Abstract

Design considerations for optically bistable narrow-bandpass interference filters are surveyed. Optimization for low incident laser irradiance levels is presented in terms of the reflective stacks, the spacer thickness, operational laser frequencies, the semiconductor thin-film materials, and the laser spot size. Conditions under which either thermal or electronic nonlinearities dominate are described.

© 1986 Optical Society of America

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  1. S. D. Smith, J. G. H. Mathew, M. R. Taghizadeh, A. C. Walker, B. S. Wherrett, A. Hendry, “Room temperature, visible wavelength optical bistability in ZnSe interference filters,” Opt. Commun. 51, 357–361 (1984).
    [CrossRef]
  2. I. Janossy, M. R. Taghizadeh, J. G. H. Mathew, S. D. Smith, “Thermally induced optical bistability in thin film devices,” IEEE J. Quantum Electron. QE-21, 1447–1452 (1985).
    [CrossRef]
  3. S. D. Smith, F. A. P. Tooley, A. C. Walker, J. G. H. Mathew, M. R. Taghizadeh, B. S. Wherrett, “All-optical logic gates with external switching by laser and incoherent radiation,” presented at the AGARD Conference on Digital Optical Circuit Technology, Schliersee, Federal Republic of Germany, 1984.
  4. F. V. Karpushko, G. V. Sinitsyn, “The anomalous nonlinearity and optical bistability in thin-film interference structures,” Appl. Phys. B 28, 137 (1982).
  5. S. P. Apanasevich, F. V. Karpushko, G. V. Sinitsyn, “Response time of bistable devices based on evaporated thin-film interferometers,” Sov. J. Quantum Electron. 14, 873–874 (1984).
    [CrossRef]
  6. F. V. Karpushko, G. V. Sinitsyn, “An optical logic element for integrated optics in a nonlinear semiconductor interferometer,” J. Appl. Spectrosc. USSR 29, 1323–1326 (1978).
    [CrossRef]
  7. G. R. Olbright, N. Peyghambarian, H. M. Gibbs, H. A. MacLeod, F. van Milligen, “Microsecond room temperature optical bistability and crosstalk studies in ZnS and ZnSe interference filters with visible light and milliwatt power,” Appl. Phys. Lett. 45, 10–14 (1984).
    [CrossRef]
  8. H. M. Gibbs, J. L. Jewell, Y. H. Lee, A. MacLeod, G. Olbright, S. Ovadia, N. Peyghambarian, M. C. Rushford, M. Warren, D. A. Weinberger, T. Venkatesan, “Prospects for parallel optical signal processing using GaAs etalons and ZnS interference filters,” presented at AGARD Conference on Digital Optical Circuit Technology, Schliersee, Federal Republic of Germany, 1984.
  9. D. A. B. Miller, “Refractive Fabry–Perot bistability with linear absorption: theory of operation and cavity optimisation,” IEEE J. Quantum Electron. QE-17, 306–311 (1981).
    [CrossRef]
  10. B. S. Wherrett, “Fabry–Perot bistable cavity optimization—on reflection,” IEEE J. Quantum Electron. QE-20, 646–651 (1984).
    [CrossRef]
  11. B. S. Wherrett, N. A. Higgins, “Theory of nonlinear refraction near the band edge of a semiconductor,” Proc. R. Soc. London Ser. A 379, 67–90 (1982).
    [CrossRef]
  12. D. A. B. Miller, C. T. Seaton, M. E. Prise, S. D. Smith, “Band gap resonant nonlinear refraction in III–V semiconductors,” Phys. Rev. Lett. 47, 197–199 (1981).
    [CrossRef]
  13. B. S. Wherrett, “One-electron theory of nonlinear refraction,” Phil. Trans. R. Soc. London Ser. A 313, 213–220 (1984).
    [CrossRef]
  14. A. C. Walker, F. A. P. Tooley, M. E. Prise, J. G. H. Mathew, A. K. Kar, M. R. Taghizadeh, S. D. Smith, “InSb devices: transphasors with high gain, bistable switches and sequential logic gates,” Phil. Trans. R. Soc. London Ser. A 313, 249–256 (1984).
    [CrossRef]
  15. W. J. Firth, E. Abraham, E. M. Wright, I. Galbraith, B. S. Wherrett, “Diffusion, diffraction and reflection in semiconductor O. B. devices,” Phil. Trans. R. Soc. London Ser. A 313, 299–306 (1984).
    [CrossRef]
  16. W. J. Firth, I. Galbraith, “Diffusive transverse coupling of bistable elements—switching waves and cross-talk,” IEEE J. Quantum Electron. QE-21, 1399–1403 (1985).
    [CrossRef]
  17. D. Weaire, J. P. Kermode, V. M. Dwyer, “The role of diffraction in dispersive optical bistability,” Opt. Commun. 55, 223–228 (1985).
    [CrossRef]
  18. D. Hutchings, B. S. Wherrett, “Analytical solutions for wave propagation through periodic structures,” internal report, Department of Physics (Heriot-Watt University, Edinburgh, Scotland, 1985) (unpublished).
  19. B. S. Wherrett, “Optical characteristics of nonlinear, active multi-layer dielectric thin-film structures,” internal report, Department of Physics (Heriot-Watt University, Edinburgh, Scotland, 1985)(unpublished).
  20. Landolt-Börnstein, Numerical Data and Functional Relationships in Science and Technology (Springer-Verlag, Berlin, 1982), Group III, Vols. 17a and 17b.
  21. See, for example, T. P. McLean, Prog. Semiconduct. 5, 55–80 (1960).
  22. B. S. Wherrett, “Scaling rules for multiphoton interband absorption in semiconductors,” J. Opt. Soc. Am. B 1, 67–72 (1984).
    [CrossRef]
  23. A. K. Kar, B. S. Wherrett, “Thermal dispersive optical bistability and absorptive bistability in bulk ZnSe,” J. Opt. Soc. Am. B 3, 345–350 (1986).
    [CrossRef]
  24. M. Lawandy, D. V. Plant, D. L. MacFarlane, “Optical bistability in a dissipative thermally expanding etalon,” IEEE J. Quantum Electron. QE-21, 108–110 (1985).
    [CrossRef]
  25. B. S. Wherrett, F. A. P. Tooley, S. D. Smith, “Absorption switching and bistability in InSb,” Opt. Commun. 2, 301–306 (1984).
    [CrossRef]
  26. M. R. Taghizadeh, I. Janossy, S. D. Smith, “Optical bistability in bulk ZnSe due to increasing absorption and self-focussing,” Appl. Phys. Lett. 46, 331–333 (1985).
    [CrossRef]
  27. J. G. H. Mathew, M. R. Taghizadeh, Heriot-Watt University (personal communication).
  28. B. S. Wherrett, “Optical computer architecture—a design for tackling a specific physical problem,” Appl. Opt. 24, 2876–2883 (1985).
    [CrossRef]
  29. D. Hutchings, B. S. Wherrett, “Optimisation of a Fabry–Perot optically bistable device,” internal report, Physics Department, Heriot-Watt University, 1984 (unpublished).
  30. J. Hunter, A. C. Walker, “Room temperature 10.6 μ m thermal dispersive optical bistability in InSb,” internal report, Department of Physics, Heriot-Watt University, 1984 (unpublished).
  31. A. Miller, G. Parry, R. Daley, “Low power nonlinear Fabry–Perot reflection in CdHgTe at 10 μ m,” IEEE J. Quantum Electron. QE-20, 710–715 (1984).
    [CrossRef]

1986

1985

B. S. Wherrett, “Optical computer architecture—a design for tackling a specific physical problem,” Appl. Opt. 24, 2876–2883 (1985).
[CrossRef]

I. Janossy, M. R. Taghizadeh, J. G. H. Mathew, S. D. Smith, “Thermally induced optical bistability in thin film devices,” IEEE J. Quantum Electron. QE-21, 1447–1452 (1985).
[CrossRef]

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

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

M. Lawandy, D. V. Plant, D. L. MacFarlane, “Optical bistability in a dissipative thermally expanding etalon,” IEEE J. Quantum Electron. QE-21, 108–110 (1985).
[CrossRef]

M. R. Taghizadeh, I. Janossy, S. D. Smith, “Optical bistability in bulk ZnSe due to increasing absorption and self-focussing,” Appl. Phys. Lett. 46, 331–333 (1985).
[CrossRef]

1984

A. Miller, G. Parry, R. Daley, “Low power nonlinear Fabry–Perot reflection in CdHgTe at 10 μ m,” IEEE J. Quantum Electron. QE-20, 710–715 (1984).
[CrossRef]

B. S. Wherrett, “Scaling rules for multiphoton interband absorption in semiconductors,” J. Opt. Soc. Am. B 1, 67–72 (1984).
[CrossRef]

B. S. Wherrett, F. A. P. Tooley, S. D. Smith, “Absorption switching and bistability in InSb,” Opt. Commun. 2, 301–306 (1984).
[CrossRef]

B. S. Wherrett, “One-electron theory of nonlinear refraction,” Phil. Trans. R. Soc. London Ser. A 313, 213–220 (1984).
[CrossRef]

A. C. Walker, F. A. P. Tooley, M. E. Prise, J. G. H. Mathew, A. K. Kar, M. R. Taghizadeh, S. D. Smith, “InSb devices: transphasors with high gain, bistable switches and sequential logic gates,” Phil. Trans. R. Soc. London Ser. A 313, 249–256 (1984).
[CrossRef]

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

S. P. Apanasevich, F. V. Karpushko, G. V. Sinitsyn, “Response time of bistable devices based on evaporated thin-film interferometers,” Sov. J. Quantum Electron. 14, 873–874 (1984).
[CrossRef]

G. R. Olbright, N. Peyghambarian, H. M. Gibbs, H. A. MacLeod, F. van Milligen, “Microsecond room temperature optical bistability and crosstalk studies in ZnS and ZnSe interference filters with visible light and milliwatt power,” Appl. Phys. Lett. 45, 10–14 (1984).
[CrossRef]

B. S. Wherrett, “Fabry–Perot bistable cavity optimization—on reflection,” IEEE J. Quantum Electron. QE-20, 646–651 (1984).
[CrossRef]

S. D. Smith, J. G. H. Mathew, M. R. Taghizadeh, A. C. Walker, B. S. Wherrett, A. Hendry, “Room temperature, visible wavelength optical bistability in ZnSe interference filters,” Opt. Commun. 51, 357–361 (1984).
[CrossRef]

1982

B. S. Wherrett, N. A. Higgins, “Theory of nonlinear refraction near the band edge of a semiconductor,” Proc. R. Soc. London Ser. A 379, 67–90 (1982).
[CrossRef]

F. V. Karpushko, G. V. Sinitsyn, “The anomalous nonlinearity and optical bistability in thin-film interference structures,” Appl. Phys. B 28, 137 (1982).

1981

D. A. B. Miller, C. T. Seaton, M. E. Prise, S. D. Smith, “Band gap resonant nonlinear refraction in III–V semiconductors,” Phys. Rev. Lett. 47, 197–199 (1981).
[CrossRef]

D. A. B. Miller, “Refractive Fabry–Perot bistability with linear absorption: theory of operation and cavity optimisation,” IEEE J. Quantum Electron. QE-17, 306–311 (1981).
[CrossRef]

1978

F. V. Karpushko, G. V. Sinitsyn, “An optical logic element for integrated optics in a nonlinear semiconductor interferometer,” J. Appl. Spectrosc. USSR 29, 1323–1326 (1978).
[CrossRef]

1960

See, for example, T. P. McLean, Prog. Semiconduct. 5, 55–80 (1960).

Abraham, E.

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

Apanasevich, S. P.

S. P. Apanasevich, F. V. Karpushko, G. V. Sinitsyn, “Response time of bistable devices based on evaporated thin-film interferometers,” Sov. J. Quantum Electron. 14, 873–874 (1984).
[CrossRef]

Daley, R.

A. Miller, G. Parry, R. Daley, “Low power nonlinear Fabry–Perot reflection in CdHgTe at 10 μ m,” IEEE J. Quantum Electron. QE-20, 710–715 (1984).
[CrossRef]

Dwyer, V. M.

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

Firth, W. J.

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

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

Galbraith, I.

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

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

Gibbs, H. M.

G. R. Olbright, N. Peyghambarian, H. M. Gibbs, H. A. MacLeod, F. van Milligen, “Microsecond room temperature optical bistability and crosstalk studies in ZnS and ZnSe interference filters with visible light and milliwatt power,” Appl. Phys. Lett. 45, 10–14 (1984).
[CrossRef]

H. M. Gibbs, J. L. Jewell, Y. H. Lee, A. MacLeod, G. Olbright, S. Ovadia, N. Peyghambarian, M. C. Rushford, M. Warren, D. A. Weinberger, T. Venkatesan, “Prospects for parallel optical signal processing using GaAs etalons and ZnS interference filters,” presented at AGARD Conference on Digital Optical Circuit Technology, Schliersee, Federal Republic of Germany, 1984.

Hendry, A.

S. D. Smith, J. G. H. Mathew, M. R. Taghizadeh, A. C. Walker, B. S. Wherrett, A. Hendry, “Room temperature, visible wavelength optical bistability in ZnSe interference filters,” Opt. Commun. 51, 357–361 (1984).
[CrossRef]

Higgins, N. A.

B. S. Wherrett, N. A. Higgins, “Theory of nonlinear refraction near the band edge of a semiconductor,” Proc. R. Soc. London Ser. A 379, 67–90 (1982).
[CrossRef]

Hunter, J.

J. Hunter, A. C. Walker, “Room temperature 10.6 μ m thermal dispersive optical bistability in InSb,” internal report, Department of Physics, Heriot-Watt University, 1984 (unpublished).

Hutchings, D.

D. Hutchings, B. S. Wherrett, “Optimisation of a Fabry–Perot optically bistable device,” internal report, Physics Department, Heriot-Watt University, 1984 (unpublished).

D. Hutchings, B. S. Wherrett, “Analytical solutions for wave propagation through periodic structures,” internal report, Department of Physics (Heriot-Watt University, Edinburgh, Scotland, 1985) (unpublished).

Janossy, I.

I. Janossy, M. R. Taghizadeh, J. G. H. Mathew, S. D. Smith, “Thermally induced optical bistability in thin film devices,” IEEE J. Quantum Electron. QE-21, 1447–1452 (1985).
[CrossRef]

M. R. Taghizadeh, I. Janossy, S. D. Smith, “Optical bistability in bulk ZnSe due to increasing absorption and self-focussing,” Appl. Phys. Lett. 46, 331–333 (1985).
[CrossRef]

Jewell, J. L.

H. M. Gibbs, J. L. Jewell, Y. H. Lee, A. MacLeod, G. Olbright, S. Ovadia, N. Peyghambarian, M. C. Rushford, M. Warren, D. A. Weinberger, T. Venkatesan, “Prospects for parallel optical signal processing using GaAs etalons and ZnS interference filters,” presented at AGARD Conference on Digital Optical Circuit Technology, Schliersee, Federal Republic of Germany, 1984.

Kar, A. K.

A. K. Kar, B. S. Wherrett, “Thermal dispersive optical bistability and absorptive bistability in bulk ZnSe,” J. Opt. Soc. Am. B 3, 345–350 (1986).
[CrossRef]

A. C. Walker, F. A. P. Tooley, M. E. Prise, J. G. H. Mathew, A. K. Kar, M. R. Taghizadeh, S. D. Smith, “InSb devices: transphasors with high gain, bistable switches and sequential logic gates,” Phil. Trans. R. Soc. London Ser. A 313, 249–256 (1984).
[CrossRef]

Karpushko, F. V.

S. P. Apanasevich, F. V. Karpushko, G. V. Sinitsyn, “Response time of bistable devices based on evaporated thin-film interferometers,” Sov. J. Quantum Electron. 14, 873–874 (1984).
[CrossRef]

F. V. Karpushko, G. V. Sinitsyn, “The anomalous nonlinearity and optical bistability in thin-film interference structures,” Appl. Phys. B 28, 137 (1982).

F. V. Karpushko, G. V. Sinitsyn, “An optical logic element for integrated optics in a nonlinear semiconductor interferometer,” J. Appl. Spectrosc. USSR 29, 1323–1326 (1978).
[CrossRef]

Kermode, J. P.

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

Landolt-Börnstein,

Landolt-Börnstein, Numerical Data and Functional Relationships in Science and Technology (Springer-Verlag, Berlin, 1982), Group III, Vols. 17a and 17b.

Lawandy, M.

M. Lawandy, D. V. Plant, D. L. MacFarlane, “Optical bistability in a dissipative thermally expanding etalon,” IEEE J. Quantum Electron. QE-21, 108–110 (1985).
[CrossRef]

Lee, Y. H.

H. M. Gibbs, J. L. Jewell, Y. H. Lee, A. MacLeod, G. Olbright, S. Ovadia, N. Peyghambarian, M. C. Rushford, M. Warren, D. A. Weinberger, T. Venkatesan, “Prospects for parallel optical signal processing using GaAs etalons and ZnS interference filters,” presented at AGARD Conference on Digital Optical Circuit Technology, Schliersee, Federal Republic of Germany, 1984.

MacFarlane, D. L.

M. Lawandy, D. V. Plant, D. L. MacFarlane, “Optical bistability in a dissipative thermally expanding etalon,” IEEE J. Quantum Electron. QE-21, 108–110 (1985).
[CrossRef]

MacLeod, A.

H. M. Gibbs, J. L. Jewell, Y. H. Lee, A. MacLeod, G. Olbright, S. Ovadia, N. Peyghambarian, M. C. Rushford, M. Warren, D. A. Weinberger, T. Venkatesan, “Prospects for parallel optical signal processing using GaAs etalons and ZnS interference filters,” presented at AGARD Conference on Digital Optical Circuit Technology, Schliersee, Federal Republic of Germany, 1984.

MacLeod, H. A.

G. R. Olbright, N. Peyghambarian, H. M. Gibbs, H. A. MacLeod, F. van Milligen, “Microsecond room temperature optical bistability and crosstalk studies in ZnS and ZnSe interference filters with visible light and milliwatt power,” Appl. Phys. Lett. 45, 10–14 (1984).
[CrossRef]

Mathew, J. G. H.

I. Janossy, M. R. Taghizadeh, J. G. H. Mathew, S. D. Smith, “Thermally induced optical bistability in thin film devices,” IEEE J. Quantum Electron. QE-21, 1447–1452 (1985).
[CrossRef]

S. D. Smith, J. G. H. Mathew, M. R. Taghizadeh, A. C. Walker, B. S. Wherrett, A. Hendry, “Room temperature, visible wavelength optical bistability in ZnSe interference filters,” Opt. Commun. 51, 357–361 (1984).
[CrossRef]

A. C. Walker, F. A. P. Tooley, M. E. Prise, J. G. H. Mathew, A. K. Kar, M. R. Taghizadeh, S. D. Smith, “InSb devices: transphasors with high gain, bistable switches and sequential logic gates,” Phil. Trans. R. Soc. London Ser. A 313, 249–256 (1984).
[CrossRef]

S. D. Smith, F. A. P. Tooley, A. C. Walker, J. G. H. Mathew, M. R. Taghizadeh, B. S. Wherrett, “All-optical logic gates with external switching by laser and incoherent radiation,” presented at the AGARD Conference on Digital Optical Circuit Technology, Schliersee, Federal Republic of Germany, 1984.

J. G. H. Mathew, M. R. Taghizadeh, Heriot-Watt University (personal communication).

McLean, T. P.

See, for example, T. P. McLean, Prog. Semiconduct. 5, 55–80 (1960).

Miller, A.

A. Miller, G. Parry, R. Daley, “Low power nonlinear Fabry–Perot reflection in CdHgTe at 10 μ m,” IEEE J. Quantum Electron. QE-20, 710–715 (1984).
[CrossRef]

Miller, D. A. B.

D. A. B. Miller, C. T. Seaton, M. E. Prise, S. D. Smith, “Band gap resonant nonlinear refraction in III–V semiconductors,” Phys. Rev. Lett. 47, 197–199 (1981).
[CrossRef]

D. A. B. Miller, “Refractive Fabry–Perot bistability with linear absorption: theory of operation and cavity optimisation,” IEEE J. Quantum Electron. QE-17, 306–311 (1981).
[CrossRef]

Olbright, G.

H. M. Gibbs, J. L. Jewell, Y. H. Lee, A. MacLeod, G. Olbright, S. Ovadia, N. Peyghambarian, M. C. Rushford, M. Warren, D. A. Weinberger, T. Venkatesan, “Prospects for parallel optical signal processing using GaAs etalons and ZnS interference filters,” presented at AGARD Conference on Digital Optical Circuit Technology, Schliersee, Federal Republic of Germany, 1984.

Olbright, G. R.

G. R. Olbright, N. Peyghambarian, H. M. Gibbs, H. A. MacLeod, F. van Milligen, “Microsecond room temperature optical bistability and crosstalk studies in ZnS and ZnSe interference filters with visible light and milliwatt power,” Appl. Phys. Lett. 45, 10–14 (1984).
[CrossRef]

Ovadia, S.

H. M. Gibbs, J. L. Jewell, Y. H. Lee, A. MacLeod, G. Olbright, S. Ovadia, N. Peyghambarian, M. C. Rushford, M. Warren, D. A. Weinberger, T. Venkatesan, “Prospects for parallel optical signal processing using GaAs etalons and ZnS interference filters,” presented at AGARD Conference on Digital Optical Circuit Technology, Schliersee, Federal Republic of Germany, 1984.

Parry, G.

A. Miller, G. Parry, R. Daley, “Low power nonlinear Fabry–Perot reflection in CdHgTe at 10 μ m,” IEEE J. Quantum Electron. QE-20, 710–715 (1984).
[CrossRef]

Peyghambarian, N.

G. R. Olbright, N. Peyghambarian, H. M. Gibbs, H. A. MacLeod, F. van Milligen, “Microsecond room temperature optical bistability and crosstalk studies in ZnS and ZnSe interference filters with visible light and milliwatt power,” Appl. Phys. Lett. 45, 10–14 (1984).
[CrossRef]

H. M. Gibbs, J. L. Jewell, Y. H. Lee, A. MacLeod, G. Olbright, S. Ovadia, N. Peyghambarian, M. C. Rushford, M. Warren, D. A. Weinberger, T. Venkatesan, “Prospects for parallel optical signal processing using GaAs etalons and ZnS interference filters,” presented at AGARD Conference on Digital Optical Circuit Technology, Schliersee, Federal Republic of Germany, 1984.

Plant, D. V.

M. Lawandy, D. V. Plant, D. L. MacFarlane, “Optical bistability in a dissipative thermally expanding etalon,” IEEE J. Quantum Electron. QE-21, 108–110 (1985).
[CrossRef]

Prise, M. E.

A. C. Walker, F. A. P. Tooley, M. E. Prise, J. G. H. Mathew, A. K. Kar, M. R. Taghizadeh, S. D. Smith, “InSb devices: transphasors with high gain, bistable switches and sequential logic gates,” Phil. Trans. R. Soc. London Ser. A 313, 249–256 (1984).
[CrossRef]

D. A. B. Miller, C. T. Seaton, M. E. Prise, S. D. Smith, “Band gap resonant nonlinear refraction in III–V semiconductors,” Phys. Rev. Lett. 47, 197–199 (1981).
[CrossRef]

Rushford, M. C.

H. M. Gibbs, J. L. Jewell, Y. H. Lee, A. MacLeod, G. Olbright, S. Ovadia, N. Peyghambarian, M. C. Rushford, M. Warren, D. A. Weinberger, T. Venkatesan, “Prospects for parallel optical signal processing using GaAs etalons and ZnS interference filters,” presented at AGARD Conference on Digital Optical Circuit Technology, Schliersee, Federal Republic of Germany, 1984.

Seaton, C. T.

D. A. B. Miller, C. T. Seaton, M. E. Prise, S. D. Smith, “Band gap resonant nonlinear refraction in III–V semiconductors,” Phys. Rev. Lett. 47, 197–199 (1981).
[CrossRef]

Sinitsyn, G. V.

S. P. Apanasevich, F. V. Karpushko, G. V. Sinitsyn, “Response time of bistable devices based on evaporated thin-film interferometers,” Sov. J. Quantum Electron. 14, 873–874 (1984).
[CrossRef]

F. V. Karpushko, G. V. Sinitsyn, “The anomalous nonlinearity and optical bistability in thin-film interference structures,” Appl. Phys. B 28, 137 (1982).

F. V. Karpushko, G. V. Sinitsyn, “An optical logic element for integrated optics in a nonlinear semiconductor interferometer,” J. Appl. Spectrosc. USSR 29, 1323–1326 (1978).
[CrossRef]

Smith, S. D.

I. Janossy, M. R. Taghizadeh, J. G. H. Mathew, S. D. Smith, “Thermally induced optical bistability in thin film devices,” IEEE J. Quantum Electron. QE-21, 1447–1452 (1985).
[CrossRef]

M. R. Taghizadeh, I. Janossy, S. D. Smith, “Optical bistability in bulk ZnSe due to increasing absorption and self-focussing,” Appl. Phys. Lett. 46, 331–333 (1985).
[CrossRef]

B. S. Wherrett, F. A. P. Tooley, S. D. Smith, “Absorption switching and bistability in InSb,” Opt. Commun. 2, 301–306 (1984).
[CrossRef]

S. D. Smith, J. G. H. Mathew, M. R. Taghizadeh, A. C. Walker, B. S. Wherrett, A. Hendry, “Room temperature, visible wavelength optical bistability in ZnSe interference filters,” Opt. Commun. 51, 357–361 (1984).
[CrossRef]

A. C. Walker, F. A. P. Tooley, M. E. Prise, J. G. H. Mathew, A. K. Kar, M. R. Taghizadeh, S. D. Smith, “InSb devices: transphasors with high gain, bistable switches and sequential logic gates,” Phil. Trans. R. Soc. London Ser. A 313, 249–256 (1984).
[CrossRef]

D. A. B. Miller, C. T. Seaton, M. E. Prise, S. D. Smith, “Band gap resonant nonlinear refraction in III–V semiconductors,” Phys. Rev. Lett. 47, 197–199 (1981).
[CrossRef]

S. D. Smith, F. A. P. Tooley, A. C. Walker, J. G. H. Mathew, M. R. Taghizadeh, B. S. Wherrett, “All-optical logic gates with external switching by laser and incoherent radiation,” presented at the AGARD Conference on Digital Optical Circuit Technology, Schliersee, Federal Republic of Germany, 1984.

Taghizadeh, M. R.

I. Janossy, M. R. Taghizadeh, J. G. H. Mathew, S. D. Smith, “Thermally induced optical bistability in thin film devices,” IEEE J. Quantum Electron. QE-21, 1447–1452 (1985).
[CrossRef]

M. R. Taghizadeh, I. Janossy, S. D. Smith, “Optical bistability in bulk ZnSe due to increasing absorption and self-focussing,” Appl. Phys. Lett. 46, 331–333 (1985).
[CrossRef]

S. D. Smith, J. G. H. Mathew, M. R. Taghizadeh, A. C. Walker, B. S. Wherrett, A. Hendry, “Room temperature, visible wavelength optical bistability in ZnSe interference filters,” Opt. Commun. 51, 357–361 (1984).
[CrossRef]

A. C. Walker, F. A. P. Tooley, M. E. Prise, J. G. H. Mathew, A. K. Kar, M. R. Taghizadeh, S. D. Smith, “InSb devices: transphasors with high gain, bistable switches and sequential logic gates,” Phil. Trans. R. Soc. London Ser. A 313, 249–256 (1984).
[CrossRef]

S. D. Smith, F. A. P. Tooley, A. C. Walker, J. G. H. Mathew, M. R. Taghizadeh, B. S. Wherrett, “All-optical logic gates with external switching by laser and incoherent radiation,” presented at the AGARD Conference on Digital Optical Circuit Technology, Schliersee, Federal Republic of Germany, 1984.

J. G. H. Mathew, M. R. Taghizadeh, Heriot-Watt University (personal communication).

Tooley, F. A. P.

A. C. Walker, F. A. P. Tooley, M. E. Prise, J. G. H. Mathew, A. K. Kar, M. R. Taghizadeh, S. D. Smith, “InSb devices: transphasors with high gain, bistable switches and sequential logic gates,” Phil. Trans. R. Soc. London Ser. A 313, 249–256 (1984).
[CrossRef]

B. S. Wherrett, F. A. P. Tooley, S. D. Smith, “Absorption switching and bistability in InSb,” Opt. Commun. 2, 301–306 (1984).
[CrossRef]

S. D. Smith, F. A. P. Tooley, A. C. Walker, J. G. H. Mathew, M. R. Taghizadeh, B. S. Wherrett, “All-optical logic gates with external switching by laser and incoherent radiation,” presented at the AGARD Conference on Digital Optical Circuit Technology, Schliersee, Federal Republic of Germany, 1984.

van Milligen, F.

G. R. Olbright, N. Peyghambarian, H. M. Gibbs, H. A. MacLeod, F. van Milligen, “Microsecond room temperature optical bistability and crosstalk studies in ZnS and ZnSe interference filters with visible light and milliwatt power,” Appl. Phys. Lett. 45, 10–14 (1984).
[CrossRef]

Venkatesan, T.

H. M. Gibbs, J. L. Jewell, Y. H. Lee, A. MacLeod, G. Olbright, S. Ovadia, N. Peyghambarian, M. C. Rushford, M. Warren, D. A. Weinberger, T. Venkatesan, “Prospects for parallel optical signal processing using GaAs etalons and ZnS interference filters,” presented at AGARD Conference on Digital Optical Circuit Technology, Schliersee, Federal Republic of Germany, 1984.

Walker, A. C.

S. D. Smith, J. G. H. Mathew, M. R. Taghizadeh, A. C. Walker, B. S. Wherrett, A. Hendry, “Room temperature, visible wavelength optical bistability in ZnSe interference filters,” Opt. Commun. 51, 357–361 (1984).
[CrossRef]

A. C. Walker, F. A. P. Tooley, M. E. Prise, J. G. H. Mathew, A. K. Kar, M. R. Taghizadeh, S. D. Smith, “InSb devices: transphasors with high gain, bistable switches and sequential logic gates,” Phil. Trans. R. Soc. London Ser. A 313, 249–256 (1984).
[CrossRef]

S. D. Smith, F. A. P. Tooley, A. C. Walker, J. G. H. Mathew, M. R. Taghizadeh, B. S. Wherrett, “All-optical logic gates with external switching by laser and incoherent radiation,” presented at the AGARD Conference on Digital Optical Circuit Technology, Schliersee, Federal Republic of Germany, 1984.

J. Hunter, A. C. Walker, “Room temperature 10.6 μ m thermal dispersive optical bistability in InSb,” internal report, Department of Physics, Heriot-Watt University, 1984 (unpublished).

Warren, M.

H. M. Gibbs, J. L. Jewell, Y. H. Lee, A. MacLeod, G. Olbright, S. Ovadia, N. Peyghambarian, M. C. Rushford, M. Warren, D. A. Weinberger, T. Venkatesan, “Prospects for parallel optical signal processing using GaAs etalons and ZnS interference filters,” presented at AGARD Conference on Digital Optical Circuit Technology, Schliersee, Federal Republic of Germany, 1984.

Weaire, D.

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

Weinberger, D. A.

H. M. Gibbs, J. L. Jewell, Y. H. Lee, A. MacLeod, G. Olbright, S. Ovadia, N. Peyghambarian, M. C. Rushford, M. Warren, D. A. Weinberger, T. Venkatesan, “Prospects for parallel optical signal processing using GaAs etalons and ZnS interference filters,” presented at AGARD Conference on Digital Optical Circuit Technology, Schliersee, Federal Republic of Germany, 1984.

Wherrett, B. S.

A. K. Kar, B. S. Wherrett, “Thermal dispersive optical bistability and absorptive bistability in bulk ZnSe,” J. Opt. Soc. Am. B 3, 345–350 (1986).
[CrossRef]

B. S. Wherrett, “Optical computer architecture—a design for tackling a specific physical problem,” Appl. Opt. 24, 2876–2883 (1985).
[CrossRef]

B. S. Wherrett, “Scaling rules for multiphoton interband absorption in semiconductors,” J. Opt. Soc. Am. B 1, 67–72 (1984).
[CrossRef]

S. D. Smith, J. G. H. Mathew, M. R. Taghizadeh, A. C. Walker, B. S. Wherrett, A. Hendry, “Room temperature, visible wavelength optical bistability in ZnSe interference filters,” Opt. Commun. 51, 357–361 (1984).
[CrossRef]

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

B. S. Wherrett, F. A. P. Tooley, S. D. Smith, “Absorption switching and bistability in InSb,” Opt. Commun. 2, 301–306 (1984).
[CrossRef]

B. S. Wherrett, “One-electron theory of nonlinear refraction,” Phil. Trans. R. Soc. London Ser. A 313, 213–220 (1984).
[CrossRef]

B. S. Wherrett, “Fabry–Perot bistable cavity optimization—on reflection,” IEEE J. Quantum Electron. QE-20, 646–651 (1984).
[CrossRef]

B. S. Wherrett, N. A. Higgins, “Theory of nonlinear refraction near the band edge of a semiconductor,” Proc. R. Soc. London Ser. A 379, 67–90 (1982).
[CrossRef]

D. Hutchings, B. S. Wherrett, “Analytical solutions for wave propagation through periodic structures,” internal report, Department of Physics (Heriot-Watt University, Edinburgh, Scotland, 1985) (unpublished).

B. S. Wherrett, “Optical characteristics of nonlinear, active multi-layer dielectric thin-film structures,” internal report, Department of Physics (Heriot-Watt University, Edinburgh, Scotland, 1985)(unpublished).

S. D. Smith, F. A. P. Tooley, A. C. Walker, J. G. H. Mathew, M. R. Taghizadeh, B. S. Wherrett, “All-optical logic gates with external switching by laser and incoherent radiation,” presented at the AGARD Conference on Digital Optical Circuit Technology, Schliersee, Federal Republic of Germany, 1984.

D. Hutchings, B. S. Wherrett, “Optimisation of a Fabry–Perot optically bistable device,” internal report, Physics Department, Heriot-Watt University, 1984 (unpublished).

Wright, E. M.

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

Appl. Opt.

Appl. Phys. B

F. V. Karpushko, G. V. Sinitsyn, “The anomalous nonlinearity and optical bistability in thin-film interference structures,” Appl. Phys. B 28, 137 (1982).

Appl. Phys. Lett.

G. R. Olbright, N. Peyghambarian, H. M. Gibbs, H. A. MacLeod, F. van Milligen, “Microsecond room temperature optical bistability and crosstalk studies in ZnS and ZnSe interference filters with visible light and milliwatt power,” Appl. Phys. Lett. 45, 10–14 (1984).
[CrossRef]

M. R. Taghizadeh, I. Janossy, S. D. Smith, “Optical bistability in bulk ZnSe due to increasing absorption and self-focussing,” Appl. Phys. Lett. 46, 331–333 (1985).
[CrossRef]

IEEE J. Quantum Electron.

M. Lawandy, D. V. Plant, D. L. MacFarlane, “Optical bistability in a dissipative thermally expanding etalon,” IEEE J. Quantum Electron. QE-21, 108–110 (1985).
[CrossRef]

A. Miller, G. Parry, R. Daley, “Low power nonlinear Fabry–Perot reflection in CdHgTe at 10 μ m,” IEEE J. Quantum Electron. QE-20, 710–715 (1984).
[CrossRef]

I. Janossy, M. R. Taghizadeh, J. G. H. Mathew, S. D. Smith, “Thermally induced optical bistability in thin film devices,” IEEE J. Quantum Electron. QE-21, 1447–1452 (1985).
[CrossRef]

D. A. B. Miller, “Refractive Fabry–Perot bistability with linear absorption: theory of operation and cavity optimisation,” IEEE J. Quantum Electron. QE-17, 306–311 (1981).
[CrossRef]

B. S. Wherrett, “Fabry–Perot bistable cavity optimization—on reflection,” IEEE J. Quantum Electron. QE-20, 646–651 (1984).
[CrossRef]

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

J. Appl. Spectrosc. USSR

F. V. Karpushko, G. V. Sinitsyn, “An optical logic element for integrated optics in a nonlinear semiconductor interferometer,” J. Appl. Spectrosc. USSR 29, 1323–1326 (1978).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Commun.

B. S. Wherrett, F. A. P. Tooley, S. D. Smith, “Absorption switching and bistability in InSb,” Opt. Commun. 2, 301–306 (1984).
[CrossRef]

S. D. Smith, J. G. H. Mathew, M. R. Taghizadeh, A. C. Walker, B. S. Wherrett, A. Hendry, “Room temperature, visible wavelength optical bistability in ZnSe interference filters,” Opt. Commun. 51, 357–361 (1984).
[CrossRef]

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

Phil. Trans. R. Soc. London Ser. A

B. S. Wherrett, “One-electron theory of nonlinear refraction,” Phil. Trans. R. Soc. London Ser. A 313, 213–220 (1984).
[CrossRef]

A. C. Walker, F. A. P. Tooley, M. E. Prise, J. G. H. Mathew, A. K. Kar, M. R. Taghizadeh, S. D. Smith, “InSb devices: transphasors with high gain, bistable switches and sequential logic gates,” Phil. Trans. R. Soc. London Ser. A 313, 249–256 (1984).
[CrossRef]

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

Phys. Rev. Lett.

D. A. B. Miller, C. T. Seaton, M. E. Prise, S. D. Smith, “Band gap resonant nonlinear refraction in III–V semiconductors,” Phys. Rev. Lett. 47, 197–199 (1981).
[CrossRef]

Proc. R. Soc. London Ser. A

B. S. Wherrett, N. A. Higgins, “Theory of nonlinear refraction near the band edge of a semiconductor,” Proc. R. Soc. London Ser. A 379, 67–90 (1982).
[CrossRef]

Prog. Semiconduct.

See, for example, T. P. McLean, Prog. Semiconduct. 5, 55–80 (1960).

Sov. J. Quantum Electron.

S. P. Apanasevich, F. V. Karpushko, G. V. Sinitsyn, “Response time of bistable devices based on evaporated thin-film interferometers,” Sov. J. Quantum Electron. 14, 873–874 (1984).
[CrossRef]

Other

S. D. Smith, F. A. P. Tooley, A. C. Walker, J. G. H. Mathew, M. R. Taghizadeh, B. S. Wherrett, “All-optical logic gates with external switching by laser and incoherent radiation,” presented at the AGARD Conference on Digital Optical Circuit Technology, Schliersee, Federal Republic of Germany, 1984.

H. M. Gibbs, J. L. Jewell, Y. H. Lee, A. MacLeod, G. Olbright, S. Ovadia, N. Peyghambarian, M. C. Rushford, M. Warren, D. A. Weinberger, T. Venkatesan, “Prospects for parallel optical signal processing using GaAs etalons and ZnS interference filters,” presented at AGARD Conference on Digital Optical Circuit Technology, Schliersee, Federal Republic of Germany, 1984.

D. Hutchings, B. S. Wherrett, “Analytical solutions for wave propagation through periodic structures,” internal report, Department of Physics (Heriot-Watt University, Edinburgh, Scotland, 1985) (unpublished).

B. S. Wherrett, “Optical characteristics of nonlinear, active multi-layer dielectric thin-film structures,” internal report, Department of Physics (Heriot-Watt University, Edinburgh, Scotland, 1985)(unpublished).

Landolt-Börnstein, Numerical Data and Functional Relationships in Science and Technology (Springer-Verlag, Berlin, 1982), Group III, Vols. 17a and 17b.

J. G. H. Mathew, M. R. Taghizadeh, Heriot-Watt University (personal communication).

D. Hutchings, B. S. Wherrett, “Optimisation of a Fabry–Perot optically bistable device,” internal report, Physics Department, Heriot-Watt University, 1984 (unpublished).

J. Hunter, A. C. Walker, “Room temperature 10.6 μ m thermal dispersive optical bistability in InSb,” internal report, Department of Physics, Heriot-Watt University, 1984 (unpublished).

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Figures (10)

Fig. 1
Fig. 1

The cavity factor f(RF, RB, αD) in expressions (4). Plots are for equal end-face reflectivities, RF = RB = 0.2, 0.6, 0.95, and 0.99, respectively. The curves indicate optimum αD values for minimizing the bistable critical irradiance under electronic nonlinearity conditions. The minimum values decrease as the cavity reflectivity is improved.

Fig. 2
Fig. 2

The cavity factor f/αD in expression (8), as a function of the front- and back-face reflectivities RF, RB (=0 to 1) for two spacer thicknesses such that αD = 1, 0.1. The maximum values of f/αD plotted are equal to 10 in both plots.

Fig. 3
Fig. 3

Stack reflectivities of ZnSe/ThF4 systems with material indices taken as 2.7 and 1.5, respectively. The left-hand scale refers to the dots, which show in detail the modifications in the presence of the ZnSe layer absorption for M = 3, 4, 5. Uppermost dots refer to zero absorption, through 200, 600, and 1000 cm−1 to 2000 cm−1 for the lowest dots in each M set.

Fig. 4
Fig. 4

(a) The cavity factor f/αD, applicable for thermal nonlinear refraction [expression (8)]. Plots are for equal end-face reflectivities as applicable to ZnSe interference filters with M layer-pair stacks. The optimum αD for each case gives a similar critical irradiance (apart from the natural reflectivity case, M = 0). (b) As for (a) but with a highly reflecting back face on the spacer. Such a system is more suitable for high-signal-difference switching, using reflection.10

Fig. 5
Fig. 5

Differential phase changes for reflective stacks as appropriate to (a) passive stacks tuned by varying the radiation wavelength and (b) active stacks tuned by varying the refractive coefficient of the high-index material alone. ϕM is the phase of each high-index layer in the stack.

Fig. 6
Fig. 6

Examples of interference filter optical bistability character. Irradiance levels are scaled to the inverse of the effective istics. n2T coefficient [expression (8)]. In each case the initial detuning is set at 0.5% (≃2.5 nm) away from resonance. Spacer/stack parameters correspond to (top) ZnSe/ZnSe, (middle) ZnSe/ZnS, and (bottom) ZnS/ZnSe with 2λ spacers and M = 3 stacks. The relative change of phase in the ZnSe layers is 2.5 × 10−4 between each dot. Heavier dots are markers for relative changes of 2.5 × 10−3. The bottom figure shows that for similar irradiance levels the phase change in the ZnS/ZnSe model system is dramatically lower.

Fig. 7
Fig. 7

Frequency dependence of the factor g that characterizes the coefficient ∂n/∂Eg, full line. The dashed line indicates the shape of the band edge. See also Figs. 8(a) and 8(b).

Fig. 8
Fig. 8

Empirical results for ZnSe, indicating the optimum operational wavelength for low-power optical bistability. For each wavelength the spacer optical thickness is taken to be 2λv. The filter stacks, with M = 3, are designed for peak transmission of the passive cavity.

Fig. 9
Fig. 9

Frequency optimization of the material factor in expression (8), based on the empirical edge in Fig. 8(a). Given an optimum f/αD value of 2.6, Fc ≃ 0.005 (eV) under the combined optimization condition.

Fig. 10
Fig. 10

(a) Critical detuning δc, as a function of cavity thickness for ZnSe systems and (b) the cavity factor δc/αD in the temperature rise at switch point (see text for details).

Equations (56)

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δ c π 3 2 F .
δ 2 π n 2 λ v I D ,
I c = λ v α 2 π n 2 f ( R F , R B , α D ) .
I c c e 2 [ n 2 E g 2 P 2 T 1 ] k B T J - 1 ( k B T E g - ω ) f ( R F , R B , α D ) .
P c m 3 c 3 e 2 n 2 P 2 T 1 k B T f J .
Δ n ( n / T ) Δ T .
Δ T α I D κ s D s .
Δ T α I D κ s r 0 ,
n 2 T ( n / T ) ( α D D / κ s ) ,
I c λ v α κ s 2 π ( n / T ) D [ f ( R F , R B , α D ) α D ] .
R M = | n n H 2 M - n s n L 2 M n n H 2 M + n s n L 2 M | 2 .
α D ( 1 - R F ) + ( 1 - R B ) ,
2 π n 0 D / λ v = N π - δ c .
Δ ϕ s = ( ϕ F + ϕ B ) ω ( ω - ω N ) .
ω n 2 D / c = 2 N π + ( ω - ω N ) n 0 2 D / c + ω Δ n 2 D / c .
2 N π + ( ω - ω N ) [ n 0 2 D / c + ( ϕ F + ϕ B ) / ω ] = 2 N π - 2 δ c .
Δ D p = c n 0 1 2 ( ϕ F + ϕ B ) ω .
Δ ϕ s ( ϕ F + ϕ B ) ω ( ω - ω N ) + ( ϕ F + ϕ B ) n H Δ n H .
Δ D a = Δ n H Δ n c ω 1 2 ( ϕ F + ϕ B ) n H .
Δ D p D H = n H n 0 1 2 [ ( ϕ F + ϕ B ) ϕ H ] p ;             Δ D a D H = Δ n H Δ n 1 2 [ ( ϕ F + ϕ B ) ϕ H ] a .
2 ϕ = 2 N π + 2 N π ( ω - ω N ) ω { 1 + 1 4 N [ ( ϕ F + ϕ B ) ϕ ] p } + 2 N π Δ n n 0 { 1 + 1 4 N n 0 Δ n H n H Δ n [ ( ϕ F + ϕ B ) ϕ ] a } .
R = Δ D a D n H / T n 0 / T = D H D 1 2 [ ( ϕ F + ϕ B ) ϕ H ] a n H / T n / T .
n T = n E g E g T + ( n T ) b .
α ( ω , E g ) = 2 e 2 1 / 2 n m 2 c ω ( 2 m r 2 ) 3 / 2 · p c v 2 ( ω - ω g ) 1 / 2 .
α = C 1 e 2 c E g n 0 P ( ω / E g - 1 ) 1 / 2 ω / E g .
Δ n = c π 0 α ( E g + Δ E g ) - α ( E g ) ( ω ) 2 - ω 2 d ω .
n E g = C 1 e 2 n 0 P E g g ( ω E g ) .
g ( x ) = Re [ 4 ( 2 - 1 + x - 1 - x ) + ( 1 / 1 + x - 1 / 1 - x ) x ] / 2 x 2 .
I c λ v α tail n / T λ v { α g exp [ ( ω - E g ) / E T ] + α b } [ ( E g / T ) ( n / E g ) + ( n + T ) b ] .
I c ( a ) λ v α b κ ( n / T ) b D ( f α D ) min ,
I c ( b ) λ v κ ( E g / T ) D [ e 2 c E g n 0 P g 1 ( ω E g , E T E g ) ] × [ e 2 n 0 P E g g ( ω E g , E T E g ) ] - 1 ( f α D ) min .
I c ( a ) ( ω ) - 1 ,             with ω E g ,
I c ( b ) E g g 2 ( ω / E g , E T / E g ) .
Δ T α I r 0 2 / κ ,
n 2 T ( n / T ) ( α r 0 2 / κ ) ,
I c λ v κ ( n / T ) r 0 2 f ( R F , R B , α D ) .
α D ( 2 - R F - R B ) / 4.
P c m c e 2 κ E g / T n 0 P f ( R F , R B , α D ) g ( ω / E g ) .
Δ ( n D ) = n Δ D + D Δ n ,
Δ ( n D ) n D = ( 1 D D T ) Δ T + ( 1 n n T ) Δ T + ( 1 n n n ) Δ N .
Δ T α I κ A ,             Δ N α i I T 1 ω ,
| n T A c κ | | n n T 1 ω | = | n 2 α | .
A c | n 2 κ α ( n / T ) | E g - 3 .
ω Δ n D / c = δ c .
Δ T α ( δ c / α D ) .
P c m λ v κ s ( n / T ) α r 0 .
Δ T m = λ c α ( n / T ) .
P c m [ λ v κ ( n / T ) ] f m .
F = π F 1 / 2 / 2.
δ c = 2 4 [ 3 ( F + 2 ) - d ] [ ( F + 2 ) d - ( F + 2 ) 2 - 2 F 2 ] 1 / 2 + sin - 1 [ ( 3 F + 2 - d ) 1 / 2 2 F 1 / 2 ] ,
δ c ( 3 / F ) 1 / 2 .
f = ( 1 - R α ) 2 ( 1 - R F ) ( 1 + R B e - α D ) ( 1 - e - α D ) × 2 16 [ 3 ( F + 2 ) - d ] 2 [ ( F + 2 ) d - ( F + 2 ) 2 - 2 F 2 ] 1 / 2
a = ( n 2 H + n 2 L ) / 2 n H n L , A = α H D H / 2 , b = [ ( a cosh A ) 2 - 1 ] 1 / 2 , λ ± = a cosh A ± b , r ± = exp ( A ) 2 n H n L ( n H 2 - n L 2 ) ( - a sinh A ± b ) , r s = ( n H - n S / ( n H + n S ) .
R M α = r + ( r s - r - ) λ - M - r - ( r s - r + ) λ + M ( r s - r - ) λ - M - ( r s - r + ) λ + M .
ϕ M ϕ H | p = 2 [ n L n H 4 M + 2 - n L n 2 M H ( n L n H + n S 2 ) 2 M + 1 + n S n 2 H n L 4 M ] ( n H - n L ) ( n H 4 M + 2 - n L n 4 M S 2 ) 2 n L / ( n H - n L ) for large M .
ϕ M ϕ H | a = 2 [ n L n 2 H 4 M + 2 - n L n 2 M H ( n L 2 + n S 2 ) 2 M + 2 + n S n 2 H n 2 L 4 M ] ( n H 2 - n L 2 ) ( n H 4 M + 2 - n L n 4 M S 2 ) 2 n L 2 / ( n H 2 - n L 2 ) for large M .

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