Abstract

The stability of the steady-state solutions of four-wave mixing in photorefractive crystals has been studied. Two complementary methods have been used: linear stability analysis and time integration of the nonlinear equations. A critical dependence of the stability on the phase of the perturbation has been revealed. The existence of two stable solutions for certain values of some parameters has been found. Also, regimes of unstable behavior in cases with a unique steady-state solution for large values of the pump ratio have been found.

© 1989 Optical Society of America

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  1. P. Günter, E. Voit, M. Z. Zha, and J. Albers, Opt. Commun. 55, 210 (1985);D. J. Gauthier, P. Narum, and R. W. Boyd, Phys. Rev. Lett. 58, 1640 (1987);G. Valley and G. Dunning, Opt. Lett. 9, 420 (1984).
    [CrossRef] [PubMed]
  2. D. Rytz and S. De Zhong, in Digest of Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1989),paper PD07.
  3. S. I. Stepanov and M. P. Petrov, Opt. Commun. 53, 64 (1985).
    [CrossRef]
  4. A. Błęgdowski and W. Królikowski, IEEE J. Quantum Electron. QE-24, 652 (1988).
    [CrossRef]
  5. K. Shaw and M. Cronin-Golomb, Opt. Commun. 65, 301 (1988).
    [CrossRef]
  6. B. Fischer and S. Weiss, Appl. Phys. Lett. 53, 257 (1988).
    [CrossRef]
  7. P. Gunter, Phys. Rep. 93, 199 (1982).
    [CrossRef]
  8. B. Imbert, H. Rajbenbach, S. Mallick, J. P. Herriau, and J. P. Huignard, Opt. Lett. 13, 327 (1988);A. M. Glass, A. M. Johnson, D. H. Olson, W. Simpson, and A. A. Ballman, Appl. Phys. Lett. 44, 948 (1984).
    [CrossRef] [PubMed]
  9. Ph. Refregier, L. Solymar, H. Rajbenbach, and J. P. Huignard, J. Appl. Phys. 58, 45 (1985).
    [CrossRef]
  10. N. V. Kukhtarev, V. Markov, S. Odulov, M. Soskin, and V. Vinetskii, Ferroelectrics 22, 949 (1979);N. V. Kukhtarev, V. Markov, and S. Odulov, Opt. Commun. 23, 338 (1977).
    [CrossRef]
  11. C. Pare, M. Piche, and P. A. Belanger, J. Opt. Soc. Am. B 5, 679 (1988).
    [CrossRef]
  12. M. Cronin-Golomb and K. Shaw, in XVI International Conference on Quantum Electronics Technical Digest (Japan Society of Applied Physics, Tokyo, 1988), pp. 132–133.
  13. J. M. Heaton and L. Solymar, Opt. Acta 32, 397 (1985);V. Kalinin and L. Solymar, Appl. Phys. B 45, 129 (1988).
    [CrossRef]
  14. A Błędowski, W. Królikowski, and A. Kujawski, “Temporal instabilities in single-grating photorefractive four-wave mixing,” J. Opt. Soc. Am. B (to be published).
  15. W. Królikowski, M. Belić, and A. Błędowski, Phys. Rev. A 37, 2224 (1988).
    [CrossRef] [PubMed]
  16. S. Odulov and B. Sturman, Sov. Phys. JETP 65, 1134 (1987);A. A. Zozulya and V. T. Tikhonchuk, Phys. Lett. A 135, 447 (1989).
    [CrossRef]
  17. H. G. Winful and G. D. Cooperman, Appl. Phys. Lett. 40, 298 (1982).
    [CrossRef]
  18. M. Cronin-Golomb, J. O. White, B. Fischer, and A. Yariv, IEEE J. Quantum Electron. QE-20, 12 (1984).
    [CrossRef]

1988 (6)

A. Błęgdowski and W. Królikowski, IEEE J. Quantum Electron. QE-24, 652 (1988).
[CrossRef]

K. Shaw and M. Cronin-Golomb, Opt. Commun. 65, 301 (1988).
[CrossRef]

B. Fischer and S. Weiss, Appl. Phys. Lett. 53, 257 (1988).
[CrossRef]

B. Imbert, H. Rajbenbach, S. Mallick, J. P. Herriau, and J. P. Huignard, Opt. Lett. 13, 327 (1988);A. M. Glass, A. M. Johnson, D. H. Olson, W. Simpson, and A. A. Ballman, Appl. Phys. Lett. 44, 948 (1984).
[CrossRef] [PubMed]

C. Pare, M. Piche, and P. A. Belanger, J. Opt. Soc. Am. B 5, 679 (1988).
[CrossRef]

W. Królikowski, M. Belić, and A. Błędowski, Phys. Rev. A 37, 2224 (1988).
[CrossRef] [PubMed]

1987 (1)

S. Odulov and B. Sturman, Sov. Phys. JETP 65, 1134 (1987);A. A. Zozulya and V. T. Tikhonchuk, Phys. Lett. A 135, 447 (1989).
[CrossRef]

1985 (4)

J. M. Heaton and L. Solymar, Opt. Acta 32, 397 (1985);V. Kalinin and L. Solymar, Appl. Phys. B 45, 129 (1988).
[CrossRef]

Ph. Refregier, L. Solymar, H. Rajbenbach, and J. P. Huignard, J. Appl. Phys. 58, 45 (1985).
[CrossRef]

P. Günter, E. Voit, M. Z. Zha, and J. Albers, Opt. Commun. 55, 210 (1985);D. J. Gauthier, P. Narum, and R. W. Boyd, Phys. Rev. Lett. 58, 1640 (1987);G. Valley and G. Dunning, Opt. Lett. 9, 420 (1984).
[CrossRef] [PubMed]

S. I. Stepanov and M. P. Petrov, Opt. Commun. 53, 64 (1985).
[CrossRef]

1984 (1)

M. Cronin-Golomb, J. O. White, B. Fischer, and A. Yariv, IEEE J. Quantum Electron. QE-20, 12 (1984).
[CrossRef]

1982 (2)

H. G. Winful and G. D. Cooperman, Appl. Phys. Lett. 40, 298 (1982).
[CrossRef]

P. Gunter, Phys. Rep. 93, 199 (1982).
[CrossRef]

1979 (1)

N. V. Kukhtarev, V. Markov, S. Odulov, M. Soskin, and V. Vinetskii, Ferroelectrics 22, 949 (1979);N. V. Kukhtarev, V. Markov, and S. Odulov, Opt. Commun. 23, 338 (1977).
[CrossRef]

Albers, J.

P. Günter, E. Voit, M. Z. Zha, and J. Albers, Opt. Commun. 55, 210 (1985);D. J. Gauthier, P. Narum, and R. W. Boyd, Phys. Rev. Lett. 58, 1640 (1987);G. Valley and G. Dunning, Opt. Lett. 9, 420 (1984).
[CrossRef] [PubMed]

Belanger, P. A.

Belic, M.

W. Królikowski, M. Belić, and A. Błędowski, Phys. Rev. A 37, 2224 (1988).
[CrossRef] [PubMed]

Bledowski, A

A Błędowski, W. Królikowski, and A. Kujawski, “Temporal instabilities in single-grating photorefractive four-wave mixing,” J. Opt. Soc. Am. B (to be published).

Bledowski, A.

W. Królikowski, M. Belić, and A. Błędowski, Phys. Rev. A 37, 2224 (1988).
[CrossRef] [PubMed]

Blegdowski, A.

A. Błęgdowski and W. Królikowski, IEEE J. Quantum Electron. QE-24, 652 (1988).
[CrossRef]

Cooperman, G. D.

H. G. Winful and G. D. Cooperman, Appl. Phys. Lett. 40, 298 (1982).
[CrossRef]

Cronin-Golomb, M.

K. Shaw and M. Cronin-Golomb, Opt. Commun. 65, 301 (1988).
[CrossRef]

M. Cronin-Golomb, J. O. White, B. Fischer, and A. Yariv, IEEE J. Quantum Electron. QE-20, 12 (1984).
[CrossRef]

M. Cronin-Golomb and K. Shaw, in XVI International Conference on Quantum Electronics Technical Digest (Japan Society of Applied Physics, Tokyo, 1988), pp. 132–133.

De Zhong, S.

D. Rytz and S. De Zhong, in Digest of Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1989),paper PD07.

Fischer, B.

B. Fischer and S. Weiss, Appl. Phys. Lett. 53, 257 (1988).
[CrossRef]

M. Cronin-Golomb, J. O. White, B. Fischer, and A. Yariv, IEEE J. Quantum Electron. QE-20, 12 (1984).
[CrossRef]

Gunter, P.

P. Gunter, Phys. Rep. 93, 199 (1982).
[CrossRef]

Günter, P.

P. Günter, E. Voit, M. Z. Zha, and J. Albers, Opt. Commun. 55, 210 (1985);D. J. Gauthier, P. Narum, and R. W. Boyd, Phys. Rev. Lett. 58, 1640 (1987);G. Valley and G. Dunning, Opt. Lett. 9, 420 (1984).
[CrossRef] [PubMed]

Heaton, J. M.

J. M. Heaton and L. Solymar, Opt. Acta 32, 397 (1985);V. Kalinin and L. Solymar, Appl. Phys. B 45, 129 (1988).
[CrossRef]

Herriau, J. P.

Huignard, J. P.

Imbert, B.

Królikowski, W.

W. Królikowski, M. Belić, and A. Błędowski, Phys. Rev. A 37, 2224 (1988).
[CrossRef] [PubMed]

A. Błęgdowski and W. Królikowski, IEEE J. Quantum Electron. QE-24, 652 (1988).
[CrossRef]

A Błędowski, W. Królikowski, and A. Kujawski, “Temporal instabilities in single-grating photorefractive four-wave mixing,” J. Opt. Soc. Am. B (to be published).

Kujawski, A.

A Błędowski, W. Królikowski, and A. Kujawski, “Temporal instabilities in single-grating photorefractive four-wave mixing,” J. Opt. Soc. Am. B (to be published).

Kukhtarev, N. V.

N. V. Kukhtarev, V. Markov, S. Odulov, M. Soskin, and V. Vinetskii, Ferroelectrics 22, 949 (1979);N. V. Kukhtarev, V. Markov, and S. Odulov, Opt. Commun. 23, 338 (1977).
[CrossRef]

Mallick, S.

Markov, V.

N. V. Kukhtarev, V. Markov, S. Odulov, M. Soskin, and V. Vinetskii, Ferroelectrics 22, 949 (1979);N. V. Kukhtarev, V. Markov, and S. Odulov, Opt. Commun. 23, 338 (1977).
[CrossRef]

Odulov, S.

S. Odulov and B. Sturman, Sov. Phys. JETP 65, 1134 (1987);A. A. Zozulya and V. T. Tikhonchuk, Phys. Lett. A 135, 447 (1989).
[CrossRef]

N. V. Kukhtarev, V. Markov, S. Odulov, M. Soskin, and V. Vinetskii, Ferroelectrics 22, 949 (1979);N. V. Kukhtarev, V. Markov, and S. Odulov, Opt. Commun. 23, 338 (1977).
[CrossRef]

Pare, C.

Petrov, M. P.

S. I. Stepanov and M. P. Petrov, Opt. Commun. 53, 64 (1985).
[CrossRef]

Piche, M.

Rajbenbach, H.

Refregier, Ph.

Ph. Refregier, L. Solymar, H. Rajbenbach, and J. P. Huignard, J. Appl. Phys. 58, 45 (1985).
[CrossRef]

Rytz, D.

D. Rytz and S. De Zhong, in Digest of Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1989),paper PD07.

Shaw, K.

K. Shaw and M. Cronin-Golomb, Opt. Commun. 65, 301 (1988).
[CrossRef]

M. Cronin-Golomb and K. Shaw, in XVI International Conference on Quantum Electronics Technical Digest (Japan Society of Applied Physics, Tokyo, 1988), pp. 132–133.

Solymar, L.

J. M. Heaton and L. Solymar, Opt. Acta 32, 397 (1985);V. Kalinin and L. Solymar, Appl. Phys. B 45, 129 (1988).
[CrossRef]

Ph. Refregier, L. Solymar, H. Rajbenbach, and J. P. Huignard, J. Appl. Phys. 58, 45 (1985).
[CrossRef]

Soskin, M.

N. V. Kukhtarev, V. Markov, S. Odulov, M. Soskin, and V. Vinetskii, Ferroelectrics 22, 949 (1979);N. V. Kukhtarev, V. Markov, and S. Odulov, Opt. Commun. 23, 338 (1977).
[CrossRef]

Stepanov, S. I.

S. I. Stepanov and M. P. Petrov, Opt. Commun. 53, 64 (1985).
[CrossRef]

Sturman, B.

S. Odulov and B. Sturman, Sov. Phys. JETP 65, 1134 (1987);A. A. Zozulya and V. T. Tikhonchuk, Phys. Lett. A 135, 447 (1989).
[CrossRef]

Vinetskii, V.

N. V. Kukhtarev, V. Markov, S. Odulov, M. Soskin, and V. Vinetskii, Ferroelectrics 22, 949 (1979);N. V. Kukhtarev, V. Markov, and S. Odulov, Opt. Commun. 23, 338 (1977).
[CrossRef]

Voit, E.

P. Günter, E. Voit, M. Z. Zha, and J. Albers, Opt. Commun. 55, 210 (1985);D. J. Gauthier, P. Narum, and R. W. Boyd, Phys. Rev. Lett. 58, 1640 (1987);G. Valley and G. Dunning, Opt. Lett. 9, 420 (1984).
[CrossRef] [PubMed]

Weiss, S.

B. Fischer and S. Weiss, Appl. Phys. Lett. 53, 257 (1988).
[CrossRef]

White, J. O.

M. Cronin-Golomb, J. O. White, B. Fischer, and A. Yariv, IEEE J. Quantum Electron. QE-20, 12 (1984).
[CrossRef]

Winful, H. G.

H. G. Winful and G. D. Cooperman, Appl. Phys. Lett. 40, 298 (1982).
[CrossRef]

Yariv, A.

M. Cronin-Golomb, J. O. White, B. Fischer, and A. Yariv, IEEE J. Quantum Electron. QE-20, 12 (1984).
[CrossRef]

Zha, M. Z.

P. Günter, E. Voit, M. Z. Zha, and J. Albers, Opt. Commun. 55, 210 (1985);D. J. Gauthier, P. Narum, and R. W. Boyd, Phys. Rev. Lett. 58, 1640 (1987);G. Valley and G. Dunning, Opt. Lett. 9, 420 (1984).
[CrossRef] [PubMed]

Appl. Phys. Lett. (2)

B. Fischer and S. Weiss, Appl. Phys. Lett. 53, 257 (1988).
[CrossRef]

H. G. Winful and G. D. Cooperman, Appl. Phys. Lett. 40, 298 (1982).
[CrossRef]

Ferroelectrics (1)

N. V. Kukhtarev, V. Markov, S. Odulov, M. Soskin, and V. Vinetskii, Ferroelectrics 22, 949 (1979);N. V. Kukhtarev, V. Markov, and S. Odulov, Opt. Commun. 23, 338 (1977).
[CrossRef]

IEEE J. Quantum Electron. (2)

M. Cronin-Golomb, J. O. White, B. Fischer, and A. Yariv, IEEE J. Quantum Electron. QE-20, 12 (1984).
[CrossRef]

A. Błęgdowski and W. Królikowski, IEEE J. Quantum Electron. QE-24, 652 (1988).
[CrossRef]

J. Appl. Phys. (1)

Ph. Refregier, L. Solymar, H. Rajbenbach, and J. P. Huignard, J. Appl. Phys. 58, 45 (1985).
[CrossRef]

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

Opt. Acta (1)

J. M. Heaton and L. Solymar, Opt. Acta 32, 397 (1985);V. Kalinin and L. Solymar, Appl. Phys. B 45, 129 (1988).
[CrossRef]

Opt. Commun. (3)

K. Shaw and M. Cronin-Golomb, Opt. Commun. 65, 301 (1988).
[CrossRef]

P. Günter, E. Voit, M. Z. Zha, and J. Albers, Opt. Commun. 55, 210 (1985);D. J. Gauthier, P. Narum, and R. W. Boyd, Phys. Rev. Lett. 58, 1640 (1987);G. Valley and G. Dunning, Opt. Lett. 9, 420 (1984).
[CrossRef] [PubMed]

S. I. Stepanov and M. P. Petrov, Opt. Commun. 53, 64 (1985).
[CrossRef]

Opt. Lett. (1)

Phys. Rep. (1)

P. Gunter, Phys. Rep. 93, 199 (1982).
[CrossRef]

Phys. Rev. A (1)

W. Królikowski, M. Belić, and A. Błędowski, Phys. Rev. A 37, 2224 (1988).
[CrossRef] [PubMed]

Sov. Phys. JETP (1)

S. Odulov and B. Sturman, Sov. Phys. JETP 65, 1134 (1987);A. A. Zozulya and V. T. Tikhonchuk, Phys. Lett. A 135, 447 (1989).
[CrossRef]

Other (3)

A Błędowski, W. Królikowski, and A. Kujawski, “Temporal instabilities in single-grating photorefractive four-wave mixing,” J. Opt. Soc. Am. B (to be published).

M. Cronin-Golomb and K. Shaw, in XVI International Conference on Quantum Electronics Technical Digest (Japan Society of Applied Physics, Tokyo, 1988), pp. 132–133.

D. Rytz and S. De Zhong, in Digest of Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1989),paper PD07.

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Figures (8)

Fig. 1
Fig. 1

Poles in the complex Ω plane, for two unstable steady-state solutions, γl = −5, I2l/I10 = 10, I40 = 0.01. (a) Lower unstable solution of Fig. 2(b); (b) upper unstable solution of Fig. 2(b).

Fig. 2
Fig. 2

Stability of the steady-state solutions, (a) γl = −3, I2l/I10 = 1. (b) γl = −5, I2l/I10 = 10. Solid curves, stable solution; dashed–dotted curves, conditionally stable; dashed curves, unstable.

Fig. 3
Fig. 3

Illustrating the numerical experiment in which special boundary conditions with nonzero I3l force the system to approach the second stable state (time is normalized to τ).

Fig. 4
Fig. 4

Temporal growth of self-oscillation for γl = −3 (I40 = 10−5 and I40 = 10−6) and γl = −5 (I40 = 10−6). I2l/I10 = 1 (time is normalized to τ).

Fig. 5
Fig. 5

Stability of the steady-state solution for positive coupling constant (γl = 7, I2l/I10 = 50). Solid curve, stable solutions, dashed–dotted curve, conditionally stable (CS); dashed curve, unstable (U). There are two stable solutions for 6.7 < I40 < 7.5.

Fig. 6
Fig. 6

Response of the lower steady-state solution of Fig. 5 to different perturbations: (a) intensity perturbation, (b) phase perturbation. γl = 7, I2l/I10 = 50, I4 = 10 (time is normalized to τ).

Fig. 7
Fig. 7

Same as Fig. 5 but for a large pump ratio, γl = 7. I2l/I10 = 100. For 10.0 < I40 < 12.0 the single steady-state solution is unstable.

Fig. 8
Fig. 8

Unstable behavior of the phase-conjugate intensity for a large pump ratio, γl = 7, I2l/I10 = 100, I40 = 10.1. In this case there is no steady state (time is normalized to τ).

Equations (5)

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

A 1 / z = Q A 4 ,
A 4 * / z = Q A 1 * ,
A 2 * / z = Q A 3 * ,
A 3 / z = Q A 2 ,
τ Q / t + Q = γ ( A 1 A 4 * + A 2 * A 3 ) / I 0 ,

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