Abstract

We argue that it should be possible to observe gap-soliton switching in a system composed of two channel waveguides coupled by microresonators, even when the system is only 50 µm long. We differentiate between gaps that occur because of Bragg reflection and gaps that occur because of the resonance of the microresonators. The latter are characterized by anomalously small group-velocity dispersion and therefore by smaller nonlinear switching intensities.

© 2002 Optical Society of America

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  1. S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
    [CrossRef] [PubMed]
  2. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
    [CrossRef] [PubMed]
  3. H. G. Winful, J. H. Marburger, and E. Garmire, “Theory of bistability in nonlinear distributed feedback structures,” Appl. Phys. Lett. 35, 379–381 (1979).
    [CrossRef]
  4. C. M. de Sterke and J. E. Sipe, “Gap solitons,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1994), Vol. 33, pp. 203–260.
  5. J. E. Sipe and H. G. Winful, “Nonlinear Schrödinger solitons in a periodic structure,” Opt. Lett. 13, 132–133 (1988).
    [CrossRef]
  6. C. M. de Sterke, D. G. Salinas, and J. E. Sipe, “Coupled-mode theory for light propagation through deep nonlinear gratings,” Phys. Rev. E 54, 1969–1989 (1996).
    [CrossRef]
  7. C. M. de Sterke and J. E. Sipe, “Envelope-function approach for the electrodynamics of nonlinear periodic structures,” Phys. Rev. A 38, 5149–5165 (1989).
    [CrossRef]
  8. G. P. Agrawal, Non-Linear Fiber Optics (Academic, San Diego, Calif., 1989).
  9. S. Pereira and J. E. Sipe, “Light propagation through birefringent, nonlinear media with deep gratings,” Phys. Rev. E 62, 5745–5757 (2000).
    [CrossRef]
  10. A. B. Aceves and S. Wabnitz, “Self-induced transparency solitons in nonlinear refractive periodic media,” Phys. Lett. A 141, 37–42 (1989).
    [CrossRef]
  11. U. Mohideen, R. E. Slusher, V. Mizrahi, T. Erdogan, M. Kuwata-Gonokami, P. J. Lemaire, J. E. Sipe, C. M. de Sterke, and N. G. R. Broderick, “Gap soliton propagation in optical fiber gratings,” Opt. Lett. 20, 1674–1676 (1995).
    [CrossRef] [PubMed]
  12. 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]
  13. N. G. R. Broderick, D. Taverner, D. J. Richardson, M. Ibsen, and R. I. Laming, “Experimental observation of nonlinear pulse compression in nonuniform Bragg gratings,” Opt. Lett. 22, 1837–1839 (1997).
    [CrossRef]
  14. D. Taverner, N. G. R. Broderick, D. J. Richardson, M. Ibsen, and R. I. Laming, “All-optical AND gate based on coupled gap-soliton formation in a fiber Bragg grating,” Opt. Lett. 23, 259–261 (1998).
    [CrossRef]
  15. D. Taverner, N. G. R. Broderick, D. J. Richardson, R. I. Laming, and M. Ibsen, “Nonlinear self-switching and multiple gap-soliton formation in a fiber Bragg grating,” Opt. Lett. 23, 328–331 (1998).
    [CrossRef]
  16. 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]
  17. R. E. Slusher, S. Spälter, B. J. Eggleton, S. Pereira, and J. E. Sipe, “Bragg-grating-enhanced polarization instability,” Opt. Lett. 25, 749–751 (2000).
    [CrossRef]
  18. N. D. Sankey, D. F. Prelewitz, and T. G. Brown, “All-optical switching in a nonlinear periodic-waveguide structure,” Appl. Phys. Lett. 60, 1427–1429 (1992).
    [CrossRef]
  19. 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]
  20. P. Millar, 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]
  21. N. G. R. Broderick, P. Millar, D. J. Richardson, J. S. Aitchison, R. De La Rue, and T. Krauss, “Spectral features associated with nonlinear pulse compression in Bragg gratings,” Opt. Lett. 25, 740–742 (2000).
    [CrossRef]
  22. S. Blair and K. Kagner, “(2+1)-D propagation of spatio-temporal solitary waves including higher-order corrections,” Opt. Quantum Electron. 30, 697–737 (1998).
    [CrossRef]
  23. N. Akozbek and S. John, “Optical solitary waves in two- and three-dimensional nonlinear photonic band-gap structures,” Phys. Rev. E 57, 2287–2319 (1998).
    [CrossRef]
  24. B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Hans, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si–SiO2 microring resonator optical channel dropping filters,” IEEE Photon. Technol. Lett. 10, 549–551 (1998).
    [CrossRef]
  25. B. E. Little, S. T. Chu, J. V. Hryniewicz, and P. Absil, “Filter synthesis for periodically coupled microring resonators,” Opt. Lett. 25, 344–346 (2000).
    [CrossRef]
  26. J. E. Heebner, R. W. Boyd, and Q.-H. Park, “SCISSOR solitons and other novel propagation effects in microresonator- modified waveguides,” J. Opt. Soc. Am. B 19, 722–731 (2002).
    [CrossRef]
  27. For an illustration of these oscillations and of how they can be removed by apodization see, e.g., J. E. Sipe, B. J. Eggleton, and T. A. Strasser, “Dispersion characteristics of nonuniform Bragg gratings: implications for WDM communication systems,” Opt. Commun. 152, 269–274 (1998).
    [CrossRef]
  28. N. W. Ashcroft and N. D. Mermin, Solid State Physics (Saunders, Philadelphia, Pa., 1976).
  29. N. A. R. Bhat and J. E. Sipe, “Optical pulse propagation in nonlinear photonic crystals,” Phys. Rev. E 64, 056604 (2001).
    [CrossRef]
  30. 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]
  31. C. M. de Sterke, “Propagation through apodized gratings,” Opt. Express 3, 405–410 (1998), http://www. opticsexpress.org.
    [CrossRef] [PubMed]
  32. Normalizations of fields that correspond to gmk̄(z, t) abound in the literature, and care must be taken in comparing both the equations and the effective coefficients that appear in various papers.
  33. S. Pereira, J. E. Sipe, J. E. Heebner, and R. W. Boyd, “Gap solitons in a two-channel microresonator structure,” Opt. Lett. 27, 536–538 (2002).
    [CrossRef]
  34. C. M. de Sterke and J. E. Sipe, “Self-localized light: launching of low-velocity solitons in corrugated nonlinear waveguides,” Opt. Lett. 14, 871–873 (1989).
    [CrossRef] [PubMed]
  35. P. P. Absil, J. V. Hryniewicz, B. E. Little, P. S. Cho, R. A. Wilson, L. G. Joneckis, and P.-T. Ho, “Wavelength conversion in GaAs microring resonators,” Opt. Lett. 25, 554–556 (2000).
    [CrossRef]

2002 (2)

2001 (1)

N. A. R. Bhat and J. E. Sipe, “Optical pulse propagation in nonlinear photonic crystals,” Phys. Rev. E 64, 056604 (2001).
[CrossRef]

2000 (6)

1999 (2)

1998 (7)

D. Taverner, N. G. R. Broderick, D. J. Richardson, M. Ibsen, and R. I. Laming, “All-optical AND gate based on coupled gap-soliton formation in a fiber Bragg grating,” Opt. Lett. 23, 259–261 (1998).
[CrossRef]

D. Taverner, N. G. R. Broderick, D. J. Richardson, R. I. Laming, and M. Ibsen, “Nonlinear self-switching and multiple gap-soliton formation in a fiber Bragg grating,” Opt. Lett. 23, 328–331 (1998).
[CrossRef]

C. M. de Sterke, “Propagation through apodized gratings,” Opt. Express 3, 405–410 (1998), http://www. opticsexpress.org.
[CrossRef] [PubMed]

S. Blair and K. Kagner, “(2+1)-D propagation of spatio-temporal solitary waves including higher-order corrections,” Opt. Quantum Electron. 30, 697–737 (1998).
[CrossRef]

N. Akozbek and S. John, “Optical solitary waves in two- and three-dimensional nonlinear photonic band-gap structures,” Phys. Rev. E 57, 2287–2319 (1998).
[CrossRef]

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Hans, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si–SiO2 microring resonator optical channel dropping filters,” IEEE Photon. Technol. Lett. 10, 549–551 (1998).
[CrossRef]

For an illustration of these oscillations and of how they can be removed by apodization see, e.g., J. E. Sipe, B. J. Eggleton, and T. A. Strasser, “Dispersion characteristics of nonuniform Bragg gratings: implications for WDM communication systems,” Opt. Commun. 152, 269–274 (1998).
[CrossRef]

1997 (1)

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]

C. M. de Sterke, D. G. Salinas, and J. E. Sipe, “Coupled-mode theory for light propagation through deep nonlinear gratings,” Phys. Rev. E 54, 1969–1989 (1996).
[CrossRef]

1995 (1)

1994 (1)

1992 (1)

N. D. Sankey, D. F. Prelewitz, and T. G. Brown, “All-optical switching in a nonlinear periodic-waveguide structure,” Appl. Phys. Lett. 60, 1427–1429 (1992).
[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]

C. M. de Sterke and J. E. Sipe, “Envelope-function approach for the electrodynamics of nonlinear periodic structures,” Phys. Rev. A 38, 5149–5165 (1989).
[CrossRef]

C. M. de Sterke and J. E. Sipe, “Self-localized light: launching of low-velocity solitons in corrugated nonlinear waveguides,” Opt. Lett. 14, 871–873 (1989).
[CrossRef] [PubMed]

1988 (1)

1987 (2)

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
[CrossRef] [PubMed]

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[CrossRef] [PubMed]

1979 (1)

H. G. Winful, J. H. Marburger, and E. Garmire, “Theory of bistability in nonlinear distributed feedback structures,” Appl. Phys. Lett. 35, 379–381 (1979).
[CrossRef]

Absil, P.

Absil, P. P.

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.

Akozbek, N.

N. Akozbek and S. John, “Optical solitary waves in two- and three-dimensional nonlinear photonic band-gap structures,” Phys. Rev. E 57, 2287–2319 (1998).
[CrossRef]

Bhat, N. A. R.

N. A. R. Bhat and J. E. Sipe, “Optical pulse propagation in nonlinear photonic crystals,” Phys. Rev. E 64, 056604 (2001).
[CrossRef]

Blair, S.

S. Blair and K. Kagner, “(2+1)-D propagation of spatio-temporal solitary waves including higher-order corrections,” Opt. Quantum Electron. 30, 697–737 (1998).
[CrossRef]

Boyd, R. W.

Broderick, N. G. R.

Brown, T. G.

N. D. Sankey, D. F. Prelewitz, and T. G. Brown, “All-optical switching in a nonlinear periodic-waveguide structure,” Appl. Phys. Lett. 60, 1427–1429 (1992).
[CrossRef]

Cho, P. S.

Chu, S. T.

B. E. Little, S. T. Chu, J. V. Hryniewicz, and P. Absil, “Filter synthesis for periodically coupled microring resonators,” Opt. Lett. 25, 344–346 (2000).
[CrossRef]

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Hans, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si–SiO2 microring resonator optical channel dropping filters,” IEEE Photon. Technol. Lett. 10, 549–551 (1998).
[CrossRef]

De La Rue, M.

De La Rue, R.

de Sterke, C. M.

Eggleton, B. J.

R. E. Slusher, S. Spälter, B. J. Eggleton, S. Pereira, and J. E. Sipe, “Bragg-grating-enhanced polarization instability,” Opt. Lett. 25, 749–751 (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]

For an illustration of these oscillations and of how they can be removed by apodization see, e.g., J. E. Sipe, B. J. Eggleton, and T. A. Strasser, “Dispersion characteristics of nonuniform Bragg gratings: implications for WDM communication systems,” Opt. Commun. 152, 269–274 (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]

Erdogan, T.

Foresi, J. S.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Hans, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si–SiO2 microring resonator optical channel dropping filters,” IEEE Photon. Technol. Lett. 10, 549–551 (1998).
[CrossRef]

Garmire, E.

H. G. Winful, J. H. Marburger, and E. Garmire, “Theory of bistability in nonlinear distributed feedback structures,” Appl. Phys. Lett. 35, 379–381 (1979).
[CrossRef]

Greene, W.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Hans, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si–SiO2 microring resonator optical channel dropping filters,” IEEE Photon. Technol. Lett. 10, 549–551 (1998).
[CrossRef]

Hans, H. A.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Hans, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si–SiO2 microring resonator optical channel dropping filters,” IEEE Photon. Technol. Lett. 10, 549–551 (1998).
[CrossRef]

Heebner, J. E.

Ho, P.-T.

Hryniewicz, J. V.

Ibsen, M.

Ippen, E. P.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Hans, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si–SiO2 microring resonator optical channel dropping filters,” IEEE Photon. Technol. Lett. 10, 549–551 (1998).
[CrossRef]

John, S.

N. Akozbek and S. John, “Optical solitary waves in two- and three-dimensional nonlinear photonic band-gap structures,” Phys. Rev. E 57, 2287–2319 (1998).
[CrossRef]

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
[CrossRef] [PubMed]

Joneckis, L. G.

Kagner, K.

S. Blair and K. Kagner, “(2+1)-D propagation of spatio-temporal solitary waves including higher-order corrections,” Opt. Quantum Electron. 30, 697–737 (1998).
[CrossRef]

Kimerling, L. C.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Hans, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si–SiO2 microring resonator optical channel dropping filters,” IEEE Photon. Technol. Lett. 10, 549–551 (1998).
[CrossRef]

Krauss, T.

Krauss, T. F.

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]

Kuwata-Gonokami, M.

Laming, R. I.

Lemaire, P. J.

Little, B. E.

Marburger, J. H.

H. G. Winful, J. H. Marburger, and E. Garmire, “Theory of bistability in nonlinear distributed feedback structures,” Appl. Phys. Lett. 35, 379–381 (1979).
[CrossRef]

Millar, P.

Mizrahi, V.

Mohideen, U.

Park, Q.-H.

Pereira, S.

Poladian, L.

Prelewitz, D. F.

N. D. Sankey, D. F. Prelewitz, and T. G. Brown, “All-optical switching in a nonlinear periodic-waveguide structure,” Appl. Phys. Lett. 60, 1427–1429 (1992).
[CrossRef]

Richardson, D. J.

Salinas, D. G.

C. M. de Sterke, D. G. Salinas, and J. E. Sipe, “Coupled-mode theory for light propagation through deep nonlinear gratings,” Phys. Rev. E 54, 1969–1989 (1996).
[CrossRef]

Sankey, N. D.

N. D. Sankey, D. F. Prelewitz, and T. G. Brown, “All-optical switching in a nonlinear periodic-waveguide structure,” Appl. Phys. Lett. 60, 1427–1429 (1992).
[CrossRef]

Sipe, J. E.

S. Pereira, J. E. Sipe, J. E. Heebner, and R. W. Boyd, “Gap solitons in a two-channel microresonator structure,” Opt. Lett. 27, 536–538 (2002).
[CrossRef]

N. A. R. Bhat and J. E. Sipe, “Optical pulse propagation in nonlinear photonic crystals,” Phys. Rev. E 64, 056604 (2001).
[CrossRef]

R. E. Slusher, S. Spälter, B. J. Eggleton, S. Pereira, and J. E. Sipe, “Bragg-grating-enhanced polarization instability,” Opt. Lett. 25, 749–751 (2000).
[CrossRef]

S. Pereira and J. E. Sipe, “Light propagation through birefringent, nonlinear media with deep gratings,” Phys. Rev. E 62, 5745–5757 (2000).
[CrossRef]

For an illustration of these oscillations and of how they can be removed by apodization see, e.g., J. E. Sipe, B. J. Eggleton, and T. A. Strasser, “Dispersion characteristics of nonuniform Bragg gratings: implications for WDM communication systems,” Opt. Commun. 152, 269–274 (1998).
[CrossRef]

C. M. de Sterke, D. G. Salinas, and J. E. Sipe, “Coupled-mode theory for light propagation through deep nonlinear gratings,” Phys. Rev. E 54, 1969–1989 (1996).
[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]

U. Mohideen, R. E. Slusher, V. Mizrahi, T. Erdogan, M. Kuwata-Gonokami, P. J. Lemaire, J. E. Sipe, C. M. de Sterke, and N. G. R. Broderick, “Gap soliton propagation in optical fiber gratings,” Opt. Lett. 20, 1674–1676 (1995).
[CrossRef] [PubMed]

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]

C. M. de Sterke and J. E. Sipe, “Self-localized light: launching of low-velocity solitons in corrugated nonlinear waveguides,” Opt. Lett. 14, 871–873 (1989).
[CrossRef] [PubMed]

C. M. de Sterke and J. E. Sipe, “Envelope-function approach for the electrodynamics of nonlinear periodic structures,” Phys. Rev. A 38, 5149–5165 (1989).
[CrossRef]

J. E. Sipe and H. G. Winful, “Nonlinear Schrödinger solitons in a periodic structure,” Opt. Lett. 13, 132–133 (1988).
[CrossRef]

Slusher, R. E.

Spälter, S.

Steinmeyer, G.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Hans, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si–SiO2 microring resonator optical channel dropping filters,” IEEE Photon. Technol. Lett. 10, 549–551 (1998).
[CrossRef]

Strasser, T. A.

For an illustration of these oscillations and of how they can be removed by apodization see, e.g., J. E. Sipe, B. J. Eggleton, and T. A. Strasser, “Dispersion characteristics of nonuniform Bragg gratings: implications for WDM communication systems,” Opt. Commun. 152, 269–274 (1998).
[CrossRef]

Taverner, D.

Thoen, E. R.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Hans, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si–SiO2 microring resonator optical channel dropping filters,” IEEE Photon. Technol. Lett. 10, 549–551 (1998).
[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]

Wilson, R. A.

Winful, H. G.

J. E. Sipe and H. G. Winful, “Nonlinear Schrödinger solitons in a periodic structure,” Opt. Lett. 13, 132–133 (1988).
[CrossRef]

H. G. Winful, J. H. Marburger, and E. Garmire, “Theory of bistability in nonlinear distributed feedback structures,” Appl. Phys. Lett. 35, 379–381 (1979).
[CrossRef]

Yablonovitch, E.

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[CrossRef] [PubMed]

Appl. Phys. Lett. (2)

H. G. Winful, J. H. Marburger, and E. Garmire, “Theory of bistability in nonlinear distributed feedback structures,” Appl. Phys. Lett. 35, 379–381 (1979).
[CrossRef]

N. D. Sankey, D. F. Prelewitz, and T. G. Brown, “All-optical switching in a nonlinear periodic-waveguide structure,” Appl. Phys. Lett. 60, 1427–1429 (1992).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Hans, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si–SiO2 microring resonator optical channel dropping filters,” IEEE Photon. Technol. Lett. 10, 549–551 (1998).
[CrossRef]

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

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

Opt. Commun. (1)

For an illustration of these oscillations and of how they can be removed by apodization see, e.g., J. E. Sipe, B. J. Eggleton, and T. A. Strasser, “Dispersion characteristics of nonuniform Bragg gratings: implications for WDM communication systems,” Opt. Commun. 152, 269–274 (1998).
[CrossRef]

Opt. Express (1)

Opt. Lett. (13)

S. Pereira, J. E. Sipe, J. E. Heebner, and R. W. Boyd, “Gap solitons in a two-channel microresonator structure,” Opt. Lett. 27, 536–538 (2002).
[CrossRef]

C. M. de Sterke and J. E. Sipe, “Self-localized light: launching of low-velocity solitons in corrugated nonlinear waveguides,” Opt. Lett. 14, 871–873 (1989).
[CrossRef] [PubMed]

P. P. Absil, J. V. Hryniewicz, B. E. Little, P. S. Cho, R. A. Wilson, L. G. Joneckis, and P.-T. Ho, “Wavelength conversion in GaAs microring resonators,” Opt. Lett. 25, 554–556 (2000).
[CrossRef]

B. E. Little, S. T. Chu, J. V. Hryniewicz, and P. Absil, “Filter synthesis for periodically coupled microring resonators,” Opt. Lett. 25, 344–346 (2000).
[CrossRef]

P. Millar, 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]

N. G. R. Broderick, P. Millar, D. J. Richardson, J. S. Aitchison, R. De La Rue, and T. Krauss, “Spectral features associated with nonlinear pulse compression in Bragg gratings,” Opt. Lett. 25, 740–742 (2000).
[CrossRef]

J. E. Sipe and H. G. Winful, “Nonlinear Schrödinger solitons in a periodic structure,” Opt. Lett. 13, 132–133 (1988).
[CrossRef]

U. Mohideen, R. E. Slusher, V. Mizrahi, T. Erdogan, M. Kuwata-Gonokami, P. J. Lemaire, J. E. Sipe, C. M. de Sterke, and N. G. R. Broderick, “Gap soliton propagation in optical fiber gratings,” Opt. Lett. 20, 1674–1676 (1995).
[CrossRef] [PubMed]

N. G. R. Broderick, D. Taverner, D. J. Richardson, M. Ibsen, and R. I. Laming, “Experimental observation of nonlinear pulse compression in nonuniform Bragg gratings,” Opt. Lett. 22, 1837–1839 (1997).
[CrossRef]

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G. P. Agrawal, Non-Linear Fiber Optics (Academic, San Diego, Calif., 1989).

Normalizations of fields that correspond to gmk̄(z, t) abound in the literature, and care must be taken in comparing both the equations and the effective coefficients that appear in various papers.

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Figures (9)

Fig. 1
Fig. 1

(a) Schematic of the two-channel SCISSOR structure. The curving of the waveguides is meant to suggest the apodization of the device discussed in Section 4 below. (b) One unit cell of the structure. We assume that coupling between the channel guides and the microresonator occurs only at the two points indicated by the large filled circles.

Fig. 2
Fig. 2

Dispersion relation of the two-channel SCISSOR structure in the vicinity of a resonator gap (ω/c4.11 µm-1) and a Bragg gap (ω/c4.075 µm-1). On the right-hand axis, we plot the vacuum wavelength associated with the given value of ω/c facilitate comparison with the rest of the plots in the paper.

Fig. 3
Fig. 3

Imaginary portion of the Bloch wave number, kim, plotted as a function of the vacuum wavelength for (a) a resonator gap and (b) a Bragg gap.

Fig. 4
Fig. 4

GVD as a function of the vacuum wavelength for (a) a resonator gap and (b) a Bragg gap.

Fig. 5
Fig. 5

(a) Plots of the linear cw transmission spectrum for a resonator gap in structures with 3 unit cells (dashed curve) and 20 unit cells (solid curve). Dotted lines, the transmission predicted by the dispersion relation. We also plot the simulated linear transmission of 100-ps pulses through structures with 3 cells (squares) and 20 cells (triangles) at the indicated center wavelengths. (b) Plots of the linear cw transmission spectrum for a Bragg gap in structures with 50 cells (dashed curve) and 100 cells (solid curve). Dotted lines, the linear transmission predicted by the dispersion relation. We also plot the simulated linear transmission of 100-ps pulses through structures with 50 cells (squares) and 100 cells (triangles).

Fig. 6
Fig. 6

(a) Plots of the linear cw transmission spectrum for a resonator gap in apodized structures with 3 unit cells (dashed curve) and 20 unit cells (solid curve). The apodization profile is discussed in the text. We also plot the calculated linear transmission for 100-ps pulses through apodized structures with 3 cells (squares) and 20 cells (triangles). (b) Plots of the group velocity predicted by the dispersion relation (solid curve) and the group velocity found by our simulations with 100-ps pulses for the 20-cell structure (filled circles).

Fig. 7
Fig. 7

Nonlinear switching out of a resonator gap for apodized structures with 3 cells (dotted curve) and 20 cells (solid curve) by use of 100-ps pulses. The 3-cell structure achieves 60% transmission at 45 MW/cm2; the 20-cell structure achieves 60% transmission at 20 MW/cm2.

Fig. 8
Fig. 8

Output pulses for the apodized 3-cell structure for three values of the incident intensity of the 100-ps pulses: 40 MW/cm2 (solid curve), 85 MW/cm2 (dashed–dotted curve), and 300 MW/cm2 (dotted curve), with the frequency content of the pulse inside a resonator gap. The output pulses are normalized to their incident intensities.

Fig. 9
Fig. 9

Nonlinear switching out of a Bragg gap with 200 unit cells.

Equations (36)

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Hmk(r)=hmk(r)exp(ik·r),
Emk(r)=emk(r)exp(ik·r),
μ00  drΩ Hmk*(r)·Hmk(r)=δmmδkk,
 drΩn2(r)Emk*(r)·Emk(r)=δmmδkk,
q(0+)l(a+)=σiκiκσq(0-)l(a-),
q(π+)u(a-)=σiκiκσq(π-)u(a+),
f(d)=T(ω)f(0),
f(z)=l(z)u(z),
T(ω)=1σ[(σ2+κ2)exp(2iπνρ)-1]×exp(iνd)[(σ2+κ2)exp(2iπνρ)-σ2]κ2 exp(iπνρ)-κ2 exp(iπνρ)exp(-iνd)[σ2 exp(2iπνρ)-1].
f(d)=exp[ik(ω)d]f(0),
T(ω)-exp[ik(ω)d]I=0,
k(ω)=-id ln[([T11(ω)+T22(ω)]±{[T11(ω)+T22(ω)]2-4}1/2)/2],
λr=2πneffρ,
λb2neffd.
B=×A,
D=-×N.
0n2(r) A(r, t)t=×N(r, t)+PNL(r, t),
μ0 N(r, t)t=-×A(r, t).
Ψ=ψ+(r, t)ψ-(r, t)=12 n(r)A(r, t)+i2[μ0/0n(r)]1/2N(r, t)12 n(r)A(r, t)-i2[μ0/0n(r)]1/2N(r, t),
Ψ=ψ+ψ-,Ψ¯=ψ-*ψ+*
Ψ+(r, t)=a m,k fmkΦmk(r)exp(-iωmkt),
Φmk(r)12 n(r)Emk(r)+i[μ0/0n(r)]1/2Hmk(r)n(r)Emk(r)-i[μ0/0n(r)]1/2Hmk(r).
Ψ+(r, t)=afmk¯(z, t)Φmk¯(r)+qm fqk¯(z, t)Φqk¯(r)exp(-iω¯t),
PNLi(r, t)=0χ3ijlk(r)Ej(r, t)Ek(r, t)Ek(r, t)El(r, t).
PNLi(r, t)=30χ3ijkl(r)Ej(r, t)Ek(r, t)[El(r, t)]*×exp(-iω¯t)+c.c.
PNL(r, t)=PNL(r, t)yˆ=30χ3yyyy(r)|E(r, t)|2E(r, t)exp(-iω¯t)yˆ+c.c.,
tn=ηnt,zn=ηnz,
fmk¯(zn;tn)=Fmk¯(0)(zn; tn)+ηFmk¯(1)(zn; tn)+ ,
fqk¯(zn; tn)=ηFqk¯(1)(zn; tn)+η2Fmk¯(2)(zn; tn)+ ,
gmk¯(z, t)=[2Aeffa2ω¯20]1/2fmk¯(z, t),
E=-|gmk¯(z, t)|2dz.
0=i t+iωmk¯ z+12ωmk¯ 2z2gmk¯+Γmk¯|gmk¯|2gmk¯,
Γmk¯=34 ω¯Aeff0 cell drΩcellχ3yyyy(r)|emk¯(r)|4,
γ=2πn2/λ,
A(ζ, t+δt)
=A(ζ-δζ, t)exp{i[neff+n2|A(ζ-δζ, t)|2]ω(δζ)/c}

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