Abstract

The response of photorefractive materials to sinusoidal intensity patterns of high modulation depths is considered. It is shown that the induced refractive-index response in this regime is highly nonlinear. Concise analytic expressions for the various nonlinear harmonics of the photorefractive response field are developed and compared with exact numerical solutions of the underlying charge-transport equations as well as with the results of previous theoretical models over a broad range of physical parameters. We show that the nonlinear response characteristics are strongly dependent on both the magnitude of applied electric fields and the relative concentrations of charge donor and acceptor sites in the material. For drift dominated recording, in particular, we determine analytically that in the limit of a minimal acceptor/donor ratio, the amplitudes of higher spatial harmonics of the response reach a maximum and eventually decay as functions of increasing applied field, whereas these amplitudes reach a nonzero limit in the case of a near-unity acceptor/donor ratio. Finally, we generalize our results to account for diffusion effects in an appended derivation.

© 1988 Optical Society of America

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  1. D. M. Kim, R. R. Shah, T. A. Rabson, F. K. Tittel, “Nonlinear dynamic theory for photorefractive phase hologram formation,” Appl. Phys. Lett. 28, 338–340 (1976).
    [CrossRef]
  2. G. A. Alphonse, R. C. Alig, D. L. Staebler, W. Phillips, “Time-dependent characteristics of photo-induced space-charge field and phase holograms in lithium niobate and other photorefractive media,” RCA Rev. 36, 213–228 (1975).
  3. J. Feinberg, D. Heiman, A. R. Tanguay, R. W. Hell-warth, “Photorefractive effects and light-induced charge migration in barium titanate,” J. Appl. Phys. 51, 1297–1305 (1980).
    [CrossRef]
  4. N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetskii, “Holographic storage in electrooptic crystals. I. Steady state,” Ferroelectrics 22,949–960 (1979).
    [CrossRef]
  5. M. Peltier, F. Micheron, “Volume hologram recording and charge transfer process in Bi12SiO20 and Bi12GeO20,” J. Appl. Phys. 48, 3683–3690 (1977).
    [CrossRef]
  6. M. G. Moharam, T. K. Gaylord, R. Magnusson, L. Young, “Holographic grating formation in photorefractive crystals with arbitrary electron transport lengths,” J. Appl. Phys. 50, 5642–5651 (1979).
    [CrossRef]
  7. T. J. Hall, R. Jaura, L. M. Connors, P. D. Foote, “The photorefractive effect—a review,” Prog. Quantum Electron. 10, 77–146 (1985).
    [CrossRef]
  8. M. P. Petrov, S. V. Miridonov, S. I. Stepanov, V. V. Kulikov, “Light diffraction and non-linear image processing in electro-optic Bi12SiO20crystals,” Opt. Commun. 31, 301–305 (1979).
    [CrossRef]
  9. T. Y. Chang, P. Yeh, “Observation of harmonic phase conjugation in a photorefractive medium,” J. Opt. Soc. Am. A 3(13), P33 (1986).
  10. J. P. Huignard, B. Ledu, “Collinear Bragg diffraction in photorefractive Bi12SiO20,” Opt. Lett. 7, 310–312 (1982).
    [CrossRef] [PubMed]
  11. E. Ochoa, F. Vachss, L. Hesselink, “Higher-order analysis of the photorefrctive effect for large modulation depths,” J. Opt. Soc. Am. A 3, 181–187 (1986).
    [CrossRef]
  12. P. Refregier, L. Solymar, H. Rajbenbach, J. P. Huignard, “Two-beam coupling in photorefractive Bi12SiO20crystals with moving grating: theory and experiments,” J. Appl. Phys. 58, 45–47 (1985).
    [CrossRef]
  13. G. C. Valley, “Simultaneous electron/hole transport in photorefractive materials,” J. Appl. Phys. 59, 3363–3366 (1986).
    [CrossRef]
  14. F. P. Strohkendl, J. M. C. Jonathan, R. W. Hellwarth, “Hole–electron competition in photorefractive gratings,” Opt. Lett. 11, 312–314 (1986).
    [CrossRef] [PubMed]
  15. M. B. Klein, G. C. Valley, “Beam coupling in BaTiO3at 442 nm,” J. Appl. Phys. 57, 4901–4905 (1985).
    [CrossRef]
  16. R. A. Mullen, R. W. Hellwarth, “Optical measurement of the photorefractive parameters of Bi12SiO20,” J. Appl. Phys. 58, 40–44 (1985).
    [CrossRef]
  17. P. Gunter, “Holography, coherent light amplification and optical phase conjugation with photorefractive materials,” doctoral dissertation (Swiss Federal Institute of Technology, Zurich, Switzerland, 1981).
  18. G. C. Valley, “Erase rates in photorefractive materials with two photoactive species,” Appl. Opt. 22, 3160–3164 (1983).
    [CrossRef] [PubMed]
  19. E. Ochoa, “Real-time intensity inversion using four-wave mixing in photorefractive crystals,” doctoral dissertation (Stanford University, Stanford, Calif., 1985).
  20. G. C. Valley, M. B. Klein, “Optical properties of photorefractive materials for optical data processing,” Opt. Eng. 22, 704–711 (1983).
    [CrossRef]
  21. G. C. Valley, “Competition between forward- and backward-stimulated photorefractive scattering in BaTiO3,” J. Opt. Soc. Am: B 4, 14–19 (1987).
    [CrossRef]
  22. M. Carrascosa, F. Agullo-Lopez, “Kinetics for optical erasure of sinusoidal holographic gratings in photorefractive materials,” IEEE J. Quantum Electron. QE-22, 1369–1375 (1986).Note that although the charge-transport equations given in this reference have somewhat different nomenclature, they are formally equivalent to those given by Kukhtarev et al.4
    [CrossRef]
  23. M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (Dover, New York, 1965), p. 256.
  24. P. M. Woodward, A. M. Woodward, R. Hensman, H. H. Davies, N. Gamble, “Four-figure tables of the airy functions in the complex plane,” Philos. Mag. 37, 236–261 (1946).Data are given only in the region 0 < Re(z) < 2.54, −2.54 < Im(z) < 2.54. In this region the magnitude of the difference between our approximate form and Ai(z) is less than 0.05 |Ai(z)|.

1987 (1)

G. C. Valley, “Competition between forward- and backward-stimulated photorefractive scattering in BaTiO3,” J. Opt. Soc. Am: B 4, 14–19 (1987).
[CrossRef]

1986 (5)

M. Carrascosa, F. Agullo-Lopez, “Kinetics for optical erasure of sinusoidal holographic gratings in photorefractive materials,” IEEE J. Quantum Electron. QE-22, 1369–1375 (1986).Note that although the charge-transport equations given in this reference have somewhat different nomenclature, they are formally equivalent to those given by Kukhtarev et al.4
[CrossRef]

E. Ochoa, F. Vachss, L. Hesselink, “Higher-order analysis of the photorefrctive effect for large modulation depths,” J. Opt. Soc. Am. A 3, 181–187 (1986).
[CrossRef]

G. C. Valley, “Simultaneous electron/hole transport in photorefractive materials,” J. Appl. Phys. 59, 3363–3366 (1986).
[CrossRef]

T. Y. Chang, P. Yeh, “Observation of harmonic phase conjugation in a photorefractive medium,” J. Opt. Soc. Am. A 3(13), P33 (1986).

F. P. Strohkendl, J. M. C. Jonathan, R. W. Hellwarth, “Hole–electron competition in photorefractive gratings,” Opt. Lett. 11, 312–314 (1986).
[CrossRef] [PubMed]

1985 (4)

P. Refregier, L. Solymar, H. Rajbenbach, J. P. Huignard, “Two-beam coupling in photorefractive Bi12SiO20crystals with moving grating: theory and experiments,” J. Appl. Phys. 58, 45–47 (1985).
[CrossRef]

T. J. Hall, R. Jaura, L. M. Connors, P. D. Foote, “The photorefractive effect—a review,” Prog. Quantum Electron. 10, 77–146 (1985).
[CrossRef]

M. B. Klein, G. C. Valley, “Beam coupling in BaTiO3at 442 nm,” J. Appl. Phys. 57, 4901–4905 (1985).
[CrossRef]

R. A. Mullen, R. W. Hellwarth, “Optical measurement of the photorefractive parameters of Bi12SiO20,” J. Appl. Phys. 58, 40–44 (1985).
[CrossRef]

1983 (2)

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

G. C. Valley, “Erase rates in photorefractive materials with two photoactive species,” Appl. Opt. 22, 3160–3164 (1983).
[CrossRef] [PubMed]

1982 (1)

1980 (1)

J. Feinberg, D. Heiman, A. R. Tanguay, R. W. Hell-warth, “Photorefractive effects and light-induced charge migration in barium titanate,” J. Appl. Phys. 51, 1297–1305 (1980).
[CrossRef]

1979 (3)

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

M. P. Petrov, S. V. Miridonov, S. I. Stepanov, V. V. Kulikov, “Light diffraction and non-linear image processing in electro-optic Bi12SiO20crystals,” Opt. Commun. 31, 301–305 (1979).
[CrossRef]

M. G. Moharam, T. K. Gaylord, R. Magnusson, L. Young, “Holographic grating formation in photorefractive crystals with arbitrary electron transport lengths,” J. Appl. Phys. 50, 5642–5651 (1979).
[CrossRef]

1977 (1)

M. Peltier, F. Micheron, “Volume hologram recording and charge transfer process in Bi12SiO20 and Bi12GeO20,” J. Appl. Phys. 48, 3683–3690 (1977).
[CrossRef]

1976 (1)

D. M. Kim, R. R. Shah, T. A. Rabson, F. K. Tittel, “Nonlinear dynamic theory for photorefractive phase hologram formation,” Appl. Phys. Lett. 28, 338–340 (1976).
[CrossRef]

1975 (1)

G. A. Alphonse, R. C. Alig, D. L. Staebler, W. Phillips, “Time-dependent characteristics of photo-induced space-charge field and phase holograms in lithium niobate and other photorefractive media,” RCA Rev. 36, 213–228 (1975).

1946 (1)

P. M. Woodward, A. M. Woodward, R. Hensman, H. H. Davies, N. Gamble, “Four-figure tables of the airy functions in the complex plane,” Philos. Mag. 37, 236–261 (1946).Data are given only in the region 0 < Re(z) < 2.54, −2.54 < Im(z) < 2.54. In this region the magnitude of the difference between our approximate form and Ai(z) is less than 0.05 |Ai(z)|.

Agullo-Lopez, F.

M. Carrascosa, F. Agullo-Lopez, “Kinetics for optical erasure of sinusoidal holographic gratings in photorefractive materials,” IEEE J. Quantum Electron. QE-22, 1369–1375 (1986).Note that although the charge-transport equations given in this reference have somewhat different nomenclature, they are formally equivalent to those given by Kukhtarev et al.4
[CrossRef]

Alig, R. C.

G. A. Alphonse, R. C. Alig, D. L. Staebler, W. Phillips, “Time-dependent characteristics of photo-induced space-charge field and phase holograms in lithium niobate and other photorefractive media,” RCA Rev. 36, 213–228 (1975).

Alphonse, G. A.

G. A. Alphonse, R. C. Alig, D. L. Staebler, W. Phillips, “Time-dependent characteristics of photo-induced space-charge field and phase holograms in lithium niobate and other photorefractive media,” RCA Rev. 36, 213–228 (1975).

Carrascosa, M.

M. Carrascosa, F. Agullo-Lopez, “Kinetics for optical erasure of sinusoidal holographic gratings in photorefractive materials,” IEEE J. Quantum Electron. QE-22, 1369–1375 (1986).Note that although the charge-transport equations given in this reference have somewhat different nomenclature, they are formally equivalent to those given by Kukhtarev et al.4
[CrossRef]

Chang, T. Y.

T. Y. Chang, P. Yeh, “Observation of harmonic phase conjugation in a photorefractive medium,” J. Opt. Soc. Am. A 3(13), P33 (1986).

Connors, L. M.

T. J. Hall, R. Jaura, L. M. Connors, P. D. Foote, “The photorefractive effect—a review,” Prog. Quantum Electron. 10, 77–146 (1985).
[CrossRef]

Davies, H. H.

P. M. Woodward, A. M. Woodward, R. Hensman, H. H. Davies, N. Gamble, “Four-figure tables of the airy functions in the complex plane,” Philos. Mag. 37, 236–261 (1946).Data are given only in the region 0 < Re(z) < 2.54, −2.54 < Im(z) < 2.54. In this region the magnitude of the difference between our approximate form and Ai(z) is less than 0.05 |Ai(z)|.

Feinberg, J.

J. Feinberg, D. Heiman, A. R. Tanguay, R. W. Hell-warth, “Photorefractive effects and light-induced charge migration in barium titanate,” J. Appl. Phys. 51, 1297–1305 (1980).
[CrossRef]

Foote, P. D.

T. J. Hall, R. Jaura, L. M. Connors, P. D. Foote, “The photorefractive effect—a review,” Prog. Quantum Electron. 10, 77–146 (1985).
[CrossRef]

Gamble, N.

P. M. Woodward, A. M. Woodward, R. Hensman, H. H. Davies, N. Gamble, “Four-figure tables of the airy functions in the complex plane,” Philos. Mag. 37, 236–261 (1946).Data are given only in the region 0 < Re(z) < 2.54, −2.54 < Im(z) < 2.54. In this region the magnitude of the difference between our approximate form and Ai(z) is less than 0.05 |Ai(z)|.

Gaylord, T. K.

M. G. Moharam, T. K. Gaylord, R. Magnusson, L. Young, “Holographic grating formation in photorefractive crystals with arbitrary electron transport lengths,” J. Appl. Phys. 50, 5642–5651 (1979).
[CrossRef]

Gunter, P.

P. Gunter, “Holography, coherent light amplification and optical phase conjugation with photorefractive materials,” doctoral dissertation (Swiss Federal Institute of Technology, Zurich, Switzerland, 1981).

Hall, T. J.

T. J. Hall, R. Jaura, L. M. Connors, P. D. Foote, “The photorefractive effect—a review,” Prog. Quantum Electron. 10, 77–146 (1985).
[CrossRef]

Heiman, D.

J. Feinberg, D. Heiman, A. R. Tanguay, R. W. Hell-warth, “Photorefractive effects and light-induced charge migration in barium titanate,” J. Appl. Phys. 51, 1297–1305 (1980).
[CrossRef]

Hellwarth, R. W.

F. P. Strohkendl, J. M. C. Jonathan, R. W. Hellwarth, “Hole–electron competition in photorefractive gratings,” Opt. Lett. 11, 312–314 (1986).
[CrossRef] [PubMed]

R. A. Mullen, R. W. Hellwarth, “Optical measurement of the photorefractive parameters of Bi12SiO20,” J. Appl. Phys. 58, 40–44 (1985).
[CrossRef]

Hell-warth, R. W.

J. Feinberg, D. Heiman, A. R. Tanguay, R. W. Hell-warth, “Photorefractive effects and light-induced charge migration in barium titanate,” J. Appl. Phys. 51, 1297–1305 (1980).
[CrossRef]

Hensman, R.

P. M. Woodward, A. M. Woodward, R. Hensman, H. H. Davies, N. Gamble, “Four-figure tables of the airy functions in the complex plane,” Philos. Mag. 37, 236–261 (1946).Data are given only in the region 0 < Re(z) < 2.54, −2.54 < Im(z) < 2.54. In this region the magnitude of the difference between our approximate form and Ai(z) is less than 0.05 |Ai(z)|.

Hesselink, L.

Huignard, J. P.

P. Refregier, L. Solymar, H. Rajbenbach, J. P. Huignard, “Two-beam coupling in photorefractive Bi12SiO20crystals with moving grating: theory and experiments,” J. Appl. Phys. 58, 45–47 (1985).
[CrossRef]

J. P. Huignard, B. Ledu, “Collinear Bragg diffraction in photorefractive Bi12SiO20,” Opt. Lett. 7, 310–312 (1982).
[CrossRef] [PubMed]

Jaura, R.

T. J. Hall, R. Jaura, L. M. Connors, P. D. Foote, “The photorefractive effect—a review,” Prog. Quantum Electron. 10, 77–146 (1985).
[CrossRef]

Jonathan, J. M. C.

Kim, D. M.

D. M. Kim, R. R. Shah, T. A. Rabson, F. K. Tittel, “Nonlinear dynamic theory for photorefractive phase hologram formation,” Appl. Phys. Lett. 28, 338–340 (1976).
[CrossRef]

Klein, M. B.

M. B. Klein, G. C. Valley, “Beam coupling in BaTiO3at 442 nm,” J. Appl. Phys. 57, 4901–4905 (1985).
[CrossRef]

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

Kukhtarev, N. V.

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

Kulikov, V. V.

M. P. Petrov, S. V. Miridonov, S. I. Stepanov, V. V. Kulikov, “Light diffraction and non-linear image processing in electro-optic Bi12SiO20crystals,” Opt. Commun. 31, 301–305 (1979).
[CrossRef]

Ledu, B.

Magnusson, R.

M. G. Moharam, T. K. Gaylord, R. Magnusson, L. Young, “Holographic grating formation in photorefractive crystals with arbitrary electron transport lengths,” J. Appl. Phys. 50, 5642–5651 (1979).
[CrossRef]

Markov, V. B.

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

Micheron, F.

M. Peltier, F. Micheron, “Volume hologram recording and charge transfer process in Bi12SiO20 and Bi12GeO20,” J. Appl. Phys. 48, 3683–3690 (1977).
[CrossRef]

Miridonov, S. V.

M. P. Petrov, S. V. Miridonov, S. I. Stepanov, V. V. Kulikov, “Light diffraction and non-linear image processing in electro-optic Bi12SiO20crystals,” Opt. Commun. 31, 301–305 (1979).
[CrossRef]

Moharam, M. G.

M. G. Moharam, T. K. Gaylord, R. Magnusson, L. Young, “Holographic grating formation in photorefractive crystals with arbitrary electron transport lengths,” J. Appl. Phys. 50, 5642–5651 (1979).
[CrossRef]

Mullen, R. A.

R. A. Mullen, R. W. Hellwarth, “Optical measurement of the photorefractive parameters of Bi12SiO20,” J. Appl. Phys. 58, 40–44 (1985).
[CrossRef]

Ochoa, E.

E. Ochoa, F. Vachss, L. Hesselink, “Higher-order analysis of the photorefrctive effect for large modulation depths,” J. Opt. Soc. Am. A 3, 181–187 (1986).
[CrossRef]

E. Ochoa, “Real-time intensity inversion using four-wave mixing in photorefractive crystals,” doctoral dissertation (Stanford University, Stanford, Calif., 1985).

Odulov, S. G.

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

Peltier, M.

M. Peltier, F. Micheron, “Volume hologram recording and charge transfer process in Bi12SiO20 and Bi12GeO20,” J. Appl. Phys. 48, 3683–3690 (1977).
[CrossRef]

Petrov, M. P.

M. P. Petrov, S. V. Miridonov, S. I. Stepanov, V. V. Kulikov, “Light diffraction and non-linear image processing in electro-optic Bi12SiO20crystals,” Opt. Commun. 31, 301–305 (1979).
[CrossRef]

Phillips, W.

G. A. Alphonse, R. C. Alig, D. L. Staebler, W. Phillips, “Time-dependent characteristics of photo-induced space-charge field and phase holograms in lithium niobate and other photorefractive media,” RCA Rev. 36, 213–228 (1975).

Rabson, T. A.

D. M. Kim, R. R. Shah, T. A. Rabson, F. K. Tittel, “Nonlinear dynamic theory for photorefractive phase hologram formation,” Appl. Phys. Lett. 28, 338–340 (1976).
[CrossRef]

Rajbenbach, H.

P. Refregier, L. Solymar, H. Rajbenbach, J. P. Huignard, “Two-beam coupling in photorefractive Bi12SiO20crystals with moving grating: theory and experiments,” J. Appl. Phys. 58, 45–47 (1985).
[CrossRef]

Refregier, P.

P. Refregier, L. Solymar, H. Rajbenbach, J. P. Huignard, “Two-beam coupling in photorefractive Bi12SiO20crystals with moving grating: theory and experiments,” J. Appl. Phys. 58, 45–47 (1985).
[CrossRef]

Shah, R. R.

D. M. Kim, R. R. Shah, T. A. Rabson, F. K. Tittel, “Nonlinear dynamic theory for photorefractive phase hologram formation,” Appl. Phys. Lett. 28, 338–340 (1976).
[CrossRef]

Solymar, L.

P. Refregier, L. Solymar, H. Rajbenbach, J. P. Huignard, “Two-beam coupling in photorefractive Bi12SiO20crystals with moving grating: theory and experiments,” J. Appl. Phys. 58, 45–47 (1985).
[CrossRef]

Soskin, M. S.

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

Staebler, D. L.

G. A. Alphonse, R. C. Alig, D. L. Staebler, W. Phillips, “Time-dependent characteristics of photo-induced space-charge field and phase holograms in lithium niobate and other photorefractive media,” RCA Rev. 36, 213–228 (1975).

Stepanov, S. I.

M. P. Petrov, S. V. Miridonov, S. I. Stepanov, V. V. Kulikov, “Light diffraction and non-linear image processing in electro-optic Bi12SiO20crystals,” Opt. Commun. 31, 301–305 (1979).
[CrossRef]

Strohkendl, F. P.

Tanguay, A. R.

J. Feinberg, D. Heiman, A. R. Tanguay, R. W. Hell-warth, “Photorefractive effects and light-induced charge migration in barium titanate,” J. Appl. Phys. 51, 1297–1305 (1980).
[CrossRef]

Tittel, F. K.

D. M. Kim, R. R. Shah, T. A. Rabson, F. K. Tittel, “Nonlinear dynamic theory for photorefractive phase hologram formation,” Appl. Phys. Lett. 28, 338–340 (1976).
[CrossRef]

Vachss, F.

Valley, G. C.

G. C. Valley, “Competition between forward- and backward-stimulated photorefractive scattering in BaTiO3,” J. Opt. Soc. Am: B 4, 14–19 (1987).
[CrossRef]

G. C. Valley, “Simultaneous electron/hole transport in photorefractive materials,” J. Appl. Phys. 59, 3363–3366 (1986).
[CrossRef]

M. B. Klein, G. C. Valley, “Beam coupling in BaTiO3at 442 nm,” J. Appl. Phys. 57, 4901–4905 (1985).
[CrossRef]

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

G. C. Valley, “Erase rates in photorefractive materials with two photoactive species,” Appl. Opt. 22, 3160–3164 (1983).
[CrossRef] [PubMed]

Vinetskii, V. L.

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

Woodward, A. M.

P. M. Woodward, A. M. Woodward, R. Hensman, H. H. Davies, N. Gamble, “Four-figure tables of the airy functions in the complex plane,” Philos. Mag. 37, 236–261 (1946).Data are given only in the region 0 < Re(z) < 2.54, −2.54 < Im(z) < 2.54. In this region the magnitude of the difference between our approximate form and Ai(z) is less than 0.05 |Ai(z)|.

Woodward, P. M.

P. M. Woodward, A. M. Woodward, R. Hensman, H. H. Davies, N. Gamble, “Four-figure tables of the airy functions in the complex plane,” Philos. Mag. 37, 236–261 (1946).Data are given only in the region 0 < Re(z) < 2.54, −2.54 < Im(z) < 2.54. In this region the magnitude of the difference between our approximate form and Ai(z) is less than 0.05 |Ai(z)|.

Yeh, P.

T. Y. Chang, P. Yeh, “Observation of harmonic phase conjugation in a photorefractive medium,” J. Opt. Soc. Am. A 3(13), P33 (1986).

Young, L.

M. G. Moharam, T. K. Gaylord, R. Magnusson, L. Young, “Holographic grating formation in photorefractive crystals with arbitrary electron transport lengths,” J. Appl. Phys. 50, 5642–5651 (1979).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

D. M. Kim, R. R. Shah, T. A. Rabson, F. K. Tittel, “Nonlinear dynamic theory for photorefractive phase hologram formation,” Appl. Phys. Lett. 28, 338–340 (1976).
[CrossRef]

Ferroelectrics (1)

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

IEEE J. Quantum Electron. (1)

M. Carrascosa, F. Agullo-Lopez, “Kinetics for optical erasure of sinusoidal holographic gratings in photorefractive materials,” IEEE J. Quantum Electron. QE-22, 1369–1375 (1986).Note that although the charge-transport equations given in this reference have somewhat different nomenclature, they are formally equivalent to those given by Kukhtarev et al.4
[CrossRef]

J. Appl. Phys. (7)

M. B. Klein, G. C. Valley, “Beam coupling in BaTiO3at 442 nm,” J. Appl. Phys. 57, 4901–4905 (1985).
[CrossRef]

R. A. Mullen, R. W. Hellwarth, “Optical measurement of the photorefractive parameters of Bi12SiO20,” J. Appl. Phys. 58, 40–44 (1985).
[CrossRef]

M. Peltier, F. Micheron, “Volume hologram recording and charge transfer process in Bi12SiO20 and Bi12GeO20,” J. Appl. Phys. 48, 3683–3690 (1977).
[CrossRef]

M. G. Moharam, T. K. Gaylord, R. Magnusson, L. Young, “Holographic grating formation in photorefractive crystals with arbitrary electron transport lengths,” J. Appl. Phys. 50, 5642–5651 (1979).
[CrossRef]

J. Feinberg, D. Heiman, A. R. Tanguay, R. W. Hell-warth, “Photorefractive effects and light-induced charge migration in barium titanate,” J. Appl. Phys. 51, 1297–1305 (1980).
[CrossRef]

P. Refregier, L. Solymar, H. Rajbenbach, J. P. Huignard, “Two-beam coupling in photorefractive Bi12SiO20crystals with moving grating: theory and experiments,” J. Appl. Phys. 58, 45–47 (1985).
[CrossRef]

G. C. Valley, “Simultaneous electron/hole transport in photorefractive materials,” J. Appl. Phys. 59, 3363–3366 (1986).
[CrossRef]

J. Opt. Soc. Am. A (2)

E. Ochoa, F. Vachss, L. Hesselink, “Higher-order analysis of the photorefrctive effect for large modulation depths,” J. Opt. Soc. Am. A 3, 181–187 (1986).
[CrossRef]

T. Y. Chang, P. Yeh, “Observation of harmonic phase conjugation in a photorefractive medium,” J. Opt. Soc. Am. A 3(13), P33 (1986).

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

G. C. Valley, “Competition between forward- and backward-stimulated photorefractive scattering in BaTiO3,” J. Opt. Soc. Am: B 4, 14–19 (1987).
[CrossRef]

Opt. Commun. (1)

M. P. Petrov, S. V. Miridonov, S. I. Stepanov, V. V. Kulikov, “Light diffraction and non-linear image processing in electro-optic Bi12SiO20crystals,” Opt. Commun. 31, 301–305 (1979).
[CrossRef]

Opt. Eng. (1)

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

Opt. Lett. (2)

Philos. Mag. (1)

P. M. Woodward, A. M. Woodward, R. Hensman, H. H. Davies, N. Gamble, “Four-figure tables of the airy functions in the complex plane,” Philos. Mag. 37, 236–261 (1946).Data are given only in the region 0 < Re(z) < 2.54, −2.54 < Im(z) < 2.54. In this region the magnitude of the difference between our approximate form and Ai(z) is less than 0.05 |Ai(z)|.

Prog. Quantum Electron. (1)

T. J. Hall, R. Jaura, L. M. Connors, P. D. Foote, “The photorefractive effect—a review,” Prog. Quantum Electron. 10, 77–146 (1985).
[CrossRef]

RCA Rev. (1)

G. A. Alphonse, R. C. Alig, D. L. Staebler, W. Phillips, “Time-dependent characteristics of photo-induced space-charge field and phase holograms in lithium niobate and other photorefractive media,” RCA Rev. 36, 213–228 (1975).

Other (3)

M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (Dover, New York, 1965), p. 256.

P. Gunter, “Holography, coherent light amplification and optical phase conjugation with photorefractive materials,” doctoral dissertation (Swiss Federal Institute of Technology, Zurich, Switzerland, 1981).

E. Ochoa, “Real-time intensity inversion using four-wave mixing in photorefractive crystals,” doctoral dissertation (Stanford University, Stanford, Calif., 1985).

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Figures (7)

Fig. 1
Fig. 1

The incident sinusoidal intensity pattern I(x) and the resultant response field E(x) are shown for the cases of (a) weak modulation (low m) and (b) strong modulation (high m) and the resulting nonlinear response. Below E(x) the total charge density ρ(x) is shown, along with the charge densities in the donor band ND and the acceptor band NA. Occupied neutral donor sites are denoted by (+−), and positively charged vacant donor sites are denoted by (+).

Fig. 2
Fig. 2

Approximate moduli of the first four harmonics e1 (top) through e4 (bottom) derived from relation (35), versus the applied field for 0 ≤ eA ≤ 1.5.

Fig. 3
Fig. 3

(a) Numerical solutions to Eq. (16) (solid lines) plotted with the approximate response from relation (35) (dashed lines): (a) for m = 1.0 and eA = 0.5 (bottom), eA = 0.8 (center), and eA = 1.3 (top) and for −πu ≤ 2π; (b) for m = 0.8 and eA = 0.5 (bottom), eA = 0.8 (center), and eA = 1.3 (top) for −πu ≤ 2π.

Fig. 4
Fig. 4

Exact solution to Eq. (41) for m = 1.0 and eA = 1.0, π, 5.0. Note that the amplitude of the cycles remains constant for eAπ, with only the spatial average e0 increasing.

Fig. 5
Fig. 5

Moduli of the first four harmonics e1 (top) through e4 (bottom) derived from relation (47) versus the applied field for 0 ≤ eA ≤ 4.0.

Fig. 6
Fig. 6

Numerical solutions to relation (41) (solid lines) plotted with the approximate response from relation (47) (dashed lines) for m = 0.8 and eA = 0.25 (bottom), eA = 0.5 (center), and eA = 1.0 (top) for −πu ≤ 2π.

Fig. 7
Fig. 7

Exact numerical solutions to Eq. (12) plotted above the approximate response from relation (A5) and the low-E results of Eq. (38a) for (a) m = 0.9 and (b) m = 0.95 at applied field strengths of eA = 0.0,0.1,0.2,0.5,1.0 for −πu ≤ 2π. Note that the vertical scale is 2 units in the top two sets of plots in (a) and (b), whereas the scales of the bottom plots are 6 and 8 units, respectively.

Equations (58)

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I ( r ) = I 1 + I 2 + 2 ( I 1 I 2 ) 1 / 2 cos ( k G x ) I 0 [ 1 + m cos ( k G x ) ] ,
m = 2 r 1 + r 2 .
[ s I ( x ) + β ] ( N D N D + ) = γ R N D + n ,
J = μ E ê n + μ k T d n d x = const . ,
d E d x = 4 π ê ( N D + n N A ) ,
E I ( u ) ( N D N D + ) N D + + E D d d u [ I ( u ) ( N D N D + ) N D + ] = const . ,
d E d u = 4 π ê k G ( N D + N A ) ,
E D k B T e k G
E q 4 π ê N A ( N D N A ) k G N D ,
e I ( u ) ( 1 d ) d + e D d d u [ I ( u ) ( 1 d ) d ] = const .
R [ 1 + ( 1 R ) d e d u ] = d ,
e I ( u ) ( 1 R d e d u ) 1 + ( 1 R ) d e d u + e D d d u [ I ( u ) ( 1 R d e d u ) 1 + ( 1 R ) d e d u ] = const .
0 L E ( x ) d x = V E A L ,
1 k G L 0 k G L e ( u ) d u = E A E q ,
1 2 π 0 2 π e ( u ) d u = E A E q e A
1 1 R d e d u 1 R .
d e d u + 1 = C e [ 1 + m cos ( u ) ]
e ( u ) = exp { C [ u + m sin ( u ) ] } u exp { C [ t + m sin ( t ) ] } d t ,
e ( u ) = n = e n exp ( inu ) ,
e n = 1 2 π 0 2 π e ( u ) exp ( inu ) d u .
e n = 1 2 π 0 2 π d u exp ( inu ) 0 d ν exp { C [ υ + 2 m sin ( ν 2 ) × cos ( x + ν 2 ) ] } = 0 d ν exp [ ( C i n 2 ) ν ] I n [ 2 m C sin ( ν 2 ) ] ,
e n = 2 π ( i ) n exp ( C π ) 1 exp ( 2 C π ) I i C ( m C ) I n + i C ( m C ) for n 0 ,
e A = π sinh ( C π ) | I i C ( m C ) | 2 ,
I i C ( z ) = [ I i C ( z ) ] * ,
e n = ( i ) n e A I n + i C ( m C ) I i C ( m C ) .
I i C ( m C ) = { ( m C 2 ) i C 1 Γ ( 1 + i C ) [ 1 + O ( m 2 C 2 ) ] , C 1 ( 2 i C ) 1 / 3 exp ( C π / 2 ) [ 1 + δ 5 O ( δ 2 ) ] { Ai [ 2 1 / 3 δ ( i C ) 2 / 3 ] + O 1 C } , C 1 ,
e A { 1 C , C 1 2 π ( 2 C ) 2 / 3 ( 1 + 0.4 δ ) | Ai [ 2 1 / 3 δ ( i C ) 2 / 3 ] | 2 , C 1 ,
Γ ( 1 + i C ) Γ ( 1 i C ) = π C sinh ( π C )
Ai ( z ) 0.28 ( 0.34 + z ) 1 / 4 exp [ ( 2 z 3 / 2 / 3 ) ] .
e A { 1 C , C 1 1.26 ( 1 + 0.4 δ ) C 2 / 3 [ 1 + 3.7 δ C 2 / 3 + ( 3.7 δ C 2 / 3 ) 2 ] 1 / 4 , C 1 .
( C 2 / 3 ) 4 [ 1 + 3.7 δ C 2 / 3 + ( 3.7 δ C 2 / 3 ) 2 ] = [ 1.26 ( 1 + 0.4 δ ) e A ] 4 .
C 1.41 ( 1 + 0.6 δ ) e A ( e A + 4.67 δ ) 1 / 2
I ν 1 ( z ) I ν + 1 ( z ) = 2 ν z I ν ( z )
b n + 1 2 b n + b n 1 = 2 i m C ( n + i δ C ) b n ,
d 2 b d n 2 = 2 i m C ( n + i δ C ) b ( n ) ,
b n b ( n ) = c 1 Ai [ ( 2 i m C ) 1 / 3 ( n + i δ C ) ] + c 2 Bi [ ( 2 i m C ) 1 / 3 ( n + i δ C ) ] ,
b n e A Ai [ ( 2 i m C ) 1 / 3 ( n + i δ C ) ] Ai [ ( 2 i m C ) 1 / 3 i δ C ] ,
e n ( 1 ) n e A [ 0.34 + ( 2 i m C ) 1 / 3 i δ C 0.34 + ( 2 i m C ) 1 / 3 ( n + i δ C ) ] 1 / 4 × exp { 2 3 ( 2 i m C ) 1 / 2 [ ( i δ C ) 3 / 2 ( n + i δ C ) 3 / 2 ] } ,
e A exp [ ( 1 i ) n 3 / 2 e A 3 / 4 / m 1 / 2 ] ,
e n ( 1 ) n e A { exp [ ( 3 / 2 ) ( 2 δ ) 1 / 2 ] } n
e n = ( 1 ) n ( e A 2 + e D 2 ) 1 / 2 r n
= ( 1 ) n e A r n in the diffusion - free case ,
e n = ( i ) n e A ( m C 2 ) n 1 n ! [ 1 + O ( C ) ] ( i ) n e A ( m 2 e A ) n 1 n ! for e A 1 ,
e A MAX 1.47 m 2 / 3 n 2
e ( 1 d e d u ) = C 1 + m cos ( u ) .
0 2 π e ( u ) d u = C 0 2 π d u 1 + m cos ( u ) = 2 π C ( 1 m 2 ) 1 / 2 ,
C = e A ( 1 m 2 ) 1 / 2 .
lim m 1 C 1 + m cos ( u ) = e A lim m 1 ( 1 m 2 ) 1 / 2 1 + m cos ( u ) = 2 π e A δ ( u π ) for 0 u 2 π ,
x 0 ( 4 π e A ) 1 / 2 .
e n = { ( 1 ) n e A 2 ( n x 0 ) 2 [ 1 + i n x 0 exp ( i n x 0 ) ] , e A π ( 1 ) n i n , e A > π ,
e n ( 1 ) n r n { 2 e A ( n x 0 ) 2 [ 1 + i n x 0 exp ( i n x 0 ) ] , e A π i n , e A > π .
e A Plateau = π k 2 n 2
lim e A e n R n
e ( u ) = e D m sin ( u ) 1 + m cos ( u ) .
e ( u ) e D m sin ( u ) 1 + a cos ( u ) ,
e n ( 1 ) n i e D m a [ 1 ( 1 a 2 ) 1 / 2 a ] n for n > 0 ,
a = m e D ( 2 m 1 ) + O ( e D 2 ) .
e n ( 1 ) n { e A r n 2 e A ( n x 0 ) 2 [ 1 + i n x 0 exp ( i n x 0 ) ] + i m a e D 1 + e A / 2 e D [ 1 ( 1 a 2 ) 1 / 2 a ] n }

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