Abstract

The response of a nonlinear Bragg grating under bidirectional illumination is studied. By analytical and numerical techniques it is found that the threshold for bistability, pulsation development, and spatial transverse instability is strongly influenced by the phase difference between fields incident from opposite directions. Breakup of pulsations that are symmetrical for both directions into nonsymmetrical pulsations is demonstrated. Solutions for focusing and defocusing nonlinearities are found to be related by means of changes in phase of the incident wave. A change of phase of π may lead to the emergence of a regular spatiotemporal light structure or may cause a transverse static pattern to become turbulent.

© 1999 Optical Society of America

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    [CrossRef] [PubMed]
  2. 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]
  3. 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, 328–330 (1998).
    [CrossRef]
  4. 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]
  5. D. Campi, C. Coriasso, A. Stano, L. Faustini, C. Cacciatore, C. Rigo, and G. Meneghini, “Nonlinear contradirectional coupler,” Appl. Phys. Lett. 72, 537–539 (1998).
    [CrossRef]
  6. N. D. Sankey, D. F. Prelewitz, and T. G. Brown, “All-optical switching in a nonlinear periodic structures,” Appl. Phys. Lett. 60, 1427–1429 (1992).
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  7. C. J. Herbert and M. S. Malcuit, “Optical bistability in nonlinear periodic structures,” Opt. Lett. 18, 1783–1785 (1993).
    [CrossRef] [PubMed]
  8. J. He, M. Cada, M.-A. Dupertuis, D. Martin, F. Morier-Genoud, C. Rolland, and A. J. Springthorpe, “All-optical bistable switching and signal regeneration in a semiconductor layered distributed-feedback/Fabry–Perot structure,” Appl. Phys. Lett. 63, 866–868 (1993).
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    [CrossRef]
  11. H. G. Winful, R. Zamir, and S. Feldman, “Modulation instability in nonlinear periodic structures: implications for ‘gap solitons,’” Appl. Phys. Lett. 58, 1001–1003 (1991).
    [CrossRef]
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    [CrossRef] [PubMed]
  13. W. Chen and D. L. Mills, “Gap solitons and the nonlinear optical response of superlattices,” Phys. Rev. Lett. 58, 160–163 (1987).
    [CrossRef] [PubMed]
  14. D. N. Christodoulides and R. I. Joseph, “Slow Bragg solitons in nonlinear periodic structures,” Phys. Rev. Lett. 62, 1746–1749 (1989).
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  19. C. Conti, S. Trillo, and G. Assanto, “Trapping of slowly moving or stationary two-color gap solitons,” Opt. Lett. 23, 334–336 (1998).
    [CrossRef]
  20. J. B. Geddes, R. A. Indik, J. V. Moloney, and W. J. Firth, “Hexagons and squares in a passive nonlinear optical system,” Phys. Rev. A 50, 3471–3485 (1994).
    [CrossRef] [PubMed]
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  23. M. Wegener and C. Klingshirn, “Self-oscillations of an induced absorber (CdS) in a hybrid ring resonator,” Phys. Rev. A 35, 1740–1752 (1987).
    [CrossRef] [PubMed]
  24. I. Galbraith and H. Haug, “Spatio-temporal chaos in a ring cavity,” J. Opt. Soc. Am. B 4, 1116–1123 (1987).
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  25. Yu. A. Logvin and A. M. Samson, “Map limit dynamics of the chain of optically bistable thin films,” Opt. Commun. 96, 107–112 (1992); Yu. A. Logvin, A. M. Samson, and V. M. Volkov, “Spatio-temporal light structures in a chain of bistable two-level medium thin films,” Solitons Chaos Fractals 4, 1451–1460 (1994).
    [CrossRef]
  26. T. Peschel, U. Peschel, and F. Lederer, “Bistability and symmetry breaking in distributed coupling of counterpropagating beams into nonlinear waveguides,” Phys. Rev. A 50, 5153–5163 (1994).
    [CrossRef] [PubMed]
  27. I. V. Babushkin, Yu. A. Logvin, and N. A. Loiko, “Symmetry breaking bifurcation in light dynamics of two bistable thin films,” Kvantovaya Elekron. (Moscow) 25, 110–115 (1998) [Quantum Electron. 28, 105–109 (1998)].
    [CrossRef]
  28. Yu. A. Logvin, “Nonreciprocal optical patterns due to symmetry breaking,” Phys. Rev. A 57, 1219–1222 (1998).
    [CrossRef]
  29. J. Zhon, M. Cada, J. He, and T. Makino, “Analysis and design of combined distributed feedback/Fabry Perot structures for surface emitting semiconductor lasers,” IEEE J. Quantum Electron. 32, 417–423 (1996).
    [CrossRef]
  30. A. A. Afanas’ev, V. M. Volkov, and T. S. Efendiev, “Spectrum of the transverse modes of a laser with static distributed feedback by a phase grating,” Kvantovaya Elektron. (Moscow) 24, 528–530 (1997) [Quantum Electron. 27, 514–516 (1997)].
    [CrossRef]
  31. Yu. A. Logvin, B. A. Samson, A. A. Afanas’ev, A. M. Samson, and N. A. Loiko, “Triadic Hopf-static structures in two-dimensional optical pattern formation,” Phys. Rev. E 54, R4548–R4551 (1996).
    [CrossRef]
  32. Yu. A. Logvin, T. Ackemann, and W. Lange, “Winking hexagons,” Europhys. Lett. 38, 583–588 (1997).
    [CrossRef]
  33. A. De Wit, D. Lima, G. Dewel, and P. Borckmans, “Spatio-temporal dynamics near a codimension-two point,” Phys. Rev. E 54, 261–271 (1996).
    [CrossRef]

1998 (6)

D. Campi, C. Coriasso, A. Stano, L. Faustini, C. Cacciatore, C. Rigo, and G. Meneghini, “Nonlinear contradirectional coupler,” Appl. Phys. Lett. 72, 537–539 (1998).
[CrossRef]

I. V. Babushkin, Yu. A. Logvin, and N. A. Loiko, “Symmetry breaking bifurcation in light dynamics of two bistable thin films,” Kvantovaya Elekron. (Moscow) 25, 110–115 (1998) [Quantum Electron. 28, 105–109 (1998)].
[CrossRef]

Yu. A. Logvin, “Nonreciprocal optical patterns due to symmetry breaking,” Phys. Rev. A 57, 1219–1222 (1998).
[CrossRef]

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, 328–330 (1998).
[CrossRef]

C. Conti, S. Trillo, and G. Assanto, “Trapping of slowly moving or stationary two-color gap solitons,” Opt. Lett. 23, 334–336 (1998).
[CrossRef]

1997 (3)

A. A. Afanas’ev, V. M. Volkov, and T. S. Efendiev, “Spectrum of the transverse modes of a laser with static distributed feedback by a phase grating,” Kvantovaya Elektron. (Moscow) 24, 528–530 (1997) [Quantum Electron. 27, 514–516 (1997)].
[CrossRef]

Yu. A. Logvin, T. Ackemann, and W. Lange, “Winking hexagons,” Europhys. Lett. 38, 583–588 (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 (4)

A. De Wit, D. Lima, G. Dewel, and P. Borckmans, “Spatio-temporal dynamics near a codimension-two point,” Phys. Rev. E 54, 261–271 (1996).
[CrossRef]

Yu. A. Logvin, B. A. Samson, A. A. Afanas’ev, A. M. Samson, and N. A. Loiko, “Triadic Hopf-static structures in two-dimensional optical pattern formation,” Phys. Rev. E 54, R4548–R4551 (1996).
[CrossRef]

J. Zhon, M. Cada, J. He, and T. Makino, “Analysis and design of combined distributed feedback/Fabry Perot structures for surface emitting semiconductor lasers,” IEEE J. Quantum Electron. 32, 417–423 (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]

1995 (1)

S. Radic, N. Georges, and G. P. Agrawal, “Analysis of nonuniform nonlinear distributed feedback structures: generalized transfer matrix method,” IEEE J. Quantum Electron. 31, 1326–1336 (1995).
[CrossRef]

1994 (2)

J. B. Geddes, R. A. Indik, J. V. Moloney, and W. J. Firth, “Hexagons and squares in a passive nonlinear optical system,” Phys. Rev. A 50, 3471–3485 (1994).
[CrossRef] [PubMed]

T. Peschel, U. Peschel, and F. Lederer, “Bistability and symmetry breaking in distributed coupling of counterpropagating beams into nonlinear waveguides,” Phys. Rev. A 50, 5153–5163 (1994).
[CrossRef] [PubMed]

1993 (3)

J. He, M. Cada, M.-A. Dupertuis, D. Martin, F. Morier-Genoud, C. Rolland, and A. J. Springthorpe, “All-optical bistable switching and signal regeneration in a semiconductor layered distributed-feedback/Fabry–Perot structure,” Appl. Phys. Lett. 63, 866–868 (1993).
[CrossRef]

J. Feng and F. K. Kneubühl, “Solitons in a periodic structure with Kerr nonlinearity,” IEEE J. Quantum Electron. 29, 590–597 (1993).
[CrossRef]

C. J. Herbert and M. S. Malcuit, “Optical bistability in nonlinear periodic structures,” Opt. Lett. 18, 1783–1785 (1993).
[CrossRef] [PubMed]

1992 (1)

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

1991 (1)

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

1990 (1)

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

1989 (2)

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]

1987 (3)

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

M. Wegener and C. Klingshirn, “Self-oscillations of an induced absorber (CdS) in a hybrid ring resonator,” Phys. Rev. A 35, 1740–1752 (1987).
[CrossRef] [PubMed]

I. Galbraith and H. Haug, “Spatio-temporal chaos in a ring cavity,” J. Opt. Soc. Am. B 4, 1116–1123 (1987).
[CrossRef]

1982 (1)

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]

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]

1972 (1)

H. Kogelnik and C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
[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]

Ackemann, T.

Yu. A. Logvin, T. Ackemann, and W. Lange, “Winking hexagons,” Europhys. Lett. 38, 583–588 (1997).
[CrossRef]

Afanas’ev, A. A.

A. A. Afanas’ev, V. M. Volkov, and T. S. Efendiev, “Spectrum of the transverse modes of a laser with static distributed feedback by a phase grating,” Kvantovaya Elektron. (Moscow) 24, 528–530 (1997) [Quantum Electron. 27, 514–516 (1997)].
[CrossRef]

Yu. A. Logvin, B. A. Samson, A. A. Afanas’ev, A. M. Samson, and N. A. Loiko, “Triadic Hopf-static structures in two-dimensional optical pattern formation,” Phys. Rev. E 54, R4548–R4551 (1996).
[CrossRef]

Agrawal, G. P.

S. Radic, N. Georges, and G. P. Agrawal, “Analysis of nonuniform nonlinear distributed feedback structures: generalized transfer matrix method,” IEEE J. Quantum Electron. 31, 1326–1336 (1995).
[CrossRef]

Assanto, G.

Babushkin, I. V.

I. V. Babushkin, Yu. A. Logvin, and N. A. Loiko, “Symmetry breaking bifurcation in light dynamics of two bistable thin films,” Kvantovaya Elekron. (Moscow) 25, 110–115 (1998) [Quantum Electron. 28, 105–109 (1998)].
[CrossRef]

Borckmans, P.

A. De Wit, D. Lima, G. Dewel, and P. Borckmans, “Spatio-temporal dynamics near a codimension-two point,” Phys. Rev. E 54, 261–271 (1996).
[CrossRef]

Broderick, N. G. R.

Brown, T. G.

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

Cacciatore, C.

D. Campi, C. Coriasso, A. Stano, L. Faustini, C. Cacciatore, C. Rigo, and G. Meneghini, “Nonlinear contradirectional coupler,” Appl. Phys. Lett. 72, 537–539 (1998).
[CrossRef]

Cada, M.

J. Zhon, M. Cada, J. He, and T. Makino, “Analysis and design of combined distributed feedback/Fabry Perot structures for surface emitting semiconductor lasers,” IEEE J. Quantum Electron. 32, 417–423 (1996).
[CrossRef]

J. He, M. Cada, M.-A. Dupertuis, D. Martin, F. Morier-Genoud, C. Rolland, and A. J. Springthorpe, “All-optical bistable switching and signal regeneration in a semiconductor layered distributed-feedback/Fabry–Perot structure,” Appl. Phys. Lett. 63, 866–868 (1993).
[CrossRef]

Campi, D.

D. Campi, C. Coriasso, A. Stano, L. Faustini, C. Cacciatore, C. Rigo, and G. Meneghini, “Nonlinear contradirectional coupler,” Appl. Phys. Lett. 72, 537–539 (1998).
[CrossRef]

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]

Conti, C.

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]

Coriasso, C.

D. Campi, C. Coriasso, A. Stano, L. Faustini, C. Cacciatore, C. Rigo, and G. Meneghini, “Nonlinear contradirectional coupler,” Appl. Phys. Lett. 72, 537–539 (1998).
[CrossRef]

de Sterke, C. M.

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]

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

De Wit, A.

A. De Wit, D. Lima, G. Dewel, and P. Borckmans, “Spatio-temporal dynamics near a codimension-two point,” Phys. Rev. E 54, 261–271 (1996).
[CrossRef]

Dewel, G.

A. De Wit, D. Lima, G. Dewel, and P. Borckmans, “Spatio-temporal dynamics near a codimension-two point,” Phys. Rev. E 54, 261–271 (1996).
[CrossRef]

Dupertuis, M.-A.

J. He, M. Cada, M.-A. Dupertuis, D. Martin, F. Morier-Genoud, C. Rolland, and A. J. Springthorpe, “All-optical bistable switching and signal regeneration in a semiconductor layered distributed-feedback/Fabry–Perot structure,” Appl. Phys. Lett. 63, 866–868 (1993).
[CrossRef]

Efendiev, T. S.

A. A. Afanas’ev, V. M. Volkov, and T. S. Efendiev, “Spectrum of the transverse modes of a laser with static distributed feedback by a phase grating,” Kvantovaya Elektron. (Moscow) 24, 528–530 (1997) [Quantum Electron. 27, 514–516 (1997)].
[CrossRef]

Eggleton, B. J.

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]

Faustini, L.

D. Campi, C. Coriasso, A. Stano, L. Faustini, C. Cacciatore, C. Rigo, and G. Meneghini, “Nonlinear contradirectional coupler,” Appl. Phys. Lett. 72, 537–539 (1998).
[CrossRef]

Feldman, S.

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

Feng, J.

J. Feng and F. K. Kneubühl, “Solitons in a periodic structure with Kerr nonlinearity,” IEEE J. Quantum Electron. 29, 590–597 (1993).
[CrossRef]

Firth, W. J.

J. B. Geddes, R. A. Indik, J. V. Moloney, and W. J. Firth, “Hexagons and squares in a passive nonlinear optical system,” Phys. Rev. A 50, 3471–3485 (1994).
[CrossRef] [PubMed]

Galbraith, I.

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]

Geddes, J. B.

J. B. Geddes, R. A. Indik, J. V. Moloney, and W. J. Firth, “Hexagons and squares in a passive nonlinear optical system,” Phys. Rev. A 50, 3471–3485 (1994).
[CrossRef] [PubMed]

Georges, N.

S. Radic, N. Georges, and G. P. Agrawal, “Analysis of nonuniform nonlinear distributed feedback structures: generalized transfer matrix method,” IEEE J. Quantum Electron. 31, 1326–1336 (1995).
[CrossRef]

Haug, H.

He, J.

J. Zhon, M. Cada, J. He, and T. Makino, “Analysis and design of combined distributed feedback/Fabry Perot structures for surface emitting semiconductor lasers,” IEEE J. Quantum Electron. 32, 417–423 (1996).
[CrossRef]

J. He, M. Cada, M.-A. Dupertuis, D. Martin, F. Morier-Genoud, C. Rolland, and A. J. Springthorpe, “All-optical bistable switching and signal regeneration in a semiconductor layered distributed-feedback/Fabry–Perot structure,” Appl. Phys. Lett. 63, 866–868 (1993).
[CrossRef]

Herbert, C. J.

Ibsen, M.

Indik, R. A.

J. B. Geddes, R. A. Indik, J. V. Moloney, and W. J. Firth, “Hexagons and squares in a passive nonlinear optical system,” Phys. Rev. A 50, 3471–3485 (1994).
[CrossRef] [PubMed]

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]

Klingshirn, C.

M. Wegener and C. Klingshirn, “Self-oscillations of an induced absorber (CdS) in a hybrid ring resonator,” Phys. Rev. A 35, 1740–1752 (1987).
[CrossRef] [PubMed]

Kneubühl, F. K.

J. Feng and F. K. Kneubühl, “Solitons in a periodic structure with Kerr nonlinearity,” IEEE J. Quantum Electron. 29, 590–597 (1993).
[CrossRef]

Kogelnik, H.

H. Kogelnik and C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
[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]

Laming, R. I.

Lange, W.

Yu. A. Logvin, T. Ackemann, and W. Lange, “Winking hexagons,” Europhys. Lett. 38, 583–588 (1997).
[CrossRef]

Lederer, F.

T. Peschel, U. Peschel, and F. Lederer, “Bistability and symmetry breaking in distributed coupling of counterpropagating beams into nonlinear waveguides,” Phys. Rev. A 50, 5153–5163 (1994).
[CrossRef] [PubMed]

Lima, D.

A. De Wit, D. Lima, G. Dewel, and P. Borckmans, “Spatio-temporal dynamics near a codimension-two point,” Phys. Rev. E 54, 261–271 (1996).
[CrossRef]

Logvin, Yu. A.

I. V. Babushkin, Yu. A. Logvin, and N. A. Loiko, “Symmetry breaking bifurcation in light dynamics of two bistable thin films,” Kvantovaya Elekron. (Moscow) 25, 110–115 (1998) [Quantum Electron. 28, 105–109 (1998)].
[CrossRef]

Yu. A. Logvin, “Nonreciprocal optical patterns due to symmetry breaking,” Phys. Rev. A 57, 1219–1222 (1998).
[CrossRef]

Yu. A. Logvin, T. Ackemann, and W. Lange, “Winking hexagons,” Europhys. Lett. 38, 583–588 (1997).
[CrossRef]

Yu. A. Logvin, B. A. Samson, A. A. Afanas’ev, A. M. Samson, and N. A. Loiko, “Triadic Hopf-static structures in two-dimensional optical pattern formation,” Phys. Rev. E 54, R4548–R4551 (1996).
[CrossRef]

Loiko, N. A.

I. V. Babushkin, Yu. A. Logvin, and N. A. Loiko, “Symmetry breaking bifurcation in light dynamics of two bistable thin films,” Kvantovaya Elekron. (Moscow) 25, 110–115 (1998) [Quantum Electron. 28, 105–109 (1998)].
[CrossRef]

Yu. A. Logvin, B. A. Samson, A. A. Afanas’ev, A. M. Samson, and N. A. Loiko, “Triadic Hopf-static structures in two-dimensional optical pattern formation,” Phys. Rev. E 54, R4548–R4551 (1996).
[CrossRef]

Makino, T.

J. Zhon, M. Cada, J. He, and T. Makino, “Analysis and design of combined distributed feedback/Fabry Perot structures for surface emitting semiconductor lasers,” IEEE J. Quantum Electron. 32, 417–423 (1996).
[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.

J. He, M. Cada, M.-A. Dupertuis, D. Martin, F. Morier-Genoud, C. Rolland, and A. J. Springthorpe, “All-optical bistable switching and signal regeneration in a semiconductor layered distributed-feedback/Fabry–Perot structure,” Appl. Phys. Lett. 63, 866–868 (1993).
[CrossRef]

Meneghini, G.

D. Campi, C. Coriasso, A. Stano, L. Faustini, C. Cacciatore, C. Rigo, and G. Meneghini, “Nonlinear contradirectional coupler,” Appl. Phys. Lett. 72, 537–539 (1998).
[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]

Moloney, J. V.

J. B. Geddes, R. A. Indik, J. V. Moloney, and W. J. Firth, “Hexagons and squares in a passive nonlinear optical system,” Phys. Rev. A 50, 3471–3485 (1994).
[CrossRef] [PubMed]

Morier-Genoud, F.

J. He, M. Cada, M.-A. Dupertuis, D. Martin, F. Morier-Genoud, C. Rolland, and A. J. Springthorpe, “All-optical bistable switching and signal regeneration in a semiconductor layered distributed-feedback/Fabry–Perot structure,” Appl. Phys. Lett. 63, 866–868 (1993).
[CrossRef]

Peschel, T.

T. Peschel, U. Peschel, and F. Lederer, “Bistability and symmetry breaking in distributed coupling of counterpropagating beams into nonlinear waveguides,” Phys. Rev. A 50, 5153–5163 (1994).
[CrossRef] [PubMed]

Peschel, U.

T. Peschel, U. Peschel, and F. Lederer, “Bistability and symmetry breaking in distributed coupling of counterpropagating beams into nonlinear waveguides,” Phys. Rev. A 50, 5153–5163 (1994).
[CrossRef] [PubMed]

Prelewitz, D. F.

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

Radic, S.

S. Radic, N. Georges, and G. P. Agrawal, “Analysis of nonuniform nonlinear distributed feedback structures: generalized transfer matrix method,” IEEE J. Quantum Electron. 31, 1326–1336 (1995).
[CrossRef]

Richardson, D. J.

Rigo, C.

D. Campi, C. Coriasso, A. Stano, L. Faustini, C. Cacciatore, C. Rigo, and G. Meneghini, “Nonlinear contradirectional coupler,” Appl. Phys. Lett. 72, 537–539 (1998).
[CrossRef]

Rolland, C.

J. He, M. Cada, M.-A. Dupertuis, D. Martin, F. Morier-Genoud, C. Rolland, and A. J. Springthorpe, “All-optical bistable switching and signal regeneration in a semiconductor layered distributed-feedback/Fabry–Perot structure,” Appl. Phys. Lett. 63, 866–868 (1993).
[CrossRef]

Samson, A. M.

Yu. A. Logvin, B. A. Samson, A. A. Afanas’ev, A. M. Samson, and N. A. Loiko, “Triadic Hopf-static structures in two-dimensional optical pattern formation,” Phys. Rev. E 54, R4548–R4551 (1996).
[CrossRef]

Samson, B. A.

Yu. A. Logvin, B. A. Samson, A. A. Afanas’ev, A. M. Samson, and N. A. Loiko, “Triadic Hopf-static structures in two-dimensional optical pattern formation,” Phys. Rev. E 54, R4548–R4551 (1996).
[CrossRef]

Sankey, N. D.

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

Shank, C. V.

H. Kogelnik and C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
[CrossRef]

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, “Switching dynamics of finite periodic nonlinear media: a numeric study,” Phys. Rev. A 42, 2858–2869 (1990).
[CrossRef] [PubMed]

Slusher, R. E.

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]

Springthorpe, A. J.

J. He, M. Cada, M.-A. Dupertuis, D. Martin, F. Morier-Genoud, C. Rolland, and A. J. Springthorpe, “All-optical bistable switching and signal regeneration in a semiconductor layered distributed-feedback/Fabry–Perot structure,” Appl. Phys. Lett. 63, 866–868 (1993).
[CrossRef]

Stano, A.

D. Campi, C. Coriasso, A. Stano, L. Faustini, C. Cacciatore, C. Rigo, and G. Meneghini, “Nonlinear contradirectional coupler,” Appl. Phys. Lett. 72, 537–539 (1998).
[CrossRef]

Taverner, D.

Trillo, S.

Volkov, V. M.

A. A. Afanas’ev, V. M. Volkov, and T. S. Efendiev, “Spectrum of the transverse modes of a laser with static distributed feedback by a phase grating,” Kvantovaya Elektron. (Moscow) 24, 528–530 (1997) [Quantum Electron. 27, 514–516 (1997)].
[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]

Wegener, M.

M. Wegener and C. Klingshirn, “Self-oscillations of an induced absorber (CdS) in a hybrid ring resonator,” Phys. Rev. A 35, 1740–1752 (1987).
[CrossRef] [PubMed]

Winful, H. G.

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

Zhon, J.

J. Zhon, M. Cada, J. He, and T. Makino, “Analysis and design of combined distributed feedback/Fabry Perot structures for surface emitting semiconductor lasers,” IEEE J. Quantum Electron. 32, 417–423 (1996).
[CrossRef]

Appl. Phys. Lett. (6)

J. He, M. Cada, M.-A. Dupertuis, D. Martin, F. Morier-Genoud, C. Rolland, and A. J. Springthorpe, “All-optical bistable switching and signal regeneration in a semiconductor layered distributed-feedback/Fabry–Perot structure,” Appl. Phys. Lett. 63, 866–868 (1993).
[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]

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, “Modulation instability in nonlinear periodic structures: implications for ‘gap solitons,’” Appl. Phys. Lett. 58, 1001–1003 (1991).
[CrossRef]

D. Campi, C. Coriasso, A. Stano, L. Faustini, C. Cacciatore, C. Rigo, and G. Meneghini, “Nonlinear contradirectional coupler,” Appl. Phys. Lett. 72, 537–539 (1998).
[CrossRef]

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

Europhys. Lett. (1)

Yu. A. Logvin, T. Ackemann, and W. Lange, “Winking hexagons,” Europhys. Lett. 38, 583–588 (1997).
[CrossRef]

IEEE J. Quantum Electron. (3)

J. Zhon, M. Cada, J. He, and T. Makino, “Analysis and design of combined distributed feedback/Fabry Perot structures for surface emitting semiconductor lasers,” IEEE J. Quantum Electron. 32, 417–423 (1996).
[CrossRef]

J. Feng and F. K. Kneubühl, “Solitons in a periodic structure with Kerr nonlinearity,” IEEE J. Quantum Electron. 29, 590–597 (1993).
[CrossRef]

S. Radic, N. Georges, and G. P. Agrawal, “Analysis of nonuniform nonlinear distributed feedback structures: generalized transfer matrix method,” IEEE J. Quantum Electron. 31, 1326–1336 (1995).
[CrossRef]

J. Appl. Phys. (1)

H. Kogelnik and C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
[CrossRef]

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

Opt. Lett. (4)

Phys. Lett. A (1)

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

Phys. Rev. A (5)

Yu. A. Logvin, “Nonreciprocal optical patterns due to symmetry breaking,” Phys. Rev. A 57, 1219–1222 (1998).
[CrossRef]

J. B. Geddes, R. A. Indik, J. V. Moloney, and W. J. Firth, “Hexagons and squares in a passive nonlinear optical system,” Phys. Rev. A 50, 3471–3485 (1994).
[CrossRef] [PubMed]

M. Wegener and C. Klingshirn, “Self-oscillations of an induced absorber (CdS) in a hybrid ring resonator,” Phys. Rev. A 35, 1740–1752 (1987).
[CrossRef] [PubMed]

T. Peschel, U. Peschel, and F. Lederer, “Bistability and symmetry breaking in distributed coupling of counterpropagating beams into nonlinear waveguides,” Phys. Rev. A 50, 5153–5163 (1994).
[CrossRef] [PubMed]

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

Phys. Rev. E (2)

A. De Wit, D. Lima, G. Dewel, and P. Borckmans, “Spatio-temporal dynamics near a codimension-two point,” Phys. Rev. E 54, 261–271 (1996).
[CrossRef]

Yu. A. Logvin, B. A. Samson, A. A. Afanas’ev, A. M. Samson, and N. A. Loiko, “Triadic Hopf-static structures in two-dimensional optical pattern formation,” Phys. Rev. E 54, R4548–R4551 (1996).
[CrossRef]

Phys. Rev. Lett. (3)

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]

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]

Quantum Electron. (2)

I. V. Babushkin, Yu. A. Logvin, and N. A. Loiko, “Symmetry breaking bifurcation in light dynamics of two bistable thin films,” Kvantovaya Elekron. (Moscow) 25, 110–115 (1998) [Quantum Electron. 28, 105–109 (1998)].
[CrossRef]

A. A. Afanas’ev, V. M. Volkov, and T. S. Efendiev, “Spectrum of the transverse modes of a laser with static distributed feedback by a phase grating,” Kvantovaya Elektron. (Moscow) 24, 528–530 (1997) [Quantum Electron. 27, 514–516 (1997)].
[CrossRef]

Other (3)

Yu. A. Logvin and A. M. Samson, “Map limit dynamics of the chain of optically bistable thin films,” Opt. Commun. 96, 107–112 (1992); Yu. A. Logvin, A. M. Samson, and V. M. Volkov, “Spatio-temporal light structures in a chain of bistable two-level medium thin films,” Solitons Chaos Fractals 4, 1451–1460 (1994).
[CrossRef]

I. S. Gradstein and I. M. Ryzhik, Tables of Integrals, Sums, Series and Products (Nauka, Moscow, 1971).

C. M. de Sterke and J. E. Sipe, in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1994), Vol. XXXIII, p. 203.

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Figures (10)

Fig. 1
Fig. 1

Nonlinear Bragg grating of length L illuminated from opposite sides by light waves EF and EB.

Fig. 2
Fig. 2

Light intensity transmitted to the right IT versus intensity incident from the left I0 at κL=2, γL=2/3, δ=0 and parameters of the wave incident from the right: a, IB=0.01, θ=π; b, IB=0.0; c, IB=0.01, θ=0.

Fig. 3
Fig. 3

Intensity pulsations for waves transmitted through grating IT (solid curves) and reflected from grating IR (dashed curves) from the cases a–c depicted by the corresponding arrows in Fig. 2.

Fig. 4
Fig. 4

(a) Light intensity in the middle of the grating and (b) constant C2 versus the input intensity at equal amplitudes of incident waves at κL=2, γL=2/3, δ=0. Solid (dashed) curves correspond to in-phase (out-of-phase) incident waves.

Fig. 5
Fig. 5

(a) Two steady distributions of light intensity inside the grating at I0=1.5; other parameters as in Fig. 2. Solid (dashed) curves correspond to in-phase (out-of-phase) incident waves. (b), (c) Other intensity distributions for θ.

Fig. 6
Fig. 6

Spatiotemporal patterns illustrating pulsations for (a) forward and (b) backward waves at I0=3 and θ=0. Time is scaled in units of td=Ln0/c. Dark (light) color corresponds to low (high) level of intensity.

Fig. 7
Fig. 7

Illustration of the symmetry-breaking instability at θ=π. (a) Symmetrical output pulsations (F2-B2) at I0=6.4. (b) Nonsymmetrical pulsations at I0=6.8.

Fig. 8
Fig. 8

Static pattern showing the forward wave intensity distribution inside the grating for the parameters of Fig. 4 and IF=IB=1.5, θ=0, d=6L/k.

Fig. 9
Fig. 9

(a) Space–time plot characterizing turbulent distribution of the transmitted light intensity for the parameters of Fig. 8; the only difference is that here θ=π.

Fig. 10
Fig. 10

(a) Space–time plot characterizing transverse distribution of the forward wave intensity at z=L and IF=IB=1.5, θ=π, d=6L/k, γL=-2/3 (defocusing nonlinearity). Other parameters as in Fig. 4. (b) Snapshot of the pattern of light intensity inside the grating for the instant of time corresponding to the upper edge of the picture in (a).

Equations (10)

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EFz+n0cEFt=iκEB exp(-i2δz)+iγ(|EF|2+2|EB|2)EF,
EBz-n0cEBt=-iκEF exp(i2δz)-iγ(|EB|2+2|EF|2)EB,
EF(z=0, t)=F0,EB(z=L, t)=B0 exp(iθ)
C1=F2-B2,
C2=2κLFB cos ψ+2δLF2+3γLF2B2,
L2dIFdz2=(κL)2IF(IF-C1)-[C2/2-δLIF-IF(IF-C1)]2.
I(z/L)=I2(I1-I3)+I3(I1-I2)sn2[(z-0.5)σ/L,m]I1-I3+(I1-I2)sn2[(z-0.5)σ/L,m],
I(z/L)=I1(I2-I4)+I4(I1-I2)sn2[(0.5-z)σ/L,m]I2-I4+(I1-I2)sn2[(0.5-z)σ/L,m],
EFz+n0cEFt=i12kΔEF+iκEB exp(-i2δz)+iγ(|EF|2+2|EB|2)EF,
EBz-n0cEBt=-i12kΔEB-iκEF exp(i2δz)-iγ(|EB|2+2|EF|2)EB,

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