Abstract

A linear stability analysis to investigate the instabilities in nonlinear distributed-feedback gratings with a finite material response time is presented. The amplification (or attenuation) rate and the frequency of sinusoidal perturbations generated in the grating are calculated for different values of the material response time, detuning, and coupling strength. To give the full picture of the stability boundaries, stability maps are plotted in the two cases of a weak grating and a strong grating. The stable region can be enlarged by increasing a response time of the nonlinearity. The required response time increases with the grating strength. A comparison with numerical simulations of the coupled-mode equations is done to confirm the validity of the stability analysis.

© 2000 Optical Society of America

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  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]
  2. M. Cada, J. He, B. Acklin, M. Proctor, D. Martin, F. Morier-Genoud, M.-A. Dupertuis, and J. M. Glinski, “All-optical reflectivity tuning and logic gating in a GaAs/AlAs periodic layered structure,” Appl. Phys. Lett. 60, 404–406 (1992).
    [CrossRef]
  3. 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]
  4. C. J. Herbert, W. S. Capinski, and M. S. Malcuit, “Optical power limiting with nonlinear periodic structures,” Opt. Lett. 17, 1037–1039 (1992).
    [CrossRef] [PubMed]
  5. C. J. Herbert and M. S. Malcuit, “Optical bistability in nonlinear periodic structures,” Opt. Lett. 18, 1783–1785 (1993).
    [CrossRef] [PubMed]
  6. H. G. Winful and G. D. Cooperman, “Self-pulsing and chaos in distributed feedback bistable optical devices,” Appl. Phys. Lett. 40, 298–300 (1982).
    [CrossRef]
  7. H. G. Winful, R. Zamir, and S. Feldman, “Modulational instability in nonlinear periodic structures: implications for ‘gap solitons’,” Appl. Phys. Lett. 58, 1001–1003 (1991).
    [CrossRef]
  8. W. Chen and D. L. Mills, “Gap solitons and the nonlinear optical response of superlattices,” Phys. Rev. Lett. 58, 160–163 (1987).
    [CrossRef] [PubMed]
  9. C. M. de Sterke, “Stability analysis of nonlinear periodic media,” Phys. Rev. A 45, 8252–8258 (1992).
    [CrossRef] [PubMed]
  10. C. M. de Sterke and J. E. Sipe, “Switching dynamics of finite periodic nonlinear media: a numerical study,” Phys. Rev. A 42, 2858–2869 (1990).
    [CrossRef] [PubMed]
  11. C. M. de Sterke, “Simulations of gap-soliton generation,” Phys. Rev. A 45, 2012–2018 (1992).
    [CrossRef] [PubMed]
  12. B. J. Eggleton, C. M. de Sterke, R. E. Slusher, and J. E. Sipe, “Distributed feedback pulse generator based on nonlinear fiber grating,” Electron. Lett. 32, 2341–2342 (1996).
    [CrossRef]
  13. 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]
  14. B. J. Eggleton, C. M. de Sterke, and R. E. Slusher, “Nonlinear pulse propagation in Bragg gratings,” J. Opt. Soc. Am. B 14, 2980–2993 (1997).
    [CrossRef]
  15. B. J. Eggleton, C. M. de Sterke, A. B. Aceves, J. E. Sipe, T. A. Strasser, and R. E. Slusher, “Modulational instability and tunable multiple soliton generation in apodized fiber gratings,” Opt. Commun. 149, 267–271 (1998).
    [CrossRef]
  16. W. J. Firth, “Stability of nonlinear Fabry–Perot resonators,” Opt. Commun. 39, 343–346 (1981).
    [CrossRef]
  17. K. Ogusu, H. Li, and T. Kamizono, “Analysis of transient optical bistability and stability in a nonlinear fiber Fabry– Perot resonator based on an iterative method,” Opt. Rev. 5, 185–190 (1998).
    [CrossRef]
  18. A. L. Steele, S. Lynch, and J. E. Hoad, “Analysis of optical instabilities and bistability in a nonlinear optical fiber loop mirror with feedback,” Opt. Commun. 137, 136–142 (1997).
    [CrossRef]
  19. H. Li and K. Ogusu, “Analysis of optical instability in a double-coupler nonlinear fiber ring resonator,” Opt. Commun. 157, 27–32 (1998).
    [CrossRef]
  20. K. Ogusu and T. Kamizono, “Effect of material response time on optical bistability in nonlinear distributed feedback gratings,” Opt. Rev. 7, 83–88 (2000).
    [CrossRef]

2000

K. Ogusu and T. Kamizono, “Effect of material response time on optical bistability in nonlinear distributed feedback gratings,” Opt. Rev. 7, 83–88 (2000).
[CrossRef]

1998

H. Li and K. Ogusu, “Analysis of optical instability in a double-coupler nonlinear fiber ring resonator,” Opt. Commun. 157, 27–32 (1998).
[CrossRef]

B. J. Eggleton, C. M. de Sterke, A. B. Aceves, J. E. Sipe, T. A. Strasser, and R. E. Slusher, “Modulational instability and tunable multiple soliton generation in apodized fiber gratings,” Opt. Commun. 149, 267–271 (1998).
[CrossRef]

K. Ogusu, H. Li, and T. Kamizono, “Analysis of transient optical bistability and stability in a nonlinear fiber Fabry– Perot resonator based on an iterative method,” Opt. Rev. 5, 185–190 (1998).
[CrossRef]

1997

A. L. Steele, S. Lynch, and J. E. Hoad, “Analysis of optical instabilities and bistability in a nonlinear optical fiber loop mirror with feedback,” Opt. Commun. 137, 136–142 (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, 2980–2993 (1997).
[CrossRef]

1996

B. J. Eggleton, C. M. de Sterke, R. E. Slusher, and J. E. Sipe, “Distributed feedback pulse generator based on nonlinear fiber grating,” Electron. Lett. 32, 2341–2342 (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]

1993

1992

M. Cada, J. He, B. Acklin, M. Proctor, D. Martin, F. Morier-Genoud, M.-A. Dupertuis, and J. M. Glinski, “All-optical reflectivity tuning and logic gating in a GaAs/AlAs periodic layered structure,” Appl. Phys. Lett. 60, 404–406 (1992).
[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]

C. J. Herbert, W. S. Capinski, and M. S. Malcuit, “Optical power limiting with nonlinear periodic structures,” Opt. Lett. 17, 1037–1039 (1992).
[CrossRef] [PubMed]

C. M. de Sterke, “Stability analysis of nonlinear periodic media,” Phys. Rev. A 45, 8252–8258 (1992).
[CrossRef] [PubMed]

C. M. de Sterke, “Simulations of gap-soliton generation,” Phys. Rev. A 45, 2012–2018 (1992).
[CrossRef] [PubMed]

1991

H. G. Winful, R. Zamir, and S. Feldman, “Modulational instability in nonlinear periodic structures: implications for ‘gap solitons’,” Appl. Phys. Lett. 58, 1001–1003 (1991).
[CrossRef]

1990

C. M. de Sterke and J. E. Sipe, “Switching dynamics of finite periodic nonlinear media: a numerical study,” Phys. Rev. A 42, 2858–2869 (1990).
[CrossRef] [PubMed]

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]

1982

H. G. Winful and G. D. Cooperman, “Self-pulsing and chaos in distributed feedback bistable optical devices,” Appl. Phys. Lett. 40, 298–300 (1982).
[CrossRef]

1981

W. J. Firth, “Stability of nonlinear Fabry–Perot resonators,” Opt. Commun. 39, 343–346 (1981).
[CrossRef]

1979

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]

Aceves, A. B.

B. J. Eggleton, C. M. de Sterke, A. B. Aceves, J. E. Sipe, T. A. Strasser, and R. E. Slusher, “Modulational instability and tunable multiple soliton generation in apodized fiber gratings,” Opt. Commun. 149, 267–271 (1998).
[CrossRef]

Acklin, B.

M. Cada, J. He, B. Acklin, M. Proctor, D. Martin, F. Morier-Genoud, M.-A. Dupertuis, and J. M. Glinski, “All-optical reflectivity tuning and logic gating in a GaAs/AlAs periodic layered structure,” Appl. Phys. Lett. 60, 404–406 (1992).
[CrossRef]

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]

Cada, M.

M. Cada, J. He, B. Acklin, M. Proctor, D. Martin, F. Morier-Genoud, M.-A. Dupertuis, and J. M. Glinski, “All-optical reflectivity tuning and logic gating in a GaAs/AlAs periodic layered structure,” Appl. Phys. Lett. 60, 404–406 (1992).
[CrossRef]

Capinski, W. S.

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]

Cooperman, G. D.

H. G. Winful and G. D. Cooperman, “Self-pulsing and chaos in distributed feedback bistable optical devices,” Appl. Phys. Lett. 40, 298–300 (1982).
[CrossRef]

de Sterke, C. M.

B. J. Eggleton, C. M. de Sterke, A. B. Aceves, J. E. Sipe, T. A. Strasser, and R. E. Slusher, “Modulational instability and tunable multiple soliton generation in apodized fiber gratings,” Opt. Commun. 149, 267–271 (1998).
[CrossRef]

B. J. Eggleton, C. M. de Sterke, and R. E. Slusher, “Nonlinear pulse propagation in Bragg gratings,” J. Opt. Soc. Am. B 14, 2980–2993 (1997).
[CrossRef]

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

B. J. Eggleton, C. M. de Sterke, R. E. Slusher, and J. E. Sipe, “Distributed feedback pulse generator based on nonlinear fiber grating,” Electron. Lett. 32, 2341–2342 (1996).
[CrossRef]

C. M. de Sterke, “Simulations of gap-soliton generation,” Phys. Rev. A 45, 2012–2018 (1992).
[CrossRef] [PubMed]

C. M. de Sterke, “Stability analysis of nonlinear periodic media,” Phys. Rev. A 45, 8252–8258 (1992).
[CrossRef] [PubMed]

C. M. de Sterke and J. E. Sipe, “Switching dynamics of finite periodic nonlinear media: a numerical study,” Phys. Rev. A 42, 2858–2869 (1990).
[CrossRef] [PubMed]

Dupertuis, M.-A.

M. Cada, J. He, B. Acklin, M. Proctor, D. Martin, F. Morier-Genoud, M.-A. Dupertuis, and J. M. Glinski, “All-optical reflectivity tuning and logic gating in a GaAs/AlAs periodic layered structure,” Appl. Phys. Lett. 60, 404–406 (1992).
[CrossRef]

Eggleton, B. J.

B. J. Eggleton, C. M. de Sterke, A. B. Aceves, J. E. Sipe, T. A. Strasser, and R. E. Slusher, “Modulational instability and tunable multiple soliton generation in apodized fiber gratings,” Opt. Commun. 149, 267–271 (1998).
[CrossRef]

B. J. Eggleton, C. M. de Sterke, and R. E. Slusher, “Nonlinear pulse propagation in Bragg gratings,” J. Opt. Soc. Am. B 14, 2980–2993 (1997).
[CrossRef]

B. J. Eggleton, C. M. de Sterke, R. E. Slusher, and J. E. Sipe, “Distributed feedback pulse generator based on nonlinear fiber grating,” Electron. Lett. 32, 2341–2342 (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]

Feldman, S.

H. G. Winful, R. Zamir, and S. Feldman, “Modulational instability in nonlinear periodic structures: implications for ‘gap solitons’,” Appl. Phys. Lett. 58, 1001–1003 (1991).
[CrossRef]

Firth, W. J.

W. J. Firth, “Stability of nonlinear Fabry–Perot resonators,” Opt. Commun. 39, 343–346 (1981).
[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]

Glinski, J. M.

M. Cada, J. He, B. Acklin, M. Proctor, D. Martin, F. Morier-Genoud, M.-A. Dupertuis, and J. M. Glinski, “All-optical reflectivity tuning and logic gating in a GaAs/AlAs periodic layered structure,” Appl. Phys. Lett. 60, 404–406 (1992).
[CrossRef]

He, J.

M. Cada, J. He, B. Acklin, M. Proctor, D. Martin, F. Morier-Genoud, M.-A. Dupertuis, and J. M. Glinski, “All-optical reflectivity tuning and logic gating in a GaAs/AlAs periodic layered structure,” Appl. Phys. Lett. 60, 404–406 (1992).
[CrossRef]

Herbert, C. J.

Hoad, J. E.

A. L. Steele, S. Lynch, and J. E. Hoad, “Analysis of optical instabilities and bistability in a nonlinear optical fiber loop mirror with feedback,” Opt. Commun. 137, 136–142 (1997).
[CrossRef]

Kamizono, T.

K. Ogusu and T. Kamizono, “Effect of material response time on optical bistability in nonlinear distributed feedback gratings,” Opt. Rev. 7, 83–88 (2000).
[CrossRef]

K. Ogusu, H. Li, and T. Kamizono, “Analysis of transient optical bistability and stability in a nonlinear fiber Fabry– Perot resonator based on an iterative method,” Opt. Rev. 5, 185–190 (1998).
[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]

Li, H.

K. Ogusu, H. Li, and T. Kamizono, “Analysis of transient optical bistability and stability in a nonlinear fiber Fabry– Perot resonator based on an iterative method,” Opt. Rev. 5, 185–190 (1998).
[CrossRef]

H. Li and K. Ogusu, “Analysis of optical instability in a double-coupler nonlinear fiber ring resonator,” Opt. Commun. 157, 27–32 (1998).
[CrossRef]

Lynch, S.

A. L. Steele, S. Lynch, and J. E. Hoad, “Analysis of optical instabilities and bistability in a nonlinear optical fiber loop mirror with feedback,” Opt. Commun. 137, 136–142 (1997).
[CrossRef]

Malcuit, M. S.

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]

Martin, D.

M. Cada, J. He, B. Acklin, M. Proctor, D. Martin, F. Morier-Genoud, M.-A. Dupertuis, and J. M. Glinski, “All-optical reflectivity tuning and logic gating in a GaAs/AlAs periodic layered structure,” Appl. Phys. Lett. 60, 404–406 (1992).
[CrossRef]

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]

Morier-Genoud, F.

M. Cada, J. He, B. Acklin, M. Proctor, D. Martin, F. Morier-Genoud, M.-A. Dupertuis, and J. M. Glinski, “All-optical reflectivity tuning and logic gating in a GaAs/AlAs periodic layered structure,” Appl. Phys. Lett. 60, 404–406 (1992).
[CrossRef]

Ogusu, K.

K. Ogusu and T. Kamizono, “Effect of material response time on optical bistability in nonlinear distributed feedback gratings,” Opt. Rev. 7, 83–88 (2000).
[CrossRef]

H. Li and K. Ogusu, “Analysis of optical instability in a double-coupler nonlinear fiber ring resonator,” Opt. Commun. 157, 27–32 (1998).
[CrossRef]

K. Ogusu, H. Li, and T. Kamizono, “Analysis of transient optical bistability and stability in a nonlinear fiber Fabry– Perot resonator based on an iterative method,” Opt. Rev. 5, 185–190 (1998).
[CrossRef]

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]

Proctor, M.

M. Cada, J. He, B. Acklin, M. Proctor, D. Martin, F. Morier-Genoud, M.-A. Dupertuis, and J. M. Glinski, “All-optical reflectivity tuning and logic gating in a GaAs/AlAs periodic layered structure,” Appl. Phys. Lett. 60, 404–406 (1992).
[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.

B. J. Eggleton, C. M. de Sterke, A. B. Aceves, J. E. Sipe, T. A. Strasser, and R. E. Slusher, “Modulational instability and tunable multiple soliton generation in apodized fiber gratings,” Opt. Commun. 149, 267–271 (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]

B. J. Eggleton, C. M. de Sterke, R. E. Slusher, and J. E. Sipe, “Distributed feedback pulse generator based on nonlinear fiber grating,” Electron. Lett. 32, 2341–2342 (1996).
[CrossRef]

C. M. de Sterke and J. E. Sipe, “Switching dynamics of finite periodic nonlinear media: a numerical study,” Phys. Rev. A 42, 2858–2869 (1990).
[CrossRef] [PubMed]

Slusher, R. E.

B. J. Eggleton, C. M. de Sterke, A. B. Aceves, J. E. Sipe, T. A. Strasser, and R. E. Slusher, “Modulational instability and tunable multiple soliton generation in apodized fiber gratings,” Opt. Commun. 149, 267–271 (1998).
[CrossRef]

B. J. Eggleton, C. M. de Sterke, and R. E. Slusher, “Nonlinear pulse propagation in Bragg gratings,” J. Opt. Soc. Am. B 14, 2980–2993 (1997).
[CrossRef]

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

B. J. Eggleton, C. M. de Sterke, R. E. Slusher, and J. E. Sipe, “Distributed feedback pulse generator based on nonlinear fiber grating,” Electron. Lett. 32, 2341–2342 (1996).
[CrossRef]

Steele, A. L.

A. L. Steele, S. Lynch, and J. E. Hoad, “Analysis of optical instabilities and bistability in a nonlinear optical fiber loop mirror with feedback,” Opt. Commun. 137, 136–142 (1997).
[CrossRef]

Strasser, T. A.

B. J. Eggleton, C. M. de Sterke, A. B. Aceves, J. E. Sipe, T. A. Strasser, and R. E. Slusher, “Modulational instability and tunable multiple soliton generation in apodized fiber gratings,” Opt. Commun. 149, 267–271 (1998).
[CrossRef]

Winful, H. G.

H. G. Winful, R. Zamir, and S. Feldman, “Modulational instability in nonlinear periodic structures: implications for ‘gap solitons’,” Appl. Phys. Lett. 58, 1001–1003 (1991).
[CrossRef]

H. G. Winful and G. D. Cooperman, “Self-pulsing and chaos in distributed feedback bistable optical devices,” Appl. Phys. Lett. 40, 298–300 (1982).
[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]

Zamir, R.

H. G. Winful, R. Zamir, and S. Feldman, “Modulational instability in nonlinear periodic structures: implications for ‘gap solitons’,” Appl. Phys. Lett. 58, 1001–1003 (1991).
[CrossRef]

Appl. Phys. Lett.

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]

M. Cada, J. He, B. Acklin, M. Proctor, D. Martin, F. Morier-Genoud, M.-A. Dupertuis, and J. M. Glinski, “All-optical reflectivity tuning and logic gating in a GaAs/AlAs periodic layered structure,” Appl. Phys. Lett. 60, 404–406 (1992).
[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]

H. G. Winful and G. D. Cooperman, “Self-pulsing and chaos in distributed feedback bistable optical devices,” Appl. Phys. Lett. 40, 298–300 (1982).
[CrossRef]

H. G. Winful, R. Zamir, and S. Feldman, “Modulational instability in nonlinear periodic structures: implications for ‘gap solitons’,” Appl. Phys. Lett. 58, 1001–1003 (1991).
[CrossRef]

Electron. Lett.

B. J. Eggleton, C. M. de Sterke, R. E. Slusher, and J. E. Sipe, “Distributed feedback pulse generator based on nonlinear fiber grating,” Electron. Lett. 32, 2341–2342 (1996).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Commun.

B. J. Eggleton, C. M. de Sterke, A. B. Aceves, J. E. Sipe, T. A. Strasser, and R. E. Slusher, “Modulational instability and tunable multiple soliton generation in apodized fiber gratings,” Opt. Commun. 149, 267–271 (1998).
[CrossRef]

W. J. Firth, “Stability of nonlinear Fabry–Perot resonators,” Opt. Commun. 39, 343–346 (1981).
[CrossRef]

A. L. Steele, S. Lynch, and J. E. Hoad, “Analysis of optical instabilities and bistability in a nonlinear optical fiber loop mirror with feedback,” Opt. Commun. 137, 136–142 (1997).
[CrossRef]

H. Li and K. Ogusu, “Analysis of optical instability in a double-coupler nonlinear fiber ring resonator,” Opt. Commun. 157, 27–32 (1998).
[CrossRef]

Opt. Lett.

Opt. Rev.

K. Ogusu and T. Kamizono, “Effect of material response time on optical bistability in nonlinear distributed feedback gratings,” Opt. Rev. 7, 83–88 (2000).
[CrossRef]

K. Ogusu, H. Li, and T. Kamizono, “Analysis of transient optical bistability and stability in a nonlinear fiber Fabry– Perot resonator based on an iterative method,” Opt. Rev. 5, 185–190 (1998).
[CrossRef]

Phys. Rev. A

C. M. de Sterke, “Stability analysis of nonlinear periodic media,” Phys. Rev. A 45, 8252–8258 (1992).
[CrossRef] [PubMed]

C. M. de Sterke and J. E. Sipe, “Switching dynamics of finite periodic nonlinear media: a numerical study,” Phys. Rev. A 42, 2858–2869 (1990).
[CrossRef] [PubMed]

C. M. de Sterke, “Simulations of gap-soliton generation,” Phys. Rev. A 45, 2012–2018 (1992).
[CrossRef] [PubMed]

Phys. Rev. Lett.

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

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

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Figures (9)

Fig. 1
Fig. 1

Schematic of a nonlinear Bragg grating and the generation of fluctuations within the cavity.

Fig. 2
Fig. 2

Typical example of the stationary input–output characteristics of a nonlinear Bragg grating.

Fig. 3
Fig. 3

Dependence of the complex frequency s/κv=(α+jσ)/κv [(a) real part and (b) imaginary part] on the incident intensity Iin for a weak Bragg grating with κL=2, ΔβL=0, and different values of material response time τ.

Fig. 4
Fig. 4

Dependence of the complex frequency s/κv=(α+jσ)/κv on the incident intensity Iin for a Bragg grating with κL=2, ΔβL=1, and different values of τ.

Fig. 5
Fig. 5

Dependence of the complex frequency s/κv=(α+jσ)/κv on the incident intensity Iin for a strong Bragg gratings with κL=5, ΔβL=4.5, and different values of τ.

Fig. 6
Fig. 6

Stability boundaries of the weak Bragg gratings with κL=2 for three values of material response time τ. S and U represent stable and unstable regions, respectively.

Fig. 7
Fig. 7

Stability boundaries of the strong Bragg gratings with κL=5 for three values of material response time τ.

Fig. 8
Fig. 8

Temporal response when an optical pulse with a peak intensity of 2 is incident into a Bragg grating with κL=2 and ΔβL=0. Material response time τ is changed: (a) τ=0, (b) τ=0.5tT, (c) τ=tT, and (d) τ=2tT.

Fig. 9
Fig. 9

Temporal response when an optical pulse with a peak intensity of 0.9 is incident into a Bragg grating with κL=5 and ΔβL=4.6. Material response time τ is changed: (a) τ=0, (b) τ=5tT, (c) τ=10tT, and (d) τ=20tT.

Equations (27)

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n=n0+δn0+(n1+δn1)cos(2β0z),
τ δnt+δn=n22||2,
(z, t)=EF(z, t)exp[j(βz-ωt)]+EB(z, t)exp[-j(βz+ωt)],
EFz+1v EFt=jκEB exp(-j2Δβz)+jγ(δn0EF+δn1EB),
EBz-1v EBt=-jκEF exp(j2Δβz)-jγ(δn0EB+δn1*EF),
τ δn0t+δn0=|EF|2+|EB|2,
τ δn1t+δn1=EFEB*,
EFz+1v EFt=jκEB exp(-j2Δβz)+jγ(|EF|2+2|EB|2)EF,
EBz-1v EBt=-jκEF exp(j2Δβz)-jγ(2|EF|2+|EB|2)EB.
F=[EFs+gF]exp[j(βz-ωt)],
B=[EBs+gB]exp[-j(βz+ωt)],
gFz+1v gFt=jκgB exp(-j2Δβz)+jγ[(|EFs|2+|EBs|2)gF+EFsδn0+EFsEBs*gB+EBsδn1],
gBz-1v gBt=-jκgF exp(j2Δβz)-jγ[(|EFs|2+|EBs|2)gB+EBsδn0+EFs*EBsgF+EFsδn1*],
τ δn0t+δn0=EFs*gF+EFsgF*+EBs*gB+EBsgB*,
τ δn1t+δn1=EBs*gF+EFsgB*.
gF=A+ exp(st)+B+* exp(s*t),
gB=A- exp(st)+B-* exp(s*t),
δn0=δn0A exp(st)+δn0B* exp(s*t),
δn1=δn1A exp(st)+δn1B* exp(s*t),
δn0A=11+τs(EFs*A++EFsB++EBs*A-+EBsB-),
δn0B*=11+τs*(EFs*B+*+EFsA+*+EBs*B-*+EBsA-*),
δn1A=11+τs(EBs*A++EFsB-),
δn1B*=11+τs*(EBs*B+*+EFsA-*).
A+z+svA+=jκ exp(-j2Δβz)A-+jγ2+sτ1+sτ(|EFs|2+|EBs|2)A++2+sτ1+sτEFsEBs*A-+11+sτEFs2B++21+sτEFsEBsB-,
B+z+svB+=-jκ exp(j2Δβz)B--jγ2+sτ1+sτ(|EFs|2+|EBs|2)B++2+sτ1+sτEFs*EBsB-+11+sτEFs*2A++21+sτEFs*EBs*A-,
A-z-svA-=-jκ exp(j2Δβz)A+-jγ2+sτ1+sτ(|EFs|2+|EBs|2)A-+2+sτ1+sτEFs*EBsA++11+sτEBs2B-+21+sτEFsEBsB+,
B-z-svB-=jκ exp(-j2Δβz)B++jγ2+sτ1+sτ(|EFs|2+|EBs|2)B-+2+sτ1+sτEFsEBs*B++11+sτEBs*2A-+21+sτEFs*EBs*A+.

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