Abstract

We investigated both analytically and numerically the simultaneous influence of higher-harmonic gratings and subharmonic gratings on the threshold for generation of the subharmonic K/2 grating. The higher-harmonic grating causes feedback to the fundamental grating, thus leading to a nonlinear correction in dissipation, and the threshold for subharmonic generation is substantially modified by the presence of more subharmonics. The numerical solution shows that the inclusion of five higher-harmonic components and four subharmonic components are sufficient to cover the entire spatial region. The discrepancy between the analytical and the numerical solution increases with increased electric-field amplitude.

© 1999 Optical Society of America

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References

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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. T. C. McClelland, D. J. Webb, B. I. Sturman, and K. H. Ringhofer, “Generation of spatial subharmonic grating in the absence of photorefractive beam coupling,” Phys. Rev. Lett. 73, 3082–3084 (1994).
    [CrossRef] [PubMed]
  3. I. Richter, A. Grunnet-Jepsen, J. Takacs, and L. Solymar, “An experimental and theoretical study of spatial subharmonics in a photorefractive Bi12SiO20 crystal induced by dc field and moving grating technique,” IEEE J. Quantum Electron. QE-30, 1645–1650 (1994).
    [CrossRef]
  4. J. Takacs and L. Solymar, “Subharmonics in Bi12SiO20 with an applied ac electric field,” Opt. Lett. 17, 247–248 (1992).
    [CrossRef] [PubMed]
  5. 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]
  6. 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]
  7. H. C. Pedersen and P. M. Johansen, “Observation of nondegenerate photorefractive parametric amplification,” Phys. Rev. Lett. 76, 4159–4162 (1996).
    [CrossRef] [PubMed]
  8. K. H. Ringhofer and L. Solymar, “New gain mechanism for wave amplification in photorefractive materials,” Appl. Phys. Lett. 53, 1039–1040 (1988).
    [CrossRef]
  9. B. I. Sturman, M. Mann, and K. H. Ringhofer, “Instability of moving gratings in photorefractive crystals,” Appl. Phys. A 55, 235–241 (1992).
    [CrossRef]
  10. B. I. Sturman, M. Mann, and K. H. Ringhofer, “Instability of spatial gratings induced by ac fields in photorefractive crystals,” Opt. Lett. 17, 1620–1622 (1992).
    [CrossRef] [PubMed]
  11. 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]
  12. B. I. Sturman, M. Aguilar, F. Agullo-Lopez, and K. H. Ringhofer, “Fundamentals of the nonlinear theory of photorefractive subharmonics,” Phys. Rev. E 55, 6072–6083 (1997).
    [CrossRef]
  13. E. V. Podivilov, H. C. Pedersen, P. M. Johansen, and B. I. Sturman, “Transversal parametric oscillation and its external stability in photorefractive sillenite crystals,” Phys. Rev. E 57, 6112–6126 (1998).
    [CrossRef]
  14. P. M. Johansen, H. C. Pedersen, E. V. Podivilov, and B. I. Sturman, “Steady state analysis of AC subharmonic generation in photorefractive sillenite crystals,” Phys. Rev. A 58, 1601–1604 (1998).
    [CrossRef]
  15. P. M. Johansen, H. C. Pedersen, and E. V. Podivilov, “Running grating induced space-charge field in the limit of quadratic recombination,” J. Opt. Soc. Am. B (to be published).
  16. 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]
  17. H. C. Pedersen, P. M. Johansen, E. V. Podivilov, and D. J. Webb, “Excitation of higher harmonic gratings in AC-field biased photorefractive crystals,” Opt. Commun. 154, 93–99 (1998).
    [CrossRef]
  18. V. E. Zakharov, V. S. L’Vov, and G. Fal’kovich, Kolmogorov Spectra of Turbulence: Wave Turbulence, Springer Series in Nonlinear Dynamics (Springer-Verlag, Berlin, 1992).
    [CrossRef]
  19. P. M. Johansen, “Vectorial solution to the photorefractive band transport model in the spatial and temporal Fourier transformed domain,” IEEE J. Quantum Electron. QE-25, 530–539 (1989).
    [CrossRef]

1998 (3)

E. V. Podivilov, H. C. Pedersen, P. M. Johansen, and B. I. Sturman, “Transversal parametric oscillation and its external stability in photorefractive sillenite crystals,” Phys. Rev. E 57, 6112–6126 (1998).
[CrossRef]

P. M. Johansen, H. C. Pedersen, E. V. Podivilov, and B. I. Sturman, “Steady state analysis of AC subharmonic generation in photorefractive sillenite crystals,” Phys. Rev. A 58, 1601–1604 (1998).
[CrossRef]

H. C. Pedersen, P. M. Johansen, E. V. Podivilov, and D. J. Webb, “Excitation of higher harmonic gratings in AC-field biased photorefractive crystals,” Opt. Commun. 154, 93–99 (1998).
[CrossRef]

1997 (1)

B. I. Sturman, M. Aguilar, F. Agullo-Lopez, and K. H. Ringhofer, “Fundamentals of the nonlinear theory of photorefractive subharmonics,” Phys. Rev. E 55, 6072–6083 (1997).
[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, “Observation of nondegenerate photorefractive parametric amplification,” Phys. Rev. Lett. 76, 4159–4162 (1996).
[CrossRef] [PubMed]

1994 (2)

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

I. Richter, A. Grunnet-Jepsen, J. Takacs, and L. Solymar, “An experimental and theoretical study of spatial subharmonics in a photorefractive Bi12SiO20 crystal induced by dc field and moving grating technique,” IEEE J. Quantum Electron. QE-30, 1645–1650 (1994).
[CrossRef]

1993 (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]

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]

1992 (3)

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. QE-25, 530–539 (1989).
[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 materials,” Appl. Phys. Lett. 53, 1039–1040 (1988).
[CrossRef]

1985 (1)

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]

Aguilar, M.

B. I. Sturman, M. Aguilar, F. Agullo-Lopez, and K. H. Ringhofer, “Fundamentals of the nonlinear theory of photorefractive subharmonics,” Phys. Rev. E 55, 6072–6083 (1997).
[CrossRef]

Agullo-Lopez, F.

B. I. Sturman, M. Aguilar, F. Agullo-Lopez, and K. H. Ringhofer, “Fundamentals of the nonlinear theory of photorefractive subharmonics,” Phys. Rev. E 55, 6072–6083 (1997).
[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]

Grunnet-Jepsen, A.

I. Richter, A. Grunnet-Jepsen, J. Takacs, and L. Solymar, “An experimental and theoretical study of spatial subharmonics in a photorefractive Bi12SiO20 crystal induced by dc field and moving grating technique,” IEEE J. Quantum Electron. QE-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]

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.

E. V. Podivilov, H. C. Pedersen, P. M. Johansen, and B. I. Sturman, “Transversal parametric oscillation and its external stability in photorefractive sillenite crystals,” Phys. Rev. E 57, 6112–6126 (1998).
[CrossRef]

H. C. Pedersen, P. M. Johansen, E. V. Podivilov, and D. J. Webb, “Excitation of higher harmonic gratings in AC-field biased photorefractive crystals,” Opt. Commun. 154, 93–99 (1998).
[CrossRef]

P. M. Johansen, H. C. Pedersen, E. V. Podivilov, and B. I. Sturman, “Steady state analysis of AC subharmonic generation in photorefractive sillenite crystals,” Phys. Rev. A 58, 1601–1604 (1998).
[CrossRef]

H. C. Pedersen and P. M. Johansen, “Observation of nondegenerate photorefractive parametric amplification,” Phys. Rev. Lett. 76, 4159–4162 (1996).
[CrossRef] [PubMed]

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]

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

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]

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.

McClelland, T. C.

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

Otten, J.

Pedersen, H. C.

E. V. Podivilov, H. C. Pedersen, P. M. Johansen, and B. I. Sturman, “Transversal parametric oscillation and its external stability in photorefractive sillenite crystals,” Phys. Rev. E 57, 6112–6126 (1998).
[CrossRef]

P. M. Johansen, H. C. Pedersen, E. V. Podivilov, and B. I. Sturman, “Steady state analysis of AC subharmonic generation in photorefractive sillenite crystals,” Phys. Rev. A 58, 1601–1604 (1998).
[CrossRef]

H. C. Pedersen, P. M. Johansen, E. V. Podivilov, and D. J. Webb, “Excitation of higher harmonic gratings in AC-field biased photorefractive crystals,” Opt. Commun. 154, 93–99 (1998).
[CrossRef]

H. C. Pedersen and P. M. Johansen, “Observation of nondegenerate photorefractive parametric amplification,” Phys. Rev. Lett. 76, 4159–4162 (1996).
[CrossRef] [PubMed]

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]

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]

Podivilov, E. V.

E. V. Podivilov, H. C. Pedersen, P. M. Johansen, and B. I. Sturman, “Transversal parametric oscillation and its external stability in photorefractive sillenite crystals,” Phys. Rev. E 57, 6112–6126 (1998).
[CrossRef]

P. M. Johansen, H. C. Pedersen, E. V. Podivilov, and B. I. Sturman, “Steady state analysis of AC subharmonic generation in photorefractive sillenite crystals,” Phys. Rev. A 58, 1601–1604 (1998).
[CrossRef]

H. C. Pedersen, P. M. Johansen, E. V. Podivilov, and D. J. Webb, “Excitation of higher harmonic gratings in AC-field biased photorefractive crystals,” Opt. Commun. 154, 93–99 (1998).
[CrossRef]

Richter, I.

I. Richter, A. Grunnet-Jepsen, J. Takacs, and L. Solymar, “An experimental and theoretical study of spatial subharmonics in a photorefractive Bi12SiO20 crystal induced by dc field and moving grating technique,” IEEE J. Quantum Electron. QE-30, 1645–1650 (1994).
[CrossRef]

Ringhofer, K. H.

B. I. Sturman, M. Aguilar, F. Agullo-Lopez, and K. H. Ringhofer, “Fundamentals of the nonlinear theory of photorefractive subharmonics,” Phys. Rev. E 55, 6072–6083 (1997).
[CrossRef]

T. C. McClelland, D. J. Webb, B. I. Sturman, and K. H. Ringhofer, “Generation of spatial subharmonic grating 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, M. Mann, and K. H. Ringhofer, “Instability of spatial gratings induced by ac fields in photorefractive crystals,” Opt. Lett. 17, 1620–1622 (1992).
[CrossRef] [PubMed]

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

Solymar, L.

I. Richter, A. Grunnet-Jepsen, J. Takacs, and L. Solymar, “An experimental and theoretical study of spatial subharmonics in a photorefractive Bi12SiO20 crystal induced by dc field and moving grating technique,” IEEE J. Quantum Electron. QE-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]

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

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

E. V. Podivilov, H. C. Pedersen, P. M. Johansen, and B. I. Sturman, “Transversal parametric oscillation and its external stability in photorefractive sillenite crystals,” Phys. Rev. E 57, 6112–6126 (1998).
[CrossRef]

P. M. Johansen, H. C. Pedersen, E. V. Podivilov, and B. I. Sturman, “Steady state analysis of AC subharmonic generation in photorefractive sillenite crystals,” Phys. Rev. A 58, 1601–1604 (1998).
[CrossRef]

B. I. Sturman, M. Aguilar, F. Agullo-Lopez, and K. H. Ringhofer, “Fundamentals of the nonlinear theory of photorefractive subharmonics,” Phys. Rev. E 55, 6072–6083 (1997).
[CrossRef]

T. C. McClelland, D. J. Webb, B. I. Sturman, and K. H. Ringhofer, “Generation of spatial subharmonic grating 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, M. Mann, and K. H. Ringhofer, “Instability of spatial gratings induced by ac fields in photorefractive crystals,” Opt. Lett. 17, 1620–1622 (1992).
[CrossRef] [PubMed]

Takacs, J.

I. Richter, A. Grunnet-Jepsen, J. Takacs, and L. Solymar, “An experimental and theoretical study of spatial subharmonics in a photorefractive Bi12SiO20 crystal induced by dc field and moving grating technique,” IEEE J. Quantum Electron. QE-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]

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

Webb, D. J.

H. C. Pedersen, P. M. Johansen, E. V. Podivilov, and D. J. Webb, “Excitation of higher harmonic gratings in AC-field biased photorefractive crystals,” Opt. Commun. 154, 93–99 (1998).
[CrossRef]

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

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)

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

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 Bi12SiO20 crystal induced by dc field and moving grating technique,” IEEE J. Quantum Electron. QE-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. QE-25, 530–539 (1989).
[CrossRef]

J. Appl. Phys. (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]

J. Opt. Soc. Am. B (1)

Opt. Commun. (2)

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

H. C. Pedersen, P. M. Johansen, E. V. Podivilov, and D. J. Webb, “Excitation of higher harmonic gratings in AC-field biased photorefractive crystals,” Opt. Commun. 154, 93–99 (1998).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. A (1)

P. M. Johansen, H. C. Pedersen, E. V. Podivilov, and B. I. Sturman, “Steady state analysis of AC subharmonic generation in photorefractive sillenite crystals,” Phys. Rev. A 58, 1601–1604 (1998).
[CrossRef]

Phys. Rev. E (2)

B. I. Sturman, M. Aguilar, F. Agullo-Lopez, and K. H. Ringhofer, “Fundamentals of the nonlinear theory of photorefractive subharmonics,” Phys. Rev. E 55, 6072–6083 (1997).
[CrossRef]

E. V. Podivilov, H. C. Pedersen, P. M. Johansen, and B. I. Sturman, “Transversal parametric oscillation and its external stability in photorefractive sillenite crystals,” Phys. Rev. E 57, 6112–6126 (1998).
[CrossRef]

Phys. Rev. Lett. (3)

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

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, “Observation of nondegenerate photorefractive parametric amplification,” Phys. Rev. Lett. 76, 4159–4162 (1996).
[CrossRef] [PubMed]

Other (2)

P. M. Johansen, H. C. Pedersen, and E. V. Podivilov, “Running grating induced space-charge field in the limit of quadratic recombination,” J. Opt. Soc. Am. B (to be published).

V. E. Zakharov, V. S. L’Vov, and G. Fal’kovich, Kolmogorov Spectra of Turbulence: Wave Turbulence, Springer Series in Nonlinear Dynamics (Springer-Verlag, Berlin, 1992).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Modulus of space-charge field versus modulation coefficient. The curves termed linear approximation are plotted from Eq. (13); the curves termed second-order approximation are plotted from Eqs. (14). The dashed line represents the level at which more higher harmonics are to be taken into account.

Fig. 2
Fig. 2

Normalized real parts of the increments versus the modulation coefficient.

Fig. 3
Fig. 3

Quality factor for BSO displayed as a function of fringe spacing. The two regions of validity Qk(i) and Qk(ii) are represented by the dashed curve.

Fig. 4
Fig. 4

Illustration of the convergence of the threshold modulation for the subharmonic K/2 grating by taking into account various subharmonic and superharmonic components. The parameter n denotes the number of higher-harmonic components, and the parameter p denotes the number of subharmonic components included in the calculation.

Fig. 5
Fig. 5

Threshold modulation for the subharmonic K/2 grating plotted as a function of fringe spacing Λ for different values of the applied electric field E0. The squares are produced from Eq. (16), the dashed curves are plotted with Eq. (18), and the solid curves are drawn from Eq. (19). The circle curves are produced from numerical calculations.

Tables (1)

Tables Icon

Table 1 Material Parameters Relevant to BSOa

Equations (36)

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

Eex(t)=E0p(t),
p(t)=1,nt0<t<(n+½)t0-1,(n+½)t0<t<(n+1)t0,
1μτx-E0p(t)2x2-kBTq3x3Et
=-E0p(t)x+kBTq2x2ζI0m cos(Kx)
-ζI0-kBTqω02x2-ω0E0p(t)x×1+m cos(Kx)-1ζI02Ext+ω0ExEx+E3Ex2t+ω02Ex2-ζI0xmcos(Kx),
E(x, t)=kE0ek(t)exp(ikx),
ekt=-[γk+iωkp(t)]ek+m2(Ak+B˜k,Kek-K+B˜k, -Kek+K)+kC˜k,kek-kt+ω0ek.
γk=ω0[Eq(k)+ED(k)][EM(k)+ED(k)]+E02[EM(k)+ED(k)]2+E02,
ωk=ω0[Eq(k)-EM(k)]E0[EM(k)+ED(k)]2+E02,
Ak=-iω0Eq(k)[ED(k)-iE0p(t)]E0[EM(k)+ED(k)-iE0p(t)]×[δk,K+δk, -K],
B˜k,K=iω0Eq(K)E0C˜k,K,
C˜k,k=iE0Eq(k)Eq(k)+[ED(k)-iE0p(t)]k-kkEM(k)+ED(k)-iE0p(t),
ekt+[γk+iωkp(t)]ek
=m2(Ak+Bk,Kek-K+Bk, -Kek+K)+kCk,kekek-k,
Bk,K=-iEq(K)EM(K)E0[Eq(K)-EM(K)]Ck,K,
Ck,k=-ω0Eq(k)-EM(k)EM(k)+ED(k)-iE0p(t)C˜k,k.
e¯kt+γke¯k=m2[A¯k+B¯k,Ke¯k-K+B¯k, -Ke¯k+K]+kC¯k,ke¯ke¯k-k,
A¯k=-iω0Eq(k)E0{ED(k)[EM(k)+ED(k)]+E02}{[EM(k)+ED(k)]2+E02}(δk,K+δk, -K),
B¯k,K=-iEq(K)EM(K)E0[Eq(K)-EM(K)]C¯k,K,
C¯k,k=-iω0E0Eq(k)-EM(k)Eq(k)[Eq(k)+ED(k-k)]{[EM(k)+ED(k)][EM(k)+ED(k)]-E02}{[EM(k)+ED(k)]2+E02}{[EM(k)+ED(k)]2+E02}+E02k-kk[EM(k)+ED(k)+EM(k)+ED(k)]{[EM(k)+ED(k)]2+E02}{[EM(k)+ED(k)]2+E02}.
|EM(k)+ED(k)|E0|Eq(k)|
ωk=ω0Eq(k)E0,
γk=ω0E02[Eq(k)EM(k)+Eq(k)ED(k)+E02].
e¯kt+γke¯k=-iωkm2(δk,K+δk, -K)+iωkke¯ke¯k-k.
e¯K=-im2ωKγK=-imQK2,K>0.
(γK+2ωKQ2K|e¯K|2)e¯K=-iωK(m/2),
e¯2K=iQ2Ke¯K2.
e¯K/2t+γK/2e¯K/2=2iωK/2(e¯Ke¯K/2*+e¯3K/2e¯K*+e¯2Ke¯3K/2*+e¯5K/2e¯2K*+),
e¯3K/2t+γ3K/2e¯3K/2=2iω3K/2(e¯2Ke¯K/2*+e¯Ke¯K/2+e¯K*e¯5K/2+e¯7K/2e¯2K*+),
e¯5K/2t+γ5K/2e¯5K/2=2iω5K/2(e¯2Ke¯K/2+e¯Ke¯3K/2>+e¯K*e¯7K/2+e¯9K/2e¯2K*+),
ν±=-γK/2+γ3K/22ωK/2|e¯K|+γK/2-γ3K/22ωK/2|e¯K|2+4ωK/2ω3K/2(|e¯2K|2-|e¯K|2)1/2.
|e¯K|1QK/2,
|e¯2K|Q2KQK/2214|e¯K|,
mth=2QKQK/2+4Q2KQK/23.
mth=521QKQK/2.

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