Abstract

We demonstrate that both the linear (diffraction) and the nonlinear dynamics of two-dimensional waveguide arrays are considerably more complex and versatile than their one-dimensional counterparts. The discrete diffraction properties of these arrays can be effectively altered, depending on the propagation Bloch k-vector within the first Brillouin zone of the lattice. In general, this diffraction behavior is anisotropic and therefore permits the existence of a new class of discrete elliptic solitons in the nonlinear regime.

© 2004 Optical Society of America

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  1. D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817 (2002).
    [Crossref]
  2. D. N. Christodoulides and R. I. Joseph, Opt. Lett. 13, 794 (1988).
    [Crossref] [PubMed]
  3. Y. S. Kivshar, Opt. Lett. 18, 1147 (1993).
    [Crossref]
  4. H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, Phys. Rev. Lett. 81, 3383 (1998).
    [Crossref]
  5. H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 85, 1863 (2000).
    [Crossref] [PubMed]
  6. R. Morandotti, H. S. Eisenberg, Y. Silberberg, M. Sorel, and J. S. Aitchison, Phys. Rev. Lett. 86, 3296 (2001).
    [Crossref] [PubMed]
  7. T. Pertsch, U. Peschel, F. Lederer, J. Meier, R. Schick, R. Iwanow, G. Stegeman, Y. H. Min, and W. Sohler, Nonlinear Guided Waves, Vol. 80 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2002), paper NLTuA1.
  8. N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, Phys. Rev. E 66, 046602 (2002).
    [Crossref]
  9. J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Nature 422, 147 (2003).
    [Crossref] [PubMed]
  10. D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 90, 053902 (2003).
    [Crossref]
  11. J. Meier, J. Hudock, D. Christodoulides, G. Stegemen, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 91, 143907 (2003).
    [Crossref]
  12. D. N. Christodoulides and E. D. Eugenieva, Phys. Rev. Lett. 87, 233901 (2001).
    [Crossref]

2003 (3)

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

D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 90, 053902 (2003).
[Crossref]

J. Meier, J. Hudock, D. Christodoulides, G. Stegemen, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 91, 143907 (2003).
[Crossref]

2002 (2)

D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817 (2002).
[Crossref]

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

2001 (2)

R. Morandotti, H. S. Eisenberg, Y. Silberberg, M. Sorel, and J. S. Aitchison, Phys. Rev. Lett. 86, 3296 (2001).
[Crossref] [PubMed]

D. N. Christodoulides and E. D. Eugenieva, Phys. Rev. Lett. 87, 233901 (2001).
[Crossref]

2000 (1)

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 85, 1863 (2000).
[Crossref] [PubMed]

1998 (1)

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

1993 (1)

1988 (1)

Aitchison, J. S.

D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 90, 053902 (2003).
[Crossref]

J. Meier, J. Hudock, D. Christodoulides, G. Stegemen, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 91, 143907 (2003).
[Crossref]

R. Morandotti, H. S. Eisenberg, Y. Silberberg, M. Sorel, and J. S. Aitchison, Phys. Rev. Lett. 86, 3296 (2001).
[Crossref] [PubMed]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 85, 1863 (2000).
[Crossref] [PubMed]

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

Boyd, A. R.

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

Christodoulides, D.

J. Meier, J. Hudock, D. Christodoulides, G. Stegemen, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 91, 143907 (2003).
[Crossref]

Christodoulides, D. N.

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]

D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817 (2002).
[Crossref]

D. N. Christodoulides and E. D. Eugenieva, Phys. Rev. Lett. 87, 233901 (2001).
[Crossref]

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

Efremidis, N. K.

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]

Eisenberg, H. S.

D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 90, 053902 (2003).
[Crossref]

R. Morandotti, H. S. Eisenberg, Y. Silberberg, M. Sorel, and J. S. Aitchison, Phys. Rev. Lett. 86, 3296 (2001).
[Crossref] [PubMed]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 85, 1863 (2000).
[Crossref] [PubMed]

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

Eugenieva, E. D.

D. N. Christodoulides and E. D. Eugenieva, Phys. Rev. Lett. 87, 233901 (2001).
[Crossref]

Fleischer, J. W.

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]

Hudock, J.

J. Meier, J. Hudock, D. Christodoulides, G. Stegemen, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 91, 143907 (2003).
[Crossref]

Iwanow, R.

T. Pertsch, U. Peschel, F. Lederer, J. Meier, R. Schick, R. Iwanow, G. Stegeman, Y. H. Min, and W. Sohler, Nonlinear Guided Waves, Vol. 80 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2002), paper NLTuA1.

Joseph, R. I.

Kivshar, Y. S.

Lederer, F.

D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817 (2002).
[Crossref]

T. Pertsch, U. Peschel, F. Lederer, J. Meier, R. Schick, R. Iwanow, G. Stegeman, Y. H. Min, and W. Sohler, Nonlinear Guided Waves, Vol. 80 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2002), paper NLTuA1.

Mandelik, D.

D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 90, 053902 (2003).
[Crossref]

Meier, J.

J. Meier, J. Hudock, D. Christodoulides, G. Stegemen, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 91, 143907 (2003).
[Crossref]

T. Pertsch, U. Peschel, F. Lederer, J. Meier, R. Schick, R. Iwanow, G. Stegeman, Y. H. Min, and W. Sohler, Nonlinear Guided Waves, Vol. 80 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2002), paper NLTuA1.

Min, Y. H.

T. Pertsch, U. Peschel, F. Lederer, J. Meier, R. Schick, R. Iwanow, G. Stegeman, Y. H. Min, and W. Sohler, Nonlinear Guided Waves, Vol. 80 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2002), paper NLTuA1.

Morandotti, R.

D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 90, 053902 (2003).
[Crossref]

J. Meier, J. Hudock, D. Christodoulides, G. Stegemen, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 91, 143907 (2003).
[Crossref]

R. Morandotti, H. S. Eisenberg, Y. Silberberg, M. Sorel, and J. S. Aitchison, Phys. Rev. Lett. 86, 3296 (2001).
[Crossref] [PubMed]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 85, 1863 (2000).
[Crossref] [PubMed]

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

Pertsch, T.

T. Pertsch, U. Peschel, F. Lederer, J. Meier, R. Schick, R. Iwanow, G. Stegeman, Y. H. Min, and W. Sohler, Nonlinear Guided Waves, Vol. 80 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2002), paper NLTuA1.

Peschel, U.

T. Pertsch, U. Peschel, F. Lederer, J. Meier, R. Schick, R. Iwanow, G. Stegeman, Y. H. Min, and W. Sohler, Nonlinear Guided Waves, Vol. 80 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2002), paper NLTuA1.

Schick, R.

T. Pertsch, U. Peschel, F. Lederer, J. Meier, R. Schick, R. Iwanow, G. Stegeman, Y. H. Min, and W. Sohler, Nonlinear Guided Waves, Vol. 80 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2002), paper NLTuA1.

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.

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]

Silberberg, Y.

D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 90, 053902 (2003).
[Crossref]

J. Meier, J. Hudock, D. Christodoulides, G. Stegemen, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 91, 143907 (2003).
[Crossref]

D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817 (2002).
[Crossref]

R. Morandotti, H. S. Eisenberg, Y. Silberberg, M. Sorel, and J. S. Aitchison, Phys. Rev. Lett. 86, 3296 (2001).
[Crossref] [PubMed]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 85, 1863 (2000).
[Crossref] [PubMed]

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

Sohler, W.

T. Pertsch, U. Peschel, F. Lederer, J. Meier, R. Schick, R. Iwanow, G. Stegeman, Y. H. Min, and W. Sohler, Nonlinear Guided Waves, Vol. 80 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2002), paper NLTuA1.

Sorel, M.

R. Morandotti, H. S. Eisenberg, Y. Silberberg, M. Sorel, and J. S. Aitchison, Phys. Rev. Lett. 86, 3296 (2001).
[Crossref] [PubMed]

Stegeman, G.

T. Pertsch, U. Peschel, F. Lederer, J. Meier, R. Schick, R. Iwanow, G. Stegeman, Y. H. Min, and W. Sohler, Nonlinear Guided Waves, Vol. 80 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2002), paper NLTuA1.

Stegemen, G.

J. Meier, J. Hudock, D. Christodoulides, G. Stegemen, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 91, 143907 (2003).
[Crossref]

Nature (2)

D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817 (2002).
[Crossref]

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

Opt. Lett. (2)

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. (6)

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

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 85, 1863 (2000).
[Crossref] [PubMed]

R. Morandotti, H. S. Eisenberg, Y. Silberberg, M. Sorel, and J. S. Aitchison, Phys. Rev. Lett. 86, 3296 (2001).
[Crossref] [PubMed]

D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 90, 053902 (2003).
[Crossref]

J. Meier, J. Hudock, D. Christodoulides, G. Stegemen, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 91, 143907 (2003).
[Crossref]

D. N. Christodoulides and E. D. Eugenieva, Phys. Rev. Lett. 87, 233901 (2001).
[Crossref]

Other (1)

T. Pertsch, U. Peschel, F. Lederer, J. Meier, R. Schick, R. Iwanow, G. Stegeman, Y. H. Min, and W. Sohler, Nonlinear Guided Waves, Vol. 80 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2002), paper NLTuA1.

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Figures (3)

Fig. 1
Fig. 1

(a) Schematic drawing of a square 2D waveguide array. (b) The first three bands of Eq. (1) as obtained from numerical computation. (c) Contour plot of the dispersion relation obtained from Eq. (3).

Fig. 2
Fig. 2

Discrete 2D Gaussian beam injected with p=π/2 and q=0. (a) at z=0 mm and (b) at z=8.39 mm. Note that the anisotropic diffraction considerably elongates the beam in the y direction.

Fig. 3
Fig. 3

(a) Discrete elliptic beam launched at p=0.8 and q=0. (b) In the linear regime the beam diffracts considerably. (c) When nonlinearity is present, a discrete elliptic soliton is formed.

Equations (6)

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

iϕzˆ+12k2ϕ+kδn0fX,Yϕ+kn2n0ϕ2ϕs2ϕs2+ϕ2=0,
iEm,nz+Em-1,n+Em+1,n+Em,n-1+Em,n+1+rEm-1,n-1+Em-1,n+1+Em+1,n-1+Em+1,n+1+ρEm,n21+Em,n2Em,n=0,
kz=2 cosp+2 cosq+4r cospcosq.
iΦz+vxΦx+vyΦy+αΦxx+βΦyy+γΦxy+Φ21+Φ2Φ=0,
tan2θ=4rsinpsinqcosq-cosp.
iΨz+ϵΨuu+µΨvv+Ψ21+Ψ2Ψ=0,

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