Abstract

We propose a theory of the instability of the space-charge field induced in a semiconductor by the movement of light fringes against subharmonic generation. A crucial element of this theory is taking into account three-dimensional perturbations and higher spatial harmonics of the space-charge field. It is shown that, irrespective of fringe contrast and velocity, the strongest instability corresponds to the excitation of subharmonics that are not split in the transverse direction. The theory is applied to experimental data on subharmonic generation in sillenites.

© 1999 Optical Society of America

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References

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  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]
  2. 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]
  3. 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 in sillenite crystals,” Phys. Rev. Lett. 73, 3082–3084 (1994).
    [CrossRef] [PubMed]
  4. B. I. Sturman, T. E. McClelland, D. J. Webb, E. Shamonina, and K. H. Ringhofer, “Investigation of photorefractive subharmonics in the absence of wave mixing,” J. Opt. Soc. Am. B 12, 1621–1627 (1995).
    [CrossRef]
  5. H. C. Pedersen and P. M. Johansen, “Parametric oscillation in photorefractive media,” J. Opt. Soc. Am. B 12, 1065–1073 (1995).
    [CrossRef]
  6. L. Solymar, D. J. Webb, and A. Grunnet-Jepsen, The Physics and Applications of Photorefractive Materials (Clarendon, Oxford, 1996).
  7. J. Takacs and L. Solymar, Department of Engineering Science, University of Oxford, Parks Road, Oxford OX1 3PJ, UK, “Observation of split subharmonics in a Bi12SiO20 crystal” (personal communication, 1991).
  8. 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]
  9. H. C. Pedersen and P. M. Johansen, “Longitudinal, degenerate, and transversal photorefractive parametric oscillation in photorefractive media,” Phys. Rev. Lett. 77, 3106–3109 (1996).
    [CrossRef] [PubMed]
  10. H. C. Pedersen and P. M. Johansen, “Longitudinal, degenerate, and transversal photorefractive parametric oscillation: theory and experiment,” J. Opt. Soc. Am. B 14, 1418–1427 (1997).
    [CrossRef]
  11. Note that in Ref. 12 some of the present authors reported a preliminary version of this method. The corresponding numerical calculations and their results suffer, however, from a random error in the numerical program. Now two different programs give identical results and correct limiting cases, which practically excludes any accidental error.
  12. B. I. Sturman, E. Shamonina, and K. H. Ringhofer, “Explanation of the transverse subharmonic split in sillenites,” in Photorefractive Materials, Effects and Devices, PR’97 (Waseda University, Tokyo, 1997), pp. 145–148.
  13. 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]
  14. T. E. McClelland, D. J. Webb, B. I. Sturman, E. Shamonina, M. Mann, and K. H. Ringhofer, “Excitation of higher spatial harmonics by a moving light pattern in sillenites,” Opt. Commun. 131, 315–321 (1996).
    [CrossRef]
  15. H. C. Pedersen, D. J. Webb, and P. M. Johansen, “Fundamental characteristics of space-charge waves in photorefractive sillenite crystals,” J. Opt. Soc. Am. B 15, 2573–2580 (1998).
    [CrossRef]
  16. E. Serrano, M. Carrascosa, F. A. López, and L. Solymar, “Subharmonic instability taking into account higher harmonics,” Appl. Phys. Lett. 64, 658–660 (1994).
    [CrossRef]
  17. B. I. Sturman, M. Aguilar, F. Agulló-López, and K. H. Ringhofer, “Fundamentals of the nonlinear theory of photorefractive subharmonics,” Phys. Rev. E 55, 6072–6083 (1997).
    [CrossRef]
  18. 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]

1998 (2)

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, D. J. Webb, and P. M. Johansen, “Fundamental characteristics of space-charge waves in photorefractive sillenite crystals,” J. Opt. Soc. Am. B 15, 2573–2580 (1998).
[CrossRef]

1997 (2)

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

H. C. Pedersen and P. M. Johansen, “Longitudinal, degenerate, and transversal photorefractive parametric oscillation: theory and experiment,” J. Opt. Soc. Am. B 14, 1418–1427 (1997).
[CrossRef]

1996 (2)

T. E. McClelland, D. J. Webb, B. I. Sturman, E. Shamonina, M. Mann, and K. H. Ringhofer, “Excitation of higher spatial harmonics by a moving light pattern in sillenites,” Opt. Commun. 131, 315–321 (1996).
[CrossRef]

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

1995 (3)

1994 (3)

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 in sillenite crystals,” Phys. Rev. Lett. 73, 3082–3084 (1994).
[CrossRef] [PubMed]

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

Aguilar, M.

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

Agulló-López, F.

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

Carrascosa, M.

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

Johansen, P. M.

López, F. A.

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

Mann, M.

T. E. McClelland, D. J. Webb, B. I. Sturman, E. Shamonina, M. Mann, and K. H. Ringhofer, “Excitation of higher spatial harmonics by a moving light pattern in sillenites,” Opt. Commun. 131, 315–321 (1996).
[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]

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]

McClelland, T. E.

T. E. McClelland, D. J. Webb, B. I. Sturman, E. Shamonina, M. Mann, and K. H. Ringhofer, “Excitation of higher spatial harmonics by a moving light pattern in sillenites,” Opt. Commun. 131, 315–321 (1996).
[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]

B. I. Sturman, T. E. McClelland, D. J. Webb, E. Shamonina, and K. H. Ringhofer, “Investigation of photorefractive subharmonics in the absence of wave mixing,” J. Opt. Soc. Am. B 12, 1621–1627 (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 in sillenite crystals,” Phys. Rev. Lett. 73, 3082–3084 (1994).
[CrossRef] [PubMed]

Otten, J.

Pedersen, H.

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]

H. C. Pedersen, D. J. Webb, and P. M. Johansen, “Fundamental characteristics of space-charge waves in photorefractive sillenite crystals,” J. Opt. Soc. Am. B 15, 2573–2580 (1998).
[CrossRef]

H. C. Pedersen and P. M. Johansen, “Longitudinal, degenerate, and transversal photorefractive parametric oscillation: theory and experiment,” J. Opt. Soc. Am. B 14, 1418–1427 (1997).
[CrossRef]

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

H. C. Pedersen and P. M. Johansen, “Parametric oscillation in photorefractive media,” J. Opt. Soc. Am. B 12, 1065–1073 (1995).
[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]

Ringhofer, K. H.

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

T. E. McClelland, D. J. Webb, B. I. Sturman, E. Shamonina, M. Mann, and K. H. Ringhofer, “Excitation of higher spatial harmonics by a moving light pattern in sillenites,” Opt. Commun. 131, 315–321 (1996).
[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]

B. I. Sturman, T. E. McClelland, D. J. Webb, E. Shamonina, and K. H. Ringhofer, “Investigation of photorefractive subharmonics in the absence of wave mixing,” J. Opt. Soc. Am. B 12, 1621–1627 (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 in sillenite crystals,” 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. I. Sturman, M. Mann, and K. H. Ringhofer, “Instability of moving gratings in photorefractive crystals,” Appl. Phys. A 55, 235–241 (1992).
[CrossRef]

Serrano, E.

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

Shamonina, E.

T. E. McClelland, D. J. Webb, B. I. Sturman, E. Shamonina, M. Mann, and K. H. Ringhofer, “Excitation of higher spatial harmonics by a moving light pattern in sillenites,” Opt. Commun. 131, 315–321 (1996).
[CrossRef]

B. I. Sturman, T. E. McClelland, D. J. Webb, E. Shamonina, and K. H. Ringhofer, “Investigation of photorefractive subharmonics in the absence of wave mixing,” J. Opt. Soc. Am. B 12, 1621–1627 (1995).
[CrossRef]

Solymar, L.

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

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

T. E. McClelland, D. J. Webb, B. I. Sturman, E. Shamonina, M. Mann, and K. H. Ringhofer, “Excitation of higher spatial harmonics by a moving light pattern in sillenites,” Opt. Commun. 131, 315–321 (1996).
[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]

B. I. Sturman, T. E. McClelland, D. J. Webb, E. Shamonina, and K. H. Ringhofer, “Investigation of photorefractive subharmonics in the absence of wave mixing,” J. Opt. Soc. Am. B 12, 1621–1627 (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 in sillenite crystals,” 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. I. Sturman, M. Mann, and K. H. Ringhofer, “Instability of moving gratings in photorefractive crystals,” Appl. Phys. A 55, 235–241 (1992).
[CrossRef]

Webb, D. J.

H. C. Pedersen, D. J. Webb, and P. M. Johansen, “Fundamental characteristics of space-charge waves in photorefractive sillenite crystals,” J. Opt. Soc. Am. B 15, 2573–2580 (1998).
[CrossRef]

T. E. McClelland, D. J. Webb, B. I. Sturman, E. Shamonina, M. Mann, and K. H. Ringhofer, “Excitation of higher spatial harmonics by a moving light pattern in sillenites,” Opt. Commun. 131, 315–321 (1996).
[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]

B. I. Sturman, T. E. McClelland, D. J. Webb, E. Shamonina, and K. H. Ringhofer, “Investigation of photorefractive subharmonics in the absence of wave mixing,” J. Opt. Soc. Am. B 12, 1621–1627 (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 in sillenite crystals,” 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. (1)

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

J. Opt. Soc. Am. B (5)

Opt. Commun. (2)

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, E. Shamonina, M. Mann, and K. H. Ringhofer, “Excitation of higher spatial harmonics by a moving light pattern in sillenites,” Opt. Commun. 131, 315–321 (1996).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. E (2)

B. I. Sturman, M. Aguilar, F. Agulló-López, 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. (2)

H. C. Pedersen and P. M. Johansen, “Longitudinal, degenerate, and transversal photorefractive 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 in sillenite crystals,” Phys. Rev. Lett. 73, 3082–3084 (1994).
[CrossRef] [PubMed]

Other (4)

L. Solymar, D. J. Webb, and A. Grunnet-Jepsen, The Physics and Applications of Photorefractive Materials (Clarendon, Oxford, 1996).

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

Note that in Ref. 12 some of the present authors reported a preliminary version of this method. The corresponding numerical calculations and their results suffer, however, from a random error in the numerical program. Now two different programs give identical results and correct limiting cases, which practically excludes any accidental error.

B. I. Sturman, E. Shamonina, and K. H. Ringhofer, “Explanation of the transverse subharmonic split in sillenites,” in Photorefractive Materials, Effects and Devices, PR’97 (Waseda University, Tokyo, 1997), pp. 145–148.

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Figures (5)

Fig. 1
Fig. 1

Geometrical scheme for nonsplit (K/2) and transversely split subharmonics. Points 1 and 2 indicate the ends of the wave vectors of the light waves that form the running light pattern. The frequency detuning between the light waves is Ω, and the difference of the light wave vectors, K, is parallel to the applied field, E0.

Fig. 2
Fig. 2

Comparison of the exact solution for EK (solid curve) with the result of the linearized theory (dotted curve).

Fig. 3
Fig. 3

Dependence |en|() on the number n for three values of . The lines are drawn as a guide for the eye.

Fig. 4
Fig. 4

Dependence of ΓN/Ω on for several values of N.

Fig. 5
Fig. 5

Dependence of Γ/Ω on |κ|=|k/K| for N=20 and several values of .

Equations (9)

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

I=I0[1+mcos(Kz-Ωt)],
E=EK exp(iKz-iΩt)+c.c.
Δφzt-ω0lsΔφ-1l0Δφt+ω0Δφz+lD2l0Δ2φt=-e0δgz+e01E0div(δgφ)+1E0div(Δφtφ),
E=E0n=120en exp(inξ)+c.c.,
u=sus exp(isξ),
(Csδss-Ass)us=0.
Cs=(s2+κ2)-isΓ+-s2-isω0Ω+1Kl0(Γ-is)+KlD2l0(s2+κ2)(Γ-is),
Ass=-m2(ss+κ2)(δss+1+δss-1)+es-s[(s-s)2(ss+κ2)+s(s2+κ2)(s+iΓ)],
Det(Csδss-Ass)=0,

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