Abstract

We analytically solve charge-transport equations for a photorefractive crystal with shallow and deep traps. We predict that, if shallow traps can accumulate a high density of charge, the photorefractive trap density and space-charge field will be strong functions of light intensity and the photoconductivity will scale sublinearly with intensity. We show that, depending on light intensity and grating spacing, shallow-trap charge gratings form either in phase or out of phase with the light pattern. As shallow traps thermally depopulate in the dark, the space-charge field either partially decays or partially develops for a few seconds. The amount of decay increases as grating spacing increases in Bi12SiO20 and as grating spacing decreases in BaTiO3.

© 1991 Optical Society of America

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Corrections

Parviz Tayebati and Daniel Mahgerefteh, "Theory of the photorefractive effect for Bi12SiO20 and BaTiO3 with shallow traps: errata," J. Opt. Soc. Am. B 9, 177-177 (1992)
https://www.osapublishing.org/josab/abstract.cfm?uri=josab-9-1-177

References

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  1. N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetskii, “Holographic storage in electrooptic crystal. I. Steady state,” Ferroelectrics 22, 949 (1979).
    [CrossRef]
  2. J. Feinberg, D. Heiman, A. R. Tanguay, R. W. Hellwarth, “Photorefractive effect and light-induced charge migration in BaTiO3,” J. Appl. Phys. 51, 1297 (1980); erratum 52, 537 (1981).
    [CrossRef]
  3. G. C. Valley, “Erasure rates in photorefractive materials with two photoactive species,” Appl. Opt. 22, 3160 (1983).
    [CrossRef]
  4. G. A. Brost, R. A. Motes, J. R. Rotge, “Intensity-dependent absorption and photorefractive effects in barium titanate,” J. Opt. Soc. Am. B 5, 1879 (1988).
    [CrossRef]
  5. D. A. Temple, C. Warde, “Photoinduced absorption effects in BaTiO3,” in OSA Annual Meeting, Vol. 11 of 1988 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1988), p. 137.
  6. F. P. Strohkendl, “Light-induced dark decays of photorefractive gratings and their observation in Bi12SiO20,” J. Appl. Phys. 65, 3773 (1989).
    [CrossRef]
  7. R. A. Mullen, “Measurement of bulk space-charge grating in photorefractive Bi12SiO20,” Ph.D. dissertation (University of Southern California, Los Angeles, Calif., 1984).
  8. D. Mahgerefteh, J. Feinberg, “Erasure rate and coasting in photorefractive barium titanate at high optical power,” Opt. Lett. 13, 1111 (1988).
    [CrossRef] [PubMed]
  9. S. Ducharme, “Photorefraction in BaTiO3,” Ph.D. dissertation (University of Southern California, Los Angeles, Calif., 1986).
  10. M. E. Lasher, D. M. Gookin, “Wavelength dependence of the photorefractive effect in photorefractive barium titanate,” in Conference on Lasers and Electro-Optics, Vol. 7 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), p. 92.
  11. P. Tayebati, “Characterization of the effect of shallow traps on photorefractive properties of Bi12SiO20,” submitted to J. Appl. Phys.
  12. S. Ducharme, J. Feinberg, “Speed of the photorefractive effect in a BaTiO3single crystal,” J. Appl. Phys. 56, 839 (1984).
    [CrossRef]
  13. D. Mahgerefteh, J. Feinberg, “Explanation of the apparent sublinear photoconductivity of photorefractive barium titanate,” Phys. Rev. Lett. 64, 2195 (1990).
    [CrossRef] [PubMed]
  14. D. Mahgerefteh, “The speed of the photorefractive effect, shallow traps, photogalvanic currents, and light-induced surface damage in barium titanate,” Ph.D dissertation (University of Southern California, Los Angeles, Calif., 1990).
  15. L. Holtmann, “A model for the nonlinear photoconductivity of BaTiO3,” Phys. Status Solidi A 113, K89 (1989).
    [CrossRef]
  16. R. S. Cudney, R. M. Pierce, G. D. Bacher, J. Feinberg, “Absorption grating having multiple levels,” submitted to J. Opt. Soc. Am. B.
  17. F. P. Strohkendl, P. Tayebati, R. W. Hellwarth, “Comparative study of photorefractive BSO crystals,” J. Appl. Phys. 66, 6024 (1989).
    [CrossRef]
  18. T. Y. Chang, “Nonlinear optical studies of BaTiO3,” Ph.D. dissertation (University of Southern California, Los Angeles, Calif., 1986).
  19. F. P. Strohkendl, J. M. C. Jonathan, R. W. Hellwarth, “Hole–electron competition in photorefractive gratings,” Opt. Lett. 11, 312 (1986).
    [CrossRef]
  20. G. C. Valley, “Simultaneous electron/hole transport in photorefractive materials,” J. Appl. Phys. 59, 2363 (1986).
    [CrossRef]
  21. P. Tayebati, “Characterization and modeling of the photorefractive effect in bismuth silicon oxide,” Ph.D. dissertation (University of Southern California, Los Angeles, Calif., 1989).
  22. A. L. Smirl, K. Bohnert, G. C. Valley, R. A. Mullen, T. F. Boggess, “Formation, decay, and erasure of photorefractive gratings written in barium titanate by picosecond pulses,” J. Opt. Soc. Am. B 6, 606 (1989).
    [CrossRef]
  23. P. M. Morse, H. Feshback, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

1990 (1)

D. Mahgerefteh, J. Feinberg, “Explanation of the apparent sublinear photoconductivity of photorefractive barium titanate,” Phys. Rev. Lett. 64, 2195 (1990).
[CrossRef] [PubMed]

1989 (4)

L. Holtmann, “A model for the nonlinear photoconductivity of BaTiO3,” Phys. Status Solidi A 113, K89 (1989).
[CrossRef]

F. P. Strohkendl, P. Tayebati, R. W. Hellwarth, “Comparative study of photorefractive BSO crystals,” J. Appl. Phys. 66, 6024 (1989).
[CrossRef]

F. P. Strohkendl, “Light-induced dark decays of photorefractive gratings and their observation in Bi12SiO20,” J. Appl. Phys. 65, 3773 (1989).
[CrossRef]

A. L. Smirl, K. Bohnert, G. C. Valley, R. A. Mullen, T. F. Boggess, “Formation, decay, and erasure of photorefractive gratings written in barium titanate by picosecond pulses,” J. Opt. Soc. Am. B 6, 606 (1989).
[CrossRef]

1988 (2)

1986 (2)

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

G. C. Valley, “Simultaneous electron/hole transport in photorefractive materials,” J. Appl. Phys. 59, 2363 (1986).
[CrossRef]

1984 (1)

S. Ducharme, J. Feinberg, “Speed of the photorefractive effect in a BaTiO3single crystal,” J. Appl. Phys. 56, 839 (1984).
[CrossRef]

1983 (1)

1980 (1)

J. Feinberg, D. Heiman, A. R. Tanguay, R. W. Hellwarth, “Photorefractive effect and light-induced charge migration in BaTiO3,” J. Appl. Phys. 51, 1297 (1980); erratum 52, 537 (1981).
[CrossRef]

1979 (1)

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

Bacher, G. D.

R. S. Cudney, R. M. Pierce, G. D. Bacher, J. Feinberg, “Absorption grating having multiple levels,” submitted to J. Opt. Soc. Am. B.

Boggess, T. F.

Bohnert, K.

Brost, G. A.

Chang, T. Y.

T. Y. Chang, “Nonlinear optical studies of BaTiO3,” Ph.D. dissertation (University of Southern California, Los Angeles, Calif., 1986).

Cudney, R. S.

R. S. Cudney, R. M. Pierce, G. D. Bacher, J. Feinberg, “Absorption grating having multiple levels,” submitted to J. Opt. Soc. Am. B.

Ducharme, S.

S. Ducharme, J. Feinberg, “Speed of the photorefractive effect in a BaTiO3single crystal,” J. Appl. Phys. 56, 839 (1984).
[CrossRef]

S. Ducharme, “Photorefraction in BaTiO3,” Ph.D. dissertation (University of Southern California, Los Angeles, Calif., 1986).

Feinberg, J.

D. Mahgerefteh, J. Feinberg, “Explanation of the apparent sublinear photoconductivity of photorefractive barium titanate,” Phys. Rev. Lett. 64, 2195 (1990).
[CrossRef] [PubMed]

D. Mahgerefteh, J. Feinberg, “Erasure rate and coasting in photorefractive barium titanate at high optical power,” Opt. Lett. 13, 1111 (1988).
[CrossRef] [PubMed]

S. Ducharme, J. Feinberg, “Speed of the photorefractive effect in a BaTiO3single crystal,” J. Appl. Phys. 56, 839 (1984).
[CrossRef]

J. Feinberg, D. Heiman, A. R. Tanguay, R. W. Hellwarth, “Photorefractive effect and light-induced charge migration in BaTiO3,” J. Appl. Phys. 51, 1297 (1980); erratum 52, 537 (1981).
[CrossRef]

R. S. Cudney, R. M. Pierce, G. D. Bacher, J. Feinberg, “Absorption grating having multiple levels,” submitted to J. Opt. Soc. Am. B.

Feshback, H.

P. M. Morse, H. Feshback, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

Gookin, D. M.

M. E. Lasher, D. M. Gookin, “Wavelength dependence of the photorefractive effect in photorefractive barium titanate,” in Conference on Lasers and Electro-Optics, Vol. 7 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), p. 92.

Heiman, D.

J. Feinberg, D. Heiman, A. R. Tanguay, R. W. Hellwarth, “Photorefractive effect and light-induced charge migration in BaTiO3,” J. Appl. Phys. 51, 1297 (1980); erratum 52, 537 (1981).
[CrossRef]

Hellwarth, R. W.

F. P. Strohkendl, P. Tayebati, R. W. Hellwarth, “Comparative study of photorefractive BSO crystals,” J. Appl. Phys. 66, 6024 (1989).
[CrossRef]

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

J. Feinberg, D. Heiman, A. R. Tanguay, R. W. Hellwarth, “Photorefractive effect and light-induced charge migration in BaTiO3,” J. Appl. Phys. 51, 1297 (1980); erratum 52, 537 (1981).
[CrossRef]

Holtmann, L.

L. Holtmann, “A model for the nonlinear photoconductivity of BaTiO3,” Phys. Status Solidi A 113, K89 (1989).
[CrossRef]

Jonathan, J. M. C.

Kukhtarev, N. V.

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

Lasher, M. E.

M. E. Lasher, D. M. Gookin, “Wavelength dependence of the photorefractive effect in photorefractive barium titanate,” in Conference on Lasers and Electro-Optics, Vol. 7 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), p. 92.

Mahgerefteh, D.

D. Mahgerefteh, J. Feinberg, “Explanation of the apparent sublinear photoconductivity of photorefractive barium titanate,” Phys. Rev. Lett. 64, 2195 (1990).
[CrossRef] [PubMed]

D. Mahgerefteh, J. Feinberg, “Erasure rate and coasting in photorefractive barium titanate at high optical power,” Opt. Lett. 13, 1111 (1988).
[CrossRef] [PubMed]

D. Mahgerefteh, “The speed of the photorefractive effect, shallow traps, photogalvanic currents, and light-induced surface damage in barium titanate,” Ph.D dissertation (University of Southern California, Los Angeles, Calif., 1990).

Markov, V. B.

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

Morse, P. M.

P. M. Morse, H. Feshback, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

Motes, R. A.

Mullen, R. A.

A. L. Smirl, K. Bohnert, G. C. Valley, R. A. Mullen, T. F. Boggess, “Formation, decay, and erasure of photorefractive gratings written in barium titanate by picosecond pulses,” J. Opt. Soc. Am. B 6, 606 (1989).
[CrossRef]

R. A. Mullen, “Measurement of bulk space-charge grating in photorefractive Bi12SiO20,” Ph.D. dissertation (University of Southern California, Los Angeles, Calif., 1984).

Odulov, S. G.

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

Pierce, R. M.

R. S. Cudney, R. M. Pierce, G. D. Bacher, J. Feinberg, “Absorption grating having multiple levels,” submitted to J. Opt. Soc. Am. B.

Rotge, J. R.

Smirl, A. L.

Soskin, M. S.

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

Strohkendl, F. P.

F. P. Strohkendl, “Light-induced dark decays of photorefractive gratings and their observation in Bi12SiO20,” J. Appl. Phys. 65, 3773 (1989).
[CrossRef]

F. P. Strohkendl, P. Tayebati, R. W. Hellwarth, “Comparative study of photorefractive BSO crystals,” J. Appl. Phys. 66, 6024 (1989).
[CrossRef]

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

Tanguay, A. R.

J. Feinberg, D. Heiman, A. R. Tanguay, R. W. Hellwarth, “Photorefractive effect and light-induced charge migration in BaTiO3,” J. Appl. Phys. 51, 1297 (1980); erratum 52, 537 (1981).
[CrossRef]

Tayebati, P.

F. P. Strohkendl, P. Tayebati, R. W. Hellwarth, “Comparative study of photorefractive BSO crystals,” J. Appl. Phys. 66, 6024 (1989).
[CrossRef]

P. Tayebati, “Characterization and modeling of the photorefractive effect in bismuth silicon oxide,” Ph.D. dissertation (University of Southern California, Los Angeles, Calif., 1989).

P. Tayebati, “Characterization of the effect of shallow traps on photorefractive properties of Bi12SiO20,” submitted to J. Appl. Phys.

Temple, D. A.

D. A. Temple, C. Warde, “Photoinduced absorption effects in BaTiO3,” in OSA Annual Meeting, Vol. 11 of 1988 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1988), p. 137.

Valley, G. C.

Vinetskii, V. L.

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

Warde, C.

D. A. Temple, C. Warde, “Photoinduced absorption effects in BaTiO3,” in OSA Annual Meeting, Vol. 11 of 1988 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1988), p. 137.

Appl. Opt. (1)

Ferroelectrics (1)

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

J. Appl. Phys. (5)

J. Feinberg, D. Heiman, A. R. Tanguay, R. W. Hellwarth, “Photorefractive effect and light-induced charge migration in BaTiO3,” J. Appl. Phys. 51, 1297 (1980); erratum 52, 537 (1981).
[CrossRef]

G. C. Valley, “Simultaneous electron/hole transport in photorefractive materials,” J. Appl. Phys. 59, 2363 (1986).
[CrossRef]

S. Ducharme, J. Feinberg, “Speed of the photorefractive effect in a BaTiO3single crystal,” J. Appl. Phys. 56, 839 (1984).
[CrossRef]

F. P. Strohkendl, P. Tayebati, R. W. Hellwarth, “Comparative study of photorefractive BSO crystals,” J. Appl. Phys. 66, 6024 (1989).
[CrossRef]

F. P. Strohkendl, “Light-induced dark decays of photorefractive gratings and their observation in Bi12SiO20,” J. Appl. Phys. 65, 3773 (1989).
[CrossRef]

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

Opt. Lett. (2)

Phys. Rev. Lett. (1)

D. Mahgerefteh, J. Feinberg, “Explanation of the apparent sublinear photoconductivity of photorefractive barium titanate,” Phys. Rev. Lett. 64, 2195 (1990).
[CrossRef] [PubMed]

Phys. Status Solidi A (1)

L. Holtmann, “A model for the nonlinear photoconductivity of BaTiO3,” Phys. Status Solidi A 113, K89 (1989).
[CrossRef]

Other (10)

R. S. Cudney, R. M. Pierce, G. D. Bacher, J. Feinberg, “Absorption grating having multiple levels,” submitted to J. Opt. Soc. Am. B.

T. Y. Chang, “Nonlinear optical studies of BaTiO3,” Ph.D. dissertation (University of Southern California, Los Angeles, Calif., 1986).

D. A. Temple, C. Warde, “Photoinduced absorption effects in BaTiO3,” in OSA Annual Meeting, Vol. 11 of 1988 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1988), p. 137.

D. Mahgerefteh, “The speed of the photorefractive effect, shallow traps, photogalvanic currents, and light-induced surface damage in barium titanate,” Ph.D dissertation (University of Southern California, Los Angeles, Calif., 1990).

S. Ducharme, “Photorefraction in BaTiO3,” Ph.D. dissertation (University of Southern California, Los Angeles, Calif., 1986).

M. E. Lasher, D. M. Gookin, “Wavelength dependence of the photorefractive effect in photorefractive barium titanate,” in Conference on Lasers and Electro-Optics, Vol. 7 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), p. 92.

P. Tayebati, “Characterization of the effect of shallow traps on photorefractive properties of Bi12SiO20,” submitted to J. Appl. Phys.

P. Tayebati, “Characterization and modeling of the photorefractive effect in bismuth silicon oxide,” Ph.D. dissertation (University of Southern California, Los Angeles, Calif., 1989).

P. M. Morse, H. Feshback, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

R. A. Mullen, “Measurement of bulk space-charge grating in photorefractive Bi12SiO20,” Ph.D. dissertation (University of Southern California, Los Angeles, Calif., 1984).

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Figures (11)

Fig. 1
Fig. 1

Band diagram of the shallow-trap model for n-type crystals.

Fig. 2
Fig. 2

Steady-state shallow-trap grating M1 as a function of the grating wave vector squared for three writing intensities. The parameters used are β = 10 Hz, β/sT = 100 mW/cm2, NA = 1.9 × 1015cm3, MT = 1.9 × 1016 cm−3, ND = 1019cm−3, κD2/κT2 = 0.5. (Only the ratio of the recombination rates affects the value of the steady-state charge gratings.)

Fig. 3
Fig. 3

Steady-state space-charge field E1 as a function of the grating wave vector squared for various writing intensities. (a) I = 0.001 W/cm2, (b) I = 0.01 W/cm2, (c) I = 0.1 W/cm2, (d) I = 10 W/cm2. The parameters used are the same as those for Fig. 2.

Fig. 4
Fig. 4

Grating diffraction efficiency as a function of time in the dark, normalized to its value at t = 0 for various writing intensities in BSO. The parameters used are β = 5 Hz, β/sT = 10 mW/cm2, k0A2 = 150 μm−2, k0T2 = 100 μm−2, κD2 = 0.2 μm2, κT2 = 0.02 μm−2.

Fig. 5
Fig. 5

Grating diffraction efficiency as a function of time in the dark normalized to its value at t = 0 for various grating wave vectors in BSO. The parameters used are the same as those for Fig. 4.

Fig. 6
Fig. 6

Percent coasting C as a function of the grating wave vector squared. The parameters used for BSO are the same as those for Fig. 4, and the parameters for BaTiO3 are β = 2 Hz, β/sT = 15 mW/cm2, k0A2 = 260 μm−2, k0T2 = 340 m−2, κD2 = 1200 μm−2, κT2 = 200 μm−2.

Fig. 7
Fig. 7

Initial dark decay rate R as a function of the grating wave vector. The parameters used for BSO are the same as those for Fig. 4, and the parameters for BaTiO3 are the same as those for Fig. 6.

Fig. 8
Fig. 8

Percent coasting in the dark C as a function of the writing intensity for BSO. The parameters used are the same as those for Fig. 4. The curve for BaTiO3 has the same shape.

Fig. 9
Fig. 9

Initial dark decay rate R as a function of the writing intensity for BSO. The parameters used are the same as those for Fig. 4. The curve for BaTiO3 has the same shape.

Fig. 10
Fig. 10

Grating diffraction efficiency as a function of time in the dark normalized to its value at t = 0 for various grating wave vectors in BaTiO3. The parameters used are the same as those for Fig. 6.

Fig. 11
Fig. 11

Grating diffraction efficiency as a function of time in the dark normalized to its value at t = 0 for BaTiO3, assuming that the charge gratings in the deep and the shallow traps have equal magnitudes and opposite signs at t = 0. Note that the space-charge field develops in the dark. The inset shows the decay of N1, the grating in the deep traps, and M1, the grating in the shallow traps.

Equations (49)

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N D i t = s D I ( N D - N D i ) - γ D n N D i ,
M t = - ( s T I + β ) M + γ T n ( M T - M ) ,
t ( N D i - M - n ) + 1 e · J = 0 ,
J = e μ n E + μ k B T n ,
· E = - e ɛ ( n - N D i + N A + M ) .
N D i = N A + N 0 + Re [ N 1 exp ( i k z ) ] , M = M 0 + Re [ M 1 exp ( i k z ) ] ,
n 0 N 0 ,             n 0 M 0 .
N 0 M 0 ,
n 1 N 1 - M 1 .
N 0 = M 0 = 1 2 [ ρ ( I 0 ) - 1 ] ( [ ρ ( I 0 ) ( N D A + M T ) + N A ] - { [ ρ ( I 0 ) ( N D A + M T ) + N A ] 2 - 4 ρ ( I 0 ) × [ ρ ( I 0 ) - 1 ] N D A M T } 1 / 2 ) ,
ρ ( I 0 ) = s D γ T s T γ D 1 ( 1 + β / s T I 0 ) ,             N D A N D - N A .
s D γ T s T γ D N A 2 4 M T N D A , 1             ( M 0 / N A 1 ) ,
s D γ T s T γ D N A N D A + M T ( M 0 / M T 1 ) .
M 0 = M ( 1 + β / s T I 0 )             ( if M 0 / N A 1 , M 0 / M T 1 ) ,
M = s D γ T M T N D A s T γ D N A
α traps = s D N D A + ( s T - s D ) M 0 .
n 0 = s D I 0 ( N D A - N 0 ) τ D ( I 0 ) ,
τ D ( I 0 ) = 1 γ D ( N A + N 0 )
n 0 = s D I 0 2 γ D ( N A + M T ) ( ( N D A - M T ) - N A ρ ( I 0 ) + { [ ( N D A - M T ) - N A ρ ( I 0 ) ] 2 + 4 N D A ( N A + M T ) ρ ( I 0 ) } 1 / 2 ) .
( + e ) N 1 = m e N E k 2 ( k 2 + k 0 2 ) + m e M E 1 ( 1 + s T I 0 / β ) k 0 D 2 ( k 2 + k 0 2 ) ,
( - e ) M 1 = m e M E 1 ( 1 + β / s T I 0 ) k 2 ( k 2 + k 0 2 ) - m e M E 1 ( 1 + s T I 0 / β ) k 0 D 2 ( k 2 + k 0 2 ) .
N E = ( N D A - N 0 ) ( N A + N 0 ) N D ,             M E = M 0 ( M T - M 0 ) M T ,
k 0 D 2 = e 2 N E ɛ k B T ,             k 0 T 2 = e 2 M E ɛ k B T ,
k 0 2 = k 0 D 2 + k 0 T 2
i k E 1 = - e ɛ ( N 1 - M 1 ) .
E 1 = - i m k B T e η ( I 0 ) k 1 + k 2 / k 0 2 ,
η ( I 0 ) = 1 k 0 2 ( k 0 D 2 + k 0 T 2 1 + β / s T I 0 ) .
[ M 0 ( t ) M T ] [ M 0 ( t ) + N A M T ] - ( 1 + N A / M T ) / ( 1 + γ D N A / γ T M T ) = const . exp ( - β t 1 + γ T M T / γ D N A ) ,
n 0 ( t ) = β M 0 ( t ) γ D N A + γ T M T + ( γ D - γ T ) M 0 ( t ) .
N 1 t = - ( k 2 + K T 2 ) γ D n 0 + e μ n 0 K D 2 / ɛ ( k 2 + K 2 ) N 1 - ( β + γ T n 0 - e μ n 0 / ɛ ) K D 2 ( k 2 + K 2 ) M 1 , M 1 t = - ( γ D n 0 - e μ n 0 / ɛ ) K T 2 k 2 + K 2 N 1 - ( k 2 + K D 2 ) ( β + γ T n 0 ) + e μ n 0 K T 2 / ɛ k 2 + K 2 M 1 ,
K D 2 = e γ D ( N A + N 0 ) μ k B T , K T 2 = e γ T ( M T - M 0 ) μ k B T , K 2 = K D 2 + K T 2 .
M 0 / N A 1 ,
M 0 / M T 1.
M 0 ( t ) = M 0 ( t = 0 ) exp ( - a t ) ,
a = β ( 1 + γ T M T / γ D N A )
n 0 ( t ) = β M 0 ( t = 0 ) γ D N A + γ T M T exp ( - a t ) ,
M 1 ( t ) = exp [ ( f + g - q ) ζ 0 exp ( - a t ) ] × [ A 0 ζ 0 exp ( - a t ) F ( 1 - b 2 - r 2 q , 2 - b ; 2 q ζ 0 exp ( - a t ) ) + B 0 exp ( - a b t ) F ( b 2 - r 2 q , b ; 2 q ζ 0 exp ( - a t ) ) ] ,
N 1 ( t ) = 1 c exp [ ( f + g - q ) ζ 0 exp ( - a t ) ] × { A 0 [ ( f - g - q ) ζ 0 exp ( - a t ) + 1 - b ] × F ( 1 - b 2 - r 2 q , 2 - b ; 2 q ζ 0 exp ( - a t ) ) + A 0 ζ 0 exp ( - a t ) ( q - r 2 - b ) × F ( 2 - b 2 - r 2 q , 3 - b ; 2 q ζ 0 exp ( - a t ) ) + B 0 exp ( - a b t ) ( f - g - q ) × F ( b 2 - r 2 q , b ; 2 q ζ 0 exp ( - a t ) ) + B 0 exp ( - a b t ) ( q - r b ) × F ( 1 + b 2 - r 2 q , 1 + b ; 2 q ζ 0 exp ( - a t ) ) } .
ζ 0 = M 0 ( t = 0 ) M T
b = 1 + k 2 / κ D 2 1 + k 2 / κ 2 , c = M T N A κ T 2 κ 2 ( 1 - k 0 A 2 / κ D 2 ) ( 1 + k 2 / κ 2 ) , f = 1 2 M T N A κ T 2 κ 2 1 + ( k 2 + k 0 A 2 ) / κ T 2 1 + k 2 / κ 2 , g = 1 2 κ T 2 κ 2 1 + ( k 2 + k 0 S 2 ) / κ D 2 1 + k 2 / κ 2 .
q 2 [ ( f - g ) 2 - c h ] ,             r [ b ( f - g ) - c d ] , h = - κ T 2 κ 2 ( 1 - k 0 S 2 / κ T 2 ) ( 1 + k 2 / κ 2 ) ,             d = 1 1 + k 2 / κ 2 .
k 0 A 2 = e 2 N A ɛ k B T ,             k 0 S 2 = e 2 M T ɛ k B T , κ D 2 = e γ D N A μ k B T ,             κ T 2 = e γ T M T μ k B T ,             κ 2 = κ T 2 + κ D 2 .
E 1 ( t ) = i e k ɛ [ N 1 ( t ) - M 1 ( t ) ] .
C 100 I sig ( t = 0 ) - I sig ( t 1 / a ) I sig ( t = 0 )
R 1 I sig ( d I sig d t ) t = 0
1 a N 1 t = - [ d - h ζ 0 exp ( - a t ) ] M 1 - 2 f ζ 0 exp ( - a t ) N 1 ,
1 a M 1 t = - ( b + 2 g ζ 0 exp ( - a t ) ) M 1 - c ζ 0 exp ( - a t ) N 1 .
2 M 1 x 2 - [ 2 ( f + g ) ζ 0 + b x ] M 1 x + [ ( c h + 4 f g ) ζ 0 2 + ( 2 f b - c d ) x ζ 0 + b x 2 ] M 1 = 0.
M 1 ( ζ 0 x ) = exp ( f + g - q ) ζ 0 x [ A 0 x ζ 0 F ( 1 - b 2 - r 2 q , 2 - b ; 2 q ζ 0 x ) + B 0 x b F ( b 2 - r 2 q , b ; 2 q ζ 0 x ) ] ,

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