Abstract

Fundamental and vortex solitons in a two-dimensional optically induced waveguide array are reported. In the strong localization regime the fundamental soliton is largely confined to one lattice site, whereas the vortex state comprises four fundamental modes superimposed in a square configuration with a phase structure that is topologically equivalent to the conventional vortex. However, in the weak localization regime, both the fundamental and the vortex solitons spread over many lattice sites. We further show that fundamental and the vortex solitons are stable against small perturbations in the strong localization regime.

© 2003 Optical Society of America

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  1. F. Lederer and Y. Silberberg, Opt. Photon. News 13(2), 48 (2002).
    [CrossRef]
  2. D. N. Christodoulides and R. J. Joseph, Opt. Lett. 13,794 (1988).
    [CrossRef] [PubMed]
  3. H. Eisenberg, Y. Silberberg, R. Morandotti, A. Boyd, and J. Aitchison, Phys. Rev. Lett. 81,3383 (1998).
    [CrossRef]
  4. N. Efremidis, S. Sears, D. N. Christodoulides, J. Fleischer, and M. Segev, Phys. Rev. E 66,046602 (2002).
    [CrossRef]
  5. J. Fleischer, T. Carmon, M. Segev, N. Efremidis, and D. N. Christodoulides, Phys. Rev. Lett. 90,023902 (2003).
    [CrossRef]
  6. J. Fleischer, M. Segev, N. Efremidis, and D. N. Christodoulides, Nature 422,147 (2003).
    [CrossRef] [PubMed]
  7. P. L. Kelley, Phys. Rev. Lett. 15,1005 (1965).
    [CrossRef]
  8. P. Xie, Z. Zhang, and Z. Zhang, Phys. Rev. E. 67,026607 (2003).
    [CrossRef]
  9. A. Ferrando, M. Zacares, P. Cordoba, D. Binosi, and J. A. Monsoriu, Opt. Express 11,452 (2003), http://www.opticsexpress.org.
    [CrossRef] [PubMed]
  10. B. B. Baizakov, B. A. Malomed, and M. Salerno, Europhys. Lett. 63,642 (2003).
    [CrossRef]
  11. N. G. Vakhitov and A. A. Kolokolov, Izy. Vyssh. Uchebn. Zaved Radiofiz. 16,1020 (1973).
  12. L. P. Pitaevskii, Phys. Lett. A 221, 14 (1996).
    [CrossRef]
  13. D. E. Pelinovsky, V. V. Afanasjev, and Yu. S. Kivshar, Phys. Rev. E 53,1940 (1996).
    [CrossRef]
  14. B. A. Malomed and P. G. Kevrekidis, Phys. Rev. E 64,026601 (2001).
    [CrossRef]

2003 (5)

P. Xie, Z. Zhang, and Z. Zhang, Phys. Rev. E. 67,026607 (2003).
[CrossRef]

A. Ferrando, M. Zacares, P. Cordoba, D. Binosi, and J. A. Monsoriu, Opt. Express 11,452 (2003), http://www.opticsexpress.org.
[CrossRef] [PubMed]

B. B. Baizakov, B. A. Malomed, and M. Salerno, Europhys. Lett. 63,642 (2003).
[CrossRef]

J. Fleischer, T. Carmon, M. Segev, N. Efremidis, and D. N. Christodoulides, Phys. Rev. Lett. 90,023902 (2003).
[CrossRef]

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

2002 (2)

F. Lederer and Y. Silberberg, Opt. Photon. News 13(2), 48 (2002).
[CrossRef]

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

2001 (1)

B. A. Malomed and P. G. Kevrekidis, Phys. Rev. E 64,026601 (2001).
[CrossRef]

1998 (1)

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

1996 (2)

L. P. Pitaevskii, Phys. Lett. A 221, 14 (1996).
[CrossRef]

D. E. Pelinovsky, V. V. Afanasjev, and Yu. S. Kivshar, Phys. Rev. E 53,1940 (1996).
[CrossRef]

1988 (1)

1973 (1)

N. G. Vakhitov and A. A. Kolokolov, Izy. Vyssh. Uchebn. Zaved Radiofiz. 16,1020 (1973).

1965 (1)

P. L. Kelley, Phys. Rev. Lett. 15,1005 (1965).
[CrossRef]

Afanasjev, V. V.

D. E. Pelinovsky, V. V. Afanasjev, and Yu. S. Kivshar, Phys. Rev. E 53,1940 (1996).
[CrossRef]

Aitchison, J.

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

Baizakov, B. B.

B. B. Baizakov, B. A. Malomed, and M. Salerno, Europhys. Lett. 63,642 (2003).
[CrossRef]

Binosi, D.

Boyd, A.

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

Carmon, T.

J. Fleischer, T. Carmon, M. Segev, N. Efremidis, and D. N. Christodoulides, Phys. Rev. Lett. 90,023902 (2003).
[CrossRef]

Christodoulides, D. N.

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

J. Fleischer, T. Carmon, M. Segev, N. Efremidis, and D. N. Christodoulides, Phys. Rev. Lett. 90,023902 (2003).
[CrossRef]

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

D. N. Christodoulides and R. J. Joseph, Opt. Lett. 13,794 (1988).
[CrossRef] [PubMed]

Cordoba, P.

Efremidis, N.

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

J. Fleischer, T. Carmon, M. Segev, N. Efremidis, and D. N. Christodoulides, Phys. Rev. Lett. 90,023902 (2003).
[CrossRef]

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

Eisenberg, H.

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

Ferrando, A.

Fleischer, J.

J. Fleischer, T. Carmon, M. Segev, N. Efremidis, and D. N. Christodoulides, Phys. Rev. Lett. 90,023902 (2003).
[CrossRef]

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

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

Joseph, R. J.

Kelley, P. L.

P. L. Kelley, Phys. Rev. Lett. 15,1005 (1965).
[CrossRef]

Kevrekidis, P. G.

B. A. Malomed and P. G. Kevrekidis, Phys. Rev. E 64,026601 (2001).
[CrossRef]

Kivshar, Yu. S.

D. E. Pelinovsky, V. V. Afanasjev, and Yu. S. Kivshar, Phys. Rev. E 53,1940 (1996).
[CrossRef]

Kolokolov, A. A.

N. G. Vakhitov and A. A. Kolokolov, Izy. Vyssh. Uchebn. Zaved Radiofiz. 16,1020 (1973).

Lederer, F.

F. Lederer and Y. Silberberg, Opt. Photon. News 13(2), 48 (2002).
[CrossRef]

Malomed, B. A.

B. B. Baizakov, B. A. Malomed, and M. Salerno, Europhys. Lett. 63,642 (2003).
[CrossRef]

B. A. Malomed and P. G. Kevrekidis, Phys. Rev. E 64,026601 (2001).
[CrossRef]

Monsoriu, J. A.

Morandotti, R.

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

Pelinovsky, D. E.

D. E. Pelinovsky, V. V. Afanasjev, and Yu. S. Kivshar, Phys. Rev. E 53,1940 (1996).
[CrossRef]

Pitaevskii, L. P.

L. P. Pitaevskii, Phys. Lett. A 221, 14 (1996).
[CrossRef]

Salerno, M.

B. B. Baizakov, B. A. Malomed, and M. Salerno, Europhys. Lett. 63,642 (2003).
[CrossRef]

Sears, S.

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

Segev, M.

J. Fleischer, T. Carmon, M. Segev, N. Efremidis, and D. N. Christodoulides, Phys. Rev. Lett. 90,023902 (2003).
[CrossRef]

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

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

Silberberg, Y.

F. Lederer and Y. Silberberg, Opt. Photon. News 13(2), 48 (2002).
[CrossRef]

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

Vakhitov, N. G.

N. G. Vakhitov and A. A. Kolokolov, Izy. Vyssh. Uchebn. Zaved Radiofiz. 16,1020 (1973).

Xie, P.

P. Xie, Z. Zhang, and Z. Zhang, Phys. Rev. E. 67,026607 (2003).
[CrossRef]

Zacares, M.

Zhang, Z.

P. Xie, Z. Zhang, and Z. Zhang, Phys. Rev. E. 67,026607 (2003).
[CrossRef]

P. Xie, Z. Zhang, and Z. Zhang, Phys. Rev. E. 67,026607 (2003).
[CrossRef]

Europhys. Lett. (1)

B. B. Baizakov, B. A. Malomed, and M. Salerno, Europhys. Lett. 63,642 (2003).
[CrossRef]

Izy. Vyssh. Uchebn. Zaved Radiofiz. (1)

N. G. Vakhitov and A. A. Kolokolov, Izy. Vyssh. Uchebn. Zaved Radiofiz. 16,1020 (1973).

Nature (1)

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

Opt. Express (1)

Opt. Lett. (1)

Opt. Photon. News (1)

F. Lederer and Y. Silberberg, Opt. Photon. News 13(2), 48 (2002).
[CrossRef]

Phys. Lett. A (1)

L. P. Pitaevskii, Phys. Lett. A 221, 14 (1996).
[CrossRef]

Phys. Rev. E (3)

D. E. Pelinovsky, V. V. Afanasjev, and Yu. S. Kivshar, Phys. Rev. E 53,1940 (1996).
[CrossRef]

B. A. Malomed and P. G. Kevrekidis, Phys. Rev. E 64,026601 (2001).
[CrossRef]

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

Phys. Rev. E. (1)

P. Xie, Z. Zhang, and Z. Zhang, Phys. Rev. E. 67,026607 (2003).
[CrossRef]

Phys. Rev. Lett. (3)

P. L. Kelley, Phys. Rev. Lett. 15,1005 (1965).
[CrossRef]

J. Fleischer, T. Carmon, M. Segev, N. Efremidis, and D. N. Christodoulides, Phys. Rev. Lett. 90,023902 (2003).
[CrossRef]

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

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Figures (4)

Fig. 1
Fig. 1

(a) Normalized power P (for V0=1, 1.5) and energy E (for V0=1) of fundamental solitons versus μ. (b) Unstable eigenvalues σ of these solitons. (c) and (d) Profiles of fundamental solitons at μ=0 and μ=0.88 V0=1, respectively.

Fig. 2
Fig. 2

Nonlinear evolution of fundamental solitons under perturbations [Eq. (4)] for (a) V0=1, ε=±0.01, up=1, and μ=0 and (b) μ=0.88. Snapshots of the soliton intensity corresponding to the dynamics depicted in (b) for (c) ε=0.01, z=80 and (d) ε=-0.01, z=60.

Fig. 3
Fig. 3

(a) Normalized power P (for V0=1, 1.5) and energy E (for V0=1) of vortex cells versus μ. (b), (d) Intensity plots of vortex cells with V0=1, μ=0 and μ=0.82, respectively. (c) Phase plot of the vortex cell in (b).

Fig. 4
Fig. 4

(a) Unstable eigenvalues of vortex cells for V0=1. (b)–(d) Instability development of the vortex with V0=1 and μ=0 [see Figs. 3(b) and 3(c)] when it is initially amplified by 1%; intensity plots at z values of 50, 51.5, and 52 are shown in (b), (c), and (d), respectively.

Equations (4)

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

iψz+2X2+2Y2ψ-Vψ+ψ2ψ=0,
E=--ψ2-12ψ4+Vψ2dXdY.
2uX2+2uY2-Vu+u2u=-μu.
ψX,Y,z=0=uX,Y1+εupX,Y,

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