Abstract

A numerical approach has been developed for simulating the recording and erasure kinetics in photorefractive materials for high modulation of the light-intensity pattern. The recording kinetics for the fundamental grating strongly deviates from exponential behavior, and it is much slower than the kinetics inferred from a linearized solution. The harmonics of the grating show lower growth rates for higher Fourier orders and exhibit an initial delay time. On the other hand, in the optical erasure experiment (starting from the steady-state solution), the decay rate of the fundamental harmonic is exponential and closely matches the predictions of the linear approximation. Moreover, the decay rate of the harmonics are in good agreement with the value corresponding to a fundamental grating with the same k vector. This uncoupling of the Fourier components is no longer intact when erasure is started from a nonsaturated grating. The effect of the grating vector of the light pattern and an externally applied field on the growth and the erasure kinetics have also been investigated. The applied field modifies the kinetic behavior and introduces some differential features with regard to the linear solution.

© 1994 Optical Society of America

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References

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  1. N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. Soskin, and V. I. Vinetskii, “Holographic storage in electrooptic crystals 1. Steady state,” Ferroelectrics 22, 949–961 (1979).
    [Crossref]
  2. E. Ochoa, F. Vachss, and L. Hesselink, “Higher-order analysis of the photorefractive effect for large modulation depths,” J. Opt. Soc. Am. A 3, 181–187 (1986).
    [Crossref]
  3. F. Vachss and L. Hesselink, “Nonlinear photorefractive response at high modulation depth,” J. Opt. Soc. Am. A 5, 690–701 (1988).
    [Crossref]
  4. R. Saxena and T. Y. Chang, “Perturbative analyses of higher-order photorefractive gratings,” J. Opt. Soc. Am. B 9, 1467–1472 (1992).
    [Crossref]
  5. L. B. Au and L. Solymar, “Space-charge field in photorefractive materials at large modulation,” Opt. Lett. 13, 660–662 (1988).
    [Crossref] [PubMed]
  6. L. B. Au and L. Solymar, “Highly harmonic gratings in photorefractive materials at large modulation with moving fringes,” J. Opt. Soc. Am. A 7, 1554–1561 (1990).
    [Crossref]
  7. Y. H. Lee and R. W. Hellwarth, “Spatial harmonics of photorefractive gratings in a barium titanate crystal,” J. Appl. Phys. 71, 916–923 (1992).
    [Crossref]
  8. A. Bledowski, J. Otten, and K. H. Ringhofer, “Photorefractive hologram writing with modulation I,” Opt. Lett. 16, 672–674 (1991).
    [Crossref] [PubMed]
  9. E. Serrano, V. López, M. Carrascosa, and F. Agullo-López, “Steady state photorefractive gratings in LiNbO3for strong light modulation depths,” submitted to IEEE J. Quantum Electron.
  10. G. A. Alphonse, R. C. Alig, D. L. Stabler, and W. Philips, “Time dependent characteristics of photoinduced space-charge field and phase holograms in lithium niobate and other photorefractive media,” RCA Rev. 36, 213–229 (1975).
  11. M. G. Moharam, T. K. Gaylord, and R. Magnusson, “Holographic grating formation in photorefractive crystals with arbitrary electron transport lengths,” J. Appl. Phys. 50, 5642–5651 (1979).
    [Crossref]
  12. D. A. Temple and C. Warde, “High-order anisotropic diffraction in photorefractive crystals,” J. Opt. Soc. Am. B 5, 1800–1806 (1988).
    [Crossref]
  13. G. A. Brost, “Photorefractive grating formations at large modulation with alternating electric fields,” J. Opt. Soc. Am. B 9, 1454–1460 (1992).
    [Crossref]
  14. G. A. Brost, “Numerical analysis of photorefractive grating formation dynamics at large modulation in BSO,” Opt. Commun. 96, 113–116 (1993).
    [Crossref]
  15. M. Horowitz, R. Daisy, and B. Fischer, “Signal-to-pump ratio dependence of buildup and decay rates in photorefractive nonlinear two-beam coupling,” J. Opt. Soc. Am. B 9, 1685–1688 (1992).
    [Crossref]
  16. T. Hirao and T. Sawada, “On the dependence of photorefractive response time on index grating spacing,” Opt. Commun. 82, 83–88 (1988).
    [Crossref]
  17. A. Marrakchi, J.-P. Huignard, and P. Günter, “Diffraction efficiency and energy transfer in two-wave mixing experiments with Bi12SiO20crystals,” Appl. Phys. 24, 131–138 (1981).
    [Crossref]
  18. R. A. Mullen and R. W. Hellwarth, “Optical measurement of the photorefractive parameters of Bi12SiO20,” J. Appl. Phys. 58, 40–44 (1985).
    [Crossref]
  19. S. I. Stepanov, V. V. Kulikov, and M. P. Petrov, “Running holograms in photorefractive Bi12SiO20crystals,” Opt. Commun. 44, 19–23 (1982).
    [Crossref]
  20. H. Rajbenbach, J.-P. Huignard, and B. Loiscaux, “Spatial frequency dependence of the energy transfer in two mixing experiments with BSO crystals,” Opt. Commun. 48, 247–252 (1983).
    [Crossref]
  21. C. Soutar, W. A. Gillespie, and C. M. Cartwright, “The effect of optical bias on grating formation dynamics in photorefractive BSO,” Opt. Commun. 90, 329–334 (1992).
    [Crossref]
  22. V. Fairen, V. López, and L. Conde, “Power series approximation solutions of nonlinear systems of differential equations,” Am. J. Phys. 56, 57–61 (1988).
    [Crossref]
  23. J. C. Jonathan, R. W. Hellwarth, and G. Roosen, “Effect of applied electric field on buildup and decay of photorefractive gratings,” IEEE J. Quantum Electron. QE-22, 1936–1941 (1986).
    [Crossref]
  24. N. V. Kukhtarev, “Kinetics of hologram recording and erasure in electrooptic crystals,” Sov. Tech. Phys. Lett. 2, 438–440 (1976).

1993 (1)

G. A. Brost, “Numerical analysis of photorefractive grating formation dynamics at large modulation in BSO,” Opt. Commun. 96, 113–116 (1993).
[Crossref]

1992 (5)

1991 (1)

1990 (1)

1988 (5)

1986 (2)

J. C. Jonathan, R. W. Hellwarth, and G. Roosen, “Effect of applied electric field on buildup and decay of photorefractive gratings,” IEEE J. Quantum Electron. QE-22, 1936–1941 (1986).
[Crossref]

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

1985 (1)

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

1983 (1)

H. Rajbenbach, J.-P. Huignard, and B. Loiscaux, “Spatial frequency dependence of the energy transfer in two mixing experiments with BSO crystals,” Opt. Commun. 48, 247–252 (1983).
[Crossref]

1982 (1)

S. I. Stepanov, V. V. Kulikov, and M. P. Petrov, “Running holograms in photorefractive Bi12SiO20crystals,” Opt. Commun. 44, 19–23 (1982).
[Crossref]

1981 (1)

A. Marrakchi, J.-P. Huignard, and P. Günter, “Diffraction efficiency and energy transfer in two-wave mixing experiments with Bi12SiO20crystals,” Appl. Phys. 24, 131–138 (1981).
[Crossref]

1979 (2)

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

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. Soskin, and V. I. Vinetskii, “Holographic storage in electrooptic crystals 1. Steady state,” Ferroelectrics 22, 949–961 (1979).
[Crossref]

1976 (1)

N. V. Kukhtarev, “Kinetics of hologram recording and erasure in electrooptic crystals,” Sov. Tech. Phys. Lett. 2, 438–440 (1976).

1975 (1)

G. A. Alphonse, R. C. Alig, D. L. Stabler, and W. Philips, “Time dependent characteristics of photoinduced space-charge field and phase holograms in lithium niobate and other photorefractive media,” RCA Rev. 36, 213–229 (1975).

Agullo-López, F.

E. Serrano, V. López, M. Carrascosa, and F. Agullo-López, “Steady state photorefractive gratings in LiNbO3for strong light modulation depths,” submitted to IEEE J. Quantum Electron.

Alig, R. C.

G. A. Alphonse, R. C. Alig, D. L. Stabler, and W. Philips, “Time dependent characteristics of photoinduced space-charge field and phase holograms in lithium niobate and other photorefractive media,” RCA Rev. 36, 213–229 (1975).

Alphonse, G. A.

G. A. Alphonse, R. C. Alig, D. L. Stabler, and W. Philips, “Time dependent characteristics of photoinduced space-charge field and phase holograms in lithium niobate and other photorefractive media,” RCA Rev. 36, 213–229 (1975).

Au, L. B.

Bledowski, A.

Brost, G. A.

G. A. Brost, “Numerical analysis of photorefractive grating formation dynamics at large modulation in BSO,” Opt. Commun. 96, 113–116 (1993).
[Crossref]

G. A. Brost, “Photorefractive grating formations at large modulation with alternating electric fields,” J. Opt. Soc. Am. B 9, 1454–1460 (1992).
[Crossref]

Carrascosa, M.

E. Serrano, V. López, M. Carrascosa, and F. Agullo-López, “Steady state photorefractive gratings in LiNbO3for strong light modulation depths,” submitted to IEEE J. Quantum Electron.

Cartwright, C. M.

C. Soutar, W. A. Gillespie, and C. M. Cartwright, “The effect of optical bias on grating formation dynamics in photorefractive BSO,” Opt. Commun. 90, 329–334 (1992).
[Crossref]

Chang, T. Y.

Conde, L.

V. Fairen, V. López, and L. Conde, “Power series approximation solutions of nonlinear systems of differential equations,” Am. J. Phys. 56, 57–61 (1988).
[Crossref]

Daisy, R.

Fairen, V.

V. Fairen, V. López, and L. Conde, “Power series approximation solutions of nonlinear systems of differential equations,” Am. J. Phys. 56, 57–61 (1988).
[Crossref]

Fischer, B.

Gaylord, T. K.

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

Gillespie, W. A.

C. Soutar, W. A. Gillespie, and C. M. Cartwright, “The effect of optical bias on grating formation dynamics in photorefractive BSO,” Opt. Commun. 90, 329–334 (1992).
[Crossref]

Günter, P.

A. Marrakchi, J.-P. Huignard, and P. Günter, “Diffraction efficiency and energy transfer in two-wave mixing experiments with Bi12SiO20crystals,” Appl. Phys. 24, 131–138 (1981).
[Crossref]

Hellwarth, R. W.

Y. H. Lee and R. W. Hellwarth, “Spatial harmonics of photorefractive gratings in a barium titanate crystal,” J. Appl. Phys. 71, 916–923 (1992).
[Crossref]

J. C. Jonathan, R. W. Hellwarth, and G. Roosen, “Effect of applied electric field on buildup and decay of photorefractive gratings,” IEEE J. Quantum Electron. QE-22, 1936–1941 (1986).
[Crossref]

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

Hesselink, L.

Hirao, T.

T. Hirao and T. Sawada, “On the dependence of photorefractive response time on index grating spacing,” Opt. Commun. 82, 83–88 (1988).
[Crossref]

Horowitz, M.

Huignard, J.-P.

H. Rajbenbach, J.-P. Huignard, and B. Loiscaux, “Spatial frequency dependence of the energy transfer in two mixing experiments with BSO crystals,” Opt. Commun. 48, 247–252 (1983).
[Crossref]

A. Marrakchi, J.-P. Huignard, and P. Günter, “Diffraction efficiency and energy transfer in two-wave mixing experiments with Bi12SiO20crystals,” Appl. Phys. 24, 131–138 (1981).
[Crossref]

Jonathan, J. C.

J. C. Jonathan, R. W. Hellwarth, and G. Roosen, “Effect of applied electric field on buildup and decay of photorefractive gratings,” IEEE J. Quantum Electron. QE-22, 1936–1941 (1986).
[Crossref]

Kukhtarev, N. V.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. Soskin, and V. I. Vinetskii, “Holographic storage in electrooptic crystals 1. Steady state,” Ferroelectrics 22, 949–961 (1979).
[Crossref]

N. V. Kukhtarev, “Kinetics of hologram recording and erasure in electrooptic crystals,” Sov. Tech. Phys. Lett. 2, 438–440 (1976).

Kulikov, V. V.

S. I. Stepanov, V. V. Kulikov, and M. P. Petrov, “Running holograms in photorefractive Bi12SiO20crystals,” Opt. Commun. 44, 19–23 (1982).
[Crossref]

Lee, Y. H.

Y. H. Lee and R. W. Hellwarth, “Spatial harmonics of photorefractive gratings in a barium titanate crystal,” J. Appl. Phys. 71, 916–923 (1992).
[Crossref]

Loiscaux, B.

H. Rajbenbach, J.-P. Huignard, and B. Loiscaux, “Spatial frequency dependence of the energy transfer in two mixing experiments with BSO crystals,” Opt. Commun. 48, 247–252 (1983).
[Crossref]

López, V.

V. Fairen, V. López, and L. Conde, “Power series approximation solutions of nonlinear systems of differential equations,” Am. J. Phys. 56, 57–61 (1988).
[Crossref]

E. Serrano, V. López, M. Carrascosa, and F. Agullo-López, “Steady state photorefractive gratings in LiNbO3for strong light modulation depths,” submitted to IEEE J. Quantum Electron.

Magnusson, R.

M. G. Moharam, T. K. Gaylord, and R. Magnusson, “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. Soskin, and V. I. Vinetskii, “Holographic storage in electrooptic crystals 1. Steady state,” Ferroelectrics 22, 949–961 (1979).
[Crossref]

Marrakchi, A.

A. Marrakchi, J.-P. Huignard, and P. Günter, “Diffraction efficiency and energy transfer in two-wave mixing experiments with Bi12SiO20crystals,” Appl. Phys. 24, 131–138 (1981).
[Crossref]

Moharam, M. G.

M. G. Moharam, T. K. Gaylord, and R. Magnusson, “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 and R. W. Hellwarth, “Optical measurement of the photorefractive parameters of Bi12SiO20,” J. Appl. Phys. 58, 40–44 (1985).
[Crossref]

Ochoa, E.

Odulov, S. G.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. Soskin, and V. I. Vinetskii, “Holographic storage in electrooptic crystals 1. Steady state,” Ferroelectrics 22, 949–961 (1979).
[Crossref]

Otten, J.

Petrov, M. P.

S. I. Stepanov, V. V. Kulikov, and M. P. Petrov, “Running holograms in photorefractive Bi12SiO20crystals,” Opt. Commun. 44, 19–23 (1982).
[Crossref]

Philips, W.

G. A. Alphonse, R. C. Alig, D. L. Stabler, and W. Philips, “Time dependent characteristics of photoinduced space-charge field and phase holograms in lithium niobate and other photorefractive media,” RCA Rev. 36, 213–229 (1975).

Rajbenbach, H.

H. Rajbenbach, J.-P. Huignard, and B. Loiscaux, “Spatial frequency dependence of the energy transfer in two mixing experiments with BSO crystals,” Opt. Commun. 48, 247–252 (1983).
[Crossref]

Ringhofer, K. H.

Roosen, G.

J. C. Jonathan, R. W. Hellwarth, and G. Roosen, “Effect of applied electric field on buildup and decay of photorefractive gratings,” IEEE J. Quantum Electron. QE-22, 1936–1941 (1986).
[Crossref]

Sawada, T.

T. Hirao and T. Sawada, “On the dependence of photorefractive response time on index grating spacing,” Opt. Commun. 82, 83–88 (1988).
[Crossref]

Saxena, R.

Serrano, E.

E. Serrano, V. López, M. Carrascosa, and F. Agullo-López, “Steady state photorefractive gratings in LiNbO3for strong light modulation depths,” submitted to IEEE J. Quantum Electron.

Solymar, L.

Soskin, M.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. Soskin, and V. I. Vinetskii, “Holographic storage in electrooptic crystals 1. Steady state,” Ferroelectrics 22, 949–961 (1979).
[Crossref]

Soutar, C.

C. Soutar, W. A. Gillespie, and C. M. Cartwright, “The effect of optical bias on grating formation dynamics in photorefractive BSO,” Opt. Commun. 90, 329–334 (1992).
[Crossref]

Stabler, D. L.

G. A. Alphonse, R. C. Alig, D. L. Stabler, and W. Philips, “Time dependent characteristics of photoinduced space-charge field and phase holograms in lithium niobate and other photorefractive media,” RCA Rev. 36, 213–229 (1975).

Stepanov, S. I.

S. I. Stepanov, V. V. Kulikov, and M. P. Petrov, “Running holograms in photorefractive Bi12SiO20crystals,” Opt. Commun. 44, 19–23 (1982).
[Crossref]

Temple, D. A.

Vachss, F.

Vinetskii, V. I.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. Soskin, and V. I. Vinetskii, “Holographic storage in electrooptic crystals 1. Steady state,” Ferroelectrics 22, 949–961 (1979).
[Crossref]

Warde, C.

Am. J. Phys. (1)

V. Fairen, V. López, and L. Conde, “Power series approximation solutions of nonlinear systems of differential equations,” Am. J. Phys. 56, 57–61 (1988).
[Crossref]

Appl. Phys. (1)

A. Marrakchi, J.-P. Huignard, and P. Günter, “Diffraction efficiency and energy transfer in two-wave mixing experiments with Bi12SiO20crystals,” Appl. Phys. 24, 131–138 (1981).
[Crossref]

Ferroelectrics (1)

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. Soskin, and V. I. Vinetskii, “Holographic storage in electrooptic crystals 1. Steady state,” Ferroelectrics 22, 949–961 (1979).
[Crossref]

IEEE J. Quantum Electron. (1)

J. C. Jonathan, R. W. Hellwarth, and G. Roosen, “Effect of applied electric field on buildup and decay of photorefractive gratings,” IEEE J. Quantum Electron. QE-22, 1936–1941 (1986).
[Crossref]

J. Appl. Phys. (3)

Y. H. Lee and R. W. Hellwarth, “Spatial harmonics of photorefractive gratings in a barium titanate crystal,” J. Appl. Phys. 71, 916–923 (1992).
[Crossref]

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

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

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

J. Opt. Soc. Am. B (4)

Opt. Commun. (5)

T. Hirao and T. Sawada, “On the dependence of photorefractive response time on index grating spacing,” Opt. Commun. 82, 83–88 (1988).
[Crossref]

G. A. Brost, “Numerical analysis of photorefractive grating formation dynamics at large modulation in BSO,” Opt. Commun. 96, 113–116 (1993).
[Crossref]

S. I. Stepanov, V. V. Kulikov, and M. P. Petrov, “Running holograms in photorefractive Bi12SiO20crystals,” Opt. Commun. 44, 19–23 (1982).
[Crossref]

H. Rajbenbach, J.-P. Huignard, and B. Loiscaux, “Spatial frequency dependence of the energy transfer in two mixing experiments with BSO crystals,” Opt. Commun. 48, 247–252 (1983).
[Crossref]

C. Soutar, W. A. Gillespie, and C. M. Cartwright, “The effect of optical bias on grating formation dynamics in photorefractive BSO,” Opt. Commun. 90, 329–334 (1992).
[Crossref]

Opt. Lett. (2)

RCA Rev. (1)

G. A. Alphonse, R. C. Alig, D. L. Stabler, and W. Philips, “Time dependent characteristics of photoinduced space-charge field and phase holograms in lithium niobate and other photorefractive media,” RCA Rev. 36, 213–229 (1975).

Sov. Tech. Phys. Lett. (1)

N. V. Kukhtarev, “Kinetics of hologram recording and erasure in electrooptic crystals,” Sov. Tech. Phys. Lett. 2, 438–440 (1976).

Other (1)

E. Serrano, V. López, M. Carrascosa, and F. Agullo-López, “Steady state photorefractive gratings in LiNbO3for strong light modulation depths,” submitted to IEEE J. Quantum Electron.

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Figures (9)

Fig. 1
Fig. 1

(a) Field amplitude (modules) for the first three Fourier gratings as a function of time. Modulation depth is m = 0.9, and fringe spacing Λ = 10 μm. The data for the linear solution (fundamental grating) are shown as dashed curves. (b) Plot of log(1 − E/Esat) versus t for data in (a), illustrating the saturating exponential character of the recording curves.

Fig. 2
Fig. 2

Amplitude (modules) of the first three Fourier components of the free-carrier pattern as a function of time. The data correspond to the same experimental parameters as in Fig. 1.

Fig. 3
Fig. 3

Dependence of the recording time ϒ, as defined in the text, on the modulation depth m for the first three Fourier gratings.

Fig. 4
Fig. 4

Dependence of the recording time ϒ for the three Fourier components on their grating wave vector k (m = 0.9). The data calculated with the linear solution are shown as the small-dotted curve.

Fig. 5
Fig. 5

Light-induced decay of the first three Fourier components after the fundamental gratings reach saturation. The modulation depth is m = 0.9 and Λ = 10 μm.

Fig. 6
Fig. 6

Dependence of the optical erasure time for the first three Fourier components on their grating wave vector k (m = 0.9). The data calculated with the linear solution are shown as the dotted curve.

Fig. 7
Fig. 7

Light-induced decay of the field amplitude for the first three Fourier components. The fundamental grating has been recorded for up to 50% of its saturation level.

Fig. 8
Fig. 8

(a) Recording curves for the first three Fourier gratings under an applied field A = 5 × 105 V/m (m = 0.9 and Λ = 10 μm). (b) Optical erasure curves for components of (a). The data calculated with the linear solution are shown as dotted curves.

Fig. 9
Fig. 9

Optical erasure curves for the fundamental (solid curve), second harmonic (dotted curve), and third harmonic (dashed curve) gratings with the same k vector (Λ = 18 μm). Obviously the second- and the third-harmonic gratings come from the fundamental grating of Λ = 9 μm and Λ = 6 μm, respectively.

Equations (6)

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

N t - e t ( E x ) = s I ( N D 0 - N D + ) - γ N ( N A - e E x ) ,
- t ( E x ) = e μ N E x + e μ E N x - μ k b T 2 N x 2 ,
E ( x , t ) j = - N N E j ( t ) exp ( i j k x ) , N ( x , t ) j = - N N N j ( t ) exp ( i j k x ) ,
d n j d t - i j d E j d t = s I 0 N A 2 γ ( N D 0 - N A ) ( δ j 0 + m 2 δ j 1 ) + i s I 0 N A γ m 2 [ ( j - 1 ) E j - 1 + ( j + 1 ) E j + 1 ] + i s I 0 N A γ j E j - n j + i p p E p n j - p
j d E j d t = E Q E M p p ( E p n j - p + n p E j - p ) + i E D E M j 2 n j ,
E ( t ) = E sat [ 1 - exp ( - t / τ ) ] ,

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