Abstract

The theory of phase-shifted nonlinear periodic structures operating in the stationary regime is presented. The transmissive properties of the structure are analyzed by solution of the corresponding set of nonlinear coupled-mode equations exactly. Extremely low switching intensities are found for the special case of λ/4-shifted structures. An all-optical low-intensity switching configuration that uses a wavelength-tunable source and a λ/4-shifted nonlinear structure is proposed. Advantages of the phase-shifted distributed-feedback design in all-optical switching applications are discussed.

© 1995 Optical Society of America

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  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]
  2. W. Chen and D. L. Mills, “Optical response of nonlinear multilayer structures: bilayers and superlattices,” Phys. Rev. B 36, 6269–6278 (1987).
    [Crossref]
  3. C. Martijn de Sterke and J. E. Sipe, “Extension and generalizations of an envelope-function approach for the electrodynamics of nonlinear periodic structures,” Phys. Rev. A 39, 5163–5178 (1989).
    [Crossref]
  4. 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]
  5. C. Martijn de Sterke and J. E. Sipe, “Switching dynamics of finite periodic nonlinear media: a numerical study,” Phys. Rev. A 42, 2858–2869 (1990).
    [Crossref]
  6. H. G. Winful, “Pulse compression in optical fiber filters,” Appl. Phys. Lett. 46, 527–529 (1984).
    [Crossref]
  7. H. G. Winful and G. I. Stegeman, “Applications of nonlinear periodic structures in guided wave optics,” in First International Conference on Integrated Optical Circuit Engineering, S. Sriram and D. B. Ostrowsky, eds., Proc. Soc. Photo-Opt. Instrum. Eng.517, 214–218 (1984).
    [Crossref]
  8. M. Okuda and K. Onaka, “Bistability of optical resonator with distributed Bragg-reflectors by using Kerr effect,” Jpn. J. Appl. Phys. 16, 769–773 (1977).
    [Crossref]
  9. J. He and M. Cada, “Combined distributed feedback and Fabry–Perot structures with a phase matching layer for optical bistable devices,” Appl. Phys. Lett. 61, 2150–2152 (1992).
    [Crossref]
  10. 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]
  11. D. N. Christodoulides and R. I. Joseph, “Slow Bragg solitons in nonlinear periodic structures,” Phys. Rev. Lett. 62, 1746–1749 (1989).
    [Crossref] [PubMed]
  12. C. Martijn de Sterke and J. E. Sipe, “Launching of gap solitons in nonuniform gratings,” Opt. Lett. 18, 269–271 (1993).
    [Crossref]
  13. C. Martijn de Sterke, “Simulations of gap-soliton generation,” Phys. Rev. A 45, 2012–2018 (1992).
    [Crossref]
  14. H. A. Haus, “Matching of distributed-feedback structures,” Opt. Lett. 15, 1134–1136 (1992).
    [Crossref]
  15. H. A. Haus and C. V. Shank, “Antisymmetric taper of distributed feedback lasers,” IEEE J. Quantum Electron. QE-12, 532–539 (1976).
    [Crossref]
  16. G. P. Agrawal and A. H. Bobeck, “Modeling of distributed feedback semiconductor lasers with axially-varying parameters,” IEEE J. Quantum Electron. QE-24, 2407–2414 (1988).
    [Crossref]
  17. K. Utaka, S. Akiba, K. Sakai, and Y. Matsushima, “λ/4-shifted InGaAsP DFB lasers,” IEEE J. Quantum Electron. QE-22, 1042–1051 (1987).
  18. M. Yamada and K. Sakuda, “Analysis of almost-periodic distributed feedback slab waveguides via a fundamental matrix approach,” Appl. Opt. 26, 3474–3478 (1987).
    [Crossref] [PubMed]
  19. G. P. Agrawal and S. Radic, “Phase-shifted fiber bragg gratings and their application for wavelength demultiplexing,” IEEE Photon. Technol. Lett. 6, 995–997 (1994).
    [Crossref]
  20. H. W. H. Lee, R. S. Hughes, J. E. Davis, C. F. McConaghy, A. V. Hamza, and M. Balooch, “Feasibility of fullerene thin films for high-speed all-optical switching,” in Conference on Lasers and Electro-Optics, Vol. 8 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), p. 59.
  21. W. E. Torruellas, D. Neher, R. Zanoni, G. I. Stegeman, I. Kajzar, and F. Leclerc, “Dispersion measurements of the third-order nonlinear susceptibility of polythiophene thin films,” Chem. Phys. Lett. 175, 11–16 (1990).
    [Crossref]
  22. B. S. Kawasaki, K. O. Hill, D. C. Johnson, and Y. Fujii, “Narrow-band Bragg reflectors in optical fibers,” Opt. Lett. 3, 66–68 (1978).
    [Crossref] [PubMed]
  23. K. O. Hill, B. Malo, F. Bilodeau, and D. C. Johnson, “Photosensitivity in optical fibers,” Annu. Rev. Mater. Sci. 23, 125–157 (1993).
    [Crossref]
  24. J. Stone, “Photorefractivity in GeO2-doped silica fibers,” J. Appl. Phys. 62, 4371–4374 (1987).
    [Crossref]
  25. G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, San Diego, Calif., 1995).
  26. H. Kogelnik and C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
    [Crossref]
  27. A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron. QE-9, 919–933 (1973).
    [Crossref]
  28. H. G. Winful, “Optical bistability in periodic structures and in four-wave mixing processes,” Ph.D. dissertation (University of Southern California, Los Angeles, Calif., 1980).
  29. P. F. Byrd and M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Scientists (Springer-Verlag, Berlin, 1971).
    [Crossref]
  30. H. G. Winful, “Effect of end reflection on distributed-feedback bistable optical devices,” IEEE J. Quantum Electron. QE-17, 164 (1981).
  31. S. Radic and N. George, “Ultrafast pulse propagation in periodic optical media: a generalized finite-difference time-domain approach,” Opt. Lett. 19, 998–1000 (1994).
    [Crossref]
  32. J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
    [Crossref]

1994 (2)

G. P. Agrawal and S. Radic, “Phase-shifted fiber bragg gratings and their application for wavelength demultiplexing,” IEEE Photon. Technol. Lett. 6, 995–997 (1994).
[Crossref]

S. Radic and N. George, “Ultrafast pulse propagation in periodic optical media: a generalized finite-difference time-domain approach,” Opt. Lett. 19, 998–1000 (1994).
[Crossref]

1993 (2)

K. O. Hill, B. Malo, F. Bilodeau, and D. C. Johnson, “Photosensitivity in optical fibers,” Annu. Rev. Mater. Sci. 23, 125–157 (1993).
[Crossref]

C. Martijn de Sterke and J. E. Sipe, “Launching of gap solitons in nonuniform gratings,” Opt. Lett. 18, 269–271 (1993).
[Crossref]

1992 (4)

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

H. A. Haus, “Matching of distributed-feedback structures,” Opt. Lett. 15, 1134–1136 (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]

J. He and M. Cada, “Combined distributed feedback and Fabry–Perot structures with a phase matching layer for optical bistable devices,” Appl. Phys. Lett. 61, 2150–2152 (1992).
[Crossref]

1990 (2)

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

W. E. Torruellas, D. Neher, R. Zanoni, G. I. Stegeman, I. Kajzar, and F. Leclerc, “Dispersion measurements of the third-order nonlinear susceptibility of polythiophene thin films,” Chem. Phys. Lett. 175, 11–16 (1990).
[Crossref]

1989 (2)

C. Martijn de Sterke and J. E. Sipe, “Extension and generalizations of an envelope-function approach for the electrodynamics of nonlinear periodic structures,” Phys. Rev. A 39, 5163–5178 (1989).
[Crossref]

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

1988 (1)

G. P. Agrawal and A. H. Bobeck, “Modeling of distributed feedback semiconductor lasers with axially-varying parameters,” IEEE J. Quantum Electron. QE-24, 2407–2414 (1988).
[Crossref]

1987 (4)

K. Utaka, S. Akiba, K. Sakai, and Y. Matsushima, “λ/4-shifted InGaAsP DFB lasers,” IEEE J. Quantum Electron. QE-22, 1042–1051 (1987).

M. Yamada and K. Sakuda, “Analysis of almost-periodic distributed feedback slab waveguides via a fundamental matrix approach,” Appl. Opt. 26, 3474–3478 (1987).
[Crossref] [PubMed]

W. Chen and D. L. Mills, “Optical response of nonlinear multilayer structures: bilayers and superlattices,” Phys. Rev. B 36, 6269–6278 (1987).
[Crossref]

J. Stone, “Photorefractivity in GeO2-doped silica fibers,” J. Appl. Phys. 62, 4371–4374 (1987).
[Crossref]

1984 (1)

H. G. Winful, “Pulse compression in optical fiber filters,” Appl. Phys. Lett. 46, 527–529 (1984).
[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]

1981 (1)

H. G. Winful, “Effect of end reflection on distributed-feedback bistable optical devices,” IEEE J. Quantum Electron. QE-17, 164 (1981).

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]

1978 (1)

1977 (1)

M. Okuda and K. Onaka, “Bistability of optical resonator with distributed Bragg-reflectors by using Kerr effect,” Jpn. J. Appl. Phys. 16, 769–773 (1977).
[Crossref]

1976 (1)

H. A. Haus and C. V. Shank, “Antisymmetric taper of distributed feedback lasers,” IEEE J. Quantum Electron. QE-12, 532–539 (1976).
[Crossref]

1973 (1)

A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron. QE-9, 919–933 (1973).
[Crossref]

1972 (1)

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

1962 (1)

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[Crossref]

Agrawal, G. P.

G. P. Agrawal and S. Radic, “Phase-shifted fiber bragg gratings and their application for wavelength demultiplexing,” IEEE Photon. Technol. Lett. 6, 995–997 (1994).
[Crossref]

G. P. Agrawal and A. H. Bobeck, “Modeling of distributed feedback semiconductor lasers with axially-varying parameters,” IEEE J. Quantum Electron. QE-24, 2407–2414 (1988).
[Crossref]

G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, San Diego, Calif., 1995).

Akiba, S.

K. Utaka, S. Akiba, K. Sakai, and Y. Matsushima, “λ/4-shifted InGaAsP DFB lasers,” IEEE J. Quantum Electron. QE-22, 1042–1051 (1987).

Armstrong, J. A.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[Crossref]

Balooch, M.

H. W. H. Lee, R. S. Hughes, J. E. Davis, C. F. McConaghy, A. V. Hamza, and M. Balooch, “Feasibility of fullerene thin films for high-speed all-optical switching,” in Conference on Lasers and Electro-Optics, Vol. 8 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), p. 59.

Bilodeau, F.

K. O. Hill, B. Malo, F. Bilodeau, and D. C. Johnson, “Photosensitivity in optical fibers,” Annu. Rev. Mater. Sci. 23, 125–157 (1993).
[Crossref]

Bloembergen, N.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[Crossref]

Bobeck, A. H.

G. P. Agrawal and A. H. Bobeck, “Modeling of distributed feedback semiconductor lasers with axially-varying parameters,” IEEE J. Quantum Electron. QE-24, 2407–2414 (1988).
[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]

Byrd, P. F.

P. F. Byrd and M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Scientists (Springer-Verlag, Berlin, 1971).
[Crossref]

Cada, M.

J. He and M. Cada, “Combined distributed feedback and Fabry–Perot structures with a phase matching layer for optical bistable devices,” Appl. Phys. Lett. 61, 2150–2152 (1992).
[Crossref]

Chen, W.

W. Chen and D. L. Mills, “Optical response of nonlinear multilayer structures: bilayers and superlattices,” Phys. Rev. B 36, 6269–6278 (1987).
[Crossref]

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]

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]

Davis, J. E.

H. W. H. Lee, R. S. Hughes, J. E. Davis, C. F. McConaghy, A. V. Hamza, and M. Balooch, “Feasibility of fullerene thin films for high-speed all-optical switching,” in Conference on Lasers and Electro-Optics, Vol. 8 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), p. 59.

Ducuing, J.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[Crossref]

Friedman, M. D.

P. F. Byrd and M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Scientists (Springer-Verlag, Berlin, 1971).
[Crossref]

Fujii, Y.

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]

George, N.

Hamza, A. V.

H. W. H. Lee, R. S. Hughes, J. E. Davis, C. F. McConaghy, A. V. Hamza, and M. Balooch, “Feasibility of fullerene thin films for high-speed all-optical switching,” in Conference on Lasers and Electro-Optics, Vol. 8 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), p. 59.

Haus, H. A.

H. A. Haus, “Matching of distributed-feedback structures,” Opt. Lett. 15, 1134–1136 (1992).
[Crossref]

H. A. Haus and C. V. Shank, “Antisymmetric taper of distributed feedback lasers,” IEEE J. Quantum Electron. QE-12, 532–539 (1976).
[Crossref]

He, J.

J. He and M. Cada, “Combined distributed feedback and Fabry–Perot structures with a phase matching layer for optical bistable devices,” Appl. Phys. Lett. 61, 2150–2152 (1992).
[Crossref]

Hill, K. O.

K. O. Hill, B. Malo, F. Bilodeau, and D. C. Johnson, “Photosensitivity in optical fibers,” Annu. Rev. Mater. Sci. 23, 125–157 (1993).
[Crossref]

B. S. Kawasaki, K. O. Hill, D. C. Johnson, and Y. Fujii, “Narrow-band Bragg reflectors in optical fibers,” Opt. Lett. 3, 66–68 (1978).
[Crossref] [PubMed]

Hughes, R. S.

H. W. H. Lee, R. S. Hughes, J. E. Davis, C. F. McConaghy, A. V. Hamza, and M. Balooch, “Feasibility of fullerene thin films for high-speed all-optical switching,” in Conference on Lasers and Electro-Optics, Vol. 8 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), p. 59.

Johnson, D. C.

K. O. Hill, B. Malo, F. Bilodeau, and D. C. Johnson, “Photosensitivity in optical fibers,” Annu. Rev. Mater. Sci. 23, 125–157 (1993).
[Crossref]

B. S. Kawasaki, K. O. Hill, D. C. Johnson, and Y. Fujii, “Narrow-band Bragg reflectors in optical fibers,” Opt. Lett. 3, 66–68 (1978).
[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]

Kajzar, I.

W. E. Torruellas, D. Neher, R. Zanoni, G. I. Stegeman, I. Kajzar, and F. Leclerc, “Dispersion measurements of the third-order nonlinear susceptibility of polythiophene thin films,” Chem. Phys. Lett. 175, 11–16 (1990).
[Crossref]

Kawasaki, B. S.

Kogelnik, H.

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

Leclerc, F.

W. E. Torruellas, D. Neher, R. Zanoni, G. I. Stegeman, I. Kajzar, and F. Leclerc, “Dispersion measurements of the third-order nonlinear susceptibility of polythiophene thin films,” Chem. Phys. Lett. 175, 11–16 (1990).
[Crossref]

Lee, H. W. H.

H. W. H. Lee, R. S. Hughes, J. E. Davis, C. F. McConaghy, A. V. Hamza, and M. Balooch, “Feasibility of fullerene thin films for high-speed all-optical switching,” in Conference on Lasers and Electro-Optics, Vol. 8 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), p. 59.

Malo, B.

K. O. Hill, B. Malo, F. Bilodeau, and D. C. Johnson, “Photosensitivity in optical fibers,” Annu. Rev. Mater. Sci. 23, 125–157 (1993).
[Crossref]

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]

Martijn de Sterke, C.

C. Martijn de Sterke and J. E. Sipe, “Launching of gap solitons in nonuniform gratings,” Opt. Lett. 18, 269–271 (1993).
[Crossref]

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

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

C. Martijn de Sterke and J. E. Sipe, “Extension and generalizations of an envelope-function approach for the electrodynamics of nonlinear periodic structures,” Phys. Rev. A 39, 5163–5178 (1989).
[Crossref]

Matsushima, Y.

K. Utaka, S. Akiba, K. Sakai, and Y. Matsushima, “λ/4-shifted InGaAsP DFB lasers,” IEEE J. Quantum Electron. QE-22, 1042–1051 (1987).

McConaghy, C. F.

H. W. H. Lee, R. S. Hughes, J. E. Davis, C. F. McConaghy, A. V. Hamza, and M. Balooch, “Feasibility of fullerene thin films for high-speed all-optical switching,” in Conference on Lasers and Electro-Optics, Vol. 8 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), p. 59.

Mills, D. L.

W. Chen and D. L. Mills, “Optical response of nonlinear multilayer structures: bilayers and superlattices,” Phys. Rev. B 36, 6269–6278 (1987).
[Crossref]

Neher, D.

W. E. Torruellas, D. Neher, R. Zanoni, G. I. Stegeman, I. Kajzar, and F. Leclerc, “Dispersion measurements of the third-order nonlinear susceptibility of polythiophene thin films,” Chem. Phys. Lett. 175, 11–16 (1990).
[Crossref]

Okuda, M.

M. Okuda and K. Onaka, “Bistability of optical resonator with distributed Bragg-reflectors by using Kerr effect,” Jpn. J. Appl. Phys. 16, 769–773 (1977).
[Crossref]

Onaka, K.

M. Okuda and K. Onaka, “Bistability of optical resonator with distributed Bragg-reflectors by using Kerr effect,” Jpn. J. Appl. Phys. 16, 769–773 (1977).
[Crossref]

Pershan, P. S.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[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]

Radic, S.

G. P. Agrawal and S. Radic, “Phase-shifted fiber bragg gratings and their application for wavelength demultiplexing,” IEEE Photon. Technol. Lett. 6, 995–997 (1994).
[Crossref]

S. Radic and N. George, “Ultrafast pulse propagation in periodic optical media: a generalized finite-difference time-domain approach,” Opt. Lett. 19, 998–1000 (1994).
[Crossref]

Sakai, K.

K. Utaka, S. Akiba, K. Sakai, and Y. Matsushima, “λ/4-shifted InGaAsP DFB lasers,” IEEE J. Quantum Electron. QE-22, 1042–1051 (1987).

Sakuda, K.

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]

Shank, C. V.

H. A. Haus and C. V. Shank, “Antisymmetric taper of distributed feedback lasers,” IEEE J. Quantum Electron. QE-12, 532–539 (1976).
[Crossref]

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

Sipe, J. E.

C. Martijn de Sterke and J. E. Sipe, “Launching of gap solitons in nonuniform gratings,” Opt. Lett. 18, 269–271 (1993).
[Crossref]

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

C. Martijn de Sterke and J. E. Sipe, “Extension and generalizations of an envelope-function approach for the electrodynamics of nonlinear periodic structures,” Phys. Rev. A 39, 5163–5178 (1989).
[Crossref]

Stegeman, G. I.

W. E. Torruellas, D. Neher, R. Zanoni, G. I. Stegeman, I. Kajzar, and F. Leclerc, “Dispersion measurements of the third-order nonlinear susceptibility of polythiophene thin films,” Chem. Phys. Lett. 175, 11–16 (1990).
[Crossref]

H. G. Winful and G. I. Stegeman, “Applications of nonlinear periodic structures in guided wave optics,” in First International Conference on Integrated Optical Circuit Engineering, S. Sriram and D. B. Ostrowsky, eds., Proc. Soc. Photo-Opt. Instrum. Eng.517, 214–218 (1984).
[Crossref]

Stone, J.

J. Stone, “Photorefractivity in GeO2-doped silica fibers,” J. Appl. Phys. 62, 4371–4374 (1987).
[Crossref]

Torruellas, W. E.

W. E. Torruellas, D. Neher, R. Zanoni, G. I. Stegeman, I. Kajzar, and F. Leclerc, “Dispersion measurements of the third-order nonlinear susceptibility of polythiophene thin films,” Chem. Phys. Lett. 175, 11–16 (1990).
[Crossref]

Utaka, K.

K. Utaka, S. Akiba, K. Sakai, and Y. Matsushima, “λ/4-shifted InGaAsP DFB lasers,” IEEE J. Quantum Electron. QE-22, 1042–1051 (1987).

Winful, H. G.

H. G. Winful, “Pulse compression in optical fiber filters,” Appl. Phys. Lett. 46, 527–529 (1984).
[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, “Effect of end reflection on distributed-feedback bistable optical devices,” IEEE J. Quantum Electron. QE-17, 164 (1981).

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. I. Stegeman, “Applications of nonlinear periodic structures in guided wave optics,” in First International Conference on Integrated Optical Circuit Engineering, S. Sriram and D. B. Ostrowsky, eds., Proc. Soc. Photo-Opt. Instrum. Eng.517, 214–218 (1984).
[Crossref]

H. G. Winful, “Optical bistability in periodic structures and in four-wave mixing processes,” Ph.D. dissertation (University of Southern California, Los Angeles, Calif., 1980).

Yamada, M.

Yariv, A.

A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron. QE-9, 919–933 (1973).
[Crossref]

Zanoni, R.

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W. E. Torruellas, D. Neher, R. Zanoni, G. I. Stegeman, I. Kajzar, and F. Leclerc, “Dispersion measurements of the third-order nonlinear susceptibility of polythiophene thin films,” Chem. Phys. Lett. 175, 11–16 (1990).
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Figures (12)

Fig. 1
Fig. 1

a, DFB structure with the phase shift ΔΩ at the center located at z = 0. The sinusoidal curve represents the effective linear index nL(z). Normalized intensities I0, J0, and T are described in text. b, Method of introducing the phase shift by combining two identical periodic regions. The regions are connected by the uniform phase-retarding section that introduces shift of ΔΩ. c, Phase shift introduced by interrupting the periodicity of a uniform periodic structure. Two segments are regarded as uniform periodic structures differing in phases by ΔΩ.

Fig. 2
Fig. 2

Transmittivity of linear zero-gain λ/4-shifted (solid curve) and uniform (dashed curve) DFB structures calculated by the F-matrix method. Both structures are characterized by κL = 4.

Fig. 3
Fig. 3

Loci of the real zeros of the polynomial Q(J) appearing in Eq. (13) for T = 0.5 and κL = 4. Three detuning domains corresponding to different integration procedures of Eq. (13) are bounded by dashed lines.

Fig. 4
Fig. 4

Transmittivity of the nonlinear, uniform DFB structure for three values of the normalized output T and κL = 4. The solid curve corresponds to the case T = 2.5 × 10−5, the dotted curve represents T = 0.05, and the dashed curve represents T = 0.138.

Fig. 5
Fig. 5

Loci of the real zeros of the polynomial P(I) that appears in Eq. (39) for T = 0.1 and κL = 4. The dashed curve represents the center-forward flux J0 and its relation to the zeros of the P(I). Three detuning domains indicate regions in which different solutions of Eq. (39) are valid.

Fig. 6
Fig. 6

Transmittivity of a nonlinear λ/4-shifted DFB structure in the case of zero detuning (Δ/βL = 0) for different values of the coupling strength κL. Even small variations in the input intensity (ΔI0 ∼ 0.2%) produce rapid changes in transmittivity (∼90%) for moderate coupling strengths (κL = 6).

Fig. 7
Fig. 7

Input–output characteristics of a nonlinear λ/4-shifted DFB device for κL = 4. The detuning parameter ΔβL is varied between −0.6 and 0.6.

Fig. 8
Fig. 8

Input–output characteristics of the uniform DFB device for κL = 4 when the detuning parameter ΔβL is chosen close to the stop-band edge (inset).

Fig. 9
Fig. 9

Effects of the end reflection on the device performance. a, Input–output characteristics for fixed reflectance phase (θR = π) and varying reflectivity R. b, Input–output characteristics for fixed reflectivity R = 2% and varying reflectance phase. The detuning parameter in both cases is set to ΔβL = −0.5.

Fig. 10
Fig. 10

Shifting behavior of the central transmissive peak for a λ/4-shifted DFB structure with κL = 4. The normalized output intensity T is kept as the fixed parameter across the entire detuning range. The solid curve corresponds to the normalized output intensity of T = 2.5 × 10−5, the dashed curve corresponds to T = 0.033, and the dotted curve corresponds to T = 0.1.

Fig. 11
Fig. 11

Transmittivity of a λ/4-shifted DFB structure as a function of ΔβL when the normalized input intensity I0 is kept fixed and the output T varies accordingly. The coupling strength of the structure is κL = 4.

Fig. 12
Fig. 12

Transmittivity of a λ/4-shifted DFB device as a function of ΔβL for κL = 4 and normalized input intensity I0 = 0.01. Wavelength tuning of the source with the fixed intensity I0 leads to bistable switching, as shown by dashed lines.

Equations (57)

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n L ( z ) = n 0 + Δ n cos ( 2 β B z + Ω ) .
Ω = { Ω 1 z < 0 Ω 2 z 0 .
E ( z ) = E + ( z ) exp ( i β z ) + E ( z ) exp ( i β z ) ,
d E + d z = i κ E exp [ i ( 2 Δ β z Ω ) ] + i γ ( | E + | 2 + 2 | E | 2 ) E + ,
d E d z = i κ E + exp [ i ( 2 Δ β Ω ) ] i γ ( 2 | E + | 2 + | E | 2 ) E .
E ± ( z ) = | E ± ( z ) | exp [ i ϕ ± ( z ) ] = A ± ( z ) exp [ i ϕ ± ( z ) ] ,
A T 2 = A + 2 A 2 ,
Γ i = A + ( z ) A ( z ) cos ψ i ( z ) + [ 2 Δ β + 3 γ A 2 ( z ) ] × A + 2 ( z ) / ( 2 κ ) , i = 1 , 2 .
ψ i ( z ) = 2 Δ β z + ϕ + ( z ) ϕ ( z ) Ω i .
A ( L / 2 ) = 0 ,
A T 2 = A + 2 ( L / 2 ) A 2 ( L / 2 ) = A + 2 ( L / 2 ) .
Γ 2 = A + ( z ) A ( z ) cos ψ 2 ( z ) + [ 2 Δ β + 3 γ A 2 ( z ) ] A + 2 ( z ) / ( 2 κ ) = Δ β A T 2 / κ .
4 κ 2 A + 2 [ 1 ( 1 κ A d A + d z ) 2 ] = A 2 ( 2 Δ β + 3 γ A + 2 ) 2 .
( L 2 d J d z ) 2 = ( J T ) [ ( κ L ) 2 J ( J T ) ( Δ β L + 4 J ) 2 ] = ( Q ) J .
r = E ( L / 2 ) E + ( L / 2 ) = A ( L / 2 ) A + ( L / 2 ) exp [ j ( ϕ ϕ + ) ] = R exp ( j θ R ) .
ψ 2 ( z ) = Δ β L + θ R Ω 2 .
J ( L 2 ) = T 1 R .
J 0 T d J Q ( J ) = 2 L 0 L / 2 d z .
J 0 T d J Q ( J ) = s n 1 ( sin ϕ , k ) / u ,
sin ϕ = [ ( J 1 J 3 ) ( J 0 J 2 ) ( J 1 J 2 ) ( J 0 J 3 ) ] 1 / 2 ,
u = 2 [ ( J 1 J 3 ) ( J 2 J 4 ) ] 1 / 2 ,
k = 2 [ ( J 1 J 2 ) ( J 3 J 4 ) ] 1 / 2 / u .
J 0 = J 3 J 3 J 2 1 J 1 J 2 J 1 J 3 sn 2 ( u , k ) .
J 0 = T [ 1 + nd ( κ L x , 1 / x ) ] / 2 .
J 0 T d J Q ( J ) = cn 1 ( cos ϕ , k ) / u ,
cos ϕ = ( J 1 J 0 ) B ( J 0 J 2 ) A ( J 1 J 0 ) B + ( J 0 J 2 ) A ,
u = 4 A B ,
k 2 = [ ( J 1 J 2 ) 2 ( A B ) 2 ] / ( 4 A B ) .
J 0 = J 2 + J 1 J 2 1 + | J 1 J 3 | | J 2 J 3 | 1 + cn ( u , k ) 1 cn ( u , k ) .
sin ϕ = [ ( J 1 J 3 ) ( J 0 J 4 ) ( J 3 J 4 ) ( J 1 J 0 ) ] 1 / 2 .
J 0 = J 1 J 1 J 4 1 J 3 J 4 J 3 J 1 sn 2 ( u , k ) .
E ± ( 0 ) = E ± ( 0 + ) .
A T 1 2 = A + 2 ( 0 ) A 2 ( 0 ) = A T 2 2 = A + 2 ( 0 + ) A 2 ( 0 + ) = A T 2 .
Γ 1 = A + ( 0 ) A ( 0 ) cos ψ 1 ( 0 ) + [ 2 Δ β + 3 γ A 2 ( 0 ) ] A + 2 ( 0 ) / ( 2 κ ) .
ψ 1 ( 0 ) = ψ 2 ( 0 ) Δ Ω .
Γ 1 = A c 2 J 0 J 0 T cos ψ 2 ( 0 ) + [ 2 Δ β + 3 γ A c 2 ( J 0 T ) ] A c 2 J 0 / ( 2 κ ) .
A + ( 0 ) A ( 0 ) cos ψ 2 ( 0 ) + [ 2 Δ β + 3 γ A 2 ( 0 ) ] A + 2 ( 0 ) / ( 2 κ ) = Δ β A T 2 / κ .
cos ψ 2 ( 0 ) = 1 κ L ( J 0 T J 0 ) 1 / 2 ( 4 J 0 + Δ β L ) .
Γ 1 = [ 8 J 0 ( J 0 T ) + ( Δ β L ) ( 2 J 0 T ) ] A c 2 / ( κ L ) .
4 κ 2 A 2 A + 2 [ 1 ( 1 κ A d A + d z ) 2 ] = [ 2 κ Γ 1 ( 2 Δ β + 3 γ A 2 ) 2 A + 2 ] 2 .
( L 2 d I d z ) 2 = ( κ L ) 2 I ( I T ) [ 8 J 0 ( J 0 T ) 4 I ( I T ) ] 2 = P ( I ) .
I i = T 2 { 1 ± k k [ η ± η + k 2 ] 1 / 2 } , i = 1 , 2 , 3 , 4 .
I 0 J 0 d I P ( I ) = 2 L L / 2 0 d z .
I 0 J 0 d I P ( I ) = I 0 I 2 d I P ( I ) + I 2 J 0 d I P ( I ) = I 0 I 2 d I P ( I ) + F ( ϕ , k ) / u .
I 2 I 0 d I P ( I ) = F { ϕ [ J 0 ( T ) ] , k ( T ) } / u ( T ) 1 .
I 0 = I 3 I 3 I 2 1 I 1 I 2 I 1 I 3 sn 2 ( υ , k ) .
I 0 = I 2 + I 1 I 2 1 + | I 1 I 3 | | I 2 I 3 | 1 + cn ( υ , k ) 1 cn ( υ , k ) .
I 0 = I 1 I 1 I 4 1 I 3 I 4 I 3 I 1 sn 2 ( υ , k ) .
E ± ( z ) = | E ± ( z ) | exp [ i ϕ ± ( z ) ] = A ± ( z ) exp [ i ϕ ± ( z ) ] .
d A + d z = κ A sin ψ ,
d A d z = κ A + sin ψ ,
A + d ϕ + d z = κ A cos ψ + γ ( | A + | 2 + 2 | A | 2 ) A + ,
A d ϕ d z = κ A + cos ψ + γ ( | A | 2 + 2 | A + | 2 ) A .
1 A d A + d z = 1 A + d A d z ,
A + 2 ( z ) A 2 ( z ) = const . = A T 2 .
( cot ψ ) [ ln ( A + A ) cos ψ ] z = 2 Δ β + 3 γ ( A + 2 + A 2 ) .
Γ i = A + ( z ) A ( z ) cos ψ i ( z ) + [ 2 Δ β + 3 γ A 2 ( z ) ] A + 2 ( z ) / ( 2 κ ) .

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