Abstract

In LiNbO3 at light intensities above 106 W/m2 much stronger refractive index changes can be induced, as expected from measurements at low intensities. Most experimental data have been published on iron-doped LiNbO3 crystals. We propose a two-center charge-transport model for LiNbO3:Fe that describes most results at low and high intensities quantitatively. It explains the intensity dependence of steady-state refractive index changes and enhanced holographic sensitivities at high light intensities as well as the presence of light-induced absorption changes.

© 1993 Optical Society of America

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  1. P. Günter, J.-P. Huignard, Photorefractive Materials and Their Applications I and II (Springer-Verlag, Heidelberg, 1988, 1989).
    [CrossRef]
  2. H. Kurz, E. Krätzig, W. Keune, H. Engelmann, U. Gonser, B. Dischler, A. Räuber, “Photorefractive centers in LiNbO3, studied by optical-, Mössbauer- and EPR-methods,” Appl. Phys. 12, 355 (1977).
    [CrossRef]
  3. A. M. Glass, D. von der Linde, T. J. Negran, “High-voltage bulk photovoltaic effect and the photorefractive process in LiNbO3,” Appl. Phys. Lett. 25, 233 (1974).
    [CrossRef]
  4. E. Krätzig, R. Sommerfeldt, “Influence of dopants on photorefractive properties of LiNbO3crystals,” in Nonlinear Optical Materials III, P. Günter, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1273, 58 (1990).
  5. C.-T. Chen, D. M. Kim, D. von der Linde, “Efficient hologram recording in LiNbO3:Fe using optical pulses,” Appl. Phys. Lett. 34, 321 (1979).
    [CrossRef]
  6. V. Wood, N. Hartmann, C. Verber, “Two-photon photorefractivity in pure and doped LiNbO3,” Ferroelectrics 27, 237 (1980).
    [CrossRef]
  7. F. Jermann, E. Krätzig, “Charge transport processes in LiNbO3:Fe at high intensity laser pulses,” Appl. Phys. A 55, 114 (1992).
    [CrossRef]
  8. P. Augustov, K. Shvarts, “The temperature and light intensity dependence of photorefraction in LiNbO3,” Appl. Phys. 21, 191 (1980).
    [CrossRef]
  9. O. Althoff, E. Krätzig, “Strong light-induced refractive index changes in LiNbO3,” in Nonlinear Optical Materials III, P. Günter, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1273, 12 (1990).
    [CrossRef]
  10. R. Göring, A. Rasch, W. Karthe, “Quantitative investigation of photorefractive effects in LiNbO3 channel wave-guides,” in Electro-Optic and Magneto-Optic Materials II, H. Dammann, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1274, 18 (1990).
    [CrossRef]
  11. R. Göring, Z. Yuang-Lung, S. Steinberg, “Photoconductivity and photovoltaic behavior of LiNbO3and LiNbO3wave-guides at high optical intensities,” Appl. Phys. A 55, 97 (1992).
    [CrossRef]
  12. O. Althoff, A. Erdmann, L. Wiskott, P. Hertel, “The photorefractive effect in LiNbO3at high light intensity,” Phys. Status Solidi A 128, K41 (1991).
    [CrossRef]
  13. E. Krätzig, R. Orlowski, “Light induced charge transport in doped LiNbO3and LiTaO3,” Ferroelectrics 27, 241 (1980).
    [CrossRef]
  14. J. Otten, A. Bledowski, K. Ringhofer, R. Rupp, “Dynamical holographic storage in photorefractive crystals,” Comput. Phys. Commun. 69, 187 (1992).
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  16. G. A. Brost, R. A. Motes, J. R. Rotgé, “Intensity-dependent absorption and photorefractive effects in barium titanate,” J. Opt. Soc. Am. B 5, 1879 (1988).
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    [CrossRef]
  18. F. Jermann, E. Krätzig, “Photorefractive effects in LiNbO3:Fe at high light intensities,” in Proceedings of the International Conference on Defects in Insulating Materials, O. Kauert, J.-G. Spaeth, eds. (World Scientific, Singapore, 1993), Vol. 2, p. 1133.
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  24. F. Chen, “Optically induced change of refractive indices in LiNbO3and LiTaO3,” J. Appl. Phys. 40, 3389 (1969).
    [CrossRef]
  25. I. Biaggio, M. Zgonik, P. Günther, “Photorefractive effects induced by picosecond light pulses in reduced KNbO3,” J. Opt. Soc. Am. B 9, 1480 (1992).
    [CrossRef]
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    [CrossRef]
  27. S. Abrahams, P. Marsh, “Defect structure dependence on composition in lithium niobate,” Acta Crystallogr. Sect. B 42, 61 (1986).
    [CrossRef]
  28. O. Schirmer, O. Thiemann, M. Wöhlecke, “Defects in LiNbO3–I. Experimental aspects,” J. Phys. Chem. Solids 52, 185 (1991).
    [CrossRef]
  29. O. Schirmer, S. Juppe, J. Koppitz, “ESR-, optical and photovoltaic studies of reduced undoped LiNbO3,” Cryst. Lattice Defects Amorph. Mat. 16, 353 (1987).
  30. K. Sweeney, L. Halliburton, “Oxygen vacancies in lithium niobate,” Appl. Phys. Lett. 43, 336 (1983).
    [CrossRef]
  31. I. Kanaev, S. Kostritsky, V. Malinovsky, A. Pugachev, “The influence of photoinduced mechanical tensions on photogalvanic effect and Raman scattering in LiNbO3,” Ferroelectrics 126, 45 (1992).
    [CrossRef]
  32. T. Volk, N. Rubinina, “Nonphotorefractive impurities in lithium niobate: magnesium and zinc,” Sov. Phys. Solid State 33, 674 (1991).
  33. G. Malovichko, V. Grachev, E. Kokonyan, O. Schirmer, K. Betzler, B. Gather, F. Jermann, S. Klauer, U. Schlarb, M. Wöhlecke, “Characterization of stoichiometric LiNbO3grown from melts containing K2O,” Appl. Phys. A 56, 103 (1992).
    [CrossRef]
  34. D. von der Linde, O. Schirmer, H. Kurz, “Intrinsic photorefractive effect of LiNbO3,” Appl. Phys. 15, 153 (1978).
    [CrossRef]
  35. I. Kanaev, V. Malinovsky, A. Pugachev, “Changes in photogalvanic and photorefractive characteristics of lithium niobate under the light,” Ferroelectrics 75, 209 (1987).
    [CrossRef]
  36. Y. Ohmachi, K. Sawamoto, H. Toyoda, “Dielectric properties of LiNbO3 single crystal up to 9 Gc,” Jpn. J. Appl. Phys. 6, 1467 (1967).
    [CrossRef]
  37. S. Shandarov, “Influence of piezoelectric effect on photorefractive gratings in electro-optic crystals,” Appl. Phys. A 55, 91 (1992).
    [CrossRef]

1993 (1)

1992 (7)

I. Biaggio, M. Zgonik, P. Günther, “Photorefractive effects induced by picosecond light pulses in reduced KNbO3,” J. Opt. Soc. Am. B 9, 1480 (1992).
[CrossRef]

S. Shandarov, “Influence of piezoelectric effect on photorefractive gratings in electro-optic crystals,” Appl. Phys. A 55, 91 (1992).
[CrossRef]

F. Jermann, E. Krätzig, “Charge transport processes in LiNbO3:Fe at high intensity laser pulses,” Appl. Phys. A 55, 114 (1992).
[CrossRef]

R. Göring, Z. Yuang-Lung, S. Steinberg, “Photoconductivity and photovoltaic behavior of LiNbO3and LiNbO3wave-guides at high optical intensities,” Appl. Phys. A 55, 97 (1992).
[CrossRef]

J. Otten, A. Bledowski, K. Ringhofer, R. Rupp, “Dynamical holographic storage in photorefractive crystals,” Comput. Phys. Commun. 69, 187 (1992).
[CrossRef]

I. Kanaev, S. Kostritsky, V. Malinovsky, A. Pugachev, “The influence of photoinduced mechanical tensions on photogalvanic effect and Raman scattering in LiNbO3,” Ferroelectrics 126, 45 (1992).
[CrossRef]

G. Malovichko, V. Grachev, E. Kokonyan, O. Schirmer, K. Betzler, B. Gather, F. Jermann, S. Klauer, U. Schlarb, M. Wöhlecke, “Characterization of stoichiometric LiNbO3grown from melts containing K2O,” Appl. Phys. A 56, 103 (1992).
[CrossRef]

1991 (4)

T. Volk, N. Rubinina, “Nonphotorefractive impurities in lithium niobate: magnesium and zinc,” Sov. Phys. Solid State 33, 674 (1991).

O. Schirmer, O. Thiemann, M. Wöhlecke, “Defects in LiNbO3–I. Experimental aspects,” J. Phys. Chem. Solids 52, 185 (1991).
[CrossRef]

O. Althoff, A. Erdmann, L. Wiskott, P. Hertel, “The photorefractive effect in LiNbO3at high light intensity,” Phys. Status Solidi A 128, K41 (1991).
[CrossRef]

L. Holtmann, K. Buse, G. Kuper, A. Groll, H. Hesse, E. Krätzig, “Photoconductivity and light-induced absorption in KNbO3:Fe,” Appl. Phys. A 53, 81 (1991).
[CrossRef]

1988 (2)

R. Sommerfeldt, L. Holtmann, E. Krätzig, B. C. Grabmeier, “Influence of Mg doping and composition on the light-induced charge transport in LiNbO3,” Phys. Status Solidi A 106, 89 (1988).
[CrossRef]

G. A. Brost, R. A. Motes, J. R. Rotgé, “Intensity-dependent absorption and photorefractive effects in barium titanate,” J. Opt. Soc. Am. B 5, 1879 (1988).
[CrossRef]

1987 (2)

I. Kanaev, V. Malinovsky, A. Pugachev, “Changes in photogalvanic and photorefractive characteristics of lithium niobate under the light,” Ferroelectrics 75, 209 (1987).
[CrossRef]

O. Schirmer, S. Juppe, J. Koppitz, “ESR-, optical and photovoltaic studies of reduced undoped LiNbO3,” Cryst. Lattice Defects Amorph. Mat. 16, 353 (1987).

1986 (1)

S. Abrahams, P. Marsh, “Defect structure dependence on composition in lithium niobate,” Acta Crystallogr. Sect. B 42, 61 (1986).
[CrossRef]

1985 (1)

T. J. Hall, R. Jaura, L. M. Connors, P. D. Foote, “The photorefractive effect—a review,” Prog. Quantum Electron. 10, 77 (1985).
[CrossRef]

1983 (1)

K. Sweeney, L. Halliburton, “Oxygen vacancies in lithium niobate,” Appl. Phys. Lett. 43, 336 (1983).
[CrossRef]

1980 (3)

E. Krätzig, R. Orlowski, “Light induced charge transport in doped LiNbO3and LiTaO3,” Ferroelectrics 27, 241 (1980).
[CrossRef]

P. Augustov, K. Shvarts, “The temperature and light intensity dependence of photorefraction in LiNbO3,” Appl. Phys. 21, 191 (1980).
[CrossRef]

V. Wood, N. Hartmann, C. Verber, “Two-photon photorefractivity in pure and doped LiNbO3,” Ferroelectrics 27, 237 (1980).
[CrossRef]

1979 (2)

C.-T. Chen, D. M. Kim, D. von der Linde, “Efficient hologram recording in LiNbO3:Fe using optical pulses,” Appl. Phys. Lett. 34, 321 (1979).
[CrossRef]

P. Augustov, K. Shvarts, “Surface recombination and photorefraction in LiNbO3:Fe crystals,” Appl. Phys. 18, 399 (1979).
[CrossRef]

1978 (2)

O. Schirmer, D. von der Linde, “Two-photon and x-ray-induced Nb4+ and O− small polarons in LiNbO3,” Appl. Phys. Lett. 33, 35 (1978).
[CrossRef]

D. von der Linde, O. Schirmer, H. Kurz, “Intrinsic photorefractive effect of LiNbO3,” Appl. Phys. 15, 153 (1978).
[CrossRef]

1977 (1)

H. Kurz, E. Krätzig, W. Keune, H. Engelmann, U. Gonser, B. Dischler, A. Räuber, “Photorefractive centers in LiNbO3, studied by optical-, Mössbauer- and EPR-methods,” Appl. Phys. 12, 355 (1977).
[CrossRef]

1976 (1)

Y. Ohmori, M. Yamaguchi, K. Yoshino, Y. Inuishi, “Electron Hall mobility in reduced LiNbO3,” Jpn. J. Appl. Phys. 15, 2263 (1976).
[CrossRef]

1974 (1)

A. M. Glass, D. von der Linde, T. J. Negran, “High-voltage bulk photovoltaic effect and the photorefractive process in LiNbO3,” Appl. Phys. Lett. 25, 233 (1974).
[CrossRef]

1969 (1)

F. Chen, “Optically induced change of refractive indices in LiNbO3and LiTaO3,” J. Appl. Phys. 40, 3389 (1969).
[CrossRef]

1967 (1)

Y. Ohmachi, K. Sawamoto, H. Toyoda, “Dielectric properties of LiNbO3 single crystal up to 9 Gc,” Jpn. J. Appl. Phys. 6, 1467 (1967).
[CrossRef]

Abrahams, S.

S. Abrahams, P. Marsh, “Defect structure dependence on composition in lithium niobate,” Acta Crystallogr. Sect. B 42, 61 (1986).
[CrossRef]

Althoff, O.

O. Althoff, A. Erdmann, L. Wiskott, P. Hertel, “The photorefractive effect in LiNbO3at high light intensity,” Phys. Status Solidi A 128, K41 (1991).
[CrossRef]

O. Althoff, E. Krätzig, “Strong light-induced refractive index changes in LiNbO3,” in Nonlinear Optical Materials III, P. Günter, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1273, 12 (1990).
[CrossRef]

Augustov, P.

P. Augustov, K. Shvarts, “The temperature and light intensity dependence of photorefraction in LiNbO3,” Appl. Phys. 21, 191 (1980).
[CrossRef]

P. Augustov, K. Shvarts, “Surface recombination and photorefraction in LiNbO3:Fe crystals,” Appl. Phys. 18, 399 (1979).
[CrossRef]

Betzler, K.

G. Malovichko, V. Grachev, E. Kokonyan, O. Schirmer, K. Betzler, B. Gather, F. Jermann, S. Klauer, U. Schlarb, M. Wöhlecke, “Characterization of stoichiometric LiNbO3grown from melts containing K2O,” Appl. Phys. A 56, 103 (1992).
[CrossRef]

Biaggio, I.

Bledowski, A.

J. Otten, A. Bledowski, K. Ringhofer, R. Rupp, “Dynamical holographic storage in photorefractive crystals,” Comput. Phys. Commun. 69, 187 (1992).
[CrossRef]

Brost, G. A.

Buse, K.

K. Buse, “Thermal gratings and pyroelectrically produced charge redistribution in BaTiO3and KNbO3,” J. Opt. Soc. Am. B 10, 1266 (1993).
[CrossRef]

L. Holtmann, K. Buse, G. Kuper, A. Groll, H. Hesse, E. Krätzig, “Photoconductivity and light-induced absorption in KNbO3:Fe,” Appl. Phys. A 53, 81 (1991).
[CrossRef]

Chen, C.-T.

C.-T. Chen, D. M. Kim, D. von der Linde, “Efficient hologram recording in LiNbO3:Fe using optical pulses,” Appl. Phys. Lett. 34, 321 (1979).
[CrossRef]

Chen, F.

F. Chen, “Optically induced change of refractive indices in LiNbO3and LiTaO3,” J. Appl. Phys. 40, 3389 (1969).
[CrossRef]

Connors, L. M.

T. J. Hall, R. Jaura, L. M. Connors, P. D. Foote, “The photorefractive effect—a review,” Prog. Quantum Electron. 10, 77 (1985).
[CrossRef]

Dischler, B.

H. Kurz, E. Krätzig, W. Keune, H. Engelmann, U. Gonser, B. Dischler, A. Räuber, “Photorefractive centers in LiNbO3, studied by optical-, Mössbauer- and EPR-methods,” Appl. Phys. 12, 355 (1977).
[CrossRef]

Engelmann, H.

H. Kurz, E. Krätzig, W. Keune, H. Engelmann, U. Gonser, B. Dischler, A. Räuber, “Photorefractive centers in LiNbO3, studied by optical-, Mössbauer- and EPR-methods,” Appl. Phys. 12, 355 (1977).
[CrossRef]

Erdmann, A.

O. Althoff, A. Erdmann, L. Wiskott, P. Hertel, “The photorefractive effect in LiNbO3at high light intensity,” Phys. Status Solidi A 128, K41 (1991).
[CrossRef]

Foote, P. D.

T. J. Hall, R. Jaura, L. M. Connors, P. D. Foote, “The photorefractive effect—a review,” Prog. Quantum Electron. 10, 77 (1985).
[CrossRef]

Gather, B.

G. Malovichko, V. Grachev, E. Kokonyan, O. Schirmer, K. Betzler, B. Gather, F. Jermann, S. Klauer, U. Schlarb, M. Wöhlecke, “Characterization of stoichiometric LiNbO3grown from melts containing K2O,” Appl. Phys. A 56, 103 (1992).
[CrossRef]

Glass, A. M.

A. M. Glass, D. von der Linde, T. J. Negran, “High-voltage bulk photovoltaic effect and the photorefractive process in LiNbO3,” Appl. Phys. Lett. 25, 233 (1974).
[CrossRef]

Gonser, U.

H. Kurz, E. Krätzig, W. Keune, H. Engelmann, U. Gonser, B. Dischler, A. Räuber, “Photorefractive centers in LiNbO3, studied by optical-, Mössbauer- and EPR-methods,” Appl. Phys. 12, 355 (1977).
[CrossRef]

Göring, R.

R. Göring, Z. Yuang-Lung, S. Steinberg, “Photoconductivity and photovoltaic behavior of LiNbO3and LiNbO3wave-guides at high optical intensities,” Appl. Phys. A 55, 97 (1992).
[CrossRef]

R. Göring, A. Rasch, W. Karthe, “Quantitative investigation of photorefractive effects in LiNbO3 channel wave-guides,” in Electro-Optic and Magneto-Optic Materials II, H. Dammann, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1274, 18 (1990).
[CrossRef]

Grabmeier, B. C.

R. Sommerfeldt, L. Holtmann, E. Krätzig, B. C. Grabmeier, “Influence of Mg doping and composition on the light-induced charge transport in LiNbO3,” Phys. Status Solidi A 106, 89 (1988).
[CrossRef]

Grachev, V.

G. Malovichko, V. Grachev, E. Kokonyan, O. Schirmer, K. Betzler, B. Gather, F. Jermann, S. Klauer, U. Schlarb, M. Wöhlecke, “Characterization of stoichiometric LiNbO3grown from melts containing K2O,” Appl. Phys. A 56, 103 (1992).
[CrossRef]

Groll, A.

L. Holtmann, K. Buse, G. Kuper, A. Groll, H. Hesse, E. Krätzig, “Photoconductivity and light-induced absorption in KNbO3:Fe,” Appl. Phys. A 53, 81 (1991).
[CrossRef]

Günter, P.

P. Günter, J.-P. Huignard, Photorefractive Materials and Their Applications I and II (Springer-Verlag, Heidelberg, 1988, 1989).
[CrossRef]

Günther, P.

Hall, T. J.

T. J. Hall, R. Jaura, L. M. Connors, P. D. Foote, “The photorefractive effect—a review,” Prog. Quantum Electron. 10, 77 (1985).
[CrossRef]

Halliburton, L.

K. Sweeney, L. Halliburton, “Oxygen vacancies in lithium niobate,” Appl. Phys. Lett. 43, 336 (1983).
[CrossRef]

Hartmann, N.

V. Wood, N. Hartmann, C. Verber, “Two-photon photorefractivity in pure and doped LiNbO3,” Ferroelectrics 27, 237 (1980).
[CrossRef]

Hertel, P.

O. Althoff, A. Erdmann, L. Wiskott, P. Hertel, “The photorefractive effect in LiNbO3at high light intensity,” Phys. Status Solidi A 128, K41 (1991).
[CrossRef]

Hesse, H.

L. Holtmann, K. Buse, G. Kuper, A. Groll, H. Hesse, E. Krätzig, “Photoconductivity and light-induced absorption in KNbO3:Fe,” Appl. Phys. A 53, 81 (1991).
[CrossRef]

Holtmann, L.

L. Holtmann, K. Buse, G. Kuper, A. Groll, H. Hesse, E. Krätzig, “Photoconductivity and light-induced absorption in KNbO3:Fe,” Appl. Phys. A 53, 81 (1991).
[CrossRef]

R. Sommerfeldt, L. Holtmann, E. Krätzig, B. C. Grabmeier, “Influence of Mg doping and composition on the light-induced charge transport in LiNbO3,” Phys. Status Solidi A 106, 89 (1988).
[CrossRef]

Huignard, J.-P.

P. Günter, J.-P. Huignard, Photorefractive Materials and Their Applications I and II (Springer-Verlag, Heidelberg, 1988, 1989).
[CrossRef]

Inuishi, Y.

Y. Ohmori, M. Yamaguchi, K. Yoshino, Y. Inuishi, “Electron Hall mobility in reduced LiNbO3,” Jpn. J. Appl. Phys. 15, 2263 (1976).
[CrossRef]

Jaura, R.

T. J. Hall, R. Jaura, L. M. Connors, P. D. Foote, “The photorefractive effect—a review,” Prog. Quantum Electron. 10, 77 (1985).
[CrossRef]

Jermann, F.

G. Malovichko, V. Grachev, E. Kokonyan, O. Schirmer, K. Betzler, B. Gather, F. Jermann, S. Klauer, U. Schlarb, M. Wöhlecke, “Characterization of stoichiometric LiNbO3grown from melts containing K2O,” Appl. Phys. A 56, 103 (1992).
[CrossRef]

F. Jermann, E. Krätzig, “Charge transport processes in LiNbO3:Fe at high intensity laser pulses,” Appl. Phys. A 55, 114 (1992).
[CrossRef]

F. Jermann, E. Krätzig, “Photorefractive effects in LiNbO3:Fe at high light intensities,” in Proceedings of the International Conference on Defects in Insulating Materials, O. Kauert, J.-G. Spaeth, eds. (World Scientific, Singapore, 1993), Vol. 2, p. 1133.

Juppe, S.

O. Schirmer, S. Juppe, J. Koppitz, “ESR-, optical and photovoltaic studies of reduced undoped LiNbO3,” Cryst. Lattice Defects Amorph. Mat. 16, 353 (1987).

Kanaev, I.

I. Kanaev, S. Kostritsky, V. Malinovsky, A. Pugachev, “The influence of photoinduced mechanical tensions on photogalvanic effect and Raman scattering in LiNbO3,” Ferroelectrics 126, 45 (1992).
[CrossRef]

I. Kanaev, V. Malinovsky, A. Pugachev, “Changes in photogalvanic and photorefractive characteristics of lithium niobate under the light,” Ferroelectrics 75, 209 (1987).
[CrossRef]

Karthe, W.

R. Göring, A. Rasch, W. Karthe, “Quantitative investigation of photorefractive effects in LiNbO3 channel wave-guides,” in Electro-Optic and Magneto-Optic Materials II, H. Dammann, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1274, 18 (1990).
[CrossRef]

Keune, W.

H. Kurz, E. Krätzig, W. Keune, H. Engelmann, U. Gonser, B. Dischler, A. Räuber, “Photorefractive centers in LiNbO3, studied by optical-, Mössbauer- and EPR-methods,” Appl. Phys. 12, 355 (1977).
[CrossRef]

Kim, D. M.

C.-T. Chen, D. M. Kim, D. von der Linde, “Efficient hologram recording in LiNbO3:Fe using optical pulses,” Appl. Phys. Lett. 34, 321 (1979).
[CrossRef]

Klauer, S.

G. Malovichko, V. Grachev, E. Kokonyan, O. Schirmer, K. Betzler, B. Gather, F. Jermann, S. Klauer, U. Schlarb, M. Wöhlecke, “Characterization of stoichiometric LiNbO3grown from melts containing K2O,” Appl. Phys. A 56, 103 (1992).
[CrossRef]

Kokonyan, E.

G. Malovichko, V. Grachev, E. Kokonyan, O. Schirmer, K. Betzler, B. Gather, F. Jermann, S. Klauer, U. Schlarb, M. Wöhlecke, “Characterization of stoichiometric LiNbO3grown from melts containing K2O,” Appl. Phys. A 56, 103 (1992).
[CrossRef]

Koppitz, J.

O. Schirmer, S. Juppe, J. Koppitz, “ESR-, optical and photovoltaic studies of reduced undoped LiNbO3,” Cryst. Lattice Defects Amorph. Mat. 16, 353 (1987).

Kostritsky, S.

I. Kanaev, S. Kostritsky, V. Malinovsky, A. Pugachev, “The influence of photoinduced mechanical tensions on photogalvanic effect and Raman scattering in LiNbO3,” Ferroelectrics 126, 45 (1992).
[CrossRef]

Krätzig, E.

F. Jermann, E. Krätzig, “Charge transport processes in LiNbO3:Fe at high intensity laser pulses,” Appl. Phys. A 55, 114 (1992).
[CrossRef]

L. Holtmann, K. Buse, G. Kuper, A. Groll, H. Hesse, E. Krätzig, “Photoconductivity and light-induced absorption in KNbO3:Fe,” Appl. Phys. A 53, 81 (1991).
[CrossRef]

R. Sommerfeldt, L. Holtmann, E. Krätzig, B. C. Grabmeier, “Influence of Mg doping and composition on the light-induced charge transport in LiNbO3,” Phys. Status Solidi A 106, 89 (1988).
[CrossRef]

E. Krätzig, R. Orlowski, “Light induced charge transport in doped LiNbO3and LiTaO3,” Ferroelectrics 27, 241 (1980).
[CrossRef]

H. Kurz, E. Krätzig, W. Keune, H. Engelmann, U. Gonser, B. Dischler, A. Räuber, “Photorefractive centers in LiNbO3, studied by optical-, Mössbauer- and EPR-methods,” Appl. Phys. 12, 355 (1977).
[CrossRef]

E. Krätzig, R. Sommerfeldt, “Influence of dopants on photorefractive properties of LiNbO3crystals,” in Nonlinear Optical Materials III, P. Günter, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1273, 58 (1990).

O. Althoff, E. Krätzig, “Strong light-induced refractive index changes in LiNbO3,” in Nonlinear Optical Materials III, P. Günter, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1273, 12 (1990).
[CrossRef]

F. Jermann, E. Krätzig, “Photorefractive effects in LiNbO3:Fe at high light intensities,” in Proceedings of the International Conference on Defects in Insulating Materials, O. Kauert, J.-G. Spaeth, eds. (World Scientific, Singapore, 1993), Vol. 2, p. 1133.

Kuper, G.

L. Holtmann, K. Buse, G. Kuper, A. Groll, H. Hesse, E. Krätzig, “Photoconductivity and light-induced absorption in KNbO3:Fe,” Appl. Phys. A 53, 81 (1991).
[CrossRef]

Kurz, H.

D. von der Linde, O. Schirmer, H. Kurz, “Intrinsic photorefractive effect of LiNbO3,” Appl. Phys. 15, 153 (1978).
[CrossRef]

H. Kurz, E. Krätzig, W. Keune, H. Engelmann, U. Gonser, B. Dischler, A. Räuber, “Photorefractive centers in LiNbO3, studied by optical-, Mössbauer- and EPR-methods,” Appl. Phys. 12, 355 (1977).
[CrossRef]

Malinovsky, V.

I. Kanaev, S. Kostritsky, V. Malinovsky, A. Pugachev, “The influence of photoinduced mechanical tensions on photogalvanic effect and Raman scattering in LiNbO3,” Ferroelectrics 126, 45 (1992).
[CrossRef]

I. Kanaev, V. Malinovsky, A. Pugachev, “Changes in photogalvanic and photorefractive characteristics of lithium niobate under the light,” Ferroelectrics 75, 209 (1987).
[CrossRef]

Malovichko, G.

G. Malovichko, V. Grachev, E. Kokonyan, O. Schirmer, K. Betzler, B. Gather, F. Jermann, S. Klauer, U. Schlarb, M. Wöhlecke, “Characterization of stoichiometric LiNbO3grown from melts containing K2O,” Appl. Phys. A 56, 103 (1992).
[CrossRef]

Marsh, P.

S. Abrahams, P. Marsh, “Defect structure dependence on composition in lithium niobate,” Acta Crystallogr. Sect. B 42, 61 (1986).
[CrossRef]

Motes, R. A.

Negran, T. J.

A. M. Glass, D. von der Linde, T. J. Negran, “High-voltage bulk photovoltaic effect and the photorefractive process in LiNbO3,” Appl. Phys. Lett. 25, 233 (1974).
[CrossRef]

Ohmachi, Y.

Y. Ohmachi, K. Sawamoto, H. Toyoda, “Dielectric properties of LiNbO3 single crystal up to 9 Gc,” Jpn. J. Appl. Phys. 6, 1467 (1967).
[CrossRef]

Ohmori, Y.

Y. Ohmori, M. Yamaguchi, K. Yoshino, Y. Inuishi, “Electron Hall mobility in reduced LiNbO3,” Jpn. J. Appl. Phys. 15, 2263 (1976).
[CrossRef]

Orlowski, R.

E. Krätzig, R. Orlowski, “Light induced charge transport in doped LiNbO3and LiTaO3,” Ferroelectrics 27, 241 (1980).
[CrossRef]

Otten, J.

J. Otten, A. Bledowski, K. Ringhofer, R. Rupp, “Dynamical holographic storage in photorefractive crystals,” Comput. Phys. Commun. 69, 187 (1992).
[CrossRef]

Pugachev, A.

I. Kanaev, S. Kostritsky, V. Malinovsky, A. Pugachev, “The influence of photoinduced mechanical tensions on photogalvanic effect and Raman scattering in LiNbO3,” Ferroelectrics 126, 45 (1992).
[CrossRef]

I. Kanaev, V. Malinovsky, A. Pugachev, “Changes in photogalvanic and photorefractive characteristics of lithium niobate under the light,” Ferroelectrics 75, 209 (1987).
[CrossRef]

Rasch, A.

R. Göring, A. Rasch, W. Karthe, “Quantitative investigation of photorefractive effects in LiNbO3 channel wave-guides,” in Electro-Optic and Magneto-Optic Materials II, H. Dammann, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1274, 18 (1990).
[CrossRef]

Räuber, A.

H. Kurz, E. Krätzig, W. Keune, H. Engelmann, U. Gonser, B. Dischler, A. Räuber, “Photorefractive centers in LiNbO3, studied by optical-, Mössbauer- and EPR-methods,” Appl. Phys. 12, 355 (1977).
[CrossRef]

Ringhofer, K.

J. Otten, A. Bledowski, K. Ringhofer, R. Rupp, “Dynamical holographic storage in photorefractive crystals,” Comput. Phys. Commun. 69, 187 (1992).
[CrossRef]

Rotgé, J. R.

Rubinina, N.

T. Volk, N. Rubinina, “Nonphotorefractive impurities in lithium niobate: magnesium and zinc,” Sov. Phys. Solid State 33, 674 (1991).

Rupp, R.

J. Otten, A. Bledowski, K. Ringhofer, R. Rupp, “Dynamical holographic storage in photorefractive crystals,” Comput. Phys. Commun. 69, 187 (1992).
[CrossRef]

Sawamoto, K.

Y. Ohmachi, K. Sawamoto, H. Toyoda, “Dielectric properties of LiNbO3 single crystal up to 9 Gc,” Jpn. J. Appl. Phys. 6, 1467 (1967).
[CrossRef]

Schirmer, O.

G. Malovichko, V. Grachev, E. Kokonyan, O. Schirmer, K. Betzler, B. Gather, F. Jermann, S. Klauer, U. Schlarb, M. Wöhlecke, “Characterization of stoichiometric LiNbO3grown from melts containing K2O,” Appl. Phys. A 56, 103 (1992).
[CrossRef]

O. Schirmer, O. Thiemann, M. Wöhlecke, “Defects in LiNbO3–I. Experimental aspects,” J. Phys. Chem. Solids 52, 185 (1991).
[CrossRef]

O. Schirmer, S. Juppe, J. Koppitz, “ESR-, optical and photovoltaic studies of reduced undoped LiNbO3,” Cryst. Lattice Defects Amorph. Mat. 16, 353 (1987).

O. Schirmer, D. von der Linde, “Two-photon and x-ray-induced Nb4+ and O− small polarons in LiNbO3,” Appl. Phys. Lett. 33, 35 (1978).
[CrossRef]

D. von der Linde, O. Schirmer, H. Kurz, “Intrinsic photorefractive effect of LiNbO3,” Appl. Phys. 15, 153 (1978).
[CrossRef]

Schlarb, U.

G. Malovichko, V. Grachev, E. Kokonyan, O. Schirmer, K. Betzler, B. Gather, F. Jermann, S. Klauer, U. Schlarb, M. Wöhlecke, “Characterization of stoichiometric LiNbO3grown from melts containing K2O,” Appl. Phys. A 56, 103 (1992).
[CrossRef]

Shandarov, S.

S. Shandarov, “Influence of piezoelectric effect on photorefractive gratings in electro-optic crystals,” Appl. Phys. A 55, 91 (1992).
[CrossRef]

Shvarts, K.

P. Augustov, K. Shvarts, “The temperature and light intensity dependence of photorefraction in LiNbO3,” Appl. Phys. 21, 191 (1980).
[CrossRef]

P. Augustov, K. Shvarts, “Surface recombination and photorefraction in LiNbO3:Fe crystals,” Appl. Phys. 18, 399 (1979).
[CrossRef]

Sommerfeldt, R.

R. Sommerfeldt, L. Holtmann, E. Krätzig, B. C. Grabmeier, “Influence of Mg doping and composition on the light-induced charge transport in LiNbO3,” Phys. Status Solidi A 106, 89 (1988).
[CrossRef]

R. Sommerfeldt, “The influence of further impurities on the photorefractive properties of Fe-doped LiNbO3 crystals,” Dissertation (Universität Osnabrück, Osnabrück, Germany, 1989).

E. Krätzig, R. Sommerfeldt, “Influence of dopants on photorefractive properties of LiNbO3crystals,” in Nonlinear Optical Materials III, P. Günter, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1273, 58 (1990).

Steinberg, S.

R. Göring, Z. Yuang-Lung, S. Steinberg, “Photoconductivity and photovoltaic behavior of LiNbO3and LiNbO3wave-guides at high optical intensities,” Appl. Phys. A 55, 97 (1992).
[CrossRef]

Sweeney, K.

K. Sweeney, L. Halliburton, “Oxygen vacancies in lithium niobate,” Appl. Phys. Lett. 43, 336 (1983).
[CrossRef]

Thiemann, O.

O. Schirmer, O. Thiemann, M. Wöhlecke, “Defects in LiNbO3–I. Experimental aspects,” J. Phys. Chem. Solids 52, 185 (1991).
[CrossRef]

Toyoda, H.

Y. Ohmachi, K. Sawamoto, H. Toyoda, “Dielectric properties of LiNbO3 single crystal up to 9 Gc,” Jpn. J. Appl. Phys. 6, 1467 (1967).
[CrossRef]

Verber, C.

V. Wood, N. Hartmann, C. Verber, “Two-photon photorefractivity in pure and doped LiNbO3,” Ferroelectrics 27, 237 (1980).
[CrossRef]

Volk, T.

T. Volk, N. Rubinina, “Nonphotorefractive impurities in lithium niobate: magnesium and zinc,” Sov. Phys. Solid State 33, 674 (1991).

von der Linde, D.

C.-T. Chen, D. M. Kim, D. von der Linde, “Efficient hologram recording in LiNbO3:Fe using optical pulses,” Appl. Phys. Lett. 34, 321 (1979).
[CrossRef]

O. Schirmer, D. von der Linde, “Two-photon and x-ray-induced Nb4+ and O− small polarons in LiNbO3,” Appl. Phys. Lett. 33, 35 (1978).
[CrossRef]

D. von der Linde, O. Schirmer, H. Kurz, “Intrinsic photorefractive effect of LiNbO3,” Appl. Phys. 15, 153 (1978).
[CrossRef]

A. M. Glass, D. von der Linde, T. J. Negran, “High-voltage bulk photovoltaic effect and the photorefractive process in LiNbO3,” Appl. Phys. Lett. 25, 233 (1974).
[CrossRef]

Wiskott, L.

O. Althoff, A. Erdmann, L. Wiskott, P. Hertel, “The photorefractive effect in LiNbO3at high light intensity,” Phys. Status Solidi A 128, K41 (1991).
[CrossRef]

Wöhlecke, M.

G. Malovichko, V. Grachev, E. Kokonyan, O. Schirmer, K. Betzler, B. Gather, F. Jermann, S. Klauer, U. Schlarb, M. Wöhlecke, “Characterization of stoichiometric LiNbO3grown from melts containing K2O,” Appl. Phys. A 56, 103 (1992).
[CrossRef]

O. Schirmer, O. Thiemann, M. Wöhlecke, “Defects in LiNbO3–I. Experimental aspects,” J. Phys. Chem. Solids 52, 185 (1991).
[CrossRef]

Wood, V.

V. Wood, N. Hartmann, C. Verber, “Two-photon photorefractivity in pure and doped LiNbO3,” Ferroelectrics 27, 237 (1980).
[CrossRef]

Yamaguchi, M.

Y. Ohmori, M. Yamaguchi, K. Yoshino, Y. Inuishi, “Electron Hall mobility in reduced LiNbO3,” Jpn. J. Appl. Phys. 15, 2263 (1976).
[CrossRef]

Yoshino, K.

Y. Ohmori, M. Yamaguchi, K. Yoshino, Y. Inuishi, “Electron Hall mobility in reduced LiNbO3,” Jpn. J. Appl. Phys. 15, 2263 (1976).
[CrossRef]

Yuang-Lung, Z.

R. Göring, Z. Yuang-Lung, S. Steinberg, “Photoconductivity and photovoltaic behavior of LiNbO3and LiNbO3wave-guides at high optical intensities,” Appl. Phys. A 55, 97 (1992).
[CrossRef]

Zgonik, M.

Acta Crystallogr. Sect. B (1)

S. Abrahams, P. Marsh, “Defect structure dependence on composition in lithium niobate,” Acta Crystallogr. Sect. B 42, 61 (1986).
[CrossRef]

Appl. Phys. (4)

H. Kurz, E. Krätzig, W. Keune, H. Engelmann, U. Gonser, B. Dischler, A. Räuber, “Photorefractive centers in LiNbO3, studied by optical-, Mössbauer- and EPR-methods,” Appl. Phys. 12, 355 (1977).
[CrossRef]

P. Augustov, K. Shvarts, “The temperature and light intensity dependence of photorefraction in LiNbO3,” Appl. Phys. 21, 191 (1980).
[CrossRef]

D. von der Linde, O. Schirmer, H. Kurz, “Intrinsic photorefractive effect of LiNbO3,” Appl. Phys. 15, 153 (1978).
[CrossRef]

P. Augustov, K. Shvarts, “Surface recombination and photorefraction in LiNbO3:Fe crystals,” Appl. Phys. 18, 399 (1979).
[CrossRef]

Appl. Phys. A (5)

S. Shandarov, “Influence of piezoelectric effect on photorefractive gratings in electro-optic crystals,” Appl. Phys. A 55, 91 (1992).
[CrossRef]

L. Holtmann, K. Buse, G. Kuper, A. Groll, H. Hesse, E. Krätzig, “Photoconductivity and light-induced absorption in KNbO3:Fe,” Appl. Phys. A 53, 81 (1991).
[CrossRef]

G. Malovichko, V. Grachev, E. Kokonyan, O. Schirmer, K. Betzler, B. Gather, F. Jermann, S. Klauer, U. Schlarb, M. Wöhlecke, “Characterization of stoichiometric LiNbO3grown from melts containing K2O,” Appl. Phys. A 56, 103 (1992).
[CrossRef]

F. Jermann, E. Krätzig, “Charge transport processes in LiNbO3:Fe at high intensity laser pulses,” Appl. Phys. A 55, 114 (1992).
[CrossRef]

R. Göring, Z. Yuang-Lung, S. Steinberg, “Photoconductivity and photovoltaic behavior of LiNbO3and LiNbO3wave-guides at high optical intensities,” Appl. Phys. A 55, 97 (1992).
[CrossRef]

Appl. Phys. Lett. (4)

O. Schirmer, D. von der Linde, “Two-photon and x-ray-induced Nb4+ and O− small polarons in LiNbO3,” Appl. Phys. Lett. 33, 35 (1978).
[CrossRef]

A. M. Glass, D. von der Linde, T. J. Negran, “High-voltage bulk photovoltaic effect and the photorefractive process in LiNbO3,” Appl. Phys. Lett. 25, 233 (1974).
[CrossRef]

C.-T. Chen, D. M. Kim, D. von der Linde, “Efficient hologram recording in LiNbO3:Fe using optical pulses,” Appl. Phys. Lett. 34, 321 (1979).
[CrossRef]

K. Sweeney, L. Halliburton, “Oxygen vacancies in lithium niobate,” Appl. Phys. Lett. 43, 336 (1983).
[CrossRef]

Comput. Phys. Commun. (1)

J. Otten, A. Bledowski, K. Ringhofer, R. Rupp, “Dynamical holographic storage in photorefractive crystals,” Comput. Phys. Commun. 69, 187 (1992).
[CrossRef]

Cryst. Lattice Defects Amorph. Mat. (1)

O. Schirmer, S. Juppe, J. Koppitz, “ESR-, optical and photovoltaic studies of reduced undoped LiNbO3,” Cryst. Lattice Defects Amorph. Mat. 16, 353 (1987).

Ferroelectrics (4)

I. Kanaev, S. Kostritsky, V. Malinovsky, A. Pugachev, “The influence of photoinduced mechanical tensions on photogalvanic effect and Raman scattering in LiNbO3,” Ferroelectrics 126, 45 (1992).
[CrossRef]

I. Kanaev, V. Malinovsky, A. Pugachev, “Changes in photogalvanic and photorefractive characteristics of lithium niobate under the light,” Ferroelectrics 75, 209 (1987).
[CrossRef]

E. Krätzig, R. Orlowski, “Light induced charge transport in doped LiNbO3and LiTaO3,” Ferroelectrics 27, 241 (1980).
[CrossRef]

V. Wood, N. Hartmann, C. Verber, “Two-photon photorefractivity in pure and doped LiNbO3,” Ferroelectrics 27, 237 (1980).
[CrossRef]

J. Appl. Phys. (1)

F. Chen, “Optically induced change of refractive indices in LiNbO3and LiTaO3,” J. Appl. Phys. 40, 3389 (1969).
[CrossRef]

J. Opt. Soc. Am. B (3)

J. Phys. Chem. Solids (1)

O. Schirmer, O. Thiemann, M. Wöhlecke, “Defects in LiNbO3–I. Experimental aspects,” J. Phys. Chem. Solids 52, 185 (1991).
[CrossRef]

Jpn. J. Appl. Phys. (2)

Y. Ohmori, M. Yamaguchi, K. Yoshino, Y. Inuishi, “Electron Hall mobility in reduced LiNbO3,” Jpn. J. Appl. Phys. 15, 2263 (1976).
[CrossRef]

Y. Ohmachi, K. Sawamoto, H. Toyoda, “Dielectric properties of LiNbO3 single crystal up to 9 Gc,” Jpn. J. Appl. Phys. 6, 1467 (1967).
[CrossRef]

Phys. Status Solidi A (2)

R. Sommerfeldt, L. Holtmann, E. Krätzig, B. C. Grabmeier, “Influence of Mg doping and composition on the light-induced charge transport in LiNbO3,” Phys. Status Solidi A 106, 89 (1988).
[CrossRef]

O. Althoff, A. Erdmann, L. Wiskott, P. Hertel, “The photorefractive effect in LiNbO3at high light intensity,” Phys. Status Solidi A 128, K41 (1991).
[CrossRef]

Prog. Quantum Electron. (1)

T. J. Hall, R. Jaura, L. M. Connors, P. D. Foote, “The photorefractive effect—a review,” Prog. Quantum Electron. 10, 77 (1985).
[CrossRef]

Sov. Phys. Solid State (1)

T. Volk, N. Rubinina, “Nonphotorefractive impurities in lithium niobate: magnesium and zinc,” Sov. Phys. Solid State 33, 674 (1991).

Other (6)

F. Jermann, E. Krätzig, “Photorefractive effects in LiNbO3:Fe at high light intensities,” in Proceedings of the International Conference on Defects in Insulating Materials, O. Kauert, J.-G. Spaeth, eds. (World Scientific, Singapore, 1993), Vol. 2, p. 1133.

R. Sommerfeldt, “The influence of further impurities on the photorefractive properties of Fe-doped LiNbO3 crystals,” Dissertation (Universität Osnabrück, Osnabrück, Germany, 1989).

E. Krätzig, R. Sommerfeldt, “Influence of dopants on photorefractive properties of LiNbO3crystals,” in Nonlinear Optical Materials III, P. Günter, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1273, 58 (1990).

P. Günter, J.-P. Huignard, Photorefractive Materials and Their Applications I and II (Springer-Verlag, Heidelberg, 1988, 1989).
[CrossRef]

O. Althoff, E. Krätzig, “Strong light-induced refractive index changes in LiNbO3,” in Nonlinear Optical Materials III, P. Günter, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1273, 12 (1990).
[CrossRef]

R. Göring, A. Rasch, W. Karthe, “Quantitative investigation of photorefractive effects in LiNbO3 channel wave-guides,” in Electro-Optic and Magneto-Optic Materials II, H. Dammann, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1274, 18 (1990).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Fe2+/Fe3+ charge-transport model modified by additional X centers (=two-center model). Arrows indicate possible transitions. In the dark, X centers are empty because the Fermi level is associated with the energy level of iron ions.

Fig. 2
Fig. 2

Light-induced absorption coefficient αlio (ordinary polarization) versus pulse intensity I for different LiNbO3:Fe crystals. The symbols are explained in Table 1. The curves are calculated according to Eqs. (7) and (12) with the parameters given in Table 2.

Fig. 3
Fig. 3

Photoconductivity σ0 versus I for illumination with laser pulses. Crystals with different concentration ratios of Fe2+ and Fe3+ ions are investigated (see Table 2). The curves are fits according to Eq. (20).

Fig. 4
Fig. 4

Increase of Δno/(mΔt)|t→0 (o denotes ordinary polarization) versus intensity I for LiNbO3:Fe crystals with different Fe2+ concentrations. The symbols are explained in Table 1. The curves are calculated according to Eq. (21).

Fig. 5
Fig. 5

Saturation value Δnso of amplitude of refractive index change (ordinary polarization) versus pulse intensity I for different LiNbO3:Fe crystals. The symbols are explained in Table 1. The curves are model predictions according to Eq. (22) using the parameter set given in Table 2.

Fig. 6
Fig. 6

Saturation value Δnse of refractive index change (extraordinary polarization) versus intensity I of a focused cw beam.

Tables (3)

Tables Icon

Table 1 Total Iron Concentrations cFe = NFe and Fe2+ Contents cFe2+ = NFeNC of the Investigated LiNbO3:Fe Crystalsa

Tables Icon

Table 2 Parameters Used for Model Calculationsa

Tables Icon

Table 3 Comparison of the Results of cw Laser Measurements and Model Calculations (Ref. 20)a

Equations (31)

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

N Fe + t = ( q Fe s Fe + q FeX s FeX N X + ) I ( N Fe N Fe + ) [ γ Fe n + γ XFe ( N X N X + ) ] N Fe + ,
N X + t = ( β X + q X s X I + γ XFe N Fe + ) ( N X N X + ) [ γ X n + q FeX s FeX I ( N Fe N Fe + ) ] N X + ,
n t + 1 e j z = N Fe + t + N X + t ,
j = e μ n E + κ Fe I ( N Fe N Fe + ) + κ X I ( N X N X + ) ,
n = q Fe s Fe I ( N Fe N Fe + ) + ( β X + q X s X I ) ( N X N X + ) γ Fe N Fe + + γ X N X + .
γ Fe N Fe + γ X N X +
α li p ( I ) = h ν p [ s X p s Fe p s FeX p N X + ( I ) ] [ N X N X + ( I ) ] ,
N X + t = ξ 1 N X + 2 + ξ 2 N X + + ξ 3 ,
ξ 1 = γ XFe q FeX s FeX I ,
ξ 2 = β X q X s X I γ XFe ( N C + 2 N X ) q FeX s FeX I ( N Fe N C N X ) ,
ξ 3 = ( β X + q X s X I ) N X + γ XFe ( N C + N X ) N X .
N X + = 1 ξ 1 τ X tanh ( t / τ X ) ( ξ 1 N X + ξ 2 / 2 ) τ X ( ξ 1 N X + ξ 2 / 2 ) τ X tanh ( t / τ X ) 1 ξ 2 2 ξ 1 ,
τ X = 2 ( ξ 2 2 4 ξ 1 ξ 3 ) 1 / 2 .
N X + = N X q FeX s FeX I ( N Fe N C ) N X β X + γ XFe N C + [ q X s X + q FeX s FeX ( N Fe N C ) ] I × [ 1 exp ( { β X + γ XFe N C + [ q X s X + q FeX s FeX ( N Fe N C ) ] I } t ) ] .
N X + ¯ = N X N ¯ , N Fe + ¯ = N C + N ¯ ,
δ N ¯ = g eff ( N Fe N C ) I , g eff = 1 2 f b q FeX s FeX N X t p .
Q ( z , t ) = Q 0 ( t ) + 1 2 [ Q 1 ( t ) exp ( i K z ) + Q 1 * ( t ) exp ( i K z ) ] .
0 E 1 t = j 1 = σ 0 ( E 1 + E 1 Phv , Fe + E 1 Phv , X ) .
E 1 Phv , Fe = ( κ Fe I 1 / σ 0 ) ( N Fe N C 2 δ N 0 ¯ ) ,
E 1 Phv , X = 2 ( κ X I 1 / σ 0 ) δ N 0 ¯ .
σ 0 = e μ ( N Fe N C ) [ q Fe s Fe I 0 + ( q X s X q Fe s Fe ) g eff I 0 2 ] γ Fe [ N C + g eff ( N Fe N C ) I 0 ] .
1 m Δ n Δ t | t 0 = n 3 r 2 0 ( N Fe N C ) [ κ Fe I 0 + 2 g eff ( κ X κ Fe ) I 0 2 ] ,
E 1 S m = E 1 Phv , Fe + E 1 Phv , X m = N Fe N C σ 0 [ κ Fe I 0 + 2 ( κ X κ Fe ) g eff I 0 2 ] = γ Fe e μ N C ( 1 + N Fe N C N C g eff I 0 ) × κ Fe + 2 ( κ X κ Fe ) g eff I 0 q Fe s Fe + ( q X s X q Fe s Fe ) g eff I 0 .
N X f = N X ξ 2 + ( ξ 2 2 4 ξ 1 ξ 3 ) 1 / 2 2 ξ 1 ,
σ 0 = e μ q Fe s Fe I ( N Fe N C ) + [ β X + ( q X s X q Fe s Fe ) I ] N X f γ Fe ( N C + N X f ) .
0 E 0 t = j 0 = [ σ 0 E 0 + κ Fe I ( N Fe N C ) + ( κ X κ Fe ) I N X f ] .
Δ n Δ t | t 0 = n 3 r 2 0 [ κ Fe I ( N Fe N C ) + ( κ X κ Fe ) I N X f ] ,
Δ n s = 1 2 n 3 r κ Fe I ( N Fe N C ) + ( κ X κ Fe ) i N X f σ 0 .
N X f = q FeX s FeX N X β X + γ FeX N C ( N Fe N C ) I .
σ 0 = σ l I + σ q I 2 ,
Δ n Δ t | t 0 , = S l I + S q I 2 ,

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