Abstract

Rigorous coupled-wave diffraction theory is used to analyze two-wave and multiwave mixing in diffusion-controlled photorefractive barium titanate, which is modeled by the Kukhtarev equations. These equations are partially decoupled to yield a system of two equations between the electron density and the electrostatic field (and hence the induced refractive index profile). The transmitted and reflected optical fields, the dielectric modulation, the electrostatic field, and the electron density are studied for the cases in which the interfering, incident optical fields have equal and unequal amplitudes for different values of the linear refractive index mismatch and for different values of photorefractive crystal length. In each case the exact longitudinal inhomogeneity in the photorefractive medium is analyzed with the use of rigorous coupled-wave diffraction theory and an exact Kukhtarev analysis. We compare the evolution of the diffracted orders for different sample lengths to show that the nature of the steady state (oscillatory or nonoscillatory) critically depends on the sample length. Our computations study in BaTiO3 the conditions for temporal instability resulting in self-pulsation and for anisotropic diffraction contributing to significant generation of higher orders (assuming that two plane waves are incident on the photorefractive material).

© 1996 Optical Society of America

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  3. J. Feinberg, “Self-pumped continuous-wave conjugator using internal reflections,” Opt. Lett. 7, 486–488 (1982).
    [CrossRef] [PubMed]
  4. J. O. White, A. Yariv, “Real-time image processing via four-wave mixing in a photorefractive medium,” Appl. Phys. Lett. 37, 5–7 (1980).
    [CrossRef]
  5. P. Yeh, A. E. T. Chiou, “Real-time contrast reversal via four-wave mixing nonlinear media,” Opt. Commun. 64, 160–162 (1987).
    [CrossRef]
  6. N. V. Kukhtarev, M. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetsky, “Holographic storage in electrooptic crystals,” Ferroelectrics 22, 949–964 (1979).
    [CrossRef]
  7. P. Yeh, Introduction to Photorefractive Nonlinear Optics (Wiley, New York, 1993).
  8. J. E. Millerd, E. M. Garmire, M. B. Klein, B. A. Wechsler, F. P. Strohkendl, G. A. Brost, “Photorefractive response at high modulation depths in Bi12TiO20,” J. Opt. Soc. Am. B 8, 1449–1453 (1991).
  9. R. A. Vazquez, F. R. Vachss, R. R. Neurgaonkar, D. Ewbank, “Large photorefractive coupling coefficient in a thin cerbium-doped strontium barium niobate crystal,” J. Opt. Soc. Am. B 9, 1932–1941 (1992).
  10. M. R. Belic, M. Petrovic, “Unified method for solution of wave equations in photorefractive media,” J. Opt. Soc. Am. B 11, 481–485 (1994).
    [CrossRef]
  11. P. P. Banerjee, J. Jarem, “Transient wave mixing and recording kinetics in photorefractive barium titanate: a nonlinear coupled mode approach,” Opt. Eng. 34, 2254–2260 (1995).
    [CrossRef]
  12. J. Jarem, P. P. Banerjee, “A nonlinear, transient analysis of two and multi-wave mixing in a photorefractive material using rigorous coupled-mode diffraction theory,” Opt. Commun. (to be published).
  13. J. Jarem, P. P. Banerjee, “Rigorous coupled wave dynamical analysis of beam coupling in photorefractive materials,” in Annual Meeting/ILS-X (Optical Society of America, Washington, D.C., 1994), p. 162.
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    [CrossRef]
  15. K. Rokushima, J. Yamakita, S. Mori, K. Tominaga, “Unified approach to wave diffraction by space–time periodic anisotropic media,” IEEE Trans. Microwave Theory Tech. MTT-35, 937–945 (1987).
    [CrossRef]
  16. E. N. Glytsis, T. K. Gaylord, “Rigorous three-dimensional coupled-wave diffraction analysis of single cascaded anisotropic gratings,” J. Opt. Soc. Am. B 4, 2061–2080 (1987).
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  17. J. M. Heaton, L. Solymar, “Transient energy transfer during hologram formation in photorefractive crystals,” Opt. Acta 32, 397–408 (1985).
    [CrossRef]
  18. J. C. Kralik, M. S. Malcuit, “Transient oscillations in nondegenerate two-beam coupling,” Opt. Commun. 107, 401–405 (1994).
    [CrossRef]
  19. P. Gunter, E. Voit, M. Z. Zha, J. Albers, “Self-pulsation and optical chaos in self-pumped photorefractive BaTiO3,” Opt. Commun. 55, 210–214 (1985).
    [CrossRef]
  20. A Nowak, T. R. Moore, R. A. Fisher, “Observation of internal beam production in BaTiO3phase conjugator,” J. Opt. Soc. Am. B 5, 1864–1878 (1988).
    [CrossRef]
  21. P. M. Jeffrey, R. W. Eason, “Lyapunov exponent analysis of irregular fluctuations in a self-pumped BaTiO3phase conjugate mirror,” J. Opt. Soc. Am. B 11, 476–480 (1994).
    [CrossRef]
  22. D. Wang, Z. Zhang, X. Wu, P. Ye, “Instabilities in a mutually pumped phase conjugator of BaTiO3,” J. Opt. Soc. Am. B 7, 2289–2293 (1990).
    [CrossRef]
  23. L.-K. Dai, C. Gu, P. Yeh, “Effect of position-dependent time-constant on photorefractive two-wave mixing,” J. Opt. Soc. Am. B 9, 1693–1697 (1992).
    [CrossRef]
  24. M. Snowbell, M. Horowitz, B. Fischer, “Dynamics of multiple two-wave mixing and fanning in photorefractive materials,” J. Opt. Soc. Am. B 9, 1972–1982 (1994).
    [CrossRef]
  25. E. Serrano, V. Lopez, M. Carrascosa, F. Agrillo-Lopez, “Recording and erasure kinetics in photorefractive materials at large modulation depths,” J. Opt. Soc. Am. B 11, 670–675 (1994).
    [CrossRef]
  26. W. P. Brown, G. C. Valley, “Kinky beam paths inside photorefractive crystals,” J. Opt. Soc. Am. B 10, 1901–1906 (1993).
    [CrossRef]
  27. L. B. Au, L. Solymar, “Higher diffraction orders in photorefractive materials,” IEEE J. Quantum Electron 24, 162–168 (1987).
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  28. J. Ma, B. Catanzaro, J. E. Ford, Y. Fainman, S. H. Lee, “Photorefractive holographic lenses and applications for dynamic focusing and dynamic image shifting,” J. Opt. Soc. Am. A 11, 2471–2480 (1994).
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  29. J. H. Kulick, J. M. Jarem, R. G. Lindquist, S. T. Kowel, M. W. Friends, T. M. Leslie, “Electrostatic and diffraction analysis of a liquid crystal device utilizing fringing fields: applications to 3-D displays,” Appl. Opt. 34, 1901–1922 (1995).
    [CrossRef] [PubMed]
  30. P. P. Banerjee, J. J. Liu, “Perturbational analysis of steady-state and transient-beam fanning in thin and thick photorefractive media,” J. Opt. Soc. Am. B 10, 1417–1423 (1993).
    [CrossRef]
  31. P. P. Banerjee, R. M. Misra, “Dependence of photorefractive beam fanning on beam parameters,” Opt. Commun. 100, 166–172 (1993).
    [CrossRef]
  32. A. Korpel, Acousto-Optics (Marcel Dekker, New York, 1988).

1995 (2)

1994 (6)

1993 (3)

1992 (2)

L.-K. Dai, C. Gu, P. Yeh, “Effect of position-dependent time-constant on photorefractive two-wave mixing,” J. Opt. Soc. Am. B 9, 1693–1697 (1992).
[CrossRef]

R. A. Vazquez, F. R. Vachss, R. R. Neurgaonkar, D. Ewbank, “Large photorefractive coupling coefficient in a thin cerbium-doped strontium barium niobate crystal,” J. Opt. Soc. Am. B 9, 1932–1941 (1992).

1991 (1)

J. E. Millerd, E. M. Garmire, M. B. Klein, B. A. Wechsler, F. P. Strohkendl, G. A. Brost, “Photorefractive response at high modulation depths in Bi12TiO20,” J. Opt. Soc. Am. B 8, 1449–1453 (1991).

1990 (1)

1988 (1)

1987 (4)

L. B. Au, L. Solymar, “Higher diffraction orders in photorefractive materials,” IEEE J. Quantum Electron 24, 162–168 (1987).
[CrossRef]

K. Rokushima, J. Yamakita, S. Mori, K. Tominaga, “Unified approach to wave diffraction by space–time periodic anisotropic media,” IEEE Trans. Microwave Theory Tech. MTT-35, 937–945 (1987).
[CrossRef]

E. N. Glytsis, T. K. Gaylord, “Rigorous three-dimensional coupled-wave diffraction analysis of single cascaded anisotropic gratings,” J. Opt. Soc. Am. B 4, 2061–2080 (1987).
[CrossRef]

P. Yeh, A. E. T. Chiou, “Real-time contrast reversal via four-wave mixing nonlinear media,” Opt. Commun. 64, 160–162 (1987).
[CrossRef]

1985 (2)

J. M. Heaton, L. Solymar, “Transient energy transfer during hologram formation in photorefractive crystals,” Opt. Acta 32, 397–408 (1985).
[CrossRef]

P. Gunter, E. Voit, M. Z. Zha, J. Albers, “Self-pulsation and optical chaos in self-pumped photorefractive BaTiO3,” Opt. Commun. 55, 210–214 (1985).
[CrossRef]

1982 (1)

1981 (2)

J.-P. Huignard, A. Marrakchi, “Coherent signal beam amplification in two-wave mixing experiments with photorefractive Bi12SiO20crystals,” Opt. Commun. 38, 249–258 (1981).
[CrossRef]

M. G. Moharam, T. K. Gaylord, “Rigorous coupled-wave analysis of planar grating diffraction,” J. Opt. Soc. Am. 71, 811–818 (1981).
[CrossRef]

1980 (1)

J. O. White, A. Yariv, “Real-time image processing via four-wave mixing in a photorefractive medium,” Appl. Phys. Lett. 37, 5–7 (1980).
[CrossRef]

1979 (1)

N. V. Kukhtarev, M. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetsky, “Holographic storage in electrooptic crystals,” Ferroelectrics 22, 949–964 (1979).
[CrossRef]

Agrillo-Lopez, F.

Albers, J.

P. Gunter, E. Voit, M. Z. Zha, J. Albers, “Self-pulsation and optical chaos in self-pumped photorefractive BaTiO3,” Opt. Commun. 55, 210–214 (1985).
[CrossRef]

Au, L. B.

L. B. Au, L. Solymar, “Higher diffraction orders in photorefractive materials,” IEEE J. Quantum Electron 24, 162–168 (1987).
[CrossRef]

Banerjee, P. P.

P. P. Banerjee, J. Jarem, “Transient wave mixing and recording kinetics in photorefractive barium titanate: a nonlinear coupled mode approach,” Opt. Eng. 34, 2254–2260 (1995).
[CrossRef]

P. P. Banerjee, J. J. Liu, “Perturbational analysis of steady-state and transient-beam fanning in thin and thick photorefractive media,” J. Opt. Soc. Am. B 10, 1417–1423 (1993).
[CrossRef]

P. P. Banerjee, R. M. Misra, “Dependence of photorefractive beam fanning on beam parameters,” Opt. Commun. 100, 166–172 (1993).
[CrossRef]

J. Jarem, P. P. Banerjee, “A nonlinear, transient analysis of two and multi-wave mixing in a photorefractive material using rigorous coupled-mode diffraction theory,” Opt. Commun. (to be published).

J. Jarem, P. P. Banerjee, “Rigorous coupled wave dynamical analysis of beam coupling in photorefractive materials,” in Annual Meeting/ILS-X (Optical Society of America, Washington, D.C., 1994), p. 162.

Belic, M. R.

Brost, G. A.

J. E. Millerd, E. M. Garmire, M. B. Klein, B. A. Wechsler, F. P. Strohkendl, G. A. Brost, “Photorefractive response at high modulation depths in Bi12TiO20,” J. Opt. Soc. Am. B 8, 1449–1453 (1991).

Brown, W. P.

Carrascosa, M.

Catanzaro, B.

Chiou, A. E. T.

P. Yeh, A. E. T. Chiou, “Real-time contrast reversal via four-wave mixing nonlinear media,” Opt. Commun. 64, 160–162 (1987).
[CrossRef]

Dai, L.-K.

Eason, R. W.

Ewbank, D.

R. A. Vazquez, F. R. Vachss, R. R. Neurgaonkar, D. Ewbank, “Large photorefractive coupling coefficient in a thin cerbium-doped strontium barium niobate crystal,” J. Opt. Soc. Am. B 9, 1932–1941 (1992).

Fainman, Y.

Feinberg, J.

Fischer, B.

M. Snowbell, M. Horowitz, B. Fischer, “Dynamics of multiple two-wave mixing and fanning in photorefractive materials,” J. Opt. Soc. Am. B 9, 1972–1982 (1994).
[CrossRef]

Fisher, R. A.

Ford, J. E.

Friends, M. W.

Garmire, E. M.

J. E. Millerd, E. M. Garmire, M. B. Klein, B. A. Wechsler, F. P. Strohkendl, G. A. Brost, “Photorefractive response at high modulation depths in Bi12TiO20,” J. Opt. Soc. Am. B 8, 1449–1453 (1991).

Gaylord, T. K.

E. N. Glytsis, T. K. Gaylord, “Rigorous three-dimensional coupled-wave diffraction analysis of single cascaded anisotropic gratings,” J. Opt. Soc. Am. B 4, 2061–2080 (1987).
[CrossRef]

M. G. Moharam, T. K. Gaylord, “Rigorous coupled-wave analysis of planar grating diffraction,” J. Opt. Soc. Am. 71, 811–818 (1981).
[CrossRef]

Glytsis, E. N.

E. N. Glytsis, T. K. Gaylord, “Rigorous three-dimensional coupled-wave diffraction analysis of single cascaded anisotropic gratings,” J. Opt. Soc. Am. B 4, 2061–2080 (1987).
[CrossRef]

Gu, C.

Gunter, P.

P. Gunter, E. Voit, M. Z. Zha, J. Albers, “Self-pulsation and optical chaos in self-pumped photorefractive BaTiO3,” Opt. Commun. 55, 210–214 (1985).
[CrossRef]

Heaton, J. M.

J. M. Heaton, L. Solymar, “Transient energy transfer during hologram formation in photorefractive crystals,” Opt. Acta 32, 397–408 (1985).
[CrossRef]

Horowitz, M.

M. Snowbell, M. Horowitz, B. Fischer, “Dynamics of multiple two-wave mixing and fanning in photorefractive materials,” J. Opt. Soc. Am. B 9, 1972–1982 (1994).
[CrossRef]

Huignard, J.-P.

J.-P. Huignard, A. Marrakchi, “Coherent signal beam amplification in two-wave mixing experiments with photorefractive Bi12SiO20crystals,” Opt. Commun. 38, 249–258 (1981).
[CrossRef]

Jarem, J.

P. P. Banerjee, J. Jarem, “Transient wave mixing and recording kinetics in photorefractive barium titanate: a nonlinear coupled mode approach,” Opt. Eng. 34, 2254–2260 (1995).
[CrossRef]

J. Jarem, P. P. Banerjee, “A nonlinear, transient analysis of two and multi-wave mixing in a photorefractive material using rigorous coupled-mode diffraction theory,” Opt. Commun. (to be published).

J. Jarem, P. P. Banerjee, “Rigorous coupled wave dynamical analysis of beam coupling in photorefractive materials,” in Annual Meeting/ILS-X (Optical Society of America, Washington, D.C., 1994), p. 162.

Jarem, J. M.

Jeffrey, P. M.

Klein, M. B.

J. E. Millerd, E. M. Garmire, M. B. Klein, B. A. Wechsler, F. P. Strohkendl, G. A. Brost, “Photorefractive response at high modulation depths in Bi12TiO20,” J. Opt. Soc. Am. B 8, 1449–1453 (1991).

Korpel, A.

A. Korpel, Acousto-Optics (Marcel Dekker, New York, 1988).

Kowel, S. T.

Kralik, J. C.

J. C. Kralik, M. S. Malcuit, “Transient oscillations in nondegenerate two-beam coupling,” Opt. Commun. 107, 401–405 (1994).
[CrossRef]

Kukhtarev, N. V.

N. V. Kukhtarev, M. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetsky, “Holographic storage in electrooptic crystals,” Ferroelectrics 22, 949–964 (1979).
[CrossRef]

Kulick, J. H.

Lee, S. H.

Leslie, T. M.

Lindquist, R. G.

Liu, J. J.

Lopez, V.

Ma, J.

Malcuit, M. S.

J. C. Kralik, M. S. Malcuit, “Transient oscillations in nondegenerate two-beam coupling,” Opt. Commun. 107, 401–405 (1994).
[CrossRef]

Markov, M. B.

N. V. Kukhtarev, M. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetsky, “Holographic storage in electrooptic crystals,” Ferroelectrics 22, 949–964 (1979).
[CrossRef]

Marrakchi, A.

J.-P. Huignard, A. Marrakchi, “Coherent signal beam amplification in two-wave mixing experiments with photorefractive Bi12SiO20crystals,” Opt. Commun. 38, 249–258 (1981).
[CrossRef]

Millerd, J. E.

J. E. Millerd, E. M. Garmire, M. B. Klein, B. A. Wechsler, F. P. Strohkendl, G. A. Brost, “Photorefractive response at high modulation depths in Bi12TiO20,” J. Opt. Soc. Am. B 8, 1449–1453 (1991).

Misra, R. M.

P. P. Banerjee, R. M. Misra, “Dependence of photorefractive beam fanning on beam parameters,” Opt. Commun. 100, 166–172 (1993).
[CrossRef]

Moharam, M. G.

Moore, T. R.

Mori, S.

K. Rokushima, J. Yamakita, S. Mori, K. Tominaga, “Unified approach to wave diffraction by space–time periodic anisotropic media,” IEEE Trans. Microwave Theory Tech. MTT-35, 937–945 (1987).
[CrossRef]

Neurgaonkar, R. R.

R. A. Vazquez, F. R. Vachss, R. R. Neurgaonkar, D. Ewbank, “Large photorefractive coupling coefficient in a thin cerbium-doped strontium barium niobate crystal,” J. Opt. Soc. Am. B 9, 1932–1941 (1992).

Nowak, A

Odulov, S. G.

N. V. Kukhtarev, M. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetsky, “Holographic storage in electrooptic crystals,” Ferroelectrics 22, 949–964 (1979).
[CrossRef]

Petrovic, M.

Rokushima, K.

K. Rokushima, J. Yamakita, S. Mori, K. Tominaga, “Unified approach to wave diffraction by space–time periodic anisotropic media,” IEEE Trans. Microwave Theory Tech. MTT-35, 937–945 (1987).
[CrossRef]

Serrano, E.

Snowbell, M.

M. Snowbell, M. Horowitz, B. Fischer, “Dynamics of multiple two-wave mixing and fanning in photorefractive materials,” J. Opt. Soc. Am. B 9, 1972–1982 (1994).
[CrossRef]

Solymar, L.

L. B. Au, L. Solymar, “Higher diffraction orders in photorefractive materials,” IEEE J. Quantum Electron 24, 162–168 (1987).
[CrossRef]

J. M. Heaton, L. Solymar, “Transient energy transfer during hologram formation in photorefractive crystals,” Opt. Acta 32, 397–408 (1985).
[CrossRef]

Soskin, M. S.

N. V. Kukhtarev, M. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetsky, “Holographic storage in electrooptic crystals,” Ferroelectrics 22, 949–964 (1979).
[CrossRef]

Strohkendl, F. P.

J. E. Millerd, E. M. Garmire, M. B. Klein, B. A. Wechsler, F. P. Strohkendl, G. A. Brost, “Photorefractive response at high modulation depths in Bi12TiO20,” J. Opt. Soc. Am. B 8, 1449–1453 (1991).

Tominaga, K.

K. Rokushima, J. Yamakita, S. Mori, K. Tominaga, “Unified approach to wave diffraction by space–time periodic anisotropic media,” IEEE Trans. Microwave Theory Tech. MTT-35, 937–945 (1987).
[CrossRef]

Vachss, F. R.

R. A. Vazquez, F. R. Vachss, R. R. Neurgaonkar, D. Ewbank, “Large photorefractive coupling coefficient in a thin cerbium-doped strontium barium niobate crystal,” J. Opt. Soc. Am. B 9, 1932–1941 (1992).

Valley, G. C.

Vazquez, R. A.

R. A. Vazquez, F. R. Vachss, R. R. Neurgaonkar, D. Ewbank, “Large photorefractive coupling coefficient in a thin cerbium-doped strontium barium niobate crystal,” J. Opt. Soc. Am. B 9, 1932–1941 (1992).

Vinetsky, V. L.

N. V. Kukhtarev, M. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetsky, “Holographic storage in electrooptic crystals,” Ferroelectrics 22, 949–964 (1979).
[CrossRef]

Voit, E.

P. Gunter, E. Voit, M. Z. Zha, J. Albers, “Self-pulsation and optical chaos in self-pumped photorefractive BaTiO3,” Opt. Commun. 55, 210–214 (1985).
[CrossRef]

Wang, D.

Wechsler, B. A.

J. E. Millerd, E. M. Garmire, M. B. Klein, B. A. Wechsler, F. P. Strohkendl, G. A. Brost, “Photorefractive response at high modulation depths in Bi12TiO20,” J. Opt. Soc. Am. B 8, 1449–1453 (1991).

White, J. O.

J. O. White, A. Yariv, “Real-time image processing via four-wave mixing in a photorefractive medium,” Appl. Phys. Lett. 37, 5–7 (1980).
[CrossRef]

Wu, X.

Yamakita, J.

K. Rokushima, J. Yamakita, S. Mori, K. Tominaga, “Unified approach to wave diffraction by space–time periodic anisotropic media,” IEEE Trans. Microwave Theory Tech. MTT-35, 937–945 (1987).
[CrossRef]

Yariv, A.

J. O. White, A. Yariv, “Real-time image processing via four-wave mixing in a photorefractive medium,” Appl. Phys. Lett. 37, 5–7 (1980).
[CrossRef]

Ye, P.

Yeh, P.

L.-K. Dai, C. Gu, P. Yeh, “Effect of position-dependent time-constant on photorefractive two-wave mixing,” J. Opt. Soc. Am. B 9, 1693–1697 (1992).
[CrossRef]

P. Yeh, A. E. T. Chiou, “Real-time contrast reversal via four-wave mixing nonlinear media,” Opt. Commun. 64, 160–162 (1987).
[CrossRef]

P. Yeh, Introduction to Photorefractive Nonlinear Optics (Wiley, New York, 1993).

Zha, M. Z.

P. Gunter, E. Voit, M. Z. Zha, J. Albers, “Self-pulsation and optical chaos in self-pumped photorefractive BaTiO3,” Opt. Commun. 55, 210–214 (1985).
[CrossRef]

Zhang, Z.

Appl. Opt. (1)

Appl. Phys. Lett. (1)

J. O. White, A. Yariv, “Real-time image processing via four-wave mixing in a photorefractive medium,” Appl. Phys. Lett. 37, 5–7 (1980).
[CrossRef]

Ferroelectrics (1)

N. V. Kukhtarev, M. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetsky, “Holographic storage in electrooptic crystals,” Ferroelectrics 22, 949–964 (1979).
[CrossRef]

IEEE J. Quantum Electron (1)

L. B. Au, L. Solymar, “Higher diffraction orders in photorefractive materials,” IEEE J. Quantum Electron 24, 162–168 (1987).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

K. Rokushima, J. Yamakita, S. Mori, K. Tominaga, “Unified approach to wave diffraction by space–time periodic anisotropic media,” IEEE Trans. Microwave Theory Tech. MTT-35, 937–945 (1987).
[CrossRef]

J. Opt. Soc. Am. (1)

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

J. Opt. Soc. Am. B (12)

P. P. Banerjee, J. J. Liu, “Perturbational analysis of steady-state and transient-beam fanning in thin and thick photorefractive media,” J. Opt. Soc. Am. B 10, 1417–1423 (1993).
[CrossRef]

A Nowak, T. R. Moore, R. A. Fisher, “Observation of internal beam production in BaTiO3phase conjugator,” J. Opt. Soc. Am. B 5, 1864–1878 (1988).
[CrossRef]

P. M. Jeffrey, R. W. Eason, “Lyapunov exponent analysis of irregular fluctuations in a self-pumped BaTiO3phase conjugate mirror,” J. Opt. Soc. Am. B 11, 476–480 (1994).
[CrossRef]

D. Wang, Z. Zhang, X. Wu, P. Ye, “Instabilities in a mutually pumped phase conjugator of BaTiO3,” J. Opt. Soc. Am. B 7, 2289–2293 (1990).
[CrossRef]

L.-K. Dai, C. Gu, P. Yeh, “Effect of position-dependent time-constant on photorefractive two-wave mixing,” J. Opt. Soc. Am. B 9, 1693–1697 (1992).
[CrossRef]

M. Snowbell, M. Horowitz, B. Fischer, “Dynamics of multiple two-wave mixing and fanning in photorefractive materials,” J. Opt. Soc. Am. B 9, 1972–1982 (1994).
[CrossRef]

E. Serrano, V. Lopez, M. Carrascosa, F. Agrillo-Lopez, “Recording and erasure kinetics in photorefractive materials at large modulation depths,” J. Opt. Soc. Am. B 11, 670–675 (1994).
[CrossRef]

W. P. Brown, G. C. Valley, “Kinky beam paths inside photorefractive crystals,” J. Opt. Soc. Am. B 10, 1901–1906 (1993).
[CrossRef]

E. N. Glytsis, T. K. Gaylord, “Rigorous three-dimensional coupled-wave diffraction analysis of single cascaded anisotropic gratings,” J. Opt. Soc. Am. B 4, 2061–2080 (1987).
[CrossRef]

J. E. Millerd, E. M. Garmire, M. B. Klein, B. A. Wechsler, F. P. Strohkendl, G. A. Brost, “Photorefractive response at high modulation depths in Bi12TiO20,” J. Opt. Soc. Am. B 8, 1449–1453 (1991).

R. A. Vazquez, F. R. Vachss, R. R. Neurgaonkar, D. Ewbank, “Large photorefractive coupling coefficient in a thin cerbium-doped strontium barium niobate crystal,” J. Opt. Soc. Am. B 9, 1932–1941 (1992).

M. R. Belic, M. Petrovic, “Unified method for solution of wave equations in photorefractive media,” J. Opt. Soc. Am. B 11, 481–485 (1994).
[CrossRef]

Opt. Acta (1)

J. M. Heaton, L. Solymar, “Transient energy transfer during hologram formation in photorefractive crystals,” Opt. Acta 32, 397–408 (1985).
[CrossRef]

Opt. Commun. (5)

J. C. Kralik, M. S. Malcuit, “Transient oscillations in nondegenerate two-beam coupling,” Opt. Commun. 107, 401–405 (1994).
[CrossRef]

P. Gunter, E. Voit, M. Z. Zha, J. Albers, “Self-pulsation and optical chaos in self-pumped photorefractive BaTiO3,” Opt. Commun. 55, 210–214 (1985).
[CrossRef]

P. Yeh, A. E. T. Chiou, “Real-time contrast reversal via four-wave mixing nonlinear media,” Opt. Commun. 64, 160–162 (1987).
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[CrossRef]

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P. P. Banerjee, J. Jarem, “Transient wave mixing and recording kinetics in photorefractive barium titanate: a nonlinear coupled mode approach,” Opt. Eng. 34, 2254–2260 (1995).
[CrossRef]

Opt. Lett. (1)

Other (5)

J. Jarem, P. P. Banerjee, “A nonlinear, transient analysis of two and multi-wave mixing in a photorefractive material using rigorous coupled-mode diffraction theory,” Opt. Commun. (to be published).

J. Jarem, P. P. Banerjee, “Rigorous coupled wave dynamical analysis of beam coupling in photorefractive materials,” in Annual Meeting/ILS-X (Optical Society of America, Washington, D.C., 1994), p. 162.

P. Gunter, J.-P. Huignard, eds., Photorefractive Materials and Their Applications, I & II (Springer, Berlin, 1989).

P. Yeh, Introduction to Photorefractive Nonlinear Optics (Wiley, New York, 1993).

A. Korpel, Acousto-Optics (Marcel Dekker, New York, 1988).

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Figures (7)

Fig. 1
Fig. 1

Part (a) shows the optical power intensity (normalized to the dark current C) in the grating when the grating period is Λ = 1λ, 2λ, 5λ, and 10λ as a function of the normalized grating distance xN = x/Λ. The incident power (evaluated at y = −L/2) was adjusted in order that the intensity profile for each differently sized grating period would have the same peak intensity. Part (b) shows the steady-state dielectric modulation function Δ (also evaluated at y = −L/2) that results when the intensity profiles of (a) were used to determine Δ. Because the PR grating was thin, the intensity profiles of (a) were not assumed to change with time as the Δ profiles reached steady state. All grating parameters used in the simulation not listed in the figure are given in Section 4.

Fig. 2
Fig. 2

Dielectric perturbation function Δ and the power transmitted in the T1, T0, R1, and R0 directions when the regions (regions 1 and 3) bounding the PR crystal are indexed matched to the PR crystal. All grating parameters used in the simulation not listed in the figure are given in Section 4.

Fig. 3
Fig. 3

The power transmitted in the T0, T1, R0, and R1 directions when the regions (regions 1 and 3) bounding the PR crystal are not index matched to the PR crystal [see Fig. 2(a)] [air was assumed (1 = 3 = 1)] is shown in (a), (b), (c), and (d), respectively, for six slightly different PR crystal lengths L. All grating parameters used in the simulation not listed in the figure are given in Section 4. The starred line on the L5 curve of (a) used 320 longitudinal layers, whereas all other simulation runs in Fig. 3 used 160 layers. Note that the number of longitudinal divisions made virtually no difference in the simulation.

Fig. 4
Fig. 4

Parts (a) and (b) show the dielectric perturbation function Δ that results in the index-mismatched case of Fig. 3 when the PR crystal length is L5 = 1530λ and, respectively, t = 56 ms and t = 90 ms (oscillatory steady state; see Fig. 3). Part (c) shows the dielectric perturbation function Δ that results in the index-mismatched case of Fig. 3 when the PR crystal length is L3 = 1510.875λ and t = 90 ms (nonoscillatory steady state; see Fig. 3). Parts (b)–(d) are drawn to the same scale. Part (d) shows the RMS Δ [(Δrms)2 is also proportional to the electrostatic energy stored in a grating period] as a function of time step when the crystal length is L3 = 1510.875λ and when the crystal length is L5 = 1530λ.

Fig. 5
Fig. 5

Numerical PR mode coupling and diffraction that occur (with the use of the Kukhtarev equations and RCWT; see Section 3) when two interfering plane waves whose amplitudes are E0 and E1 = 0.4E0 are incident on a PR crystal (λ = 0.633 μm, L = 1530λ = L5) that is not index matched to the surrounding medium [free space is assumed to surround the PR crystal (1 = 3 = 1)]. Part (a) shows the dielectric modulation function Δ that results when NL = 160 layers is used, and part (b) shows the results when NL = 640 layers is used. Part (c) shows the transmitted and the reflected power that is diffracted in the zero and first orders when NL = 640. Part (d) shows the transmitted power that is transmitted in the second order when NL = 160, 320, and 640 layers. The dashed line in (d) shows the transmitted power that is diffracted in the second order when NL = 160 layers and MT = 3.

Fig. 6
Fig. 6

(a) Plot of the normalized electron density ñ = n/NA that results from the simulation shown in the matched case of Fig. 2 (t = 113 ms, y = −L/2, L = 1530λ, λ = 0.633 μm, Λ = 5λ) and the mismatched case of Figs. 3 and 4 (t = 200 ms, y = −L/2, L = L5 = 1530λ, λ = 0.633 μm, Λ = 5λ), (b) plot of the dielectric modulation function Δ obtained in the same location as that of the electron densities of (a).

Fig. 7
Fig. 7

Simulations similar to that shown in Fig. 2 (matched case) except that the grating period was chosen to have the very small value of Λ = 2λ rather than Λ = 5λ.

Equations (32)

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( N D + - n ) t = - 1 e J x ,
N D + t = ( N D - N D + ) ( s I + β ) - γ R N D + n ,
J = e D s n x + e μ n E s ,
s E s x = e ( N D + - N A - n ) ,
n t = N D + t + μ n E s x + D s 2 n x 2 .
N D + = s e E s x + N A + n
N D + t = s e 2 E s x t + n t ,
s e 2 E s x t + μ ( n E s ) x + D s 2 n x 2 = 0
E s t + μ e s n E s = - D s e s n x .
Δ ( x , y , t ) = n O 2 n E 2 r 42 E s ( x , y , t ) .
Δ t + μ e s ( n Δ ) = - D s e s r 42 n O 2 n E 2 n x .
s e 2 E s x t + n t = ( s I + β ) ( N D - N D + ) - γ R n N D + .
- μ n E s x - D s 2 n x 2 + n t = ( s I + β ) ( N D - s e E s x - N A - n ) - γ R n ( s e E s x + N A + n ) .
Γ 2 2 n ˜ x ˜ 2 + Γ 1 Δ n ˜ x ˜ + [ - Γ 3 ( 1 + I C ) + ( Γ 1 - Γ 4 ) Δ x ˜ - Γ 3 Γ 4 ( 1 + n ˜ ) ] n ˜ = Γ 3 [ n ˜ t ˜ - ( 1 + I C ) ( N D N A - 1 - 1 Γ 3 Δ x ˜ ) ] ,
Γ 1 = μ e N A β s , Γ 2 = e k 0 D s N A n O 2 n E 2 r 42 β s , Γ 3 = e N A n O 2 n E 2 r 42 s k 0 , Γ 4 = γ R N A β .
Δ t ˜ + Γ 1 n ˜ Δ = - Γ 2 n ˜ x ˜ .
H 1 = ( H z inc + H z ref ) a ^ z , H z 1 inc = E 0 η 0 exp [ - j ( k x 0 x - k y 10 y ) ] δ i 0 , H z 1 ref = E 0 η 0 i r i exp [ - j ( k x i x + k y 1 i y ) ] .
H 2 = 1 η 0 ( n C n H z n e ) a ˜ z , H z n e = i U z i n exp ( q n y - j k x i x ) ,
V y = AV ,             A = [ A 11 A 12 A 21 A 22 ] ,             V = ( S x U z ) ,
A 11 = j k 0 K x y y - 1 y x , A 12 = j k 0 ( - K x y y - 1 K x + I ) , A 21 = j k 0 ( x x - x y y y - 1 y x ) , A 22 = j k 0 ( x y y y - 1 K x ) ,
H 3 = H z 3 tr a ^ z , H z 3 tr = E 0 η 0 i t i exp [ - j k x i x + j k y 3 i ( y + L ) ] .
2 E 0 n 1 ( cos θ ) δ i 0 = n C n ( k y 1 i r 1 U z i n + S x i n ) , 0 = n C n exp ( - q n L ) ( - k y 3 i r 3 U z i n + S x i n ) .
x x = n C O 2 cos 2 θ c + n C E 2 sin 2 θ c + Δ ( x , y , t ) F x x , x y = ( n C O 2 - n C E 2 ) sin θ c cos θ c + Δ ( x , y , t ) F x y , y x = x y , y y = n C O 2 sin 2 θ c + n C E 2 cos 2 θ c + Δ ( x , y , t ) F y y ,
F x x = - F O E ( r 13 n C O 2 r 42 n C E 2 sin θ c cos 2 θ c + 2 sin θ c cos 2 θ c + r 33 n C E 2 r 42 n C O 2 sin 3 θ c ) , F x y = - F O E [ r 13 n C O 2 r 42 n C E 2 sin 2 θ c cos θ c + ( cos θ c ) ( - cos 2 θ c + sin 2 θ c ) - r 33 n C E 2 r 42 n C O 2 sin 2 θ c cos θ c ] , F y y = - F O E ( r 13 n C O 2 r 42 n C E 2 sin 3 θ c - 2 sin θ c cos 2 θ c + r 33 n C E 2 r 42 n C O 2 cos 2 θ c sin θ c ) , F O E = n C O 2 n C E 2 n O 2 n E 2 , n C O 2 = n O 2 - j O , n C E 2 = n E 2 - j E ,
Δ ( x ˜ , y , t ˜ + Δ t ˜ ) = Δ ( x ˜ , y , t ˜ ) + ( - Γ 1 n ˜ Δ - Γ 2 n ˜ x ˜ ) t ˜ .
2 n ˜ x ˜ 2 | x ˜ p n ˜ ( p + 1 ) - 2 n ˜ ( p ) + n ˜ ( p - 1 ) ( δ x ˜ ) 2 , n ˜ x ˜ | x ˜ p n ˜ ( p + 1 ) - n ˜ ( p - 1 ) 2 δ x ˜ .
P ( p ) = Γ 1 Δ ( p ) , H ( p ) = - Γ 3 [ 1 + I ( p ) C ] + ( Γ 1 + Γ 4 ) Δ x ˜ | x ˜ = x ˜ p - Γ 3 Γ 4 [ 1 + n ˜ ( p ) ] , S ( p ) = Γ 3 { n ˜ ( p ) t ˜ - [ 1 + I ( p ) C ] × [ N D N A - 1 - 1 Γ 3 Δ x | x ˜ = x ˜ p ] } ,
Γ 2 [ n ˜ ( p + 1 ) - 2 n ˜ ( p ) + n ˜ ( p - 1 ) ( δ x ˜ ) 2 ] + P ( p ) [ n ˜ ( p + 1 ) - n ˜ ( p - 1 ) 2 δ x ˜ ] + H ( p ) n ˜ ( p ) = S ( p ) .
R + ( p ) n ˜ ( p + 1 ) + R ( p ) n ˜ ( p ) + R - ( p ) n ˜ ( p - 1 ) = S ( p ) ,
L = [ R ( 1 ) R + ( 1 ) 0 0 0 0 0 0 0 R - ( 1 ) R - ( 2 ) R ( 2 ) R + ( 2 ) 0 0 0 0 0 0 0 0 R - ( 3 ) R ( 3 ) R + ( 3 ) 0 0 0 0 0 0 0 0 . . . 0 0 0 0 0 0 0 0 . . . 0 0 0 0 0 0 0 0 . . . 0 0 0 0 0 0 0 0 . . . 0 0 0 0 0 0 0 0 R - ( N p - 2 ) R ( N p - 2 ) R + ( N p - 2 ) 0 0 0 0 0 0 0 0 R - ( N p - 1 ) R ( N p - 1 ) R + ( N p - 1 ) R + ( N p ) 0 0 0 0 0 0 0 R - ( N p ) R ( N p ) ] , S = [ S ( 1 ) S ( 2 ) S ( N p ) ] tr , n ˜ = [ n ˜ ( 1 ) n ˜ ( 2 ) n ˜ ( N p ) ] tr , L n ˜ = S .
n ˜ = L - 1 S ,
I 2 ~ ν 2 Q 2 sin 2 ν ,

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