Abstract

Propagation of a spatial soliton in a dissipative modulated Bessel optical lattice is investigated, both analytically and numerically. The dynamic evolution equations for beam width, amplitude, and curvature wavefront are obtained by a variational approach. It is shown that by properly increasing the modulation depth of refractive index of the optical lattice, the loss effect can be compensated exactly to fulfill a stable spatial soliton propagation.

© 2007 Optical Society of America

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  1. A. S. Davydov, "The theory contraction of proteins under their excitation," J. Theor. Biol. 38, 559-569 (1973).
    [CrossRef] [PubMed]
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    [CrossRef]
  3. D. N. Christodes and R. I. Joseph, "Discrete self-focusing in nonlinear arrays of coupled waveguides," Opt. Lett. 13, 794-796 (1988).
    [CrossRef]
  4. A. Trombettoni and A. Smerzi, "Discrete solitons and breathers with dilute Bose-Einstin condensates," Phys. Rev. Lett. 86, 2353-2356 (2001).
    [CrossRef] [PubMed]
  5. P. Pedri, L. Santos, P. Öhberg, and S. Stringari, "Violation of self-similarity in the expansion of a one-dimensional Bose gas," Phys. Rev. A 68, 043601 (2003).
    [CrossRef]
  6. J.-P. Martikainen and H. T. C. Stoof, "Excitations of a Bose-Einstein condensate in a one-dimensional optical lattice," Phys. Rev. A 68, 013610 (2003).
    [CrossRef]
  7. R. Scharf and A. R. Bishop, "Length-scale competition for the one-dimensional nonlinear Schrödinger equation with spatially periodic potentials," Phys. Rev. E 47, 1375-1383 (1993).
    [CrossRef]
  8. D. Mihalache, D. Mazilu, F. Lederer, B. A. Malomed, Y. V. Kartashov, L.-C. Crasovan, and L. Torner, "Stable spatiotemporal solitons in Bessel optical lattices," Phys. Rev. Lett. 95, 023902 (2005).
    [CrossRef] [PubMed]
  9. N. K. Efremidis and D. N. Christodoulides, "Lattice solitons in Bose-Einstein condensates," Phys. Rev. A 67, 063608 (2003).
    [CrossRef]
  10. M. B. Dahan, E. Peik, J. Reichel, Y. Castin, and C. Salomon, "Bloch oscillations of atoms in an optical potential," Phys. Rev. Lett. 76, 4508-4511 (1996).
    [CrossRef] [PubMed]
  11. B. P. Anderson and M. A. Kasevich, "Macroscopic quantum interference from atomic tunnel arrays," Science 282, 1686-1689 (1998).
    [CrossRef] [PubMed]
  12. H. Buljan, O. Cohen, J. W. Fleischer, T. Schwartz, M. Segev, Z. H. Musslimani, N. K. Efremidis, and D. N. Christodoulides, "Random-phase solitons in nonlinear periodic lattices," Phys. Rev. Lett. 92, 223901 (2004).
    [CrossRef] [PubMed]
  13. L. Pricoupenko, "Variational approach for the two-dimensional trapped Bose-Einstein condensate," Phys. Rev. A 70, 013601 (2004).
    [CrossRef]
  14. D. E. Pelinovsky, A. A. Sukhorukov, and Y. S. Kivshar, "Bifurcations and stability of gap solitons in periodic potentials," Phys. Rev. E 70, 036618 (2004).
    [CrossRef]
  15. D. N. Christodoulides, F. Lederer, and Y. Silberberg, "Discretizing light behaviour in linear and nonlinear waveguide lattices," Nature 424, 817-823 (2003).
    [CrossRef] [PubMed]
  16. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, "Soliton control in chirped photonic lattices," J. Opt. Soc. Am. B 22, 1356-1359 (2005).
    [CrossRef]
  17. Y. V. Kartashov, A. S. Zelenina, L. Torner, and V. A. Vysloukh, "Spatial soliton switching in quasi-continuous optical arrays," Opt. Lett. 29, 766-768 (2004).
    [CrossRef] [PubMed]
  18. Z. Xu, Y. V. Kartashov, and L. Torner, "Soliton mobility in nonlocal optical lattices," Phys. Rev. Lett. 95, 113901 (2005).
    [CrossRef] [PubMed]
  19. V. M. Pérez-García, V. V. Konotop, and J. J. García-Ripoll, "Dynamics of quasicollapse in nonlinear Schrödinger systems with nonlocal interactions," Phys. Rev. E 62, 4300-4308 (2000).
    [CrossRef]
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    [CrossRef] [PubMed]
  21. H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, "Diffraction management," Phys. Rev. Lett. 85, 1863-1866 (2000).
    [CrossRef] [PubMed]
  22. T. Pertsch, T. Zentgraf, U. Peschel, A. Bräer, and F. Lederer, "Anomalous refraction and diffraction in discrete optical systems," Phys. Rev. Lett. 88, 093901 (2002).
    [CrossRef] [PubMed]
  23. D. Anderson, "Variational approach to nonlinear pulse propagation in optical fibers," Phys. Rev. A 27, 3135-3145 (1983).
    [CrossRef]
  24. C. P. Jisha, V. C. Kuriakose, and K. Porsezian, "Variational approach to spatial optical solitons in bulk cubic-quintic media stabilized by self-induced multiphoton ionization," Phys. Rev. E 71, 056615 (2005).
    [CrossRef]
  25. D. P. Caetano, S. B. Cavalcanti, J. M. Hickmann, A. M. Kamchatnov, R. A. Kraenkel, and E. A. Makarova, "Soliton propagation in a medium with Kerr nonlinearity and resonant impurities: a variational approach," Phys. Rev. E 67, 046615 (2003).
    [CrossRef]
  26. D. Anderson and M. Bonnedal, "Variational approach to nonlinear self-focusing of Gaussian laser beams," Phys. Fluids 22, 105-109 (1979).
    [CrossRef]
  27. D. Anderson, M. Bonnedal, and M. Lisak, "Self-trapped cylindrical laser beams," Phys. Fluids 22, 1838-1840 (1979).
    [CrossRef]

2005 (4)

D. Mihalache, D. Mazilu, F. Lederer, B. A. Malomed, Y. V. Kartashov, L.-C. Crasovan, and L. Torner, "Stable spatiotemporal solitons in Bessel optical lattices," Phys. Rev. Lett. 95, 023902 (2005).
[CrossRef] [PubMed]

Z. Xu, Y. V. Kartashov, and L. Torner, "Soliton mobility in nonlocal optical lattices," Phys. Rev. Lett. 95, 113901 (2005).
[CrossRef] [PubMed]

C. P. Jisha, V. C. Kuriakose, and K. Porsezian, "Variational approach to spatial optical solitons in bulk cubic-quintic media stabilized by self-induced multiphoton ionization," Phys. Rev. E 71, 056615 (2005).
[CrossRef]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, "Soliton control in chirped photonic lattices," J. Opt. Soc. Am. B 22, 1356-1359 (2005).
[CrossRef]

2004 (4)

Y. V. Kartashov, A. S. Zelenina, L. Torner, and V. A. Vysloukh, "Spatial soliton switching in quasi-continuous optical arrays," Opt. Lett. 29, 766-768 (2004).
[CrossRef] [PubMed]

H. Buljan, O. Cohen, J. W. Fleischer, T. Schwartz, M. Segev, Z. H. Musslimani, N. K. Efremidis, and D. N. Christodoulides, "Random-phase solitons in nonlinear periodic lattices," Phys. Rev. Lett. 92, 223901 (2004).
[CrossRef] [PubMed]

L. Pricoupenko, "Variational approach for the two-dimensional trapped Bose-Einstein condensate," Phys. Rev. A 70, 013601 (2004).
[CrossRef]

D. E. Pelinovsky, A. A. Sukhorukov, and Y. S. Kivshar, "Bifurcations and stability of gap solitons in periodic potentials," Phys. Rev. E 70, 036618 (2004).
[CrossRef]

2003 (5)

D. N. Christodoulides, F. Lederer, and Y. Silberberg, "Discretizing light behaviour in linear and nonlinear waveguide lattices," Nature 424, 817-823 (2003).
[CrossRef] [PubMed]

N. K. Efremidis and D. N. Christodoulides, "Lattice solitons in Bose-Einstein condensates," Phys. Rev. A 67, 063608 (2003).
[CrossRef]

P. Pedri, L. Santos, P. Öhberg, and S. Stringari, "Violation of self-similarity in the expansion of a one-dimensional Bose gas," Phys. Rev. A 68, 043601 (2003).
[CrossRef]

J.-P. Martikainen and H. T. C. Stoof, "Excitations of a Bose-Einstein condensate in a one-dimensional optical lattice," Phys. Rev. A 68, 013610 (2003).
[CrossRef]

D. P. Caetano, S. B. Cavalcanti, J. M. Hickmann, A. M. Kamchatnov, R. A. Kraenkel, and E. A. Makarova, "Soliton propagation in a medium with Kerr nonlinearity and resonant impurities: a variational approach," Phys. Rev. E 67, 046615 (2003).
[CrossRef]

2002 (1)

T. Pertsch, T. Zentgraf, U. Peschel, A. Bräer, and F. Lederer, "Anomalous refraction and diffraction in discrete optical systems," Phys. Rev. Lett. 88, 093901 (2002).
[CrossRef] [PubMed]

2001 (1)

A. Trombettoni and A. Smerzi, "Discrete solitons and breathers with dilute Bose-Einstin condensates," Phys. Rev. Lett. 86, 2353-2356 (2001).
[CrossRef] [PubMed]

2000 (2)

V. M. Pérez-García, V. V. Konotop, and J. J. García-Ripoll, "Dynamics of quasicollapse in nonlinear Schrödinger systems with nonlocal interactions," Phys. Rev. E 62, 4300-4308 (2000).
[CrossRef]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, "Diffraction management," Phys. Rev. Lett. 85, 1863-1866 (2000).
[CrossRef] [PubMed]

1998 (1)

B. P. Anderson and M. A. Kasevich, "Macroscopic quantum interference from atomic tunnel arrays," Science 282, 1686-1689 (1998).
[CrossRef] [PubMed]

1996 (1)

M. B. Dahan, E. Peik, J. Reichel, Y. Castin, and C. Salomon, "Bloch oscillations of atoms in an optical potential," Phys. Rev. Lett. 76, 4508-4511 (1996).
[CrossRef] [PubMed]

1993 (1)

R. Scharf and A. R. Bishop, "Length-scale competition for the one-dimensional nonlinear Schrödinger equation with spatially periodic potentials," Phys. Rev. E 47, 1375-1383 (1993).
[CrossRef]

1988 (1)

1987 (1)

1983 (1)

D. Anderson, "Variational approach to nonlinear pulse propagation in optical fibers," Phys. Rev. A 27, 3135-3145 (1983).
[CrossRef]

1979 (3)

W. P. Su, J. R. Schieffer, and A. J. Heeger, "Solitons in polyacetylene," Phys. Rev. Lett. 42, 1698-1701 (1979).
[CrossRef]

D. Anderson and M. Bonnedal, "Variational approach to nonlinear self-focusing of Gaussian laser beams," Phys. Fluids 22, 105-109 (1979).
[CrossRef]

D. Anderson, M. Bonnedal, and M. Lisak, "Self-trapped cylindrical laser beams," Phys. Fluids 22, 1838-1840 (1979).
[CrossRef]

1973 (1)

A. S. Davydov, "The theory contraction of proteins under their excitation," J. Theor. Biol. 38, 559-569 (1973).
[CrossRef] [PubMed]

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

J. Theor. Biol. (1)

A. S. Davydov, "The theory contraction of proteins under their excitation," J. Theor. Biol. 38, 559-569 (1973).
[CrossRef] [PubMed]

Nature (1)

D. N. Christodoulides, F. Lederer, and Y. Silberberg, "Discretizing light behaviour in linear and nonlinear waveguide lattices," Nature 424, 817-823 (2003).
[CrossRef] [PubMed]

Opt. Lett. (3)

Phys. Fluids (2)

D. Anderson and M. Bonnedal, "Variational approach to nonlinear self-focusing of Gaussian laser beams," Phys. Fluids 22, 105-109 (1979).
[CrossRef]

D. Anderson, M. Bonnedal, and M. Lisak, "Self-trapped cylindrical laser beams," Phys. Fluids 22, 1838-1840 (1979).
[CrossRef]

Phys. Rev. A (5)

L. Pricoupenko, "Variational approach for the two-dimensional trapped Bose-Einstein condensate," Phys. Rev. A 70, 013601 (2004).
[CrossRef]

D. Anderson, "Variational approach to nonlinear pulse propagation in optical fibers," Phys. Rev. A 27, 3135-3145 (1983).
[CrossRef]

P. Pedri, L. Santos, P. Öhberg, and S. Stringari, "Violation of self-similarity in the expansion of a one-dimensional Bose gas," Phys. Rev. A 68, 043601 (2003).
[CrossRef]

J.-P. Martikainen and H. T. C. Stoof, "Excitations of a Bose-Einstein condensate in a one-dimensional optical lattice," Phys. Rev. A 68, 013610 (2003).
[CrossRef]

N. K. Efremidis and D. N. Christodoulides, "Lattice solitons in Bose-Einstein condensates," Phys. Rev. A 67, 063608 (2003).
[CrossRef]

Phys. Rev. E (5)

R. Scharf and A. R. Bishop, "Length-scale competition for the one-dimensional nonlinear Schrödinger equation with spatially periodic potentials," Phys. Rev. E 47, 1375-1383 (1993).
[CrossRef]

C. P. Jisha, V. C. Kuriakose, and K. Porsezian, "Variational approach to spatial optical solitons in bulk cubic-quintic media stabilized by self-induced multiphoton ionization," Phys. Rev. E 71, 056615 (2005).
[CrossRef]

D. P. Caetano, S. B. Cavalcanti, J. M. Hickmann, A. M. Kamchatnov, R. A. Kraenkel, and E. A. Makarova, "Soliton propagation in a medium with Kerr nonlinearity and resonant impurities: a variational approach," Phys. Rev. E 67, 046615 (2003).
[CrossRef]

V. M. Pérez-García, V. V. Konotop, and J. J. García-Ripoll, "Dynamics of quasicollapse in nonlinear Schrödinger systems with nonlocal interactions," Phys. Rev. E 62, 4300-4308 (2000).
[CrossRef]

D. E. Pelinovsky, A. A. Sukhorukov, and Y. S. Kivshar, "Bifurcations and stability of gap solitons in periodic potentials," Phys. Rev. E 70, 036618 (2004).
[CrossRef]

Phys. Rev. Lett. (8)

W. P. Su, J. R. Schieffer, and A. J. Heeger, "Solitons in polyacetylene," Phys. Rev. Lett. 42, 1698-1701 (1979).
[CrossRef]

Z. Xu, Y. V. Kartashov, and L. Torner, "Soliton mobility in nonlocal optical lattices," Phys. Rev. Lett. 95, 113901 (2005).
[CrossRef] [PubMed]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, "Diffraction management," Phys. Rev. Lett. 85, 1863-1866 (2000).
[CrossRef] [PubMed]

T. Pertsch, T. Zentgraf, U. Peschel, A. Bräer, and F. Lederer, "Anomalous refraction and diffraction in discrete optical systems," Phys. Rev. Lett. 88, 093901 (2002).
[CrossRef] [PubMed]

D. Mihalache, D. Mazilu, F. Lederer, B. A. Malomed, Y. V. Kartashov, L.-C. Crasovan, and L. Torner, "Stable spatiotemporal solitons in Bessel optical lattices," Phys. Rev. Lett. 95, 023902 (2005).
[CrossRef] [PubMed]

M. B. Dahan, E. Peik, J. Reichel, Y. Castin, and C. Salomon, "Bloch oscillations of atoms in an optical potential," Phys. Rev. Lett. 76, 4508-4511 (1996).
[CrossRef] [PubMed]

A. Trombettoni and A. Smerzi, "Discrete solitons and breathers with dilute Bose-Einstin condensates," Phys. Rev. Lett. 86, 2353-2356 (2001).
[CrossRef] [PubMed]

H. Buljan, O. Cohen, J. W. Fleischer, T. Schwartz, M. Segev, Z. H. Musslimani, N. K. Efremidis, and D. N. Christodoulides, "Random-phase solitons in nonlinear periodic lattices," Phys. Rev. Lett. 92, 223901 (2004).
[CrossRef] [PubMed]

Science (1)

B. P. Anderson and M. A. Kasevich, "Macroscopic quantum interference from atomic tunnel arrays," Science 282, 1686-1689 (1998).
[CrossRef] [PubMed]

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Figures (3)

Fig. 1
Fig. 1

Variation of the potential function with the normalized beam width for (i) the initial state z = 0 (solid curve); (ii) h = 0.5 m 1 (dotted curve), and (iii) h varying with propagation distance for loss compensation (dashed curve).

Fig. 2
Fig. 2

Propagation of light beam in the Bessel optical lattice δ = 0.3 . (a) α = 0 m 1 and h = 0.5 m 1 ; (b) α = 3.5 m 1 and h = 0.5 m 1 ; (c) α = 3.5 m 1 and h ( ξ ) varies to compensate the loss of medium.

Fig. 3
Fig. 3

Propagation of light beam in the Bessel optical lattice for incident angle δ = 0.3 . (a) α = 0 m 1 and h = 0.5 m 1 ; (b) α = 3.5 m 1 and h = 0.5 m 1 ; (c) α = 3.5 m 1 and h ( ξ ) varies to compensate the loss of medium.

Equations (11)

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i ψ z + i 2 α ψ + 1 2 k 2 ψ x 2 + γ ψ 2 ψ + h p ( x ) ψ = 0 ,
L g = [ i 2 ( ψ ψ * z ψ * ψ z ) + 1 2 k ψ x 2 γ 2 ψ 4 h p ( x ) ψ 2 ] exp ( α z ) .
d 2 a d z 2 = 1 k 2 a 3 γ E 0 2 k a 2 e α z + h a 2 k β 2 ( I 1 I 0 ) exp [ a 2 8 β 2 ] ,
V ( y ) = μ y 2 + υ p ( y ) w f ( y ) + m ,
d ϕ d z = 1 2 k a 2 + 5 γ E 0 4 2 a e α z + h a 2 8 β 2 [ ( 1 + 8 β 2 a 2 ) I 0 I 1 ] exp ( a 2 8 β 2 ) .
V ( 1 ) = 2 μ + υ e α z w h ( z ) [ I 1 ( τ 2 8 ) I 0 ( τ 2 8 ) ] exp ( τ 2 8 ) .
h ( z ) = [ 2 γ E 0 k a 0 exp ( α z ) 2 ] exp ( τ 2 8 ) k ( I 1 I 0 ) τ 2 a 0 2 .
ϕ ( z ) = 1 2 k a 0 2 z 5 γ E 0 exp ( α z ) 4 2 α a 0 2 z 2 α γ E 0 k n 0 exp ( α z ) 8 k ( I 1 I 0 ) a 0 2 ,
ψ ( z , x ) = A 0 exp ( α z ) exp [ x 2 2 a 0 2 + i ϕ ( z ) ] .
γ ( z ) = [ 4 + h k τ 2 a 0 2 ( I 1 I 0 ) exp ( τ 2 8 ) ] exp ( α z ) 2 E 0 k a 0 .
i U ξ + 1 2 2 U η 2 + N 2 U 2 U + 1 2 i α k a 0 2 U + h k a 0 2 p ( a 0 β η ) U = 0 .

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