Abstract

Bistable responses of Fabry–Perot cavities and optical arrays in the presence of diffraction and diffusion are studied both analytically and numerically. The model is a pair of nonlinear Schrödinger equations coupled through a diffusion equation. The numerical computations are based on a split-step method, with three distinct characteristics. In these diffusion-dominated arrays with weak diffraction, this study demonstrates that focusing nonlinearity can improve the response characteristics significantly. The primary results of the study are that (1) for diffusion-dominated media a small amount of diffraction is sufficient to alter optical bistability significantly; (2) focusing nonlinearities enhance optical bistability in comparison with defocusing nonlinearities; (3) in diffusion-dominated media these focusing–defocusing effects are quite distinct from self-focusing behavior in Kerr media; (4) in the presence of diffraction the response of the array can be described analytically by a reduced map, whose derivation can be viewed as an extension of Firth’s diffusive model to include weak diffraction; (5) this map is used to explain analytically certain qualitative features of bistability, as observed in the numerical experiments; and (6) optimal spacing predictions are made with a reduced map and verified with numerical simulations of small all-optical arrays.

© 1999 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. Szoke, V. Daneu, J. Goldnar, and N. A. Kurnit, “Bistable optical element and its applications,” Appl. Phys. Lett. 15, 376–379 (1969).
    [CrossRef]
  2. M. I. Nathan, J. C. Marinace, R. F. Rutz, A. E. Michel, and G. J. Lasher, “GaAs injection laser with novel mode control and switching properties,” J. Appl. Phys. 36, 473–480 (1965).
    [CrossRef]
  3. N. B. Abraham and W. J. Firth, “Overview of transverse effects in nonlinear-optical systems,” J. Opt. Soc. Am. B 7, 951–962 (1990).
    [CrossRef]
  4. H. M. Gibbs, Optical Bistability: Controlling Light with Light (Academic, London 1985).
  5. S. L. McCall, H. M. Gibbs, G. G. Churchill, and T. N. C. Venkatesan, “Optical transistor and bistability,” Bull. Am. Phys. Soc. 20, 636 (1975).
  6. S. L. McCall, H. M. Gibbs, and T. N. C. Venkatesan, “Optical transistor and bistability,” J. Opt. Soc. Am. 65, 1184 (1975).
  7. W. J. Firth, “Optically bistable arrays and chaotic dynamics,” Phys. Lett. A 125, 375–379 (1987).
    [CrossRef]
  8. W. J. Firth, “Optical memory and spatial chaos,” Phys. Rev. Lett. 61, 329–332 (1988).
    [CrossRef] [PubMed]
  9. K. D. Stephen and W. J. Firth, “Absorptive optically bistable arrays,” J. Mod. Opt. 37, 627–638 (1990).
    [CrossRef]
  10. W. J. Firth, E. Abraham, E. M. Wright, I. Galbraith, and B. S. Wherrett, “Diffusion, diffraction and reflection in semiconductor O.B. devices,” Philos. Trans. R. Soc. London, Ser. A 313, 299–306 (1984).
    [CrossRef]
  11. J. V. Moloney and H. M. Gibbs, “Role of diffractive coupling and self-focusing or defocusing in the dynamical switching of a bistable optical cavity,” Phys. Rev. Lett. 48, 1607–1609 (1982).
    [CrossRef]
  12. Y. Chen, “Diffraction effects on diffusive optical bistability and optical memory,” Ph.D. dissertation (Princeton University, Princeton, N.J., 1998).
  13. W. J. Firth, I. Galbraith, and E. M. Wright, “Diffusion and diffraction in dispersive optical bistability,” J. Opt. Soc. Am. B 2, 1005–1009 (1985).
    [CrossRef]
  14. Y. Chen and D. W. McLaughlin, “Diffraction effects on diffusive bistable optical arrays and optical memory,” submitted to Physica D.
  15. D. Gottlieb and S. A. Orszag, Numerical Analysis of Spectral Methods: Theory and Application (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1977).
  16. J. V. Moloney, M. R. Belic, and H. M. Gibbs, “Calculation of transverse effects in optical bistability using fast Fourier transform techniques,” Opt. Commun. 41, 379–382 (1982).
    [CrossRef]
  17. J. V. Moloney, M. Sargent III, and H. M. Gibbs, “Transverse effects in absorptive optical bistability,” Opt. Commun. 44, 289–292 (1983).
    [CrossRef]
  18. D. Weaire and J. P. Kermode, “Dispersive optical bistability: numerical methods and definitive results,” J. Opt. Soc. Am. B 3, 1706–1711 (1986).
    [CrossRef]
  19. D. Weaire, J. P. Kermode, and V. M. Dwyer, “The role of diffraction in dispersive optical bistability,” Opt. Commun. 55, 223–228 (1985).
    [CrossRef]
  20. D. Weaire, C. O’Carroll, and C. Wickham, “Dispersive optical bistability with diffusion: a scaling law,” Europhys. Lett. 8, 25–28 (1989).
    [CrossRef]
  21. D. J. Hagan, H. A. MacKenzie, H. A. Al-Atlar, and W. J. Firth, “Carrier diffusion measurements in InSb by the angular dependence of degenerate four-wave mixing,” Opt. Lett. 10, 187–189 (1985).
    [CrossRef] [PubMed]
  22. D. J. Hagan, H. A. MacKenzie, J. J. E. Reid, A. C. Walker, and F. A. P. Tooley, “Spot size dependence of switching power for an optically bistable InSb element,” Appl. Phys. Lett. 47, 203–205 (1985).
    [CrossRef]
  23. G. S. McDonald and W. J. Firth, “All-optical switching in a nonlinear resonator,” J. Mod. Opt. 37, 613–626 (1990).
    [CrossRef]
  24. G. S. McDonald and W. J. Firth, “Spatial solitary-wave optical memory,” J. Opt. Soc. Am. B 7, 1328–1335 (1990).
    [CrossRef]
  25. G. S. McDonald, “Spatial solitary wave optical memory,” Ph.D. dissertation (Strathclyde University, Glasgow, Scotland, 1989).
  26. W. J. Firth and I. Galbraith, “Diffusive transverse coupling of bistable elements-switching waves and crosstalk,” IEEE J. Quantum Electron. QE-21, 1399–1403 (1985).
    [CrossRef]
  27. G. S. McDonald and W. J. Firth, “Switching dynamics of spatial solitary wave pixels,” J. Opt. Soc. Am. B 10, 1081–1089 (1993).
    [CrossRef]

1993 (1)

1990 (4)

N. B. Abraham and W. J. Firth, “Overview of transverse effects in nonlinear-optical systems,” J. Opt. Soc. Am. B 7, 951–962 (1990).
[CrossRef]

G. S. McDonald and W. J. Firth, “Spatial solitary-wave optical memory,” J. Opt. Soc. Am. B 7, 1328–1335 (1990).
[CrossRef]

K. D. Stephen and W. J. Firth, “Absorptive optically bistable arrays,” J. Mod. Opt. 37, 627–638 (1990).
[CrossRef]

G. S. McDonald and W. J. Firth, “All-optical switching in a nonlinear resonator,” J. Mod. Opt. 37, 613–626 (1990).
[CrossRef]

1989 (1)

D. Weaire, C. O’Carroll, and C. Wickham, “Dispersive optical bistability with diffusion: a scaling law,” Europhys. Lett. 8, 25–28 (1989).
[CrossRef]

1988 (1)

W. J. Firth, “Optical memory and spatial chaos,” Phys. Rev. Lett. 61, 329–332 (1988).
[CrossRef] [PubMed]

1987 (1)

W. J. Firth, “Optically bistable arrays and chaotic dynamics,” Phys. Lett. A 125, 375–379 (1987).
[CrossRef]

1986 (1)

1985 (5)

D. Weaire, J. P. Kermode, and V. M. Dwyer, “The role of diffraction in dispersive optical bistability,” Opt. Commun. 55, 223–228 (1985).
[CrossRef]

W. J. Firth and I. Galbraith, “Diffusive transverse coupling of bistable elements-switching waves and crosstalk,” IEEE J. Quantum Electron. QE-21, 1399–1403 (1985).
[CrossRef]

W. J. Firth, I. Galbraith, and E. M. Wright, “Diffusion and diffraction in dispersive optical bistability,” J. Opt. Soc. Am. B 2, 1005–1009 (1985).
[CrossRef]

D. J. Hagan, H. A. MacKenzie, J. J. E. Reid, A. C. Walker, and F. A. P. Tooley, “Spot size dependence of switching power for an optically bistable InSb element,” Appl. Phys. Lett. 47, 203–205 (1985).
[CrossRef]

D. J. Hagan, H. A. MacKenzie, H. A. Al-Atlar, and W. J. Firth, “Carrier diffusion measurements in InSb by the angular dependence of degenerate four-wave mixing,” Opt. Lett. 10, 187–189 (1985).
[CrossRef] [PubMed]

1984 (1)

W. J. Firth, E. Abraham, E. M. Wright, I. Galbraith, and B. S. Wherrett, “Diffusion, diffraction and reflection in semiconductor O.B. devices,” Philos. Trans. R. Soc. London, Ser. A 313, 299–306 (1984).
[CrossRef]

1983 (1)

J. V. Moloney, M. Sargent III, and H. M. Gibbs, “Transverse effects in absorptive optical bistability,” Opt. Commun. 44, 289–292 (1983).
[CrossRef]

1982 (2)

J. V. Moloney, M. R. Belic, and H. M. Gibbs, “Calculation of transverse effects in optical bistability using fast Fourier transform techniques,” Opt. Commun. 41, 379–382 (1982).
[CrossRef]

J. V. Moloney and H. M. Gibbs, “Role of diffractive coupling and self-focusing or defocusing in the dynamical switching of a bistable optical cavity,” Phys. Rev. Lett. 48, 1607–1609 (1982).
[CrossRef]

1975 (2)

S. L. McCall, H. M. Gibbs, G. G. Churchill, and T. N. C. Venkatesan, “Optical transistor and bistability,” Bull. Am. Phys. Soc. 20, 636 (1975).

S. L. McCall, H. M. Gibbs, and T. N. C. Venkatesan, “Optical transistor and bistability,” J. Opt. Soc. Am. 65, 1184 (1975).

1969 (1)

A. Szoke, V. Daneu, J. Goldnar, and N. A. Kurnit, “Bistable optical element and its applications,” Appl. Phys. Lett. 15, 376–379 (1969).
[CrossRef]

1965 (1)

M. I. Nathan, J. C. Marinace, R. F. Rutz, A. E. Michel, and G. J. Lasher, “GaAs injection laser with novel mode control and switching properties,” J. Appl. Phys. 36, 473–480 (1965).
[CrossRef]

Abraham, E.

W. J. Firth, E. Abraham, E. M. Wright, I. Galbraith, and B. S. Wherrett, “Diffusion, diffraction and reflection in semiconductor O.B. devices,” Philos. Trans. R. Soc. London, Ser. A 313, 299–306 (1984).
[CrossRef]

Abraham, N. B.

Al-Atlar, H. A.

Belic, M. R.

J. V. Moloney, M. R. Belic, and H. M. Gibbs, “Calculation of transverse effects in optical bistability using fast Fourier transform techniques,” Opt. Commun. 41, 379–382 (1982).
[CrossRef]

Churchill, G. G.

S. L. McCall, H. M. Gibbs, G. G. Churchill, and T. N. C. Venkatesan, “Optical transistor and bistability,” Bull. Am. Phys. Soc. 20, 636 (1975).

Daneu, V.

A. Szoke, V. Daneu, J. Goldnar, and N. A. Kurnit, “Bistable optical element and its applications,” Appl. Phys. Lett. 15, 376–379 (1969).
[CrossRef]

Dwyer, V. M.

D. Weaire, J. P. Kermode, and V. M. Dwyer, “The role of diffraction in dispersive optical bistability,” Opt. Commun. 55, 223–228 (1985).
[CrossRef]

Firth, W. J.

G. S. McDonald and W. J. Firth, “Switching dynamics of spatial solitary wave pixels,” J. Opt. Soc. Am. B 10, 1081–1089 (1993).
[CrossRef]

N. B. Abraham and W. J. Firth, “Overview of transverse effects in nonlinear-optical systems,” J. Opt. Soc. Am. B 7, 951–962 (1990).
[CrossRef]

G. S. McDonald and W. J. Firth, “Spatial solitary-wave optical memory,” J. Opt. Soc. Am. B 7, 1328–1335 (1990).
[CrossRef]

G. S. McDonald and W. J. Firth, “All-optical switching in a nonlinear resonator,” J. Mod. Opt. 37, 613–626 (1990).
[CrossRef]

K. D. Stephen and W. J. Firth, “Absorptive optically bistable arrays,” J. Mod. Opt. 37, 627–638 (1990).
[CrossRef]

W. J. Firth, “Optical memory and spatial chaos,” Phys. Rev. Lett. 61, 329–332 (1988).
[CrossRef] [PubMed]

W. J. Firth, “Optically bistable arrays and chaotic dynamics,” Phys. Lett. A 125, 375–379 (1987).
[CrossRef]

D. J. Hagan, H. A. MacKenzie, H. A. Al-Atlar, and W. J. Firth, “Carrier diffusion measurements in InSb by the angular dependence of degenerate four-wave mixing,” Opt. Lett. 10, 187–189 (1985).
[CrossRef] [PubMed]

W. J. Firth and I. Galbraith, “Diffusive transverse coupling of bistable elements-switching waves and crosstalk,” IEEE J. Quantum Electron. QE-21, 1399–1403 (1985).
[CrossRef]

W. J. Firth, I. Galbraith, and E. M. Wright, “Diffusion and diffraction in dispersive optical bistability,” J. Opt. Soc. Am. B 2, 1005–1009 (1985).
[CrossRef]

W. J. Firth, E. Abraham, E. M. Wright, I. Galbraith, and B. S. Wherrett, “Diffusion, diffraction and reflection in semiconductor O.B. devices,” Philos. Trans. R. Soc. London, Ser. A 313, 299–306 (1984).
[CrossRef]

Galbraith, I.

W. J. Firth, I. Galbraith, and E. M. Wright, “Diffusion and diffraction in dispersive optical bistability,” J. Opt. Soc. Am. B 2, 1005–1009 (1985).
[CrossRef]

W. J. Firth and I. Galbraith, “Diffusive transverse coupling of bistable elements-switching waves and crosstalk,” IEEE J. Quantum Electron. QE-21, 1399–1403 (1985).
[CrossRef]

W. J. Firth, E. Abraham, E. M. Wright, I. Galbraith, and B. S. Wherrett, “Diffusion, diffraction and reflection in semiconductor O.B. devices,” Philos. Trans. R. Soc. London, Ser. A 313, 299–306 (1984).
[CrossRef]

Gibbs, H. M.

J. V. Moloney, M. Sargent III, and H. M. Gibbs, “Transverse effects in absorptive optical bistability,” Opt. Commun. 44, 289–292 (1983).
[CrossRef]

J. V. Moloney and H. M. Gibbs, “Role of diffractive coupling and self-focusing or defocusing in the dynamical switching of a bistable optical cavity,” Phys. Rev. Lett. 48, 1607–1609 (1982).
[CrossRef]

J. V. Moloney, M. R. Belic, and H. M. Gibbs, “Calculation of transverse effects in optical bistability using fast Fourier transform techniques,” Opt. Commun. 41, 379–382 (1982).
[CrossRef]

S. L. McCall, H. M. Gibbs, and T. N. C. Venkatesan, “Optical transistor and bistability,” J. Opt. Soc. Am. 65, 1184 (1975).

S. L. McCall, H. M. Gibbs, G. G. Churchill, and T. N. C. Venkatesan, “Optical transistor and bistability,” Bull. Am. Phys. Soc. 20, 636 (1975).

Goldnar, J.

A. Szoke, V. Daneu, J. Goldnar, and N. A. Kurnit, “Bistable optical element and its applications,” Appl. Phys. Lett. 15, 376–379 (1969).
[CrossRef]

Hagan, D. J.

D. J. Hagan, H. A. MacKenzie, J. J. E. Reid, A. C. Walker, and F. A. P. Tooley, “Spot size dependence of switching power for an optically bistable InSb element,” Appl. Phys. Lett. 47, 203–205 (1985).
[CrossRef]

D. J. Hagan, H. A. MacKenzie, H. A. Al-Atlar, and W. J. Firth, “Carrier diffusion measurements in InSb by the angular dependence of degenerate four-wave mixing,” Opt. Lett. 10, 187–189 (1985).
[CrossRef] [PubMed]

Kermode, J. P.

D. Weaire and J. P. Kermode, “Dispersive optical bistability: numerical methods and definitive results,” J. Opt. Soc. Am. B 3, 1706–1711 (1986).
[CrossRef]

D. Weaire, J. P. Kermode, and V. M. Dwyer, “The role of diffraction in dispersive optical bistability,” Opt. Commun. 55, 223–228 (1985).
[CrossRef]

Kurnit, N. A.

A. Szoke, V. Daneu, J. Goldnar, and N. A. Kurnit, “Bistable optical element and its applications,” Appl. Phys. Lett. 15, 376–379 (1969).
[CrossRef]

Lasher, G. J.

M. I. Nathan, J. C. Marinace, R. F. Rutz, A. E. Michel, and G. J. Lasher, “GaAs injection laser with novel mode control and switching properties,” J. Appl. Phys. 36, 473–480 (1965).
[CrossRef]

MacKenzie, H. A.

D. J. Hagan, H. A. MacKenzie, J. J. E. Reid, A. C. Walker, and F. A. P. Tooley, “Spot size dependence of switching power for an optically bistable InSb element,” Appl. Phys. Lett. 47, 203–205 (1985).
[CrossRef]

D. J. Hagan, H. A. MacKenzie, H. A. Al-Atlar, and W. J. Firth, “Carrier diffusion measurements in InSb by the angular dependence of degenerate four-wave mixing,” Opt. Lett. 10, 187–189 (1985).
[CrossRef] [PubMed]

Marinace, J. C.

M. I. Nathan, J. C. Marinace, R. F. Rutz, A. E. Michel, and G. J. Lasher, “GaAs injection laser with novel mode control and switching properties,” J. Appl. Phys. 36, 473–480 (1965).
[CrossRef]

McCall, S. L.

S. L. McCall, H. M. Gibbs, G. G. Churchill, and T. N. C. Venkatesan, “Optical transistor and bistability,” Bull. Am. Phys. Soc. 20, 636 (1975).

S. L. McCall, H. M. Gibbs, and T. N. C. Venkatesan, “Optical transistor and bistability,” J. Opt. Soc. Am. 65, 1184 (1975).

McDonald, G. S.

Michel, A. E.

M. I. Nathan, J. C. Marinace, R. F. Rutz, A. E. Michel, and G. J. Lasher, “GaAs injection laser with novel mode control and switching properties,” J. Appl. Phys. 36, 473–480 (1965).
[CrossRef]

Moloney, J. V.

J. V. Moloney, M. Sargent III, and H. M. Gibbs, “Transverse effects in absorptive optical bistability,” Opt. Commun. 44, 289–292 (1983).
[CrossRef]

J. V. Moloney and H. M. Gibbs, “Role of diffractive coupling and self-focusing or defocusing in the dynamical switching of a bistable optical cavity,” Phys. Rev. Lett. 48, 1607–1609 (1982).
[CrossRef]

J. V. Moloney, M. R. Belic, and H. M. Gibbs, “Calculation of transverse effects in optical bistability using fast Fourier transform techniques,” Opt. Commun. 41, 379–382 (1982).
[CrossRef]

Nathan, M. I.

M. I. Nathan, J. C. Marinace, R. F. Rutz, A. E. Michel, and G. J. Lasher, “GaAs injection laser with novel mode control and switching properties,” J. Appl. Phys. 36, 473–480 (1965).
[CrossRef]

O’Carroll, C.

D. Weaire, C. O’Carroll, and C. Wickham, “Dispersive optical bistability with diffusion: a scaling law,” Europhys. Lett. 8, 25–28 (1989).
[CrossRef]

Reid, J. J. E.

D. J. Hagan, H. A. MacKenzie, J. J. E. Reid, A. C. Walker, and F. A. P. Tooley, “Spot size dependence of switching power for an optically bistable InSb element,” Appl. Phys. Lett. 47, 203–205 (1985).
[CrossRef]

Rutz, R. F.

M. I. Nathan, J. C. Marinace, R. F. Rutz, A. E. Michel, and G. J. Lasher, “GaAs injection laser with novel mode control and switching properties,” J. Appl. Phys. 36, 473–480 (1965).
[CrossRef]

Sargent III, M.

J. V. Moloney, M. Sargent III, and H. M. Gibbs, “Transverse effects in absorptive optical bistability,” Opt. Commun. 44, 289–292 (1983).
[CrossRef]

Stephen, K. D.

K. D. Stephen and W. J. Firth, “Absorptive optically bistable arrays,” J. Mod. Opt. 37, 627–638 (1990).
[CrossRef]

Szoke, A.

A. Szoke, V. Daneu, J. Goldnar, and N. A. Kurnit, “Bistable optical element and its applications,” Appl. Phys. Lett. 15, 376–379 (1969).
[CrossRef]

Tooley, F. A. P.

D. J. Hagan, H. A. MacKenzie, J. J. E. Reid, A. C. Walker, and F. A. P. Tooley, “Spot size dependence of switching power for an optically bistable InSb element,” Appl. Phys. Lett. 47, 203–205 (1985).
[CrossRef]

Venkatesan, T. N. C.

S. L. McCall, H. M. Gibbs, and T. N. C. Venkatesan, “Optical transistor and bistability,” J. Opt. Soc. Am. 65, 1184 (1975).

S. L. McCall, H. M. Gibbs, G. G. Churchill, and T. N. C. Venkatesan, “Optical transistor and bistability,” Bull. Am. Phys. Soc. 20, 636 (1975).

Walker, A. C.

D. J. Hagan, H. A. MacKenzie, J. J. E. Reid, A. C. Walker, and F. A. P. Tooley, “Spot size dependence of switching power for an optically bistable InSb element,” Appl. Phys. Lett. 47, 203–205 (1985).
[CrossRef]

Weaire, D.

D. Weaire, C. O’Carroll, and C. Wickham, “Dispersive optical bistability with diffusion: a scaling law,” Europhys. Lett. 8, 25–28 (1989).
[CrossRef]

D. Weaire and J. P. Kermode, “Dispersive optical bistability: numerical methods and definitive results,” J. Opt. Soc. Am. B 3, 1706–1711 (1986).
[CrossRef]

D. Weaire, J. P. Kermode, and V. M. Dwyer, “The role of diffraction in dispersive optical bistability,” Opt. Commun. 55, 223–228 (1985).
[CrossRef]

Wherrett, B. S.

W. J. Firth, E. Abraham, E. M. Wright, I. Galbraith, and B. S. Wherrett, “Diffusion, diffraction and reflection in semiconductor O.B. devices,” Philos. Trans. R. Soc. London, Ser. A 313, 299–306 (1984).
[CrossRef]

Wickham, C.

D. Weaire, C. O’Carroll, and C. Wickham, “Dispersive optical bistability with diffusion: a scaling law,” Europhys. Lett. 8, 25–28 (1989).
[CrossRef]

Wright, E. M.

W. J. Firth, I. Galbraith, and E. M. Wright, “Diffusion and diffraction in dispersive optical bistability,” J. Opt. Soc. Am. B 2, 1005–1009 (1985).
[CrossRef]

W. J. Firth, E. Abraham, E. M. Wright, I. Galbraith, and B. S. Wherrett, “Diffusion, diffraction and reflection in semiconductor O.B. devices,” Philos. Trans. R. Soc. London, Ser. A 313, 299–306 (1984).
[CrossRef]

Appl. Phys. Lett. (2)

A. Szoke, V. Daneu, J. Goldnar, and N. A. Kurnit, “Bistable optical element and its applications,” Appl. Phys. Lett. 15, 376–379 (1969).
[CrossRef]

D. J. Hagan, H. A. MacKenzie, J. J. E. Reid, A. C. Walker, and F. A. P. Tooley, “Spot size dependence of switching power for an optically bistable InSb element,” Appl. Phys. Lett. 47, 203–205 (1985).
[CrossRef]

Bull. Am. Phys. Soc. (1)

S. L. McCall, H. M. Gibbs, G. G. Churchill, and T. N. C. Venkatesan, “Optical transistor and bistability,” Bull. Am. Phys. Soc. 20, 636 (1975).

Europhys. Lett. (1)

D. Weaire, C. O’Carroll, and C. Wickham, “Dispersive optical bistability with diffusion: a scaling law,” Europhys. Lett. 8, 25–28 (1989).
[CrossRef]

IEEE J. Quantum Electron. (1)

W. J. Firth and I. Galbraith, “Diffusive transverse coupling of bistable elements-switching waves and crosstalk,” IEEE J. Quantum Electron. QE-21, 1399–1403 (1985).
[CrossRef]

J. Appl. Phys. (1)

M. I. Nathan, J. C. Marinace, R. F. Rutz, A. E. Michel, and G. J. Lasher, “GaAs injection laser with novel mode control and switching properties,” J. Appl. Phys. 36, 473–480 (1965).
[CrossRef]

J. Mod. Opt. (2)

K. D. Stephen and W. J. Firth, “Absorptive optically bistable arrays,” J. Mod. Opt. 37, 627–638 (1990).
[CrossRef]

G. S. McDonald and W. J. Firth, “All-optical switching in a nonlinear resonator,” J. Mod. Opt. 37, 613–626 (1990).
[CrossRef]

J. Opt. Soc. Am. (1)

S. L. McCall, H. M. Gibbs, and T. N. C. Venkatesan, “Optical transistor and bistability,” J. Opt. Soc. Am. 65, 1184 (1975).

J. Opt. Soc. Am. B (5)

Opt. Commun. (3)

D. Weaire, J. P. Kermode, and V. M. Dwyer, “The role of diffraction in dispersive optical bistability,” Opt. Commun. 55, 223–228 (1985).
[CrossRef]

J. V. Moloney, M. R. Belic, and H. M. Gibbs, “Calculation of transverse effects in optical bistability using fast Fourier transform techniques,” Opt. Commun. 41, 379–382 (1982).
[CrossRef]

J. V. Moloney, M. Sargent III, and H. M. Gibbs, “Transverse effects in absorptive optical bistability,” Opt. Commun. 44, 289–292 (1983).
[CrossRef]

Opt. Lett. (1)

Philos. Trans. R. Soc. London, Ser. A (1)

W. J. Firth, E. Abraham, E. M. Wright, I. Galbraith, and B. S. Wherrett, “Diffusion, diffraction and reflection in semiconductor O.B. devices,” Philos. Trans. R. Soc. London, Ser. A 313, 299–306 (1984).
[CrossRef]

Phys. Lett. A (1)

W. J. Firth, “Optically bistable arrays and chaotic dynamics,” Phys. Lett. A 125, 375–379 (1987).
[CrossRef]

Phys. Rev. Lett. (2)

W. J. Firth, “Optical memory and spatial chaos,” Phys. Rev. Lett. 61, 329–332 (1988).
[CrossRef] [PubMed]

J. V. Moloney and H. M. Gibbs, “Role of diffractive coupling and self-focusing or defocusing in the dynamical switching of a bistable optical cavity,” Phys. Rev. Lett. 48, 1607–1609 (1982).
[CrossRef]

Other (5)

Y. Chen, “Diffraction effects on diffusive optical bistability and optical memory,” Ph.D. dissertation (Princeton University, Princeton, N.J., 1998).

Y. Chen and D. W. McLaughlin, “Diffraction effects on diffusive bistable optical arrays and optical memory,” submitted to Physica D.

D. Gottlieb and S. A. Orszag, Numerical Analysis of Spectral Methods: Theory and Application (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1977).

H. M. Gibbs, Optical Bistability: Controlling Light with Light (Academic, London 1985).

G. S. McDonald, “Spatial solitary wave optical memory,” Ph.D. dissertation (Strathclyde University, Glasgow, Scotland, 1989).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (16)

Fig. 1
Fig. 1

A defocusing array of 20 pixels with pixel separation S=3 is not able to address the pattern 01011110101110010110.

Fig. 2
Fig. 2

The same array as in Fig. 1 with focusing nonlinearity can address the pattern 01011110101110010110.

Fig. 3
Fig. 3

Bistable states of a nonlinear Fabry–Perot cavity with plane-wave input. δ=π, rf=0.8, rb=0.5, r=0.4. The solid, dotted, and dashed curves are for Iin equal to 0.5625, 0.4415, and 0.2, respectively.

Fig. 4
Fig. 4

Bistable solutions of a diffusive cavity with a narrow Gaussian input of width w=0.2. Dotted curves, the solutions with a δ-function input. Horizontal dashed-line, the detuning.

Fig. 5
Fig. 5

Hysteresis curves of a diffusive cavity for inputs of various widths. Left-most curve, a δ-function input. (Throughout the paper, computed points are linked by straight lines to construct curves for ease of visualization.)

Fig. 6
Fig. 6

Bistable solutions of a cavity for both focusing (positive Γ) and defocusing (negative Γ) nonlinearities. The beam width w=0.2, and the input power Pin=1.125.

Fig. 7
Fig. 7

Adiabatic solutions that mimic hysteresis loops. In each case the power Pin of the Gaussian input is linearly (in time) ramped up from 0 to 2.5 during a time interval of 500 round trips and then linearly ramped down to 0 during the same period of time. The solid curve has self-focusing nonlinearity, and the dashed curve has self-defocusing nonlinearity, both with Γ=0.005. Dotted curve, diffractionless.

Fig. 8
Fig. 8

Solutions of a self-focusing cavity for different levels of diffusion. The input is distributed as a hyperbolic secant function: Fin=(Pin/2w)1/2sech(x/w), where the width w=0.9 and the power Pin=2.20. Γ=0.02 for all curves.

Fig. 9
Fig. 9

Bistable solutions as functions of diffraction for a cavity with a wide input. The beam width w=1.414.

Fig. 10
Fig. 10

Theoretical estimates for curves in Fig. 6. The marked points are numerical results for inputs with width w=0.05.

Fig. 11
Fig. 11

Region in which bistability exists.

Fig. 12
Fig. 12

Solution representing the pattern 101 of a diffusive three-pixel array. Dotted curve, the initial condition constructed by concatenating two on-state solutions and one off-state solution of Fig. 3. Pixel separation S=4.

Fig. 13
Fig. 13

Bifurcation of the pattern 101 of a diffusive three-pixel array as packing density increases. Curves are for interpixel spacing of S=4, 3.5, 3, 2.77, 2.73.

Fig. 14
Fig. 14

Bifurcation of the pattern 101 of a three-pixel array with self-defocusing nonlinearity. Curves are for interpixel spacing of S=5, 4, 3.733, 3.7. Γ=-0.005 for all curves.

Fig. 15
Fig. 15

Bifurcation of the pattern 101 of a three-pixel array with self-focusing nonlinearity. Curves are for interpixel spacing of S=4, 3, 2.4, 2.367. Γ=0.005 for all curves.

Fig. 16
Fig. 16

Addressing the pattern 101 of a three-pixel array when S is near its critical value. S=3.733 and Γ=-0.005.

Equations (31)

Equations on this page are rendered with MathJax. Learn more.

τϕt=-ϕ+δ+2x2ϕ+2(1+rb2)f(ϕ)Iin.
Ft+Fz-iΓ2x2F=ihF,
Bt-Bz-iΓ2x2B=ihB,
τht-2x2h+h=n2(|F|2+|B|2),
F=rfB+Fin,z=0,
B=rb exp(iδ)F,z=1.
h=(1+rb2)1+r2-2r cos(δ+2h)Iin,
ϕ-δ=2(1+rb2)f(ϕ)Iin.
f(ϕ)=11+r2-2r cos(ϕ)
g1(ϕ)=ϕ-δ,
g2(ϕ)=2(1+rb2)f(ϕ)Iin.
d2dx2h(x)-h(x)+(1+rb2)f[δ+2h(x)]Iin(x)=0.
An+1=2cAn-s(1+rb2)f(δ+2An)P0-Bn,
Bn+1=An,
Fin=Pinπw21/2 exp[-(x2/w2)],
Pin=Iin(x)dx,
Fz-iΓ2x2F=ihF,
lD22x2h-h+|F|2=0.
h(x)=-+G(x-x)|F(x)|2dx,
Fz-iΓ2x2F=i-+G(x-x)|F(x)|2dxF.
2x2h(x, z)-h(x, z)+(1+rb2)P0nf(δ+2hn)
×δ(x-n)=0,
f(δ+2hn)=k=0+{r exp[i(δ+2hn)]}k[πw2(1+4ik)]1/2×exp-u2w2(1+4ik)2du.
An+1=2cAn-s(1+rb2)f(δ+2An)P0-Bn,
Bn+1=An.
h0=(1/2)[(1+rb2)f(δ+2h0)P0],
ϕ0-δ=(1+rb2)f(ϕ0)P0,
r exp[i(δ+2hn)]/(1+4i)1/2.
r=r/[1+(4)2]1/4,
δ=δ-½ arctan(4).
k=0+1[πw2(1+4ik)]1/2exp-x2w2(1+4ik)2dx=k=0+1[π(1+4ik)]1/2exp-x2(1+4ik)2dx,

Metrics