Abstract

In this article, we investigate the N-soliton bound state (BS) motions in photonic lattices. BSs split into their constituent single-solitons when they are incident normally into lattices. The splitting is sensitive to the incident position with respect to the lattices. In such a process, the lattice profile not only provides an antisymmetric frequency modulation perturbation and causes the BS to split but also acts as an external potential which affects the propagating dynamics of each single-soliton.

© 2010 Optical Society of America

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  1. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystal: Molding the Flow of light (Princeton U. Press, 1995).
  2. A. Hasegawa and Y. Kodama, Solitons in Optical Communications (Oxford U. Press, 1994).
  3. Y. Kivshar and G. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, 2003).
  4. G. Agrawal, Nonlinear Fiber Optics (Academic, 2001).
  5. P. K. A. Wai, C. R. Menyuk, Y. C. Lee, and H. H. Chen, “Nonlinear pulse propagation in the neighborhood of the zero-dispersion wavelength of monomode optical fibers,” Opt. Lett. 11, 464–466 (1986).
    [CrossRef] [PubMed]
  6. M. Golles, I. M. Uzunov, and F. Lederer, “Break up of N-soliton bound states due to intrapulse Raman scattering and third-order dispersion—an eigenvalue analysis,” Phys. Lett. A 231, 195–200 (1997).
    [CrossRef]
  7. K. Tai, A. Hasegawa, and N. Bekki, “Fission of optical solitons induced by stimulated Raman effect,” Opt. Lett. 13, 392–394 (1988).
    [CrossRef] [PubMed]
  8. V. V. Afanasyev, V. A. Vysloukh, and V. N. Serkin, “Decay and interaction of femtosecond optical solitons induced by the Raman self-scattering effect,” Opt. Lett. 15, 489–491 (1990).
    [CrossRef] [PubMed]
  9. Y. Silberberg, “Solitons and two-photon absorption,” Opt. Lett. 15, 1005–1007 (1990).
    [CrossRef] [PubMed]
  10. L. Torner, J. P. Torres, and C. R. Menyuk, “Fission and self-deflection of spatial solitons by cascading,” Opt. Lett. 21, 462–464 (1996).
    [CrossRef]
  11. S. R. Friberg and K. W. DeLong, “Breakup of bound higher-order solitons,” Opt. Lett. 17, 979–981 (1992).
    [CrossRef] [PubMed]
  12. A. V. Buryak and N. N. Akhmediev, “Internal friction between solitons in near-integrable systems,” Phys. Rev. E 50, 3126–3133 (1994).
    [CrossRef]
  13. D. Artigas, L. Torner, J. P. Torres, and N. N. Akhmediev, “Asymmetrical splitting of higher-order optical solitons induced by quintic nonlinearity,” Opt. Commun. 143, 322–328 (1997).
    [CrossRef]
  14. V. A. Aleshkevich, Y. V. Kartashov, A. S. Zelenina, V. A. Vysloukh, J. P. Torres, and L. Torner, “Eigenvalue control and switching by fission of multisoliton bound states in planar waveguides,” Opt. Lett. 29, 483–485 (2004).
    [CrossRef] [PubMed]
  15. J. E. Prilepsky and S. A. Derevyanko, “Breakup of a multisoliton state of the linearly damped nonlinear Schrödinger equation,” Phys. Rev. E 75, 036616 (2007).
    [CrossRef]
  16. Y. V. Kartashov, L. Carsovan, A. S. Zelenina, V. A. Vysloukh, A. S. Sampera, M. Lewenstein, and L. Torner, “Soliton eigenvalue control in optical lattices,” Phys. Rev. Lett. 93, 143902 (2004).
    [CrossRef] [PubMed]
  17. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Soliton control in fading optical lattices,” Opt. Lett. 31, 2181–2183 (2006).
    [CrossRef] [PubMed]
  18. 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]
  19. R. Scharf and A. R. Bishop, “Length-scale competition for the one-dimensional Schrödinger equation with spatially periodic potentials,” Phys. Rev. E 47, 1375–1383 (1993).
    [CrossRef]
  20. D. Liang-Wei, J. Hong-Zhen, and W. Hui, “Oscillation of spatial solitons in a waveguide with a symmetrical refractive index profile,” Chin. Phys. 16, 3004–3008 (2007).
    [CrossRef]

2007 (2)

J. E. Prilepsky and S. A. Derevyanko, “Breakup of a multisoliton state of the linearly damped nonlinear Schrödinger equation,” Phys. Rev. E 75, 036616 (2007).
[CrossRef]

D. Liang-Wei, J. Hong-Zhen, and W. Hui, “Oscillation of spatial solitons in a waveguide with a symmetrical refractive index profile,” Chin. Phys. 16, 3004–3008 (2007).
[CrossRef]

2006 (1)

2004 (3)

1997 (2)

D. Artigas, L. Torner, J. P. Torres, and N. N. Akhmediev, “Asymmetrical splitting of higher-order optical solitons induced by quintic nonlinearity,” Opt. Commun. 143, 322–328 (1997).
[CrossRef]

M. Golles, I. M. Uzunov, and F. Lederer, “Break up of N-soliton bound states due to intrapulse Raman scattering and third-order dispersion—an eigenvalue analysis,” Phys. Lett. A 231, 195–200 (1997).
[CrossRef]

1996 (1)

1994 (1)

A. V. Buryak and N. N. Akhmediev, “Internal friction between solitons in near-integrable systems,” Phys. Rev. E 50, 3126–3133 (1994).
[CrossRef]

1993 (1)

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

1992 (1)

1990 (2)

1988 (1)

1986 (1)

Afanasyev, V. V.

Agrawal, G.

Y. Kivshar and G. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, 2003).

G. Agrawal, Nonlinear Fiber Optics (Academic, 2001).

Akhmediev, N. N.

D. Artigas, L. Torner, J. P. Torres, and N. N. Akhmediev, “Asymmetrical splitting of higher-order optical solitons induced by quintic nonlinearity,” Opt. Commun. 143, 322–328 (1997).
[CrossRef]

A. V. Buryak and N. N. Akhmediev, “Internal friction between solitons in near-integrable systems,” Phys. Rev. E 50, 3126–3133 (1994).
[CrossRef]

Aleshkevich, V. A.

Artigas, D.

D. Artigas, L. Torner, J. P. Torres, and N. N. Akhmediev, “Asymmetrical splitting of higher-order optical solitons induced by quintic nonlinearity,” Opt. Commun. 143, 322–328 (1997).
[CrossRef]

Bekki, N.

Bishop, A. R.

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

Buryak, A. V.

A. V. Buryak and N. N. Akhmediev, “Internal friction between solitons in near-integrable systems,” Phys. Rev. E 50, 3126–3133 (1994).
[CrossRef]

Carsovan, L.

Y. V. Kartashov, L. Carsovan, A. S. Zelenina, V. A. Vysloukh, A. S. Sampera, M. Lewenstein, and L. Torner, “Soliton eigenvalue control in optical lattices,” Phys. Rev. Lett. 93, 143902 (2004).
[CrossRef] [PubMed]

Chen, H. H.

DeLong, K. W.

Derevyanko, S. A.

J. E. Prilepsky and S. A. Derevyanko, “Breakup of a multisoliton state of the linearly damped nonlinear Schrödinger equation,” Phys. Rev. E 75, 036616 (2007).
[CrossRef]

Friberg, S. R.

Golles, M.

M. Golles, I. M. Uzunov, and F. Lederer, “Break up of N-soliton bound states due to intrapulse Raman scattering and third-order dispersion—an eigenvalue analysis,” Phys. Lett. A 231, 195–200 (1997).
[CrossRef]

Hasegawa, A.

Hong-Zhen, J.

D. Liang-Wei, J. Hong-Zhen, and W. Hui, “Oscillation of spatial solitons in a waveguide with a symmetrical refractive index profile,” Chin. Phys. 16, 3004–3008 (2007).
[CrossRef]

Hui, W.

D. Liang-Wei, J. Hong-Zhen, and W. Hui, “Oscillation of spatial solitons in a waveguide with a symmetrical refractive index profile,” Chin. Phys. 16, 3004–3008 (2007).
[CrossRef]

Joannopoulos, J. D.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystal: Molding the Flow of light (Princeton U. Press, 1995).

Kartashov, Y. V.

Kivshar, Y.

Y. Kivshar and G. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, 2003).

Kodama, Y.

A. Hasegawa and Y. Kodama, Solitons in Optical Communications (Oxford U. Press, 1994).

Lederer, F.

M. Golles, I. M. Uzunov, and F. Lederer, “Break up of N-soliton bound states due to intrapulse Raman scattering and third-order dispersion—an eigenvalue analysis,” Phys. Lett. A 231, 195–200 (1997).
[CrossRef]

Lee, Y. C.

Lewenstein, M.

Y. V. Kartashov, L. Carsovan, A. S. Zelenina, V. A. Vysloukh, A. S. Sampera, M. Lewenstein, and L. Torner, “Soliton eigenvalue control in optical lattices,” Phys. Rev. Lett. 93, 143902 (2004).
[CrossRef] [PubMed]

Liang-Wei, D.

D. Liang-Wei, J. Hong-Zhen, and W. Hui, “Oscillation of spatial solitons in a waveguide with a symmetrical refractive index profile,” Chin. Phys. 16, 3004–3008 (2007).
[CrossRef]

Meade, R. D.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystal: Molding the Flow of light (Princeton U. Press, 1995).

Menyuk, C. R.

Prilepsky, J. E.

J. E. Prilepsky and S. A. Derevyanko, “Breakup of a multisoliton state of the linearly damped nonlinear Schrödinger equation,” Phys. Rev. E 75, 036616 (2007).
[CrossRef]

Sampera, A. S.

Y. V. Kartashov, L. Carsovan, A. S. Zelenina, V. A. Vysloukh, A. S. Sampera, M. Lewenstein, and L. Torner, “Soliton eigenvalue control in optical lattices,” Phys. Rev. Lett. 93, 143902 (2004).
[CrossRef] [PubMed]

Scharf, R.

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

Serkin, V. N.

Silberberg, Y.

Tai, K.

Torner, L.

Torres, J. P.

Uzunov, I. M.

M. Golles, I. M. Uzunov, and F. Lederer, “Break up of N-soliton bound states due to intrapulse Raman scattering and third-order dispersion—an eigenvalue analysis,” Phys. Lett. A 231, 195–200 (1997).
[CrossRef]

Vysloukh, V. A.

Wai, P. K. A.

Winn, J. N.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystal: Molding the Flow of light (Princeton U. Press, 1995).

Zelenina, A. S.

Chin. Phys. (1)

D. Liang-Wei, J. Hong-Zhen, and W. Hui, “Oscillation of spatial solitons in a waveguide with a symmetrical refractive index profile,” Chin. Phys. 16, 3004–3008 (2007).
[CrossRef]

Opt. Commun. (1)

D. Artigas, L. Torner, J. P. Torres, and N. N. Akhmediev, “Asymmetrical splitting of higher-order optical solitons induced by quintic nonlinearity,” Opt. Commun. 143, 322–328 (1997).
[CrossRef]

Opt. Lett. (9)

V. A. Aleshkevich, Y. V. Kartashov, A. S. Zelenina, V. A. Vysloukh, J. P. Torres, and L. Torner, “Eigenvalue control and switching by fission of multisoliton bound states in planar waveguides,” Opt. Lett. 29, 483–485 (2004).
[CrossRef] [PubMed]

P. K. A. Wai, C. R. Menyuk, Y. C. Lee, and H. H. Chen, “Nonlinear pulse propagation in the neighborhood of the zero-dispersion wavelength of monomode optical fibers,” Opt. Lett. 11, 464–466 (1986).
[CrossRef] [PubMed]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Soliton control in fading optical lattices,” Opt. Lett. 31, 2181–2183 (2006).
[CrossRef] [PubMed]

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]

K. Tai, A. Hasegawa, and N. Bekki, “Fission of optical solitons induced by stimulated Raman effect,” Opt. Lett. 13, 392–394 (1988).
[CrossRef] [PubMed]

V. V. Afanasyev, V. A. Vysloukh, and V. N. Serkin, “Decay and interaction of femtosecond optical solitons induced by the Raman self-scattering effect,” Opt. Lett. 15, 489–491 (1990).
[CrossRef] [PubMed]

Y. Silberberg, “Solitons and two-photon absorption,” Opt. Lett. 15, 1005–1007 (1990).
[CrossRef] [PubMed]

L. Torner, J. P. Torres, and C. R. Menyuk, “Fission and self-deflection of spatial solitons by cascading,” Opt. Lett. 21, 462–464 (1996).
[CrossRef]

S. R. Friberg and K. W. DeLong, “Breakup of bound higher-order solitons,” Opt. Lett. 17, 979–981 (1992).
[CrossRef] [PubMed]

Phys. Lett. A (1)

M. Golles, I. M. Uzunov, and F. Lederer, “Break up of N-soliton bound states due to intrapulse Raman scattering and third-order dispersion—an eigenvalue analysis,” Phys. Lett. A 231, 195–200 (1997).
[CrossRef]

Phys. Rev. E (3)

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

J. E. Prilepsky and S. A. Derevyanko, “Breakup of a multisoliton state of the linearly damped nonlinear Schrödinger equation,” Phys. Rev. E 75, 036616 (2007).
[CrossRef]

A. V. Buryak and N. N. Akhmediev, “Internal friction between solitons in near-integrable systems,” Phys. Rev. E 50, 3126–3133 (1994).
[CrossRef]

Phys. Rev. Lett. (1)

Y. V. Kartashov, L. Carsovan, A. S. Zelenina, V. A. Vysloukh, A. S. Sampera, M. Lewenstein, and L. Torner, “Soliton eigenvalue control in optical lattices,” Phys. Rev. Lett. 93, 143902 (2004).
[CrossRef] [PubMed]

Other (4)

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystal: Molding the Flow of light (Princeton U. Press, 1995).

A. Hasegawa and Y. Kodama, Solitons in Optical Communications (Oxford U. Press, 1994).

Y. Kivshar and G. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, 2003).

G. Agrawal, Nonlinear Fiber Optics (Academic, 2001).

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Figures (4)

Fig. 1
Fig. 1

Dynamics of BS in photonic lattices. The modulation frequency is Ω = 4 , modulation depth is p = 0.1 , and initial beam tilt is zero. (a)–(c) are the situations for N = 2 BS, and the incident positions are η 0 = 0 , 0.2 T , and 0.5 T . (d)–(f) are the situations for N = 3 BS, and the incident positions are η 0 = 0 , 0.2 T , and 0.5 T .

Fig. 2
Fig. 2

(a) The output positions of χ 1 constituent versus input position for two soliton BSs. (b) The output positions of soliton constituents for three soliton BSs, where circles and stars correspond to the χ 1 = 1 and χ 2 = 3 constituents, respectively. (c) The output profiles of two soliton BSs at η 0 = 0.02 T (solid line) and η 0 = 0.25 T (dashed line). (d) The output profiles of three soliton BSs at η 0 = 0.03 T (solid line), 0.07 T (dashed line), and 0.1 T (dashed-dotted line). Other parameters are p = 0.1 , T = π / 2 , α = 0 , ξ max = 40 . The photonic lattices are shown in the same plots in arbitrary unit.

Fig. 3
Fig. 3

Two examples of three soliton BSs in photonic lattices. The parameters are p = 0.1 , η 0 = 0.4 T , and α = 0 ; the lattice periods are T = π / 2 and π / 3 , respectively, for (a) and (b).

Fig. 4
Fig. 4

Dynamics of two soliton BSs in photonic lattices with different initial tilts and incident positions: BSs in (a)–(c) are incident with initial tilt of α = 0.3 , (d)–(f) are incident with initial tilt of α = 0.3 , and (d)–(f) are incident with initial tilt of α = 0.3 , and (g)–(i) are incident with initial tilt of α = 0.4 . The incident positions are labeled in each plot in the figure. Other parameters are p = 0.1 , T = π / 2 , ξ max = 40 .

Equations (10)

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i q ξ + 1 2 2 q η 2 + q | q | 2 + p R ( η ) q = 0 ,
q ( η , ξ = 0 ) = N   sech ( η η 0 ) ,
q j = χ j   sech ( χ j η ) exp ( i χ j 2 ξ / 2 ) .
q ( η , Δ ξ ) = N   sech ( η η 0 ) exp [ i p Δ ξ   cos ( 2 π η / T ) ] = N   sech ( η ) exp [ i p Δ ξ   cos ( 2 π ( η + η 0 ) / T ) ] = N   sech ( η ) F ( η ) G ( η ) ,
F ( η ) = exp [ i ϵ f   cos ( 2 π η / T ) ] ,
G ( η ) = exp [ i ϵ g   sin ( 2 π η / T ) ] ,
d 2 d ξ 2 η = p U + | q | 2 d R d η d η = 2 π p T π 2 / T χ sinh ( π 2 / T χ ) sin ( 2 π η T ) ,
α cr = [ 2 p π 2 T χ [ 1 + cos ( 2 π η 0 / T ) ] sinh 1 ( π 2 T χ ) ] 1 / 2 .
q ( η ) ξ = 0 = N   sech ( η η 0 ) exp ( i α η ) .
η out = η out A η out + A η | q ( η , ξ max ) | 2 d η η out A η out + A | q ( η , ξ max ) | 2 d η ,

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