Abstract

Solutions to the band-transport equations are developed to include effects on the amplitude, dynamics, and spatial-frequency response of the space-charge field when periodic or dc electric fields are applied to a photorefractive material. Analysis of the effects of both idealized sinusoidal and square ac waveforms are given and compared with the case of resonant space-charge field enhancement with applied dc fields and moving fringes. Consideration is given to the effect of finite slew rate, uneven mark–space ratio, and other approximations to square ac waveforms. It is shown that the ideal square waveform provides the greatest enhancement of the space-charge field. It is also shown that in absorbing materials resonant enhancement with dc fields and moving fringes is localized to thin regions within the material, as the resonance condition is intensity dependent. This reduces the effective enhancement. Illustrative experimental results are presented for Bi12SiO20 and GaAs.

© 1990 Optical Society of America

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References

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  1. Ph. Refregier, L. Solymar, H. Rajbenbach, and J. P. Huignard, “Two-beam coupling in photorefractive Bi12SiO20 crystals with moving grating: theory and experiments,” J. Appl. Phys. 58, 45–57 (1985).
    [CrossRef]
  2. S. I. Stepanov and M. P. Petrov, “Efficient unstationary holographic recording in photorefractive crystals under an external alternating electric field,” Opt. Commun. 53, 292–295 (1985), and references therein.
    [CrossRef]
  3. J. Kumar, G. Albanese, W. H. Steier, and M. Ziari, “Enhanced two-beam mixing gain in photorefractive GaAs using alternating electric fields,” Opt. Lett. 12, 120–122 (1987).
    [CrossRef] [PubMed]
  4. X. Gan, S. Ye, and Y. Sun, “Alternating electric field enhancement of two-wave mixing gain in photorefractive Bi12SiO20,” Opt. Commun. 66, 155–160 (1988).
    [CrossRef]
  5. J. Kumar, G. Albanese, and W. H. Steier, “Photorefractive two-beam coupling with applied radio-frequency fields: theory and experiment,” J. Opt. Soc. Am. B 4, 1079–1082 (1987).
    [CrossRef]
  6. G. Lesaux, J. C. Launay, and A. Brun, “Transient photocurrent induced by nanosecond light pulses in Bi12SiO20 and Bi12GeO20,” Opt. Commun. 57, 166–167 (1986).
    [CrossRef]
  7. G. C. Valley and A. L. Smirl, “Theory of transient energy transfer in gallium arsenide,” IEEE J. Quantum Electron. 24, 304–310 (1988).
    [CrossRef]
  8. N. V. Kukhtarev, M. V. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electro-optic crystals. I: Steady state,” Ferroelectrics 22, 949–960 (1979), and references therein.
    [CrossRef]
  9. T. J. Hall, R. Jaura, L. M. Connors, and P. D. Foote, “The photorefractive effect—a review,” Prog. Quantum Electron. 10, 77–146 (1985).
    [CrossRef]
  10. K. Walsh and T. J. Hall, “Gain exceeding absorptive losses in photorefractive GaAs,” Appl. Opt. 28, 16–17 (1989).
    [CrossRef] [PubMed]
  11. K. Walsh, T. J. Hall, and R. E. Burge, “Influence of polarization state and absorption gratings on two-wave mixing in GaAs,” Opt. Lett. 12, 1026–1028 (1987).
    [CrossRef] [PubMed]
  12. P. Refrégier, L. Solymar, H. Rajbenbach, and J. P. Huignard, “Large-signal effects in an optical Bi12SiO20 amplifier,” Electron. Lett. 20, 656–657 (1984).
    [CrossRef]
  13. L. Solymar and D. J. Cooke, Volume Holography and Volume Gratings (Academic, London, 1981).
  14. J. L. de Bougrenet de la Tocnaye, P. Pellat-Finet, and J. P. Huignard, “Effect of using a Bi12SiO20 light amplifier on the formation and competition of modes in optical resonators,” J. Opt. Soc. Am. B 3, 315–319 (1986).
    [CrossRef]
  15. H. Rajbenbach, J. P. Huignard, and Ph. Refrégier, “Amplified phase-conjugate beam reflection by four-wave mixing with photorefractive Bi12SiO20 crystals,” Opt. Lett. 9, 558–560 (1984).
    [CrossRef] [PubMed]
  16. J. Kumar, G. Albanese, and W. H. Steier, “Measurement of two-wave mixing gain in GaAs with a moving grating,” Opt. Commun. 63, 191–193 (1987).
    [CrossRef]
  17. B. Imbert, H. Rajbenbach, S. Mallick, J. P. Herriau, and J. P. Huignard, “High photorefractive gain in two-beam coupling with moving fringes in GaAs:Cr crystals,” Opt. Lett. 13, 327–329 (1988).
    [CrossRef] [PubMed]
  18. S. I. Stepanov, V. V. Kulikov, and M. P. Petrov, “‘Running’ holograms in photorefractive Bi12SiO20 crystals,” Opt. Commun. 44, 19–23 (1982).
    [CrossRef]
  19. P. Yeh, “Fundamental limit of the speed of photorefractive effect and its impact on device applications and material research,” Appl. Opt. 26, 602–604 (1987);A. M. Glass, M. B. Klein, and G. C. Valley, “Fundamental limit of the speed of the photorefractive effect and its impact on device applications and material 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 material research: author’s reply to comment,” Appl. Opt. 26, 3190–3191 (1987).
    [CrossRef] [PubMed]
  20. P. Bayvel, M. McCall, and R. V. Wright, “Continuous method for measuring the electro-optic coefficient in Bi12SiO20 and Bi12GeO20,” Opt. Lett. 13, 27–29 (1988).
    [CrossRef] [PubMed]
  21. P. D. Foote, “Optically induced anisotropic light diffraction in photorefractive crystals,” doctoral dissertation (University of London, London, 1987).
  22. D. Dascǎlu, Electronic Processes in Unipolar Solid-State Devices (Abacus, Tunbridge Wells, England, 1977).
  23. C. Stace, A. K. Powell, K. Walsh, and T. J. Hall, “Coupling modulation in photorefractive materials by applying a.c. electric fields,” Opt. Commun. 70, 509–514 (1989).
    [CrossRef]
  24. S. L. Sochava, S. I. Stepanov, and M. P. Petrov, “Ring oscillator using a photorefractive Bi12TiO20 crystal,” Sov. Tech. Phys. Lett. 13, 274–275 (1987).
  25. H. Rajbenbach, B. Imbert, J. P. Huignard, and S. Mallick, “Near-infrared four-wave mixing with gain and self-starting oscillators with photorefractive GaAs,” Opt. Lett. 14, 78–80 (1989).
    [CrossRef] [PubMed]

1989 (3)

1988 (4)

P. Bayvel, M. McCall, and R. V. Wright, “Continuous method for measuring the electro-optic coefficient in Bi12SiO20 and Bi12GeO20,” Opt. Lett. 13, 27–29 (1988).
[CrossRef] [PubMed]

B. Imbert, H. Rajbenbach, S. Mallick, J. P. Herriau, and J. P. Huignard, “High photorefractive gain in two-beam coupling with moving fringes in GaAs:Cr crystals,” Opt. Lett. 13, 327–329 (1988).
[CrossRef] [PubMed]

X. Gan, S. Ye, and Y. Sun, “Alternating electric field enhancement of two-wave mixing gain in photorefractive Bi12SiO20,” Opt. Commun. 66, 155–160 (1988).
[CrossRef]

G. C. Valley and A. L. Smirl, “Theory of transient energy transfer in gallium arsenide,” IEEE J. Quantum Electron. 24, 304–310 (1988).
[CrossRef]

1987 (6)

J. Kumar, G. Albanese, W. H. Steier, and M. Ziari, “Enhanced two-beam mixing gain in photorefractive GaAs using alternating electric fields,” Opt. Lett. 12, 120–122 (1987).
[CrossRef] [PubMed]

J. Kumar, G. Albanese, and W. H. Steier, “Photorefractive two-beam coupling with applied radio-frequency fields: theory and experiment,” J. Opt. Soc. Am. B 4, 1079–1082 (1987).
[CrossRef]

J. Kumar, G. Albanese, and W. H. Steier, “Measurement of two-wave mixing gain in GaAs with a moving grating,” Opt. Commun. 63, 191–193 (1987).
[CrossRef]

P. Yeh, “Fundamental limit of the speed of photorefractive effect and its impact on device applications and material research,” Appl. Opt. 26, 602–604 (1987);A. M. Glass, M. B. Klein, and G. C. Valley, “Fundamental limit of the speed of the photorefractive effect and its impact on device applications and material 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 material research: author’s reply to comment,” Appl. Opt. 26, 3190–3191 (1987).
[CrossRef] [PubMed]

K. Walsh, T. J. Hall, and R. E. Burge, “Influence of polarization state and absorption gratings on two-wave mixing in GaAs,” Opt. Lett. 12, 1026–1028 (1987).
[CrossRef] [PubMed]

S. L. Sochava, S. I. Stepanov, and M. P. Petrov, “Ring oscillator using a photorefractive Bi12TiO20 crystal,” Sov. Tech. Phys. Lett. 13, 274–275 (1987).

1986 (2)

J. L. de Bougrenet de la Tocnaye, P. Pellat-Finet, and J. P. Huignard, “Effect of using a Bi12SiO20 light amplifier on the formation and competition of modes in optical resonators,” J. Opt. Soc. Am. B 3, 315–319 (1986).
[CrossRef]

G. Lesaux, J. C. Launay, and A. Brun, “Transient photocurrent induced by nanosecond light pulses in Bi12SiO20 and Bi12GeO20,” Opt. Commun. 57, 166–167 (1986).
[CrossRef]

1985 (3)

Ph. Refregier, L. Solymar, H. Rajbenbach, and J. P. Huignard, “Two-beam coupling in photorefractive Bi12SiO20 crystals with moving grating: theory and experiments,” J. Appl. Phys. 58, 45–57 (1985).
[CrossRef]

S. I. Stepanov and M. P. Petrov, “Efficient unstationary holographic recording in photorefractive crystals under an external alternating electric field,” Opt. Commun. 53, 292–295 (1985), and references therein.
[CrossRef]

T. J. Hall, R. Jaura, L. M. Connors, and P. D. Foote, “The photorefractive effect—a review,” Prog. Quantum Electron. 10, 77–146 (1985).
[CrossRef]

1984 (2)

H. Rajbenbach, J. P. Huignard, and Ph. Refrégier, “Amplified phase-conjugate beam reflection by four-wave mixing with photorefractive Bi12SiO20 crystals,” Opt. Lett. 9, 558–560 (1984).
[CrossRef] [PubMed]

P. Refrégier, L. Solymar, H. Rajbenbach, and J. P. Huignard, “Large-signal effects in an optical Bi12SiO20 amplifier,” Electron. Lett. 20, 656–657 (1984).
[CrossRef]

1982 (1)

S. I. Stepanov, V. V. Kulikov, and M. P. Petrov, “‘Running’ holograms in photorefractive Bi12SiO20 crystals,” Opt. Commun. 44, 19–23 (1982).
[CrossRef]

1979 (1)

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

Albanese, G.

Bayvel, P.

Brun, A.

G. Lesaux, J. C. Launay, and A. Brun, “Transient photocurrent induced by nanosecond light pulses in Bi12SiO20 and Bi12GeO20,” Opt. Commun. 57, 166–167 (1986).
[CrossRef]

Burge, R. E.

Connors, L. M.

T. J. Hall, R. Jaura, L. M. Connors, and P. D. Foote, “The photorefractive effect—a review,” Prog. Quantum Electron. 10, 77–146 (1985).
[CrossRef]

Cooke, D. J.

L. Solymar and D. J. Cooke, Volume Holography and Volume Gratings (Academic, London, 1981).

Dascalu, D.

D. Dascǎlu, Electronic Processes in Unipolar Solid-State Devices (Abacus, Tunbridge Wells, England, 1977).

de Bougrenet de la Tocnaye, J. L.

Foote, P. D.

T. J. Hall, R. Jaura, L. M. Connors, and P. D. Foote, “The photorefractive effect—a review,” Prog. Quantum Electron. 10, 77–146 (1985).
[CrossRef]

P. D. Foote, “Optically induced anisotropic light diffraction in photorefractive crystals,” doctoral dissertation (University of London, London, 1987).

Gan, X.

X. Gan, S. Ye, and Y. Sun, “Alternating electric field enhancement of two-wave mixing gain in photorefractive Bi12SiO20,” Opt. Commun. 66, 155–160 (1988).
[CrossRef]

Hall, T. J.

C. Stace, A. K. Powell, K. Walsh, and T. J. Hall, “Coupling modulation in photorefractive materials by applying a.c. electric fields,” Opt. Commun. 70, 509–514 (1989).
[CrossRef]

K. Walsh and T. J. Hall, “Gain exceeding absorptive losses in photorefractive GaAs,” Appl. Opt. 28, 16–17 (1989).
[CrossRef] [PubMed]

K. Walsh, T. J. Hall, and R. E. Burge, “Influence of polarization state and absorption gratings on two-wave mixing in GaAs,” Opt. Lett. 12, 1026–1028 (1987).
[CrossRef] [PubMed]

T. J. Hall, R. Jaura, L. M. Connors, and P. D. Foote, “The photorefractive effect—a review,” Prog. Quantum Electron. 10, 77–146 (1985).
[CrossRef]

Herriau, J. P.

Huignard, J. P.

Imbert, B.

Jaura, R.

T. J. Hall, R. Jaura, L. M. Connors, and P. D. Foote, “The photorefractive effect—a review,” Prog. Quantum Electron. 10, 77–146 (1985).
[CrossRef]

Kukhtarev, N. V.

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

Kulikov, V. V.

S. I. Stepanov, V. V. Kulikov, and M. P. Petrov, “‘Running’ holograms in photorefractive Bi12SiO20 crystals,” Opt. Commun. 44, 19–23 (1982).
[CrossRef]

Kumar, J.

Launay, J. C.

G. Lesaux, J. C. Launay, and A. Brun, “Transient photocurrent induced by nanosecond light pulses in Bi12SiO20 and Bi12GeO20,” Opt. Commun. 57, 166–167 (1986).
[CrossRef]

Lesaux, G.

G. Lesaux, J. C. Launay, and A. Brun, “Transient photocurrent induced by nanosecond light pulses in Bi12SiO20 and Bi12GeO20,” Opt. Commun. 57, 166–167 (1986).
[CrossRef]

Mallick, S.

Markov, M. V.

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

McCall, M.

Odulov, S. G.

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

Pellat-Finet, P.

Petrov, M. P.

S. L. Sochava, S. I. Stepanov, and M. P. Petrov, “Ring oscillator using a photorefractive Bi12TiO20 crystal,” Sov. Tech. Phys. Lett. 13, 274–275 (1987).

S. I. Stepanov and M. P. Petrov, “Efficient unstationary holographic recording in photorefractive crystals under an external alternating electric field,” Opt. Commun. 53, 292–295 (1985), and references therein.
[CrossRef]

S. I. Stepanov, V. V. Kulikov, and M. P. Petrov, “‘Running’ holograms in photorefractive Bi12SiO20 crystals,” Opt. Commun. 44, 19–23 (1982).
[CrossRef]

Powell, A. K.

C. Stace, A. K. Powell, K. Walsh, and T. J. Hall, “Coupling modulation in photorefractive materials by applying a.c. electric fields,” Opt. Commun. 70, 509–514 (1989).
[CrossRef]

Rajbenbach, H.

Refregier, Ph.

Ph. Refregier, L. Solymar, H. Rajbenbach, and J. P. Huignard, “Two-beam coupling in photorefractive Bi12SiO20 crystals with moving grating: theory and experiments,” J. Appl. Phys. 58, 45–57 (1985).
[CrossRef]

Refrégier, P.

P. Refrégier, L. Solymar, H. Rajbenbach, and J. P. Huignard, “Large-signal effects in an optical Bi12SiO20 amplifier,” Electron. Lett. 20, 656–657 (1984).
[CrossRef]

Refrégier, Ph.

Smirl, A. L.

G. C. Valley and A. L. Smirl, “Theory of transient energy transfer in gallium arsenide,” IEEE J. Quantum Electron. 24, 304–310 (1988).
[CrossRef]

Sochava, S. L.

S. L. Sochava, S. I. Stepanov, and M. P. Petrov, “Ring oscillator using a photorefractive Bi12TiO20 crystal,” Sov. Tech. Phys. Lett. 13, 274–275 (1987).

Solymar, L.

Ph. Refregier, L. Solymar, H. Rajbenbach, and J. P. Huignard, “Two-beam coupling in photorefractive Bi12SiO20 crystals with moving grating: theory and experiments,” J. Appl. Phys. 58, 45–57 (1985).
[CrossRef]

P. Refrégier, L. Solymar, H. Rajbenbach, and J. P. Huignard, “Large-signal effects in an optical Bi12SiO20 amplifier,” Electron. Lett. 20, 656–657 (1984).
[CrossRef]

L. Solymar and D. J. Cooke, Volume Holography and Volume Gratings (Academic, London, 1981).

Soskin, M. S.

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

Stace, C.

C. Stace, A. K. Powell, K. Walsh, and T. J. Hall, “Coupling modulation in photorefractive materials by applying a.c. electric fields,” Opt. Commun. 70, 509–514 (1989).
[CrossRef]

Steier, W. H.

Stepanov, S. I.

S. L. Sochava, S. I. Stepanov, and M. P. Petrov, “Ring oscillator using a photorefractive Bi12TiO20 crystal,” Sov. Tech. Phys. Lett. 13, 274–275 (1987).

S. I. Stepanov and M. P. Petrov, “Efficient unstationary holographic recording in photorefractive crystals under an external alternating electric field,” Opt. Commun. 53, 292–295 (1985), and references therein.
[CrossRef]

S. I. Stepanov, V. V. Kulikov, and M. P. Petrov, “‘Running’ holograms in photorefractive Bi12SiO20 crystals,” Opt. Commun. 44, 19–23 (1982).
[CrossRef]

Sun, Y.

X. Gan, S. Ye, and Y. Sun, “Alternating electric field enhancement of two-wave mixing gain in photorefractive Bi12SiO20,” Opt. Commun. 66, 155–160 (1988).
[CrossRef]

Valley, G. C.

G. C. Valley and A. L. Smirl, “Theory of transient energy transfer in gallium arsenide,” IEEE J. Quantum Electron. 24, 304–310 (1988).
[CrossRef]

Vinetskii, V. L.

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

Walsh, K.

Wright, R. V.

Ye, S.

X. Gan, S. Ye, and Y. Sun, “Alternating electric field enhancement of two-wave mixing gain in photorefractive Bi12SiO20,” Opt. Commun. 66, 155–160 (1988).
[CrossRef]

Yeh, P.

Ziari, M.

Appl. Opt. (2)

Electron. Lett. (1)

P. Refrégier, L. Solymar, H. Rajbenbach, and J. P. Huignard, “Large-signal effects in an optical Bi12SiO20 amplifier,” Electron. Lett. 20, 656–657 (1984).
[CrossRef]

Ferroelectrics (1)

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

IEEE J. Quantum Electron. (1)

G. C. Valley and A. L. Smirl, “Theory of transient energy transfer in gallium arsenide,” IEEE J. Quantum Electron. 24, 304–310 (1988).
[CrossRef]

J. Appl. Phys. (1)

Ph. Refregier, L. Solymar, H. Rajbenbach, and J. P. Huignard, “Two-beam coupling in photorefractive Bi12SiO20 crystals with moving grating: theory and experiments,” J. Appl. Phys. 58, 45–57 (1985).
[CrossRef]

J. Opt. Soc. Am. B (2)

Opt. Commun. (6)

J. Kumar, G. Albanese, and W. H. Steier, “Measurement of two-wave mixing gain in GaAs with a moving grating,” Opt. Commun. 63, 191–193 (1987).
[CrossRef]

G. Lesaux, J. C. Launay, and A. Brun, “Transient photocurrent induced by nanosecond light pulses in Bi12SiO20 and Bi12GeO20,” Opt. Commun. 57, 166–167 (1986).
[CrossRef]

S. I. Stepanov and M. P. Petrov, “Efficient unstationary holographic recording in photorefractive crystals under an external alternating electric field,” Opt. Commun. 53, 292–295 (1985), and references therein.
[CrossRef]

X. Gan, S. Ye, and Y. Sun, “Alternating electric field enhancement of two-wave mixing gain in photorefractive Bi12SiO20,” Opt. Commun. 66, 155–160 (1988).
[CrossRef]

C. Stace, A. K. Powell, K. Walsh, and T. J. Hall, “Coupling modulation in photorefractive materials by applying a.c. electric fields,” Opt. Commun. 70, 509–514 (1989).
[CrossRef]

S. I. Stepanov, V. V. Kulikov, and M. P. Petrov, “‘Running’ holograms in photorefractive Bi12SiO20 crystals,” Opt. Commun. 44, 19–23 (1982).
[CrossRef]

Opt. Lett. (6)

Prog. Quantum Electron. (1)

T. J. Hall, R. Jaura, L. M. Connors, and P. D. Foote, “The photorefractive effect—a review,” Prog. Quantum Electron. 10, 77–146 (1985).
[CrossRef]

Sov. Tech. Phys. Lett. (1)

S. L. Sochava, S. I. Stepanov, and M. P. Petrov, “Ring oscillator using a photorefractive Bi12TiO20 crystal,” Sov. Tech. Phys. Lett. 13, 274–275 (1987).

Other (3)

P. D. Foote, “Optically induced anisotropic light diffraction in photorefractive crystals,” doctoral dissertation (University of London, London, 1987).

D. Dascǎlu, Electronic Processes in Unipolar Solid-State Devices (Abacus, Tunbridge Wells, England, 1977).

L. Solymar and D. J. Cooke, Volume Holography and Volume Gratings (Academic, London, 1981).

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Figures (20)

Fig. 1
Fig. 1

Examples of symmetric ac field waveforms.

Fig. 2
Fig. 2

Comparison of enhancement factors (η) versus P parameter for (a) square waveform and (b) sinusoidal waveform.

Fig. 3
Fig. 3

Two-wave-coupling experimental points for Bi12SiO20 and theoretical curves versus peak applied cosinusoidal field for several photorefractive grating spacings. The frequency of the applied field was 50 Hz, the intensity was less than 2 mW cm−2 to ensure that the grating time constant was greater than the period of the field, the beam ratio was 200:1, and the interaction length was 2.7 mm. The wavelength was 514.5 nm, for which the absorption was 2.2 cm−1 and the optical activity was 0.6 rad mm−1. The space-charge field and the applied electric field were aligned along the [110] direction.

Fig. 4
Fig. 4

Two-wave mixing in GaAs:Cr. The space-charge field and the applied electric field were both in the [110] direction. The frequency of the applied cosinusoidal electric field was 780 kHz. The beam ratio was 30:1, and the intensity was 10 mW cm−2. The grating period was 44.9 μm, the absorption was 2.2 cm−1, and the laser wavelength was 1.09 μm. Comprehensive details of the experiment are presented in Refs. 10 and 11.

Fig. 5
Fig. 5

Experimental points comparing the coupling versus peak applied voltage for square and sinusoidal waveforms at two grating spacings in Bi12SiO20. The dashed and solid curves show the trends of the square and sinusoid experimental points and are not theoretical curves. The frequency of the applied field is 50 Hz, the input intensity is less than 2 mW cm−2, the beam ratio is 200:1, and the interaction length is 2.7 mm.

Fig. 6
Fig. 6

Exponential-edge waveform q(t). This is a practical limitation for an applied square waveform.

Fig. 7
Fig. 7

Waveform shape function q(t) versus time for several values of the exponential time constant τ.

Fig. 8
Fig. 8

Enhancement factor η versus the P parameter for varied values of the exponential time constant τ.

Fig. 9
Fig. 9

Slew-rate-limited square waveform.

Fig. 10
Fig. 10

Waveform shape function q(t) for varied slew-rate parameter R.

Fig. 11
Fig. 11

Enhancement factor η versus the P parameter for varied values of the slew-rate parameter R.

Fig. 12
Fig. 12

Unequal mark–space ratio waveform defined by the u parameter.

Fig. 13
Fig. 13

Enhancement factor η (real component) versus the P parameter for varied values of the mark–space ratio parameter u. u varies from 0.5 (perfect square waveform) to 0.7 in steps of 0.02.

Fig. 14
Fig. 14

Variation of the maximum two-wave mixing gain enhancement ξmax as a function of ωgτg, showing the effects of increasing the crystal absorption αL.

Fig. 15
Fig. 15

Two-wave mixing in GaAs with an applied dc field of 10 kV cm−1 and a resonantly moving grating. The signal beam is monitored by a photodetector as the direction of fringe movement is periodically reversed. Crystal orientation and absorption coefficient are as in Fig. 4. The interaction length was 4 mm. The beam ratio was 60:1, and the incident intensity was 20 mW cm−2.

Fig. 16
Fig. 16

Experimental results and theoretical predictions for the grating time constants in Bi12SiO20 for varied applied sinusoidal fields. Three spatial frequencies are shown.

Fig. 17
Fig. 17

Schematic diagram of space-charge field magnitude versus spatial frequency of photorefractive grating for the case of no electric field (bounded by ED and ES) and applied dc field cases (bounded by EA and ES). The form of the space-charge field is shown for a large applied field (EA2) and for a small applied field (EA1).

Fig. 18
Fig. 18

Schematic diagram of space-charge field magnitude versus spatial frequency for no applied electric field (bounded by ED and ES) and for an applied ac field (bounded by ED and Es).

Fig. 19
Fig. 19

Comparison of enhancement sensitivity of the space-charge field with spatial frequency for sinusoidal and square applied waveforms. The gain ΓL shown is a theoretical prediction for Bi12SiO20.

Fig. 20
Fig. 20

Enhancement factor as a function of spatial frequency for several values of the transport length parameter P0. P0 = 4, 6, …, 20. Note the narrow range of spatial frequencies for which enhancement is significant.

Equations (139)

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I = I 0 { 1 + m cos [ K X + ϕ ( t ) ] } ,
N D + t = ( N D N D + ) σ I = γ R N D + n ,
n t = N D + t = ( D n + μ n E ) ,
( E ) = q ( N D + N A n ) ,
W = N D N D + N D .
W c = N D N A N D n c N D ,
E D = D μ K ,
E s = W c ( 1 W c ) q N d c K ,
K c 2 = μ D W c ( 1 W c ) q N D c
E c = E D ( K c ) = E S ( K C ) .
ν = | K | K c ,
W W c { 1 + m cos [ ξ + ϕ ( t ) ] } ( 1 W ) ( 1 W C ) n θ 2 ν 2 E ξ t = 0 ,
ξ ( ν n ξ + n E + E t ) = 0 ,
ν E ξ + W W c W c ( 1 W c ) = 0 ,
E ( 0 ) = E A ,
E ( 1 ) = E 1 exp ( i ξ ) + E 1 * exp ( i ξ ) ,
( t + s ) E 1 = s F exp [ i ϕ ( t ) ] ,
F = 1 2 [ i ν + E A 1 i ν ( E A + i ν ) ] ,
s = 1 i ν ( i ν + E A ) 1 i Θ 2 ν ( i ν + E A ) = ( 1 τ g + i ω g ) .
E 1 = F = 1 2 [ i ν + E A 1 i ν ( E A + i ν ) ] .
F ( E A ) = F * ( E A ) ,
s ( E A ) = s * ( E A ) ,
q ( t ) = 1 , E 1 t + s E 1 = s F , T 2 t < 0 ,
q ( t ) = 1 , E 1 t + s * E 1 = s * F * , 0 t < T 2 .
E 1 ( T 2 ) = E 1 ( + T 2 ) ,
E 1 ( 0 ) = E 1 ( 0 + ) .
E 1 = ( F + F * ) [ exp ( s * T / 2 ) 1 ] exp ( s * T / 2 ) + exp ( s T / 2 ) exp ( s t ) + F , T 2 < t < 0 ,
E 1 = ( F + F * ) [ exp ( s T / 2 ) 1 ] exp ( s * T / 2 ) + exp ( s T / 2 ) exp ( s * t ) F * , 0 < t < T 2 .
E 1 = s F s * F * s + s * .
E 1 t + s E 1 = s F
s = ½ ( s + s * )
s F = ½ ( s F s * F * ) ,
s = 1 + ν 2 i ν Ê A q ( t ) ( 1 + Θ 2 ν 2 ) [ 1 i P q ( t ) ] ,
P = Θ 2 ν Ê A 1 + Θ 2 ν 2 ,
P = L Ê A K 1 + L D 2 K 2 ,
L D = ( D τ R ) 1 / 2 ,
L E A = μ τ R E A ,
s = 1 + ν 2 1 + Θ 2 ν 2 1 P I 0 ( P ) + 1 Θ 2 I 1 ( P ) ,
s F = 1 2 [ i ν 1 + Θ 2 ν 2 1 P I 0 ( P ) + i Θ 2 ν I 1 ( P ) ] ,
I 0 ( P ) = 1 T T / 2 T / 2 P 1 i P q ( t ) d t
I 1 ( P ) = 1 T T / 2 T / 2 i P q ( t ) 1 i P q ( t ) d t = 1 I 0 ( P ) P .
E 1 = 1 2 [ i ν + E A 1 i ν ( i ν + E A ) ] .
E A = i I 1 ( P ) I 0 ( P ) Ê A = i η Ê A ,
η = I 1 ( P ) I 0 ( P ) = 1 I 0 ( P ) 1 P .
q ( t ) = q ( t ) , T 2 t 0
q ( t ) = q ( t + T 2 ) , T 2 t 0 ,
I 0 ( P ) = 2 T 0 T / 2 P 1 + P 2 q 2 ( t ) d t ,
q ( t ) = 1 , T 2 t < 0 = + 1 0 < t T 2 .
I 0 ( P ) = 2 T 0 T / 2 P 1 + P 2 d t = P 1 + P 2 .
η = P ,
E A = i P Ê A .
q ( t ) = cos ( t ) .
I 0 ( P ) = 1 π 0 π P 1 + P 2 cos 2 ( t ) d t = P [ 1 + P 2 ] 1 / 2 ,
E A = i 1 P ( [ P 2 + 1 ] 1 / 2 1 ) Ê A .
E A = i P 2 Ê A ;
E A = i Ê A ,
τ q t + q = + 1 , 0 t 1 2 , = 1 , ½ t 0 .
q ( t ) = 1 2 { 1 exp [ 1 / ( 2 τ ) ] } exp [ 1 / ( 2 τ ) ] exp [ 1 / ( 2 τ ) ] exp ( t / τ ) , 0 t 1 2 .
I 0 ( P ) = P 1 + P 2 + 2 τ P 2 1 + P 2 × ( tan 1 ( P { 1 A exp [ 1 / ( 2 τ ) ] } tan 1 [ P ( 1 A ) ] ) + τ P ( 1 + P 2 ) ln ( 1 + P 2 { 1 A exp [ 1 / ( 2 τ ) ] } 2 1 + P 2 ( 1 A ) 2 ) ,
A = 2 { exp [ 1 / ( 2 τ ) ] } exp [ 1 / ( 2 τ ) ] exp [ 1 / ( 2 τ ) ] .
I 0 ( P ) = 1 P + 2 τ π ,
η = 1 1 P + 2 τ π 1 P .
q ( t ) = 2 t R , 0 < t < R 2 , q ( t ) = 1 , R 2 < t < 1 R 2 , q ( t ) = 2 R ( 1 t ) , 1 R 2 < t < 1
I 0 ( P ) = R tan 1 ( P ) + P 1 + P 2 ( 1 R ) .
E A = i ( 1 tan 1 P 1 P ) Ê A .
η = ( 1 + P 2 ) P [ 1 + i P ( 2 U 1 ) ] 1 P .
I 0 ( P ) = 2 T 0 T / 2 P 1 + P 2 q 2 ( t ) d t
η = 1 I 0 ( P ) 1 P ,
q 2 ( t ) = 1 .
E 1 = s ( s i ω ) F ,
s = 1 τ g + i ω g .
s i ω = s i ω g = 1 / τ g
E 1 = ( 1 + i ω g τ g ) F .
F ½ E A .
ω g τ g L E K 1 + ( L D K ) 2 ,
s = 1 2 ( s + s * ) = Re ( s ) = 1 τ g ,
E 1 = s F ½ ( s + s * ) .
Im ( E 1 ) = s F s * F * s + s * ,
s ( z ) = s ( 0 ) exp ( α z ) .
Im [ E 1 ] = Im [ s F s i ω ] .
I 1 ( L ) = I 1 ( 0 ) exp ( Γ ¯ l ) ,
Γ ¯ ( z ) = 1 L 0 L Γ ( z ) d z
ξ = Im [ 1 L 0 L s ( 0 ) exp ( α z ) s ( 0 ) exp ( α z ) i ω d z ] .
ξ = 1 α L ( tan 1 [ ( ω g ω ) τ g ] + tan 1 { [ ω ω g exp ( α L ) ] τ g exp ( α L ) } ) .
ω m τ g = [ 1 + ( ω g τ g ) 2 ] 1 / 2 exp ( α L 2 ) .
ω m = ω g exp ( α L 2 ) ,
ξ max = 1 α L [ tan 1 ( A ) + tan 1 ( b ) ] ,
A = ω g τ g [ 1 exp ( α L 2 ) ]
B = ω g τ g [ exp ( α L 2 ) 1 ] .
ξ max = 2 α L tan 1 [ ω g τ g ( α L 2 ) ] .
ξ max ω g τ g ,
ξ max π / α L .
ω g τ g L E k 1 + ( L D k ) 2 ,
ξ max ω g τ g = π α Δ L .
Q = 2 π λ L Λ 2 < 1 ,
1 exp [ s ( z ) t ] ,
s ( z ) = s ( 0 ) exp ( α z ) i ω m
= 1 τ g exp ( α z ) + i [ ω g exp ( α z ) ω m ] .
s ( E A = 0 ) = 1 + ν 2 1 + θ 2 ν 2 ,
τ g = 1 + θ 2 ν 2 1 + ν 2 ,
s = 1 τ g + i ω g = 1 1 + P 2 ( 1 + ν 2 1 + Θ 2 ν 2 + P 2 Θ 2 ) + i P 1 + P 2 ( 1 + ν 2 1 + Θ 2 ν 2 1 Θ 2 ) .
τ g = ( 1 + P 2 ) 1 ( 1 + ν 2 1 + Θ 2 ν 2 + P 2 Θ 2 ) ,
τ g = ( 1 + P 2 ) 2 1 { 1 + ν 2 1 + Θ 2 ν 2 + [ ( 1 + P 2 ) 1 / 2 1 Θ 2 } ,
τ g = Θ 2 .
τ g = Θ 2 τ c
( 1 W c ) σ I 0 ,
E s = W c ( 1 W c ) q W D K ,
E D = D K μ ,
E D = L D 2 K μ τ R ,
L D = ( D τ R ) 1 / 2 .
| E 1 | = ½ [ E A 2 + E D 2 ] 1 / 2 .
| E 1 | ½ E A .
E 1 = 1 2 ( E A + i E D 1 + E D E S i E A E S ) .
E 1 = i 2 E D 1 + E D E S .
E 1 = i 2 ( η Ê A 1 + η Ê A / E S ) .
E D = η Ê A .
E D = P Ê A .
P L Ê A K .
E D = L Ê A 2 k μ τ R .
ξ = s s i ω = 1 + i ω g τ g 1 + i ( ω g ω ) τ g .
ζ = K / K 0 .
P 0 = L E K 0 .
P = L E K ,
P = P 0 K K 0 = P 0 ζ .
τ g = 1 + ( L E K ) 2 ,
ω g = 1 τ g L E K = L E K 1 + ( L E K ) 2 .
ω = P 0 1 + P 0 2 .
| ξ | 2 = 1 + P 0 2 1 + [ P 0 ζ ( 1 + P 0 2 ) P 0 ( 1 + P 0 2 ζ 2 ) 1 + P 0 2 ] 2 .
τ f t + g f = h .
g ( t ) = g ( χ , ψ ) | χ = t τ , ψ = t T ,
h ( t ) = h ( χ , ψ ) | χ = t τ , ψ = t T ,
f ( t ) = f ( χ , ψ , ρ ) | χ = t τ , ψ = t τ ,
f χ + ρ 1 f ψ + g f = h .
f = f 0 + f 1 ρ + O ( ρ 2 ) .
f 0 ψ = 0
f 0 χ + f 1 ψ + g f 0 = h .
P ( ) = 1 / 2 + 1 / 2 ( ) d ψ .
f 0 χ + P ( g ) f 0 = P ( h ) .
f 0 = P ( h ) P ( g ) ,

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