Abstract

We study the dynamics of 2D gap solitons (GSs) in Bragg resonant nonlinear (photonic) gratings in the presence of localized defects. Previous work [Stud. Appl. Math. 115, 209 (2005) ] explains the mechanism of trapping the GS-carried energy at a defect via a resonant energy transfer from the GS into defect modes. We derive a finite-dimensional model that describes the evolution of the defect-trapped state as an interaction of linear defect modes and show that this model approximates the full dynamics very well in the regime when moderate amounts of GS energy are trapped.

© 2006 Optical Society of America

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  1. R.E.Slusher and B.J.Eggleton, eds., Nonlinear Photonic Crystals (Springer-Verlag, 2003).
  2. S. F. Mingaleev and Y. S. Kivshar, Phys. Rev. Lett. 86, 5474 (2001).
    [CrossRef] [PubMed]
  3. D. N. Christodoulides and R. I. Joseph, Opt. Lett. 13, 794 (1988).
    [CrossRef] [PubMed]
  4. A. B. Aceves and S. Wabnitz, Phys. Lett. A 141, 37 (1989).
    [CrossRef]
  5. B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, Phys. Rev. Lett. 76, 1627 (1996).
    [CrossRef] [PubMed]
  6. T. Dohnal and A. B. Aceves, Stud. Appl. Math. 115, 209 (2005).
    [CrossRef]
  7. N. Aközbek and S. John, Phys. Rev. E 57, 2287 (1998).
    [CrossRef]
  8. A. B. Aceves, G. Fibich, and B. Ilan, Physica D 189, 277 (2004).
    [CrossRef]
  9. R. Morandotti, H. S. Eisenberg, D. Mandelik, Y. Silberberg, D. Modotto, M. Sorel, C. R. Stanley, and J. S. Aitchinson, Opt. Lett. 28, 834 (2003).
    [CrossRef] [PubMed]
  10. R. H. Goodman, R. E. Slusher, and M. I. Weinstein, J. Opt. Soc. Am. B 19, 1635 (2002).
    [CrossRef]
  11. < f,g > = int omegag*fdxdz and ||f||2L2(omega = < f,f gt;. For numerical purposes, Ω is large enough to enclose the defect, the supports of its defect modes, as well as of the trapped pulse.
  12. G. Gregori and S. Wabnitz, Phys. Rev. Lett. 56, 600 (1986).
    [CrossRef] [PubMed]
  13. C. A. Kennedy and M. H. Carpenter, Technical Report NASA-2001-TM211038 (NASA Langley, 2001).
  14. T. Dohnal and T. Hagstrom, "Perfectly matched layers in photonics computations: 1D and 2D nonlinear coupled mode equations" (submitted to J. Comput. Phys.) .
  15. T. Hagstrom, "New results on absorbing layers and radiation boundary conditions," in Topics in Computational Wave Propagation, M.Ainsworth, P.Davies, D.Duncan, P.Martin, and B.Rynne, eds. (Springer-Verlag, 2003), Vol. 1.
    [CrossRef]
  16. J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Nature 422, 147 (2003).
    [CrossRef] [PubMed]

2005 (1)

T. Dohnal and A. B. Aceves, Stud. Appl. Math. 115, 209 (2005).
[CrossRef]

2004 (1)

A. B. Aceves, G. Fibich, and B. Ilan, Physica D 189, 277 (2004).
[CrossRef]

2003 (2)

2002 (1)

2001 (1)

S. F. Mingaleev and Y. S. Kivshar, Phys. Rev. Lett. 86, 5474 (2001).
[CrossRef] [PubMed]

1998 (1)

N. Aközbek and S. John, Phys. Rev. E 57, 2287 (1998).
[CrossRef]

1996 (1)

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, Phys. Rev. Lett. 76, 1627 (1996).
[CrossRef] [PubMed]

1989 (1)

A. B. Aceves and S. Wabnitz, Phys. Lett. A 141, 37 (1989).
[CrossRef]

1988 (1)

1986 (1)

G. Gregori and S. Wabnitz, Phys. Rev. Lett. 56, 600 (1986).
[CrossRef] [PubMed]

Aceves, A. B.

T. Dohnal and A. B. Aceves, Stud. Appl. Math. 115, 209 (2005).
[CrossRef]

A. B. Aceves, G. Fibich, and B. Ilan, Physica D 189, 277 (2004).
[CrossRef]

A. B. Aceves and S. Wabnitz, Phys. Lett. A 141, 37 (1989).
[CrossRef]

Aitchinson, J. S.

Aközbek, N.

N. Aközbek and S. John, Phys. Rev. E 57, 2287 (1998).
[CrossRef]

Carpenter, M. H.

C. A. Kennedy and M. H. Carpenter, Technical Report NASA-2001-TM211038 (NASA Langley, 2001).

Christodoulides, D. N.

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

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

de Sterke, C. M.

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, Phys. Rev. Lett. 76, 1627 (1996).
[CrossRef] [PubMed]

Dohnal, T.

T. Dohnal and A. B. Aceves, Stud. Appl. Math. 115, 209 (2005).
[CrossRef]

T. Dohnal and T. Hagstrom, "Perfectly matched layers in photonics computations: 1D and 2D nonlinear coupled mode equations" (submitted to J. Comput. Phys.) .

Efremidis, N. K.

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

Eggleton, B. J.

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, Phys. Rev. Lett. 76, 1627 (1996).
[CrossRef] [PubMed]

Eisenberg, H. S.

Fibich, G.

A. B. Aceves, G. Fibich, and B. Ilan, Physica D 189, 277 (2004).
[CrossRef]

Fleischer, J. W.

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

Goodman, R. H.

Gregori, G.

G. Gregori and S. Wabnitz, Phys. Rev. Lett. 56, 600 (1986).
[CrossRef] [PubMed]

Hagstrom, T.

T. Dohnal and T. Hagstrom, "Perfectly matched layers in photonics computations: 1D and 2D nonlinear coupled mode equations" (submitted to J. Comput. Phys.) .

T. Hagstrom, "New results on absorbing layers and radiation boundary conditions," in Topics in Computational Wave Propagation, M.Ainsworth, P.Davies, D.Duncan, P.Martin, and B.Rynne, eds. (Springer-Verlag, 2003), Vol. 1.
[CrossRef]

Ilan, B.

A. B. Aceves, G. Fibich, and B. Ilan, Physica D 189, 277 (2004).
[CrossRef]

John, S.

N. Aközbek and S. John, Phys. Rev. E 57, 2287 (1998).
[CrossRef]

Joseph, R. I.

Kennedy, C. A.

C. A. Kennedy and M. H. Carpenter, Technical Report NASA-2001-TM211038 (NASA Langley, 2001).

Kivshar, Y. S.

S. F. Mingaleev and Y. S. Kivshar, Phys. Rev. Lett. 86, 5474 (2001).
[CrossRef] [PubMed]

Krug, P. A.

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, Phys. Rev. Lett. 76, 1627 (1996).
[CrossRef] [PubMed]

Mandelik, D.

Mingaleev, S. F.

S. F. Mingaleev and Y. S. Kivshar, Phys. Rev. Lett. 86, 5474 (2001).
[CrossRef] [PubMed]

Modotto, D.

Morandotti, R.

Segev, M.

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

Silberberg, Y.

Sipe, J. E.

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, Phys. Rev. Lett. 76, 1627 (1996).
[CrossRef] [PubMed]

Slusher, R. E.

R. H. Goodman, R. E. Slusher, and M. I. Weinstein, J. Opt. Soc. Am. B 19, 1635 (2002).
[CrossRef]

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, Phys. Rev. Lett. 76, 1627 (1996).
[CrossRef] [PubMed]

Sorel, M.

Stanley, C. R.

Wabnitz, S.

A. B. Aceves and S. Wabnitz, Phys. Lett. A 141, 37 (1989).
[CrossRef]

G. Gregori and S. Wabnitz, Phys. Rev. Lett. 56, 600 (1986).
[CrossRef] [PubMed]

Weinstein, M. I.

J. Opt. Soc. Am. B (1)

Nature (1)

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

Opt. Lett. (2)

Phys. Lett. A (1)

A. B. Aceves and S. Wabnitz, Phys. Lett. A 141, 37 (1989).
[CrossRef]

Phys. Rev. E (1)

N. Aközbek and S. John, Phys. Rev. E 57, 2287 (1998).
[CrossRef]

Phys. Rev. Lett. (3)

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, Phys. Rev. Lett. 76, 1627 (1996).
[CrossRef] [PubMed]

G. Gregori and S. Wabnitz, Phys. Rev. Lett. 56, 600 (1986).
[CrossRef] [PubMed]

S. F. Mingaleev and Y. S. Kivshar, Phys. Rev. Lett. 86, 5474 (2001).
[CrossRef] [PubMed]

Physica D (1)

A. B. Aceves, G. Fibich, and B. Ilan, Physica D 189, 277 (2004).
[CrossRef]

Stud. Appl. Math. (1)

T. Dohnal and A. B. Aceves, Stud. Appl. Math. 115, 209 (2005).
[CrossRef]

Other (5)

< f,g > = int omegag*fdxdz and ||f||2L2(omega = < f,f gt;. For numerical purposes, Ω is large enough to enclose the defect, the supports of its defect modes, as well as of the trapped pulse.

C. A. Kennedy and M. H. Carpenter, Technical Report NASA-2001-TM211038 (NASA Langley, 2001).

T. Dohnal and T. Hagstrom, "Perfectly matched layers in photonics computations: 1D and 2D nonlinear coupled mode equations" (submitted to J. Comput. Phys.) .

T. Hagstrom, "New results on absorbing layers and radiation boundary conditions," in Topics in Computational Wave Propagation, M.Ainsworth, P.Davies, D.Duncan, P.Martin, and B.Rynne, eds. (Springer-Verlag, 2003), Vol. 1.
[CrossRef]

R.E.Slusher and B.J.Eggleton, eds., Nonlinear Photonic Crystals (Springer-Verlag, 2003).

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Figures (3)

Fig. 1
Fig. 1

Waveguide.

Fig. 2
Fig. 2

Example 1: Top and bottom rows ω 0 0.931 and 0.959, respectively. Column (a) E ( 0 , z , t ) ( E + qualitatively the same); (b) (—) N ( t ) , (--) N dm ( t ) ; (c) Eq. (1) vs. Eq. (3). Regular lines, A k ( t ) ; bold lines, a k ( t ) .

Fig. 3
Fig. 3

Example 2. Top and bottom rows, v = 0.2 and 0.6, respectively. Column (a) E ( 0 , z , t ) ( E + ); (b) (—) N ( t ) , (--) N dm ( t ) ; (c) Eq. (1) vs. Eq. (3), regular lines: A k ( t ) , bold lines: a k ( t ) .

Equations (4)

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

[ i t ± i z + x 2 + V ( x , z ) ] E ± + κ ( x , z ) E + N ± = 0 ,
[ E + ( x , z , t ) E ( x , z , t ) ] k = 1 M a k ( t ) e i η L k t [ ψ + k ( x , z ) ψ k ( x , z ) ] .
i a ̃ 1 Δ η a ̃ 1 + μ 1 a ̃ 1 2 a ̃ 1 + 2 ν a ̃ 2 2 a ̃ 1 + ν a ̃ 2 2 a ̃ 1 * = 0 ,
i a ̃ 2 + Δ η a ̃ 2 + μ 2 a ̃ 2 2 a ̃ 2 + 2 ν a ̃ 1 2 a ̃ 2 + ν a ̃ 1 2 a ̃ 2 * = 0 ,

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