Abstract

We present a theoretical model of photorefractive parametric oscillation that is capable of explaining the occurrence of so-called longitudinal, degenerate, and transversal parametric oscillation in photorefractive crystals. It appears that transversal parametric oscillation occurs in a parameter region that, until now, has been overlooked in the literature. Moreover, we present experiments on the different parametric processes that qualitatively verify our theoretical results.

© 1997 Optical Society of America

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References

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  1. R. Saxena and T. Y. Chang, “Perturbative analysis of higher-order photorefractive gratings,” J. Opt. Soc. Am. B 9, 1467–1472 (1992).
    [CrossRef]
  2. P. E. Andersen, P. Buchhave, P. M. Petersen, and M. V. Vasnetsov, “Nonlinear combinations of gratings in Bi12SiO20: theory and experiments,” J. Opt. Soc. Am. B 12, 1422–1433 (1995).
    [CrossRef]
  3. 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]
  4. 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]
  5. J. Takacs and L. Solymar, Department of Engineering Science, University of Oxford, Parks Road, Oxford OX1 3P2, UK, “Observation of split subharmonics in a Bi12SiO20 crystal (personal communication, 1991).
  6. H. C. Pedersen and P. M. Johansen, “Observation of angularly tilted subharmonic gratings in photorefractive BSO,” Opt. Lett. 19, 1418–1420 (1994).
    [CrossRef] [PubMed]
  7. B. I. Sturman, M. Mann, and K. H. Ringhofer, “Instability of moving gratings in photorefractive crystals,” Appl. Phys. A 55, 235–241 (1992).
    [CrossRef]
  8. 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]
  9. H. C. Pedersen and P. M. Johansen, “Parametric oscillation in photorefractive media,” J. Opt. Soc. Am. B 12, 1065–1073 (1995).
    [CrossRef]
  10. H. C. Pedersen and P. M. Johansen, “Longitudinal, degenerate, and transversal parametric oscillation in photorefractive media,” Phys. Rev. Lett. 77, 3106–3109 (1996).
    [CrossRef] [PubMed]
  11. Ph. Refregier, L. Solymar, H. Rajbenbach, and J. P. Huignard, “Two-beam coupling in photorefractive Bi12SiO20 crystals with moving grating: theory and experiments,” J. Appl. Phys. 58, 45–57 (1985).
    [CrossRef]
  12. 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]
  13. D. J. Webb and L. Solymar, “Observation of spatial subharmonics arising during two-wave mixing in BSO,” Opt. Commun. 74, 386–388 (1990).
    [CrossRef]
  14. C. H. Kwak, J. Takacs, and L. Solymar, “Spatial subharmonic instability in photorefractive Bi12SiO20 crystal,” Electron. Lett. 28, 530–531 (1992).
    [CrossRef]
  15. 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]
  16. 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]
  17. H. C. Pedersen and P. M. Johansen, “Incoherent enhancement of the photorefractive response in Bi12SiO20 by subharmonic interaction,” Opt. Lett. 20, 689–691 (1995).
    [CrossRef] [PubMed]
  18. 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]
  19. 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]
  20. Ye. P. Shershakov and O. P. Nestiorkin, “Nondegenerate spatial subharmonic generation in photorefractive crystal,” Opt. Commun. 96, 271–277 (1993).
    [CrossRef]
  21. H. C. Pedersen and P. M. Johansen, “Degenerate parametric amplification in photorefractive media: theoretical analysis,” J. Opt. Soc. Am. B 13, 590–600 (1996).
    [CrossRef]
  22. E. Serrano, M. Carrascosa, F. Agulló-López, and L. Solymar, “Subharmonic instability taking into account higher harmonics,” Appl. Phys. Lett. 64, 658–660 (1994).
    [CrossRef]

1996 (2)

H. C. Pedersen and P. M. Johansen, “Longitudinal, degenerate, and transversal parametric oscillation in photorefractive media,” Phys. Rev. Lett. 77, 3106–3109 (1996).
[CrossRef] [PubMed]

H. C. Pedersen and P. M. Johansen, “Degenerate parametric amplification in photorefractive media: theoretical analysis,” J. Opt. Soc. Am. B 13, 590–600 (1996).
[CrossRef]

1995 (4)

1994 (4)

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

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]

E. Serrano, M. Carrascosa, F. Agulló-López, and L. Solymar, “Subharmonic instability taking into account higher harmonics,” Appl. Phys. Lett. 64, 658–660 (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]

1993 (3)

Ye. P. Shershakov and O. P. Nestiorkin, “Nondegenerate spatial subharmonic generation in photorefractive crystal,” Opt. Commun. 96, 271–277 (1993).
[CrossRef]

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]

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]

1992 (4)

R. Saxena and T. Y. Chang, “Perturbative analysis of higher-order photorefractive gratings,” J. Opt. Soc. Am. B 9, 1467–1472 (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]

C. H. Kwak, J. Takacs, and L. Solymar, “Spatial subharmonic instability in photorefractive Bi12SiO20 crystal,” Electron. Lett. 28, 530–531 (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]

1990 (1)

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

1989 (1)

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)

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

Agulló-López, F.

E. Serrano, M. Carrascosa, F. Agulló-López, and L. Solymar, “Subharmonic instability taking into account higher harmonics,” Appl. Phys. Lett. 64, 658–660 (1994).
[CrossRef]

Andersen, P. E.

Bledowski, A.

Buchhave, P.

Carrascosa, M.

E. Serrano, M. Carrascosa, F. Agulló-López, and L. Solymar, “Subharmonic instability taking into account higher harmonics,” Appl. Phys. Lett. 64, 658–660 (1994).
[CrossRef]

Chang, T. Y.

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]

Grunnet-Jepsen, A.

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]

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]

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

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]

Johansen, P. M.

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 instability in photorefractive Bi12SiO20 crystal,” Electron. Lett. 28, 530–531 (1992).
[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.

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 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]

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]

Nestiorkin, O. P.

Ye. P. Shershakov and O. P. Nestiorkin, “Nondegenerate spatial subharmonic generation in photorefractive crystal,” Opt. Commun. 96, 271–277 (1993).
[CrossRef]

Otten, J.

Pedersen, H. C.

Petersen, P. M.

Rajbenbach, H.

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

Refregier, Ph.

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

Richter, I.

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.

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

Saxena, R.

Serrano, E.

E. Serrano, M. Carrascosa, F. Agulló-López, and L. Solymar, “Subharmonic instability taking into account higher harmonics,” Appl. Phys. Lett. 64, 658–660 (1994).
[CrossRef]

Shamonin, M.

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]

Shershakov, Ye. P.

Ye. P. Shershakov and O. P. Nestiorkin, “Nondegenerate spatial subharmonic generation in photorefractive crystal,” Opt. Commun. 96, 271–277 (1993).
[CrossRef]

Solymar, L.

E. Serrano, M. Carrascosa, F. Agulló-López, and L. Solymar, “Subharmonic instability taking into account higher harmonics,” Appl. Phys. Lett. 64, 658–660 (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]

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 instability in photorefractive Bi12SiO20 crystal,” Electron. Lett. 28, 530–531 (1992).
[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]

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

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

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 instability in photorefractive Bi12SiO20 crystal,” Electron. Lett. 28, 530–531 (1992).
[CrossRef]

Vasnetsov, M. V.

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]

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 (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)

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]

E. Serrano, M. Carrascosa, F. Agulló-López, and L. Solymar, “Subharmonic instability taking into account higher harmonics,” Appl. Phys. Lett. 64, 658–660 (1994).
[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)

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]

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

J. Opt. Soc. Am. B (6)

Opt. Commun. (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]

Ye. P. Shershakov and O. P. Nestiorkin, “Nondegenerate spatial subharmonic generation in photorefractive crystal,” Opt. Commun. 96, 271–277 (1993).
[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]

Opt. Lett. (2)

Phys. Rev. Lett. (2)

H. C. Pedersen and P. M. Johansen, “Longitudinal, degenerate, and transversal parametric oscillation in photorefractive media,” Phys. Rev. Lett. 77, 3106–3109 (1996).
[CrossRef] [PubMed]

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]

Other (1)

J. Takacs and L. Solymar, Department of Engineering Science, University of Oxford, Parks Road, Oxford OX1 3P2, UK, “Observation of split subharmonics in a Bi12SiO20 crystal (personal communication, 1991).

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Figures (8)

Fig. 1
Fig. 1

Schematic diagram of the setup used for photorefractive parametric oscillation.

Fig. 2
Fig. 2

Contour plots of the real part of s+ versus X (abscissa) and Y (ordinate) for different values of ε. In all cases the light areas represent unstable regions and the black areas represent stable regions. The contours outlined by dashed curves in each figure represent Re(s+)=0; the other contours represent Re(s+) of 5, 10, 15, and 20 s-1, respectively. The plots are generated from Eq. (19) with the following experimental parameters inserted: E0=14 kV/cm, I0=40.7 mW/cm2, kp=2π/30 µm-1, and m=1.

Fig. 3
Fig. 3

Experimental setup.

Fig. 4
Fig. 4

Central part of the experimental setup seen in the direction of the x axis.

Fig. 5
Fig. 5

Diffraction patterns obtained on the screen for various values of ε. In all cases the powerful spot at the left is the directly transmitted spot (zeroth order), whereas the spot at the right is the first-order spot. All spots between stem from diffraction in secondary gratings that arise because of parametric oscillation.

Fig. 6
Fig. 6

Exponential profile of the dc intensity in the crystal.

Fig. 7
Fig. 7

Contour plot of the real part of s+ versus ε and r for X =0.5 and Y=0. In white region (1) DPO is possible, whereas in black region (2) TPO is possible. The same experimental parameters as in Fig. 2 have been used.

Fig. 8
Fig. 8

Contour plot of Re{s+} versus X and ε for Y=0. The various light regions represent unstable regions; the black area represents stable regions. The contours outlined by dashed curves represent Re(s+)=0; the lighter shaded contours represent Re(s+) of 5, 10, 15, and 20 s-1. White curve (1) marks the maximum of the landscape; black-and-white dashed curve (2) represents the values of X and ε for which the secondary waves fulfill the dispersion relation. The black circles show, for ε =0.22, the X values for the signal and idler waves in the case of eigenwave excitation; the white circles show an example of noneigenwave excitation. The same experimental parameters as in Fig. 2 have been used.

Tables (1)

Tables Icon

Table 1 Crystal Parameters Relevant to BSOa

Equations (28)

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

ND+t=-·μnE+μkBTqn,
ND+t=NDsI-γRnND+,
·E=q0S(ND+-NA),
I=I0+mI0 cos(kPx-Ωt)=I0+I1,
ND+=ND0++ND+,n=n0+n1,
E=E0+E1,
E0·(2E˙1)+kBTq·(2E˙1)+ω0E0·(2E1)+kBTqω0·(2E1)-ζI0·E1-1μτ·E˙1
=ζE0·(I1)+kBTqζ2I1+ζ·(I1E1)̲̲-ω0·[E1(·E1)]̲̲-·[E1(·E˙1)]̲̲,
E1=E10 exp(ik·r-iω˜t),
ω˜=ω0Eq,k+ED,k-iE0E0+iED,k+iEM,kωk-iγk,
Eq,k=qNAε0εSk,ED,k=kBTk2qk,EM,k=1μτk,
E1=xˆEP exp(ikPx-iΩt),
EP=1/2mω0Eq,kPE0+iED,kPΩ(E0+iED,kP+iEM,kP)-ω0(Eq,kP+ED,kP-iE0)=1/2mω0Eq,kPE0+i(ED,kP+EM,kP)E0+iED,kPΩ-ωkP+iγkP.
E1=xˆEP exp(ikPx-iΩt)+ES(t) exp(ikS·r)+EI(t) exp(ikI·r)+c.c.,
kS+kI=kP.
E˙S(t)+ASES(t)=BSE˜I*(t)+CSE˜˙I*(t),
E˜˙I*(t)+AI*E˜I*(t)=BI*ES(t)+CI*E˙S(t),
ES(t)=kˆSES(t),EI*(t)=kˆIE˜I*(t) exp(-iΩt)
AS=γkS+iωkS,AI*=γkI-i(ωkI-Ω),
BS=-1/2mω0Eq,kSν1+[(Ω+iω0)X-1ν1+(Ω-iω0)ν2]EPED,kS+EM,kS-iE0,
BI*=-1/2mω0Eq,kIν1+[(Ω-iω0)(1-X)-1ν1+iω0ν2-1]EP*ED,kI+EM,kI+iE0,
CS=-iEPED,kS+EM,kS-iE0ν2,CI*=iEP*ED,kI+EM,kI+iE0ν2-1,
ν1=X(1-X)-Y2(X2+Y2)1/2[(1-X)2+Y2]1/2,
ν2=[(1-X)2+Y2]1/2(X2+Y2)1/2,
X=kSkP,Y=kSkP.
s±=-Q1±(Q12-Q2)1/2,
Q1=12AS+AI*-BSCI*-BI*CS1-CSCI*,
Q2=ASAI*-BSBI*1-CSCI*.

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