermann, and F. Lederer, Phys. Rev. Lett. 93, 053901 (2004).
    [CrossRef] [PubMed]
  10. T. Schwartz, G. Bartal, S. Fishman, and M. Segev, Nature 446, 52 (2007).
    [CrossRef] [PubMed]
  11. Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, Phys. Rev. Lett. 100, 013906 (2008).
    [CrossRef] [PubMed]
  12. G. Kopidakis and S. Aubry, Phys. Rev. Lett. 84, 3236 (2000).
    [CrossRef] [PubMed]
  13. N. K. Efremidis and K. Hizanidis, Phys. Rev. Lett. 101, 143903 (2008).
    [CrossRef] [PubMed]
  14. N. K. Efremidis, J. Hudock, D. N. Christodoulides, J. W. Fleischer, O. Cohen, and M. Segev, Phys. Rev. Lett. 91, 213906 (2003).
    [CrossRef] [PubMed]
  15. N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, Phys. Rev. E 66, 046602 (2002).
    [CrossRef]
  16. J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Nature 422, 147 (2003).
    [CrossRef] [PubMed]
  17. T. Pertsch, U. Peschel, F. Lederer, J. Burghoff, M. Will, S. Nolte, and A. Tünnermann, Opt. Lett. 29, 468 (2004).
    [CrossRef] [PubMed]

2008 (3)

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

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, Phys. Rev. Lett. 100, 013906 (2008).
[CrossRef] [PubMed]

N. K. Efremidis and K. Hizanidis, Phys. Rev. Lett. 101, 143903 (2008).
[CrossRef] [PubMed]

2007 (1)

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, Nature 446, 52 (2007).
[CrossRef] [PubMed]

2004 (2)

T. Pertsch, U. Peschel, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tünnermann, and F. Lederer, Phys. Rev. Lett. 93, 053901 (2004).
[CrossRef] [PubMed]

T. Pertsch, U. Peschel, F. Lederer, J. Burghoff, M. Will, S. Nolte, and A. Tünnermann, Opt. Lett. 29, 468 (2004).
[CrossRef] [PubMed]

2003 (3)

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

N. K. Efremidis, J. Hudock, D. N. Christodoulides, J. W. Fleischer, O. Cohen, and M. Segev, Phys. Rev. Lett. 91, 213906 (2003).
[CrossRef] [PubMed]

D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817 (2003).
[CrossRef] [PubMed]

2002 (1)

N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, Phys. Rev. E 66, 046602 (2002).
[CrossRef]

2001 (1)

J. D. Maynard, Rev. Mod. Phys. 73, 401 (2001).
[CrossRef]

2000 (1)

G. Kopidakis and S. Aubry, Phys. Rev. Lett. 84, 3236 (2000).
[CrossRef] [PubMed]

1994 (1)

D. Belitz and T. R. Kirkpatrick, Rev. Mod. Phys. 66, 261 (1994).
[CrossRef]

1985 (1)

P. A. Lee and T. V. Ramakrishnan, Rev. Mod. Phys. 57, 287 (1985).
[CrossRef]

1958 (1)

P. Anderson, Phys. Rev. 109, 1492 (1958).
[CrossRef]

Anderson, P.

P. Anderson, Phys. Rev. 109, 1492 (1958).
[CrossRef]

Assanto, G.

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

Aubry, S.

G. Kopidakis and S. Aubry, Phys. Rev. Lett. 84, 3236 (2000).
[CrossRef] [PubMed]

Avidan, A.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, Phys. Rev. Lett. 100, 013906 (2008).
[CrossRef] [PubMed]

Bartal, G.

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, Nature 446, 52 (2007).
[CrossRef] [PubMed]

Bartelt, H.

T. Pertsch, U. Peschel, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tünnermann, and F. Lederer, Phys. Rev. Lett. 93, 053901 (2004).
[CrossRef] [PubMed]

Belitz, D.

D. Belitz and T. R. Kirkpatrick, Rev. Mod. Phys. 66, 261 (1994).
[CrossRef]

Burghoff, J.

Christodoulides, D. N.

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

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, Phys. Rev. Lett. 100, 013906 (2008).
[CrossRef] [PubMed]

D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817 (2003).
[CrossRef] [PubMed]

N. K. Efremidis, J. Hudock, D. N. Christodoulides, J. W. Fleischer, O. Cohen, and M. Segev, Phys. Rev. Lett. 91, 213906 (2003).
[CrossRef] [PubMed]

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

N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, Phys. Rev. E 66, 046602 (2002).
[CrossRef]

Cohen, O.

N. K. Efremidis, J. Hudock, D. N. Christodoulides, J. W. Fleischer, O. Cohen, and M. Segev, Phys. Rev. Lett. 91, 213906 (2003).
[CrossRef] [PubMed]

Efremidis, N. K.

N. K. Efremidis and K. Hizanidis, Phys. Rev. Lett. 101, 143903 (2008).
[CrossRef] [PubMed]

N. K. Efremidis, J. Hudock, D. N. Christodoulides, J. W. Fleischer, O. Cohen, and M. Segev, Phys. Rev. Lett. 91, 213906 (2003).
[CrossRef] [PubMed]

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

N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, Phys. Rev. E 66, 046602 (2002).
[CrossRef]

Fishman, S.

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, Nature 446, 52 (2007).
[CrossRef] [PubMed]

Fleischer, J. W.

N. K. Efremidis, J. Hudock, D. N. Christodoulides, J. W. Fleischer, O. Cohen, and M. Segev, Phys. Rev. Lett. 91, 213906 (2003).
[CrossRef] [PubMed]

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

N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, Phys. Rev. E 66, 046602 (2002).
[CrossRef]

Hizanidis, K.

N. K. Efremidis and K. Hizanidis, Phys. Rev. Lett. 101, 143903 (2008).
[CrossRef] [PubMed]

Hudock, J.

N. K. Efremidis, J. Hudock, D. N. Christodoulides, J. W. Fleischer, O. Cohen, and M. Segev, Phys. Rev. Lett. 91, 213906 (2003).
[CrossRef] [PubMed]

Kirkpatrick, T. R.

D. Belitz and T. R. Kirkpatrick, Rev. Mod. Phys. 66, 261 (1994).
[CrossRef]

Kittel, C.

C. Kittel, Introduction to Solid State Physics (Wiley, 1986).

Kobelke, J.

T. Pertsch, U. Peschel, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tünnermann, and F. Lederer, Phys. Rev. Lett. 93, 053901 (2004).
[CrossRef] [PubMed]

Kopidakis, G.

G. Kopidakis and S. Aubry, Phys. Rev. Lett. 84, 3236 (2000).
[CrossRef] [PubMed]

Lahini, Y.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, Phys. Rev. Lett. 100, 013906 (2008).
[CrossRef] [PubMed]

Lederer, F.

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

T. Pertsch, U. Peschel, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tünnermann, and F. Lederer, Phys. Rev. Lett. 93, 053901 (2004).
[CrossRef] [PubMed]

T. Pertsch, U. Peschel, F. Lederer, J. Burghoff, M. Will, S. Nolte, and A. Tünnermann, Opt. Lett. 29, 468 (2004).
[CrossRef] [PubMed]

D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817 (2003).
[CrossRef] [PubMed]

Lee, P. A.

P. A. Lee and T. V. Ramakrishnan, Rev. Mod. Phys. 57, 287 (1985).
[CrossRef]

Maynard, J. D.

J. D. Maynard, Rev. Mod. Phys. 73, 401 (2001).
[CrossRef]

Morandotti, R.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, Phys. Rev. Lett. 100, 013906 (2008).
[CrossRef] [PubMed]

Nolte, S.

T. Pertsch, U. Peschel, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tünnermann, and F. Lederer, Phys. Rev. Lett. 93, 053901 (2004).
[CrossRef] [PubMed]

T. Pertsch, U. Peschel, F. Lederer, J. Burghoff, M. Will, S. Nolte, and A. Tünnermann, Opt. Lett. 29, 468 (2004).
[CrossRef] [PubMed]

Pertsch, T.

T. Pertsch, U. Peschel, F. Lederer, J. Burghoff, M. Will, S. Nolte, and A. Tünnermann, Opt. Lett. 29, 468 (2004).
[CrossRef] [PubMed]

T. Pertsch, U. Peschel, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tünnermann, and F. Lederer, Phys. Rev. Lett. 93, 053901 (2004).
[CrossRef] [PubMed]

Peschel, U.

T. Pertsch, U. Peschel, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tünnermann, and F. Lederer, Phys. Rev. Lett. 93, 053901 (2004).
[CrossRef] [PubMed]

T. Pertsch, U. Peschel, F. Lederer, J. Burghoff, M. Will, S. Nolte, and A. Tünnermann, Opt. Lett. 29, 468 (2004).
[CrossRef] [PubMed]

Pozzi, F.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, Phys. Rev. Lett. 100, 013906 (2008).
[CrossRef] [PubMed]

Ramakrishnan, T. V.

P. A. Lee and T. V. Ramakrishnan, Rev. Mod. Phys. 57, 287 (1985).
[CrossRef]

Schuster, K.

T. Pertsch, U. Peschel, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tünnermann, and F. Lederer, Phys. Rev. Lett. 93, 053901 (2004).
[CrossRef] [PubMed]

Schwartz, T.

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, Nature 446, 52 (2007).
[CrossRef] [PubMed]

Sears, S.

N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, Phys. Rev. E 66, 046602 (2002).
[CrossRef]

Segev, M.

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

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, Nature 446, 52 (2007).
[CrossRef] [PubMed]

N. K. Efremidis, J. Hudock, D. N. Christodoulides, J. W. Fleischer, O. Cohen, and M. Segev, Phys. Rev. Lett. 91, 213906 (2003).
[CrossRef] [PubMed]

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

N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, Phys. Rev. E 66, 046602 (2002).
[CrossRef]

Sheng, P.

P. Sheng, Introduction to Wave Scattering, Localization and Mesoscopic Phenomena (Springer, 2006).

Silberberg, Y.

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

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, Phys. Rev. Lett. 100, 013906 (2008).
[CrossRef] [PubMed]

D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817 (2003).
[CrossRef] [PubMed]

Sorel, M.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, Phys. Rev. Lett. 100, 013906 (2008).
[CrossRef] [PubMed]

Stegeman, G. I.

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

Tünnermann, A.

T. Pertsch, U. Peschel, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tünnermann, and F. Lederer, Phys. Rev. Lett. 93, 053901 (2004).
[CrossRef] [PubMed]

T. Pertsch, U. Peschel, F. Lederer, J. Burghoff, M. Will, S. Nolte, and A. Tünnermann, Opt. Lett. 29, 468 (2004).
[CrossRef] [PubMed]

Will, M.

Nature (3)

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, Nature 446, 52 (2007).
[CrossRef] [PubMed]

D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817 (2003).
[CrossRef] [PubMed]

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

Opt. Lett. (1)

Phys. Rep. (1)

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

Phys. Rev. (1)

P. Anderson, Phys. Rev. 109, 1492 (1958).
[CrossRef]

Phys. Rev. E (1)

N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, Phys. Rev. E 66, 046602 (2002).
[CrossRef]

Phys. Rev. Lett. (5)

T. Pertsch, U. Peschel, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tünnermann, and F. Lederer, Phys. Rev. Lett. 93, 053901 (2004).
[CrossRef] [PubMed]

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, Phys. Rev. Lett. 100, 013906 (2008).
[CrossRef] [PubMed]

G. Kopidakis and S. Aubry, Phys. Rev. Lett. 84, 3236 (2000).
[CrossRef] [PubMed]

N. K. Efremidis and K. Hizanidis, Phys. Rev. Lett. 101, 143903 (2008).
[CrossRef] [PubMed]

N. K. Efremidis, J. Hudock, D. N. Christodoulides, J. W. Fleischer, O. Cohen, and M. Segev, Phys. Rev. Lett. 91, 213906 (2003).
[CrossRef] [PubMed]

Rev. Mod. Phys. (3)

P. A. Lee and T. V. Ramakrishnan, Rev. Mod. Phys. 57, 287 (1985).
[CrossRef]

D. Belitz and T. R. Kirkpatrick, Rev. Mod. Phys. 66, 261 (1994).
[CrossRef]

J. D. Maynard, Rev. Mod. Phys. 73, 401 (2001).
[CrossRef]

Other (2)

P. Sheng, Introduction to Wave Scattering, Localization and Mesoscopic Phenomena (Springer, 2006).

C. Kittel, Introduction to Solid State Physics (Wiley, 1986).

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Figures (5)

Fig. 1
Fig. 1

(a) Disordered index potential. (b) Eigenvalues of the linear modes (enumerated in increasing E order) forming a band structure. (c) Regions of bands (shaded, green online) and bandgaps (white) as a function of the disorder parameter r. (d)–(f) Linear modes 1, 38, and 64 with E 1 = 6.67 , E 38 = 8.69 , and E 64 = 9.88 , respectively.

Fig. 2
Fig. 2

Self-focusing DLS highly confined in a single waveguide of the lattice. (a) Typical P E curve and (b) corresponding field amplitude for E = 5 . (c) P E curves of all 64 families.

Fig. 3
Fig. 3

Defocusing family of DLS highly confined in a single waveguide of the lattice (a) P E curve. (b) and (c) Field amplitude for E = 10.6 , 13 .

Fig. 4
Fig. 4

Family of self-focusing DLS originating from linear mode 4. (a) P E curve, (b) linear mode with E 4 = 7.017 , and (c)–(f) DLS for E = 7.007 , 6.947 , 6.727 , 6.572 , respectively.

Fig. 5
Fig. 5

Self-focusing family of DLS originating from linear mode 64. (a) P E curve. (b) and (c) Nonlinear modes for E = 9.85 and 9.2 , respectively.

Equations (3)

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i ψ z + ψ x x + ψ y y + V ( x , y ) ψ + γ ψ 2 ψ = 0 ,
V p = ( V 0 2 ) ( cos 2 ( π x 2 ) + cos 2 ( π y 2 ) )
V d = 1 V m j 1 , j 2 = 1 10 α j 1 , j 2 sin ( ( 2 π j 1 x L ) + ϕ x , j 1 , j 2 ) sin ( ( 2 π j 2 y L ) + ϕ y , j 1 , j 2 )

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