Abstract

We investigate discrete soliton propagation in a waveguide array with periodically modulated coupling constants. If the modulation wave vector coincides with that of an internal mode, the respective bound state of the soliton is excited. Eventually, growing internal vibrations may result in a transverse motion or in a resonant splitting of the soliton.

© 2002 Optical Society of America

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  1. D. N. Christodoulides and R. I. Joseph, “Discrete self-focusing in nonlinear arrays of coupled waveguides,” Opt. Lett. 13, 794–798 (1988).
    [CrossRef] [PubMed]
  2. W. Krolikowski, U. Trutschel, M. Cronin-Golomb, and C. Schmidt-Hattenberger, “Soliton-like optical switching in a circular fiber array,” Opt. Lett. 19, 320–322 (1994).
    [CrossRef]
  3. A. B. Aceves, C. DeAngelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, “Discrete self-trapping, soliton interactions, and beam steering in nonlinear waveguide arrays,” Phys. Rev. E 53, 1172–1189 (1996).
    [CrossRef]
  4. S. Flach and C. R. Willis, “Discrete breathers,” Phys. Rep. 295, 182–264 (1998).
    [CrossRef]
  5. H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. Boyd, and J. S. Aitchison, “Discrete spatial solitons in waveguide arrays,” Phys. Rev. Lett. 81, 3383–3386 (1998).
    [CrossRef]
  6. R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Dynamics of discrete solitons in optical waveguide arrays,” Phys. Rev. Lett. 83, 2726–2729 (1999).
    [CrossRef]
  7. W. Krolikowski and Y. S. Kivshar, “Soliton-based optical switching in waveguide arrays,” J. Opt. Soc. Am. B 13, 876–887 (1996).
    [CrossRef]
  8. P. L. Chu, B. A. Malomed, G. D. Peng, and I. M. Skinner, “Soliton dynamics in periodically modulated directional couplers,” Phys. Rev. E 49, 5763–5767 (1994).
    [CrossRef]
  9. B. A. Malomed, “Resonant transmission of a chirped soliton in a long optical fiber with periodic amplification,” J. Opt. Soc. Am. B 13, 677–686 (1996).
    [CrossRef]
  10. T. S. Yang and W. L. Kath, “Radiation loss of dispersion-managed solitons in optical fibers,” Physica D 149, 80–94 (2001).
    [CrossRef]
  11. B. J. Eggleton, R. E. Slusher, C. M. deSterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. 76, 1627–1630 (1996).
    [CrossRef] [PubMed]
  12. Y. S. Kivshar, A. R. Champneys, D. Cai, and A. R. Bishop, “Multiple states of intrinsic localized modes,” Phys. Rev. B 58, 5423–5428 (1998).
    [CrossRef]

2001

T. S. Yang and W. L. Kath, “Radiation loss of dispersion-managed solitons in optical fibers,” Physica D 149, 80–94 (2001).
[CrossRef]

1999

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Dynamics of discrete solitons in optical waveguide arrays,” Phys. Rev. Lett. 83, 2726–2729 (1999).
[CrossRef]

1998

S. Flach and C. R. Willis, “Discrete breathers,” Phys. Rep. 295, 182–264 (1998).
[CrossRef]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. Boyd, and J. S. Aitchison, “Discrete spatial solitons in waveguide arrays,” Phys. Rev. Lett. 81, 3383–3386 (1998).
[CrossRef]

Y. S. Kivshar, A. R. Champneys, D. Cai, and A. R. Bishop, “Multiple states of intrinsic localized modes,” Phys. Rev. B 58, 5423–5428 (1998).
[CrossRef]

1996

B. J. Eggleton, R. E. Slusher, C. M. deSterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. 76, 1627–1630 (1996).
[CrossRef] [PubMed]

A. B. Aceves, C. DeAngelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, “Discrete self-trapping, soliton interactions, and beam steering in nonlinear waveguide arrays,” Phys. Rev. E 53, 1172–1189 (1996).
[CrossRef]

B. A. Malomed, “Resonant transmission of a chirped soliton in a long optical fiber with periodic amplification,” J. Opt. Soc. Am. B 13, 677–686 (1996).
[CrossRef]

W. Krolikowski and Y. S. Kivshar, “Soliton-based optical switching in waveguide arrays,” J. Opt. Soc. Am. B 13, 876–887 (1996).
[CrossRef]

1994

W. Krolikowski, U. Trutschel, M. Cronin-Golomb, and C. Schmidt-Hattenberger, “Soliton-like optical switching in a circular fiber array,” Opt. Lett. 19, 320–322 (1994).
[CrossRef]

P. L. Chu, B. A. Malomed, G. D. Peng, and I. M. Skinner, “Soliton dynamics in periodically modulated directional couplers,” Phys. Rev. E 49, 5763–5767 (1994).
[CrossRef]

1988

Aceves, A. B.

A. B. Aceves, C. DeAngelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, “Discrete self-trapping, soliton interactions, and beam steering in nonlinear waveguide arrays,” Phys. Rev. E 53, 1172–1189 (1996).
[CrossRef]

Aitchison, J. S.

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Dynamics of discrete solitons in optical waveguide arrays,” Phys. Rev. Lett. 83, 2726–2729 (1999).
[CrossRef]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. Boyd, and J. S. Aitchison, “Discrete spatial solitons in waveguide arrays,” Phys. Rev. Lett. 81, 3383–3386 (1998).
[CrossRef]

Bishop, A. R.

Y. S. Kivshar, A. R. Champneys, D. Cai, and A. R. Bishop, “Multiple states of intrinsic localized modes,” Phys. Rev. B 58, 5423–5428 (1998).
[CrossRef]

Boyd, A.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. Boyd, and J. S. Aitchison, “Discrete spatial solitons in waveguide arrays,” Phys. Rev. Lett. 81, 3383–3386 (1998).
[CrossRef]

Cai, D.

Y. S. Kivshar, A. R. Champneys, D. Cai, and A. R. Bishop, “Multiple states of intrinsic localized modes,” Phys. Rev. B 58, 5423–5428 (1998).
[CrossRef]

Champneys, A. R.

Y. S. Kivshar, A. R. Champneys, D. Cai, and A. R. Bishop, “Multiple states of intrinsic localized modes,” Phys. Rev. B 58, 5423–5428 (1998).
[CrossRef]

Christodoulides, D. N.

Chu, P. L.

P. L. Chu, B. A. Malomed, G. D. Peng, and I. M. Skinner, “Soliton dynamics in periodically modulated directional couplers,” Phys. Rev. E 49, 5763–5767 (1994).
[CrossRef]

Cronin-Golomb, M.

DeAngelis, C.

A. B. Aceves, C. DeAngelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, “Discrete self-trapping, soliton interactions, and beam steering in nonlinear waveguide arrays,” Phys. Rev. E 53, 1172–1189 (1996).
[CrossRef]

deSterke, C. M.

B. J. Eggleton, R. E. Slusher, C. M. deSterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. 76, 1627–1630 (1996).
[CrossRef] [PubMed]

Eggleton, B. J.

B. J. Eggleton, R. E. Slusher, C. M. deSterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. 76, 1627–1630 (1996).
[CrossRef] [PubMed]

Eisenberg, H. S.

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Dynamics of discrete solitons in optical waveguide arrays,” Phys. Rev. Lett. 83, 2726–2729 (1999).
[CrossRef]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. Boyd, and J. S. Aitchison, “Discrete spatial solitons in waveguide arrays,” Phys. Rev. Lett. 81, 3383–3386 (1998).
[CrossRef]

Flach, S.

S. Flach and C. R. Willis, “Discrete breathers,” Phys. Rep. 295, 182–264 (1998).
[CrossRef]

Joseph, R. I.

Kath, W. L.

T. S. Yang and W. L. Kath, “Radiation loss of dispersion-managed solitons in optical fibers,” Physica D 149, 80–94 (2001).
[CrossRef]

Kivshar, Y. S.

Y. S. Kivshar, A. R. Champneys, D. Cai, and A. R. Bishop, “Multiple states of intrinsic localized modes,” Phys. Rev. B 58, 5423–5428 (1998).
[CrossRef]

W. Krolikowski and Y. S. Kivshar, “Soliton-based optical switching in waveguide arrays,” J. Opt. Soc. Am. B 13, 876–887 (1996).
[CrossRef]

Krolikowski, W.

Krug, P. A.

B. J. Eggleton, R. E. Slusher, C. M. deSterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. 76, 1627–1630 (1996).
[CrossRef] [PubMed]

Lederer, F.

A. B. Aceves, C. DeAngelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, “Discrete self-trapping, soliton interactions, and beam steering in nonlinear waveguide arrays,” Phys. Rev. E 53, 1172–1189 (1996).
[CrossRef]

Malomed, B. A.

B. A. Malomed, “Resonant transmission of a chirped soliton in a long optical fiber with periodic amplification,” J. Opt. Soc. Am. B 13, 677–686 (1996).
[CrossRef]

P. L. Chu, B. A. Malomed, G. D. Peng, and I. M. Skinner, “Soliton dynamics in periodically modulated directional couplers,” Phys. Rev. E 49, 5763–5767 (1994).
[CrossRef]

Morandotti, R.

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Dynamics of discrete solitons in optical waveguide arrays,” Phys. Rev. Lett. 83, 2726–2729 (1999).
[CrossRef]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. Boyd, and J. S. Aitchison, “Discrete spatial solitons in waveguide arrays,” Phys. Rev. Lett. 81, 3383–3386 (1998).
[CrossRef]

Muschall, R.

A. B. Aceves, C. DeAngelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, “Discrete self-trapping, soliton interactions, and beam steering in nonlinear waveguide arrays,” Phys. Rev. E 53, 1172–1189 (1996).
[CrossRef]

Peng, G. D.

P. L. Chu, B. A. Malomed, G. D. Peng, and I. M. Skinner, “Soliton dynamics in periodically modulated directional couplers,” Phys. Rev. E 49, 5763–5767 (1994).
[CrossRef]

Peschel, T.

A. B. Aceves, C. DeAngelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, “Discrete self-trapping, soliton interactions, and beam steering in nonlinear waveguide arrays,” Phys. Rev. E 53, 1172–1189 (1996).
[CrossRef]

Peschel, U.

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Dynamics of discrete solitons in optical waveguide arrays,” Phys. Rev. Lett. 83, 2726–2729 (1999).
[CrossRef]

Schmidt-Hattenberger, C.

Silberberg, Y.

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Dynamics of discrete solitons in optical waveguide arrays,” Phys. Rev. Lett. 83, 2726–2729 (1999).
[CrossRef]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. Boyd, and J. S. Aitchison, “Discrete spatial solitons in waveguide arrays,” Phys. Rev. Lett. 81, 3383–3386 (1998).
[CrossRef]

Sipe, J. E.

B. J. Eggleton, R. E. Slusher, C. M. deSterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. 76, 1627–1630 (1996).
[CrossRef] [PubMed]

Skinner, I. M.

P. L. Chu, B. A. Malomed, G. D. Peng, and I. M. Skinner, “Soliton dynamics in periodically modulated directional couplers,” Phys. Rev. E 49, 5763–5767 (1994).
[CrossRef]

Slusher, R. E.

B. J. Eggleton, R. E. Slusher, C. M. deSterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. 76, 1627–1630 (1996).
[CrossRef] [PubMed]

Trillo, S.

A. B. Aceves, C. DeAngelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, “Discrete self-trapping, soliton interactions, and beam steering in nonlinear waveguide arrays,” Phys. Rev. E 53, 1172–1189 (1996).
[CrossRef]

Trutschel, U.

Wabnitz, S.

A. B. Aceves, C. DeAngelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, “Discrete self-trapping, soliton interactions, and beam steering in nonlinear waveguide arrays,” Phys. Rev. E 53, 1172–1189 (1996).
[CrossRef]

Willis, C. R.

S. Flach and C. R. Willis, “Discrete breathers,” Phys. Rep. 295, 182–264 (1998).
[CrossRef]

Yang, T. S.

T. S. Yang and W. L. Kath, “Radiation loss of dispersion-managed solitons in optical fibers,” Physica D 149, 80–94 (2001).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Lett.

Phys. Rep.

S. Flach and C. R. Willis, “Discrete breathers,” Phys. Rep. 295, 182–264 (1998).
[CrossRef]

Phys. Rev. B

Y. S. Kivshar, A. R. Champneys, D. Cai, and A. R. Bishop, “Multiple states of intrinsic localized modes,” Phys. Rev. B 58, 5423–5428 (1998).
[CrossRef]

Phys. Rev. E

P. L. Chu, B. A. Malomed, G. D. Peng, and I. M. Skinner, “Soliton dynamics in periodically modulated directional couplers,” Phys. Rev. E 49, 5763–5767 (1994).
[CrossRef]

A. B. Aceves, C. DeAngelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, “Discrete self-trapping, soliton interactions, and beam steering in nonlinear waveguide arrays,” Phys. Rev. E 53, 1172–1189 (1996).
[CrossRef]

Phys. Rev. Lett.

B. J. Eggleton, R. E. Slusher, C. M. deSterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. 76, 1627–1630 (1996).
[CrossRef] [PubMed]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. Boyd, and J. S. Aitchison, “Discrete spatial solitons in waveguide arrays,” Phys. Rev. Lett. 81, 3383–3386 (1998).
[CrossRef]

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Dynamics of discrete solitons in optical waveguide arrays,” Phys. Rev. Lett. 83, 2726–2729 (1999).
[CrossRef]

Physica D

T. S. Yang and W. L. Kath, “Radiation loss of dispersion-managed solitons in optical fibers,” Physica D 149, 80–94 (2001).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Schematics of the configuration under investigation with (a) a symmetric and (b) an asymmetric variation of the coupling constant.

Fig. 2
Fig. 2

Discrete soliton and its nontrivial bound states: (a) soliton, (b) quasi-translational mode, (c) symmetric nontrivial internal mode. β=2.53. Every circle corresponds to the amplitude in a guide: filled circles, real part; open circles, imaginary part.

Fig. 3
Fig. 3

Properties of discrete solitons as a function of the propagation constant. (a) Soliton power (dashed curve, approximation for high soliton power; Psolitonβ-2). (b) The shaded area represents the continuous spectrum, where solid curves denote eigenvalues of bound states.

Fig. 4
Fig. 4

Evolution of the field distribution and of the peak power for fixed initial conditions and varying strengths of symmetric periodic modulation (parameter of the excitation, A0=1.6; W0=1.4; period of modulation, L0=2).

Fig. 5
Fig. 5

Field evolution for several power levels and symmetric periodic modulation of the coupling constant (width of the excitation, W0=1.4; modulation, L0=2; δC=0.5).

Fig. 6
Fig. 6

Field distribution at the output facet for various input power levels for three periods Z of symmetric modulation (width of the excitation, W0=1.4; strength of modulation, δC=0.5).

Fig. 7
Fig. 7

Transmission through channels as functions of input power (same parameters as in Fig. 6; L0=2).

Fig. 8
Fig. 8

Field evolution for three power levels and an asymmetric periodic modulation (width of the excitation, W0=1.4; modulation, L0=1; δC=0.8).

Equations (16)

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

idandz+γ|an|2an+C0{[1+δC(z)nn+1]an+1+[1+δC(z)nn-1]an-1}=0,
idAndZ+|An|2An+[1+δC(z)nn+1]An+1+[1+δC(z)nn-1]An-1=0.
Z=zZ0,An=anP0,
An(Z)=An0(β)exp(iβZ),
Psol(β)=n|An0(β)|2.
An(Z)=[An0+δAn(Z)]exp(iβZ).
ddZA=L(A, β),L[A0(β), β]=0.
δA=k ckek exp(iλkZ).
Jˆ(β)=L(β)AA0.
Jˆek=iλkek
β-2<|λk|<β+2
|λk|<β-2
δCnn+1(Z)=δCnn-1(Z)=δC0 cos2πL0Z.
An(Z=0)=A0 exp-n2W2,
λkresβ-2Psol-2.
δCnn+1(Z)=δCn+1n(Z)=(-1)nδC0 cos2πL0Z,

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