Abstract

We report results of the investigation of gap solitons (GSs) in the generic model of a periodically modulated Bragg grating (BG), which includes periodic modulation of the BG chirp or local refractive index, and periodic variation of the local reflectivity. We demonstrate that, while the previously studied reflectivity modulation strongly destabilizes all solitons, the periodic chirp modulation, which is a novel feature, stabilizes a new family of double-peak fundamental BGs in the side bandgap at negative frequencies (gap No. -1), and keeps solitons stable in the central bandgap (No. 0). The two soliton families demonstrate bistability, coexisting at equal values of energy. In addition, stable 4-peak bound states are formed by pairs of fundamental GSs in bandgap -1. Self-trapping and mobility of the solitons are studied too.

©2008 Optical Society of America

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  1. P. St. J. Russell, “Optical superlattices for modulation and deflection of light,” J. Appl. Phys. 59, 3344 (1986).
    [Crossref]
  2. B. J. Eggleton, P. A. Krug, L. Poladian, and F. Ouellette, “Long periodic superstructure Bragg gratings in optical fibres,” Electron. Lett. 30, 1620 (1994).
    [Crossref]
  3. N. G. R. Broderick and C. M. de Sterke, “Theory of grating superstructures,” Phys. Rev. E 55, 3634 (1997).
    [Crossref]
  4. P. J. Y. Louis, E. A. Ostrovskaya, and Y. S. Kivshar, “Dispersion control for matter waves and gap solitons in optical superlattices,” Phys. Rev. A 71, 032612 (2005).
    [Crossref]
  5. A. B. Aceves and S. Wabnitz, “Self-induced transparency solitons in nonlinear refractive periodic media,” Phys. Lett. A 141, 37 (1989).
    [Crossref]
  6. D. N. Christodoulides and R. I. Joseph, “Slow Bragg solitons in nonlinear periodic structures,” Phys. Rev. Lett. 62, 1746 (1989).
    [Crossref] [PubMed]
  7. C. M. de Sterke and J. E. Sipe, “Gap solitons,” Prog. Opt. 33, 203 (1994).
  8. J. E. Sipe, L. Poladian, and C. M. de Sterke, “Propagation through nonuniform grating structures,” J. Opt. Soc. Am. A 11, 1307 (1994).
    [Crossref]
  9. T. Iizuka and C. M. de Sterke, “Corrections to coupled mode theory for deep gratings,” Phys. Rev. E 61, 4491 (2000).
    [Crossref]
  10. J. B. Khurgin, “Light slowing down in Moiré fiber gratings and its implications for nonlinear optics,” Phys. Rev. A 62, 013821 (2000).
    [Crossref]
  11. R. Shimada, T. Koda, T. Ueta, and K. Ohtaka, “Strong localization of Bloch photons in dual-periodic dielectric multilayer structures,” J. Appl. Phys. 90, 3905 (2001).
    [Crossref]
  12. D. Janner, G. Galzerano, G. Della Valle, P. Laporta, S. Longhi, and M. Belmonte, “Slow light in periodic super-structure Bragg gratings,” Phys. Rev. E 72, 056605 (2005).
    [Crossref]
  13. A. Melloni, F. Morichetti, and M. Martinelli, “Linear and nonlinear pulse propagation in coupled resonator slow-wave optical structures,” Opt. Quantum Electron. 35, 365 (2003).
    [Crossref]
  14. J. K. S. Poon, J. Scheuer, S. Mookherjea, G. Paloczi, Y. Huang, and A. Yariv, “Matrix analysis of microring coupled-resonator optical waveguides,” Opt. Express 12, 90 (2004).
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  15. K. Levy and B. A. Malomed, “Stability and collisions of traveling solitons in Bragg-grating superstructures,” J. Opt. Soc. Am. B 25, 302 (2008).
    [Crossref]
  16. N. Groothoff, J. Canning, E. Buckley, K. Lyttikainen, and J. Zagari, “Bragg gratings in air-silica structured fibers,” Opt. Lett. 28, 233–235 (2003).
    [Crossref] [PubMed]
  17. Y. N. Zhu, P. Shum, J. H. Chong, M. K. Rao, and C. Lu, “Deep-notch, ultracompact long-period grating in a large-mode-area photonic crystal fiber,” Opt. Lett. 28, 2467–2469 (2003).
    [Crossref] [PubMed]
  18. J. H. Lim, K. S. Lee, J. C. Kim, and B. H. Lee, “Tunable fiber gratings fabricated in photonic crystal fiber by use of mechanical pressure,” Opt. Lett. 29, 331–333 (2004).
    [Crossref] [PubMed]
  19. 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]
  20. I. V. Barashenkov, D. E. Pelinovsky, and E. V. Zemlyanaya, “Vibrations and Oscillatory Instabilities of Gap Solitons,” Phys. Rev. Lett. 80, 5117 (1998).
    [Crossref]
  21. A. De Rossi, C. Conti, and S. Trillo, “Stability, Multistability, andWobbling of Optical Gap Solitons,” Phys. Rev. Lett. 81, 85 (1998).
    [Crossref]
  22. K. Yagasaki, I. M. Merhasin, B. A. Malomed, T. Wagenknecht, and A. R. Champneys, “Gap solitons in Bragg gratings with a harmonic superlattice,” Europhys. Lett. 74, 1006–1012 (2006).
    [Crossref]
  23. 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]
  24. E. N. Tsoy and C. M. de Sterke, “Propagation of nonlinear pulses in chirped fiber gratings,” Phys. Rev. E 62, 2882–2890 (2000).
    [Crossref]
  25. J. Feng, “Alternative scheme for studying gap solitons in an infinite periodic Kerr medium,” Opt. Lett. 18, 1302–1304 (1993).
    [Crossref] [PubMed]
  26. R. F. Nabiev, P. Yeh, and D. Botez, “Spatial gap solitons in periodic nonlinear structures,” Opt. Lett. 18, 1612–1614 (1993).
    [Crossref] [PubMed]
  27. W. C. K. Mak, B. A. Malomed, and P. L. Chu, “Three-wave gap solitons in waveguides with quadratic nonlinearity,” Phys. Rev. E 58, 6708–6722 (1998).
    [Crossref]
  28. Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic Press: Boston, 2003).
  29. F. Biancalana, A. Amann, and E. P. O’Reilly, “Gap solitons in spatiotemporal photonic crystals,” Phys. Rev. A 77, 011801(R) (2008).
    [Crossref]
  30. 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]
  31. B. J. Eggleton, C. M. de Sterke, and R. E. Slusher, “Bragg solitons in the nonlinear Schrödinger limit: Experiment and theory,” J. Opt. Soc. Am. B 16, 587–599 (1999).
    [Crossref]
  32. J. T. Mok, C. M. de Sterke, I. C. M. Littler, and B. J. Eggleton, “Dispersionless slow light using gap solitons,” Nat. Physics 2, 775–780 (2006).
    [Crossref]
  33. B. Deconinck, F. Kiyak, J. D. Carter, and J. N. Kutz, “SpectrUW: A laboratory for the numerical exploration of spectra of linear operators,” Math. Comput. Simul. 74, 370–378 (2007).
    [Crossref]
  34. W. C. K. Mak, B. A. Malomed, and P. L. Chu, “Slowdown and Splitting of Gap Solitons in Apodized Bragg Gratings,” J. Mod. Opt. 51, 2141–2158 (2004).
    [Crossref]
  35. W. C. K. Mak, B. A. Malomed, and P. L. Chu, “Formation of a standing-light pulse through collision of gap solitons,” Phys. Rev. E 68, 026609 (2003).
    [Crossref]

2008 (2)

K. Levy and B. A. Malomed, “Stability and collisions of traveling solitons in Bragg-grating superstructures,” J. Opt. Soc. Am. B 25, 302 (2008).
[Crossref]

F. Biancalana, A. Amann, and E. P. O’Reilly, “Gap solitons in spatiotemporal photonic crystals,” Phys. Rev. A 77, 011801(R) (2008).
[Crossref]

2007 (1)

B. Deconinck, F. Kiyak, J. D. Carter, and J. N. Kutz, “SpectrUW: A laboratory for the numerical exploration of spectra of linear operators,” Math. Comput. Simul. 74, 370–378 (2007).
[Crossref]

2006 (2)

J. T. Mok, C. M. de Sterke, I. C. M. Littler, and B. J. Eggleton, “Dispersionless slow light using gap solitons,” Nat. Physics 2, 775–780 (2006).
[Crossref]

K. Yagasaki, I. M. Merhasin, B. A. Malomed, T. Wagenknecht, and A. R. Champneys, “Gap solitons in Bragg gratings with a harmonic superlattice,” Europhys. Lett. 74, 1006–1012 (2006).
[Crossref]

2005 (2)

D. Janner, G. Galzerano, G. Della Valle, P. Laporta, S. Longhi, and M. Belmonte, “Slow light in periodic super-structure Bragg gratings,” Phys. Rev. E 72, 056605 (2005).
[Crossref]

P. J. Y. Louis, E. A. Ostrovskaya, and Y. S. Kivshar, “Dispersion control for matter waves and gap solitons in optical superlattices,” Phys. Rev. A 71, 032612 (2005).
[Crossref]

2004 (3)

2003 (4)

W. C. K. Mak, B. A. Malomed, and P. L. Chu, “Formation of a standing-light pulse through collision of gap solitons,” Phys. Rev. E 68, 026609 (2003).
[Crossref]

N. Groothoff, J. Canning, E. Buckley, K. Lyttikainen, and J. Zagari, “Bragg gratings in air-silica structured fibers,” Opt. Lett. 28, 233–235 (2003).
[Crossref] [PubMed]

Y. N. Zhu, P. Shum, J. H. Chong, M. K. Rao, and C. Lu, “Deep-notch, ultracompact long-period grating in a large-mode-area photonic crystal fiber,” Opt. Lett. 28, 2467–2469 (2003).
[Crossref] [PubMed]

A. Melloni, F. Morichetti, and M. Martinelli, “Linear and nonlinear pulse propagation in coupled resonator slow-wave optical structures,” Opt. Quantum Electron. 35, 365 (2003).
[Crossref]

2001 (2)

R. Shimada, T. Koda, T. Ueta, and K. Ohtaka, “Strong localization of Bloch photons in dual-periodic dielectric multilayer structures,” J. Appl. Phys. 90, 3905 (2001).
[Crossref]

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]

2000 (3)

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

T. Iizuka and C. M. de Sterke, “Corrections to coupled mode theory for deep gratings,” Phys. Rev. E 61, 4491 (2000).
[Crossref]

J. B. Khurgin, “Light slowing down in Moiré fiber gratings and its implications for nonlinear optics,” Phys. Rev. A 62, 013821 (2000).
[Crossref]

1999 (1)

1998 (3)

W. C. K. Mak, B. A. Malomed, and P. L. Chu, “Three-wave gap solitons in waveguides with quadratic nonlinearity,” Phys. Rev. E 58, 6708–6722 (1998).
[Crossref]

I. V. Barashenkov, D. E. Pelinovsky, and E. V. Zemlyanaya, “Vibrations and Oscillatory Instabilities of Gap Solitons,” Phys. Rev. Lett. 80, 5117 (1998).
[Crossref]

A. De Rossi, C. Conti, and S. Trillo, “Stability, Multistability, andWobbling of Optical Gap Solitons,” Phys. Rev. Lett. 81, 85 (1998).
[Crossref]

1997 (1)

N. G. R. Broderick and C. M. de Sterke, “Theory of grating superstructures,” Phys. Rev. E 55, 3634 (1997).
[Crossref]

1996 (1)

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]

1994 (4)

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]

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

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

B. J. Eggleton, P. A. Krug, L. Poladian, and F. Ouellette, “Long periodic superstructure Bragg gratings in optical fibres,” Electron. Lett. 30, 1620 (1994).
[Crossref]

1993 (2)

1989 (2)

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

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

1986 (1)

P. St. J. Russell, “Optical superlattices for modulation and deflection of light,” J. Appl. Phys. 59, 3344 (1986).
[Crossref]

Aceves, A. B.

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

Agrawal, G. P.

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

Amann, A.

F. Biancalana, A. Amann, and E. P. O’Reilly, “Gap solitons in spatiotemporal photonic crystals,” Phys. Rev. A 77, 011801(R) (2008).
[Crossref]

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 (1998).
[Crossref]

Belmonte, M.

D. Janner, G. Galzerano, G. Della Valle, P. Laporta, S. Longhi, and M. Belmonte, “Slow light in periodic super-structure Bragg gratings,” Phys. Rev. E 72, 056605 (2005).
[Crossref]

Biancalana, F.

F. Biancalana, A. Amann, and E. P. O’Reilly, “Gap solitons in spatiotemporal photonic crystals,” Phys. Rev. A 77, 011801(R) (2008).
[Crossref]

Botez, D.

Broderick, N. G. R.

N. G. R. Broderick and C. M. de Sterke, “Theory of grating superstructures,” Phys. Rev. E 55, 3634 (1997).
[Crossref]

Buckley, E.

Canning, J.

Carter, J. D.

B. Deconinck, F. Kiyak, J. D. Carter, and J. N. Kutz, “SpectrUW: A laboratory for the numerical exploration of spectra of linear operators,” Math. Comput. Simul. 74, 370–378 (2007).
[Crossref]

Champneys, A. R.

K. Yagasaki, I. M. Merhasin, B. A. Malomed, T. Wagenknecht, and A. R. Champneys, “Gap solitons in Bragg gratings with a harmonic superlattice,” Europhys. Lett. 74, 1006–1012 (2006).
[Crossref]

Chong, J. H.

Christodoulides, D. N.

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

Chu, P. L.

W. C. K. Mak, B. A. Malomed, and P. L. Chu, “Slowdown and Splitting of Gap Solitons in Apodized Bragg Gratings,” J. Mod. Opt. 51, 2141–2158 (2004).
[Crossref]

W. C. K. Mak, B. A. Malomed, and P. L. Chu, “Formation of a standing-light pulse through collision of gap solitons,” Phys. Rev. E 68, 026609 (2003).
[Crossref]

W. C. K. Mak, B. A. Malomed, and P. L. Chu, “Three-wave gap solitons in waveguides with quadratic nonlinearity,” Phys. Rev. E 58, 6708–6722 (1998).
[Crossref]

Conti, C.

A. De Rossi, C. Conti, and S. Trillo, “Stability, Multistability, andWobbling of Optical Gap Solitons,” Phys. Rev. Lett. 81, 85 (1998).
[Crossref]

De Rossi, A.

A. De Rossi, C. Conti, and S. Trillo, “Stability, Multistability, andWobbling of Optical Gap Solitons,” Phys. Rev. Lett. 81, 85 (1998).
[Crossref]

de Sterke, C. M.

J. T. Mok, C. M. de Sterke, I. C. M. Littler, and B. J. Eggleton, “Dispersionless slow light using gap solitons,” Nat. Physics 2, 775–780 (2006).
[Crossref]

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]

T. Iizuka and C. M. de Sterke, “Corrections to coupled mode theory for deep gratings,” Phys. Rev. E 61, 4491 (2000).
[Crossref]

B. J. Eggleton, C. M. de Sterke, and R. E. Slusher, “Bragg solitons in the nonlinear Schrödinger limit: Experiment and theory,” J. Opt. Soc. Am. B 16, 587–599 (1999).
[Crossref]

N. G. R. Broderick and C. M. de Sterke, “Theory of grating superstructures,” Phys. Rev. E 55, 3634 (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 (1994).

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

Deconinck, B.

B. Deconinck, F. Kiyak, J. D. Carter, and J. N. Kutz, “SpectrUW: A laboratory for the numerical exploration of spectra of linear operators,” Math. Comput. Simul. 74, 370–378 (2007).
[Crossref]

Eggleton, B. J.

J. T. Mok, C. M. de Sterke, I. C. M. Littler, and B. J. Eggleton, “Dispersionless slow light using gap solitons,” Nat. Physics 2, 775–780 (2006).
[Crossref]

B. J. Eggleton, C. M. de Sterke, and R. E. Slusher, “Bragg solitons in the nonlinear Schrödinger limit: Experiment and theory,” J. Opt. Soc. Am. B 16, 587–599 (1999).
[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, P. A. Krug, L. Poladian, and F. Ouellette, “Long periodic superstructure Bragg gratings in optical fibres,” Electron. Lett. 30, 1620 (1994).
[Crossref]

Feng, J.

Galzerano, G.

D. Janner, G. Galzerano, G. Della Valle, P. Laporta, S. Longhi, and M. Belmonte, “Slow light in periodic super-structure Bragg gratings,” Phys. Rev. E 72, 056605 (2005).
[Crossref]

Groothoff, N.

Huang, Y.

Iizuka, T.

T. Iizuka and C. M. de Sterke, “Corrections to coupled mode theory for deep gratings,” Phys. Rev. E 61, 4491 (2000).
[Crossref]

Janner, D.

D. Janner, G. Galzerano, G. Della Valle, P. Laporta, S. Longhi, and M. Belmonte, “Slow light in periodic super-structure Bragg gratings,” Phys. Rev. E 72, 056605 (2005).
[Crossref]

Joseph, R. I.

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

Khurgin, J. B.

J. B. Khurgin, “Light slowing down in Moiré fiber gratings and its implications for nonlinear optics,” Phys. Rev. A 62, 013821 (2000).
[Crossref]

Kim, J. C.

Kivshar, Y. S.

P. J. Y. Louis, E. A. Ostrovskaya, and Y. S. Kivshar, “Dispersion control for matter waves and gap solitons in optical superlattices,” Phys. Rev. A 71, 032612 (2005).
[Crossref]

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

Kiyak, F.

B. Deconinck, F. Kiyak, J. D. Carter, and J. N. Kutz, “SpectrUW: A laboratory for the numerical exploration of spectra of linear operators,” Math. Comput. Simul. 74, 370–378 (2007).
[Crossref]

Koda, T.

R. Shimada, T. Koda, T. Ueta, and K. Ohtaka, “Strong localization of Bloch photons in dual-periodic dielectric multilayer structures,” J. Appl. Phys. 90, 3905 (2001).
[Crossref]

Krug, P. A.

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, P. A. Krug, L. Poladian, and F. Ouellette, “Long periodic superstructure Bragg gratings in optical fibres,” Electron. Lett. 30, 1620 (1994).
[Crossref]

Kutz, J. N.

B. Deconinck, F. Kiyak, J. D. Carter, and J. N. Kutz, “SpectrUW: A laboratory for the numerical exploration of spectra of linear operators,” Math. Comput. Simul. 74, 370–378 (2007).
[Crossref]

Laporta, P.

D. Janner, G. Galzerano, G. Della Valle, P. Laporta, S. Longhi, and M. Belmonte, “Slow light in periodic super-structure Bragg gratings,” Phys. Rev. E 72, 056605 (2005).
[Crossref]

Lee, B. H.

Lee, K. S.

Levy, K.

Lim, J. H.

Littler, I. C. M.

J. T. Mok, C. M. de Sterke, I. C. M. Littler, and B. J. Eggleton, “Dispersionless slow light using gap solitons,” Nat. Physics 2, 775–780 (2006).
[Crossref]

Longhi, S.

D. Janner, G. Galzerano, G. Della Valle, P. Laporta, S. Longhi, and M. Belmonte, “Slow light in periodic super-structure Bragg gratings,” Phys. Rev. E 72, 056605 (2005).
[Crossref]

Louis, P. J. Y.

P. J. Y. Louis, E. A. Ostrovskaya, and Y. S. Kivshar, “Dispersion control for matter waves and gap solitons in optical superlattices,” Phys. Rev. A 71, 032612 (2005).
[Crossref]

Lu, C.

Lyttikainen, K.

Mak, W. C. K.

W. C. K. Mak, B. A. Malomed, and P. L. Chu, “Slowdown and Splitting of Gap Solitons in Apodized Bragg Gratings,” J. Mod. Opt. 51, 2141–2158 (2004).
[Crossref]

W. C. K. Mak, B. A. Malomed, and P. L. Chu, “Formation of a standing-light pulse through collision of gap solitons,” Phys. Rev. E 68, 026609 (2003).
[Crossref]

W. C. K. Mak, B. A. Malomed, and P. L. Chu, “Three-wave gap solitons in waveguides with quadratic nonlinearity,” Phys. Rev. E 58, 6708–6722 (1998).
[Crossref]

Malomed, B. A.

K. Levy and B. A. Malomed, “Stability and collisions of traveling solitons in Bragg-grating superstructures,” J. Opt. Soc. Am. B 25, 302 (2008).
[Crossref]

K. Yagasaki, I. M. Merhasin, B. A. Malomed, T. Wagenknecht, and A. R. Champneys, “Gap solitons in Bragg gratings with a harmonic superlattice,” Europhys. Lett. 74, 1006–1012 (2006).
[Crossref]

W. C. K. Mak, B. A. Malomed, and P. L. Chu, “Slowdown and Splitting of Gap Solitons in Apodized Bragg Gratings,” J. Mod. Opt. 51, 2141–2158 (2004).
[Crossref]

W. C. K. Mak, B. A. Malomed, and P. L. Chu, “Formation of a standing-light pulse through collision of gap solitons,” Phys. Rev. E 68, 026609 (2003).
[Crossref]

W. C. K. Mak, B. A. Malomed, and P. L. Chu, “Three-wave gap solitons in waveguides with quadratic nonlinearity,” Phys. Rev. E 58, 6708–6722 (1998).
[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]

Martinelli, M.

A. Melloni, F. Morichetti, and M. Martinelli, “Linear and nonlinear pulse propagation in coupled resonator slow-wave optical structures,” Opt. Quantum Electron. 35, 365 (2003).
[Crossref]

Melloni, A.

A. Melloni, F. Morichetti, and M. Martinelli, “Linear and nonlinear pulse propagation in coupled resonator slow-wave optical structures,” Opt. Quantum Electron. 35, 365 (2003).
[Crossref]

Merhasin, I. M.

K. Yagasaki, I. M. Merhasin, B. A. Malomed, T. Wagenknecht, and A. R. Champneys, “Gap solitons in Bragg gratings with a harmonic superlattice,” Europhys. Lett. 74, 1006–1012 (2006).
[Crossref]

Mok, J. T.

J. T. Mok, C. M. de Sterke, I. C. M. Littler, and B. J. Eggleton, “Dispersionless slow light using gap solitons,” Nat. Physics 2, 775–780 (2006).
[Crossref]

Mookherjea, S.

Morichetti, F.

A. Melloni, F. Morichetti, and M. Martinelli, “Linear and nonlinear pulse propagation in coupled resonator slow-wave optical structures,” Opt. Quantum Electron. 35, 365 (2003).
[Crossref]

Nabiev, R. F.

O’Reilly, E. P.

F. Biancalana, A. Amann, and E. P. O’Reilly, “Gap solitons in spatiotemporal photonic crystals,” Phys. Rev. A 77, 011801(R) (2008).
[Crossref]

Ohtaka, K.

R. Shimada, T. Koda, T. Ueta, and K. Ohtaka, “Strong localization of Bloch photons in dual-periodic dielectric multilayer structures,” J. Appl. Phys. 90, 3905 (2001).
[Crossref]

Ostrovskaya, E. A.

P. J. Y. Louis, E. A. Ostrovskaya, and Y. S. Kivshar, “Dispersion control for matter waves and gap solitons in optical superlattices,” Phys. Rev. A 71, 032612 (2005).
[Crossref]

Ouellette, F.

B. J. Eggleton, P. A. Krug, L. Poladian, and F. Ouellette, “Long periodic superstructure Bragg gratings in optical fibres,” Electron. Lett. 30, 1620 (1994).
[Crossref]

Paloczi, G.

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 (1998).
[Crossref]

Poladian, L.

B. J. Eggleton, P. A. Krug, L. Poladian, and F. Ouellette, “Long periodic superstructure Bragg gratings in optical fibres,” Electron. Lett. 30, 1620 (1994).
[Crossref]

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

Poon, J. K. S.

Rao, M. K.

Russell, P. St. J.

P. St. J. Russell, “Optical superlattices for modulation and deflection of light,” J. Appl. Phys. 59, 3344 (1986).
[Crossref]

Scheuer, J.

Shimada, R.

R. Shimada, T. Koda, T. Ueta, and K. Ohtaka, “Strong localization of Bloch photons in dual-periodic dielectric multilayer structures,” J. Appl. Phys. 90, 3905 (2001).
[Crossref]

Shum, P.

Sipe, J. E.

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 (1994).

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

Slusher, R. E.

B. J. Eggleton, C. M. de Sterke, and R. E. Slusher, “Bragg solitons in the nonlinear Schrödinger limit: Experiment and theory,” J. Opt. Soc. Am. B 16, 587–599 (1999).
[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]

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]

Trillo, S.

A. De Rossi, C. Conti, and S. Trillo, “Stability, Multistability, andWobbling of Optical Gap Solitons,” Phys. Rev. Lett. 81, 85 (1998).
[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]

Ueta, T.

R. Shimada, T. Koda, T. Ueta, and K. Ohtaka, “Strong localization of Bloch photons in dual-periodic dielectric multilayer structures,” J. Appl. Phys. 90, 3905 (2001).
[Crossref]

Valle, G. Della

D. Janner, G. Galzerano, G. Della Valle, P. Laporta, S. Longhi, and M. Belmonte, “Slow light in periodic super-structure Bragg gratings,” Phys. Rev. E 72, 056605 (2005).
[Crossref]

Wabnitz, S.

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

Wagenknecht, T.

K. Yagasaki, I. M. Merhasin, B. A. Malomed, T. Wagenknecht, and A. R. Champneys, “Gap solitons in Bragg gratings with a harmonic superlattice,” Europhys. Lett. 74, 1006–1012 (2006).
[Crossref]

Yagasaki, K.

K. Yagasaki, I. M. Merhasin, B. A. Malomed, T. Wagenknecht, and A. R. Champneys, “Gap solitons in Bragg gratings with a harmonic superlattice,” Europhys. Lett. 74, 1006–1012 (2006).
[Crossref]

Yariv, A.

Yeh, P.

Zagari, J.

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 (1998).
[Crossref]

Zhu, Y. N.

Electron. Lett. (1)

B. J. Eggleton, P. A. Krug, L. Poladian, and F. Ouellette, “Long periodic superstructure Bragg gratings in optical fibres,” Electron. Lett. 30, 1620 (1994).
[Crossref]

Europhys. Lett. (1)

K. Yagasaki, I. M. Merhasin, B. A. Malomed, T. Wagenknecht, and A. R. Champneys, “Gap solitons in Bragg gratings with a harmonic superlattice,” Europhys. Lett. 74, 1006–1012 (2006).
[Crossref]

J. Appl. Phys. (2)

P. St. J. Russell, “Optical superlattices for modulation and deflection of light,” J. Appl. Phys. 59, 3344 (1986).
[Crossref]

R. Shimada, T. Koda, T. Ueta, and K. Ohtaka, “Strong localization of Bloch photons in dual-periodic dielectric multilayer structures,” J. Appl. Phys. 90, 3905 (2001).
[Crossref]

J. Mod. Opt. (1)

W. C. K. Mak, B. A. Malomed, and P. L. Chu, “Slowdown and Splitting of Gap Solitons in Apodized Bragg Gratings,” J. Mod. Opt. 51, 2141–2158 (2004).
[Crossref]

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

J. Opt. Soc. Am. B (3)

Math. Comput. Simul. (1)

B. Deconinck, F. Kiyak, J. D. Carter, and J. N. Kutz, “SpectrUW: A laboratory for the numerical exploration of spectra of linear operators,” Math. Comput. Simul. 74, 370–378 (2007).
[Crossref]

Nat. Physics (1)

J. T. Mok, C. M. de Sterke, I. C. M. Littler, and B. J. Eggleton, “Dispersionless slow light using gap solitons,” Nat. Physics 2, 775–780 (2006).
[Crossref]

Opt. Express (1)

Opt. Lett. (5)

Opt. Quantum Electron. (1)

A. Melloni, F. Morichetti, and M. Martinelli, “Linear and nonlinear pulse propagation in coupled resonator slow-wave optical structures,” Opt. Quantum Electron. 35, 365 (2003).
[Crossref]

Phys. Lett. A (1)

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

Phys. Rev. A (3)

P. J. Y. Louis, E. A. Ostrovskaya, and Y. S. Kivshar, “Dispersion control for matter waves and gap solitons in optical superlattices,” Phys. Rev. A 71, 032612 (2005).
[Crossref]

J. B. Khurgin, “Light slowing down in Moiré fiber gratings and its implications for nonlinear optics,” Phys. Rev. A 62, 013821 (2000).
[Crossref]

F. Biancalana, A. Amann, and E. P. O’Reilly, “Gap solitons in spatiotemporal photonic crystals,” Phys. Rev. A 77, 011801(R) (2008).
[Crossref]

Phys. Rev. E (7)

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

W. C. K. Mak, B. A. Malomed, and P. L. Chu, “Three-wave gap solitons in waveguides with quadratic nonlinearity,” Phys. Rev. E 58, 6708–6722 (1998).
[Crossref]

W. C. K. Mak, B. A. Malomed, and P. L. Chu, “Formation of a standing-light pulse through collision of gap solitons,” Phys. Rev. E 68, 026609 (2003).
[Crossref]

D. Janner, G. Galzerano, G. Della Valle, P. Laporta, S. Longhi, and M. Belmonte, “Slow light in periodic super-structure Bragg gratings,” Phys. Rev. E 72, 056605 (2005).
[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]

N. G. R. Broderick and C. M. de Sterke, “Theory of grating superstructures,” Phys. Rev. E 55, 3634 (1997).
[Crossref]

T. Iizuka and C. M. de Sterke, “Corrections to coupled mode theory for deep gratings,” Phys. Rev. E 61, 4491 (2000).
[Crossref]

Phys. Rev. Lett. (4)

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

I. V. Barashenkov, D. E. Pelinovsky, and E. V. Zemlyanaya, “Vibrations and Oscillatory Instabilities of Gap Solitons,” Phys. Rev. Lett. 80, 5117 (1998).
[Crossref]

A. De Rossi, C. Conti, and S. Trillo, “Stability, Multistability, andWobbling of Optical Gap Solitons,” Phys. Rev. Lett. 81, 85 (1998).
[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]

Prog. Opt. (1)

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

Other (1)

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

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Figures (8)

Fig. 1.
Fig. 1. The bandgap structure found from the linearization of Eq. (2) for (a) ε=0, (b) ε=0.5, (c) ε=1 and (d) μ=0. Shaded areas are occupied by Bloch bands. Five gaps are displayed: the central one (No. 0) and two side bandgaps, ±1 and ±2 (gaps ±2 are not labeled). Stable solitons are found in gaps 0 and -1, where borders between stability and instability areas are shown by dashed lines. Note that all solitons are unstable for ε=1.

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Fig. 2.
Fig. 2. Gap-soliton families, shown in the form of energy E versus intrinsic frequency ω, for (a) μ=0.5, ε=0, (b) μ=ε=0.5, (c) μ=-ε=0.5. Stable and unstable portions of the families are depicted by continuous and dashed lines, respectively. The upper bold curve in gap -1 in (a) represents the family of 4-peak bound states of fundamental solitons. Two different curves in (b) and (c), in gaps -1 and +1, respectively, pertain to two regions in which these gaps exist, cf. Fig. 1(b). Recall that, for ε<0 [as in (c)], the bandgap structure is obtained from that for -ε as the mirror image, with ω→-ω.

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Fig. 3.
Fig. 3. (a) A stable double-peak soliton found in gap -1, for μ=0.5, ε=0, and ω=-1. The energy of this soliton is E=1.83. (b) A stable single-peak soliton in gap 0, for μ=0.5, ε=0, and ω=0.6. Its energy is E=0.96.

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Fig. 4.
Fig. 4. (a) A stable bound state of two fundamental twin-peak solitons in gap -1, for μ=0.5, ε=0, and ω=-1.0. The energy of this state is 3.76, while the energy of each constituent soliton is 1.83. (b) A stable bound state of three single-peak solitons for μ=0, ε=0.3, and ω=-1. The energy of the bound state is 1.30, the energy of each constituent being 0.35.

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Fig. 5.
Fig. 5. Self-trapping of an input pulse of the forward wave (u), at initial velocity c=0.2, into a quiescent (c=0) soliton with residual internal vibrations, which falls into the central bandgap, in the model with μ=0.5 and ε=0. The inset in (b) illustrates the initial growth of field ν, which is absent in the input, at the soliton’s center. The energy of the input pulse is E=2.28, of which 40% is kept by the established soliton.

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Fig. 6.
Fig. 6. Self-trapping of a two-component input pulse, moving at velocity c=0.2, into a standing double-peak soliton, in the model with μ=0.5 and ε=0. This case is relevant to the spatial-domain model, see text. The input energy is E=3.04, about 60% of which is kept by the emerging double-peak soliton.

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Fig. 7.
Fig. 7. (a) Formation of a breather from an unstable double-peak soliton in gap -1, for μ=0.1, ε=0 and ω=-1.12, E=2.05. Note the leap of the breather from the original position. (b) The transformation of an unstable double-peak soliton, with μ=0.9, ε=0, ω=-1.1 and E=5, into a stable gap soliton of the same type, with energy E=3.3. In (a) and (b), only the u component is shown, as the evolution of field ν is quite similar.

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Fig. 8.
Fig. 8. Depinning of a soliton with energy E=3.00, which belongs to the central bandgap (ω=0.40) in the model with μ=0.03 and ε=0, by the kick with c 0=0.31, (this value only slightly exceeds the depinning threshold). The motion of the soliton is shown by means of contour plots of |u|2.

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Equations (5)

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i ∂ u ∂ t + i ∂ u ∂ x + [ 1 − ε cos ⁡ ( kx ) ] v + μ cos ⁡ ( kx ) · u + ( ∣ v ∣ 2 + 1 2 ∣ u ∣ 2 ) u = 0 ,
i ∂ v ∂ t − i ∂ v ∂ x + [ 1 − ε cos ⁡ ( kx ) ] u + μ cos ⁡ ( kx ) · v + ( ∣ u ∣ 2 + 1 2 ∣ v ∣ 2 ) v = 0 .
+ i d U d x + [ ω + μ cos ⁡ ( kx ) ] U + [ 1 − ε cos ⁡ ( kx ) ] V + [ ( ∣ V ∣ 2 + 1 2 ∣ U ∣ 2 ) ] U = 0 ,
− i d V d x + [ ω + μ cos ⁡ ( kx ) ] V + [ 1 − ε cos ⁡ ( kx ) ] U + [ ( ∣ U ∣ 2 + 1 2 ∣ V ∣ 2 ) ] V = 0 .
E = ∫ − ∞ + ∞ ( ∣ u ∣ 2 + ∣ v ∣ 2 ) dx ,

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