Abstract

A novel method for obtaining analytical solitary wave solutions of the nonlinear Kronig–Penney model in periodic photonic structures with self-defocusing nonlinearity is applied for providing generic families of solutions corresponding to the gaps of the linear band structure. Characteristic cases are shown to be quite robust under propagation.

© 2006 Optical Society of America

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References

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  1. A. A. Sukhorukov, Y. S. Kivshar, H. S. Eisenberg, and Y. Silberberg, IEEE J. Quantum Electron. 39, 31 (2003).
    [CrossRef]
  2. J. W. Fleischer, G. Bartal, O. Cohen, T. Schwartz, O. Manela, B. Freedman, M. Segev, H. Buljan, and N. K. Efremidis, Opt. Express 13, 1780 (2005).
    [CrossRef] [PubMed]
  3. D. Mandelik, R. Morandotti, J. S. Aitchison, and Y. Silberberg, Phys. Rev. Lett. 92, 093904 (2004).
    [CrossRef] [PubMed]
  4. R. A. Vicencio, M. I. Molina, and Y. S. Kivshar, Opt. Lett. 28, 1942 (2003).
    [CrossRef] [PubMed]
  5. R. A. Vicencio, M. I. Molina, and Y. S. Kivshar, Opt. Lett. 29, 2905 (2004).
    [CrossRef]
  6. Y. V. Kartashov, L. Torner, and V. A. Vysloukh, Opt. Lett. 29, 1102 (2004).
    [CrossRef] [PubMed]
  7. F. Kh. Abdullaev, B. B. Baizakov, S. A. Darmanyan, V. V. Konotop, and M. Salerno, Phys. Rev. A 64, 043606 (2001).
    [CrossRef]
  8. G. L. Alfimov, V. V. Konotop, and M. Salerno, Europhys. Lett. 58, 7 (2002).
    [CrossRef]
  9. V. V. Konotop and M. Salerno, Phys. Rev. A 65, 021602(R) (2002).
    [CrossRef]
  10. N. K. Efremidis and D. N. Christodoulides, Phys. Rev. A 67, 063608 (2003).
    [CrossRef]
  11. P. J. Y. Louis, E. A. Ostrovskaya, and Y. S. Kivshar, J. Opt. B 6, S309 (2004).
    [CrossRef]
  12. Y. S. Kivshar, W. Krolikowsi, and O. A. Chubykalo, Phys. Rev. E 50, 5020 (1994).
    [CrossRef]
  13. M. Johansson and Y. S. Kivshar, Phys. Rev. Lett. 82, 85 (1999).
    [CrossRef]
  14. V. V. Konotop and S. Takeno, Phys. Rev. E 60, 1001 (1999).
    [CrossRef]
  15. B. Sanchez-Rey and M. Johansson, Phys. Rev. E 71, 036627 (2005).
    [CrossRef]
  16. H. Susanto and M. Johansson, Phys. Rev. E 72, 016605 (2005).
    [CrossRef]
  17. S. Theodorakis and E. Leontidis, J. Phys. A 30, 4835 (1997).
    [CrossRef]
  18. Y. Kominis, Phys. Rev. E 73, 066619 (2006).
    [CrossRef]
  19. C. K. R. T Jones, T. Kupper, and K. Schaffner, Z. Angew. Math. Phys. 52, 859 (2001).
    [CrossRef]
  20. R. K. Jackson and M. I. Weinstein, J. Stat. Phys. 116, 881 (2004).
    [CrossRef]
  21. P. J. Y. Louis, E. A. Ostrovskaya, C. M. Savage, and Y. S. Kivshar, Phys. Rev. A 67, 013602 (2003).
    [CrossRef]
  22. Y. S. Kivshar, D. E. Pelinovsky, T. Cretegny, M. Peyrard, Phys. Rev. Lett. 80, 5032 (1998).
    [CrossRef]
  23. A. A. Sukhorukov and Y. S. Kivshar, Phys. Rev. Lett. 87, 083901 (2001).
    [CrossRef] [PubMed]
  24. D. E. Pelinovsky, A. A. Sukhorukov, and Y. S. Kivshar, Phys. Rev. E 70, 036618 (2004).
    [CrossRef]
  25. J. Yang and Z. Chen, Phys. Rev. E 73, 026609 (2006).
    [CrossRef]

2006 (2)

Y. Kominis, Phys. Rev. E 73, 066619 (2006).
[CrossRef]

J. Yang and Z. Chen, Phys. Rev. E 73, 026609 (2006).
[CrossRef]

2005 (3)

J. W. Fleischer, G. Bartal, O. Cohen, T. Schwartz, O. Manela, B. Freedman, M. Segev, H. Buljan, and N. K. Efremidis, Opt. Express 13, 1780 (2005).
[CrossRef] [PubMed]

B. Sanchez-Rey and M. Johansson, Phys. Rev. E 71, 036627 (2005).
[CrossRef]

H. Susanto and M. Johansson, Phys. Rev. E 72, 016605 (2005).
[CrossRef]

2004 (6)

R. K. Jackson and M. I. Weinstein, J. Stat. Phys. 116, 881 (2004).
[CrossRef]

D. E. Pelinovsky, A. A. Sukhorukov, and Y. S. Kivshar, Phys. Rev. E 70, 036618 (2004).
[CrossRef]

P. J. Y. Louis, E. A. Ostrovskaya, and Y. S. Kivshar, J. Opt. B 6, S309 (2004).
[CrossRef]

Y. V. Kartashov, L. Torner, and V. A. Vysloukh, Opt. Lett. 29, 1102 (2004).
[CrossRef] [PubMed]

R. A. Vicencio, M. I. Molina, and Y. S. Kivshar, Opt. Lett. 29, 2905 (2004).
[CrossRef]

D. Mandelik, R. Morandotti, J. S. Aitchison, and Y. Silberberg, Phys. Rev. Lett. 92, 093904 (2004).
[CrossRef] [PubMed]

2003 (4)

R. A. Vicencio, M. I. Molina, and Y. S. Kivshar, Opt. Lett. 28, 1942 (2003).
[CrossRef] [PubMed]

A. A. Sukhorukov, Y. S. Kivshar, H. S. Eisenberg, and Y. Silberberg, IEEE J. Quantum Electron. 39, 31 (2003).
[CrossRef]

N. K. Efremidis and D. N. Christodoulides, Phys. Rev. A 67, 063608 (2003).
[CrossRef]

P. J. Y. Louis, E. A. Ostrovskaya, C. M. Savage, and Y. S. Kivshar, Phys. Rev. A 67, 013602 (2003).
[CrossRef]

2002 (2)

G. L. Alfimov, V. V. Konotop, and M. Salerno, Europhys. Lett. 58, 7 (2002).
[CrossRef]

V. V. Konotop and M. Salerno, Phys. Rev. A 65, 021602(R) (2002).
[CrossRef]

2001 (3)

F. Kh. Abdullaev, B. B. Baizakov, S. A. Darmanyan, V. V. Konotop, and M. Salerno, Phys. Rev. A 64, 043606 (2001).
[CrossRef]

A. A. Sukhorukov and Y. S. Kivshar, Phys. Rev. Lett. 87, 083901 (2001).
[CrossRef] [PubMed]

C. K. R. T Jones, T. Kupper, and K. Schaffner, Z. Angew. Math. Phys. 52, 859 (2001).
[CrossRef]

1999 (2)

M. Johansson and Y. S. Kivshar, Phys. Rev. Lett. 82, 85 (1999).
[CrossRef]

V. V. Konotop and S. Takeno, Phys. Rev. E 60, 1001 (1999).
[CrossRef]

1998 (1)

Y. S. Kivshar, D. E. Pelinovsky, T. Cretegny, M. Peyrard, Phys. Rev. Lett. 80, 5032 (1998).
[CrossRef]

1997 (1)

S. Theodorakis and E. Leontidis, J. Phys. A 30, 4835 (1997).
[CrossRef]

1994 (1)

Y. S. Kivshar, W. Krolikowsi, and O. A. Chubykalo, Phys. Rev. E 50, 5020 (1994).
[CrossRef]

Abdullaev, F. Kh.

F. Kh. Abdullaev, B. B. Baizakov, S. A. Darmanyan, V. V. Konotop, and M. Salerno, Phys. Rev. A 64, 043606 (2001).
[CrossRef]

Aitchison, J. S.

D. Mandelik, R. Morandotti, J. S. Aitchison, and Y. Silberberg, Phys. Rev. Lett. 92, 093904 (2004).
[CrossRef] [PubMed]

Alfimov, G. L.

G. L. Alfimov, V. V. Konotop, and M. Salerno, Europhys. Lett. 58, 7 (2002).
[CrossRef]

Baizakov, B. B.

F. Kh. Abdullaev, B. B. Baizakov, S. A. Darmanyan, V. V. Konotop, and M. Salerno, Phys. Rev. A 64, 043606 (2001).
[CrossRef]

Bartal, G.

Buljan, H.

Chen, Z.

J. Yang and Z. Chen, Phys. Rev. E 73, 026609 (2006).
[CrossRef]

Christodoulides, D. N.

N. K. Efremidis and D. N. Christodoulides, Phys. Rev. A 67, 063608 (2003).
[CrossRef]

Chubykalo, O. A.

Y. S. Kivshar, W. Krolikowsi, and O. A. Chubykalo, Phys. Rev. E 50, 5020 (1994).
[CrossRef]

Cohen, O.

Cretegny, T.

Y. S. Kivshar, D. E. Pelinovsky, T. Cretegny, M. Peyrard, Phys. Rev. Lett. 80, 5032 (1998).
[CrossRef]

Darmanyan, S. A.

F. Kh. Abdullaev, B. B. Baizakov, S. A. Darmanyan, V. V. Konotop, and M. Salerno, Phys. Rev. A 64, 043606 (2001).
[CrossRef]

Efremidis, N. K.

Eisenberg, H. S.

A. A. Sukhorukov, Y. S. Kivshar, H. S. Eisenberg, and Y. Silberberg, IEEE J. Quantum Electron. 39, 31 (2003).
[CrossRef]

Fleischer, J. W.

Freedman, B.

Jackson, R. K.

R. K. Jackson and M. I. Weinstein, J. Stat. Phys. 116, 881 (2004).
[CrossRef]

Johansson, M.

H. Susanto and M. Johansson, Phys. Rev. E 72, 016605 (2005).
[CrossRef]

B. Sanchez-Rey and M. Johansson, Phys. Rev. E 71, 036627 (2005).
[CrossRef]

M. Johansson and Y. S. Kivshar, Phys. Rev. Lett. 82, 85 (1999).
[CrossRef]

Jones, C. K. R. T

C. K. R. T Jones, T. Kupper, and K. Schaffner, Z. Angew. Math. Phys. 52, 859 (2001).
[CrossRef]

Kartashov, Y. V.

Kivshar, Y. S.

R. A. Vicencio, M. I. Molina, and Y. S. Kivshar, Opt. Lett. 29, 2905 (2004).
[CrossRef]

P. J. Y. Louis, E. A. Ostrovskaya, and Y. S. Kivshar, J. Opt. B 6, S309 (2004).
[CrossRef]

D. E. Pelinovsky, A. A. Sukhorukov, and Y. S. Kivshar, Phys. Rev. E 70, 036618 (2004).
[CrossRef]

A. A. Sukhorukov, Y. S. Kivshar, H. S. Eisenberg, and Y. Silberberg, IEEE J. Quantum Electron. 39, 31 (2003).
[CrossRef]

R. A. Vicencio, M. I. Molina, and Y. S. Kivshar, Opt. Lett. 28, 1942 (2003).
[CrossRef] [PubMed]

P. J. Y. Louis, E. A. Ostrovskaya, C. M. Savage, and Y. S. Kivshar, Phys. Rev. A 67, 013602 (2003).
[CrossRef]

A. A. Sukhorukov and Y. S. Kivshar, Phys. Rev. Lett. 87, 083901 (2001).
[CrossRef] [PubMed]

M. Johansson and Y. S. Kivshar, Phys. Rev. Lett. 82, 85 (1999).
[CrossRef]

Y. S. Kivshar, D. E. Pelinovsky, T. Cretegny, M. Peyrard, Phys. Rev. Lett. 80, 5032 (1998).
[CrossRef]

Y. S. Kivshar, W. Krolikowsi, and O. A. Chubykalo, Phys. Rev. E 50, 5020 (1994).
[CrossRef]

Kominis, Y.

Y. Kominis, Phys. Rev. E 73, 066619 (2006).
[CrossRef]

Konotop, V. V.

G. L. Alfimov, V. V. Konotop, and M. Salerno, Europhys. Lett. 58, 7 (2002).
[CrossRef]

V. V. Konotop and M. Salerno, Phys. Rev. A 65, 021602(R) (2002).
[CrossRef]

F. Kh. Abdullaev, B. B. Baizakov, S. A. Darmanyan, V. V. Konotop, and M. Salerno, Phys. Rev. A 64, 043606 (2001).
[CrossRef]

V. V. Konotop and S. Takeno, Phys. Rev. E 60, 1001 (1999).
[CrossRef]

Krolikowsi, W.

Y. S. Kivshar, W. Krolikowsi, and O. A. Chubykalo, Phys. Rev. E 50, 5020 (1994).
[CrossRef]

Kupper, T.

C. K. R. T Jones, T. Kupper, and K. Schaffner, Z. Angew. Math. Phys. 52, 859 (2001).
[CrossRef]

Leontidis, E.

S. Theodorakis and E. Leontidis, J. Phys. A 30, 4835 (1997).
[CrossRef]

Louis, P. J. Y.

P. J. Y. Louis, E. A. Ostrovskaya, and Y. S. Kivshar, J. Opt. B 6, S309 (2004).
[CrossRef]

P. J. Y. Louis, E. A. Ostrovskaya, C. M. Savage, and Y. S. Kivshar, Phys. Rev. A 67, 013602 (2003).
[CrossRef]

Mandelik, D.

D. Mandelik, R. Morandotti, J. S. Aitchison, and Y. Silberberg, Phys. Rev. Lett. 92, 093904 (2004).
[CrossRef] [PubMed]

Manela, O.

Molina, M. I.

Morandotti, R

D. Mandelik, R. Morandotti, J. S. Aitchison, and Y. Silberberg, Phys. Rev. Lett. 92, 093904 (2004).
[CrossRef] [PubMed]

Ostrovskaya, E. A.

P. J. Y. Louis, E. A. Ostrovskaya, and Y. S. Kivshar, J. Opt. B 6, S309 (2004).
[CrossRef]

P. J. Y. Louis, E. A. Ostrovskaya, C. M. Savage, and Y. S. Kivshar, Phys. Rev. A 67, 013602 (2003).
[CrossRef]

Pelinovsky, D. E.

D. E. Pelinovsky, A. A. Sukhorukov, and Y. S. Kivshar, Phys. Rev. E 70, 036618 (2004).
[CrossRef]

Y. S. Kivshar, D. E. Pelinovsky, T. Cretegny, M. Peyrard, Phys. Rev. Lett. 80, 5032 (1998).
[CrossRef]

Peyrard, M.

Y. S. Kivshar, D. E. Pelinovsky, T. Cretegny, M. Peyrard, Phys. Rev. Lett. 80, 5032 (1998).
[CrossRef]

Salerno, M.

G. L. Alfimov, V. V. Konotop, and M. Salerno, Europhys. Lett. 58, 7 (2002).
[CrossRef]

V. V. Konotop and M. Salerno, Phys. Rev. A 65, 021602(R) (2002).
[CrossRef]

F. Kh. Abdullaev, B. B. Baizakov, S. A. Darmanyan, V. V. Konotop, and M. Salerno, Phys. Rev. A 64, 043606 (2001).
[CrossRef]

Sanchez-Rey, B.

B. Sanchez-Rey and M. Johansson, Phys. Rev. E 71, 036627 (2005).
[CrossRef]

Savage, C. M.

P. J. Y. Louis, E. A. Ostrovskaya, C. M. Savage, and Y. S. Kivshar, Phys. Rev. A 67, 013602 (2003).
[CrossRef]

Schaffner, K.

C. K. R. T Jones, T. Kupper, and K. Schaffner, Z. Angew. Math. Phys. 52, 859 (2001).
[CrossRef]

Schwartz, T.

Segev, M.

Silberberg, Y.

D. Mandelik, R. Morandotti, J. S. Aitchison, and Y. Silberberg, Phys. Rev. Lett. 92, 093904 (2004).
[CrossRef] [PubMed]

A. A. Sukhorukov, Y. S. Kivshar, H. S. Eisenberg, and Y. Silberberg, IEEE J. Quantum Electron. 39, 31 (2003).
[CrossRef]

Sukhorukov, A. A.

D. E. Pelinovsky, A. A. Sukhorukov, and Y. S. Kivshar, Phys. Rev. E 70, 036618 (2004).
[CrossRef]

A. A. Sukhorukov, Y. S. Kivshar, H. S. Eisenberg, and Y. Silberberg, IEEE J. Quantum Electron. 39, 31 (2003).
[CrossRef]

A. A. Sukhorukov and Y. S. Kivshar, Phys. Rev. Lett. 87, 083901 (2001).
[CrossRef] [PubMed]

Susanto, H.

H. Susanto and M. Johansson, Phys. Rev. E 72, 016605 (2005).
[CrossRef]

Takeno, S.

V. V. Konotop and S. Takeno, Phys. Rev. E 60, 1001 (1999).
[CrossRef]

Theodorakis, S.

S. Theodorakis and E. Leontidis, J. Phys. A 30, 4835 (1997).
[CrossRef]

Torner, L.

Vicencio, R. A.

Vysloukh, V. A.

Weinstein, M. I.

R. K. Jackson and M. I. Weinstein, J. Stat. Phys. 116, 881 (2004).
[CrossRef]

Yang, J.

J. Yang and Z. Chen, Phys. Rev. E 73, 026609 (2006).
[CrossRef]

Europhys. Lett. (1)

G. L. Alfimov, V. V. Konotop, and M. Salerno, Europhys. Lett. 58, 7 (2002).
[CrossRef]

IEEE J. Quantum Electron. (1)

A. A. Sukhorukov, Y. S. Kivshar, H. S. Eisenberg, and Y. Silberberg, IEEE J. Quantum Electron. 39, 31 (2003).
[CrossRef]

J. Opt. B (1)

P. J. Y. Louis, E. A. Ostrovskaya, and Y. S. Kivshar, J. Opt. B 6, S309 (2004).
[CrossRef]

J. Phys. A (1)

S. Theodorakis and E. Leontidis, J. Phys. A 30, 4835 (1997).
[CrossRef]

J. Stat. Phys. (1)

R. K. Jackson and M. I. Weinstein, J. Stat. Phys. 116, 881 (2004).
[CrossRef]

Opt. Express (1)

Opt. Lett. (3)

Phys. Rev. A (4)

F. Kh. Abdullaev, B. B. Baizakov, S. A. Darmanyan, V. V. Konotop, and M. Salerno, Phys. Rev. A 64, 043606 (2001).
[CrossRef]

P. J. Y. Louis, E. A. Ostrovskaya, C. M. Savage, and Y. S. Kivshar, Phys. Rev. A 67, 013602 (2003).
[CrossRef]

V. V. Konotop and M. Salerno, Phys. Rev. A 65, 021602(R) (2002).
[CrossRef]

N. K. Efremidis and D. N. Christodoulides, Phys. Rev. A 67, 063608 (2003).
[CrossRef]

Phys. Rev. E (7)

Y. S. Kivshar, W. Krolikowsi, and O. A. Chubykalo, Phys. Rev. E 50, 5020 (1994).
[CrossRef]

Y. Kominis, Phys. Rev. E 73, 066619 (2006).
[CrossRef]

V. V. Konotop and S. Takeno, Phys. Rev. E 60, 1001 (1999).
[CrossRef]

B. Sanchez-Rey and M. Johansson, Phys. Rev. E 71, 036627 (2005).
[CrossRef]

H. Susanto and M. Johansson, Phys. Rev. E 72, 016605 (2005).
[CrossRef]

D. E. Pelinovsky, A. A. Sukhorukov, and Y. S. Kivshar, Phys. Rev. E 70, 036618 (2004).
[CrossRef]

J. Yang and Z. Chen, Phys. Rev. E 73, 026609 (2006).
[CrossRef]

Phys. Rev. Lett. (4)

D. Mandelik, R. Morandotti, J. S. Aitchison, and Y. Silberberg, Phys. Rev. Lett. 92, 093904 (2004).
[CrossRef] [PubMed]

Y. S. Kivshar, D. E. Pelinovsky, T. Cretegny, M. Peyrard, Phys. Rev. Lett. 80, 5032 (1998).
[CrossRef]

A. A. Sukhorukov and Y. S. Kivshar, Phys. Rev. Lett. 87, 083901 (2001).
[CrossRef] [PubMed]

M. Johansson and Y. S. Kivshar, Phys. Rev. Lett. 82, 85 (1999).
[CrossRef]

Z. Angew. Math. Phys. (1)

C. K. R. T Jones, T. Kupper, and K. Schaffner, Z. Angew. Math. Phys. 52, 859 (2001).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Geometry of the periodic structure consisting of linear ( ϵ L ) and nonlinear ( ϵ N ) layers.

Fig. 2
Fig. 2

Phase space representation of the stationary solutions (thick curve) for the case of n, odd (left); and n, even (right). The phase space of the linear system (elliptic curves) and the heteroclinic orbit connecting the saddle points ( x ) of the nonlinear system, are shown.

Fig. 3
Fig. 3

Field profiles of the analytically obtained stationary solutions for a periodic structure with L = 4 π , N = π , and ϵ N = 0 . (left column) ϵ L = 0.1 , n = 2 ; (middle column), ϵ L = 0.5 , n = 1 ; and (right column) ϵ L = 0.5 , n = 2 . Top and bottom rows, respectively, correspond to x 0 = 0 , N 2 .

Fig. 4
Fig. 4

(Top row) Contrast versus x 0 , and (bottom row) band structure (propagation constant β versus Bloch wavenumber q) and location of the propagation constants β n of the analytically obtained stationary solutions (x: no analytical solution exists). The parameters are (left column) L = 4 π , N = π , ϵ N = 0 , and ϵ L = 0.1 ; (right column) ϵ L = 0.5 .

Fig. 5
Fig. 5

Propagation of the solitary waves corresponding to Fig. 3 (top row, left column) and Fig. 3 (bottom row, middle column) from left to right, respectively.

Equations (5)

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

i ψ z + 2 ψ x 2 + ϵ ( x ) ψ + g ( x , ψ 2 ) ψ = 0 ,
d 2 u d x 2 + [ ϵ ( x ) β ] u + g ( x , u 2 ) u = 0 ,
d 2 u d x 2 + ( ϵ N β ) u 2 u 3 = 0 , x U N ,
d 2 u d x 2 + ( ϵ L β ) u = 0 , x U L .
β n = ϵ L ( n π L ) 2 , n = 1 , 2 , ,

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