Abstract

Gap solitons are localized nonlinear coherent states that have been shown both theoretically and experimentally to propagate in periodic structures. Although theory allows for their propagation at any speed v, 0vc, they have been observed in experiments at speeds of approximately 50% of c. It is of scientific and technological interest to trap gap solitons. We first introduce an explicit multiparameter family of periodic structures with localized defects, which support linear defect modes. These linear defect modes are shown to persist into the nonlinear regime, as nonlinear defect modes. Using mathematical analysis and numerical simulations, we then investigate the capture of an incident gap soliton by these defects. The mechanism of capture of a gap soliton is resonant transfer of its energy to nonlinear defect modes. We introduce a useful bifurcation diagram from which information on the parameter regimes of gap-soliton capture, reflection, and transmission can be obtained by simple conservation of energy and resonant energy transfer principles.

© 2002 Optical Society of America

Full Article  |  PDF Article

Errata

Roy H. Goodman, Richard E. Slusher, and Michael I. Weinstein, "Stopping light on a defect: erratum," J. Opt. Soc. Am. B 19, 2737-2737 (2002)
https://www.osapublishing.org/josab/abstract.cfm?uri=josab-19-11-2737

References

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  1. A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion,” Appl. Phys. Lett. 23, 142–144 (1971).
    [CrossRef]
  2. L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Experimental observation of picosecond pulse narrowing and solitons in optical fibers,” Phys. Rev. Lett. 45, 1095–1098 (1980).
    [CrossRef]
  3. G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic, San Diego, 2001).
  4. G. P. Agrawal, Fiber-Optic Communication Systems (Wiley, New York, 1997).
  5. W. Chen and D. L. Mills, “Gap solitons and the nonlinear optical response of superlattices,” Phys. Rev. Lett. 58, 160–163 (1987).
    [CrossRef] [PubMed]
  6. D. N. Christodoulides and R. I. Joseph, “Slow Bragg solitons in nonlinear periodic structures,” Phys. Rev. Lett. 62, 1746–1749 (1989).
    [CrossRef] [PubMed]
  7. A. B. Aceves and S. Wabnitz, “Self induced transparency solitons in nonlinear refractive periodic media,” Phys. Lett. A 141, 37–42 (1989).
    [CrossRef]
  8. C. M. de Sterke and J. E. Sipe, “Gap solitons,” Prog. Opt. 33, 203–260 (1994).
    [CrossRef]
  9. B. J. Eggleton, C. M. de Sterke, and R. E. Slusher, “Nonlinear pulse propagation in Bragg gratings,” J. Opt. Soc. Am. B 14, 2980–2992 (1997).
    [CrossRef]
  10. P. Millar, R. M. De La Rue, T. F. Krauss, J. S. Aitchison, N. G. R. Broderick, and D. J. Richardson, “Nonlinear propagation effects in an AlGaAs Bragg grating filter,” Opt. Lett. 24, 685–687 (1999).
    [CrossRef]
  11. N. G. R. Broderick, D. J. Richardson, and M. Ibsen, “Nonlinear switching in a 20-cm-long fiber Bragg grating,” Opt. Lett. 25, 536–538 (2000).
    [CrossRef]
  12. Experimentally it is hard to get to low velocities because most of the incident energy on the grating is reflected by the abrupt change from uniform to modulated index. Simulations show that for very gradually apodized gratings, over many Bragg lengths, one can hope to easily get down to 1/10 c/n, even with intensities that do not destroy the grating. Ideas like Raman gap solitons13 would allow generation of the gap soliton in the grating, and one would avoid these impedance-mismatch problems.
  13. H. G. Winful and V. Perlin, “Raman gap solitons,” Phys. Rev. Lett. 84, 3586–3589 (1999).
    [CrossRef]
  14. N. G. R. Broderick and C. M. de Sterke, “Approximate method for gap soliton propagation in nonuniform Bragg gratings,” Phys. Rev. E 58, 7941–7950 (1998).
    [CrossRef]
  15. R. H. Goodman, P. J. Holmes, and M. I. Weinstein, “Interaction of sine-Gordon kinks with defects: phase space transport in a two-mode model,” Physica D 161, 21–41 (2002).
    [CrossRef]
  16. R. H. Goodman, P. J. Holmes, and M. I. Weinstein, “Interaction of NLS solitons with defects,” http://arXiv.org/abs/nlin/0203057 (to be published).
  17. P. J. Holmes, R. H. Goodman, and M. I. Weinstein, “Trapping of kinks and solitons by defects: phase space transport in finite dimensional models,” presented at the International Conference on Progress in Nonlinear Science dedicated to Alexander Andronov, Nizhny Novgorod, Russia, July 2001.
  18. J. E. Sipe, L. Poladian, and C. M. de Sterke, “Propagation through nonuniform grating structures,” J. Opt. Soc. Am. A 11, 1307–1320 (1994).
    [CrossRef]
  19. C. M. de Sterke, “Wave propagation through nonuniform gratings with slowly varying parameters,” J. Lightwave Technol. 17, 2405–2411 (1999).
    [CrossRef]
  20. E. N. Tsoy and C. M. de Sterke, “Soliton dynamics in nonuniform fiber Bragg gratings,” J. Opt. Soc. Am. B 18, 1–6 (2001).
    [CrossRef]
  21. R. E. Slusher, B. J. Eggleton, T. A. Strasser, and C. M. de Sterke, “Nonlinear pulse reflections from chirped fiber gratings,” Opt. Express 3, 465–475 (1998).
    [CrossRef] [PubMed]
  22. E. N. Tsoy and C. M. de Sterke, “Propagation of nonlinear pulses in chirped fiber gratings,” Phys. Rev. E 62, 2882–2890 (2000).
    [CrossRef]
  23. C. M. de Sterke, B. J. Eggleton, and P. A. Krug, “High-intensity pulse propagation in uniform gratings and grating superstructures,” J. Lighwave Technol. 15, 1494–1502 (1997).
    [CrossRef]
  24. G. Lenz and B. J. Eggleton, “Adiabatic Bragg soliton compression in nonuniform grating structures,” J. Opt. Soc. Am. B 15, 2979–2985 (1998).
    [CrossRef]
  25. We work with the model (2.6)–(2.7) since it yields a simple derivation of the envelope equations (2.10). The situation is, however, a bit more complicated. Although model (2.6)–(2.7) incorporates the effects of photonic band dispersion, this alone is insufficient to arrest optical carrier shock formation on the relevant temporal and spatial scales.27 In fact, a valid envelope description in the absence of material dispersion would require the incorporation of coupling to all higher harmonics since they are in resonance.
  26. R. H. Goodman, M. I. Weinstein, and P. J. Holmes, “Nonlinear propagation of light in one-dimensional periodic structures,” J. Nonlinear Sci. 11, 123–168 (2001).
    [CrossRef]
  27. It is common to slightly redefine κ˜ by κ˜(z)=ηπΔnν(z)/λB, where 0<η<1 is defined as an overlap integral of the radial variation of the forward and backward modes and represents the fraction of total energy in the core of the fiber.
  28. 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]
  29. G.-H. Song and S.-Y. Shin, “Design of corrugated waveguide filters by the Gel’fand–Levitan–Marchenko inverse scattering method,” J. Opt. Soc. Am. A 2, 1905–1915 (1985).
    [CrossRef]
  30. M. I. Weinstein, “Notes on wave propagation in 1-d periodic media with defects,” Bell Labs Technical Memorandum (Lucent Technologies, Murray Hill, N.J., 1999).
  31. V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).
  32. V. E. Zakharov and A. B. Shabat, “Interaction between solitons in a stable medium,” Sov. Phys. JETP 37, 823–828 (1973).
  33. H. Rose and M. I. Weinstein, “On the bound states of the nonlinear Schrödinger equation with a linear potential,” Physica D 30, 207–218 (1988).
    [CrossRef]
  34. A. Soffer and M. I. Weinstein, “Selection of the ground state in nonlinear Schrödinger equations,” 2001 preprint (to be published).
  35. A. A. Sukhorukov and Y. S. Kivshar, “Nonlinear localized waves in a periodic medium,” Phys. Rev. Lett. 87, 083901 (2001).
    [CrossRef] [PubMed]
  36. Z. Fei, Y. S. Kivshar, and L. Vázquez, “Resonant kink-impurity interactions in the ø4 model,” Phys. Rev. A 46, 5214–5220 (1992).
    [CrossRef] [PubMed]
  37. Z. Fei, Y. S. Kivshar, and L. Vázquez, “Resonant kink-impurity interactions in the sine-Gordon model,” Phys. Rev. A 45, 6019–6030 (1992).
    [CrossRef]
  38. K. Forinash, M. Peyrard, and B. Malomed, “Interaction of discrete breathers with impurity modes,” Phys. Rev. E 49, 3400–3411 (1994).
    [CrossRef]
  39. B. A. Malomed, “Inelastic interactions of solitons in nearly integrable systems 2,” Physica D 15, 385–401 (1985).
    [CrossRef]
  40. E. C. Titchmarsh, Eigenfunction Expansions Associated with Second-Order Differential Equations. Part I (Clarendon, Oxford, 1962).

2002

R. H. Goodman, P. J. Holmes, and M. I. Weinstein, “Interaction of sine-Gordon kinks with defects: phase space transport in a two-mode model,” Physica D 161, 21–41 (2002).
[CrossRef]

2001

E. N. Tsoy and C. M. de Sterke, “Soliton dynamics in nonuniform fiber Bragg gratings,” J. Opt. Soc. Am. B 18, 1–6 (2001).
[CrossRef]

R. H. Goodman, M. I. Weinstein, and P. J. Holmes, “Nonlinear propagation of light in one-dimensional periodic structures,” J. Nonlinear Sci. 11, 123–168 (2001).
[CrossRef]

A. A. Sukhorukov and Y. S. Kivshar, “Nonlinear localized waves in a periodic medium,” Phys. Rev. Lett. 87, 083901 (2001).
[CrossRef] [PubMed]

2000

E. N. Tsoy and C. M. de Sterke, “Propagation of nonlinear pulses in chirped fiber gratings,” Phys. Rev. E 62, 2882–2890 (2000).
[CrossRef]

N. G. R. Broderick, D. J. Richardson, and M. Ibsen, “Nonlinear switching in a 20-cm-long fiber Bragg grating,” Opt. Lett. 25, 536–538 (2000).
[CrossRef]

1999

1998

G. Lenz and B. J. Eggleton, “Adiabatic Bragg soliton compression in nonuniform grating structures,” J. Opt. Soc. Am. B 15, 2979–2985 (1998).
[CrossRef]

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]

N. G. R. Broderick and C. M. de Sterke, “Approximate method for gap soliton propagation in nonuniform Bragg gratings,” Phys. Rev. E 58, 7941–7950 (1998).
[CrossRef]

R. E. Slusher, B. J. Eggleton, T. A. Strasser, and C. M. de Sterke, “Nonlinear pulse reflections from chirped fiber gratings,” Opt. Express 3, 465–475 (1998).
[CrossRef] [PubMed]

1997

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

B. J. Eggleton, C. M. de Sterke, and R. E. Slusher, “Nonlinear pulse propagation in Bragg gratings,” J. Opt. Soc. Am. B 14, 2980–2992 (1997).
[CrossRef]

1994

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

J. E. Sipe, L. Poladian, and C. M. de Sterke, “Propagation through nonuniform grating structures,” J. Opt. Soc. Am. A 11, 1307–1320 (1994).
[CrossRef]

K. Forinash, M. Peyrard, and B. Malomed, “Interaction of discrete breathers with impurity modes,” Phys. Rev. E 49, 3400–3411 (1994).
[CrossRef]

1992

Z. Fei, Y. S. Kivshar, and L. Vázquez, “Resonant kink-impurity interactions in the ø4 model,” Phys. Rev. A 46, 5214–5220 (1992).
[CrossRef] [PubMed]

Z. Fei, Y. S. Kivshar, and L. Vázquez, “Resonant kink-impurity interactions in the sine-Gordon model,” Phys. Rev. A 45, 6019–6030 (1992).
[CrossRef]

1989

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

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

1988

H. Rose and M. I. Weinstein, “On the bound states of the nonlinear Schrödinger equation with a linear potential,” Physica D 30, 207–218 (1988).
[CrossRef]

1987

W. Chen and D. L. Mills, “Gap solitons and the nonlinear optical response of superlattices,” Phys. Rev. Lett. 58, 160–163 (1987).
[CrossRef] [PubMed]

1985

1980

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Experimental observation of picosecond pulse narrowing and solitons in optical fibers,” Phys. Rev. Lett. 45, 1095–1098 (1980).
[CrossRef]

1973

V. E. Zakharov and A. B. Shabat, “Interaction between solitons in a stable medium,” Sov. Phys. JETP 37, 823–828 (1973).

1972

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).

1971

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion,” Appl. Phys. Lett. 23, 142–144 (1971).
[CrossRef]

Aceves, A. B.

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

Aitchison, J. S.

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]

Broderick, N. G. R.

Chen, W.

W. Chen and D. L. Mills, “Gap solitons and the nonlinear optical response of superlattices,” Phys. Rev. Lett. 58, 160–163 (1987).
[CrossRef] [PubMed]

Christodoulides, D. N.

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

De La Rue, R. M.

de Sterke, C. M.

Eggleton, B. J.

Fei, Z.

Z. Fei, Y. S. Kivshar, and L. Vázquez, “Resonant kink-impurity interactions in the ø4 model,” Phys. Rev. A 46, 5214–5220 (1992).
[CrossRef] [PubMed]

Z. Fei, Y. S. Kivshar, and L. Vázquez, “Resonant kink-impurity interactions in the sine-Gordon model,” Phys. Rev. A 45, 6019–6030 (1992).
[CrossRef]

Forinash, K.

K. Forinash, M. Peyrard, and B. Malomed, “Interaction of discrete breathers with impurity modes,” Phys. Rev. E 49, 3400–3411 (1994).
[CrossRef]

Goodman, R. H.

R. H. Goodman, P. J. Holmes, and M. I. Weinstein, “Interaction of sine-Gordon kinks with defects: phase space transport in a two-mode model,” Physica D 161, 21–41 (2002).
[CrossRef]

R. H. Goodman, M. I. Weinstein, and P. J. Holmes, “Nonlinear propagation of light in one-dimensional periodic structures,” J. Nonlinear Sci. 11, 123–168 (2001).
[CrossRef]

Gordon, J. P.

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Experimental observation of picosecond pulse narrowing and solitons in optical fibers,” Phys. Rev. Lett. 45, 1095–1098 (1980).
[CrossRef]

Hasegawa, A.

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion,” Appl. Phys. Lett. 23, 142–144 (1971).
[CrossRef]

Holmes, P. J.

R. H. Goodman, P. J. Holmes, and M. I. Weinstein, “Interaction of sine-Gordon kinks with defects: phase space transport in a two-mode model,” Physica D 161, 21–41 (2002).
[CrossRef]

R. H. Goodman, M. I. Weinstein, and P. J. Holmes, “Nonlinear propagation of light in one-dimensional periodic structures,” J. Nonlinear Sci. 11, 123–168 (2001).
[CrossRef]

Ibsen, M.

Joseph, R. I.

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

Kivshar, Y. S.

A. A. Sukhorukov and Y. S. Kivshar, “Nonlinear localized waves in a periodic medium,” Phys. Rev. Lett. 87, 083901 (2001).
[CrossRef] [PubMed]

Z. Fei, Y. S. Kivshar, and L. Vázquez, “Resonant kink-impurity interactions in the sine-Gordon model,” Phys. Rev. A 45, 6019–6030 (1992).
[CrossRef]

Z. Fei, Y. S. Kivshar, and L. Vázquez, “Resonant kink-impurity interactions in the ø4 model,” Phys. Rev. A 46, 5214–5220 (1992).
[CrossRef] [PubMed]

Krauss, T. F.

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. Lighwave Technol. 15, 1494–1502 (1997).
[CrossRef]

Lenz, G.

Malomed, B.

K. Forinash, M. Peyrard, and B. Malomed, “Interaction of discrete breathers with impurity modes,” Phys. Rev. E 49, 3400–3411 (1994).
[CrossRef]

Malomed, B. A.

B. A. Malomed, “Inelastic interactions of solitons in nearly integrable systems 2,” Physica D 15, 385–401 (1985).
[CrossRef]

Millar, P.

Mills, D. L.

W. Chen and D. L. Mills, “Gap solitons and the nonlinear optical response of superlattices,” Phys. Rev. Lett. 58, 160–163 (1987).
[CrossRef] [PubMed]

Mollenauer, L. F.

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Experimental observation of picosecond pulse narrowing and solitons in optical fibers,” Phys. Rev. Lett. 45, 1095–1098 (1980).
[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]

Perlin, V.

H. G. Winful and V. Perlin, “Raman gap solitons,” Phys. Rev. Lett. 84, 3586–3589 (1999).
[CrossRef]

Peyrard, M.

K. Forinash, M. Peyrard, and B. Malomed, “Interaction of discrete breathers with impurity modes,” Phys. Rev. E 49, 3400–3411 (1994).
[CrossRef]

Poladian, L.

Richardson, D. J.

Rose, H.

H. Rose and M. I. Weinstein, “On the bound states of the nonlinear Schrödinger equation with a linear potential,” Physica D 30, 207–218 (1988).
[CrossRef]

Shabat, A. B.

V. E. Zakharov and A. B. Shabat, “Interaction between solitons in a stable medium,” Sov. Phys. JETP 37, 823–828 (1973).

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).

Shin, S.-Y.

Sipe, J. E.

Slusher, R. E.

Song, G.-H.

Stolen, R. H.

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Experimental observation of picosecond pulse narrowing and solitons in optical fibers,” Phys. Rev. Lett. 45, 1095–1098 (1980).
[CrossRef]

Strasser, T. A.

Sukhorukov, A. A.

A. A. Sukhorukov and Y. S. Kivshar, “Nonlinear localized waves in a periodic medium,” Phys. Rev. Lett. 87, 083901 (2001).
[CrossRef] [PubMed]

Tappert, F.

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion,” Appl. Phys. Lett. 23, 142–144 (1971).
[CrossRef]

Tsoy, E. N.

E. N. Tsoy and C. M. de Sterke, “Soliton dynamics in nonuniform fiber Bragg gratings,” J. Opt. Soc. Am. B 18, 1–6 (2001).
[CrossRef]

E. N. Tsoy and C. M. de Sterke, “Propagation of nonlinear pulses in chirped fiber gratings,” Phys. Rev. E 62, 2882–2890 (2000).
[CrossRef]

Vázquez, L.

Z. Fei, Y. S. Kivshar, and L. Vázquez, “Resonant kink-impurity interactions in the ø4 model,” Phys. Rev. A 46, 5214–5220 (1992).
[CrossRef] [PubMed]

Z. Fei, Y. S. Kivshar, and L. Vázquez, “Resonant kink-impurity interactions in the sine-Gordon model,” Phys. Rev. A 45, 6019–6030 (1992).
[CrossRef]

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]

Weinstein, M. I.

R. H. Goodman, P. J. Holmes, and M. I. Weinstein, “Interaction of sine-Gordon kinks with defects: phase space transport in a two-mode model,” Physica D 161, 21–41 (2002).
[CrossRef]

R. H. Goodman, M. I. Weinstein, and P. J. Holmes, “Nonlinear propagation of light in one-dimensional periodic structures,” J. Nonlinear Sci. 11, 123–168 (2001).
[CrossRef]

H. Rose and M. I. Weinstein, “On the bound states of the nonlinear Schrödinger equation with a linear potential,” Physica D 30, 207–218 (1988).
[CrossRef]

Winful, H. G.

H. G. Winful and V. Perlin, “Raman gap solitons,” Phys. Rev. Lett. 84, 3586–3589 (1999).
[CrossRef]

Zakharov, V. E.

V. E. Zakharov and A. B. Shabat, “Interaction between solitons in a stable medium,” Sov. Phys. JETP 37, 823–828 (1973).

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).

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]

Appl. Phys. Lett.

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion,” Appl. Phys. Lett. 23, 142–144 (1971).
[CrossRef]

J. Lightwave Technol.

J. Lighwave Technol.

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

J. Nonlinear Sci.

R. H. Goodman, M. I. Weinstein, and P. J. Holmes, “Nonlinear propagation of light in one-dimensional periodic structures,” J. Nonlinear Sci. 11, 123–168 (2001).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Opt. Express

Opt. Lett.

Phys. Lett. A

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. A

Z. Fei, Y. S. Kivshar, and L. Vázquez, “Resonant kink-impurity interactions in the ø4 model,” Phys. Rev. A 46, 5214–5220 (1992).
[CrossRef] [PubMed]

Z. Fei, Y. S. Kivshar, and L. Vázquez, “Resonant kink-impurity interactions in the sine-Gordon model,” Phys. Rev. A 45, 6019–6030 (1992).
[CrossRef]

Phys. Rev. E

K. Forinash, M. Peyrard, and B. Malomed, “Interaction of discrete breathers with impurity modes,” Phys. Rev. E 49, 3400–3411 (1994).
[CrossRef]

N. G. R. Broderick and C. M. de Sterke, “Approximate method for gap soliton propagation in nonuniform Bragg gratings,” Phys. Rev. E 58, 7941–7950 (1998).
[CrossRef]

E. N. Tsoy and C. M. de Sterke, “Propagation of nonlinear pulses in chirped fiber gratings,” Phys. Rev. E 62, 2882–2890 (2000).
[CrossRef]

Phys. Rev. Lett.

H. G. Winful and V. Perlin, “Raman gap solitons,” Phys. Rev. Lett. 84, 3586–3589 (1999).
[CrossRef]

W. Chen and D. L. Mills, “Gap solitons and the nonlinear optical response of superlattices,” Phys. Rev. Lett. 58, 160–163 (1987).
[CrossRef] [PubMed]

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

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Experimental observation of picosecond pulse narrowing and solitons in optical fibers,” Phys. Rev. Lett. 45, 1095–1098 (1980).
[CrossRef]

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]

A. A. Sukhorukov and Y. S. Kivshar, “Nonlinear localized waves in a periodic medium,” Phys. Rev. Lett. 87, 083901 (2001).
[CrossRef] [PubMed]

Physica D

B. A. Malomed, “Inelastic interactions of solitons in nearly integrable systems 2,” Physica D 15, 385–401 (1985).
[CrossRef]

R. H. Goodman, P. J. Holmes, and M. I. Weinstein, “Interaction of sine-Gordon kinks with defects: phase space transport in a two-mode model,” Physica D 161, 21–41 (2002).
[CrossRef]

H. Rose and M. I. Weinstein, “On the bound states of the nonlinear Schrödinger equation with a linear potential,” Physica D 30, 207–218 (1988).
[CrossRef]

Prog. Opt.

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

Sov. Phys. JETP

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).

V. E. Zakharov and A. B. Shabat, “Interaction between solitons in a stable medium,” Sov. Phys. JETP 37, 823–828 (1973).

Other

A. Soffer and M. I. Weinstein, “Selection of the ground state in nonlinear Schrödinger equations,” 2001 preprint (to be published).

M. I. Weinstein, “Notes on wave propagation in 1-d periodic media with defects,” Bell Labs Technical Memorandum (Lucent Technologies, Murray Hill, N.J., 1999).

It is common to slightly redefine κ˜ by κ˜(z)=ηπΔnν(z)/λB, where 0<η<1 is defined as an overlap integral of the radial variation of the forward and backward modes and represents the fraction of total energy in the core of the fiber.

We work with the model (2.6)–(2.7) since it yields a simple derivation of the envelope equations (2.10). The situation is, however, a bit more complicated. Although model (2.6)–(2.7) incorporates the effects of photonic band dispersion, this alone is insufficient to arrest optical carrier shock formation on the relevant temporal and spatial scales.27 In fact, a valid envelope description in the absence of material dispersion would require the incorporation of coupling to all higher harmonics since they are in resonance.

G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic, San Diego, 2001).

G. P. Agrawal, Fiber-Optic Communication Systems (Wiley, New York, 1997).

R. H. Goodman, P. J. Holmes, and M. I. Weinstein, “Interaction of NLS solitons with defects,” http://arXiv.org/abs/nlin/0203057 (to be published).

P. J. Holmes, R. H. Goodman, and M. I. Weinstein, “Trapping of kinks and solitons by defects: phase space transport in finite dimensional models,” presented at the International Conference on Progress in Nonlinear Science dedicated to Alexander Andronov, Nizhny Novgorod, Russia, July 2001.

Experimentally it is hard to get to low velocities because most of the incident energy on the grating is reflected by the abrupt change from uniform to modulated index. Simulations show that for very gradually apodized gratings, over many Bragg lengths, one can hope to easily get down to 1/10 c/n, even with intensities that do not destroy the grating. Ideas like Raman gap solitons13 would allow generation of the gap soliton in the grating, and one would avoid these impedance-mismatch problems.

E. C. Titchmarsh, Eigenfunction Expansions Associated with Second-Order Differential Equations. Part I (Clarendon, Oxford, 1962).

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Figures (22)

Fig. 1
Fig. 1

Solid and dashed curves are two different periodic index profiles with localized defects having the same “spectral characteristics” (see Subsection 4.B).

Fig. 2
Fig. 2

Intensity of the stationary gap soliton as a function of its frequency.

Fig. 3
Fig. 3

Dispersion relation (4.7), showing a bandgap.

Fig. 4
Fig. 4

Three eigenmodes for the defects (4.18) and (4.19) with (ω, k, n)=(-1, 2, 2). In each plot, E+ is in the left column and E- is in the right. Solid and dashed curves correspond to real and imaginary parts, respectively.

Fig. 5
Fig. 5

Intensity versus frequency for the gap soliton (bold) and a nonlinear defect mode with parameters (ω0, k)=(-1, 4).

Fig. 6
Fig. 6

(Experiment 1.1) Initial value of |E+|2+|E-|2 (solid curve), which gives the approximate strength of the nonlinear forcing, and of the defect κ(Z)-κ (dashed), which gives the forcing due to the defect.

Fig. 7
Fig. 7

(Experiment 1.1) (a) Gap soliton with v=0.2565 is reflected by a defect at Z=0. (b) Slightly faster gap soliton with v=0.257 is transmitted.

Fig. 8
Fig. 8

(Experiment 1.2) With δ=2, a defect mode of significant intensity remains behind after the soliton passes through. (Due to the absorbing boundary conditions used in the simulations, the gap soliton dissipates as it approaches the edge of the computational domain).

Fig. 9
Fig. 9

Intensity versus frequency for the gap soliton (bold) and a nonlinear mode for defect with parameters (ω0, k)=(1, 4). Although the defect is the same as that used in Fig. 5 excepting a sign in the definition of V(Z), the defect-mode curve is further to the right and closer to the gap-soliton curve, predicting greatly improved trapping.

Fig. 10
Fig. 10

Intensity versus frequency for the gap soliton (bold) and the three nonlinear modes for the defect with parameters (ω0, k, n)=(-1, 2, 2). Trapping is possible for frequencies on the thickened section of the gap-soliton curve.

Fig. 11
Fig. 11

(Experiment 2.1a) Typical picture of the capture of a gap soliton by a defect centered at Z=0.

Fig. 12
Fig. 12

(Experiment 2.1a) The position versus time of an escaping and a captured gap soliton. Note that the instantaneous velocity increases when the gap soliton is in the defect region (Z near zero). δ=0.9 and v near vc0.103, the potential as described in Subsection 6.B.

Fig. 13
Fig. 13

(Experiment 2.1a) Initial velocity (vi) versus final velocity (vf) of the gap soliton; parameters are as in Fig. 12 with variable v.

Fig. 14
Fig. 14

Decay of the local L2 norm (normalized) for Experiment 2.1a.

Fig. 15
Fig. 15

(Experiment 2.1b) Partial capture: an incident gap soliton results in part of its energy captured in the defect and part transmitted as a lower-energy gap soliton.

Fig. 16
Fig. 16

(Experiment 2.2a) Capture of a gap soliton with δ=0.45 by a wide well.

Fig. 17
Fig. 17

Local L2 norm of the solution (bold), the projection onto the ω+2 eigenmode (solid), and the ω+1 mode (dashed) for (a) Experiment 2.2a and (b) 2.2b.

Fig. 18
Fig. 18

(Experiment 3) κ(Z) and V(Z) for an array of two defects.

Fig. 19
Fig. 19

(Experiment 3) Position versus time for a gap soliton incident on an array of defects. Defect positions are given by dashed lines.

Fig. 20
Fig. 20

(Experiment 4) Modified defect described in Subsection 6.D.

Fig. 21
Fig. 21

(Experiment 4) Local L2 norm as a function of time for a gap soliton with v=0.2625 (solid) and v=0.3 (dashed).

Fig. 22
Fig. 22

In the presence of the nonlinear damping, a gap soliton may be (a) damped before reaching the defect, (b) captured, or (c) transmitted. In all cases shown, the soliton would have been transmitted without the damping.

Tables (2)

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Table 1 Experimental Parameters Describing the Defects in Numerical Simulations

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Table 2 Experimental Parameters Describing the Gap Solitons in Numerical Simulations

Equations (131)

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t2[n2(z)E(z, t)]=c2z2E,
n=n¯+Δn12W(z)+ν(z)cos[2kBz+2Φ(z)].
d=π/kB
ν(z)1,zΦ(z)0,zW(z)0,as|z|.
ω=±ckn¯.
λB=2n¯d.
PNL=0χ(3)E3.
n2(z, E2)=n¯2+n¯ΔnW(z)+2n¯Δnν(z)×cos[2kBz+2Φ(z)]+χ(3)E2.
t2[n2(z, E2)E]=c2z2E.
Δn=O(ε),
zW=O(ε),zν=O(ε),zΦ=O(ε),z2Φ=O(ε2).
E=e+(z, t)exp[i(kBz+Φ-ωBt)]+e-(z, t)exp[-i(kBz+Φ+ωBt)]+E1,
χ(3)|E|2=O(ε).
te±=O(ε),ze±=O(ε),
t2e±=O(2),z2e±=O(2).
E1/e±=O(ε).
i n¯cte++ize++κ˜(z)e-+V˜(z)e+
+Γ˜(|e+|2+2|e-|2)e+=0,
i n¯cte--ize-+κ˜(z)e++V˜(z)e-
+Γ˜(|e-|2+2|e+|2)e-=0.
κ˜(z)=πΔnλB ν(z),V˜(z)=πΔnλBW(z)-Φ(z),
Γ˜=3πχ(3)nλ¯B.
V˜(z)0,κ˜(z)κ˜πΔnλB.
z=ZZ,t=Zn¯cT,e±=EE±.
iTE++iZE++κ(Z)E-+V(Z)E+
+Γ(|E+|2+2|E-|2)E+=0,
iTE--izE-+κ(Z)E++V(Z)E-
+Γ(|E-|2+2|E+|2)E-=0,
κ(Z)=Zκ˜(ZZ),V(Z)=ZV˜(ZZ),
Γ=ZE2Γ˜.
E+=sα exp(iη)κ2Γ 1Δ(sin δ)exp(isσ)×sech(θ-iδ/2),
E-=-α exp(iη)κ2ΓΔ(sin δ)exp(isσ)×sech(θ+iδ/2)
γ=11-v2,Δ=1-v1+v1/4,
θ=γκ(sin δ)(Z-vT),
σ=γκ(cos δ)(vZ-T),
α=2(1-v2)3-v2,s=sign(κΓ),
exp(iη)=-exp(2θ)+exp(-iδ)exp(2θ)+exp(iδ)2v/(3-v2).
Imax=maxZ(|E+|2+|E-|2)=8κ1-v2Γ(3-v2) sin2 δ2,
Itot=-(|E+|2+|E-|2)dZ=4(1-v2)δΓ(3-v2),
FWHM=21-v2κ sin δ cosh-1 1+cos2 δ2.
ω(δ)=κ cos δ,Itot(δ)=4δ3Γforδ[0, π].
 iTE++iZE++κ(Z)E-+V(Z)E+=0,
iTE--iZE-+κ(Z)E++V(Z)E-=0.
[iT+iσ3Z+V(Z)+κ(Z)σ1]E=0,
E=E+E-,σ1=0110,σ3=100-1.
E(Z)=exp(-iωT)expiσ30ZV(ζ)dζF(Z)
ZF=iωu(Z)u(Z)¯-iωF,
u(Z)=iκ(Z)exp-2i0ZV(ζ)dζ.
u(Z)ρ exp(iθ±),Z±,
ZF=iωρ exp(iθ±)ρ exp(-iθ±)-iωF.
ω2=ρ2+Q2.
iτu-Z2u+2|u|2u.
ρ=|ω+ik|.
u(Z)=-exp(iϕ)[ω-ik tanh(kZ)],
F(Z)=1iexp(-iϕ)sech(kZ).
E+E-=expi2 arctan k tanh(kZ)ωi exp(-iϕ)exp-i2 arctan k tanh(kZ)ω×exp(-iωt)sech(kZ).
iκ(Z)exp-2i0ZV(s)ds=exp(iϕ)[ω-ik tanh(kZ)].
κ(Z)=[ω2+k2 tanh2(kZ)]1/2,
V(Z)=12k2ω[ω2+k2 tanh2(kZ)]-1 sech2(kZ),
i exp(-iϕ)=1
κ(z)=|k tanh(kZ)|,
V(Z)=±π2δ(Z),
κ(Z)=-k tanh kZ,
F(Z)=1±isech kZ.
κ(Z)=ω2+n2k2 tanh2(kZ),
V(Z)=ωnk2 sech2(kZ)2[ω2+n2k2 tanh2(kZ)]. 
E+E-=expi2 arctan nk tanh(kZ)ωexp-i2 arctan nk tanh(kZ)ω×exp(-iωt)sechn(kZ).
κ=lim|Z| κ(Z)=ω2+n2k2.
Δ*=ω2+n2k2-|ω|.
FWHM=2k tanh-12|ω|ω2+n2k2+n2k2-2ω22nk
[i(T+σ3z)+σ1κ(Z)+V(Z)]E+ΓN(E, E*)E=0,
N(E, E*)=|E+|2+2|E-|200|E-|2+2|E+|2.
E(Z, T)=exp(-iωT)E(Z),
[ω+iσ3z+σ1κ(Z)+V(Z)]E+ΓN(E, E*)E=0.
[ω(0)+iσ3z+σ1κ(Z)+V(Z)]E=0,
E(Z)=α[E0(Z)+|α|2E1(Z)+O(|α|4)],
ω=ω(0)+ω(1)|α|2+O(|α|4),
 O(1) : L0E0=0,
O(|α|2) : L0E1=-ω(1)E0-ΓN(E0, E0*)E0,
L0=ω(0)+iσ3z+σ1κ+V
E0|ω(1)E0+ΓN(E0, E0*)E0=0.
ω(1)=-Γ E0|N(E0, E0*)E0E0|E0.
E(Z, T)=exp(-iωT)α[E0(Z)+|α|2E1(Z)+O(|α|4)],
ω=ω(0)+|α|2ω(1)+O(|α|4).
localL2 norm=D|E+|2+|E-|2dZ1/2,
n2=3χ(3)/40cn¯2,
Γ˜=4π0cn¯n2λB.
λB=1053nm,
n¯=1.45,
Δn=3×10-4,
n2=2.3×10-20 m2/W,
c=2.98×108 m/s,
0=8.85×10-12 CNm2.
kB=2π n¯λB=8.7×106 m-1,
κ˜=900m-1,
Γ˜=1.06×10-15 C2N2m.
Z=κκ˜=κλBπΔn,T=κλBπΔn n¯c,
E2=Γκ˜Γ˜κ=ΓΔn4κ0cn¯n2,
E=e+ exp[ikB(z-ct/n¯)]+e- exp[-ikB(z+ct/n¯)]+cc,
|E|2=2(|e+|2+|e-|2)=2E2(|E+|2+|E-|2),
I=120cn¯ max|E|2=0cn¯E2Imax.
I=2Δn1-v2n2(3-v2) sin2 δ2.
dimensionalFWHM
=λBπΔn 21-v2sin δ cosh-11+cos2 δ2.
dimensionalFWHMtemporal
=nλ¯BπcΔn 21-v2v sin δ cosh-1 1+cos2 δ2.
Δ*=πΔnλB 1-|ω|ω2+n2k2.
FWHM
=λBπΔn 2ω2+k2n2k×tanh-12|ω|ω2+n2k2+n2k2-2ω22nk.
E+E-=exp(-iωT)exp[iσ3Θ(Z)]f(Z)v+v-,
E+Z=exp(-iωT)(f+iVf)exp[iΘ(Z)]v+.
E±Z=(g±iV)E+.
(ω±ig)E±+κE=0.
Lv=(ω+ig)exp(iΘ)κ exp(-iΘ)κ exp(iΘ)(ω-ig)exp(-iΘ)v+v-=0.
det L=ω2+g2-κ2=0.
κ(Z)=ω2+g2(Z).
κ=limZ κ(Z)=ω2+k2.
v-v+=-exp(2iΘ)ω+igκ.
ω+igκ=1
v+v-=exp(iα),αreal.
2Θ=arg -ω+igκ.
Θ=-12 arctan gω.
V=-ωg2(ω2+g2).
κ(Z)=±g(Z),
v-=iv+.
f=sechn(kZ).
κ(Z)=ω2+n2k2 tanh2(kZ);
Θ(Z)=12 arctan nk tanh(kZ)ω;
V(Z)=ωnk2 sech2(kZ)2[ω2+n2k2 tanh2(kZ)].
E+E-=exp(-iωt)exp±i2 arctan nk tanh(kZ)ω×sechn(kZ).
ω±j=±ω2+(2nj-j2)k2

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