Abstract

The erasure kinetics of holographic gratings has been theoretically studied for a material containing two photorefractive species. The approach is an extension of the method developed by Carrascosa and Agullo-Lopez for a simple photorefractive center. The erasure of the grating involves the transfer of electronic charge between the two photorefractive systems together with a spatial transport of the charge. Both processes may have, in general, comparable time constants leading to a more complicated formalism than that for a single species. The electronic exchange between two photoactive centers has been first solved analytically. Then, the erasure kinetics of a sinusoidal grating, including charge exchange, has been formulated under the short transport length approximation. The coupled equations governing the decay of grating amplitude and the velocity of fringes have been numerically solved (a) after neglecting diffusion and (b) in the general case. The solution for the time dependence of grating amplitude is nonexponential. The particular situation where the electronic exchange process is very fast in comparison to grating erasure has been solved assuming arbitrary transport lengths. The decay of grating amplitude consists of two exponential curves if the photovoltaic drift is ignored and it is nonexponential if it is included. For short transport lengths, the decay reduces to a single exponential.

© 1988 Optical Society of America

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References

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  1. D. L. Staebler, J. J. Amodei, “Coupled-Wave Analysis of Holographic Storage in LiNbO3,” J. Appl. Phys. 43, 1042 (1972).
    [CrossRef]
  2. G. A. Alphonse, R. C. Alig, D. L. Staebler, W. Phillips, “Time Dependent Characteristics of Photoinduced Space-Charge Field and Phase Holograms in Lithium Niobate and Other Photorefractive Media,” RCA Rev. 36, 213 (1975).
  3. D. W. Vahey, “A Non-Linear Coupled-Wave Theory of Holographic Storage in Ferroelectric Materials,” J. Appl. Phys. 46, 3510 (1975).
    [CrossRef]
  4. 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 (1976).
    [CrossRef]
  5. N. V. Kukhtarev, “Kinetics of Hologram Recording and Erasure in Electrooptic Crystals,” Sov. Tech. Phys. Lett. 2, 438 (1976).
  6. N. Kukhtarev, V. Markov, S. Odulov, “Transient Energy Transfer During Hologram Formation in LiNbO3 in External Electric Field,” Opt. Commun. 23, 338 (1977).
    [CrossRef]
  7. M. G. Moharam, L. Young, “Hologram Writing by the Photorefractive Effect,” J. Appl. Phys. 48, 3230 (1977).
    [CrossRef]
  8. 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 (1979).
    [CrossRef]
  9. 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 (1979).
    [CrossRef]
  10. J. Feinberg, D. Heinman, A. R. Tanguay, R. W. Hellwarth, “Photorefractive Effects and Light-Induced Charge Migration in Barium Titanate,” J. Appl. Phys. 51, 1297 (1980).
    [CrossRef]
  11. M. Carrascosa, F. Agulló-López, “Kinetics for Optical Erasure of Sinusoidal Holographic Gratings in Photorefractive Materials,” IEEE J. Quantum Electron. QE-22, 1369 (1986).
    [CrossRef]
  12. G. C. Valley, “Erase Rates in Photorefractive Materials with Two Photoactive Species,” Appl. Opt. 22, 3160 (1983).
    [CrossRef] [PubMed]
  13. F. P. Strohkendl, J. M. C. Jonathan, R. W. Hellwarth, “Hole–Electron Competition in Photorefractive Gratings,” Opt. Lett. 11, 312 (1986).
    [CrossRef] [PubMed]
  14. G. C. Valley, “Simultaneous Electron–Hole Transport in Photorefractive Materials,” J. Appl. Phys. 59, 3363 (1986).
    [CrossRef]
  15. J. Baquedano, M. Carrascosa, L. Arizmendi, J. M. Cabrera, “Erasure Kinetics and Spectral Dependence of the Photorefractive Effect in Fe:LiNbO3,” J. Opt. Soc. Am. B 4, 309 (1987).
    [CrossRef]

1987 (1)

1986 (3)

F. P. Strohkendl, J. M. C. Jonathan, R. W. Hellwarth, “Hole–Electron Competition in Photorefractive Gratings,” Opt. Lett. 11, 312 (1986).
[CrossRef] [PubMed]

G. C. Valley, “Simultaneous Electron–Hole Transport in Photorefractive Materials,” J. Appl. Phys. 59, 3363 (1986).
[CrossRef]

M. Carrascosa, F. Agulló-López, “Kinetics for Optical Erasure of Sinusoidal Holographic Gratings in Photorefractive Materials,” IEEE J. Quantum Electron. QE-22, 1369 (1986).
[CrossRef]

1983 (1)

1980 (1)

J. Feinberg, D. Heinman, A. R. Tanguay, R. W. Hellwarth, “Photorefractive Effects and Light-Induced Charge Migration in Barium Titanate,” J. Appl. Phys. 51, 1297 (1980).
[CrossRef]

1979 (2)

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 (1979).
[CrossRef]

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 (1979).
[CrossRef]

1977 (2)

N. Kukhtarev, V. Markov, S. Odulov, “Transient Energy Transfer During Hologram Formation in LiNbO3 in External Electric Field,” Opt. Commun. 23, 338 (1977).
[CrossRef]

M. G. Moharam, L. Young, “Hologram Writing by the Photorefractive Effect,” J. Appl. Phys. 48, 3230 (1977).
[CrossRef]

1976 (2)

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 (1976).
[CrossRef]

N. V. Kukhtarev, “Kinetics of Hologram Recording and Erasure in Electrooptic Crystals,” Sov. Tech. Phys. Lett. 2, 438 (1976).

1975 (2)

G. A. Alphonse, R. C. Alig, D. L. Staebler, W. Phillips, “Time Dependent Characteristics of Photoinduced Space-Charge Field and Phase Holograms in Lithium Niobate and Other Photorefractive Media,” RCA Rev. 36, 213 (1975).

D. W. Vahey, “A Non-Linear Coupled-Wave Theory of Holographic Storage in Ferroelectric Materials,” J. Appl. Phys. 46, 3510 (1975).
[CrossRef]

1972 (1)

D. L. Staebler, J. J. Amodei, “Coupled-Wave Analysis of Holographic Storage in LiNbO3,” J. Appl. Phys. 43, 1042 (1972).
[CrossRef]

Agulló-López, F.

M. Carrascosa, F. Agulló-López, “Kinetics for Optical Erasure of Sinusoidal Holographic Gratings in Photorefractive Materials,” IEEE J. Quantum Electron. QE-22, 1369 (1986).
[CrossRef]

Alig, R. C.

G. A. Alphonse, R. C. Alig, D. L. Staebler, W. Phillips, “Time Dependent Characteristics of Photoinduced Space-Charge Field and Phase Holograms in Lithium Niobate and Other Photorefractive Media,” RCA Rev. 36, 213 (1975).

Alphonse, G. A.

G. A. Alphonse, R. C. Alig, D. L. Staebler, W. Phillips, “Time Dependent Characteristics of Photoinduced Space-Charge Field and Phase Holograms in Lithium Niobate and Other Photorefractive Media,” RCA Rev. 36, 213 (1975).

Amodei, J. J.

D. L. Staebler, J. J. Amodei, “Coupled-Wave Analysis of Holographic Storage in LiNbO3,” J. Appl. Phys. 43, 1042 (1972).
[CrossRef]

Arizmendi, L.

Baquedano, J.

Cabrera, J. M.

Carrascosa, M.

J. Baquedano, M. Carrascosa, L. Arizmendi, J. M. Cabrera, “Erasure Kinetics and Spectral Dependence of the Photorefractive Effect in Fe:LiNbO3,” J. Opt. Soc. Am. B 4, 309 (1987).
[CrossRef]

M. Carrascosa, F. Agulló-López, “Kinetics for Optical Erasure of Sinusoidal Holographic Gratings in Photorefractive Materials,” IEEE J. Quantum Electron. QE-22, 1369 (1986).
[CrossRef]

Feinberg, J.

J. Feinberg, D. Heinman, A. R. Tanguay, R. W. Hellwarth, “Photorefractive Effects and Light-Induced Charge Migration in Barium Titanate,” J. Appl. Phys. 51, 1297 (1980).
[CrossRef]

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 (1979).
[CrossRef]

Heinman, D.

J. Feinberg, D. Heinman, A. R. Tanguay, R. W. Hellwarth, “Photorefractive Effects and Light-Induced Charge Migration in Barium Titanate,” J. Appl. Phys. 51, 1297 (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 (1986).
[CrossRef] [PubMed]

J. Feinberg, D. Heinman, A. R. Tanguay, R. W. Hellwarth, “Photorefractive Effects and Light-Induced Charge Migration in Barium Titanate,” J. Appl. Phys. 51, 1297 (1980).
[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 (1976).
[CrossRef]

Kukhtarev, N.

N. Kukhtarev, V. Markov, S. Odulov, “Transient Energy Transfer During Hologram Formation in LiNbO3 in External Electric Field,” Opt. Commun. 23, 338 (1977).
[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 (1979).
[CrossRef]

N. V. Kukhtarev, “Kinetics of Hologram Recording and Erasure in Electrooptic Crystals,” Sov. Tech. Phys. Lett. 2, 438 (1976).

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 (1979).
[CrossRef]

Markov, V.

N. Kukhtarev, V. Markov, S. Odulov, “Transient Energy Transfer During Hologram Formation in LiNbO3 in External Electric Field,” Opt. Commun. 23, 338 (1977).
[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 (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 (1979).
[CrossRef]

M. G. Moharam, L. Young, “Hologram Writing by the Photorefractive Effect,” J. Appl. Phys. 48, 3230 (1977).
[CrossRef]

Odulov, S.

N. Kukhtarev, V. Markov, S. Odulov, “Transient Energy Transfer During Hologram Formation in LiNbO3 in External Electric Field,” Opt. Commun. 23, 338 (1977).
[CrossRef]

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 (1979).
[CrossRef]

Phillips, W.

G. A. Alphonse, R. C. Alig, D. L. Staebler, W. Phillips, “Time Dependent Characteristics of Photoinduced Space-Charge Field and Phase Holograms in Lithium Niobate and Other Photorefractive Media,” RCA Rev. 36, 213 (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 (1976).
[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 (1976).
[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 (1979).
[CrossRef]

Staebler, D. L.

G. A. Alphonse, R. C. Alig, D. L. Staebler, W. Phillips, “Time Dependent Characteristics of Photoinduced Space-Charge Field and Phase Holograms in Lithium Niobate and Other Photorefractive Media,” RCA Rev. 36, 213 (1975).

D. L. Staebler, J. J. Amodei, “Coupled-Wave Analysis of Holographic Storage in LiNbO3,” J. Appl. Phys. 43, 1042 (1972).
[CrossRef]

Strohkendl, F. P.

Tanguay, A. R.

J. Feinberg, D. Heinman, A. R. Tanguay, R. W. Hellwarth, “Photorefractive Effects and Light-Induced Charge Migration in Barium Titanate,” J. Appl. Phys. 51, 1297 (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 (1976).
[CrossRef]

Vahey, D. W.

D. W. Vahey, “A Non-Linear Coupled-Wave Theory of Holographic Storage in Ferroelectric Materials,” J. Appl. Phys. 46, 3510 (1975).
[CrossRef]

Valley, G. C.

G. C. Valley, “Simultaneous Electron–Hole Transport in Photorefractive Materials,” J. Appl. Phys. 59, 3363 (1986).
[CrossRef]

G. C. Valley, “Erase Rates in Photorefractive Materials with Two Photoactive Species,” Appl. Opt. 22, 3160 (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 (1979).
[CrossRef]

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 (1979).
[CrossRef]

M. G. Moharam, L. Young, “Hologram Writing by the Photorefractive Effect,” J. Appl. Phys. 48, 3230 (1977).
[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 (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 (1979).
[CrossRef]

IEEE J. Quantum Electron. (1)

M. Carrascosa, F. Agulló-López, “Kinetics for Optical Erasure of Sinusoidal Holographic Gratings in Photorefractive Materials,” IEEE J. Quantum Electron. QE-22, 1369 (1986).
[CrossRef]

J. Appl. Phys. (6)

G. C. Valley, “Simultaneous Electron–Hole Transport in Photorefractive Materials,” J. Appl. Phys. 59, 3363 (1986).
[CrossRef]

J. Feinberg, D. Heinman, A. R. Tanguay, R. W. Hellwarth, “Photorefractive Effects and Light-Induced Charge Migration in Barium Titanate,” J. Appl. Phys. 51, 1297 (1980).
[CrossRef]

D. L. Staebler, J. J. Amodei, “Coupled-Wave Analysis of Holographic Storage in LiNbO3,” J. Appl. Phys. 43, 1042 (1972).
[CrossRef]

D. W. Vahey, “A Non-Linear Coupled-Wave Theory of Holographic Storage in Ferroelectric Materials,” J. Appl. Phys. 46, 3510 (1975).
[CrossRef]

M. G. Moharam, L. Young, “Hologram Writing by the Photorefractive Effect,” J. Appl. Phys. 48, 3230 (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 (1979).
[CrossRef]

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

Opt. Commun. (1)

N. Kukhtarev, V. Markov, S. Odulov, “Transient Energy Transfer During Hologram Formation in LiNbO3 in External Electric Field,” Opt. Commun. 23, 338 (1977).
[CrossRef]

Opt. Lett. (1)

RCA Rev. (1)

G. A. Alphonse, R. C. Alig, D. L. Staebler, W. Phillips, “Time Dependent Characteristics of Photoinduced Space-Charge Field and Phase Holograms in Lithium Niobate and Other Photorefractive Media,” RCA Rev. 36, 213 (1975).

Sov. Tech. Phys. Lett. (1)

N. V. Kukhtarev, “Kinetics of Hologram Recording and Erasure in Electrooptic Crystals,” Sov. Tech. Phys. Lett. 2, 438 (1976).

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Figures (6)

Fig. 1
Fig. 1

Schematic drawing to illustrate the ionization and trapping processes leading to exchange of charge between donors D1 and D2 and acceptors A1 and A2.

Fig. 2
Fig. 2

Time evolution of the donor concentrations D1 and D2 due to light-induced exchange of carriers between the two impurity systems.

Fig. 3
Fig. 3

Evolution of the free carrier density n corresponding to the same physical situation depicted in Fig. 2.

Fig. 4
Fig. 4

Decay for the amplitude of the total donor grating in a semilogarithmic plot. The dashed line corresponds to the case where diffusion is neglected. The two full lines represent the general case and two different grating spacings.

Fig. 5
Fig. 5

Time evolution of the grating phase (in radians) for (a) no applied field and (b) applied field of 104 V/cm (Λ = 0.4 μm).

Fig. 6
Fig. 6

Time evolution for the contribution of the two donor species to the total grating amplitude (Λ = 0.4 μm).

Equations (57)

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n t = σ 1 I D 1 + I D 2 - γ 1 A 1 n - γ 2 A 2 ,
D 1 t = γ 1 A 1 n - σ 1 I D 1 ,
D 2 t = γ 2 A 2 n - σ 2 I D 2 ,
D 1 + D 2 = D ,             A 1 + A 2 = A .
{ n ( t ) = σ 1 I D 1 + σ 2 I D 2 γ A ( γ 1 = γ 2 = γ ) .
D ˙ 1 = - σ 1 I D 1 + ( N - D 1 ) σ 1 I D 1 + σ 2 I D 2 A ,
D ˙ 2 = - σ 2 I D 2 + ( N - D 2 ) σ 1 I D 1 + σ 2 I D 2 A .
D ˙ 1 = G D 1 2 + B D 1 + C ,
B = ( - Δ σ A N 1 + Δ σ - σ 2 A N ) I , G = Δ σ A I , C = σ 2 D N 1 I A , Δ σ = σ 2 - σ 1 .
D 1 ( t ) = 1 k exp ( - F t ) - G F + H ,
k = G { H - D 1 ( 0 ) } - F F { H - D 1 ( 0 ) } ,
F = 2 H G + B ,
H = - B + B 2 - 4 G C 2 G .
n ( ) = σ 1 D 1 ( ) + σ 2 D 2 ( ) γ A I .
n t = σ 1 I D 1 + σ 2 I D 2 - γ n A - 1 e j x ,
D 1 t = - σ 1 I D 1 + γ n A 1 ,
D 2 t = - σ 2 I D 2 + γ n A 2 ,
ρ ( x , t ) = e { D 1 ( x , t ) - D 1 ( x , 0 ) } + e { D 2 ( x , t ) - D 2 ( x , 0 ) } ,
j ( x , t ) = μ n e E - e D c n x + e χ 1 I D 1 + e χ 2 I D 2 ,
E ρ x = E x = ρ ,
( D 1 + D 2 ) t = - 1 e j x .
D 1 = D ¯ 1 ( 0 ) + M 1 ( 0 ) cos [ k x + ϕ ( 0 ) ] ,
D 2 = D ¯ 2 ( 0 ) + M 2 ( 0 ) cos [ k x + ϕ ( 0 ) ]
A 1 = A ¯ 1 ( 0 ) - M 1 ( 0 ) cos [ k x + ϕ ( 0 ) ] ,
A 2 = A ¯ 2 ( 0 ) - M 2 ( 0 ) cos [ k x + ϕ ( 0 ) ]
ρ ( x , t ) = e { M 1 ( t ) + M 2 ( t ) } cos ( k x + ϕ ) .
E ρ = e ( M 1 + M 2 ) x sin ( k x + ϕ ) .
n ( x , t ) = n 0 ( t ) + n 1 ( t ) cos ( k x + ϕ ) ,
n 0 ( t ) = σ 1 D ¯ 1 ( t ) + σ 2 D ¯ 2 ( t ) γ A I ,
n 1 ( t ) = σ 1 M 1 ( t ) + σ 2 M 2 ( t ) γ A + M 1 ( t ) + M 2 ( t ) A n 0 ( t ) .
e ( M ˙ 1 + M ˙ 2 ) cos ( k x + ϕ ) - e ( M 1 + M 2 ) ϕ ˙ sin ( k x + ϕ ) = - e μ x { [ n 0 + n 1 cos ( k x + ϕ ) ] [ E a + e k ( M 1 + M 2 ) sin ( k x + ϕ ) - D c e n 1 k 2 cos ( k x + ϕ ) - χ 1 I k M 1 sin ( k x + ϕ ) - χ 2 I k M 2 sin ( k x + ϕ ) .
M ˙ 1 + M ˙ 2 = - e μ n 0 ( M 1 + M 2 ) - D c n 1 k 2 ,
( M 1 + M 2 ) ϕ ˙ = - μ n 1 E a k - [ χ 1 M 1 + χ 2 M 2 ] I k ,
D ¯ 1 + M ˙ 1 cos ( k x + ϕ ) = - σ 1 I D 1 + γ n 0 ( N 1 - D ¯ 1 ) + [ σ 1 I M 1 - γ n 0 M 1 + γ n 1 ( N 1 - D ¯ 1 ) ] cos ( k x + ϕ ) ,
D ¯ ˙ 1 = - σ 1 I D ¯ 1 + γ n 0 ( N 1 - D ¯ 1 ) ,
M ˙ 1 = [ - σ 1 I - γ n 0 ] M 1 + γ ( N 1 - D ¯ 1 ) n 1 .
M ˙ 1 + M ˙ 2 = - e μ n 0 ( t ) ( M 1 + M 2 ) ,
n = σ 1 I D 1 γ A 1 = σ 2 I D 2 γ A 2 .
n 0 = σ 1 I D ¯ 1 γ A ¯ 1 = σ 2 I D ¯ 2 γ A ¯ 2 ,
n 1 = β 1 n 0 M 1 = β 2 n 0 M 2 ,
M 2 = β 1 β 2 M 1 .
M 1 ( 1 + β 1 β 2 ) = - e μ n 0 ( 1 + β 1 β 2 ) M 1 - D c k 2 n 0 β 1 M 1 , M 2 ( 1 + β 1 β 2 ) = - μ E a β 1 n 0 k M 1 - ( χ 1 + β 1 β 2 χ 2 ) M 1 k I ,
M ˙ 1 = - ( e μ + D c k 2 β 1 β 2 β 1 + β 2 ) n 0 M 1 ,
ϕ ˙ = - μ E a k n 0 β 1 β 2 β 1 + β 2 - χ 1 β 1 + χ 2 β 2 β 1 + β 2 k I .
n 0 = σ 1 I D ¯ 1 γ A ¯ 1 = σ 2 I D ¯ 2 γ A ¯ 2 .
n = n 0 + n 1 cos ( k x + ϕ + φ ) ,
tan φ = - μ k E a γ A + D c k 2 ,
n 1 = σ 1 I M 1 + σ 2 I M 2 + ( γ n 0 - μ e n 0 ) M ( γ A + D c k 2 ) cos φ - μ k E a sin φ .
M ˙ = - μ n 0 e M + μ k E a n 1 sin φ - D c k 2 n 1 cos φ ,
M ˙ 1 = - ( σ 1 I + γ n 0 ) M 1 + γ ( N - D 1 ) n 1 cos φ .
n 1 cos φ = γ A + D c k 2 ( γ A + D c k 2 ) 2 + ( μ k E a ) 2 × [ Δ σ I M 1 + ( σ 2 I + γ n 0 - μ n 0 e ) M ] ,
n 1 sin φ = - μ k E a ( γ A + D c k 2 ) 2 + ( μ k E a ) 2 × [ Δ σ I M 1 + ( σ 2 I + γ n 0 - μ n 0 e ) M ] .
M ˙ = P M + Q M 1 ,
M ˙ 1 = R M 1 + S M ,
M ¨ - ( P + R ) M ˙ + ( Q S - R P ) M = 0 ,
M = C 1 exp ( - α 1 t ) + C 2 exp ( - α 2 t ) ,
α 1 , 2 = - ( P + R ) ± ( P + R ) 2 - 4 ( Q S - P R ) 2 .

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