Abstract

It is shown that weakly damped space-charge waves exist in photorefractive crystals with a sufficiently large lifetime–mobility product. The dispersion law, the damping constant, and the quality factor of these waves and their dependence on the crystal parameters and on the experimental conditions are found. The instability of the fundamental grating against excitation of these waves is investigated; the fundamental grating is excited either by a moving interference pattern or by a standing interference pattern plus an oscillating external electric field. The dependence of the threshold and the characteristic exponent of the instability on the wave vectors of the exciting waves and on the experimental parameters is found for the three-dimensional case. It turns out that the strongest instabilities correspond to the cases of optimal enhancement of photoinduced gratings. The theory is compared with the experimental data for BSO crystals.

© 1993 Optical Society of America

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  1. P. Günter and J.-P. Huignard, eds., Photorefractive Materials and Their Applications I and II, Vols. 61 and 62 of Topics in Applied Physics (Springer-Verlag, Berlin, 1988 and 1989).
    [CrossRef]
  2. N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals,” Ferroelectrics 22, 949–960 (1979); Ferroelectrics961–964 (1979).
    [CrossRef]
  3. A. Temple and C. Warde, “Anisotropic scattering in photorefractive crystals,” J. Opt. Soc. Am. B 3, 337–341 (1986).
    [CrossRef]
  4. B. I. Sturman and V. M. Fridkin, The Photovoltaic and Photorefractive Effect in Noncentrosymmetric Materials (Gordon & Breach, New York, 1992).
  5. J.-P. Huignard and A. Marrakchi, “Coherent signal beam amplification in two-wave-mixing experiments with photorefractive Bi12SiO20crystals,” Opt. Commun. 38, 249–254 (1981).
    [CrossRef]
  6. P. Refregier, L. Solymar, H. Rajbenbach, and J.-P. Huignard, “Two-beam coupling in photorefractive Bi12SiO20crystals with moving grating: theory and experiments,” J. Appl. Phys. 58, 45–57 (1985).
    [CrossRef]
  7. 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).
    [CrossRef]
  8. S. Mallick, B. Imbert, H. Ducollet, J. P. Herriau, and J.-P. Huignard, “Generation of spatial subharmonics by two-wave mixing in a nonlinear photorefractive medium,” J. Appl. Phys. 63, 5660–5663 (1988).
    [CrossRef]
  9. D. J. Webb and L. Solymar, “Observations of spatial subharmonics arising during two-wave mixing in BSO,” Opt. Commun. 74, 386–389 (1990).
    [CrossRef]
  10. D. J. Webb, L. B. Au, D. C. Jones, and L. Solymar, “Onset of subharmonics generated by forward wave interactions in Bi12SiO20,” Appl. Phys. Lett. 57, 1602–1604 (1990).
    [CrossRef]
  11. J. Takacs, M. Schaub, and L. Solymar, “Subharmonics in photorefractive Bi12TiO20crystals,” Opt. Commun. 91, 252–254 (1992).
    [CrossRef]
  12. J. Takacs and L. Solymar, “Subharmonics in Bi12SiO20 with an applied a.c. electric field,” Opt. Lett. 17, 247–248 (1992).
    [CrossRef] [PubMed]
  13. K. H. Ringhofer and L. Solymar, “New gain mechanism for wave amplification in photorefractive crystals,” Appl. Phys. Lett. 53, 1039–1040 (1988).
    [CrossRef]
  14. L. B. Au, L. Solymar, and K. H. Ringhofer, “Subharmonics in BSO,” in Technical Digest on Photorefractive Materials, Effects and Devices II (Société Francais d’Optique, Aussois, France, 1990), pp. 87–91.
  15. O. P. Nestiorkin, “Instability of spatial subharmonics under hologram recording in a photorefractive crystal,” Opt. Commun. 81, 315–320 (1991).
    [CrossRef]
  16. A. Bledowski, B. Sturman, J. Otten, and K. H. Ringhofer, “Analytic and Numeric Results in Subharmonics in BSO,” in Photorefractive Materials, Effects and Devices, Vol. 14 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), pp. 436–439.
  17. B. Sturman, A. Bledowski, J. Otten, and K. H. Ringhofer, “Spatial subharmonics in photorefractive crystals,” J. Opt. Soc. Am. B 9, 672–681 (1992).
    [CrossRef]
  18. B. Sturman, A. Bledowski, J. Otten, and K. H. Ringhofer, “Subharmonics in photorefractive crystals,” Sov. Phys. JETP 102, 215–224 (1992).
  19. B. I. Sturman, M. Mann, J. Otten, and K. H. Ringhofer, “Subharmonic generation in photorefractive crystals: Application of the theory to experiment,” Appl. Phys. A 55, 55–60 (1992).
    [CrossRef]
  20. B. I. Sturman, M. Mann, and K. H. Ringhofer, “Instability of moving gratings in photorefractive crystals,” Appl. Phys. A 55, 235–241 (1992).
    [CrossRef]
  21. Such waves and the instabilities connected with them are also known in the theory of semiconductors; see, for example, R. F. Kazarinov, R. A. Suris, and B. I. Fuks, “Instability with respect to waves of spatial charge-exchange in compensated semiconductors,” Sov. Phys. Semicond. 7, 480 (1973); V. N. Alimpiev and I. R. Gural’nik, “Parametric instability in a photosensitive semiconductor due to a travelling illumination intensity grating,” Sov. Phys. Semicond. 20, 512–514 (1986).
  22. C. S. K. Walsh, A. K. Powell, and T. J. Hall, “Techniques for the enhancement of space-charge fields in photorefractive materials,” J. Opt. Soc. Am. B 7, 288–303 (1990).
    [CrossRef]
  23. G. Pauliat, A. Villing, J. C. Launau, and G. Roosen, “Optical measurements of charge-carrier mobilities in photorefractive sillenite crystals,” J. Opt. Soc. Am. B 7, 1481–1990 (1990).
    [CrossRef]
  24. H. G. Festl, P. Hertel, E. Krätzig, and R. von Baltz, “Investigations of the photovoltaic tensor in doped LiNbO3,” Phys. Status Solidi B 113, 157–164 (1982).
    [CrossRef]
  25. M. B. Klein and R. N. Schwartz, “Photorefractive effect in BaTiO3: microscopic origins,” J. Opt. Soc. Am. 3, 293–305 (1986).
    [CrossRef]
  26. M. Peltier and F. Micheron, “Volume hologram recording and charge transfer process in Bi12SiO20and Bi12GeO20,” J. Appl. Phys. 48, 3683–3690 (1977).
    [CrossRef]
  27. G. C. Valley, H. Rajbenbach, and H. J. von Bardeleben, “Mobility-lifetime product of photoexcited electrons in GaAs,” Appl. Phys. Lett. 56, 364–366 (1990).
    [CrossRef]
  28. V. E. Zakharov, S. L. Musher, and A. M. Rubenchik, “Hamiltonian approach to the description of non-linear plasma phenomena,” Phys. Rep. 129, 285–366 (1985).
    [CrossRef]
  29. V. E. Zakharov, V. S. L’vov, and G. Falkovich, Kolmogorov Spectra of Turbulence I: Wave Turbulence (Springer-Verlag, Berlin, 1992).
    [CrossRef]
  30. In the three-dimensional case the simplification of Eq. (31) is best performed in the k representation.
  31. E. L. Ince, Ordinary Differential Equations (Dover, New York, 1956).
  32. In particular, an inhomogeneity such as this can suppress the resonance excitation of a moving grating, since the resonance condition |Ω − ωK| ≲ γK is fulfilled only in a very narrow layer of the crystal.

1992 (6)

B. Sturman, A. Bledowski, J. Otten, and K. H. Ringhofer, “Spatial subharmonics in photorefractive crystals,” J. Opt. Soc. Am. B 9, 672–681 (1992).
[CrossRef]

B. Sturman, A. Bledowski, J. Otten, and K. H. Ringhofer, “Subharmonics in photorefractive crystals,” Sov. Phys. JETP 102, 215–224 (1992).

B. I. Sturman, M. Mann, J. Otten, and K. H. Ringhofer, “Subharmonic generation in photorefractive crystals: Application of the theory to experiment,” Appl. Phys. A 55, 55–60 (1992).
[CrossRef]

B. I. Sturman, M. Mann, and K. H. Ringhofer, “Instability of moving gratings in photorefractive crystals,” Appl. Phys. A 55, 235–241 (1992).
[CrossRef]

J. Takacs, M. Schaub, and L. Solymar, “Subharmonics in photorefractive Bi12TiO20crystals,” Opt. Commun. 91, 252–254 (1992).
[CrossRef]

J. Takacs and L. Solymar, “Subharmonics in Bi12SiO20 with an applied a.c. electric field,” Opt. Lett. 17, 247–248 (1992).
[CrossRef] [PubMed]

1991 (1)

O. P. Nestiorkin, “Instability of spatial subharmonics under hologram recording in a photorefractive crystal,” Opt. Commun. 81, 315–320 (1991).
[CrossRef]

1990 (5)

G. C. Valley, H. Rajbenbach, and H. J. von Bardeleben, “Mobility-lifetime product of photoexcited electrons in GaAs,” Appl. Phys. Lett. 56, 364–366 (1990).
[CrossRef]

C. S. K. Walsh, A. K. Powell, and T. J. Hall, “Techniques for the enhancement of space-charge fields in photorefractive materials,” J. Opt. Soc. Am. B 7, 288–303 (1990).
[CrossRef]

G. Pauliat, A. Villing, J. C. Launau, and G. Roosen, “Optical measurements of charge-carrier mobilities in photorefractive sillenite crystals,” J. Opt. Soc. Am. B 7, 1481–1990 (1990).
[CrossRef]

D. J. Webb and L. Solymar, “Observations of spatial subharmonics arising during two-wave mixing in BSO,” Opt. Commun. 74, 386–389 (1990).
[CrossRef]

D. J. Webb, L. B. Au, D. C. Jones, and L. Solymar, “Onset of subharmonics generated by forward wave interactions in Bi12SiO20,” Appl. Phys. Lett. 57, 1602–1604 (1990).
[CrossRef]

1988 (2)

S. Mallick, B. Imbert, H. Ducollet, J. P. Herriau, and J.-P. Huignard, “Generation of spatial subharmonics by two-wave mixing in a nonlinear photorefractive medium,” J. Appl. Phys. 63, 5660–5663 (1988).
[CrossRef]

K. H. Ringhofer and L. Solymar, “New gain mechanism for wave amplification in photorefractive crystals,” Appl. Phys. Lett. 53, 1039–1040 (1988).
[CrossRef]

1986 (2)

A. Temple and C. Warde, “Anisotropic scattering in photorefractive crystals,” J. Opt. Soc. Am. B 3, 337–341 (1986).
[CrossRef]

M. B. Klein and R. N. Schwartz, “Photorefractive effect in BaTiO3: microscopic origins,” J. Opt. Soc. Am. 3, 293–305 (1986).
[CrossRef]

1985 (3)

V. E. Zakharov, S. L. Musher, and A. M. Rubenchik, “Hamiltonian approach to the description of non-linear plasma phenomena,” Phys. Rep. 129, 285–366 (1985).
[CrossRef]

P. Refregier, L. Solymar, H. Rajbenbach, and J.-P. Huignard, “Two-beam coupling in photorefractive Bi12SiO20crystals 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).
[CrossRef]

1982 (1)

H. G. Festl, P. Hertel, E. Krätzig, and R. von Baltz, “Investigations of the photovoltaic tensor in doped LiNbO3,” Phys. Status Solidi B 113, 157–164 (1982).
[CrossRef]

1981 (1)

J.-P. Huignard and A. Marrakchi, “Coherent signal beam amplification in two-wave-mixing experiments with photorefractive Bi12SiO20crystals,” Opt. Commun. 38, 249–254 (1981).
[CrossRef]

1979 (1)

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals,” Ferroelectrics 22, 949–960 (1979); Ferroelectrics961–964 (1979).
[CrossRef]

1977 (1)

M. Peltier and F. Micheron, “Volume hologram recording and charge transfer process in Bi12SiO20and Bi12GeO20,” J. Appl. Phys. 48, 3683–3690 (1977).
[CrossRef]

1973 (1)

Such waves and the instabilities connected with them are also known in the theory of semiconductors; see, for example, R. F. Kazarinov, R. A. Suris, and B. I. Fuks, “Instability with respect to waves of spatial charge-exchange in compensated semiconductors,” Sov. Phys. Semicond. 7, 480 (1973); V. N. Alimpiev and I. R. Gural’nik, “Parametric instability in a photosensitive semiconductor due to a travelling illumination intensity grating,” Sov. Phys. Semicond. 20, 512–514 (1986).

Au, L. B.

D. J. Webb, L. B. Au, D. C. Jones, and L. Solymar, “Onset of subharmonics generated by forward wave interactions in Bi12SiO20,” Appl. Phys. Lett. 57, 1602–1604 (1990).
[CrossRef]

L. B. Au, L. Solymar, and K. H. Ringhofer, “Subharmonics in BSO,” in Technical Digest on Photorefractive Materials, Effects and Devices II (Société Francais d’Optique, Aussois, France, 1990), pp. 87–91.

Bledowski, A.

B. Sturman, A. Bledowski, J. Otten, and K. H. Ringhofer, “Spatial subharmonics in photorefractive crystals,” J. Opt. Soc. Am. B 9, 672–681 (1992).
[CrossRef]

B. Sturman, A. Bledowski, J. Otten, and K. H. Ringhofer, “Subharmonics in photorefractive crystals,” Sov. Phys. JETP 102, 215–224 (1992).

A. Bledowski, B. Sturman, J. Otten, and K. H. Ringhofer, “Analytic and Numeric Results in Subharmonics in BSO,” in Photorefractive Materials, Effects and Devices, Vol. 14 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), pp. 436–439.

Ducollet, H.

S. Mallick, B. Imbert, H. Ducollet, J. P. Herriau, and J.-P. Huignard, “Generation of spatial subharmonics by two-wave mixing in a nonlinear photorefractive medium,” J. Appl. Phys. 63, 5660–5663 (1988).
[CrossRef]

Falkovich, G.

V. E. Zakharov, V. S. L’vov, and G. Falkovich, Kolmogorov Spectra of Turbulence I: Wave Turbulence (Springer-Verlag, Berlin, 1992).
[CrossRef]

Festl, H. G.

H. G. Festl, P. Hertel, E. Krätzig, and R. von Baltz, “Investigations of the photovoltaic tensor in doped LiNbO3,” Phys. Status Solidi B 113, 157–164 (1982).
[CrossRef]

Fridkin, V. M.

B. I. Sturman and V. M. Fridkin, The Photovoltaic and Photorefractive Effect in Noncentrosymmetric Materials (Gordon & Breach, New York, 1992).

Fuks, B. I.

Such waves and the instabilities connected with them are also known in the theory of semiconductors; see, for example, R. F. Kazarinov, R. A. Suris, and B. I. Fuks, “Instability with respect to waves of spatial charge-exchange in compensated semiconductors,” Sov. Phys. Semicond. 7, 480 (1973); V. N. Alimpiev and I. R. Gural’nik, “Parametric instability in a photosensitive semiconductor due to a travelling illumination intensity grating,” Sov. Phys. Semicond. 20, 512–514 (1986).

Hall, T. J.

Herriau, J. P.

S. Mallick, B. Imbert, H. Ducollet, J. P. Herriau, and J.-P. Huignard, “Generation of spatial subharmonics by two-wave mixing in a nonlinear photorefractive medium,” J. Appl. Phys. 63, 5660–5663 (1988).
[CrossRef]

Hertel, P.

H. G. Festl, P. Hertel, E. Krätzig, and R. von Baltz, “Investigations of the photovoltaic tensor in doped LiNbO3,” Phys. Status Solidi B 113, 157–164 (1982).
[CrossRef]

Huignard, J.-P.

S. Mallick, B. Imbert, H. Ducollet, J. P. Herriau, and J.-P. Huignard, “Generation of spatial subharmonics by two-wave mixing in a nonlinear photorefractive medium,” J. Appl. Phys. 63, 5660–5663 (1988).
[CrossRef]

P. Refregier, L. Solymar, H. Rajbenbach, and J.-P. Huignard, “Two-beam coupling in photorefractive Bi12SiO20crystals with moving grating: theory and experiments,” J. Appl. Phys. 58, 45–57 (1985).
[CrossRef]

J.-P. Huignard and A. Marrakchi, “Coherent signal beam amplification in two-wave-mixing experiments with photorefractive Bi12SiO20crystals,” Opt. Commun. 38, 249–254 (1981).
[CrossRef]

Imbert, B.

S. Mallick, B. Imbert, H. Ducollet, J. P. Herriau, and J.-P. Huignard, “Generation of spatial subharmonics by two-wave mixing in a nonlinear photorefractive medium,” J. Appl. Phys. 63, 5660–5663 (1988).
[CrossRef]

Ince, E. L.

E. L. Ince, Ordinary Differential Equations (Dover, New York, 1956).

Jones, D. C.

D. J. Webb, L. B. Au, D. C. Jones, and L. Solymar, “Onset of subharmonics generated by forward wave interactions in Bi12SiO20,” Appl. Phys. Lett. 57, 1602–1604 (1990).
[CrossRef]

Kazarinov, R. F.

Such waves and the instabilities connected with them are also known in the theory of semiconductors; see, for example, R. F. Kazarinov, R. A. Suris, and B. I. Fuks, “Instability with respect to waves of spatial charge-exchange in compensated semiconductors,” Sov. Phys. Semicond. 7, 480 (1973); V. N. Alimpiev and I. R. Gural’nik, “Parametric instability in a photosensitive semiconductor due to a travelling illumination intensity grating,” Sov. Phys. Semicond. 20, 512–514 (1986).

Klein, M. B.

M. B. Klein and R. N. Schwartz, “Photorefractive effect in BaTiO3: microscopic origins,” J. Opt. Soc. Am. 3, 293–305 (1986).
[CrossRef]

Krätzig, E.

H. G. Festl, P. Hertel, E. Krätzig, and R. von Baltz, “Investigations of the photovoltaic tensor in doped LiNbO3,” Phys. Status Solidi B 113, 157–164 (1982).
[CrossRef]

Kukhtarev, N. V.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals,” Ferroelectrics 22, 949–960 (1979); Ferroelectrics961–964 (1979).
[CrossRef]

L’vov, V. S.

V. E. Zakharov, V. S. L’vov, and G. Falkovich, Kolmogorov Spectra of Turbulence I: Wave Turbulence (Springer-Verlag, Berlin, 1992).
[CrossRef]

Launau, J. C.

Mallick, S.

S. Mallick, B. Imbert, H. Ducollet, J. P. Herriau, and J.-P. Huignard, “Generation of spatial subharmonics by two-wave mixing in a nonlinear photorefractive medium,” J. Appl. Phys. 63, 5660–5663 (1988).
[CrossRef]

Mann, M.

B. I. Sturman, M. Mann, and K. H. Ringhofer, “Instability of moving gratings in photorefractive crystals,” Appl. Phys. A 55, 235–241 (1992).
[CrossRef]

B. I. Sturman, M. Mann, J. Otten, and K. H. Ringhofer, “Subharmonic generation in photorefractive crystals: Application of the theory to experiment,” Appl. Phys. A 55, 55–60 (1992).
[CrossRef]

Markov, V. B.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals,” Ferroelectrics 22, 949–960 (1979); Ferroelectrics961–964 (1979).
[CrossRef]

Marrakchi, A.

J.-P. Huignard and A. Marrakchi, “Coherent signal beam amplification in two-wave-mixing experiments with photorefractive Bi12SiO20crystals,” Opt. Commun. 38, 249–254 (1981).
[CrossRef]

Micheron, F.

M. Peltier and F. Micheron, “Volume hologram recording and charge transfer process in Bi12SiO20and Bi12GeO20,” J. Appl. Phys. 48, 3683–3690 (1977).
[CrossRef]

Musher, S. L.

V. E. Zakharov, S. L. Musher, and A. M. Rubenchik, “Hamiltonian approach to the description of non-linear plasma phenomena,” Phys. Rep. 129, 285–366 (1985).
[CrossRef]

Nestiorkin, O. P.

O. P. Nestiorkin, “Instability of spatial subharmonics under hologram recording in a photorefractive crystal,” Opt. Commun. 81, 315–320 (1991).
[CrossRef]

Odulov, S. G.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals,” Ferroelectrics 22, 949–960 (1979); Ferroelectrics961–964 (1979).
[CrossRef]

Otten, J.

B. Sturman, A. Bledowski, J. Otten, and K. H. Ringhofer, “Subharmonics in photorefractive crystals,” Sov. Phys. JETP 102, 215–224 (1992).

B. Sturman, A. Bledowski, J. Otten, and K. H. Ringhofer, “Spatial subharmonics in photorefractive crystals,” J. Opt. Soc. Am. B 9, 672–681 (1992).
[CrossRef]

B. I. Sturman, M. Mann, J. Otten, and K. H. Ringhofer, “Subharmonic generation in photorefractive crystals: Application of the theory to experiment,” Appl. Phys. A 55, 55–60 (1992).
[CrossRef]

A. Bledowski, B. Sturman, J. Otten, and K. H. Ringhofer, “Analytic and Numeric Results in Subharmonics in BSO,” in Photorefractive Materials, Effects and Devices, Vol. 14 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), pp. 436–439.

Pauliat, G.

Peltier, M.

M. Peltier and F. Micheron, “Volume hologram recording and charge transfer process in Bi12SiO20and Bi12GeO20,” J. Appl. Phys. 48, 3683–3690 (1977).
[CrossRef]

Petrov, M. P.

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).
[CrossRef]

Powell, A. K.

Rajbenbach, H.

G. C. Valley, H. Rajbenbach, and H. J. von Bardeleben, “Mobility-lifetime product of photoexcited electrons in GaAs,” Appl. Phys. Lett. 56, 364–366 (1990).
[CrossRef]

P. Refregier, L. Solymar, H. Rajbenbach, and J.-P. Huignard, “Two-beam coupling in photorefractive Bi12SiO20crystals with moving grating: theory and experiments,” J. Appl. Phys. 58, 45–57 (1985).
[CrossRef]

Refregier, P.

P. Refregier, L. Solymar, H. Rajbenbach, and J.-P. Huignard, “Two-beam coupling in photorefractive Bi12SiO20crystals with moving grating: theory and experiments,” J. Appl. Phys. 58, 45–57 (1985).
[CrossRef]

Ringhofer, K. H.

B. I. Sturman, M. Mann, J. Otten, and K. H. Ringhofer, “Subharmonic generation in photorefractive crystals: Application of the theory to experiment,” Appl. Phys. A 55, 55–60 (1992).
[CrossRef]

B. I. Sturman, M. Mann, and K. H. Ringhofer, “Instability of moving gratings in photorefractive crystals,” Appl. Phys. A 55, 235–241 (1992).
[CrossRef]

B. Sturman, A. Bledowski, J. Otten, and K. H. Ringhofer, “Spatial subharmonics in photorefractive crystals,” J. Opt. Soc. Am. B 9, 672–681 (1992).
[CrossRef]

B. Sturman, A. Bledowski, J. Otten, and K. H. Ringhofer, “Subharmonics in photorefractive crystals,” Sov. Phys. JETP 102, 215–224 (1992).

K. H. Ringhofer and L. Solymar, “New gain mechanism for wave amplification in photorefractive crystals,” Appl. Phys. Lett. 53, 1039–1040 (1988).
[CrossRef]

A. Bledowski, B. Sturman, J. Otten, and K. H. Ringhofer, “Analytic and Numeric Results in Subharmonics in BSO,” in Photorefractive Materials, Effects and Devices, Vol. 14 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), pp. 436–439.

L. B. Au, L. Solymar, and K. H. Ringhofer, “Subharmonics in BSO,” in Technical Digest on Photorefractive Materials, Effects and Devices II (Société Francais d’Optique, Aussois, France, 1990), pp. 87–91.

Roosen, G.

Rubenchik, A. M.

V. E. Zakharov, S. L. Musher, and A. M. Rubenchik, “Hamiltonian approach to the description of non-linear plasma phenomena,” Phys. Rep. 129, 285–366 (1985).
[CrossRef]

Schaub, M.

J. Takacs, M. Schaub, and L. Solymar, “Subharmonics in photorefractive Bi12TiO20crystals,” Opt. Commun. 91, 252–254 (1992).
[CrossRef]

Schwartz, R. N.

M. B. Klein and R. N. Schwartz, “Photorefractive effect in BaTiO3: microscopic origins,” J. Opt. Soc. Am. 3, 293–305 (1986).
[CrossRef]

Solymar, L.

J. Takacs, M. Schaub, and L. Solymar, “Subharmonics in photorefractive Bi12TiO20crystals,” Opt. Commun. 91, 252–254 (1992).
[CrossRef]

J. Takacs and L. Solymar, “Subharmonics in Bi12SiO20 with an applied a.c. electric field,” Opt. Lett. 17, 247–248 (1992).
[CrossRef] [PubMed]

D. J. Webb and L. Solymar, “Observations of spatial subharmonics arising during two-wave mixing in BSO,” Opt. Commun. 74, 386–389 (1990).
[CrossRef]

D. J. Webb, L. B. Au, D. C. Jones, and L. Solymar, “Onset of subharmonics generated by forward wave interactions in Bi12SiO20,” Appl. Phys. Lett. 57, 1602–1604 (1990).
[CrossRef]

K. H. Ringhofer and L. Solymar, “New gain mechanism for wave amplification in photorefractive crystals,” Appl. Phys. Lett. 53, 1039–1040 (1988).
[CrossRef]

P. Refregier, L. Solymar, H. Rajbenbach, and J.-P. Huignard, “Two-beam coupling in photorefractive Bi12SiO20crystals with moving grating: theory and experiments,” J. Appl. Phys. 58, 45–57 (1985).
[CrossRef]

L. B. Au, L. Solymar, and K. H. Ringhofer, “Subharmonics in BSO,” in Technical Digest on Photorefractive Materials, Effects and Devices II (Société Francais d’Optique, Aussois, France, 1990), pp. 87–91.

Soskin, M. S.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals,” Ferroelectrics 22, 949–960 (1979); Ferroelectrics961–964 (1979).
[CrossRef]

Stepanov, S. I.

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).
[CrossRef]

Sturman, B.

B. Sturman, A. Bledowski, J. Otten, and K. H. Ringhofer, “Spatial subharmonics in photorefractive crystals,” J. Opt. Soc. Am. B 9, 672–681 (1992).
[CrossRef]

B. Sturman, A. Bledowski, J. Otten, and K. H. Ringhofer, “Subharmonics in photorefractive crystals,” Sov. Phys. JETP 102, 215–224 (1992).

A. Bledowski, B. Sturman, J. Otten, and K. H. Ringhofer, “Analytic and Numeric Results in Subharmonics in BSO,” in Photorefractive Materials, Effects and Devices, Vol. 14 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), pp. 436–439.

Sturman, B. I.

B. I. Sturman, M. Mann, J. Otten, and K. H. Ringhofer, “Subharmonic generation in photorefractive crystals: Application of the theory to experiment,” Appl. Phys. A 55, 55–60 (1992).
[CrossRef]

B. I. Sturman, M. Mann, and K. H. Ringhofer, “Instability of moving gratings in photorefractive crystals,” Appl. Phys. A 55, 235–241 (1992).
[CrossRef]

B. I. Sturman and V. M. Fridkin, The Photovoltaic and Photorefractive Effect in Noncentrosymmetric Materials (Gordon & Breach, New York, 1992).

Suris, R. A.

Such waves and the instabilities connected with them are also known in the theory of semiconductors; see, for example, R. F. Kazarinov, R. A. Suris, and B. I. Fuks, “Instability with respect to waves of spatial charge-exchange in compensated semiconductors,” Sov. Phys. Semicond. 7, 480 (1973); V. N. Alimpiev and I. R. Gural’nik, “Parametric instability in a photosensitive semiconductor due to a travelling illumination intensity grating,” Sov. Phys. Semicond. 20, 512–514 (1986).

Takacs, J.

J. Takacs and L. Solymar, “Subharmonics in Bi12SiO20 with an applied a.c. electric field,” Opt. Lett. 17, 247–248 (1992).
[CrossRef] [PubMed]

J. Takacs, M. Schaub, and L. Solymar, “Subharmonics in photorefractive Bi12TiO20crystals,” Opt. Commun. 91, 252–254 (1992).
[CrossRef]

Temple, A.

Valley, G. C.

G. C. Valley, H. Rajbenbach, and H. J. von Bardeleben, “Mobility-lifetime product of photoexcited electrons in GaAs,” Appl. Phys. Lett. 56, 364–366 (1990).
[CrossRef]

Villing, A.

Vinetskii, V. L.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals,” Ferroelectrics 22, 949–960 (1979); Ferroelectrics961–964 (1979).
[CrossRef]

von Baltz, R.

H. G. Festl, P. Hertel, E. Krätzig, and R. von Baltz, “Investigations of the photovoltaic tensor in doped LiNbO3,” Phys. Status Solidi B 113, 157–164 (1982).
[CrossRef]

von Bardeleben, H. J.

G. C. Valley, H. Rajbenbach, and H. J. von Bardeleben, “Mobility-lifetime product of photoexcited electrons in GaAs,” Appl. Phys. Lett. 56, 364–366 (1990).
[CrossRef]

Walsh, C. S. K.

Warde, C.

Webb, D. J.

D. J. Webb and L. Solymar, “Observations of spatial subharmonics arising during two-wave mixing in BSO,” Opt. Commun. 74, 386–389 (1990).
[CrossRef]

D. J. Webb, L. B. Au, D. C. Jones, and L. Solymar, “Onset of subharmonics generated by forward wave interactions in Bi12SiO20,” Appl. Phys. Lett. 57, 1602–1604 (1990).
[CrossRef]

Zakharov, V. E.

V. E. Zakharov, S. L. Musher, and A. M. Rubenchik, “Hamiltonian approach to the description of non-linear plasma phenomena,” Phys. Rep. 129, 285–366 (1985).
[CrossRef]

V. E. Zakharov, V. S. L’vov, and G. Falkovich, Kolmogorov Spectra of Turbulence I: Wave Turbulence (Springer-Verlag, Berlin, 1992).
[CrossRef]

Appl. Phys. A (2)

B. I. Sturman, M. Mann, J. Otten, and K. H. Ringhofer, “Subharmonic generation in photorefractive crystals: Application of the theory to experiment,” Appl. Phys. A 55, 55–60 (1992).
[CrossRef]

B. I. Sturman, M. Mann, and K. H. Ringhofer, “Instability of moving gratings in photorefractive crystals,” Appl. Phys. A 55, 235–241 (1992).
[CrossRef]

Appl. Phys. Lett. (3)

K. H. Ringhofer and L. Solymar, “New gain mechanism for wave amplification in photorefractive crystals,” Appl. Phys. Lett. 53, 1039–1040 (1988).
[CrossRef]

D. J. Webb, L. B. Au, D. C. Jones, and L. Solymar, “Onset of subharmonics generated by forward wave interactions in Bi12SiO20,” Appl. Phys. Lett. 57, 1602–1604 (1990).
[CrossRef]

G. C. Valley, H. Rajbenbach, and H. J. von Bardeleben, “Mobility-lifetime product of photoexcited electrons in GaAs,” Appl. Phys. Lett. 56, 364–366 (1990).
[CrossRef]

Ferroelectrics (1)

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals,” Ferroelectrics 22, 949–960 (1979); Ferroelectrics961–964 (1979).
[CrossRef]

J. Appl. Phys. (3)

M. Peltier and F. Micheron, “Volume hologram recording and charge transfer process in Bi12SiO20and Bi12GeO20,” J. Appl. Phys. 48, 3683–3690 (1977).
[CrossRef]

S. Mallick, B. Imbert, H. Ducollet, J. P. Herriau, and J.-P. Huignard, “Generation of spatial subharmonics by two-wave mixing in a nonlinear photorefractive medium,” J. Appl. Phys. 63, 5660–5663 (1988).
[CrossRef]

P. Refregier, L. Solymar, H. Rajbenbach, and J.-P. Huignard, “Two-beam coupling in photorefractive Bi12SiO20crystals with moving grating: theory and experiments,” J. Appl. Phys. 58, 45–57 (1985).
[CrossRef]

J. Opt. Soc. Am. (1)

M. B. Klein and R. N. Schwartz, “Photorefractive effect in BaTiO3: microscopic origins,” J. Opt. Soc. Am. 3, 293–305 (1986).
[CrossRef]

J. Opt. Soc. Am. B (4)

Opt. Commun. (5)

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).
[CrossRef]

J.-P. Huignard and A. Marrakchi, “Coherent signal beam amplification in two-wave-mixing experiments with photorefractive Bi12SiO20crystals,” Opt. Commun. 38, 249–254 (1981).
[CrossRef]

D. J. Webb and L. Solymar, “Observations of spatial subharmonics arising during two-wave mixing in BSO,” Opt. Commun. 74, 386–389 (1990).
[CrossRef]

J. Takacs, M. Schaub, and L. Solymar, “Subharmonics in photorefractive Bi12TiO20crystals,” Opt. Commun. 91, 252–254 (1992).
[CrossRef]

O. P. Nestiorkin, “Instability of spatial subharmonics under hologram recording in a photorefractive crystal,” Opt. Commun. 81, 315–320 (1991).
[CrossRef]

Opt. Lett. (1)

Phys. Rep. (1)

V. E. Zakharov, S. L. Musher, and A. M. Rubenchik, “Hamiltonian approach to the description of non-linear plasma phenomena,” Phys. Rep. 129, 285–366 (1985).
[CrossRef]

Phys. Status Solidi B (1)

H. G. Festl, P. Hertel, E. Krätzig, and R. von Baltz, “Investigations of the photovoltaic tensor in doped LiNbO3,” Phys. Status Solidi B 113, 157–164 (1982).
[CrossRef]

Sov. Phys. JETP (1)

B. Sturman, A. Bledowski, J. Otten, and K. H. Ringhofer, “Subharmonics in photorefractive crystals,” Sov. Phys. JETP 102, 215–224 (1992).

Sov. Phys. Semicond. (1)

Such waves and the instabilities connected with them are also known in the theory of semiconductors; see, for example, R. F. Kazarinov, R. A. Suris, and B. I. Fuks, “Instability with respect to waves of spatial charge-exchange in compensated semiconductors,” Sov. Phys. Semicond. 7, 480 (1973); V. N. Alimpiev and I. R. Gural’nik, “Parametric instability in a photosensitive semiconductor due to a travelling illumination intensity grating,” Sov. Phys. Semicond. 20, 512–514 (1986).

Other (8)

A. Bledowski, B. Sturman, J. Otten, and K. H. Ringhofer, “Analytic and Numeric Results in Subharmonics in BSO,” in Photorefractive Materials, Effects and Devices, Vol. 14 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), pp. 436–439.

L. B. Au, L. Solymar, and K. H. Ringhofer, “Subharmonics in BSO,” in Technical Digest on Photorefractive Materials, Effects and Devices II (Société Francais d’Optique, Aussois, France, 1990), pp. 87–91.

B. I. Sturman and V. M. Fridkin, The Photovoltaic and Photorefractive Effect in Noncentrosymmetric Materials (Gordon & Breach, New York, 1992).

V. E. Zakharov, V. S. L’vov, and G. Falkovich, Kolmogorov Spectra of Turbulence I: Wave Turbulence (Springer-Verlag, Berlin, 1992).
[CrossRef]

In the three-dimensional case the simplification of Eq. (31) is best performed in the k representation.

E. L. Ince, Ordinary Differential Equations (Dover, New York, 1956).

In particular, an inhomogeneity such as this can suppress the resonance excitation of a moving grating, since the resonance condition |Ω − ωK| ≲ γK is fulfilled only in a very narrow layer of the crystal.

P. Günter and J.-P. Huignard, eds., Photorefractive Materials and Their Applications I and II, Vols. 61 and 62 of Topics in Applied Physics (Springer-Verlag, Berlin, 1988 and 1989).
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

Geometrical scheme of a two-beam experiment.

Fig. 2
Fig. 2

Lines of constant quality factor Q(E0, 2π/kz) = 1, 2, … for typical parameters of BSO (see Table 1) and μτ = 6 × 10−11 m2/V, k = 0. The filled squares represent the experimental values of Ref. 12.

Fig. 3
Fig. 3

Dependence of the dimensionless space-charge field |E1/(E0m)| on the frequency t0−1 of the external field for α = 150 m−1, μτ = 6 × 10−11 m2/V, I0 = 40 W/m2, Λ = 15 μm, and E0 = 700 kV/m. Other parameters for BSO are given in Table 1.

Fig. 4
Fig. 4

Geometrical scheme for the decay of the fundamental grating. The dashed lines show the decay surface. The point marks the position of the vectors k1,2 that correspond to the first subharmonic ( = 1/4).

Fig. 5
Fig. 5

Lines of constant modulation threshold mth(Ω/αI0, Λ) = 0.5, 0,6, … in the case of a moving grating for E0 = 700 kV/m and typical parameters of BSO (see Fig. 2). The curves marked = 1/4 and = 1 correspond to the boundaries of the decay regions of the fundamental gratings EK and E2K, respectively.

Fig. 6
Fig. 6

Curves of constant s times modulation threshold (sm)th[Ω/(αI0), Λ] = 0.4, 0.5, … in the case of a weakly oscillating external field for typical parameters of BSO (see Fig. 2).

Fig. 7
Fig. 7

Dependence of the modulation-threshold mth (k1/K) in the limit of a rapidly oscillating external field for μτ equal to the values indicated.

Fig. 8
Fig. 8

Dependence of the characteristic exponent Γ on the frequency of an ac field for k1/K = 1/2, 1/3, 1/4 (first, second, and third subharmonics) and m = 0.05.

Fig. 9
Fig. 9

Dependence of the characteristic exponent Γ on k1/K for m = 0.05. Curves 1, 2, 3, and 4 correspond to t0−1 = 50, 25, 15, and 10 Hz, respectively. Peripherally oscillating parts of the curves have not been drawn.

Tables (1)

Tables Icon

Table 1 Typical Values for Some Photorefractive Crystals

Equations (61)

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div E = e 0 ¯ ( N D + N A n ) , N D + t = s i I ( N D N D + ) s r n N D + , s i I ( N D N D + ) + s r n N D + = μ div [ n ( E ex + E ) + k B T e n ] .
E ¯ = 0 , N D + ¯ n ¯ = N A .
s i ( N D N A ) = α ћ ω , s r N A = 1 τ ,
I = I 0 [ 1 + m cos ( K · x Ω t ) ] ,
E = φ .
E ex = E 0 ( 1 + s cos Ω 0 t ) , s 1 ,
E ex = E 0 p ( t ) , p ( t ) = { + 1 for 0 t < t 0 / 2 1 for t 0 / 2 t < t 0 .
φ , δ n , δ N + exp ( i k · x i ω k t γ k t ) .
ω k i γ k = ω 0 E q + E D i E 0 E 0 + i ( E D + E m ) ,
ω 0 = α I 0 ћ ω N , E q = e N 0 ¯ k z , E D = k 2 k B T e k z , E M = 1 k z μ τ ,
E q E 0 , E 0 E M , E 0 E D .
ω k = ω 0 E q E 0 e 0 ¯ E 0 α I 0 ћ ω 1 k z , γ k = ω 0 ( 1 + E q E M E 0 2 + E D E q E 0 2 ) α I 0 ћ ω ( 1 N + e 0 ¯ E 0 2 μ τ 1 k z 2 + k B T 0 ¯ E 0 2 k 2 k z 2 ) .
γ k = γ k , γ k ( E 0 ) = γ k ( E 0 ) ω k = ω k , ω k ( E 0 ) = ω k ( E 0 ) .
Q k = ( E 0 E q + E M E 0 + E D E 0 ) 1 .
Q max = 1 2 ( E q E M ) 1 / 2 ( N e μ τ 4 0 ¯ ) 1 / 2 ,
E 0 = E q E M = 1 k z ( e N 0 ¯ μ τ ) 1 / 2 , k z ( 2 e k B T μ τ ) 1 / 2 .
( N e μ τ 4 0 ¯ ) 1 / 2 1 ,
k 2 φ k = e 0 ¯ δ N k + , δ n k = i ω k τ δ N k + , i k z E 0 δ n k = n 0 k 2 φ k .
exp { 0 t [ i ω k ( E ex ) + γ k ( E ex ) ] d t } .
( 1 + i s ω k ω 0 sin ω 0 t ) exp ( i ω k t γ k t ) ,
exp ( γ k t i Φ k ) ,
E = E K exp ( i K z ) + c . c .
( t + i ω K + γ K ) E K = i 2 m ω K E ex exp ( i Ω t ) .
E K E 0 = m 2 ω K exp ( i Ω t ) Ω ω K + i γ K ,
E K E 0 = m 2 ( 1 + i s ω K Ω 0 sin Ω 0 t ) ,
E K E 0 = 2 i m γ K t 0 sin ( ω K t 0 4 ) exp [ i ( Φ K ω K t 0 / 4 ) ] .
E K i m E 0 Q K / 2 .
E 2 K E 0 E K 2 3 E 0 2 ,
Ω = ω k 1 + ω k 2 , K = k 1 + k 2 .
k 1 z = K 2 ( 1 ± 1 4 ) , k 2 z = K 2 ( 1 1 4 ) , k 1 = k 2 ,
= e 0 ¯ K Ω E 0 α I 0 ћ ω ω K Ω .
k 1 = k 3 + k 4 , ω k 1 = ω k 3 + ω k 4 .
Δ φ ¯ + e ¯ 0 δ N + ¯ = 0 , δ N t + ¯ + 1 τ δ n ¯ + ω 0 δ N + α δ I ћ ω + [ s i δ I δ N + ] + [ s r δ n δ N + ] = 0 , 1 μ δ N t + + E ex δ n z ¯ n 0 Δ φ ¯ + k B T e Δ δ n div ( δ n φ ) = 0 ,
Δ φ z t ω 0 l s Δ φ + ω 0 Δ φ z 1 l 0 Δ φ t + l D 2 l 0 Δ 2 φ t = e 0 ¯ α δ I z ћ ω + e 0 ¯ α ћ ω 1 E ex div ( δ I φ ) + 1 E ex div ( Δ φ t φ ) ,
l s = 0 ¯ E ex e N , l 0 = μ τ E ex , l D = ( μ τ k B T e ) 1 / 2 .
( E z t ω 0 l s E ) z ω 0 l 0 l s E + ω 0 ( 1 + l D 2 l 0 l s ) E z z = ω 0 E ex l s ( δ I I 0 E 2 E ex 2 ) z .
φ = φ K exp ( i K · x ) + φ 1 exp ( i K 1 · x ) + φ 2 exp ( i k 2 · x ) + c . c . ,
( t + i ω k 1 + γ k 1 ) φ 1 = i E K E ex ω 0 l s f ( k 1 , k 2 ) φ 2 * , ( t i ω k 2 + γ k 2 ) φ 2 * = i E K * E ex ω 0 l s f ( k 2 , k 1 ) φ 1 ,
f ( k 1 , k 2 ) = 2 k 1 z k 2 z 2 + k 2 ( k 1 z k 2 z ) k 1 z k 2 z k 1 2 ,
φ 1 exp ( i ω k 1 t + Γ t ) , φ 2 * exp ( i ω k 2 t + Γ t ) ,
Γ = 1 2 ( γ k 1 + γ k 2 ) ± [ 1 4 ( γ k 1 γ k 2 ) 2 + Γ 0 2 ] 1 / 2 ,
Γ 0 = m 2 | Ω 1 | F ( , k 2 K 2 ) , F ( , x ) = [ 4 3 + ( 4 1 ) x ( x 2 ) 2 + ( 1 2 ) x + x 2 ] 1 / 2 .
Γ 0 2 = γ k 1 γ k 2 .
m th = 1 Q k 1 z Q k 2 z .
( E K E 0 ) th 1 Q .
Γ 0 = m 1 ω K .
Γ 0 = m 2 ω K 3 6 [ ( Ω ω K ) 2 + γ K 2 ] .
Γ 0 = 1 6 m 2 Q K 2 ω K , m th = 6 Q K 3 / 2 .
( E K E 0 ) th 1 Q K .
Γ ˜ 0 = m s Ω 0 4 F ( , k 2 / K 2 ) .
( m s ) th = 2 Q k 1 Q k 2 .
Γ m s 2 w K Ω 0 .
φ 1 = A 1 k 1 3 / 2 exp ( i Φ k 1 ) , φ 2 * = A 2 * k 2 3 / 2 exp ( i Φ k 2 ) .
( d d t + γ k 1 ) A 1 = κ ω K exp ( i ψ ) A 2 * , ( d d t + γ k 2 ) A 2 * = κ ω K exp ( i ψ ) A 1 .
κ = 2 K k 1 k 2 | E K E 0 | , d ψ d t = 2 p ω K Δ , Δ = 1 2 ( K 2 k 1 k 2 1 ) .
Γ = t 0 1 ln | Z | .
[ A 1 ( t 0 / 2 ) A 2 * ( t 0 / 2 ) ] = T ˆ 1 [ A 1 ( 0 ) A 2 * ( 0 ) ] .
| T ˆ Z E ˆ | = 0 ,
Z = 1 + 2 S ± 2 S + S 2 , S = κ 2 ( Δ 2 κ 2 ) sin 2 ( ω K t 0 2 Δ 2 κ 2 ) .
Γ 0 m ω K Q K K k 1 k 2 .
m th = Q K 1 ( Q k 1 Q k 2 ) 1 / 2 .

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