Abstract

Using a new optical configuration free from the influence of photorefractive optical nonlinearity, we investigate the main characteristics of the spatial subharmonic K/2 excited in a Bi12SiO20 crystal by a light-intensity pattern with wave vector K and frequency Ω. It is shown that in a large region of intensity and applied electric field the optimum value Ω of the frequency corresponds to the conditions of parametric excitation of the weakly damped eigenmodes of the medium: the space-charge waves. The threshold and above-threshold characteristics of the subharmonic regime are in good agreement with the theory.

© 1995 Optical Society of America

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  1. 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]
  2. D. C. Jones and L. Solymar, "Competition between subharmonic and resonating beams for photorefractive gain in bismuth silicon oxide," Opt. Lett. 14, 743–744 (1989).
    [CrossRef] [PubMed]
  3. D. J. Webb and L. Solymar, "Observation of spatial subharmonics arising during two-wave mixing in BSO," Opt. Commun. 74, 386–389 (1990).
    [CrossRef]
  4. 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]
  5. J. Takacs and L. Solymar, "Subharmonics in Bi12SiO20 with an applied ac electric field," Opt. Lett. 17, 247–248 (1992).
    [CrossRef] [PubMed]
  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]
  7. A. Bledowski, J. Otten, K. H. Ringhofer, and B. Sturman, "Subharmonics in photorefractive crystals," Sov. Phys. JETP 75, 215–224 (1992).
  8. B. I. Sturman, M. Mann, and K. H. Ringhofer, "Instability of moving gratings in photorefractive crystals," Appl. Phys. A 55, 235–241 (1992).
    [CrossRef]
  9. B. I. Sturman, M. Mann, J. Otten, and K. H. Ringhofer, "Space-charge waves and their parametric excitation," J. Opt. Soc. Am. B 10, 1919–1932 (1993).
    [CrossRef]
  10. J. Richter, A. Grunnet-Jepsen, J. Takacs, and L. Solymar, "An experimental and theoretical study of spatial subharmonics in a photorefractive Bi12GeO20 crystal induced by a dc-field and by the moving-grating technique," IEEE J. Quantum Electron. 30, 1645–1650 (1994).
    [CrossRef]
  11. T. E. McClelland, D. J. Webb, B. I. Sturman, and K. H. Ringhofer, "Generation of spatial subharmonic gratings in the absence of photorefractive beam coupling," Phys. Rev. Lett. 73, 3082–3084 (1994).
    [CrossRef] [PubMed]
  12. V. E. Zakharov, V. S. L'vov, and G. Falkovich, Kolmogorov Spectra of Turbulence I: Wave Turbulence, Springer Series in Nonlinear Dynamics (Springer-Verlag, Berlin, 1992), Chap. 5, p. 207.
    [CrossRef]
  13. D. R. Nicholson, Introduction to Plasma Theory (Wiley-Interscience, New York, 1983), Chap. 3, p. 137.
  14. Y. R. Shen, The Principles of Nonlinear Optics (Wiley-Interscience, New York, 1984), Chap. 28, p. 541.
  15. 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), Chap. 2, p. 7.
    [CrossRef]
  16. A. Marrakchi, R. V. Johnson, and J. A. R. Tanguay, "Polarization properties of photorefractive diffraction in electrooptic and optically active sillenite crystals (Bragg regime)," J. Opt. Soc. Am. B 3, 321–336 (1986).
    [CrossRef]
  17. M. P. Petrov, S. I. Stepanov, and A. V. Khomenko, Photorefractive Crystals in Coherent Optical Systems, Vol. 59 of Springer Series in Optical Sciences (Springer-Verlag, Berlin, 1991), Chap. 7, p. 133.
    [CrossRef]
  18. P. Réfrégier, 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]
  19. G. A. Brost, K. M. Madge, J. J. Larkin, and T. Harris, "Modulation dependence of the photorefractive response with moving gratings: numerical analysis and experiment," J. Opt. Soc. Am. B 11, 1764–1772 (1994).
    [CrossRef]
  20. T. E. McClelland, D. J. Webb, B. I. Sturman, M. Mann, and K. H. Ringhofer, "Low frequency peculiarities of the photorefractive response in sillenites," Opt. Commun. 113, 371–377 (1995).
    [CrossRef]
  21. S. Mallick, D. Rouède, and A. G. Apostolidis, "Efficiency and polarization characteristics of photorefractive diffraction in a Bi12SiO20 crystal," J. Opt. Soc. Am. B 4, 1247–1259 (1987).
    [CrossRef]
  22. B. I. Sturman, D. J. Webb, R. Kowarschik, E. Shamonina, and K. H. Ringhofer, "Exact solution of the Bragg diffraction problem in sillenites," J. Opt. Soc. Am. B 11, 1813–1819 (1994).
    [CrossRef]
  23. F. Vachss and L. Hesselink, "Holographic beam coupling in anisotropic photorefractive media," J. Opt. Soc. Am. A 4, 325–339 (1987).
    [CrossRef]
  24. P. Pauliat, J. M. Cohen-Jonathan, M. Allain, J. C. Launay, and G. Roosen, "Determinations of the photorefractive parameters of Bi12GeO20 crystals using transient grating analysis," Opt. Commun. 59, 266–271 (1986).
    [CrossRef]
  25. N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, "Holographic storage in electrooptic crystals," Ferroelectrics 22, 949–964 (1979).
    [CrossRef]
  26. F. Vachss and L. Hesselink, "Selective enhancement of spatial harmonics of a photorefractive grating," J. Opt. Soc. Am. B 5, 1814–1821 (1988).
    [CrossRef]
  27. R. Grousson, M. Henry, and S. Mallick, "Transport properties of photoelectrons in Bi12SiO20," J. Appl. Phys. 56, 224–229 (1984).
    [CrossRef]
  28. R. A. Mullen and R. W. Hellwarth, "Optical measurement of the photorefractive parameters of Bi12SiO20," J. Appl. Phys. 58, 40–44 (1985).
    [CrossRef]
  29. J. P. Partanen, J. M. C. Jonathan, and R. W. Hellwarth, "Direct determination of electron mobility in photorefractive Bi12SiO20 by a holographic time-of-flight technique," Appl. Phys. Lett. 57, 2404–2406 (1990).
    [CrossRef]
  30. G. Pauliat, A. Villing, J. C. Launay, and G. Roosen, "Optical measurements of charge-carrier mobilities in photorefractive sillenite crystals," J. Opt. Soc. Am. B 7, 1481–1486 (1990).
    [CrossRef]
  31. H. Pedersen and P. M. Johansen, "Observation of angularly tilted subharmonic gratings in photorefractive bismuth silicon oxide," Opt. Lett. 19, 1418–1420 (1994).
    [CrossRef] [PubMed]
  32. 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]
  33. 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]
  34. Y. Belaud, P. Delaye, J.-C. Launay, and G. Roosen, "Photorefractive response of CdTe:V under ac electric field from 1 to 1.5 μm," Opt. Commun. 105, 204–208 (1994).
    [CrossRef]

1995 (1)

T. E. McClelland, D. J. Webb, B. I. Sturman, M. Mann, and K. H. Ringhofer, "Low frequency peculiarities of the photorefractive response in sillenites," Opt. Commun. 113, 371–377 (1995).
[CrossRef]

1994 (6)

J. Richter, A. Grunnet-Jepsen, J. Takacs, and L. Solymar, "An experimental and theoretical study of spatial subharmonics in a photorefractive Bi12GeO20 crystal induced by a dc-field and by the moving-grating technique," IEEE J. Quantum Electron. 30, 1645–1650 (1994).
[CrossRef]

T. E. McClelland, D. J. Webb, B. I. Sturman, and K. H. Ringhofer, "Generation of spatial subharmonic gratings in the absence of photorefractive beam coupling," Phys. Rev. Lett. 73, 3082–3084 (1994).
[CrossRef] [PubMed]

Y. Belaud, P. Delaye, J.-C. Launay, and G. Roosen, "Photorefractive response of CdTe:V under ac electric field from 1 to 1.5 μm," Opt. Commun. 105, 204–208 (1994).
[CrossRef]

G. A. Brost, K. M. Madge, J. J. Larkin, and T. Harris, "Modulation dependence of the photorefractive response with moving gratings: numerical analysis and experiment," J. Opt. Soc. Am. B 11, 1764–1772 (1994).
[CrossRef]

B. I. Sturman, D. J. Webb, R. Kowarschik, E. Shamonina, and K. H. Ringhofer, "Exact solution of the Bragg diffraction problem in sillenites," J. Opt. Soc. Am. B 11, 1813–1819 (1994).
[CrossRef]

H. Pedersen and P. M. Johansen, "Observation of angularly tilted subharmonic gratings in photorefractive bismuth silicon oxide," Opt. Lett. 19, 1418–1420 (1994).
[CrossRef] [PubMed]

1993 (1)

1992 (4)

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

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]

A. Bledowski, J. Otten, K. H. Ringhofer, and B. Sturman, "Subharmonics in photorefractive crystals," Sov. Phys. JETP 75, 215–224 (1992).

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

1990 (4)

D. J. Webb and L. Solymar, "Observation 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]

J. P. Partanen, J. M. C. Jonathan, and R. W. Hellwarth, "Direct determination of electron mobility in photorefractive Bi12SiO20 by a holographic time-of-flight technique," Appl. Phys. Lett. 57, 2404–2406 (1990).
[CrossRef]

G. Pauliat, A. Villing, J. C. Launay, and G. Roosen, "Optical measurements of charge-carrier mobilities in photorefractive sillenite crystals," J. Opt. Soc. Am. B 7, 1481–1486 (1990).
[CrossRef]

1989 (1)

1988 (3)

1987 (3)

1986 (2)

P. Pauliat, J. M. Cohen-Jonathan, M. Allain, J. C. Launay, and G. Roosen, "Determinations of the photorefractive parameters of Bi12GeO20 crystals using transient grating analysis," Opt. Commun. 59, 266–271 (1986).
[CrossRef]

A. Marrakchi, R. V. Johnson, and J. A. R. Tanguay, "Polarization properties of photorefractive diffraction in electrooptic and optically active sillenite crystals (Bragg regime)," J. Opt. Soc. Am. B 3, 321–336 (1986).
[CrossRef]

1985 (2)

R. A. Mullen and R. W. Hellwarth, "Optical measurement of the photorefractive parameters of Bi12SiO20," J. Appl. Phys. 58, 40–44 (1985).
[CrossRef]

P. Réfrégier, 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]

1984 (1)

R. Grousson, M. Henry, and S. Mallick, "Transport properties of photoelectrons in Bi12SiO20," J. Appl. Phys. 56, 224–229 (1984).
[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, 949–964 (1979).
[CrossRef]

Albanese, G.

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]

Allain, M.

P. Pauliat, J. M. Cohen-Jonathan, M. Allain, J. C. Launay, and G. Roosen, "Determinations of the photorefractive parameters of Bi12GeO20 crystals using transient grating analysis," Opt. Commun. 59, 266–271 (1986).
[CrossRef]

Apostolidis, A. G.

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]

Belaud, Y.

Y. Belaud, P. Delaye, J.-C. Launay, and G. Roosen, "Photorefractive response of CdTe:V under ac electric field from 1 to 1.5 μm," Opt. Commun. 105, 204–208 (1994).
[CrossRef]

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]

A. Bledowski, J. Otten, K. H. Ringhofer, and B. Sturman, "Subharmonics in photorefractive crystals," Sov. Phys. JETP 75, 215–224 (1992).

Brost, G. A.

Cohen-Jonathan, J. M.

P. Pauliat, J. M. Cohen-Jonathan, M. Allain, J. C. Launay, and G. Roosen, "Determinations of the photorefractive parameters of Bi12GeO20 crystals using transient grating analysis," Opt. Commun. 59, 266–271 (1986).
[CrossRef]

Delaye, P.

Y. Belaud, P. Delaye, J.-C. Launay, and G. Roosen, "Photorefractive response of CdTe:V under ac electric field from 1 to 1.5 μm," Opt. Commun. 105, 204–208 (1994).
[CrossRef]

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 Series in Nonlinear Dynamics (Springer-Verlag, Berlin, 1992), Chap. 5, p. 207.
[CrossRef]

Grousson, R.

R. Grousson, M. Henry, and S. Mallick, "Transport properties of photoelectrons in Bi12SiO20," J. Appl. Phys. 56, 224–229 (1984).
[CrossRef]

Grunnet-Jepsen, A.

J. Richter, A. Grunnet-Jepsen, J. Takacs, and L. Solymar, "An experimental and theoretical study of spatial subharmonics in a photorefractive Bi12GeO20 crystal induced by a dc-field and by the moving-grating technique," IEEE J. Quantum Electron. 30, 1645–1650 (1994).
[CrossRef]

Harris, T.

Hellwarth, R. W.

J. P. Partanen, J. M. C. Jonathan, and R. W. Hellwarth, "Direct determination of electron mobility in photorefractive Bi12SiO20 by a holographic time-of-flight technique," Appl. Phys. Lett. 57, 2404–2406 (1990).
[CrossRef]

R. A. Mullen and R. W. Hellwarth, "Optical measurement of the photorefractive parameters of Bi12SiO20," J. Appl. Phys. 58, 40–44 (1985).
[CrossRef]

Henry, M.

R. Grousson, M. Henry, and S. Mallick, "Transport properties of photoelectrons in Bi12SiO20," J. Appl. Phys. 56, 224–229 (1984).
[CrossRef]

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]

Herriau, J.-P.

Hesselink, L.

Huignard, J.-P.

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]

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. Réfrégier, 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]

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]

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]

Johansen, P. M.

Johnson, R. V.

Jonathan, J. M. C.

J. P. Partanen, J. M. C. Jonathan, and R. W. Hellwarth, "Direct determination of electron mobility in photorefractive Bi12SiO20 by a holographic time-of-flight technique," Appl. Phys. Lett. 57, 2404–2406 (1990).
[CrossRef]

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]

D. C. Jones and L. Solymar, "Competition between subharmonic and resonating beams for photorefractive gain in bismuth silicon oxide," Opt. Lett. 14, 743–744 (1989).
[CrossRef] [PubMed]

Khomenko, A. V.

M. P. Petrov, S. I. Stepanov, and A. V. Khomenko, Photorefractive Crystals in Coherent Optical Systems, Vol. 59 of Springer Series in Optical Sciences (Springer-Verlag, Berlin, 1991), Chap. 7, p. 133.
[CrossRef]

Kowarschik, R.

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, 949–964 (1979).
[CrossRef]

Kumar, J.

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]

Larkin, J. J.

Launay, J. C.

G. Pauliat, A. Villing, J. C. Launay, and G. Roosen, "Optical measurements of charge-carrier mobilities in photorefractive sillenite crystals," J. Opt. Soc. Am. B 7, 1481–1486 (1990).
[CrossRef]

P. Pauliat, J. M. Cohen-Jonathan, M. Allain, J. C. Launay, and G. Roosen, "Determinations of the photorefractive parameters of Bi12GeO20 crystals using transient grating analysis," Opt. Commun. 59, 266–271 (1986).
[CrossRef]

Launay, J.-C.

Y. Belaud, P. Delaye, J.-C. Launay, and G. Roosen, "Photorefractive response of CdTe:V under ac electric field from 1 to 1.5 μm," Opt. Commun. 105, 204–208 (1994).
[CrossRef]

L'vov, V. S.

V. E. Zakharov, V. S. L'vov, and G. Falkovich, Kolmogorov Spectra of Turbulence I: Wave Turbulence, Springer Series in Nonlinear Dynamics (Springer-Verlag, Berlin, 1992), Chap. 5, p. 207.
[CrossRef]

Madge, K. M.

Mallick, S.

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]

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]

S. Mallick, D. Rouède, and A. G. Apostolidis, "Efficiency and polarization characteristics of photorefractive diffraction in a Bi12SiO20 crystal," J. Opt. Soc. Am. B 4, 1247–1259 (1987).
[CrossRef]

R. Grousson, M. Henry, and S. Mallick, "Transport properties of photoelectrons in Bi12SiO20," J. Appl. Phys. 56, 224–229 (1984).
[CrossRef]

Mann, M.

T. E. McClelland, D. J. Webb, B. I. Sturman, M. Mann, and K. H. Ringhofer, "Low frequency peculiarities of the photorefractive response in sillenites," Opt. Commun. 113, 371–377 (1995).
[CrossRef]

B. I. Sturman, M. Mann, J. Otten, and K. H. Ringhofer, "Space-charge waves and their parametric excitation," J. Opt. Soc. Am. B 10, 1919–1932 (1993).
[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]

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, 949–964 (1979).
[CrossRef]

Marrakchi, A.

McClelland, T. E.

T. E. McClelland, D. J. Webb, B. I. Sturman, M. Mann, and K. H. Ringhofer, "Low frequency peculiarities of the photorefractive response in sillenites," Opt. Commun. 113, 371–377 (1995).
[CrossRef]

T. E. McClelland, D. J. Webb, B. I. Sturman, and K. H. Ringhofer, "Generation of spatial subharmonic gratings in the absence of photorefractive beam coupling," Phys. Rev. Lett. 73, 3082–3084 (1994).
[CrossRef] [PubMed]

Mullen, R. A.

R. A. Mullen and R. W. Hellwarth, "Optical measurement of the photorefractive parameters of Bi12SiO20," J. Appl. Phys. 58, 40–44 (1985).
[CrossRef]

Nicholson, D. R.

D. R. Nicholson, Introduction to Plasma Theory (Wiley-Interscience, New York, 1983), Chap. 3, p. 137.

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, 949–964 (1979).
[CrossRef]

Otten, J.

Partanen, J. P.

J. P. Partanen, J. M. C. Jonathan, and R. W. Hellwarth, "Direct determination of electron mobility in photorefractive Bi12SiO20 by a holographic time-of-flight technique," Appl. Phys. Lett. 57, 2404–2406 (1990).
[CrossRef]

Pauliat, G.

Pauliat, P.

P. Pauliat, J. M. Cohen-Jonathan, M. Allain, J. C. Launay, and G. Roosen, "Determinations of the photorefractive parameters of Bi12GeO20 crystals using transient grating analysis," Opt. Commun. 59, 266–271 (1986).
[CrossRef]

Pedersen, H.

Petrov, M. P.

M. P. Petrov, S. I. Stepanov, and A. V. Khomenko, Photorefractive Crystals in Coherent Optical Systems, Vol. 59 of Springer Series in Optical Sciences (Springer-Verlag, Berlin, 1991), Chap. 7, p. 133.
[CrossRef]

Rajbenbach, H.

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]

P. Réfrégier, 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]

Réfrégier, P.

P. Réfrégier, 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]

Richter, J.

J. Richter, A. Grunnet-Jepsen, J. Takacs, and L. Solymar, "An experimental and theoretical study of spatial subharmonics in a photorefractive Bi12GeO20 crystal induced by a dc-field and by the moving-grating technique," IEEE J. Quantum Electron. 30, 1645–1650 (1994).
[CrossRef]

Ringhofer, K. H.

T. E. McClelland, D. J. Webb, B. I. Sturman, M. Mann, and K. H. Ringhofer, "Low frequency peculiarities of the photorefractive response in sillenites," Opt. Commun. 113, 371–377 (1995).
[CrossRef]

B. I. Sturman, D. J. Webb, R. Kowarschik, E. Shamonina, and K. H. Ringhofer, "Exact solution of the Bragg diffraction problem in sillenites," J. Opt. Soc. Am. B 11, 1813–1819 (1994).
[CrossRef]

T. E. McClelland, D. J. Webb, B. I. Sturman, and K. H. Ringhofer, "Generation of spatial subharmonic gratings in the absence of photorefractive beam coupling," Phys. Rev. Lett. 73, 3082–3084 (1994).
[CrossRef] [PubMed]

B. I. Sturman, M. Mann, J. Otten, and K. H. Ringhofer, "Space-charge waves and their parametric excitation," J. Opt. Soc. Am. B 10, 1919–1932 (1993).
[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]

A. Bledowski, J. Otten, K. H. Ringhofer, and B. Sturman, "Subharmonics in photorefractive crystals," Sov. Phys. JETP 75, 215–224 (1992).

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

Roosen, G.

Y. Belaud, P. Delaye, J.-C. Launay, and G. Roosen, "Photorefractive response of CdTe:V under ac electric field from 1 to 1.5 μm," Opt. Commun. 105, 204–208 (1994).
[CrossRef]

G. Pauliat, A. Villing, J. C. Launay, and G. Roosen, "Optical measurements of charge-carrier mobilities in photorefractive sillenite crystals," J. Opt. Soc. Am. B 7, 1481–1486 (1990).
[CrossRef]

P. Pauliat, J. M. Cohen-Jonathan, M. Allain, J. C. Launay, and G. Roosen, "Determinations of the photorefractive parameters of Bi12GeO20 crystals using transient grating analysis," Opt. Commun. 59, 266–271 (1986).
[CrossRef]

Rouède, D.

Shamonina, E.

Shen, Y. R.

Y. R. Shen, The Principles of Nonlinear Optics (Wiley-Interscience, New York, 1984), Chap. 28, p. 541.

Solymar, L.

J. Richter, A. Grunnet-Jepsen, J. Takacs, and L. Solymar, "An experimental and theoretical study of spatial subharmonics in a photorefractive Bi12GeO20 crystal induced by a dc-field and by the moving-grating technique," IEEE J. Quantum Electron. 30, 1645–1650 (1994).
[CrossRef]

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

D. J. Webb and L. Solymar, "Observation 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]

D. C. Jones and L. Solymar, "Competition between subharmonic and resonating beams for photorefractive gain in bismuth silicon oxide," Opt. Lett. 14, 743–744 (1989).
[CrossRef] [PubMed]

P. Réfrégier, 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]

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, 949–964 (1979).
[CrossRef]

Steier, W. H.

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]

Stepanov, S. I.

M. P. Petrov, S. I. Stepanov, and A. V. Khomenko, Photorefractive Crystals in Coherent Optical Systems, Vol. 59 of Springer Series in Optical Sciences (Springer-Verlag, Berlin, 1991), Chap. 7, p. 133.
[CrossRef]

Sturman, B.

A. Bledowski, J. Otten, K. H. Ringhofer, and B. Sturman, "Subharmonics in photorefractive crystals," Sov. Phys. JETP 75, 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]

Sturman, B. I.

T. E. McClelland, D. J. Webb, B. I. Sturman, M. Mann, and K. H. Ringhofer, "Low frequency peculiarities of the photorefractive response in sillenites," Opt. Commun. 113, 371–377 (1995).
[CrossRef]

T. E. McClelland, D. J. Webb, B. I. Sturman, and K. H. Ringhofer, "Generation of spatial subharmonic gratings in the absence of photorefractive beam coupling," Phys. Rev. Lett. 73, 3082–3084 (1994).
[CrossRef] [PubMed]

B. I. Sturman, D. J. Webb, R. Kowarschik, E. Shamonina, and K. H. Ringhofer, "Exact solution of the Bragg diffraction problem in sillenites," J. Opt. Soc. Am. B 11, 1813–1819 (1994).
[CrossRef]

B. I. Sturman, M. Mann, J. Otten, and K. H. Ringhofer, "Space-charge waves and their parametric excitation," J. Opt. Soc. Am. B 10, 1919–1932 (1993).
[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]

Takacs, J.

J. Richter, A. Grunnet-Jepsen, J. Takacs, and L. Solymar, "An experimental and theoretical study of spatial subharmonics in a photorefractive Bi12GeO20 crystal induced by a dc-field and by the moving-grating technique," IEEE J. Quantum Electron. 30, 1645–1650 (1994).
[CrossRef]

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

Tanguay, J. A. R.

Vachss, F.

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, 949–964 (1979).
[CrossRef]

Webb, D. J.

T. E. McClelland, D. J. Webb, B. I. Sturman, M. Mann, and K. H. Ringhofer, "Low frequency peculiarities of the photorefractive response in sillenites," Opt. Commun. 113, 371–377 (1995).
[CrossRef]

T. E. McClelland, D. J. Webb, B. I. Sturman, and K. H. Ringhofer, "Generation of spatial subharmonic gratings in the absence of photorefractive beam coupling," Phys. Rev. Lett. 73, 3082–3084 (1994).
[CrossRef] [PubMed]

B. I. Sturman, D. J. Webb, R. Kowarschik, E. Shamonina, and K. H. Ringhofer, "Exact solution of the Bragg diffraction problem in sillenites," J. Opt. Soc. Am. B 11, 1813–1819 (1994).
[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]

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

Zakharov, V. E.

V. E. Zakharov, V. S. L'vov, and G. Falkovich, Kolmogorov Spectra of Turbulence I: Wave Turbulence, Springer Series in Nonlinear Dynamics (Springer-Verlag, Berlin, 1992), Chap. 5, p. 207.
[CrossRef]

Appl. Phys. A (1)

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. (2)

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]

J. P. Partanen, J. M. C. Jonathan, and R. W. Hellwarth, "Direct determination of electron mobility in photorefractive Bi12SiO20 by a holographic time-of-flight technique," Appl. Phys. Lett. 57, 2404–2406 (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, 949–964 (1979).
[CrossRef]

IEEE J. Quantum Electron. (1)

J. Richter, A. Grunnet-Jepsen, J. Takacs, and L. Solymar, "An experimental and theoretical study of spatial subharmonics in a photorefractive Bi12GeO20 crystal induced by a dc-field and by the moving-grating technique," IEEE J. Quantum Electron. 30, 1645–1650 (1994).
[CrossRef]

J. Appl. Phys. (4)

R. Grousson, M. Henry, and S. Mallick, "Transport properties of photoelectrons in Bi12SiO20," J. Appl. Phys. 56, 224–229 (1984).
[CrossRef]

R. A. Mullen and R. W. Hellwarth, "Optical measurement of the photorefractive parameters of Bi12SiO20," J. Appl. Phys. 58, 40–44 (1985).
[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. Réfrégier, 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. A (1)

J. Opt. Soc. Am. B (8)

A. Marrakchi, R. V. Johnson, and J. A. R. Tanguay, "Polarization properties of photorefractive diffraction in electrooptic and optically active sillenite crystals (Bragg regime)," J. Opt. Soc. Am. B 3, 321–336 (1986).
[CrossRef]

F. Vachss and L. Hesselink, "Selective enhancement of spatial harmonics of a photorefractive grating," J. Opt. Soc. Am. B 5, 1814–1821 (1988).
[CrossRef]

G. A. Brost, K. M. Madge, J. J. Larkin, and T. Harris, "Modulation dependence of the photorefractive response with moving gratings: numerical analysis and experiment," J. Opt. Soc. Am. B 11, 1764–1772 (1994).
[CrossRef]

B. I. Sturman, D. J. Webb, R. Kowarschik, E. Shamonina, and K. H. Ringhofer, "Exact solution of the Bragg diffraction problem in sillenites," J. Opt. Soc. Am. B 11, 1813–1819 (1994).
[CrossRef]

B. I. Sturman, M. Mann, J. Otten, and K. H. Ringhofer, "Space-charge waves and their parametric excitation," J. Opt. Soc. Am. B 10, 1919–1932 (1993).
[CrossRef]

S. Mallick, D. Rouède, and A. G. Apostolidis, "Efficiency and polarization characteristics of photorefractive diffraction in a Bi12SiO20 crystal," J. Opt. Soc. Am. B 4, 1247–1259 (1987).
[CrossRef]

G. Pauliat, A. Villing, J. C. Launay, and G. Roosen, "Optical measurements of charge-carrier mobilities in photorefractive sillenite crystals," J. Opt. Soc. Am. B 7, 1481–1486 (1990).
[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]

Opt. Commun. (5)

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

T. E. McClelland, D. J. Webb, B. I. Sturman, M. Mann, and K. H. Ringhofer, "Low frequency peculiarities of the photorefractive response in sillenites," Opt. Commun. 113, 371–377 (1995).
[CrossRef]

P. Pauliat, J. M. Cohen-Jonathan, M. Allain, J. C. Launay, and G. Roosen, "Determinations of the photorefractive parameters of Bi12GeO20 crystals using transient grating analysis," Opt. Commun. 59, 266–271 (1986).
[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]

Y. Belaud, P. Delaye, J.-C. Launay, and G. Roosen, "Photorefractive response of CdTe:V under ac electric field from 1 to 1.5 μm," Opt. Commun. 105, 204–208 (1994).
[CrossRef]

Opt. Lett. (4)

Phys. Rev. Lett. (1)

T. E. McClelland, D. J. Webb, B. I. Sturman, and K. H. Ringhofer, "Generation of spatial subharmonic gratings in the absence of photorefractive beam coupling," Phys. Rev. Lett. 73, 3082–3084 (1994).
[CrossRef] [PubMed]

Sov. Phys. JETP (1)

A. Bledowski, J. Otten, K. H. Ringhofer, and B. Sturman, "Subharmonics in photorefractive crystals," Sov. Phys. JETP 75, 215–224 (1992).

Other (5)

V. E. Zakharov, V. S. L'vov, and G. Falkovich, Kolmogorov Spectra of Turbulence I: Wave Turbulence, Springer Series in Nonlinear Dynamics (Springer-Verlag, Berlin, 1992), Chap. 5, p. 207.
[CrossRef]

D. R. Nicholson, Introduction to Plasma Theory (Wiley-Interscience, New York, 1983), Chap. 3, p. 137.

Y. R. Shen, The Principles of Nonlinear Optics (Wiley-Interscience, New York, 1984), Chap. 28, p. 541.

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), Chap. 2, p. 7.
[CrossRef]

M. P. Petrov, S. I. Stepanov, and A. V. Khomenko, Photorefractive Crystals in Coherent Optical Systems, Vol. 59 of Springer Series in Optical Sciences (Springer-Verlag, Berlin, 1991), Chap. 7, p. 133.
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

Experimental setup.

Fig. 2
Fig. 2

Diffraction efficiency of (a) the fundamental and (b) the subharmonic grating as a function of the frequency detuning between the pump beams.

Fig. 3
Fig. 3

Dependence of the optimum frequency detuning on the pump intensity for an external field E0 = 7 kV cm−1.

Fig. 4
Fig. 4

Dependence of the optimum frequency detuning on the applied field for a pump intensity I0 = 19.3 mW cm−2.

Fig. 5
Fig. 5

Dependence of the optimum frequency detunings on the vertical shift of the readout beams d for β = 0.85, I0 = 17.7 mW cm−2, E0 = 7 kV cm−1.

Fig. 6
Fig. 6

Dependence of the diffraction efficiency of the subharmonic grating on the frequency detuning for different intensity ratios of the pump beams; d = 2.9 cm, I0 = 20 mW cm−2, E0 = 7 kV cm−1.

Fig. 7
Fig. 7

Peak amplitude values of the fundamental and subharmonic gratings as a function of the external field, calculated from the experimental data obtained for m ≃ 1, I0 ≃ 19.3 mW cm−2.

Fig. 8
Fig. 8

Contour lines of the quality factor Q(k, E0) = 1, … 7 for Nt = 1016 cm−3, μτ = 8 × 10−7 cm2 V−1.

Fig. 9
Fig. 9

Dependence of the subharmonic amplitude on the detuning frequency calculated for different values of the modulation coefficient.

Equations (4)

Equations on this page are rendered with MathJax. Learn more.

η = ( s E g 2 κ E ) 2 ( s 2 l 2 + ρ 2 κ 2 sin 2 κ l ) .
Ω f = ω K ,             Ω s = 2 ω K / 2 .
ω k = e 0 k E α I ω ,
m th = 3 Q K / 2 ,             Ω ± - Ω s = Ω s m 2 - m th 2 .

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