Abstract

A general formalism is developed to study intracavity phase conjugation through resonant degenerate four-wave mixing. The effects of spatial hole burning, arising from interference of the counterpropagating pump waves, are fully incorporated. The analysis is applied to discuss intracavity phase conjugation in nonlinear media that can be modeled using either a two-level system or a Λ-type three-level system. In both cases, the phase-conjugate reflectivity exhibits bistability for an appropriate choice of the input parameters. For a two-level system, the tuning characteristics of the Fabry–Perot cavity significantly affect the phase-conjugate spectrum that displays hysteresis when the cavity is detuned from the applied laser frequency.

© 1983 Optical Society of America

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  1. For a review see, for example, D. M. Pepper, “Nonlinear optical phase conjugation,” Opt. Eng. 21, 155–183 (1982); see also R. A. Fisher, ed., Optical Phase Conjugation (Academic, New York, 1983).
    [Crossref]
  2. P. F. Liao, D. M. Bloom, and N. P. Economou, “Cw optical wave-front conjugation by saturated absorption in atomic sodium vapor,” Appl. Phys. Lett. 32, 813–815 (1978).
    [Crossref]
  3. R. L. Abrams and R. C. Lind, “Degenerate four-wave mixing in absorbing media,” Opt. Lett. 2, 94–96 (1978); errata, Opt. Lett.3, 205 (1978).
    [Crossref] [PubMed]
  4. T. Fu and M. Sargent, “Theory of two-photon phase conjugation,” Opt. Lett. 5, 433–435 (1980).
    [Crossref] [PubMed]
  5. G. P. Agrawal, “Phase conjugation and degenerate four-wave mixing in three-level systems,” IEEE J. Quantum Electron. QE-17, 2335–2340 (1981).
    [Crossref]
  6. In some of the early experiments on phase conjugation, the laser cavity of the pump wave itself was used. See E. F. Bergmann, I. J. Bigio, B. J. Feldman, and R. A. Fisher, “High-efficiency pulsed 10.6-μ m phase-conjugate reflection via degenerate four-wave mixing,” Opt. Lett. 3, 82–84 (1978).
    [Crossref]
  7. H. M. Gibbs, S. L. McCall, and T. N. C. Venkatesan, “Differential gain and bistability using a sodium-filled Fabry–Perot interferometer,” Phys. Rev. Lett. 36, 1135–1138 (1976).
    [Crossref]
  8. R. Bonifacio and L. A. Lugiato, “Optical bistability and cooperative effects in resonance fluorescence,” Phys. Rev. A 18, 1129–1144 (1978).
    [Crossref]
  9. G. P. Agrawal and H. J. Carmichael, “Optical bistability through nonlinear dispersion and absorption,” Phys. Rev. A 19, 2074–2086 (1979).
    [Crossref]
  10. G. P. Agrawal and C. Flytzanis, “Bistability and hysteresis in phase-conjugated reflectivity,” IEEE J. Quantum Electron. QE-14, 374–380 (1981).
    [Crossref]
  11. G. P. Agrawal, C. Flytzanis, R. Frey, and F. Pradere, “Bistable reflectivity of phase-conjugated signal through intracavity degenerate four-wave mixing,” Appl. Phys. Lett. 38, 492–494 (1981).
    [Crossref]
  12. L. Fu-Li, J. A. Hermann, and J. N. Elgin, “Effects of two-photon optical bistability upon phase conjugation,” Opt. Commun. 40, 446–450 (1982).
    [Crossref]
  13. C. Flytzanis and C. L. Tang, “Light-induced critical behavior in four-wave interaction in nonlinear systems,” Phys. Rev. Lett. 45, 441–444 (1980).
    [Crossref]
  14. H. G. Winful and J. H. Marburger, “Hysteresis and optical bistability in degenerate four-wave mixing,” Appl. Phys. Lett. 36, 613–614 (1980).
    [Crossref]
  15. C. Flytzanis, G. P. Agrawal, and C. L. Tang, “Critical behavior in optical phase conjugation,” in Lasers and Their Applications, W. O. N. Guimaraes, C.-T. Lin, and A. Mooradian, eds. (Springer-Verlag, Berlin, 1981).
    [Crossref]
  16. A. Borshch, M. Brodin, V. Volkov, and N. Kukhtarev, “Optical bistability and hysteresis in phase conjugation by degenerate six-photon mixing,” Opt. Commun. 41, 213–215 (1982).
    [Crossref]
  17. G. P. Agrawal and H. J. Carmichael, “Inhomogeneous broadening and the mean-field approximation for optical bistability in a Fabry–Perot,” Opt. Acta. 27, 651–660 (1980).
    [Crossref]
  18. R. K. Raj, D. Bloch, J. J. Snyder, G. Camy, and M. Ducloy, “High-frequency optically heterodyned saturation spectroscopy via resonant degenerate four-wave mixing,” Phys. Rev. Lett. 44, 1251–1254 (1980).
    [Crossref]
  19. P. Aubourg, J. P. Bettini, G. P. Agrawal, P. Cottin, D. Guérin, O. Meunier, and J. L. Boulnois, “Doppler-free cw phase-conjugate spectrum of SF6by resonant degenerate four-wave mixing at 10.6 μ m,” Opt. Lett. 6, 383–385 (1981).
    [Crossref] [PubMed]
  20. D. G. Steel and J. F. Lam, “Saturation effects and inhomogeneous broadening in Doppler-free degenerate four-wave mixing,” Opt. Commun. 40, 77–80 (1981).
    [Crossref]
  21. For the case of pulsed radiation, the analysis can be applied if the medium-response time is shorter than the duration of the optical pulse.
  22. In the literature on optical bistability, this is often referred to as the mean-field approximation (see Refs. 8, 9, and 17).
  23. G. P. Agrawal, J. L. Boulnois, P. Aubourg, and A. van Lerberghe, “Saturation splitting in the spectrum of resonant degenerate four-wave mixing,” Opt. Lett. 7, 540–542 (1982).
    [Crossref] [PubMed]
  24. D. F. Walls and P. Zoller, “A coherent nonlinear mechanism for optical bistability from three-level atoms,” Opt. Commun. 34, 260–264 (1980).
    [Crossref]
  25. R. G. Brewer and E. L. Hahn, “Coherent two-photon processes: transient and steady-state cases,” Phys. Rev. A 11, 1641–1649 (1975).
    [Crossref]
  26. H. R. Gray, R. M. Whitley, and C. R. Stroud, “Coherent trapping of atomic populations,” Opt. Lett. 3, 218–220 (1978).
    [Crossref] [PubMed]
  27. It is not appropriate to use the term reflectivity spectrum because the laser frequency is kept fixed.
  28. J. Mlynek, F. Mitschke, R. Deserno, and W. Lange, “Optical bistability by transverse optical pumping,” Appl. Phys. B 28, 135 (1982).

1982 (5)

For a review see, for example, D. M. Pepper, “Nonlinear optical phase conjugation,” Opt. Eng. 21, 155–183 (1982); see also R. A. Fisher, ed., Optical Phase Conjugation (Academic, New York, 1983).
[Crossref]

L. Fu-Li, J. A. Hermann, and J. N. Elgin, “Effects of two-photon optical bistability upon phase conjugation,” Opt. Commun. 40, 446–450 (1982).
[Crossref]

A. Borshch, M. Brodin, V. Volkov, and N. Kukhtarev, “Optical bistability and hysteresis in phase conjugation by degenerate six-photon mixing,” Opt. Commun. 41, 213–215 (1982).
[Crossref]

G. P. Agrawal, J. L. Boulnois, P. Aubourg, and A. van Lerberghe, “Saturation splitting in the spectrum of resonant degenerate four-wave mixing,” Opt. Lett. 7, 540–542 (1982).
[Crossref] [PubMed]

J. Mlynek, F. Mitschke, R. Deserno, and W. Lange, “Optical bistability by transverse optical pumping,” Appl. Phys. B 28, 135 (1982).

1981 (5)

P. Aubourg, J. P. Bettini, G. P. Agrawal, P. Cottin, D. Guérin, O. Meunier, and J. L. Boulnois, “Doppler-free cw phase-conjugate spectrum of SF6by resonant degenerate four-wave mixing at 10.6 μ m,” Opt. Lett. 6, 383–385 (1981).
[Crossref] [PubMed]

D. G. Steel and J. F. Lam, “Saturation effects and inhomogeneous broadening in Doppler-free degenerate four-wave mixing,” Opt. Commun. 40, 77–80 (1981).
[Crossref]

G. P. Agrawal, “Phase conjugation and degenerate four-wave mixing in three-level systems,” IEEE J. Quantum Electron. QE-17, 2335–2340 (1981).
[Crossref]

G. P. Agrawal and C. Flytzanis, “Bistability and hysteresis in phase-conjugated reflectivity,” IEEE J. Quantum Electron. QE-14, 374–380 (1981).
[Crossref]

G. P. Agrawal, C. Flytzanis, R. Frey, and F. Pradere, “Bistable reflectivity of phase-conjugated signal through intracavity degenerate four-wave mixing,” Appl. Phys. Lett. 38, 492–494 (1981).
[Crossref]

1980 (6)

T. Fu and M. Sargent, “Theory of two-photon phase conjugation,” Opt. Lett. 5, 433–435 (1980).
[Crossref] [PubMed]

D. F. Walls and P. Zoller, “A coherent nonlinear mechanism for optical bistability from three-level atoms,” Opt. Commun. 34, 260–264 (1980).
[Crossref]

G. P. Agrawal and H. J. Carmichael, “Inhomogeneous broadening and the mean-field approximation for optical bistability in a Fabry–Perot,” Opt. Acta. 27, 651–660 (1980).
[Crossref]

R. K. Raj, D. Bloch, J. J. Snyder, G. Camy, and M. Ducloy, “High-frequency optically heterodyned saturation spectroscopy via resonant degenerate four-wave mixing,” Phys. Rev. Lett. 44, 1251–1254 (1980).
[Crossref]

C. Flytzanis and C. L. Tang, “Light-induced critical behavior in four-wave interaction in nonlinear systems,” Phys. Rev. Lett. 45, 441–444 (1980).
[Crossref]

H. G. Winful and J. H. Marburger, “Hysteresis and optical bistability in degenerate four-wave mixing,” Appl. Phys. Lett. 36, 613–614 (1980).
[Crossref]

1979 (1)

G. P. Agrawal and H. J. Carmichael, “Optical bistability through nonlinear dispersion and absorption,” Phys. Rev. A 19, 2074–2086 (1979).
[Crossref]

1978 (5)

1976 (1)

H. M. Gibbs, S. L. McCall, and T. N. C. Venkatesan, “Differential gain and bistability using a sodium-filled Fabry–Perot interferometer,” Phys. Rev. Lett. 36, 1135–1138 (1976).
[Crossref]

1975 (1)

R. G. Brewer and E. L. Hahn, “Coherent two-photon processes: transient and steady-state cases,” Phys. Rev. A 11, 1641–1649 (1975).
[Crossref]

Abrams, R. L.

Agrawal, G. P.

G. P. Agrawal, J. L. Boulnois, P. Aubourg, and A. van Lerberghe, “Saturation splitting in the spectrum of resonant degenerate four-wave mixing,” Opt. Lett. 7, 540–542 (1982).
[Crossref] [PubMed]

P. Aubourg, J. P. Bettini, G. P. Agrawal, P. Cottin, D. Guérin, O. Meunier, and J. L. Boulnois, “Doppler-free cw phase-conjugate spectrum of SF6by resonant degenerate four-wave mixing at 10.6 μ m,” Opt. Lett. 6, 383–385 (1981).
[Crossref] [PubMed]

G. P. Agrawal, “Phase conjugation and degenerate four-wave mixing in three-level systems,” IEEE J. Quantum Electron. QE-17, 2335–2340 (1981).
[Crossref]

G. P. Agrawal and C. Flytzanis, “Bistability and hysteresis in phase-conjugated reflectivity,” IEEE J. Quantum Electron. QE-14, 374–380 (1981).
[Crossref]

G. P. Agrawal, C. Flytzanis, R. Frey, and F. Pradere, “Bistable reflectivity of phase-conjugated signal through intracavity degenerate four-wave mixing,” Appl. Phys. Lett. 38, 492–494 (1981).
[Crossref]

G. P. Agrawal and H. J. Carmichael, “Inhomogeneous broadening and the mean-field approximation for optical bistability in a Fabry–Perot,” Opt. Acta. 27, 651–660 (1980).
[Crossref]

G. P. Agrawal and H. J. Carmichael, “Optical bistability through nonlinear dispersion and absorption,” Phys. Rev. A 19, 2074–2086 (1979).
[Crossref]

C. Flytzanis, G. P. Agrawal, and C. L. Tang, “Critical behavior in optical phase conjugation,” in Lasers and Their Applications, W. O. N. Guimaraes, C.-T. Lin, and A. Mooradian, eds. (Springer-Verlag, Berlin, 1981).
[Crossref]

Aubourg, P.

Bergmann, E. F.

Bettini, J. P.

Bigio, I. J.

Bloch, D.

R. K. Raj, D. Bloch, J. J. Snyder, G. Camy, and M. Ducloy, “High-frequency optically heterodyned saturation spectroscopy via resonant degenerate four-wave mixing,” Phys. Rev. Lett. 44, 1251–1254 (1980).
[Crossref]

Bloom, D. M.

P. F. Liao, D. M. Bloom, and N. P. Economou, “Cw optical wave-front conjugation by saturated absorption in atomic sodium vapor,” Appl. Phys. Lett. 32, 813–815 (1978).
[Crossref]

Bonifacio, R.

R. Bonifacio and L. A. Lugiato, “Optical bistability and cooperative effects in resonance fluorescence,” Phys. Rev. A 18, 1129–1144 (1978).
[Crossref]

Borshch, A.

A. Borshch, M. Brodin, V. Volkov, and N. Kukhtarev, “Optical bistability and hysteresis in phase conjugation by degenerate six-photon mixing,” Opt. Commun. 41, 213–215 (1982).
[Crossref]

Boulnois, J. L.

Brewer, R. G.

R. G. Brewer and E. L. Hahn, “Coherent two-photon processes: transient and steady-state cases,” Phys. Rev. A 11, 1641–1649 (1975).
[Crossref]

Brodin, M.

A. Borshch, M. Brodin, V. Volkov, and N. Kukhtarev, “Optical bistability and hysteresis in phase conjugation by degenerate six-photon mixing,” Opt. Commun. 41, 213–215 (1982).
[Crossref]

Camy, G.

R. K. Raj, D. Bloch, J. J. Snyder, G. Camy, and M. Ducloy, “High-frequency optically heterodyned saturation spectroscopy via resonant degenerate four-wave mixing,” Phys. Rev. Lett. 44, 1251–1254 (1980).
[Crossref]

Carmichael, H. J.

G. P. Agrawal and H. J. Carmichael, “Inhomogeneous broadening and the mean-field approximation for optical bistability in a Fabry–Perot,” Opt. Acta. 27, 651–660 (1980).
[Crossref]

G. P. Agrawal and H. J. Carmichael, “Optical bistability through nonlinear dispersion and absorption,” Phys. Rev. A 19, 2074–2086 (1979).
[Crossref]

Cottin, P.

Deserno, R.

J. Mlynek, F. Mitschke, R. Deserno, and W. Lange, “Optical bistability by transverse optical pumping,” Appl. Phys. B 28, 135 (1982).

Ducloy, M.

R. K. Raj, D. Bloch, J. J. Snyder, G. Camy, and M. Ducloy, “High-frequency optically heterodyned saturation spectroscopy via resonant degenerate four-wave mixing,” Phys. Rev. Lett. 44, 1251–1254 (1980).
[Crossref]

Economou, N. P.

P. F. Liao, D. M. Bloom, and N. P. Economou, “Cw optical wave-front conjugation by saturated absorption in atomic sodium vapor,” Appl. Phys. Lett. 32, 813–815 (1978).
[Crossref]

Elgin, J. N.

L. Fu-Li, J. A. Hermann, and J. N. Elgin, “Effects of two-photon optical bistability upon phase conjugation,” Opt. Commun. 40, 446–450 (1982).
[Crossref]

Feldman, B. J.

Fisher, R. A.

Flytzanis, C.

G. P. Agrawal, C. Flytzanis, R. Frey, and F. Pradere, “Bistable reflectivity of phase-conjugated signal through intracavity degenerate four-wave mixing,” Appl. Phys. Lett. 38, 492–494 (1981).
[Crossref]

G. P. Agrawal and C. Flytzanis, “Bistability and hysteresis in phase-conjugated reflectivity,” IEEE J. Quantum Electron. QE-14, 374–380 (1981).
[Crossref]

C. Flytzanis and C. L. Tang, “Light-induced critical behavior in four-wave interaction in nonlinear systems,” Phys. Rev. Lett. 45, 441–444 (1980).
[Crossref]

C. Flytzanis, G. P. Agrawal, and C. L. Tang, “Critical behavior in optical phase conjugation,” in Lasers and Their Applications, W. O. N. Guimaraes, C.-T. Lin, and A. Mooradian, eds. (Springer-Verlag, Berlin, 1981).
[Crossref]

Frey, R.

G. P. Agrawal, C. Flytzanis, R. Frey, and F. Pradere, “Bistable reflectivity of phase-conjugated signal through intracavity degenerate four-wave mixing,” Appl. Phys. Lett. 38, 492–494 (1981).
[Crossref]

Fu, T.

Fu-Li, L.

L. Fu-Li, J. A. Hermann, and J. N. Elgin, “Effects of two-photon optical bistability upon phase conjugation,” Opt. Commun. 40, 446–450 (1982).
[Crossref]

Gibbs, H. M.

H. M. Gibbs, S. L. McCall, and T. N. C. Venkatesan, “Differential gain and bistability using a sodium-filled Fabry–Perot interferometer,” Phys. Rev. Lett. 36, 1135–1138 (1976).
[Crossref]

Gray, H. R.

Guérin, D.

Hahn, E. L.

R. G. Brewer and E. L. Hahn, “Coherent two-photon processes: transient and steady-state cases,” Phys. Rev. A 11, 1641–1649 (1975).
[Crossref]

Hermann, J. A.

L. Fu-Li, J. A. Hermann, and J. N. Elgin, “Effects of two-photon optical bistability upon phase conjugation,” Opt. Commun. 40, 446–450 (1982).
[Crossref]

Kukhtarev, N.

A. Borshch, M. Brodin, V. Volkov, and N. Kukhtarev, “Optical bistability and hysteresis in phase conjugation by degenerate six-photon mixing,” Opt. Commun. 41, 213–215 (1982).
[Crossref]

Lam, J. F.

D. G. Steel and J. F. Lam, “Saturation effects and inhomogeneous broadening in Doppler-free degenerate four-wave mixing,” Opt. Commun. 40, 77–80 (1981).
[Crossref]

Lange, W.

J. Mlynek, F. Mitschke, R. Deserno, and W. Lange, “Optical bistability by transverse optical pumping,” Appl. Phys. B 28, 135 (1982).

Liao, P. F.

P. F. Liao, D. M. Bloom, and N. P. Economou, “Cw optical wave-front conjugation by saturated absorption in atomic sodium vapor,” Appl. Phys. Lett. 32, 813–815 (1978).
[Crossref]

Lind, R. C.

Lugiato, L. A.

R. Bonifacio and L. A. Lugiato, “Optical bistability and cooperative effects in resonance fluorescence,” Phys. Rev. A 18, 1129–1144 (1978).
[Crossref]

Marburger, J. H.

H. G. Winful and J. H. Marburger, “Hysteresis and optical bistability in degenerate four-wave mixing,” Appl. Phys. Lett. 36, 613–614 (1980).
[Crossref]

McCall, S. L.

H. M. Gibbs, S. L. McCall, and T. N. C. Venkatesan, “Differential gain and bistability using a sodium-filled Fabry–Perot interferometer,” Phys. Rev. Lett. 36, 1135–1138 (1976).
[Crossref]

Meunier, O.

Mitschke, F.

J. Mlynek, F. Mitschke, R. Deserno, and W. Lange, “Optical bistability by transverse optical pumping,” Appl. Phys. B 28, 135 (1982).

Mlynek, J.

J. Mlynek, F. Mitschke, R. Deserno, and W. Lange, “Optical bistability by transverse optical pumping,” Appl. Phys. B 28, 135 (1982).

Pepper, D. M.

For a review see, for example, D. M. Pepper, “Nonlinear optical phase conjugation,” Opt. Eng. 21, 155–183 (1982); see also R. A. Fisher, ed., Optical Phase Conjugation (Academic, New York, 1983).
[Crossref]

Pradere, F.

G. P. Agrawal, C. Flytzanis, R. Frey, and F. Pradere, “Bistable reflectivity of phase-conjugated signal through intracavity degenerate four-wave mixing,” Appl. Phys. Lett. 38, 492–494 (1981).
[Crossref]

Raj, R. K.

R. K. Raj, D. Bloch, J. J. Snyder, G. Camy, and M. Ducloy, “High-frequency optically heterodyned saturation spectroscopy via resonant degenerate four-wave mixing,” Phys. Rev. Lett. 44, 1251–1254 (1980).
[Crossref]

Sargent, M.

Snyder, J. J.

R. K. Raj, D. Bloch, J. J. Snyder, G. Camy, and M. Ducloy, “High-frequency optically heterodyned saturation spectroscopy via resonant degenerate four-wave mixing,” Phys. Rev. Lett. 44, 1251–1254 (1980).
[Crossref]

Steel, D. G.

D. G. Steel and J. F. Lam, “Saturation effects and inhomogeneous broadening in Doppler-free degenerate four-wave mixing,” Opt. Commun. 40, 77–80 (1981).
[Crossref]

Stroud, C. R.

Tang, C. L.

C. Flytzanis and C. L. Tang, “Light-induced critical behavior in four-wave interaction in nonlinear systems,” Phys. Rev. Lett. 45, 441–444 (1980).
[Crossref]

C. Flytzanis, G. P. Agrawal, and C. L. Tang, “Critical behavior in optical phase conjugation,” in Lasers and Their Applications, W. O. N. Guimaraes, C.-T. Lin, and A. Mooradian, eds. (Springer-Verlag, Berlin, 1981).
[Crossref]

van Lerberghe, A.

Venkatesan, T. N. C.

H. M. Gibbs, S. L. McCall, and T. N. C. Venkatesan, “Differential gain and bistability using a sodium-filled Fabry–Perot interferometer,” Phys. Rev. Lett. 36, 1135–1138 (1976).
[Crossref]

Volkov, V.

A. Borshch, M. Brodin, V. Volkov, and N. Kukhtarev, “Optical bistability and hysteresis in phase conjugation by degenerate six-photon mixing,” Opt. Commun. 41, 213–215 (1982).
[Crossref]

Walls, D. F.

D. F. Walls and P. Zoller, “A coherent nonlinear mechanism for optical bistability from three-level atoms,” Opt. Commun. 34, 260–264 (1980).
[Crossref]

Whitley, R. M.

Winful, H. G.

H. G. Winful and J. H. Marburger, “Hysteresis and optical bistability in degenerate four-wave mixing,” Appl. Phys. Lett. 36, 613–614 (1980).
[Crossref]

Zoller, P.

D. F. Walls and P. Zoller, “A coherent nonlinear mechanism for optical bistability from three-level atoms,” Opt. Commun. 34, 260–264 (1980).
[Crossref]

Appl. Phys. B (1)

J. Mlynek, F. Mitschke, R. Deserno, and W. Lange, “Optical bistability by transverse optical pumping,” Appl. Phys. B 28, 135 (1982).

Appl. Phys. Lett. (3)

P. F. Liao, D. M. Bloom, and N. P. Economou, “Cw optical wave-front conjugation by saturated absorption in atomic sodium vapor,” Appl. Phys. Lett. 32, 813–815 (1978).
[Crossref]

G. P. Agrawal, C. Flytzanis, R. Frey, and F. Pradere, “Bistable reflectivity of phase-conjugated signal through intracavity degenerate four-wave mixing,” Appl. Phys. Lett. 38, 492–494 (1981).
[Crossref]

H. G. Winful and J. H. Marburger, “Hysteresis and optical bistability in degenerate four-wave mixing,” Appl. Phys. Lett. 36, 613–614 (1980).
[Crossref]

IEEE J. Quantum Electron. (2)

G. P. Agrawal, “Phase conjugation and degenerate four-wave mixing in three-level systems,” IEEE J. Quantum Electron. QE-17, 2335–2340 (1981).
[Crossref]

G. P. Agrawal and C. Flytzanis, “Bistability and hysteresis in phase-conjugated reflectivity,” IEEE J. Quantum Electron. QE-14, 374–380 (1981).
[Crossref]

Opt. Acta. (1)

G. P. Agrawal and H. J. Carmichael, “Inhomogeneous broadening and the mean-field approximation for optical bistability in a Fabry–Perot,” Opt. Acta. 27, 651–660 (1980).
[Crossref]

Opt. Commun. (4)

A. Borshch, M. Brodin, V. Volkov, and N. Kukhtarev, “Optical bistability and hysteresis in phase conjugation by degenerate six-photon mixing,” Opt. Commun. 41, 213–215 (1982).
[Crossref]

L. Fu-Li, J. A. Hermann, and J. N. Elgin, “Effects of two-photon optical bistability upon phase conjugation,” Opt. Commun. 40, 446–450 (1982).
[Crossref]

D. G. Steel and J. F. Lam, “Saturation effects and inhomogeneous broadening in Doppler-free degenerate four-wave mixing,” Opt. Commun. 40, 77–80 (1981).
[Crossref]

D. F. Walls and P. Zoller, “A coherent nonlinear mechanism for optical bistability from three-level atoms,” Opt. Commun. 34, 260–264 (1980).
[Crossref]

Opt. Eng. (1)

For a review see, for example, D. M. Pepper, “Nonlinear optical phase conjugation,” Opt. Eng. 21, 155–183 (1982); see also R. A. Fisher, ed., Optical Phase Conjugation (Academic, New York, 1983).
[Crossref]

Opt. Lett. (6)

Phys. Rev. A (3)

R. G. Brewer and E. L. Hahn, “Coherent two-photon processes: transient and steady-state cases,” Phys. Rev. A 11, 1641–1649 (1975).
[Crossref]

R. Bonifacio and L. A. Lugiato, “Optical bistability and cooperative effects in resonance fluorescence,” Phys. Rev. A 18, 1129–1144 (1978).
[Crossref]

G. P. Agrawal and H. J. Carmichael, “Optical bistability through nonlinear dispersion and absorption,” Phys. Rev. A 19, 2074–2086 (1979).
[Crossref]

Phys. Rev. Lett. (3)

H. M. Gibbs, S. L. McCall, and T. N. C. Venkatesan, “Differential gain and bistability using a sodium-filled Fabry–Perot interferometer,” Phys. Rev. Lett. 36, 1135–1138 (1976).
[Crossref]

R. K. Raj, D. Bloch, J. J. Snyder, G. Camy, and M. Ducloy, “High-frequency optically heterodyned saturation spectroscopy via resonant degenerate four-wave mixing,” Phys. Rev. Lett. 44, 1251–1254 (1980).
[Crossref]

C. Flytzanis and C. L. Tang, “Light-induced critical behavior in four-wave interaction in nonlinear systems,” Phys. Rev. Lett. 45, 441–444 (1980).
[Crossref]

Other (4)

C. Flytzanis, G. P. Agrawal, and C. L. Tang, “Critical behavior in optical phase conjugation,” in Lasers and Their Applications, W. O. N. Guimaraes, C.-T. Lin, and A. Mooradian, eds. (Springer-Verlag, Berlin, 1981).
[Crossref]

For the case of pulsed radiation, the analysis can be applied if the medium-response time is shorter than the duration of the optical pulse.

In the literature on optical bistability, this is often referred to as the mean-field approximation (see Refs. 8, 9, and 17).

It is not appropriate to use the term reflectivity spectrum because the laser frequency is kept fixed.

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Figures (9)

Fig. 1
Fig. 1

Schematic illustration of the geometry for intracavity degenerate four-wave mixing. In practice, the angle between the waves E1 and E3 is kept small (≈1°) in order to increase the interaction volume.

Fig. 2
Fig. 2

Reflectivity variation with the incident intensity I0 (measured in units of saturation intensity Is) for several values of α0L. The other parameters are Rm= 0.99, Δ = 0, and ϕ = 0. The dashed portion of each reflectivity curve corresponds to the unstable branch of the intracavity pump intensity and is experimentally inaccessible. For α0L = 0.05, corresponding to the bistability parameter C = 5, bistability is not observed.

Fig. 3
Fig. 3

Illustrating the effects of spatial hole burning on the phase-conjugate reflectivity: (a) before and (b) after including the standing-wave effects. The other parameters are Δ = 0, ϕ = 0, α0L = 0.1, and Rm = 0.99. The dashed portion of each curve is experimentally inaccessible.

Fig. 4
Fig. 4

Illustrating the dispersive effects in degenerate four-wave mixing for α0L = 0.2 and Rm = 0.99. The atomic detuning parameter Δ = 2 and the cavity detuning parameter ϕ is varied. The dashed portion of each reflectivity curve is experimentally inaccessible: for ϕ = −2, the parameters are such that bistability disappears.

Fig. 5
Fig. 5

Spectrum of DFWM for I0 = 1, ϕ = 0, α0L = 0.2 and Rm = 0.99. Although a second solution exists for |Δ| ≤ 1.7, no bistability is observed because this branch is inaccessible.

Fig. 6
Fig. 6

Same as in Fig. 5, except that the cavity is slightly detuned, ϕ = 2. The line shape now exhibits bistability and hysteresis since the dashed portion is experimentally inaccessible.

Fig. 7
Fig. 7

Variation of the intracavity pump intensity Ip with the laser frequency for ϕ = 0 and ϕ = 2. The other parameters are identical with those in Fig. 5. For, ϕ = 0, the upper stable branch is disconnected from the lower branch, and the system does not exhibit bistability and hysteresis even though a second solution exists.

Fig. 8
Fig. 8

Variation of the phase-conjugate reflectivity R with the incident intensity I0 for a Λ-type three-level system. The parameter δ is a measure of sublevel splitting [Eq. (5.3)]. Other parameters are α0L = 0.1, Rm = 0.99, r = 1, and ϕ = 0.

Fig. 9
Fig. 9

Variation of the phase-conjugate reflectivity with the sub-level-splitting parameter δ at the incident intensity I0 = 0.1. The other parameters are same as in Fig. 8.

Equations (35)

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E = ( E 1 + E 2 ) + ( E 3 + E 4 ) = E p + E
( 2 + k 2 ) E = - k 2 χ ( E ) E ,
( 2 + k 2 ) E p = - k 2 χ ( E p ) E p ,
( 2 + k 2 ) E = - k 2 [ ( χ ( E p ) + E p χ ( E p ) E p ) E + E p χ ( E p ) E p * E * ] .
E p = E 1 + E 2 = A 1 exp ( i k z ) + A 2 exp ( - i k z )
d A 1 d z = i k 2 ( χ 0 + | A 2 A 1 | χ 1 ) A 1 ,
d A 2 d z = - i k 2 ( χ 0 + | A 1 A 2 | χ 1 ) A 2 ,
χ n = 1 2 π 0 2 π χ ( E p ) cos ( n θ ) d θ ,
A 2 ( L c ) = R m 1 / 2 exp ( 2 i k L c ) A 1 ( L c ) ,
A 1 ( 0 ) = R m 1 / 2 A 2 ( 0 ) + ( 1 - R m ) 1 / 2 E 0 ,
E 0 2 = ( 1 - R m ) A 1 2 { [ 1 + k L 1 - R m Im ( χ 0 + χ 1 ) ] 2 + [ ϕ - k L 1 - R m Re ( χ 0 + χ 1 ) ] 2 } ,
A 1 2 A 2 2 ,
ϕ = 2 m π - 2 k L c 1 - R m = ( ω L - ω ) γ c ,
E = E 3 + E 4 = A 3 exp ( i k z ) + A 4 exp ( - i k z ) ,
d A 3 d z = - α A 3 + i κ A 4 * ,
d A 4 * d z = α * A 4 * + i κ * A 3 ,
α = i k 2 ( χ 0 + χ ) , κ = k 2 χ exp [ i ( ϕ 1 + ϕ 2 ) ] ,
χ = 1 2 π 0 2 π E p 2 χ ( E p ) E p 2 d θ ,
R = | A 4 ( 0 ) A 3 ( 0 ) | 2 = | κ α R + g cot ( g L ) | 2 ,
χ ( E ) = 2 α 0 k ( Δ + i ) ( 1 + Δ 2 + E 2 / I s ) ,
E p 2 2 A 1 2 ( 1 + cos θ ) = 4 A 1 2 cos 2 ( θ / 2 ) .
I 0 = E 0 2 / I s ,             I p = 2 A 1 2 / I s ,
I 0 = ( 1 - R m ) 2 I p { [ 1 + 2 C I p ( 1 - S ) ] 2 + [ ϕ - 2 C Δ I p ( 1 - S ) ] 2 } ,
α = α 0 ( 1 - i Δ ) 1 + Δ 2 1 + Δ 2 + I p 1 + Δ 2 S 3 ,
κ = α 0 1 + Δ 2 ( Δ + i ) I p 1 + Δ 2 S 3 ,
S = ( 1 + Δ 2 1 + Δ 2 + 2 I p ) 1 / 2
C = α 0 L / ( 1 - R m ) .
χ ( E ) = 2 i α 0 k ( 1 + δ ˜ 2 + 3 2 E 2 I s ) - 1 ,
δ ˜ = δ - r 2 δ E 2 I s ,
δ = 1 2 Ω 21 T 2 ,
I 0 = ( 1 - R m ) 2 I p { [ 1 + 2 C δ 2 d r I p ( S + - S - ] 2 + ϕ 2 } ,
α = α 0 δ 2 2 d [ ( b - d + r I p / 2 ) ( b - d ) 2 S - 3 - ( b + d + r I p / 2 ) ( b + d ) 2 S + 3 ] ,
κ = α 0 δ 2 2 d r I p 2 ( S - 3 ( b - d ) 2 - S + 3 ( b + d ) 2 ) ,
a = δ ( 1 + δ 2 ) 1 / 2 ,             b = [ 3 / ( 2 r ) - 1 ] δ 2 ,
S ± = ( b ± d b ± d + r I p ) 1 / 2 .