Abstract

We present an analysis of what is called parametric oscillation in photorefractive media. It is shown theoretically that when a running grating and a constant electric field are applied to a photorefractive medium it is possible for two secondary gratings to grow spontaneously. These theoretical predictions are subsequently verified experimentally in the crystal Bi12SiO20, and excellent qualitative agreement is obtained. In previous subharmonic experiments, only grating vectors K/2, K/3, and K /4 were observed. Now, however, a whole continuum of grating vectors has been observed. Thus subharmonic generation in photorefractive materials is simply a special case of parametric oscillation in which the two secondary gratings are identical.

© 1995 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  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, 386–388 (1990).
    [CrossRef]
  4. L. B. Au, L. Solymar, and K. H. Ringhofer, "Subharmonics in BSO," in Technical Digest on Photorefractive Materials, Effects, and Devices II (Société Français d'Optique, Aussois, France, 1990), pp. 87–91.
  5. D. J. Webb, L. B. Au, D. C. Jones, and L. Solymar, "Onset of subharmonics generated by forward interactions in Bi12SiO20," Appl. Phys. Lett. 57, 1602–1604 (1990).
    [CrossRef]
  6. D. C. Jones, S. F. Lyuksyutov, and L. Solymar, "Competition between subharmonic and signal beams for photorefractive gain in BSO with two pump beams," Appl. Phys. B 52, 173–175 (1991).
    [CrossRef]
  7. C. H. Kwak, J. Takacs, and L. Solymar, "Spatial subharmonic instability in photorefractive Bi12SiO20 crystal," Electron. Lett. 28, 530–531 (1992).
    [CrossRef]
  8. C. H. Kwak, J. Takacs, and L. Solymar, "Spatial subharmonic instabilities," Opt. Commun. 96, 278–282 (1993).
    [CrossRef]
  9. J. Takacs, M. Schaub, and L. Solymar, "Subharmonics in photorefractive Bi12SiO20 crystals," Opt. Commun. 91, 252–254 (1992).
    [CrossRef]
  10. I. Richter, A. Grunnet-Jepsen, J. Takacs, and L. Solymar, "An experimental and theoretical study of spatial subharmonics in a photorefractive Bi12GeO20 crystal induced by dc field and moving grating technique," IEEE J. Quantum Electron. 30, 1645–1650 (1994).
    [CrossRef]
  11. J. Takacs and L. Solymar, "Subharmonics in Bi12SiO20 with an applied ac electric field," Opt. Lett. 17, 247–248 (1992).
    [CrossRef] [PubMed]
  12. C. H. Kwak, M. Shamonin, J. Takacs, and L. Solymar, "Spatial subharmonics in photorefractive Bi12SiO20 crystal with a square wave applied field," Appl. Phys. Lett. 62, 328–330 (1993).
    [CrossRef]
  13. A. Grunnet-Jepsen, I. Richter, M. Shamonin, and L. Solymar, "Subharmonic instabilities in photorefractive crystals for an applied alternating electric field: theoretical analysis," J. Opt. Soc. Am. B 11, 132–135 (1994).
    [CrossRef]
  14. A. Grunnet-Jepsen and L. Solymar, "Effect of subharmonics on two-wave gain in Bi12SiO20 under alternating electric fields," Opt. Lett. 19, 1299–1301 (1994).
    [CrossRef] [PubMed]
  15. B. I. Sturman, M. Mann, and K. H. Ringhofer, "Instability of moving gratings in photorefractive crystals," Appl. Phys. A 55, 235–241 (1992).
    [CrossRef]
  16. B. I. Sturman, M. Mann, J. Otten, K. H. Ringhofer, and A. Bledowski, "Subharmonic generation in photorefractive crystals: application of theory to experiment," Appl. Phys. A 55, 55–60 (1992).
    [CrossRef]
  17. B. I. 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. I. Sturman, M. Mann, and K. H. Ringhofer, "Instability of the resonance enhancement of moving photorefractive gratings," Opt. Lett. 18, 702–704 (1993).
    [CrossRef] [PubMed]
  19. B. I. Sturman, M. Mann, J. Otten, and K. H. Ringhofer, "Space-charge waves in photorefractive crystals and their parametric excitation," J. Opt. Soc. Am. B 10, 1919–1932 (1993).
    [CrossRef]
  20. O. P. Nestiorkin, "Instability of spatial subharmonics under hologram recording in a photorefractive crystal," Opt. Commun. 81, 315–320 (1991).
    [CrossRef]
  21. O. P. Nestiorkin and Ye. P. Shershakov, "Parametric generation of spatial subharmonic grating in photorefractive crystals: theory," J. Opt. Soc. Am. B 10, 1907–1918 (1993).
    [CrossRef]
  22. H. C. Pedersen and P. M. Johansen, "Observation of angularly tilted subharmonic gratings in photorefractive BSO," Opt. Lett. 19, 1418–1420 (1994).
    [CrossRef] [PubMed]
  23. R. G. Smith, J. E. Geusic, H. J. Levinstein, S. Singh, and L. G. van Uitert, "Low-threshold optical parametric oscillator using Ba2NaNb5O15," J. Appl. Phys. 39, 4030–4032 (1968).
    [CrossRef]
  24. Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984), Chap. 9.
  25. 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]
  26. See, for example, Ph. 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]
  27. P. M. Johansen, "Vectorial solution to the photorefractive band transport model in the spatial and temporal Fourier transformed domain," IEEE J. Quantum Electron. 25, 530–539 (1989).
    [CrossRef]

1994 (4)

1993 (5)

1992 (6)

J. Takacs, M. Schaub, and L. Solymar, "Subharmonics in photorefractive Bi12SiO20 crystals," Opt. Commun. 91, 252–254 (1992).
[CrossRef]

C. H. Kwak, J. Takacs, and L. Solymar, "Spatial subharmonic instability in photorefractive Bi12SiO20 crystal," Electron. Lett. 28, 530–531 (1992).
[CrossRef]

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

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, K. H. Ringhofer, and A. Bledowski, "Subharmonic generation in photorefractive crystals: application of theory to experiment," Appl. Phys. A 55, 55–60 (1992).
[CrossRef]

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

1991 (2)

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

D. C. Jones, S. F. Lyuksyutov, and L. Solymar, "Competition between subharmonic and signal beams for photorefractive gain in BSO with two pump beams," Appl. Phys. B 52, 173–175 (1991).
[CrossRef]

1990 (2)

D. J. Webb and L. Solymar, "Observation of spatial subharmonics arising during two-wave mixing in BSO," Opt. Commun. 74, 386–388 (1990).
[CrossRef]

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

1989 (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]

P. M. Johansen, "Vectorial solution to the photorefractive band transport model in the spatial and temporal Fourier transformed domain," IEEE J. Quantum Electron. 25, 530–539 (1989).
[CrossRef]

1988 (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]

1985 (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]

1968 (1)

R. G. Smith, J. E. Geusic, H. J. Levinstein, S. Singh, and L. G. van Uitert, "Low-threshold optical parametric oscillator using Ba2NaNb5O15," J. Appl. Phys. 39, 4030–4032 (1968).
[CrossRef]

Au, L. B.

D. J. Webb, L. B. Au, D. C. Jones, and L. Solymar, "Onset of subharmonics generated by forward 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 Français d'Optique, Aussois, France, 1990), pp. 87–91.

Bledowski, A.

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

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

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]

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]

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]

Geusic, J. E.

R. G. Smith, J. E. Geusic, H. J. Levinstein, S. Singh, and L. G. van Uitert, "Low-threshold optical parametric oscillator using Ba2NaNb5O15," J. Appl. Phys. 39, 4030–4032 (1968).
[CrossRef]

Grunnet-Jepsen, A.

Hall, T. J.

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.

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]

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]

See, for example, Ph. 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]

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]

Johansen, P. M.

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

P. M. Johansen, "Vectorial solution to the photorefractive band transport model in the spatial and temporal Fourier transformed domain," IEEE J. Quantum Electron. 25, 530–539 (1989).
[CrossRef]

Jones, D. C.

D. C. Jones, S. F. Lyuksyutov, and L. Solymar, "Competition between subharmonic and signal beams for photorefractive gain in BSO with two pump beams," Appl. Phys. B 52, 173–175 (1991).
[CrossRef]

D. J. Webb, L. B. Au, D. C. Jones, and L. Solymar, "Onset of subharmonics generated by forward 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]

Kwak, C. H.

C. H. Kwak, M. Shamonin, J. Takacs, and L. Solymar, "Spatial subharmonics in photorefractive Bi12SiO20 crystal with a square wave applied field," Appl. Phys. Lett. 62, 328–330 (1993).
[CrossRef]

C. H. Kwak, J. Takacs, and L. Solymar, "Spatial subharmonic instabilities," Opt. Commun. 96, 278–282 (1993).
[CrossRef]

C. H. Kwak, J. Takacs, and L. Solymar, "Spatial subharmonic instability in photorefractive Bi12SiO20 crystal," Electron. Lett. 28, 530–531 (1992).
[CrossRef]

Levinstein, H. J.

R. G. Smith, J. E. Geusic, H. J. Levinstein, S. Singh, and L. G. van Uitert, "Low-threshold optical parametric oscillator using Ba2NaNb5O15," J. Appl. Phys. 39, 4030–4032 (1968).
[CrossRef]

Lyuksyutov, S. F.

D. C. Jones, S. F. Lyuksyutov, and L. Solymar, "Competition between subharmonic and signal beams for photorefractive gain in BSO with two pump beams," Appl. Phys. B 52, 173–175 (1991).
[CrossRef]

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 the resonance enhancement of moving photorefractive gratings," Opt. Lett. 18, 702–704 (1993).
[CrossRef] [PubMed]

B. I. Sturman, M. Mann, J. Otten, and K. H. Ringhofer, "Space-charge waves in photorefractive crystals 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]

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

Nestiorkin, O. P.

O. P. Nestiorkin and Ye. P. Shershakov, "Parametric generation of spatial subharmonic grating in photorefractive crystals: theory," J. Opt. Soc. Am. B 10, 1907–1918 (1993).
[CrossRef]

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

Otten, J.

Pedersen, H. C.

Rajbenbach, H.

See, for example, Ph. 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, Ph.

See, for example, Ph. 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, I.

A. Grunnet-Jepsen, I. Richter, M. Shamonin, and L. Solymar, "Subharmonic instabilities in photorefractive crystals for an applied alternating electric field: theoretical analysis," J. Opt. Soc. Am. B 11, 132–135 (1994).
[CrossRef]

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

Ringhofer, K. H.

B. I. Sturman, M. Mann, J. Otten, and K. H. Ringhofer, "Space-charge waves in photorefractive crystals 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 the resonance enhancement of moving photorefractive gratings," Opt. Lett. 18, 702–704 (1993).
[CrossRef] [PubMed]

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, 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, K. H. Ringhofer, and A. Bledowski, "Subharmonic generation in photorefractive crystals: application of theory to experiment," Appl. Phys. A 55, 55–60 (1992).
[CrossRef]

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

Schaub, M.

J. Takacs, M. Schaub, and L. Solymar, "Subharmonics in photorefractive Bi12SiO20 crystals," Opt. Commun. 91, 252–254 (1992).
[CrossRef]

Shamonin, M.

A. Grunnet-Jepsen, I. Richter, M. Shamonin, and L. Solymar, "Subharmonic instabilities in photorefractive crystals for an applied alternating electric field: theoretical analysis," J. Opt. Soc. Am. B 11, 132–135 (1994).
[CrossRef]

C. H. Kwak, M. Shamonin, J. Takacs, and L. Solymar, "Spatial subharmonics in photorefractive Bi12SiO20 crystal with a square wave applied field," Appl. Phys. Lett. 62, 328–330 (1993).
[CrossRef]

Shen, Y. R.

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984), Chap. 9.

Shershakov, Ye. P.

Singh, S.

R. G. Smith, J. E. Geusic, H. J. Levinstein, S. Singh, and L. G. van Uitert, "Low-threshold optical parametric oscillator using Ba2NaNb5O15," J. Appl. Phys. 39, 4030–4032 (1968).
[CrossRef]

Smith, R. G.

R. G. Smith, J. E. Geusic, H. J. Levinstein, S. Singh, and L. G. van Uitert, "Low-threshold optical parametric oscillator using Ba2NaNb5O15," J. Appl. Phys. 39, 4030–4032 (1968).
[CrossRef]

Solymar, L.

A. Grunnet-Jepsen, I. Richter, M. Shamonin, and L. Solymar, "Subharmonic instabilities in photorefractive crystals for an applied alternating electric field: theoretical analysis," J. Opt. Soc. Am. B 11, 132–135 (1994).
[CrossRef]

A. Grunnet-Jepsen and L. Solymar, "Effect of subharmonics on two-wave gain in Bi12SiO20 under alternating electric fields," Opt. Lett. 19, 1299–1301 (1994).
[CrossRef] [PubMed]

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

C. H. Kwak, J. Takacs, and L. Solymar, "Spatial subharmonic instabilities," Opt. Commun. 96, 278–282 (1993).
[CrossRef]

C. H. Kwak, M. Shamonin, J. Takacs, and L. Solymar, "Spatial subharmonics in photorefractive Bi12SiO20 crystal with a square wave applied field," Appl. Phys. Lett. 62, 328–330 (1993).
[CrossRef]

J. Takacs, M. Schaub, and L. Solymar, "Subharmonics in photorefractive Bi12SiO20 crystals," Opt. Commun. 91, 252–254 (1992).
[CrossRef]

C. H. Kwak, J. Takacs, and L. Solymar, "Spatial subharmonic instability in photorefractive Bi12SiO20 crystal," Electron. Lett. 28, 530–531 (1992).
[CrossRef]

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

D. C. Jones, S. F. Lyuksyutov, and L. Solymar, "Competition between subharmonic and signal beams for photorefractive gain in BSO with two pump beams," Appl. Phys. B 52, 173–175 (1991).
[CrossRef]

D. J. Webb, L. B. Au, D. C. Jones, and L. Solymar, "Onset of subharmonics generated by forward 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, 386–388 (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]

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

See, for example, Ph. 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]

Sturman, B. I.

Takacs, J.

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

C. H. Kwak, M. Shamonin, J. Takacs, and L. Solymar, "Spatial subharmonics in photorefractive Bi12SiO20 crystal with a square wave applied field," Appl. Phys. Lett. 62, 328–330 (1993).
[CrossRef]

C. H. Kwak, J. Takacs, and L. Solymar, "Spatial subharmonic instabilities," Opt. Commun. 96, 278–282 (1993).
[CrossRef]

J. Takacs, M. Schaub, and L. Solymar, "Subharmonics in photorefractive Bi12SiO20 crystals," Opt. Commun. 91, 252–254 (1992).
[CrossRef]

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

C. H. Kwak, J. Takacs, and L. Solymar, "Spatial subharmonic instability in photorefractive Bi12SiO20 crystal," Electron. Lett. 28, 530–531 (1992).
[CrossRef]

Uitert, L. G. van

R. G. Smith, J. E. Geusic, H. J. Levinstein, S. Singh, and L. G. van Uitert, "Low-threshold optical parametric oscillator using Ba2NaNb5O15," J. Appl. Phys. 39, 4030–4032 (1968).
[CrossRef]

Webb, D. J.

D. J. Webb, L. B. Au, D. C. Jones, and L. Solymar, "Onset of subharmonics generated by forward 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, 386–388 (1990).
[CrossRef]

Appl. Phys. A (2)

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, K. H. Ringhofer, and A. Bledowski, "Subharmonic generation in photorefractive crystals: application of theory to experiment," Appl. Phys. A 55, 55–60 (1992).
[CrossRef]

Appl. Phys. B (1)

D. C. Jones, S. F. Lyuksyutov, and L. Solymar, "Competition between subharmonic and signal beams for photorefractive gain in BSO with two pump beams," Appl. Phys. B 52, 173–175 (1991).
[CrossRef]

Appl. Phys. Lett. (2)

C. H. Kwak, M. Shamonin, J. Takacs, and L. Solymar, "Spatial subharmonics in photorefractive Bi12SiO20 crystal with a square wave applied field," Appl. Phys. Lett. 62, 328–330 (1993).
[CrossRef]

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

Electron. Lett. (1)

C. H. Kwak, J. Takacs, and L. Solymar, "Spatial subharmonic instability in photorefractive Bi12SiO20 crystal," Electron. Lett. 28, 530–531 (1992).
[CrossRef]

IEEE J. Quantum Electron. (2)

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

P. M. Johansen, "Vectorial solution to the photorefractive band transport model in the spatial and temporal Fourier transformed domain," IEEE J. Quantum Electron. 25, 530–539 (1989).
[CrossRef]

J. Appl. Phys. (2)

R. G. Smith, J. E. Geusic, H. J. Levinstein, S. Singh, and L. G. van Uitert, "Low-threshold optical parametric oscillator using Ba2NaNb5O15," J. Appl. Phys. 39, 4030–4032 (1968).
[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]

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

Opt. Commun. (4)

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

D. J. Webb and L. Solymar, "Observation of spatial subharmonics arising during two-wave mixing in BSO," Opt. Commun. 74, 386–388 (1990).
[CrossRef]

C. H. Kwak, J. Takacs, and L. Solymar, "Spatial subharmonic instabilities," Opt. Commun. 96, 278–282 (1993).
[CrossRef]

J. Takacs, M. Schaub, and L. Solymar, "Subharmonics in photorefractive Bi12SiO20 crystals," Opt. Commun. 91, 252–254 (1992).
[CrossRef]

Opt. Lett. (5)

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]

Other (3)

See, for example, Ph. 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]

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984), Chap. 9.

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

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (11)

Fig. 1
Fig. 1

Schematic of parametric oscillation in (a) a nonlinear optical medium, (b) a photorefractive medium. In (a) all waves are optical waves, whereas in (b) the incident gray waves [with frequencies ω(o) and ω(o) + Ω] are optical and the internal black waves (with frequencies Ω, ωS, and ωI) are waves of the space-charge field. Frequencies of optical fields are denoted by the superscript (o) to distinguish them from the frequencies of space-charge field. The two partially reflecting mirrors (M’s) in (a) form a resonator that is needed to produce a sufficient interaction length. No resonator is present in the photorefractive case.

Fig. 2
Fig. 2

Contour plot of a typical linear space-charge field response of BSO exposed to a running interference pattern and a dc electric field. The lighter-shaded areas correspond to high response. The plot is generated from Eq. (11), into which the parameters from Table 1 have been inserted. The dashed curve shows the dispersion relation presented in expressions (14).

Fig. 3
Fig. 3

Vectorial picture of the three-wave interaction. Three cases are shown, for three different values of the fundamental frequency Ω. The frequency vectors (k, ω) of the two secondary waves (kS, ωS) and (kI, ωI) should end on the dispersion curve to yield a high response. The parameters from Table 1 have been used for the dispersion curve.

Fig. 4
Fig. 4

Top, unshaded contour of Fig. 2, sketching the limiting region where secondary waves can grow. The fundamental running grating has the frequency vector (K, Ω1). Bottom, The corresponding diffraction pattern is suggested.

Fig. 5
Fig. 5

As in Fig. 4, except that here Ω = Ω2. Two pairs (long-dashed lines and short-dashed lines) of secondary frequency vectors are shown.

Fig. 6
Fig. 6

As in Fig. 5, except that here Ω = Ω3.

Fig. 7
Fig. 7

As in Fig. 5, except that here Ω = Ω4.

Fig. 8
Fig. 8

Schematic of the experimental setup for generation of parametric oscillations. The pump beams consist of two expanded Ar+ laser beams, one of which is shifted in frequency by an amount Omega;. This frequency shift is obtained by reflection of a piezomirror. An applied dc electric field E0 and the induced fundamental grating vector K are both aligned to be along the 〈001〉 direction of the crystal. Readout is performed by a He–Ne laser beam, where the plane of incidence is perpendicular to the 〈110〉 direction. The diffraction pattern is recorded by a CCD camera.

Fig. 9
Fig. 9

Dc intensity dependence of the writing beams on crystal depth.

Fig. 10
Fig. 10

Diffraction patterns in the output plane, showing parametric oscillation. The four pictures correspond to four different values of Ω: (a) Ω = 85 s−1, (b) Ω = 156 s−1, (c) Ω = 276 s−1, (d) Ω = 348 s−1. In all four cases the circular spot at the right, centered approximately at x pixel near 212, is the directly transmitted readout beam, or zeroth-order beam. The spots at the left in (a) and (b), centered approximately at x pixel near 13, are the first-order diffracted spots that are due to diffraction in the fundamental grating. The middle spots are diffraction spots that are due to parametric oscillation.

Fig. 11
Fig. 11

Three-dimensional plot of the diffraction pattern from Fig. 10(c). Note that the height of the zeroth-order peak (the one at the right) has been damped to avoid overflow of the CCD camera.

Tables (1)

Tables Icon

Table 1 Material Parameters Relevant to the Crystal Bi12SiO20 and the Used Values of m, I0, and E0a

Equations (17)

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

N D + t = n t · ( μ n E + μ k B T q n ) ,
N D + t = ( N D N D + ) ( s I + β ) γ R n N D + ,
· E = q 0 s ( N D + N A n ) ,
N D + = N D 0 + + N D 1 + , n = n 0 + n 1 , E = E 0 + E 1 , I = I 0 + I 1 .
X Y = X 0 Y 0 + X 0 Y 1 + X 1 Y 0 + X 1 Y 1 ,
| 1 | | E 0 | , n 1 n 0 , 1 I 0 1 , N D 1 + N D 0 + 1 ,
E 0 ( · Ė 1 ) k B T q 2 Ė 1 ω 0 E 0 ( · E 1 ) k B T q ω 0 2 E 1 + ζ I 10 E 1 + 1 μ τ Ė 1 = ζ I 1 E 0 k B T q ζ I 1 ζ I 1 E 1 + ω 0 E 1 ( · E 1 ) + E 1 ( · Ė 1 ) ,
ω 0 = N D N A s I 0 , ζ = s q 0 s ( N D N A ) ,
I 1 = ½ m I 0 exp ( i K x i Ω t ) + c.c. ,
E 1 = x ̂ 1 exp ( i K x i Ω t ) + c . c . ,
1 = ½ m E q E 0 + i E D Ω ω 0 [ E 0 + i ( E D + E M ) ] ( E q + E D i E 0 ) ,
E q = N D N A N D q N A s 0 K , E D = k B T q K , E M = 1 μ τ K .
E q E 0 , E 0 E D , E 0 E M .
Ω ω 0 E 0 = E q Ω = E q E 0 ω 0 .
E 1 = 1 exp ( i K x i Ω t ) + S exp ( i k S x i ω S t ) + I exp ( i k I x i ω I t ) + c.c. ,
k S + k I = K , ω S + ω I = Ω .
m I 0 S * exp ( i k I x i ω I t ) , m I 0 I * exp ( i k S x i ω S t ) , I s * exp ( i k I x i ω I t ) , 1 I * exp ( i k S x i ω S t ) , S I exp ( i K x i Ω t ) ,

Metrics