Abstract

A mathematical model is developed to account for the fixing and developing behavior of holographic gratings in photorefractive materials such as LiNbO3. All transport processes for the carriers (electrons and protons) are considered. Moreover, unlike in previous studies, the thermal excitation of carriers is taken into account. Two possible experimental procedures that involve fixing during or after writing are theoretically described. The model is applied to simulate the kinetics, overall efficiency, and temperature dependence of the fixing process for LiNbO3:Fe, for which the most experimental information is available.

© 1990 Optical Society of America

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  1. P. Yeh, “Two-wave mixing in nonlinear media,” IEEE J. Quantum Electron. 25, 484–519 (1989).
    [CrossRef]
  2. W. J. Burke, D. L. Staebler, W. Phillips, G. A. Alphonse, “Volume phase holographic storage in ferroelectric crystals,” Opt. Eng. 17, 308–316 (1978).
    [CrossRef]
  3. G. C. Valley, M. B. Klein, “Optimal properties of photorefractive materials for optical data processing,” Opt. Eng. 22, 704–711 (1983).
    [CrossRef]
  4. J. J. Amodei, W. Phillips, D. L. Staebler, “Improved electro-optic materials and fixing techniques for holographic recording,” Appl. Opt. 11, 390–396 (1972).
    [CrossRef] [PubMed]
  5. D. L. Staebler, W. J. Burke, W. Phillips, J. J. Amodei, “Multiple storage and erasure of fixed holograms in Fe-doped LiNbO3,” Appl. Phys. Lett. 26, 182–184 (1975).
    [CrossRef]
  6. H. Vormann, G. Weber, S. Kapphan, E. Kratzig, “Hydrogen as origin of thermal fixing in LiNbO3,” Solid State Commun. 40, 543–545 (1981).
    [CrossRef]
  7. R. Sommerfeldt, R. A. Rupp, H. Vormann, E. Kratzig, “Thermal fixing of volume phase holograms in LiNbO3:Cu,” Phys. Status Solidi A 99, k15–k18 (1987).
    [CrossRef]
  8. R. Matull, R. A. Rupp, “Microphometric investigations of fixed holograms,” J. Phys. D 21, 1556–1565 (1988).
    [CrossRef]
  9. J. P. Herriau, J. P. Huignard, “Hologram fixing process at room temperature in photorefractive Bi12SiO20crystals,” Appl. Phys. Lett. 49, 1140–1142 (1986).
    [CrossRef]
  10. L. Arizmendi, “Thermal fixing of holographic gratings in Bi12SiO20,” J. Appl. Phys. 65, 423–427 (1989).
    [CrossRef]
  11. S. W. McCahon, D. Rytz, G. C. Valley, M. V. Klein, B. A. Wechsler, “Hologram fixing in Bi12TiO20using heating and an ac electric field,” Appl. Opt. 28, 1967–1969 (1989).
    [CrossRef] [PubMed]
  12. W. Meyer, P. Wurfel, R. Munser, G. Muller-Vogt, “Kinetics of fixation of phase holograms in LiNbO3,” Phys. Status Solidi A 53, 171–180 (1979).
    [CrossRef]
  13. M. Carrascosa, F. Agulló-López, “Kinetics for optical erasure of sinusoidal holographic gratings in photorefractive materials,” IEEE J. Quantum Elecctron. QE-22, 1369–1375 (1986).
    [CrossRef]
  14. P. Hertel, K. H. Ringhofer, R. Sommerfeldt, “Theory of thermal hologram fixing and application to LiNbO3:Cu,” Phys. Status Solidi A 104, 855–862 (1987).
    [CrossRef]
  15. M. G. Clark, F. J. Disalvo, A. M. Glass, G. E. Peterson, “Electronic structure and optical index damage of iron-doped lithium niobate,” J. Chem. Phys. 59, 6209–6219 (1973).
    [CrossRef]
  16. A. Zylberstejn, “Thermally activated trapping in Fe-doped LiNbO3,” Appl. Phys. 29, 778–780 (1976).
  17. N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetskii, “Holographic storage in electro-optic crystals. I. Steady state,” Ferroelectrics 22, 949–960 (1979).
    [CrossRef]
  18. W. Josch, R. Munser, W. Ruppel, P. Wurfel, “The photovoltaic effect and the charge transport in LiNbO3,” Ferroelectrics 21, 623–625 (1977).
    [CrossRef]
  19. E. Kratzig, R. Orlowski, “Light induced charge transport in doped LiNbO3and TiTaO3,” Ferroelectrics 27, 241–244 (1980).
    [CrossRef]
  20. W. Bollmann, “Diffusion of hydrogen (OH−ions) in LiNbO3crystals,” Phys. Status Solidi A 104, 643–648 (1987).
    [CrossRef]

1989

P. Yeh, “Two-wave mixing in nonlinear media,” IEEE J. Quantum Electron. 25, 484–519 (1989).
[CrossRef]

L. Arizmendi, “Thermal fixing of holographic gratings in Bi12SiO20,” J. Appl. Phys. 65, 423–427 (1989).
[CrossRef]

S. W. McCahon, D. Rytz, G. C. Valley, M. V. Klein, B. A. Wechsler, “Hologram fixing in Bi12TiO20using heating and an ac electric field,” Appl. Opt. 28, 1967–1969 (1989).
[CrossRef] [PubMed]

1988

R. Matull, R. A. Rupp, “Microphometric investigations of fixed holograms,” J. Phys. D 21, 1556–1565 (1988).
[CrossRef]

1987

R. Sommerfeldt, R. A. Rupp, H. Vormann, E. Kratzig, “Thermal fixing of volume phase holograms in LiNbO3:Cu,” Phys. Status Solidi A 99, k15–k18 (1987).
[CrossRef]

P. Hertel, K. H. Ringhofer, R. Sommerfeldt, “Theory of thermal hologram fixing and application to LiNbO3:Cu,” Phys. Status Solidi A 104, 855–862 (1987).
[CrossRef]

W. Bollmann, “Diffusion of hydrogen (OH−ions) in LiNbO3crystals,” Phys. Status Solidi A 104, 643–648 (1987).
[CrossRef]

1986

M. Carrascosa, F. Agulló-López, “Kinetics for optical erasure of sinusoidal holographic gratings in photorefractive materials,” IEEE J. Quantum Elecctron. QE-22, 1369–1375 (1986).
[CrossRef]

J. P. Herriau, J. P. Huignard, “Hologram fixing process at room temperature in photorefractive Bi12SiO20crystals,” Appl. Phys. Lett. 49, 1140–1142 (1986).
[CrossRef]

1983

G. C. Valley, M. B. Klein, “Optimal properties of photorefractive materials for optical data processing,” Opt. Eng. 22, 704–711 (1983).
[CrossRef]

1981

H. Vormann, G. Weber, S. Kapphan, E. Kratzig, “Hydrogen as origin of thermal fixing in LiNbO3,” Solid State Commun. 40, 543–545 (1981).
[CrossRef]

1980

E. Kratzig, R. Orlowski, “Light induced charge transport in doped LiNbO3and TiTaO3,” Ferroelectrics 27, 241–244 (1980).
[CrossRef]

1979

W. Meyer, P. Wurfel, R. Munser, G. Muller-Vogt, “Kinetics of fixation of phase holograms in LiNbO3,” Phys. Status Solidi A 53, 171–180 (1979).
[CrossRef]

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetskii, “Holographic storage in electro-optic crystals. I. Steady state,” Ferroelectrics 22, 949–960 (1979).
[CrossRef]

1978

W. J. Burke, D. L. Staebler, W. Phillips, G. A. Alphonse, “Volume phase holographic storage in ferroelectric crystals,” Opt. Eng. 17, 308–316 (1978).
[CrossRef]

1977

W. Josch, R. Munser, W. Ruppel, P. Wurfel, “The photovoltaic effect and the charge transport in LiNbO3,” Ferroelectrics 21, 623–625 (1977).
[CrossRef]

1976

A. Zylberstejn, “Thermally activated trapping in Fe-doped LiNbO3,” Appl. Phys. 29, 778–780 (1976).

1975

D. L. Staebler, W. J. Burke, W. Phillips, J. J. Amodei, “Multiple storage and erasure of fixed holograms in Fe-doped LiNbO3,” Appl. Phys. Lett. 26, 182–184 (1975).
[CrossRef]

1973

M. G. Clark, F. J. Disalvo, A. M. Glass, G. E. Peterson, “Electronic structure and optical index damage of iron-doped lithium niobate,” J. Chem. Phys. 59, 6209–6219 (1973).
[CrossRef]

1972

Agulló-López, F.

M. Carrascosa, F. Agulló-López, “Kinetics for optical erasure of sinusoidal holographic gratings in photorefractive materials,” IEEE J. Quantum Elecctron. QE-22, 1369–1375 (1986).
[CrossRef]

Alphonse, G. A.

W. J. Burke, D. L. Staebler, W. Phillips, G. A. Alphonse, “Volume phase holographic storage in ferroelectric crystals,” Opt. Eng. 17, 308–316 (1978).
[CrossRef]

Amodei, J. J.

D. L. Staebler, W. J. Burke, W. Phillips, J. J. Amodei, “Multiple storage and erasure of fixed holograms in Fe-doped LiNbO3,” Appl. Phys. Lett. 26, 182–184 (1975).
[CrossRef]

J. J. Amodei, W. Phillips, D. L. Staebler, “Improved electro-optic materials and fixing techniques for holographic recording,” Appl. Opt. 11, 390–396 (1972).
[CrossRef] [PubMed]

Arizmendi, L.

L. Arizmendi, “Thermal fixing of holographic gratings in Bi12SiO20,” J. Appl. Phys. 65, 423–427 (1989).
[CrossRef]

Bollmann, W.

W. Bollmann, “Diffusion of hydrogen (OH−ions) in LiNbO3crystals,” Phys. Status Solidi A 104, 643–648 (1987).
[CrossRef]

Burke, W. J.

W. J. Burke, D. L. Staebler, W. Phillips, G. A. Alphonse, “Volume phase holographic storage in ferroelectric crystals,” Opt. Eng. 17, 308–316 (1978).
[CrossRef]

D. L. Staebler, W. J. Burke, W. Phillips, J. J. Amodei, “Multiple storage and erasure of fixed holograms in Fe-doped LiNbO3,” Appl. Phys. Lett. 26, 182–184 (1975).
[CrossRef]

Carrascosa, M.

M. Carrascosa, F. Agulló-López, “Kinetics for optical erasure of sinusoidal holographic gratings in photorefractive materials,” IEEE J. Quantum Elecctron. QE-22, 1369–1375 (1986).
[CrossRef]

Clark, M. G.

M. G. Clark, F. J. Disalvo, A. M. Glass, G. E. Peterson, “Electronic structure and optical index damage of iron-doped lithium niobate,” J. Chem. Phys. 59, 6209–6219 (1973).
[CrossRef]

Disalvo, F. J.

M. G. Clark, F. J. Disalvo, A. M. Glass, G. E. Peterson, “Electronic structure and optical index damage of iron-doped lithium niobate,” J. Chem. Phys. 59, 6209–6219 (1973).
[CrossRef]

Glass, A. M.

M. G. Clark, F. J. Disalvo, A. M. Glass, G. E. Peterson, “Electronic structure and optical index damage of iron-doped lithium niobate,” J. Chem. Phys. 59, 6209–6219 (1973).
[CrossRef]

Herriau, J. P.

J. P. Herriau, J. P. Huignard, “Hologram fixing process at room temperature in photorefractive Bi12SiO20crystals,” Appl. Phys. Lett. 49, 1140–1142 (1986).
[CrossRef]

Hertel, P.

P. Hertel, K. H. Ringhofer, R. Sommerfeldt, “Theory of thermal hologram fixing and application to LiNbO3:Cu,” Phys. Status Solidi A 104, 855–862 (1987).
[CrossRef]

Huignard, J. P.

J. P. Herriau, J. P. Huignard, “Hologram fixing process at room temperature in photorefractive Bi12SiO20crystals,” Appl. Phys. Lett. 49, 1140–1142 (1986).
[CrossRef]

Josch, W.

W. Josch, R. Munser, W. Ruppel, P. Wurfel, “The photovoltaic effect and the charge transport in LiNbO3,” Ferroelectrics 21, 623–625 (1977).
[CrossRef]

Kapphan, S.

H. Vormann, G. Weber, S. Kapphan, E. Kratzig, “Hydrogen as origin of thermal fixing in LiNbO3,” Solid State Commun. 40, 543–545 (1981).
[CrossRef]

Klein, M. B.

G. C. Valley, M. B. Klein, “Optimal properties of photorefractive materials for optical data processing,” Opt. Eng. 22, 704–711 (1983).
[CrossRef]

Klein, M. V.

Kratzig, E.

R. Sommerfeldt, R. A. Rupp, H. Vormann, E. Kratzig, “Thermal fixing of volume phase holograms in LiNbO3:Cu,” Phys. Status Solidi A 99, k15–k18 (1987).
[CrossRef]

H. Vormann, G. Weber, S. Kapphan, E. Kratzig, “Hydrogen as origin of thermal fixing in LiNbO3,” Solid State Commun. 40, 543–545 (1981).
[CrossRef]

E. Kratzig, R. Orlowski, “Light induced charge transport in doped LiNbO3and TiTaO3,” Ferroelectrics 27, 241–244 (1980).
[CrossRef]

Kukhtarev, N. V.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetskii, “Holographic storage in electro-optic crystals. I. Steady state,” Ferroelectrics 22, 949–960 (1979).
[CrossRef]

Markov, V. B.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetskii, “Holographic storage in electro-optic crystals. I. Steady state,” Ferroelectrics 22, 949–960 (1979).
[CrossRef]

Matull, R.

R. Matull, R. A. Rupp, “Microphometric investigations of fixed holograms,” J. Phys. D 21, 1556–1565 (1988).
[CrossRef]

McCahon, S. W.

Meyer, W.

W. Meyer, P. Wurfel, R. Munser, G. Muller-Vogt, “Kinetics of fixation of phase holograms in LiNbO3,” Phys. Status Solidi A 53, 171–180 (1979).
[CrossRef]

Muller-Vogt, G.

W. Meyer, P. Wurfel, R. Munser, G. Muller-Vogt, “Kinetics of fixation of phase holograms in LiNbO3,” Phys. Status Solidi A 53, 171–180 (1979).
[CrossRef]

Munser, R.

W. Meyer, P. Wurfel, R. Munser, G. Muller-Vogt, “Kinetics of fixation of phase holograms in LiNbO3,” Phys. Status Solidi A 53, 171–180 (1979).
[CrossRef]

W. Josch, R. Munser, W. Ruppel, P. Wurfel, “The photovoltaic effect and the charge transport in LiNbO3,” Ferroelectrics 21, 623–625 (1977).
[CrossRef]

Odulov, S. G.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetskii, “Holographic storage in electro-optic crystals. I. Steady state,” Ferroelectrics 22, 949–960 (1979).
[CrossRef]

Orlowski, R.

E. Kratzig, R. Orlowski, “Light induced charge transport in doped LiNbO3and TiTaO3,” Ferroelectrics 27, 241–244 (1980).
[CrossRef]

Peterson, G. E.

M. G. Clark, F. J. Disalvo, A. M. Glass, G. E. Peterson, “Electronic structure and optical index damage of iron-doped lithium niobate,” J. Chem. Phys. 59, 6209–6219 (1973).
[CrossRef]

Phillips, W.

W. J. Burke, D. L. Staebler, W. Phillips, G. A. Alphonse, “Volume phase holographic storage in ferroelectric crystals,” Opt. Eng. 17, 308–316 (1978).
[CrossRef]

D. L. Staebler, W. J. Burke, W. Phillips, J. J. Amodei, “Multiple storage and erasure of fixed holograms in Fe-doped LiNbO3,” Appl. Phys. Lett. 26, 182–184 (1975).
[CrossRef]

J. J. Amodei, W. Phillips, D. L. Staebler, “Improved electro-optic materials and fixing techniques for holographic recording,” Appl. Opt. 11, 390–396 (1972).
[CrossRef] [PubMed]

Ringhofer, K. H.

P. Hertel, K. H. Ringhofer, R. Sommerfeldt, “Theory of thermal hologram fixing and application to LiNbO3:Cu,” Phys. Status Solidi A 104, 855–862 (1987).
[CrossRef]

Rupp, R. A.

R. Matull, R. A. Rupp, “Microphometric investigations of fixed holograms,” J. Phys. D 21, 1556–1565 (1988).
[CrossRef]

R. Sommerfeldt, R. A. Rupp, H. Vormann, E. Kratzig, “Thermal fixing of volume phase holograms in LiNbO3:Cu,” Phys. Status Solidi A 99, k15–k18 (1987).
[CrossRef]

Ruppel, W.

W. Josch, R. Munser, W. Ruppel, P. Wurfel, “The photovoltaic effect and the charge transport in LiNbO3,” Ferroelectrics 21, 623–625 (1977).
[CrossRef]

Rytz, D.

Sommerfeldt, R.

R. Sommerfeldt, R. A. Rupp, H. Vormann, E. Kratzig, “Thermal fixing of volume phase holograms in LiNbO3:Cu,” Phys. Status Solidi A 99, k15–k18 (1987).
[CrossRef]

P. Hertel, K. H. Ringhofer, R. Sommerfeldt, “Theory of thermal hologram fixing and application to LiNbO3:Cu,” Phys. Status Solidi A 104, 855–862 (1987).
[CrossRef]

Soskin, M. S.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetskii, “Holographic storage in electro-optic crystals. I. Steady state,” Ferroelectrics 22, 949–960 (1979).
[CrossRef]

Staebler, D. L.

W. J. Burke, D. L. Staebler, W. Phillips, G. A. Alphonse, “Volume phase holographic storage in ferroelectric crystals,” Opt. Eng. 17, 308–316 (1978).
[CrossRef]

D. L. Staebler, W. J. Burke, W. Phillips, J. J. Amodei, “Multiple storage and erasure of fixed holograms in Fe-doped LiNbO3,” Appl. Phys. Lett. 26, 182–184 (1975).
[CrossRef]

J. J. Amodei, W. Phillips, D. L. Staebler, “Improved electro-optic materials and fixing techniques for holographic recording,” Appl. Opt. 11, 390–396 (1972).
[CrossRef] [PubMed]

Valley, G. C.

S. W. McCahon, D. Rytz, G. C. Valley, M. V. Klein, B. A. Wechsler, “Hologram fixing in Bi12TiO20using heating and an ac electric field,” Appl. Opt. 28, 1967–1969 (1989).
[CrossRef] [PubMed]

G. C. Valley, M. B. Klein, “Optimal properties of photorefractive materials for optical data processing,” Opt. Eng. 22, 704–711 (1983).
[CrossRef]

Vinetskii, V. L.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetskii, “Holographic storage in electro-optic crystals. I. Steady state,” Ferroelectrics 22, 949–960 (1979).
[CrossRef]

Vormann, H.

R. Sommerfeldt, R. A. Rupp, H. Vormann, E. Kratzig, “Thermal fixing of volume phase holograms in LiNbO3:Cu,” Phys. Status Solidi A 99, k15–k18 (1987).
[CrossRef]

H. Vormann, G. Weber, S. Kapphan, E. Kratzig, “Hydrogen as origin of thermal fixing in LiNbO3,” Solid State Commun. 40, 543–545 (1981).
[CrossRef]

Weber, G.

H. Vormann, G. Weber, S. Kapphan, E. Kratzig, “Hydrogen as origin of thermal fixing in LiNbO3,” Solid State Commun. 40, 543–545 (1981).
[CrossRef]

Wechsler, B. A.

Wurfel, P.

W. Meyer, P. Wurfel, R. Munser, G. Muller-Vogt, “Kinetics of fixation of phase holograms in LiNbO3,” Phys. Status Solidi A 53, 171–180 (1979).
[CrossRef]

W. Josch, R. Munser, W. Ruppel, P. Wurfel, “The photovoltaic effect and the charge transport in LiNbO3,” Ferroelectrics 21, 623–625 (1977).
[CrossRef]

Yeh, P.

P. Yeh, “Two-wave mixing in nonlinear media,” IEEE J. Quantum Electron. 25, 484–519 (1989).
[CrossRef]

Zylberstejn, A.

A. Zylberstejn, “Thermally activated trapping in Fe-doped LiNbO3,” Appl. Phys. 29, 778–780 (1976).

Appl. Opt.

Appl. Phys.

A. Zylberstejn, “Thermally activated trapping in Fe-doped LiNbO3,” Appl. Phys. 29, 778–780 (1976).

Appl. Phys. Lett.

D. L. Staebler, W. J. Burke, W. Phillips, J. J. Amodei, “Multiple storage and erasure of fixed holograms in Fe-doped LiNbO3,” Appl. Phys. Lett. 26, 182–184 (1975).
[CrossRef]

J. P. Herriau, J. P. Huignard, “Hologram fixing process at room temperature in photorefractive Bi12SiO20crystals,” Appl. Phys. Lett. 49, 1140–1142 (1986).
[CrossRef]

Ferroelectrics

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetskii, “Holographic storage in electro-optic crystals. I. Steady state,” Ferroelectrics 22, 949–960 (1979).
[CrossRef]

W. Josch, R. Munser, W. Ruppel, P. Wurfel, “The photovoltaic effect and the charge transport in LiNbO3,” Ferroelectrics 21, 623–625 (1977).
[CrossRef]

E. Kratzig, R. Orlowski, “Light induced charge transport in doped LiNbO3and TiTaO3,” Ferroelectrics 27, 241–244 (1980).
[CrossRef]

IEEE J. Quantum Elecctron.

M. Carrascosa, F. Agulló-López, “Kinetics for optical erasure of sinusoidal holographic gratings in photorefractive materials,” IEEE J. Quantum Elecctron. QE-22, 1369–1375 (1986).
[CrossRef]

IEEE J. Quantum Electron.

P. Yeh, “Two-wave mixing in nonlinear media,” IEEE J. Quantum Electron. 25, 484–519 (1989).
[CrossRef]

J. Appl. Phys.

L. Arizmendi, “Thermal fixing of holographic gratings in Bi12SiO20,” J. Appl. Phys. 65, 423–427 (1989).
[CrossRef]

J. Chem. Phys.

M. G. Clark, F. J. Disalvo, A. M. Glass, G. E. Peterson, “Electronic structure and optical index damage of iron-doped lithium niobate,” J. Chem. Phys. 59, 6209–6219 (1973).
[CrossRef]

J. Phys. D

R. Matull, R. A. Rupp, “Microphometric investigations of fixed holograms,” J. Phys. D 21, 1556–1565 (1988).
[CrossRef]

Opt. Eng.

W. J. Burke, D. L. Staebler, W. Phillips, G. A. Alphonse, “Volume phase holographic storage in ferroelectric crystals,” Opt. Eng. 17, 308–316 (1978).
[CrossRef]

G. C. Valley, M. B. Klein, “Optimal properties of photorefractive materials for optical data processing,” Opt. Eng. 22, 704–711 (1983).
[CrossRef]

Phys. Status Solidi A

R. Sommerfeldt, R. A. Rupp, H. Vormann, E. Kratzig, “Thermal fixing of volume phase holograms in LiNbO3:Cu,” Phys. Status Solidi A 99, k15–k18 (1987).
[CrossRef]

P. Hertel, K. H. Ringhofer, R. Sommerfeldt, “Theory of thermal hologram fixing and application to LiNbO3:Cu,” Phys. Status Solidi A 104, 855–862 (1987).
[CrossRef]

W. Meyer, P. Wurfel, R. Munser, G. Muller-Vogt, “Kinetics of fixation of phase holograms in LiNbO3,” Phys. Status Solidi A 53, 171–180 (1979).
[CrossRef]

W. Bollmann, “Diffusion of hydrogen (OH−ions) in LiNbO3crystals,” Phys. Status Solidi A 104, 643–648 (1987).
[CrossRef]

Solid State Commun.

H. Vormann, G. Weber, S. Kapphan, E. Kratzig, “Hydrogen as origin of thermal fixing in LiNbO3,” Solid State Commun. 40, 543–545 (1981).
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

Time evolution of donor-grating M, proton-grating h amplitudes for thermal fixing procedure (a). Tf = 150°C, Λ = 1 μm, A = 0.9N.

Fig. 2
Fig. 2

Temperature dependence of fixing efficiency ηF for thermal fixing procedure (b). Two doping levels are considered: (a) N = 1019 cm−3 and (b) N = 1017 cm−3. In both cases H0 = 1019 cm−3, Λ = 1 μm, A = 0.9N.

Fig. 3
Fig. 3

Semilogarithmic plot of dependence of fixing efficiency ηF on proton concentration H0. N = 1019 cm−3, Λ = 1 μm, Tf = 150°C, A = 0.9N.

Tables (1)

Tables Icon

Table 1 Parameters Chosen for Simulation

Equations (52)

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

n t = ( s + σ I ) D γ n A + 1 e j e x ,
D t = A t = ( s + σ I ) D γ n A ,
E x = ρ ,
j h x = H t ,
j e = e μ e n E + D e e n x + e σ L p h D I ,
j h = e μ h H E D h e H x ,
ρ = [ ( H H 0 ) ( D D 0 ) ] e ,
I = I 0 ( 1 + m cos K x ) ,
D = D 0 + M 0 cos K x ,
A = A 0 + M 0 cos K x ,
H = H 0 ,
D ( t ) = D 0 + M ( t ) cos K x ,
A ( t ) = A 0 M ( t ) cos K x ,
H ( t ) = H 0 + h cos K x ;
ρ = ( h M ) e cos K x .
h ˙ = ( μ h H 0 e + D h K 2 ) h + μ h H 0 e M ,
M ˙ = ( μ e n 0 e + β D e K 2 n 0 ) M + μ e n 0 e h ,
n 0 = s D γ A .
h ¨ + ( Γ e + Γ h ) h ˙ + ( Γ e Γ h A e A h ) h = 0 ,
Γ e = μ e n 0 e + β D e K 2 n 0 ,
Γ h = μ h H 0 e + D h K 2 ,
A e = μ e n 0 e ,
A h = μ h H 0 e .
h ( t ) = A h M 0 α 2 α 1 [ exp ( α 1 t ) exp ( α 2 t ) ] ,
M ( t ) = M 0 α 2 α 1 [ ( α 2 Γ e ) exp ( α 1 t ) ( α 1 Γ e ) exp ( α 2 t ) ] ,
α 1 , 2 = Γ h + Γ e ± [ ( Γ h Γ e ) 2 + 4 A e A h ] 1 / 2 2 .
t s = 1 α 2 α 1 ln α 2 α 1 .
D = D 0 + M s cos ( K x + ϕ ) = D 0 + M 1 cos K x + M 2 sin K x ,
A = A 0 M s cos ( K x + ϕ ) = A 0 M 1 cos K x M 2 sin K x ,
M s = ( M 1 2 + M 2 2 ) 1 / 2
tan ϕ = M 2 M 1 .
H = H 0 + h s cos ( K x + ϕ ) = H 0 + h 1 cos K x + h 2 sin K x .
n = n 0 + ( n 0 β M 1 + n 0 L m ) cos K x + n 0 β M 2 sin K x ,
n = n 0 D + n 0 L ,
n 0 D = s D γ A ,
n 0 L = σ I D γ A .
Γ e M 1 + A e h 1 σ L ph I 0 K M 2 D e K 2 n 0 L m = 0 ,
Γ e M 2 + A e h 2 + σ L ph I 0 K M 1 + σ L ph K D 0 m = 0 ,
Γ h h 1 + A h M 1 = 0 ,
Γ h h 2 + A h M 2 = 0 ,
h s = 1 1 + ( K B T / e 2 ) ( K 2 / H 0 ) M s .
M s = m β { [ ( K B T e K ) 2 + ( L ph γ μ A 0 ) 2 ] / [ ( s + σ I σ I ) 2 × ( e K β { 1 1 1 + [ K B T K 2 / ( e H 0 ) ] } + K B T e K ) 2 + ( L ph γ μ A 0 β D 0 ) 2 ] } 1 / 2 ,
M s = m β { ( E D 2 + E ph 2 ) / [ ( s + σ I σ I ) 2 × ( E q { 1 1 1 + [ K B T K 2 / ( e H 0 ) ] } + E D ) 2 + ( E ph β D 0 ) 2 ] } 1 / 2 .
D = D 0 + M s cos ( K x + ϕ s ) ,
A = A 0 M s cos ( K x + ϕ s ) ,
H = H 0 + h s cos ( K x + ϕ s ) ,
ρ s = ( h s M s ) e cos ( K x + ϕ s ) .
n ( t ) = n 0 + n 0 M ( t ) β cos [ K x + ϕ ( t ) ] .
M ( t ) = M D + ( M s M D ) exp ( Γ e t ) ,
M D = h s 1 1 + ( β K B T K 2 / e 2 ) ,
ρ D = h s 1 1 + [ e 2 / ( β K B T K 2 ) ] .
η F = Δ n D Δ n s = ρ D ρ 0 ,

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