Abstract

The response of a photorefractive phase conjugator to time-varying signals is examined. Maxwell’s equations are coupled to the material equations followed by linearization using a strong undepleted pump approximation and simplification by the slowly varying envelope approximation. The resulting set of equations is solved by frequency-domain techniques. The solution is expressed in terms of a transfer function that relates the complex frequencies of the probe and the conjugate field. Limiting forms of the transfer function are derived, and a comparison with a Kerr material is made. The effects of various parameters on the fidelity and stability of the conjugation process are determined. Numerical results are presented showing the distortion of time-varying signals owing to the nonideal conjugation process.

© 1988 Optical Society of America

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  1. See, for example, B. Ya. Zel’dovich, N. F. Pilipetsky, V. V. Shkunov, Principles of Phase Conjugation, Vol. 42 of Springer Series in Optical Sciences (Springer-Verlag, Berlin, 1985).
    [CrossRef]
  2. D. M. Pepper, J. AuYeung, D. Fekete, A. Yariv, “Spatial convolution and correlation of optical fields via four-wave mixing,” Opt. Lett. 3, 7–9 (1978).
    [CrossRef] [PubMed]
  3. J. Feinberg, R. W. Hellwarth, “Phase-conjugating mirror with continuous-wave gain,” Opt. Lett. 5, 519–521 (1980).
    [CrossRef] [PubMed]
  4. G. J. Dunning, E. Marom, Y. Owechko, B. H. Soffer, “All-optical associative memory with shift invariance and multiple-image recall,” Opt. Lett. 12, 346–348 (1987).
    [CrossRef] [PubMed]
  5. O. Ikeda, “Low-pass, high-pass, and bandpass spatial-filtering characteristics of a BaTiO3phase conjugator,” J. Opt. Soc. Am. B 4, 1387–1391 (1987).
    [CrossRef]
  6. For review, see Optical Phase Conjugation, R. A. Fisher, ed.(Academic, New York, 1983).
  7. For reviews, see P. Gunter, “Holography, coherent light amplification and optical phase conjugation with photorefractive materials,” Phys. Rep. 93, 199–299 (1982);T. J. Hall, R. Jaura, L. M. Conners, P. D. Foote, “The photorefractive effect—a review,” Prog. Quantum Electron. 10, 77–146 (1985).
    [CrossRef]
  8. J. P. Huignard, J. P. Herriau, P. Aubourg, E. Spitz, “Phase-conjugate wavefront generation via real-time holography in BSO crystals,” Opt. Lett. 4, 21–23 (1979).
    [CrossRef]
  9. M. Cronin-Golomb, B. Fisher, J. O. White, A. Yariv, “Theory and applications of four-wave mixing in photorefractive media,” IEEE J. Quantum Electron. QE-20, 12–30 (1984).
    [CrossRef]
  10. J. H. Marburger, “Optical pulse integration and chirp reversal in degenerate four-wave mixing,” Appl. Phys. Lett. 32, 372–374 (1978).
    [CrossRef]
  11. B. Ya. Zel’dovich, M. A. Orlova, V. V. Shkunov, “Nonstationary theory of the time of establishment of four-wave wave-front reversal,” Sov. Phys. Dokl. 25, 390–391 (1980).
  12. W. W. Rigrod, R. A. Fisher, B. J. Feldman, “Transient analysis of nearly degenerate four-wave mixing,” Opt. Lett. 5, 105–107 (1980).
    [CrossRef] [PubMed]
  13. R. A. Fisher, B. R. Suydam, B. J. Feldman, “Transient analysis of Kerr-like phase conjugators using frequency-domain techniques,” Phys. Rev. A 23, 3071–3083 (1981).
    [CrossRef]
  14. N. Kukhtarev, V. Markov, S. Odulov, “Transient energy transfer during hologram formation in LiNbO3in external electric field,” Opt. Commun. 23, 338–343 (1977).
    [CrossRef]
  15. J. M. Heaton, L. Solymar, “Transient energy transfer during hologram formation in photorefractive crystals,” Opt. Acta 32, 397–408 (1985);M. Cronin-Golomb, “Analytic solution for photorefractive two-beam coupling with time-varying signals,” in Digest of Topical Meeting on Photorefractive Materials, Effects, and Devices (Optical Society of America, Washington, D.C., 1987), pp. 142–145.
    [CrossRef]
  16. F. P. Strohkendl, J. M. C. Jonathan, R. W. Hellwarth, “Hole–electron competition in photorefractive gratings,” Opt. Lett. 11, 312–314 (1986).
    [CrossRef]
  17. G. Valley, “Simultaneous electron/hole transport in photorefractive materials,” J. Appl. Phys. 59, 3363–3366 (1986).
    [CrossRef]
  18. N. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetskii, “Holographic storage in electro-optic crystals. I. Steady-state,” Ferroelectrics 22, 949–960 (1979).
    [CrossRef]
  19. G. Valley, “Short pulse grating formation in photorefractive materials,” IEEE J. Quantum Electron. QE-19, 1637–1645 (1983).
    [CrossRef]
  20. P. Refregier, L. Solymar, H. Rajbenbach, J. P. Huignard, “Two-beam coupling in photorefractive BSO crystals with moving grating: theory and experiments,” J. Appl. Phys. 58, 45–57 (1985).
    [CrossRef]
  21. J. A. Tataronis, G. C. Papen, “Transient phase conjugation in plasmas,” to be submitted to Phys. Fluids.
  22. A. L. Smirl, G. C. Valley, R. A. Mullen, K. Bohnert, C. D. Mire, T. F. Boggess, “Picosecond photorefractive effect in BaTiO3,” Opt. Lett. 12, 501–503 (1987).
    [CrossRef] [PubMed]
  23. G. C. Valley, M. B. Klein, “Optimal properties of photorefractive materials for optical data processing,” Opt. Eng. 22, 704–711 (1983).
    [CrossRef]
  24. S. Ducharme, J. Feinberg, “Speed of the photorefractive effect in a BaTiO3single crystal,” J. Appl. Phys. 56, 839–842 (1984).
    [CrossRef]
  25. S. D. Fisher, Complex Variables (Wadsworth and Brooks, Belmont, Calif., 1986).
  26. A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley-Inter-science, New York, 1984).
  27. J. F. Lam, “Spectral response of nearly degenerate four-wave mixing in photorefractive materials,” Appl. Phys. Lett. 42, 155–157 (1982).
    [CrossRef]
  28. M. B. Klein, “Beam coupling in undoped GaAs at 1.06 μ m using the photorefractive effect,” Opt. Lett. 9, 350–352 (1984).
    [CrossRef] [PubMed]
  29. A. M. Glass, A. M. Johnson, D. H. Olson, W. Simpson, A. A. Ballman, “Four-wave mixing in semi-insulating InP and GaAs using the photorefractive effect,” Appl. Phys. Lett. 44, 948–950 (1984).
    [CrossRef]
  30. P. Yeh, “Fundamental limit of the speed of the photorefractive effect and its impact on device applications and materials research,” Appl. Opt. 26, 602–604 (1987);A. M. Glass, M. B. Klein, G. C. Valley, “Fundamental limit of the speed of the photorefractive effect and its impact on device applications and materials research: comment,” Appl. Opt. 26, 3189–3190 (1987);P. Yeh, “Fundamental limit of the speed of the photorefractive effect and its impact on device applications and materials research: author’s reply to comment,” Appl. Opt. 26, 3190–3191 (1987).
    [CrossRef] [PubMed]
  31. K. MacDonald, J. Feinberg, “Enhanced four-wave mixing by use of frequency-shifted optical waves in photorefractive BaTiO3,” Phys. Rev. Lett. 55, 821–824 (1985).
    [CrossRef] [PubMed]
  32. B. Fischer, “Theory of self-frequency detuning of oscillations by wave mixing in photorefractive crystals,” Opt. Lett. 11, 236–238 (1986).
    [CrossRef] [PubMed]
  33. B. Fischer, M. Cronin-Golomb, J. O. White, A. Yariv, “Amplified reflection, transmission, and self-oscillation in real-time holography,” Opt. Lett. 6, 519–521 (1981).
    [CrossRef] [PubMed]
  34. E. O. Brighham, The Fast Fourier Transform (Prentice-Hall, Englewood Cliffs, N.J., 1971).

1987 (4)

1986 (3)

1985 (3)

J. M. Heaton, L. Solymar, “Transient energy transfer during hologram formation in photorefractive crystals,” Opt. Acta 32, 397–408 (1985);M. Cronin-Golomb, “Analytic solution for photorefractive two-beam coupling with time-varying signals,” in Digest of Topical Meeting on Photorefractive Materials, Effects, and Devices (Optical Society of America, Washington, D.C., 1987), pp. 142–145.
[CrossRef]

P. Refregier, L. Solymar, H. Rajbenbach, J. P. Huignard, “Two-beam coupling in photorefractive BSO crystals with moving grating: theory and experiments,” J. Appl. Phys. 58, 45–57 (1985).
[CrossRef]

K. MacDonald, J. Feinberg, “Enhanced four-wave mixing by use of frequency-shifted optical waves in photorefractive BaTiO3,” Phys. Rev. Lett. 55, 821–824 (1985).
[CrossRef] [PubMed]

1984 (4)

A. M. Glass, A. M. Johnson, D. H. Olson, W. Simpson, A. A. Ballman, “Four-wave mixing in semi-insulating InP and GaAs using the photorefractive effect,” Appl. Phys. Lett. 44, 948–950 (1984).
[CrossRef]

S. Ducharme, J. Feinberg, “Speed of the photorefractive effect in a BaTiO3single crystal,” J. Appl. Phys. 56, 839–842 (1984).
[CrossRef]

M. Cronin-Golomb, B. Fisher, J. O. White, A. Yariv, “Theory and applications of four-wave mixing in photorefractive media,” IEEE J. Quantum Electron. QE-20, 12–30 (1984).
[CrossRef]

M. B. Klein, “Beam coupling in undoped GaAs at 1.06 μ m using the photorefractive effect,” Opt. Lett. 9, 350–352 (1984).
[CrossRef] [PubMed]

1983 (2)

G. C. Valley, M. B. Klein, “Optimal properties of photorefractive materials for optical data processing,” Opt. Eng. 22, 704–711 (1983).
[CrossRef]

G. Valley, “Short pulse grating formation in photorefractive materials,” IEEE J. Quantum Electron. QE-19, 1637–1645 (1983).
[CrossRef]

1982 (2)

J. F. Lam, “Spectral response of nearly degenerate four-wave mixing in photorefractive materials,” Appl. Phys. Lett. 42, 155–157 (1982).
[CrossRef]

For reviews, see P. Gunter, “Holography, coherent light amplification and optical phase conjugation with photorefractive materials,” Phys. Rep. 93, 199–299 (1982);T. J. Hall, R. Jaura, L. M. Conners, P. D. Foote, “The photorefractive effect—a review,” Prog. Quantum Electron. 10, 77–146 (1985).
[CrossRef]

1981 (2)

R. A. Fisher, B. R. Suydam, B. J. Feldman, “Transient analysis of Kerr-like phase conjugators using frequency-domain techniques,” Phys. Rev. A 23, 3071–3083 (1981).
[CrossRef]

B. Fischer, M. Cronin-Golomb, J. O. White, A. Yariv, “Amplified reflection, transmission, and self-oscillation in real-time holography,” Opt. Lett. 6, 519–521 (1981).
[CrossRef] [PubMed]

1980 (3)

1979 (2)

N. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetskii, “Holographic storage in electro-optic crystals. I. Steady-state,” Ferroelectrics 22, 949–960 (1979).
[CrossRef]

J. P. Huignard, J. P. Herriau, P. Aubourg, E. Spitz, “Phase-conjugate wavefront generation via real-time holography in BSO crystals,” Opt. Lett. 4, 21–23 (1979).
[CrossRef]

1978 (2)

D. M. Pepper, J. AuYeung, D. Fekete, A. Yariv, “Spatial convolution and correlation of optical fields via four-wave mixing,” Opt. Lett. 3, 7–9 (1978).
[CrossRef] [PubMed]

J. H. Marburger, “Optical pulse integration and chirp reversal in degenerate four-wave mixing,” Appl. Phys. Lett. 32, 372–374 (1978).
[CrossRef]

1977 (1)

N. Kukhtarev, V. Markov, S. Odulov, “Transient energy transfer during hologram formation in LiNbO3in external electric field,” Opt. Commun. 23, 338–343 (1977).
[CrossRef]

Aubourg, P.

AuYeung, J.

Ballman, A. A.

A. M. Glass, A. M. Johnson, D. H. Olson, W. Simpson, A. A. Ballman, “Four-wave mixing in semi-insulating InP and GaAs using the photorefractive effect,” Appl. Phys. Lett. 44, 948–950 (1984).
[CrossRef]

Boggess, T. F.

Bohnert, K.

Brighham, E. O.

E. O. Brighham, The Fast Fourier Transform (Prentice-Hall, Englewood Cliffs, N.J., 1971).

Cronin-Golomb, M.

M. Cronin-Golomb, B. Fisher, J. O. White, A. Yariv, “Theory and applications of four-wave mixing in photorefractive media,” IEEE J. Quantum Electron. QE-20, 12–30 (1984).
[CrossRef]

B. Fischer, M. Cronin-Golomb, J. O. White, A. Yariv, “Amplified reflection, transmission, and self-oscillation in real-time holography,” Opt. Lett. 6, 519–521 (1981).
[CrossRef] [PubMed]

Ducharme, S.

S. Ducharme, J. Feinberg, “Speed of the photorefractive effect in a BaTiO3single crystal,” J. Appl. Phys. 56, 839–842 (1984).
[CrossRef]

Dunning, G. J.

Feinberg, J.

K. MacDonald, J. Feinberg, “Enhanced four-wave mixing by use of frequency-shifted optical waves in photorefractive BaTiO3,” Phys. Rev. Lett. 55, 821–824 (1985).
[CrossRef] [PubMed]

S. Ducharme, J. Feinberg, “Speed of the photorefractive effect in a BaTiO3single crystal,” J. Appl. Phys. 56, 839–842 (1984).
[CrossRef]

J. Feinberg, R. W. Hellwarth, “Phase-conjugating mirror with continuous-wave gain,” Opt. Lett. 5, 519–521 (1980).
[CrossRef] [PubMed]

Fekete, D.

Feldman, B. J.

R. A. Fisher, B. R. Suydam, B. J. Feldman, “Transient analysis of Kerr-like phase conjugators using frequency-domain techniques,” Phys. Rev. A 23, 3071–3083 (1981).
[CrossRef]

W. W. Rigrod, R. A. Fisher, B. J. Feldman, “Transient analysis of nearly degenerate four-wave mixing,” Opt. Lett. 5, 105–107 (1980).
[CrossRef] [PubMed]

Fischer, B.

Fisher, B.

M. Cronin-Golomb, B. Fisher, J. O. White, A. Yariv, “Theory and applications of four-wave mixing in photorefractive media,” IEEE J. Quantum Electron. QE-20, 12–30 (1984).
[CrossRef]

Fisher, R. A.

R. A. Fisher, B. R. Suydam, B. J. Feldman, “Transient analysis of Kerr-like phase conjugators using frequency-domain techniques,” Phys. Rev. A 23, 3071–3083 (1981).
[CrossRef]

W. W. Rigrod, R. A. Fisher, B. J. Feldman, “Transient analysis of nearly degenerate four-wave mixing,” Opt. Lett. 5, 105–107 (1980).
[CrossRef] [PubMed]

Fisher, S. D.

S. D. Fisher, Complex Variables (Wadsworth and Brooks, Belmont, Calif., 1986).

Glass, A. M.

A. M. Glass, A. M. Johnson, D. H. Olson, W. Simpson, A. A. Ballman, “Four-wave mixing in semi-insulating InP and GaAs using the photorefractive effect,” Appl. Phys. Lett. 44, 948–950 (1984).
[CrossRef]

Gunter, P.

For reviews, see P. Gunter, “Holography, coherent light amplification and optical phase conjugation with photorefractive materials,” Phys. Rep. 93, 199–299 (1982);T. J. Hall, R. Jaura, L. M. Conners, P. D. Foote, “The photorefractive effect—a review,” Prog. Quantum Electron. 10, 77–146 (1985).
[CrossRef]

Heaton, J. M.

J. M. Heaton, L. Solymar, “Transient energy transfer during hologram formation in photorefractive crystals,” Opt. Acta 32, 397–408 (1985);M. Cronin-Golomb, “Analytic solution for photorefractive two-beam coupling with time-varying signals,” in Digest of Topical Meeting on Photorefractive Materials, Effects, and Devices (Optical Society of America, Washington, D.C., 1987), pp. 142–145.
[CrossRef]

Hellwarth, R. W.

Herriau, J. P.

Huignard, J. P.

P. Refregier, L. Solymar, H. Rajbenbach, J. P. Huignard, “Two-beam coupling in photorefractive BSO crystals with moving grating: theory and experiments,” J. Appl. Phys. 58, 45–57 (1985).
[CrossRef]

J. P. Huignard, J. P. Herriau, P. Aubourg, E. Spitz, “Phase-conjugate wavefront generation via real-time holography in BSO crystals,” Opt. Lett. 4, 21–23 (1979).
[CrossRef]

Ikeda, O.

Johnson, A. M.

A. M. Glass, A. M. Johnson, D. H. Olson, W. Simpson, A. A. Ballman, “Four-wave mixing in semi-insulating InP and GaAs using the photorefractive effect,” Appl. Phys. Lett. 44, 948–950 (1984).
[CrossRef]

Jonathan, J. M. C.

Klein, M. B.

M. B. Klein, “Beam coupling in undoped GaAs at 1.06 μ m using the photorefractive effect,” Opt. Lett. 9, 350–352 (1984).
[CrossRef] [PubMed]

G. C. Valley, M. B. Klein, “Optimal properties of photorefractive materials for optical data processing,” Opt. Eng. 22, 704–711 (1983).
[CrossRef]

Kukhtarev, N.

N. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetskii, “Holographic storage in electro-optic crystals. I. Steady-state,” Ferroelectrics 22, 949–960 (1979).
[CrossRef]

N. Kukhtarev, V. Markov, S. Odulov, “Transient energy transfer during hologram formation in LiNbO3in external electric field,” Opt. Commun. 23, 338–343 (1977).
[CrossRef]

Lam, J. F.

J. F. Lam, “Spectral response of nearly degenerate four-wave mixing in photorefractive materials,” Appl. Phys. Lett. 42, 155–157 (1982).
[CrossRef]

MacDonald, K.

K. MacDonald, J. Feinberg, “Enhanced four-wave mixing by use of frequency-shifted optical waves in photorefractive BaTiO3,” Phys. Rev. Lett. 55, 821–824 (1985).
[CrossRef] [PubMed]

Marburger, J. H.

J. H. Marburger, “Optical pulse integration and chirp reversal in degenerate four-wave mixing,” Appl. Phys. Lett. 32, 372–374 (1978).
[CrossRef]

Markov, V.

N. Kukhtarev, V. Markov, S. Odulov, “Transient energy transfer during hologram formation in LiNbO3in external electric field,” Opt. Commun. 23, 338–343 (1977).
[CrossRef]

Markov, V. B.

N. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetskii, “Holographic storage in electro-optic crystals. I. Steady-state,” Ferroelectrics 22, 949–960 (1979).
[CrossRef]

Marom, E.

Mire, C. D.

Mullen, R. A.

Odulov, S.

N. Kukhtarev, V. Markov, S. Odulov, “Transient energy transfer during hologram formation in LiNbO3in external electric field,” Opt. Commun. 23, 338–343 (1977).
[CrossRef]

Odulov, S. G.

N. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetskii, “Holographic storage in electro-optic crystals. I. Steady-state,” Ferroelectrics 22, 949–960 (1979).
[CrossRef]

Olson, D. H.

A. M. Glass, A. M. Johnson, D. H. Olson, W. Simpson, A. A. Ballman, “Four-wave mixing in semi-insulating InP and GaAs using the photorefractive effect,” Appl. Phys. Lett. 44, 948–950 (1984).
[CrossRef]

Orlova, M. A.

B. Ya. Zel’dovich, M. A. Orlova, V. V. Shkunov, “Nonstationary theory of the time of establishment of four-wave wave-front reversal,” Sov. Phys. Dokl. 25, 390–391 (1980).

Owechko, Y.

Papen, G. C.

J. A. Tataronis, G. C. Papen, “Transient phase conjugation in plasmas,” to be submitted to Phys. Fluids.

Pepper, D. M.

Pilipetsky, N. F.

See, for example, B. Ya. Zel’dovich, N. F. Pilipetsky, V. V. Shkunov, Principles of Phase Conjugation, Vol. 42 of Springer Series in Optical Sciences (Springer-Verlag, Berlin, 1985).
[CrossRef]

Rajbenbach, H.

P. Refregier, L. Solymar, H. Rajbenbach, J. P. Huignard, “Two-beam coupling in photorefractive BSO crystals with moving grating: theory and experiments,” J. Appl. Phys. 58, 45–57 (1985).
[CrossRef]

Refregier, P.

P. Refregier, L. Solymar, H. Rajbenbach, J. P. Huignard, “Two-beam coupling in photorefractive BSO crystals with moving grating: theory and experiments,” J. Appl. Phys. 58, 45–57 (1985).
[CrossRef]

Rigrod, W. W.

Shkunov, V. V.

B. Ya. Zel’dovich, M. A. Orlova, V. V. Shkunov, “Nonstationary theory of the time of establishment of four-wave wave-front reversal,” Sov. Phys. Dokl. 25, 390–391 (1980).

See, for example, B. Ya. Zel’dovich, N. F. Pilipetsky, V. V. Shkunov, Principles of Phase Conjugation, Vol. 42 of Springer Series in Optical Sciences (Springer-Verlag, Berlin, 1985).
[CrossRef]

Simpson, W.

A. M. Glass, A. M. Johnson, D. H. Olson, W. Simpson, A. A. Ballman, “Four-wave mixing in semi-insulating InP and GaAs using the photorefractive effect,” Appl. Phys. Lett. 44, 948–950 (1984).
[CrossRef]

Smirl, A. L.

Soffer, B. H.

Solymar, L.

J. M. Heaton, L. Solymar, “Transient energy transfer during hologram formation in photorefractive crystals,” Opt. Acta 32, 397–408 (1985);M. Cronin-Golomb, “Analytic solution for photorefractive two-beam coupling with time-varying signals,” in Digest of Topical Meeting on Photorefractive Materials, Effects, and Devices (Optical Society of America, Washington, D.C., 1987), pp. 142–145.
[CrossRef]

P. Refregier, L. Solymar, H. Rajbenbach, J. P. Huignard, “Two-beam coupling in photorefractive BSO crystals with moving grating: theory and experiments,” J. Appl. Phys. 58, 45–57 (1985).
[CrossRef]

Soskin, M. S.

N. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetskii, “Holographic storage in electro-optic crystals. I. Steady-state,” Ferroelectrics 22, 949–960 (1979).
[CrossRef]

Spitz, E.

Strohkendl, F. P.

Suydam, B. R.

R. A. Fisher, B. R. Suydam, B. J. Feldman, “Transient analysis of Kerr-like phase conjugators using frequency-domain techniques,” Phys. Rev. A 23, 3071–3083 (1981).
[CrossRef]

Tataronis, J. A.

J. A. Tataronis, G. C. Papen, “Transient phase conjugation in plasmas,” to be submitted to Phys. Fluids.

Valley, G.

G. Valley, “Simultaneous electron/hole transport in photorefractive materials,” J. Appl. Phys. 59, 3363–3366 (1986).
[CrossRef]

G. Valley, “Short pulse grating formation in photorefractive materials,” IEEE J. Quantum Electron. QE-19, 1637–1645 (1983).
[CrossRef]

Valley, G. C.

A. L. Smirl, G. C. Valley, R. A. Mullen, K. Bohnert, C. D. Mire, T. F. Boggess, “Picosecond photorefractive effect in BaTiO3,” Opt. Lett. 12, 501–503 (1987).
[CrossRef] [PubMed]

G. C. Valley, M. B. Klein, “Optimal properties of photorefractive materials for optical data processing,” Opt. Eng. 22, 704–711 (1983).
[CrossRef]

Vinetskii, V. L.

N. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetskii, “Holographic storage in electro-optic crystals. I. Steady-state,” Ferroelectrics 22, 949–960 (1979).
[CrossRef]

White, J. O.

M. Cronin-Golomb, B. Fisher, J. O. White, A. Yariv, “Theory and applications of four-wave mixing in photorefractive media,” IEEE J. Quantum Electron. QE-20, 12–30 (1984).
[CrossRef]

B. Fischer, M. Cronin-Golomb, J. O. White, A. Yariv, “Amplified reflection, transmission, and self-oscillation in real-time holography,” Opt. Lett. 6, 519–521 (1981).
[CrossRef] [PubMed]

Yariv, A.

M. Cronin-Golomb, B. Fisher, J. O. White, A. Yariv, “Theory and applications of four-wave mixing in photorefractive media,” IEEE J. Quantum Electron. QE-20, 12–30 (1984).
[CrossRef]

B. Fischer, M. Cronin-Golomb, J. O. White, A. Yariv, “Amplified reflection, transmission, and self-oscillation in real-time holography,” Opt. Lett. 6, 519–521 (1981).
[CrossRef] [PubMed]

D. M. Pepper, J. AuYeung, D. Fekete, A. Yariv, “Spatial convolution and correlation of optical fields via four-wave mixing,” Opt. Lett. 3, 7–9 (1978).
[CrossRef] [PubMed]

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley-Inter-science, New York, 1984).

Yeh, P.

Zel’dovich, B. Ya.

B. Ya. Zel’dovich, M. A. Orlova, V. V. Shkunov, “Nonstationary theory of the time of establishment of four-wave wave-front reversal,” Sov. Phys. Dokl. 25, 390–391 (1980).

See, for example, B. Ya. Zel’dovich, N. F. Pilipetsky, V. V. Shkunov, Principles of Phase Conjugation, Vol. 42 of Springer Series in Optical Sciences (Springer-Verlag, Berlin, 1985).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (3)

A. M. Glass, A. M. Johnson, D. H. Olson, W. Simpson, A. A. Ballman, “Four-wave mixing in semi-insulating InP and GaAs using the photorefractive effect,” Appl. Phys. Lett. 44, 948–950 (1984).
[CrossRef]

J. H. Marburger, “Optical pulse integration and chirp reversal in degenerate four-wave mixing,” Appl. Phys. Lett. 32, 372–374 (1978).
[CrossRef]

J. F. Lam, “Spectral response of nearly degenerate four-wave mixing in photorefractive materials,” Appl. Phys. Lett. 42, 155–157 (1982).
[CrossRef]

Ferroelectrics (1)

N. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetskii, “Holographic storage in electro-optic crystals. I. Steady-state,” Ferroelectrics 22, 949–960 (1979).
[CrossRef]

IEEE J. Quantum Electron. (2)

G. Valley, “Short pulse grating formation in photorefractive materials,” IEEE J. Quantum Electron. QE-19, 1637–1645 (1983).
[CrossRef]

M. Cronin-Golomb, B. Fisher, J. O. White, A. Yariv, “Theory and applications of four-wave mixing in photorefractive media,” IEEE J. Quantum Electron. QE-20, 12–30 (1984).
[CrossRef]

J. Appl. Phys. (3)

G. Valley, “Simultaneous electron/hole transport in photorefractive materials,” J. Appl. Phys. 59, 3363–3366 (1986).
[CrossRef]

P. Refregier, L. Solymar, H. Rajbenbach, J. P. Huignard, “Two-beam coupling in photorefractive BSO crystals with moving grating: theory and experiments,” J. Appl. Phys. 58, 45–57 (1985).
[CrossRef]

S. Ducharme, J. Feinberg, “Speed of the photorefractive effect in a BaTiO3single crystal,” J. Appl. Phys. 56, 839–842 (1984).
[CrossRef]

J. Opt. Soc. Am. B (1)

Opt. Acta (1)

J. M. Heaton, L. Solymar, “Transient energy transfer during hologram formation in photorefractive crystals,” Opt. Acta 32, 397–408 (1985);M. Cronin-Golomb, “Analytic solution for photorefractive two-beam coupling with time-varying signals,” in Digest of Topical Meeting on Photorefractive Materials, Effects, and Devices (Optical Society of America, Washington, D.C., 1987), pp. 142–145.
[CrossRef]

Opt. Commun. (1)

N. Kukhtarev, V. Markov, S. Odulov, “Transient energy transfer during hologram formation in LiNbO3in external electric field,” Opt. Commun. 23, 338–343 (1977).
[CrossRef]

Opt. Eng. (1)

G. C. Valley, M. B. Klein, “Optimal properties of photorefractive materials for optical data processing,” Opt. Eng. 22, 704–711 (1983).
[CrossRef]

Opt. Lett. (10)

D. M. Pepper, J. AuYeung, D. Fekete, A. Yariv, “Spatial convolution and correlation of optical fields via four-wave mixing,” Opt. Lett. 3, 7–9 (1978).
[CrossRef] [PubMed]

J. P. Huignard, J. P. Herriau, P. Aubourg, E. Spitz, “Phase-conjugate wavefront generation via real-time holography in BSO crystals,” Opt. Lett. 4, 21–23 (1979).
[CrossRef]

W. W. Rigrod, R. A. Fisher, B. J. Feldman, “Transient analysis of nearly degenerate four-wave mixing,” Opt. Lett. 5, 105–107 (1980).
[CrossRef] [PubMed]

J. Feinberg, R. W. Hellwarth, “Phase-conjugating mirror with continuous-wave gain,” Opt. Lett. 5, 519–521 (1980).
[CrossRef] [PubMed]

B. Fischer, M. Cronin-Golomb, J. O. White, A. Yariv, “Amplified reflection, transmission, and self-oscillation in real-time holography,” Opt. Lett. 6, 519–521 (1981).
[CrossRef] [PubMed]

M. B. Klein, “Beam coupling in undoped GaAs at 1.06 μ m using the photorefractive effect,” Opt. Lett. 9, 350–352 (1984).
[CrossRef] [PubMed]

B. Fischer, “Theory of self-frequency detuning of oscillations by wave mixing in photorefractive crystals,” Opt. Lett. 11, 236–238 (1986).
[CrossRef] [PubMed]

F. P. Strohkendl, J. M. C. Jonathan, R. W. Hellwarth, “Hole–electron competition in photorefractive gratings,” Opt. Lett. 11, 312–314 (1986).
[CrossRef]

G. J. Dunning, E. Marom, Y. Owechko, B. H. Soffer, “All-optical associative memory with shift invariance and multiple-image recall,” Opt. Lett. 12, 346–348 (1987).
[CrossRef] [PubMed]

A. L. Smirl, G. C. Valley, R. A. Mullen, K. Bohnert, C. D. Mire, T. F. Boggess, “Picosecond photorefractive effect in BaTiO3,” Opt. Lett. 12, 501–503 (1987).
[CrossRef] [PubMed]

Phys. Rep. (1)

For reviews, see P. Gunter, “Holography, coherent light amplification and optical phase conjugation with photorefractive materials,” Phys. Rep. 93, 199–299 (1982);T. J. Hall, R. Jaura, L. M. Conners, P. D. Foote, “The photorefractive effect—a review,” Prog. Quantum Electron. 10, 77–146 (1985).
[CrossRef]

Phys. Rev. A (1)

R. A. Fisher, B. R. Suydam, B. J. Feldman, “Transient analysis of Kerr-like phase conjugators using frequency-domain techniques,” Phys. Rev. A 23, 3071–3083 (1981).
[CrossRef]

Phys. Rev. Lett. (1)

K. MacDonald, J. Feinberg, “Enhanced four-wave mixing by use of frequency-shifted optical waves in photorefractive BaTiO3,” Phys. Rev. Lett. 55, 821–824 (1985).
[CrossRef] [PubMed]

Sov. Phys. Dokl. (1)

B. Ya. Zel’dovich, M. A. Orlova, V. V. Shkunov, “Nonstationary theory of the time of establishment of four-wave wave-front reversal,” Sov. Phys. Dokl. 25, 390–391 (1980).

Other (6)

See, for example, B. Ya. Zel’dovich, N. F. Pilipetsky, V. V. Shkunov, Principles of Phase Conjugation, Vol. 42 of Springer Series in Optical Sciences (Springer-Verlag, Berlin, 1985).
[CrossRef]

S. D. Fisher, Complex Variables (Wadsworth and Brooks, Belmont, Calif., 1986).

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley-Inter-science, New York, 1984).

For review, see Optical Phase Conjugation, R. A. Fisher, ed.(Academic, New York, 1983).

J. A. Tataronis, G. C. Papen, “Transient phase conjugation in plasmas,” to be submitted to Phys. Fluids.

E. O. Brighham, The Fast Fourier Transform (Prentice-Hall, Englewood Cliffs, N.J., 1971).

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Figures (8)

Fig. 1
Fig. 1

Geometry of four-wave interaction. The z axis is chosen to lie along the line defined by the probe and conjugate wave vectors within the crystal.

Fig. 2
Fig. 2

The two grating wave vectors k32 and k42 generated by the interference of the probe and pump fields.

Fig. 3
Fig. 3

Magnitude of the transfer function plotted versus the normalized frequency (ωN = Ωτg) for different values of the ratio of the time constants; τc is the transit time through the crystal; τg is the grating buildup time. (a) The case when there is no applied or internal field (φ = π/2, δ = 0). (b) φπ/2,δ ≠ 0, and the magnitude becomes asymmetric. Note that the normalized bandwidth decreases as the ratio of the time constants increases.

Fig. 4
Fig. 4

Dependence of the magnitude of the transfer function on the normalized parameters r|γ|l, φ, and δ for τcg = 0. One of the parameters varies while the others are fixed. The case when r = 0.5, φ = 1.5, and δ = 0.1, and |γ|l = 3 is repeated for comparison.

Fig. 5
Fig. 5

Dependence of the phase of the transfer function on the normalized parameters r, |γ|l, φ and δ for τcg = 0. One of the parameters varies while the others are fixed. The case when r = 0.5, and |γ|l = 3, φ = 1.5, and δ = 0.1 is repeated for comparison.

Fig. 6
Fig. 6

The magnitude and phase of the impulse response, (a) and (b) Are for the same set of parameters that were held fixed in Figs. 4 and 5. (c) and (d) Are for the case of no applied or internal electric field.

Fig. 7
Fig. 7

The magnitude of the impulse response for different ratios of the time constants: (a) τcg = 1, (b) τcg = 10. Neither plot has an applied or internal electric field. These plots are the inverse Fourier transforms of two of the plots in Fig. 3(a). Note the difference in the time scales.

Fig. 8
Fig. 8

Profile of the conjugate wave for an input probe of unit height and normalized width, t/τg = 5. The height of the ideal conjugate output pulse was chosen to correspond to the steady-state gain and is shown in each of the figures: (a) τcg = 0, (b) τcg = 1, (c) τcg = 10, (d) the same as (c) but with |γ|l = 1 and r = 1. The decrease of |γ|l and increase of r moves a pole of the transfer function away from the real axis. The conjugator then approximates an integrator over a time interval 2τc.

Equations (51)

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J = e N D + t ( continuity equation ) ,
N D + t = s I N D γ R n N D + ( rate equation ) ,
J = e μ n E k B T μ n ( equation of motion ) ,
E = e s ( N A N D + ) ( Poission’s equation ) ,
( r , t ) = m = 1 4 A m ( r , t ) exp [ i ( k m r ω t ) ] + c . c . ,
I ( r , t ) = ( r , t ) 2 , ¯
I ( r , t ) = I 3 + I 4 + ( A 3 A 4 * ) exp [ i ( 2 k 3 r ) ] + c . c . + [ A 1 ( r , t ) A 4 * + A 2 * ( r , t ) A 3 ] × exp [ i ( k 3 k 2 ) r ] + c . c . + [ A 1 ( r , t ) A 3 * + A 2 * ( r , t ) A 4 ] × exp [ i ( k 4 k 2 ) r ] + c . c . ,
E Q = E 0 + E 34 exp [ i ( 2 k 3 r ) ] + c . c . ,
δ E = E 32 ( r , t ) exp [ i ( k 3 k 2 ) . r ] + c . c . + E 42 ( r , t ) exp [ i ( k 4 k 2 ) . r ] + c . c . ,
t E 32 ( r , t ) + g 32 E 32 ( r , t ) = h 32 [ A 1 ( r , t ) A 4 * + A 2 * ( r , t ) A 3 ] ,
t E 42 ( r , t ) + g 42 E 42 ( r , t ) = h 42 [ A 1 ( r , t ) A 3 * + A 2 * ( r , t ) A 4 ) ] ,
g 32 = 1 τ g + i ω g ,
h 32 = E 32 S g 32 I 3 + I 4 ,
2 ( r , t ) n b 2 c 2 2 t 2 ( r , t ) = 2 n b Δ n ( r , t ) c 2 2 t 2 ( r , t ) ,
Δ n ( r , t ) = 1 4 n b 3 i j R i j [ E i j ( r , t ) + E i j * ( r , t ) ] ,
k 1 A 1 ( r , t ) + n b 2 ω c 2 t A 1 ( r , t ) = i ω 2 n b 4 4 c 2 [ R 42 E 42 ( r , t ) A 3 + R 32 E 32 ( r , t ) A 4 ] ,
k 2 A 2 ( r , t ) + n b 2 ω c 2 t A 2 ( r , t ) = i ω 2 n b 4 4 c 2 [ R 41 E 41 ( r , t ) A 3 + R 31 E 31 ( r , t ) A 4 ] .
t E 32 ( z , t ) + g 32 E 32 ( z , t ) = h 32 [ A 1 ( z , t ) A 4 * + A 2 * ( z , t ) A 3 ] ,
t A 1 ( z , t ) 1 υ t A 1 ( z , t ) = i C [ R 32 E 32 ( z , t ) A 4 ] ,
t A 1 * ( z , t ) + 1 υ t A 2 * ( z , t ) = i C [ R 32 E 32 ( z , t ) A 3 * ] ,
f ( Ω ) = + F ( t ) e i Ω t d t , F ( t ) = 1 2 π + i y + + i y f ( Ω ) e i Ω t d ( Ω ) ,
t a 1 ( z , Ω ) i α 1 ( Ω ) a 1 ( z , Ω ) = i Γ ( Ω ) G ã 2 ( z , Ω ) ,
t ã 2 ( z , Ω ) + i α 2 ( Ω ) ã 2 ( z , Ω ) = i Γ ( Ω ) G * a 1 ( z , Ω ) ,
α 1 ( Ω ) = Ω υ + Γ ( Ω ) | A 4 | 2 ,
α 2 ( Ω ) = Ω υ Γ ( Ω ) | A 3 | 2 ,
Γ ( Ω ) = C R 32 h 32 g 32 + i Ω ,
G = A 3 A 4 ,
ã 2 ( 0 , Ω ) = f ( 0 , Ω ) , a 1 ( l , Ω ) = 0 ,
H ( 0 , ω N ) = i η M β cot β + i ( τ c τ g ω N + ρ M ) ,
β = [ ( τ c τ g ω N + ρ M ) 2 + ( η M ) 2 ] 1 / 2 ,
ρ = ( r 1 r + 1 ) γ l 2 ,
η = ( 2 ( r ) 1 / 2 r + 1 ) γ l 2 ,
M = 1 + i δ 1 + i ( ω N + δ ) ,
H ( 0 , 0 ) = 2 ( r ) 1 / 2 ( r 1 ) i ( r + 1 ) cot ( γ l 2 ) .
H ( 0 , Ω ) = i κ l β cot ( β ) + i Ω τ c ,
β = [ ( Ω τ c ) 2 + ( κ l ) 2 ] 1 / 2
H ( 0 , ω N ) = 2 ( r ) 1 / 2 ( r 1 ) i ( r + 1 ) cot ( M γ l 2 ) .
Re { S } < 1 ,
S = { ( 1 + i δ ) γ l m π + i ln ( r ) } , m an odd integer .
[ ln ( r ) ] 2 | γ | l ln ( r ) + π 2 > 0 .
f c ( t ) = + f ρ * ( τ ) h ( t τ ) d τ ,
E 32 S = | E 32 S | e i φ ,
| E 32 S | = E q [ E 0 2 + E D 2 E 0 2 + ( E D + E q ) 2 ] 1 / 2 ,
tan φ = E D E 0 [ 1 + E D E q + E 0 2 E D + E q ] ,
τ g = τ di ( 1 + τ R / τ D ) 2 + ( τ R / τ E ) 2 [ 1 + ( τ R τ di / τ D τ I ) ] ( 1 + τ R / τ D ) + ( τ R / τ E ) 2 ( τ di / τ I ) ,
ω g = 1 τ di ( τ R / τ E ) + ( τ di / τ I 1 ) ( 1 + τ R / τ D ) 2 + ( τ R / τ E ) 2 ,
τ di = s γ R N A e μ s N D ( I 3 + I 4 )
τ E = 1 k 32 μ E 0
τ D = e μ k B T k 32 2
τ R = 1 γ R N A
τ I = N A s N D ( I 3 + I 4 )

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