Abstract

We study the interactions of a Bragg-grating soliton with a localized defect that is a combined perturbation of the grating and the refractive index of a fiber. A family of exact analytical solutions for solitons trapped by the deltalike defect is found. Direct simulations demonstrate that, up to the available numerical accuracy, the trapped soliton is stable at a single value of its intrinsic parameter θ. Depending on the parameter values, simulations of collisions between moving solitons and the defect show that the soliton can be captured by, pass through, or even bounce off the defect. If the defect is strong and the soliton is heavy enough, it may split into three fragments: trapped, transmitted, and reflected.

© 2003 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  8. Y. J. Rao, “Recent progress in applications of in-fibre Bragg grating sensors,” Opt. Lasers Eng. 31, 297-324 (1999).
    [CrossRef]
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  12. A. B. Aceves and S. Wabnitz, “Self-induced transparency solitons in nonlinear refractive periodic media,” Phys. Lett. A 141, 37-42 (1989).
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  17. J. E. Heebner, R. W. Boyd, and Q. H. Park, “Slow light, induced dispersion, enhanced nonlinearity, and optical solitons in a resonator-array waveguide,” Phys. Rev. E 65, 036619 (2002).
    [CrossRef]
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    [CrossRef]
  22. M. G. Vakhitov and A. A. Kolokolov, “Stationary solutions of the wave equation in media with nonlinearity saturation,” Sov. J. Quantum Electron. Radiophys. 16, 783-785 (1973) [Izv. Vyssh. Zav Radiofizika 16, 1020-1022 (1973)].
  23. B. A. Malomed and R. S. Tasgal, “Vibration modes of a gap soliton in a nonlinear optical medium,” Phys. Rev. E 49, 5787-5796 (1994).
    [CrossRef]
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    [CrossRef]

2002 (4)

K. Pran, G. B. Havsga˚rd, G. Sagvolden, Ø. Farsund, and G. Wang, “Wavelength multiplexed fibre Bragg grating system for high-strain health monitoring applications,” Meas. Sci. Technol. 13, 471-476 (2002).

J. E. Heebner, R. W. Boyd, and Q. H. Park, “Slow light, induced dispersion, enhanced nonlinearity, and optical solitons in a resonator-array waveguide,” Phys. Rev. E 65, 036619 (2002).
[CrossRef]

C. M. de Sterke, E. N. Tsoy, and J. E. Sipe, “Light trapping in a fiber grating defect by four-wave mixing,” Opt. Lett. 27, 485-487 (2002).
[CrossRef]

R. H. Goodman, R. E. Slusher, and M. I. Weinstein, “Stopping light on a defect,” J. Opt. Soc. Am. B 19, 1635-1652 (2002).
[CrossRef]

2001 (2)

T. Inui, T. Komukai T, and M. Nakazawa, “Highly efficient tunable fiber Bragg grating filters using multilayer piezoelectric transducers,” Opt. Commun. 190, 1-4 (2001).
[CrossRef]

H. S. Park, S. H. Yun, I. K. Hwang, S. B. Lee, and B. Y. Kim, “All-fiber add-drop wavelength-division multiplexer based on intermodal coupling,” IEEE Photon. Technol. Lett. 13, 460-462 (2001).
[CrossRef]

2000 (1)

K. T. McDonald, “Slow light,” Am. J. Phys. 68, 293-294 (2000).
[CrossRef]

1999 (3)

Y. J. Rao, “Recent progress in applications of in-fibre Bragg grating sensors,” Opt. Lasers Eng. 31, 297-324 (1999).
[CrossRef]

J. Marangos, “Slow light in cool atoms,” Nature 397, 559-560 (1999).
[CrossRef]

N. M. Litchinitser, B. J. Eggleton, C. M. de Sterke, A. B. Aceves, and G. P. Agrawal, “Interaction of Bragg solitons in fiber gratings,” J. Opt. Soc. Am. B 16, 18-23 (1999).
[CrossRef]

1998 (3)

I. V. Barashenkov, D. E. Pelinovsky, and E. V. Zemlyanaya, “Vibrations and oscillatory instabilities of gap solitons,” Phys. Rev. Lett. 80, 5117–5120 (1998).
[CrossRef]

J. M. Senior, S. E. Moss, and S. D. Cusworth, “Multiplexing techniques for noninterferometric optical point-sensor networks: a review,” Fiber Integr. Opt. 17, 3-20 (1998).
[CrossRef]

A. S. Kewitsch, G. A. Rakuljic, P. A. Willems, and A. Yariv, “All-fiber zero-insertion-loss add-drop filter for wavelength-division multiplexing,” Opt. Lett. 23, 106-108 (1998).
[CrossRef]

1997 (1)

C. M. de Sterke, B. J. Eggleton, and P. A. Krug, “High-intensity pulse propagation in uniform gratings and grating superstructures,” J. Lightwave Technol. 15, 1494-1502 (1997).
[CrossRef]

1996 (2)

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

B. J. Eggleton, T. Stephens, P. A. Krug, G. Dhosi, Z. Brodzeli, and F. Ouellette, “Dispersion compensation using a fibre grating in transmission,” Electron. Lett. 32, 1610-1611 (1996).
[CrossRef]

1994 (2)

C. M. de Sterke and J. E. Sipe, “Gap solitons,” Prog. Opt. 33, 203-260 (1994).
[CrossRef]

B. A. Malomed and R. S. Tasgal, “Vibration modes of a gap soliton in a nonlinear optical medium,” Phys. Rev. E 49, 5787-5796 (1994).
[CrossRef]

1989 (3)

A. B. Aceves and S. Wabnitz, “Self-induced transparency solitons in nonlinear refractive periodic media,” Phys. Lett. A 141, 37-42 (1989).
[CrossRef]

D. N. Christodoulides and R. I. Joseph, “Slow Bragg soli-tons in nonlinear periodic structures,” Phys. Rev. Lett. 62, 1746-1749 (1989).
[CrossRef] [PubMed]

Y. S. Kivshar and B. A. Malomed, “Dynamics of solitons in nearly integrable systems,” Rev. Mod. Phys. 61, 763-915 (1989).
[CrossRef]

1973 (1)

M. G. Vakhitov and A. A. Kolokolov, “Stationary solutions of the wave equation in media with nonlinearity saturation,” Sov. J. Quantum Electron. Radiophys. 16, 783-785 (1973) [Izv. Vyssh. Zav Radiofizika 16, 1020-1022 (1973)].

Aceves, A. B.

N. M. Litchinitser, B. J. Eggleton, C. M. de Sterke, A. B. Aceves, and G. P. Agrawal, “Interaction of Bragg solitons in fiber gratings,” J. Opt. Soc. Am. B 16, 18-23 (1999).
[CrossRef]

A. B. Aceves and S. Wabnitz, “Self-induced transparency solitons in nonlinear refractive periodic media,” Phys. Lett. A 141, 37-42 (1989).
[CrossRef]

Agrawal, G. P.

Barashenkov, I. V.

I. V. Barashenkov, D. E. Pelinovsky, and E. V. Zemlyanaya, “Vibrations and oscillatory instabilities of gap solitons,” Phys. Rev. Lett. 80, 5117–5120 (1998).
[CrossRef]

Boyd, R. W.

J. E. Heebner, R. W. Boyd, and Q. H. Park, “Slow light, induced dispersion, enhanced nonlinearity, and optical solitons in a resonator-array waveguide,” Phys. Rev. E 65, 036619 (2002).
[CrossRef]

Brodzeli, Z.

B. J. Eggleton, T. Stephens, P. A. Krug, G. Dhosi, Z. Brodzeli, and F. Ouellette, “Dispersion compensation using a fibre grating in transmission,” Electron. Lett. 32, 1610-1611 (1996).
[CrossRef]

Christodoulides, D. N.

D. N. Christodoulides and R. I. Joseph, “Slow Bragg soli-tons in nonlinear periodic structures,” Phys. Rev. Lett. 62, 1746-1749 (1989).
[CrossRef] [PubMed]

Cusworth, S. D.

J. M. Senior, S. E. Moss, and S. D. Cusworth, “Multiplexing techniques for noninterferometric optical point-sensor networks: a review,” Fiber Integr. Opt. 17, 3-20 (1998).
[CrossRef]

de Sterke, C. M.

C. M. de Sterke, E. N. Tsoy, and J. E. Sipe, “Light trapping in a fiber grating defect by four-wave mixing,” Opt. Lett. 27, 485-487 (2002).
[CrossRef]

N. M. Litchinitser, B. J. Eggleton, C. M. de Sterke, A. B. Aceves, and G. P. Agrawal, “Interaction of Bragg solitons in fiber gratings,” J. Opt. Soc. Am. B 16, 18-23 (1999).
[CrossRef]

C. M. de Sterke, B. J. Eggleton, and P. A. Krug, “High-intensity pulse propagation in uniform gratings and grating superstructures,” J. Lightwave Technol. 15, 1494-1502 (1997).
[CrossRef]

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

C. M. de Sterke and J. E. Sipe, “Gap solitons,” Prog. Opt. 33, 203-260 (1994).
[CrossRef]

Dhosi, G.

B. J. Eggleton, T. Stephens, P. A. Krug, G. Dhosi, Z. Brodzeli, and F. Ouellette, “Dispersion compensation using a fibre grating in transmission,” Electron. Lett. 32, 1610-1611 (1996).
[CrossRef]

Eggleton, B. J.

N. M. Litchinitser, B. J. Eggleton, C. M. de Sterke, A. B. Aceves, and G. P. Agrawal, “Interaction of Bragg solitons in fiber gratings,” J. Opt. Soc. Am. B 16, 18-23 (1999).
[CrossRef]

C. M. de Sterke, B. J. Eggleton, and P. A. Krug, “High-intensity pulse propagation in uniform gratings and grating superstructures,” J. Lightwave Technol. 15, 1494-1502 (1997).
[CrossRef]

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

B. J. Eggleton, T. Stephens, P. A. Krug, G. Dhosi, Z. Brodzeli, and F. Ouellette, “Dispersion compensation using a fibre grating in transmission,” Electron. Lett. 32, 1610-1611 (1996).
[CrossRef]

Farsund, Ø.

K. Pran, G. B. Havsga˚rd, G. Sagvolden, Ø. Farsund, and G. Wang, “Wavelength multiplexed fibre Bragg grating system for high-strain health monitoring applications,” Meas. Sci. Technol. 13, 471-476 (2002).

Goodman, R. H.

Havsga°rd, G. B.

K. Pran, G. B. Havsga˚rd, G. Sagvolden, Ø. Farsund, and G. Wang, “Wavelength multiplexed fibre Bragg grating system for high-strain health monitoring applications,” Meas. Sci. Technol. 13, 471-476 (2002).

Heebner, J. E.

J. E. Heebner, R. W. Boyd, and Q. H. Park, “Slow light, induced dispersion, enhanced nonlinearity, and optical solitons in a resonator-array waveguide,” Phys. Rev. E 65, 036619 (2002).
[CrossRef]

Hwang, I. K.

H. S. Park, S. H. Yun, I. K. Hwang, S. B. Lee, and B. Y. Kim, “All-fiber add-drop wavelength-division multiplexer based on intermodal coupling,” IEEE Photon. Technol. Lett. 13, 460-462 (2001).
[CrossRef]

Inui, T.

T. Inui, T. Komukai T, and M. Nakazawa, “Highly efficient tunable fiber Bragg grating filters using multilayer piezoelectric transducers,” Opt. Commun. 190, 1-4 (2001).
[CrossRef]

Joseph, R. I.

D. N. Christodoulides and R. I. Joseph, “Slow Bragg soli-tons in nonlinear periodic structures,” Phys. Rev. Lett. 62, 1746-1749 (1989).
[CrossRef] [PubMed]

Kewitsch, A. S.

Kim, B. Y.

H. S. Park, S. H. Yun, I. K. Hwang, S. B. Lee, and B. Y. Kim, “All-fiber add-drop wavelength-division multiplexer based on intermodal coupling,” IEEE Photon. Technol. Lett. 13, 460-462 (2001).
[CrossRef]

Kivshar, Y. S.

Y. S. Kivshar and B. A. Malomed, “Dynamics of solitons in nearly integrable systems,” Rev. Mod. Phys. 61, 763-915 (1989).
[CrossRef]

Kolokolov, A. A.

M. G. Vakhitov and A. A. Kolokolov, “Stationary solutions of the wave equation in media with nonlinearity saturation,” Sov. J. Quantum Electron. Radiophys. 16, 783-785 (1973) [Izv. Vyssh. Zav Radiofizika 16, 1020-1022 (1973)].

Komukai T, T.

T. Inui, T. Komukai T, and M. Nakazawa, “Highly efficient tunable fiber Bragg grating filters using multilayer piezoelectric transducers,” Opt. Commun. 190, 1-4 (2001).
[CrossRef]

Krug, P. A.

C. M. de Sterke, B. J. Eggleton, and P. A. Krug, “High-intensity pulse propagation in uniform gratings and grating superstructures,” J. Lightwave Technol. 15, 1494-1502 (1997).
[CrossRef]

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

B. J. Eggleton, T. Stephens, P. A. Krug, G. Dhosi, Z. Brodzeli, and F. Ouellette, “Dispersion compensation using a fibre grating in transmission,” Electron. Lett. 32, 1610-1611 (1996).
[CrossRef]

Lee, S. B.

H. S. Park, S. H. Yun, I. K. Hwang, S. B. Lee, and B. Y. Kim, “All-fiber add-drop wavelength-division multiplexer based on intermodal coupling,” IEEE Photon. Technol. Lett. 13, 460-462 (2001).
[CrossRef]

Litchinitser, N. M.

Malomed, B. A.

B. A. Malomed and R. S. Tasgal, “Vibration modes of a gap soliton in a nonlinear optical medium,” Phys. Rev. E 49, 5787-5796 (1994).
[CrossRef]

Y. S. Kivshar and B. A. Malomed, “Dynamics of solitons in nearly integrable systems,” Rev. Mod. Phys. 61, 763-915 (1989).
[CrossRef]

Marangos, J.

J. Marangos, “Slow light in cool atoms,” Nature 397, 559-560 (1999).
[CrossRef]

McDonald, K. T.

K. T. McDonald, “Slow light,” Am. J. Phys. 68, 293-294 (2000).
[CrossRef]

Moss, S. E.

J. M. Senior, S. E. Moss, and S. D. Cusworth, “Multiplexing techniques for noninterferometric optical point-sensor networks: a review,” Fiber Integr. Opt. 17, 3-20 (1998).
[CrossRef]

Nakazawa, M.

T. Inui, T. Komukai T, and M. Nakazawa, “Highly efficient tunable fiber Bragg grating filters using multilayer piezoelectric transducers,” Opt. Commun. 190, 1-4 (2001).
[CrossRef]

Ouellette, F.

B. J. Eggleton, T. Stephens, P. A. Krug, G. Dhosi, Z. Brodzeli, and F. Ouellette, “Dispersion compensation using a fibre grating in transmission,” Electron. Lett. 32, 1610-1611 (1996).
[CrossRef]

Park, H. S.

H. S. Park, S. H. Yun, I. K. Hwang, S. B. Lee, and B. Y. Kim, “All-fiber add-drop wavelength-division multiplexer based on intermodal coupling,” IEEE Photon. Technol. Lett. 13, 460-462 (2001).
[CrossRef]

Park, Q. H.

J. E. Heebner, R. W. Boyd, and Q. H. Park, “Slow light, induced dispersion, enhanced nonlinearity, and optical solitons in a resonator-array waveguide,” Phys. Rev. E 65, 036619 (2002).
[CrossRef]

Pelinovsky, D. E.

I. V. Barashenkov, D. E. Pelinovsky, and E. V. Zemlyanaya, “Vibrations and oscillatory instabilities of gap solitons,” Phys. Rev. Lett. 80, 5117–5120 (1998).
[CrossRef]

Pran, K.

K. Pran, G. B. Havsga˚rd, G. Sagvolden, Ø. Farsund, and G. Wang, “Wavelength multiplexed fibre Bragg grating system for high-strain health monitoring applications,” Meas. Sci. Technol. 13, 471-476 (2002).

Rakuljic, G. A.

Rao, Y. J.

Y. J. Rao, “Recent progress in applications of in-fibre Bragg grating sensors,” Opt. Lasers Eng. 31, 297-324 (1999).
[CrossRef]

Sagvolden, G.

K. Pran, G. B. Havsga˚rd, G. Sagvolden, Ø. Farsund, and G. Wang, “Wavelength multiplexed fibre Bragg grating system for high-strain health monitoring applications,” Meas. Sci. Technol. 13, 471-476 (2002).

Senior, J. M.

J. M. Senior, S. E. Moss, and S. D. Cusworth, “Multiplexing techniques for noninterferometric optical point-sensor networks: a review,” Fiber Integr. Opt. 17, 3-20 (1998).
[CrossRef]

Sipe, J. E.

C. M. de Sterke, E. N. Tsoy, and J. E. Sipe, “Light trapping in a fiber grating defect by four-wave mixing,” Opt. Lett. 27, 485-487 (2002).
[CrossRef]

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

C. M. de Sterke and J. E. Sipe, “Gap solitons,” Prog. Opt. 33, 203-260 (1994).
[CrossRef]

Slusher, R. E.

R. H. Goodman, R. E. Slusher, and M. I. Weinstein, “Stopping light on a defect,” J. Opt. Soc. Am. B 19, 1635-1652 (2002).
[CrossRef]

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

Stephens, T.

B. J. Eggleton, T. Stephens, P. A. Krug, G. Dhosi, Z. Brodzeli, and F. Ouellette, “Dispersion compensation using a fibre grating in transmission,” Electron. Lett. 32, 1610-1611 (1996).
[CrossRef]

Tasgal, R. S.

B. A. Malomed and R. S. Tasgal, “Vibration modes of a gap soliton in a nonlinear optical medium,” Phys. Rev. E 49, 5787-5796 (1994).
[CrossRef]

Tsoy, E. N.

Vakhitov, M. G.

M. G. Vakhitov and A. A. Kolokolov, “Stationary solutions of the wave equation in media with nonlinearity saturation,” Sov. J. Quantum Electron. Radiophys. 16, 783-785 (1973) [Izv. Vyssh. Zav Radiofizika 16, 1020-1022 (1973)].

Wabnitz, S.

A. B. Aceves and S. Wabnitz, “Self-induced transparency solitons in nonlinear refractive periodic media,” Phys. Lett. A 141, 37-42 (1989).
[CrossRef]

Wang, G.

K. Pran, G. B. Havsga˚rd, G. Sagvolden, Ø. Farsund, and G. Wang, “Wavelength multiplexed fibre Bragg grating system for high-strain health monitoring applications,” Meas. Sci. Technol. 13, 471-476 (2002).

Weinstein, M. I.

Willems, P. A.

Yariv, A.

Yun, S. H.

H. S. Park, S. H. Yun, I. K. Hwang, S. B. Lee, and B. Y. Kim, “All-fiber add-drop wavelength-division multiplexer based on intermodal coupling,” IEEE Photon. Technol. Lett. 13, 460-462 (2001).
[CrossRef]

Zemlyanaya, E. V.

I. V. Barashenkov, D. E. Pelinovsky, and E. V. Zemlyanaya, “Vibrations and oscillatory instabilities of gap solitons,” Phys. Rev. Lett. 80, 5117–5120 (1998).
[CrossRef]

Am. J. Phys. (1)

K. T. McDonald, “Slow light,” Am. J. Phys. 68, 293-294 (2000).
[CrossRef]

Electron. Lett. (1)

B. J. Eggleton, T. Stephens, P. A. Krug, G. Dhosi, Z. Brodzeli, and F. Ouellette, “Dispersion compensation using a fibre grating in transmission,” Electron. Lett. 32, 1610-1611 (1996).
[CrossRef]

Fiber Integr. Opt. (1)

J. M. Senior, S. E. Moss, and S. D. Cusworth, “Multiplexing techniques for noninterferometric optical point-sensor networks: a review,” Fiber Integr. Opt. 17, 3-20 (1998).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

H. S. Park, S. H. Yun, I. K. Hwang, S. B. Lee, and B. Y. Kim, “All-fiber add-drop wavelength-division multiplexer based on intermodal coupling,” IEEE Photon. Technol. Lett. 13, 460-462 (2001).
[CrossRef]

Izv. Vyssh. Zav Radiofizika (1)

M. G. Vakhitov and A. A. Kolokolov, “Stationary solutions of the wave equation in media with nonlinearity saturation,” Sov. J. Quantum Electron. Radiophys. 16, 783-785 (1973) [Izv. Vyssh. Zav Radiofizika 16, 1020-1022 (1973)].

J. Lightwave Technol. (1)

C. M. de Sterke, B. J. Eggleton, and P. A. Krug, “High-intensity pulse propagation in uniform gratings and grating superstructures,” J. Lightwave Technol. 15, 1494-1502 (1997).
[CrossRef]

J. Opt. Soc. Am. B (2)

Meas. Sci. Technol. (1)

K. Pran, G. B. Havsga˚rd, G. Sagvolden, Ø. Farsund, and G. Wang, “Wavelength multiplexed fibre Bragg grating system for high-strain health monitoring applications,” Meas. Sci. Technol. 13, 471-476 (2002).

Nature (1)

J. Marangos, “Slow light in cool atoms,” Nature 397, 559-560 (1999).
[CrossRef]

Opt. Commun. (1)

T. Inui, T. Komukai T, and M. Nakazawa, “Highly efficient tunable fiber Bragg grating filters using multilayer piezoelectric transducers,” Opt. Commun. 190, 1-4 (2001).
[CrossRef]

Opt. Lasers Eng. (1)

Y. J. Rao, “Recent progress in applications of in-fibre Bragg grating sensors,” Opt. Lasers Eng. 31, 297-324 (1999).
[CrossRef]

Opt. Lett. (2)

Phys. Lett. A (1)

A. B. Aceves and S. Wabnitz, “Self-induced transparency solitons in nonlinear refractive periodic media,” Phys. Lett. A 141, 37-42 (1989).
[CrossRef]

Phys. Rev. E (2)

J. E. Heebner, R. W. Boyd, and Q. H. Park, “Slow light, induced dispersion, enhanced nonlinearity, and optical solitons in a resonator-array waveguide,” Phys. Rev. E 65, 036619 (2002).
[CrossRef]

B. A. Malomed and R. S. Tasgal, “Vibration modes of a gap soliton in a nonlinear optical medium,” Phys. Rev. E 49, 5787-5796 (1994).
[CrossRef]

Phys. Rev. Lett. (3)

I. V. Barashenkov, D. E. Pelinovsky, and E. V. Zemlyanaya, “Vibrations and oscillatory instabilities of gap solitons,” Phys. Rev. Lett. 80, 5117–5120 (1998).
[CrossRef]

D. N. Christodoulides and R. I. Joseph, “Slow Bragg soli-tons in nonlinear periodic structures,” Phys. Rev. Lett. 62, 1746-1749 (1989).
[CrossRef] [PubMed]

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

Prog. Opt. (1)

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[CrossRef]

Rev. Mod. Phys. (1)

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[CrossRef]

Other (2)

W. Mak, B. A. Malomed, and P. L. Chu, “Interaction of a soliton with a localized gain in a fiber Bragg grating,” submitted to Phys. Rev. E.

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Figures (7)

Fig. 1
Fig. 1

Real (even) and imaginary (odd) parts of the field U(x) for the trapped soliton, with κ=0.4 and θ=0.5π. Dashed curves depict the exact analytical solution [Eqs. (10)] obtained for the ideal δ function. Solid curves, the numerical solution, with the δ function approximated in accordance with Eq. (23) with N=2. The two pairs of curves are nearly indiscernible. We also verified that using an approximation closer to the ideal δ function brings the numerically found waveform still closer to the exact solution.

Fig. 2
Fig. 2

Typical results illustrating the stability and instability of the trapped solitons for λ=0 and κ=0.08. In each case a side view of the evolution of |U|, starting with the initial stationary trapped soliton configuration, is shown at the left. At the right (or as insets) the evolution in terms of contour plots is shown. T is the total simulation time. (a) θin=0.4π is smaller than θstab. The soliton decays into radiation. (b) θin=0.5πθstab. The field directly self-traps into a stable soliton. (c) θ=0.7π>θstab. The field evolves into a stable soliton, shedding excess energy in the form of radiation. (d) θ=0.9π. This value is much larger than θstab, and the pulse decays into radiation, generating a small residual soliton thrown away from the defect.

Fig. 3
Fig. 3

Region in the plane (κ, θin) between the two borders that gives rise to the stable soliton trapped by the local defect with λ=0. In fact, in the region of relatively large values of κ the lower border becomes fuzzy: As is explained in the text, no systematic deviation from θ=π/2 was found, but a deviation within a margin of Δθ=0.03π is not ruled out, as the instability of unstable solitons becomes slow, impeding the monitoring of their evolution up to a definite result. Initial solitons taken outside the stability region decay (in the region above the upper stability border, a residual small-amplitude soliton can be found, that has been flung away from the localized defect).

Fig. 4
Fig. 4

Collision between a moving soliton with a fixed value of θ=π/2 and the defect. In each case the lower and upper panels show, respectively, the evolution of the field |u(x, t)| and the waveforms |u(x)| (solid curve) and |v(x)| (dashed curve) at the end of the simulation. (a) A weak defect with κ=0.1 cannot capture the moving soliton (with velocity c=0.3), which passes through it. (b) A stronger defect, with κ=0.2, captures the soliton with velocity c=0.2, which is accompanied by a conspicuous emission of radiation. The soliton oscillates about the defect after being captured. (c) When the defect’s strength is larger, κ=0.3, and the soliton is faster, c=0.5, the energy splits into three parts as a result of the collision. Some energy is trapped at the defect to form a trapped soliton. A smaller amount bounces back, while a large portion passes through and self-traps into a secondary free soliton. (d) For a still stronger defect, with κ=0.6, less energy passes through, while larger shares of the energy are trapped at the defect and reflected back. (e) In the strongest defect in this series of the simulations, with κ=0.8, most energy is reflected back, while the amounts of energy trapped and transmitted through the defect are smaller.

Fig. 5
Fig. 5

Regions in the parametric plane (c, κ) with fixed θ=π/2 in which the moving soliton passes through the defect, is captured by it, or splits into three parts.

Fig. 6
Fig. 6

Collision of the moving soliton with the defect at a fixed value of the velocity, c=0.075. Upper and lower parts have the same meaning as in Fig. 4. An example of the capture of a relatively light soliton is not included, as it is similar to that in Fig. 4(b): (a) κ=0.5, θ=0.7π. The defect is repulsive, and the soliton bounces back after collision. (b) κ=0.1, θ=0.9π. The soliton overcomes the repulsive barrier and passes through the defect with a considerable loss of energy. The radiation loss is much larger than in the case depicted in Fig. 4(a), where the soliton was lighter, with θ=0.5π.

Fig. 7
Fig. 7

Summary of results obtained for the interaction of the moving soliton and the local defect when the soliton’s velocity is kept constant, c=0.075. In region (A) the defect is attractive for solitons with small θ, and the soliton is captured by it. In region (B) the defect is repulsive for solitons with larger θ, so the soliton bounces back. In region (C) the defect is weak; the soliton passes through it, regardless of whether the interaction is attractive or repulsive. In region (D) the defect is strong and, simultaneously, the soliton is heavy. The interaction combines repulsive and attractive features, and the soliton splits into three parts. One part is trapped, the remaining energy is scattered in forward and backward directions.

Equations (33)

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i ut+i ux+v+12 |u|2+|v|2u
=κδ(x)v+λδ(x)u,
i vt-i vx+u+12 |v|2+|u|2v
=κδ(x)u+λδ(x)v,
H=-+12 i-u*ux+v*vx+u u*x-v v*x-14 (|u|4+|v|4)-|u|2|v|2-(u*v+uv*)+Hint.
Hint=[κ(u*v+uv*)+λ(|u|2+|v|2)]|x=0.
u=U(x)exp(-iωt),v=V(x)exp(-iωt),
ωU+iU+V+(½|U|2+|V|2)U
=κδ(x)V+λδ(x)U,
ωV-iV+U+(½|V|2+|U|2)V
=κδ(x)U+λδ(x)V,
U(x)=2/3(sin θ)sech(x-ξ)sin θ-i2 θ,
V=-U*,
0<θ<π,
ω=cos θ.
U=23 (sin θ)sech(x+a sgn x)sin θ-i2 θ,
V=-V*,
tanh(a sin θ)=κ-λκ+λtanh(κ2-λ2/2)tan(θ/2).
tanh(a sin θ)=tanh(κ/2)tan(θ/2).
|U(x=0)|2=23sin2 θsinh2(a sin2 θ)+cos2 θ.
(Hint)trapped=-2(κ-λ)|U(x=0)|2.
tanh(a sin θ)=-λ-κλ+κtan(λ2-κ2/2)tan(θ/2),
tanh(a sin θ)=-tan(λ/2)tan(θ/2)
tanh(a sin θ)=κtan(θ/2)-λtan(θ/2).
θmin<θ<π,
θmin=2 tan-1κ-λκ+λtanh(κ2-λ2/2).
E=83θ-π2-sin-1(sech κ)sgn κ.
0<θ<θcr,θcr1.011(π/2)
Uint(ξ)=83(λ-κ)sin2 θcosh(2ξ sin θ)+cos θ
δ˜[xn-(N+1)]
A cosn-(N+1)2N+1 πn=1,, 2N+10otherwise.
A=Δx n=12N+1cosn-(N+1)2N+1 π-1,
θin(0)>[θin(0)]minsin-1(sech κ).

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