Abstract

By combining the band-transport equations with the coupled wave equations, we obtain a self-consistent set of equations governing the recording of photorefractive holograms. Using the results of this model leads to accurate predictions for the observed intensity coupling and holographic diffraction efficiency. The model also predicts complicated dynamics for long recording times. The origin of this prediction is discussed and is found to be due to a new effect, which is called transient fringe dislocation.

© 1992 Optical Society of America

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References

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  1. N. V. Kukhtarev, “Kinetics of hologram recording and erasure in electro-optic crystals,” Sov. Tech. Phys. Lett. 2, 438–440 (1976).
  2. G. C. Valley and M. B. Klein, “Optimal properties of photorefractive materials for optical data processing,” Opt. Eng. 22, 704–711 (1983).
    [CrossRef]
  3. E. S. Maniloff and K. M. Johnson, “Maximized photorefractive holographic storage,” J. Appl. Phys. 70, 4702–4707 (1991).
    [CrossRef]
  4. N. Kukhtarev, V. Markov, and S. Odulov, “Transient energy transfer during hologram formation in LiNbO3in external electric field,” Opt. Commun. 23, 338–343 (1977).
    [CrossRef]
  5. N. V. Kukhtarev, V. B. Markov, and S. G. Odulov, “Nonstationary energy exchange during interactions between two light beams in electrooptical crystals,” Sov. Phys. Tech. Phys. 25, 1109–1114 (1980).
  6. L. Solymar and J. M. Heaton, “Transient energy transfer in photorefractive materials; an analytic solution,” Opt. Commun. 51, 76–78 (1984).
    [CrossRef]
  7. J. M. Heaton and L. Solymar, “Transient energy transfer during hologram formation in photorefractive crystals,” Opt. Acta 32, 397–408 (1985).
    [CrossRef]
  8. J. M. Heaton and L. Solymar, “Transient effects during dynamic hologram formation in BSO crystals: theory and experiments,” IEEE J. Quantum Electron. 24, 558–567 (1988).
    [CrossRef]
  9. V I. Belinicher, V K. Malinovski, and B. I. Sturman, “Photo-galvanic effect in a crystal with polar axis,” Sov. Phys. JETP,  46, 362–366 (1977).
  10. V I. Belinicher and B. I. Sturman, “The photogalvanic effect in media lacking a center of symmetry,” Sov. Phys. Usp. 23, 199–223 (1980).
    [CrossRef]
  11. A. M. Glass and D. von der Linde, “Photorefractive effects for reversible holographic storage of information,” App. Phys. 8, 85–100 (1975).
    [CrossRef]
  12. A. M. Glass, “The photorefractive effect,” Opt. Eng. 17, 470–479 (1978).
    [CrossRef]
  13. C. Kittel and H. Kroemer, Thermal Physics (Freeman, San Francisco, Calif, 1980).
  14. G. C. Valley, “Short pulse grating formation in photorefractive materials,” IEEE J. Quantum Electron. QE-19, 1637–1645 (1983).
    [CrossRef]
  15. D. W. Vahey, “A nonlinear coupled-wave theory of holographic storage in ferroelectric materials,” J Appl. Phys. 46, 3510–3515 (1975).
    [CrossRef]
  16. M. Braun, Differential Equations and Their Applications (Springer-Verlag, Berlin, 1983).
  17. W. Phillips and D. L. Staebler, “Control of the Fe2+concentration of iron-doped lithium niobate,” J. Electron. Mater. 3, 601–617 (1974).
    [CrossRef]
  18. N. B. Baranova and B. Y. Zel’dovich, “Dislocations of the wave-front surface and zeros of the amplitude,” Sov. Phys. JETP,  62, 925–929 (1982).
  19. N. B. Baranova, A. V. Mamaev, N. F. Pilipetsky, V. V. Shkunov, and B. Y. Zel’dovich, “Wave-front dislocations: topological limitations for adaptive systems with phase conjugation,” J. Opt. Soc. Am. 73, 525–528 (1983).
    [CrossRef]

1991 (1)

E. S. Maniloff and K. M. Johnson, “Maximized photorefractive holographic storage,” J. Appl. Phys. 70, 4702–4707 (1991).
[CrossRef]

1988 (1)

J. M. Heaton and L. Solymar, “Transient effects during dynamic hologram formation in BSO crystals: theory and experiments,” IEEE J. Quantum Electron. 24, 558–567 (1988).
[CrossRef]

1985 (1)

J. M. Heaton and L. Solymar, “Transient energy transfer during hologram formation in photorefractive crystals,” Opt. Acta 32, 397–408 (1985).
[CrossRef]

1984 (1)

L. Solymar and J. M. Heaton, “Transient energy transfer in photorefractive materials; an analytic solution,” Opt. Commun. 51, 76–78 (1984).
[CrossRef]

1983 (3)

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

G. C. Valley, “Short pulse grating formation in photorefractive materials,” IEEE J. Quantum Electron. QE-19, 1637–1645 (1983).
[CrossRef]

N. B. Baranova, A. V. Mamaev, N. F. Pilipetsky, V. V. Shkunov, and B. Y. Zel’dovich, “Wave-front dislocations: topological limitations for adaptive systems with phase conjugation,” J. Opt. Soc. Am. 73, 525–528 (1983).
[CrossRef]

1982 (1)

N. B. Baranova and B. Y. Zel’dovich, “Dislocations of the wave-front surface and zeros of the amplitude,” Sov. Phys. JETP,  62, 925–929 (1982).

1980 (2)

V I. Belinicher and B. I. Sturman, “The photogalvanic effect in media lacking a center of symmetry,” Sov. Phys. Usp. 23, 199–223 (1980).
[CrossRef]

N. V. Kukhtarev, V. B. Markov, and S. G. Odulov, “Nonstationary energy exchange during interactions between two light beams in electrooptical crystals,” Sov. Phys. Tech. Phys. 25, 1109–1114 (1980).

1978 (1)

A. M. Glass, “The photorefractive effect,” Opt. Eng. 17, 470–479 (1978).
[CrossRef]

1977 (2)

V I. Belinicher, V K. Malinovski, and B. I. Sturman, “Photo-galvanic effect in a crystal with polar axis,” Sov. Phys. JETP,  46, 362–366 (1977).

N. Kukhtarev, V. Markov, and S. Odulov, “Transient energy transfer during hologram formation in LiNbO3in external electric field,” Opt. Commun. 23, 338–343 (1977).
[CrossRef]

1976 (1)

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

1975 (2)

A. M. Glass and D. von der Linde, “Photorefractive effects for reversible holographic storage of information,” App. Phys. 8, 85–100 (1975).
[CrossRef]

D. W. Vahey, “A nonlinear coupled-wave theory of holographic storage in ferroelectric materials,” J Appl. Phys. 46, 3510–3515 (1975).
[CrossRef]

1974 (1)

W. Phillips and D. L. Staebler, “Control of the Fe2+concentration of iron-doped lithium niobate,” J. Electron. Mater. 3, 601–617 (1974).
[CrossRef]

Baranova, N. B.

Belinicher, V I.

V I. Belinicher and B. I. Sturman, “The photogalvanic effect in media lacking a center of symmetry,” Sov. Phys. Usp. 23, 199–223 (1980).
[CrossRef]

V I. Belinicher, V K. Malinovski, and B. I. Sturman, “Photo-galvanic effect in a crystal with polar axis,” Sov. Phys. JETP,  46, 362–366 (1977).

Braun, M.

M. Braun, Differential Equations and Their Applications (Springer-Verlag, Berlin, 1983).

Glass, A. M.

A. M. Glass, “The photorefractive effect,” Opt. Eng. 17, 470–479 (1978).
[CrossRef]

A. M. Glass and D. von der Linde, “Photorefractive effects for reversible holographic storage of information,” App. Phys. 8, 85–100 (1975).
[CrossRef]

Heaton, J. M.

J. M. Heaton and L. Solymar, “Transient effects during dynamic hologram formation in BSO crystals: theory and experiments,” IEEE J. Quantum Electron. 24, 558–567 (1988).
[CrossRef]

J. M. Heaton and L. Solymar, “Transient energy transfer during hologram formation in photorefractive crystals,” Opt. Acta 32, 397–408 (1985).
[CrossRef]

L. Solymar and J. M. Heaton, “Transient energy transfer in photorefractive materials; an analytic solution,” Opt. Commun. 51, 76–78 (1984).
[CrossRef]

Johnson, K. M.

E. S. Maniloff and K. M. Johnson, “Maximized photorefractive holographic storage,” J. Appl. Phys. 70, 4702–4707 (1991).
[CrossRef]

Kittel, C.

C. Kittel and H. Kroemer, Thermal Physics (Freeman, San Francisco, Calif, 1980).

Klein, M. B.

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

Kroemer, H.

C. Kittel and H. Kroemer, Thermal Physics (Freeman, San Francisco, Calif, 1980).

Kukhtarev, N.

N. Kukhtarev, V. Markov, and S. Odulov, “Transient energy transfer during hologram formation in LiNbO3in external electric field,” Opt. Commun. 23, 338–343 (1977).
[CrossRef]

Kukhtarev, N. V.

N. V. Kukhtarev, V. B. Markov, and S. G. Odulov, “Nonstationary energy exchange during interactions between two light beams in electrooptical crystals,” Sov. Phys. Tech. Phys. 25, 1109–1114 (1980).

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

Malinovski, V K.

V I. Belinicher, V K. Malinovski, and B. I. Sturman, “Photo-galvanic effect in a crystal with polar axis,” Sov. Phys. JETP,  46, 362–366 (1977).

Mamaev, A. V.

Maniloff, E. S.

E. S. Maniloff and K. M. Johnson, “Maximized photorefractive holographic storage,” J. Appl. Phys. 70, 4702–4707 (1991).
[CrossRef]

Markov, V.

N. Kukhtarev, V. Markov, and S. Odulov, “Transient energy transfer during hologram formation in LiNbO3in external electric field,” Opt. Commun. 23, 338–343 (1977).
[CrossRef]

Markov, V. B.

N. V. Kukhtarev, V. B. Markov, and S. G. Odulov, “Nonstationary energy exchange during interactions between two light beams in electrooptical crystals,” Sov. Phys. Tech. Phys. 25, 1109–1114 (1980).

Odulov, S.

N. Kukhtarev, V. Markov, and S. Odulov, “Transient energy transfer during hologram formation in LiNbO3in external electric field,” Opt. Commun. 23, 338–343 (1977).
[CrossRef]

Odulov, S. G.

N. V. Kukhtarev, V. B. Markov, and S. G. Odulov, “Nonstationary energy exchange during interactions between two light beams in electrooptical crystals,” Sov. Phys. Tech. Phys. 25, 1109–1114 (1980).

Phillips, W.

W. Phillips and D. L. Staebler, “Control of the Fe2+concentration of iron-doped lithium niobate,” J. Electron. Mater. 3, 601–617 (1974).
[CrossRef]

Pilipetsky, N. F.

Shkunov, V. V.

Solymar, L.

J. M. Heaton and L. Solymar, “Transient effects during dynamic hologram formation in BSO crystals: theory and experiments,” IEEE J. Quantum Electron. 24, 558–567 (1988).
[CrossRef]

J. M. Heaton and L. Solymar, “Transient energy transfer during hologram formation in photorefractive crystals,” Opt. Acta 32, 397–408 (1985).
[CrossRef]

L. Solymar and J. M. Heaton, “Transient energy transfer in photorefractive materials; an analytic solution,” Opt. Commun. 51, 76–78 (1984).
[CrossRef]

Staebler, D. L.

W. Phillips and D. L. Staebler, “Control of the Fe2+concentration of iron-doped lithium niobate,” J. Electron. Mater. 3, 601–617 (1974).
[CrossRef]

Sturman, B. I.

V I. Belinicher and B. I. Sturman, “The photogalvanic effect in media lacking a center of symmetry,” Sov. Phys. Usp. 23, 199–223 (1980).
[CrossRef]

V I. Belinicher, V K. Malinovski, and B. I. Sturman, “Photo-galvanic effect in a crystal with polar axis,” Sov. Phys. JETP,  46, 362–366 (1977).

Vahey, D. W.

D. W. Vahey, “A nonlinear coupled-wave theory of holographic storage in ferroelectric materials,” J Appl. Phys. 46, 3510–3515 (1975).
[CrossRef]

Valley, G. C.

G. C. Valley, “Short pulse grating formation in photorefractive materials,” IEEE J. Quantum Electron. QE-19, 1637–1645 (1983).
[CrossRef]

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

von der Linde, D.

A. M. Glass and D. von der Linde, “Photorefractive effects for reversible holographic storage of information,” App. Phys. 8, 85–100 (1975).
[CrossRef]

Zel’dovich, B. Y.

App. Phys. (1)

A. M. Glass and D. von der Linde, “Photorefractive effects for reversible holographic storage of information,” App. Phys. 8, 85–100 (1975).
[CrossRef]

IEEE J. Quantum Electron. (2)

G. C. Valley, “Short pulse grating formation in photorefractive materials,” IEEE J. Quantum Electron. QE-19, 1637–1645 (1983).
[CrossRef]

J. M. Heaton and L. Solymar, “Transient effects during dynamic hologram formation in BSO crystals: theory and experiments,” IEEE J. Quantum Electron. 24, 558–567 (1988).
[CrossRef]

J Appl. Phys. (1)

D. W. Vahey, “A nonlinear coupled-wave theory of holographic storage in ferroelectric materials,” J Appl. Phys. 46, 3510–3515 (1975).
[CrossRef]

J. Appl. Phys. (1)

E. S. Maniloff and K. M. Johnson, “Maximized photorefractive holographic storage,” J. Appl. Phys. 70, 4702–4707 (1991).
[CrossRef]

J. Electron. Mater. (1)

W. Phillips and D. L. Staebler, “Control of the Fe2+concentration of iron-doped lithium niobate,” J. Electron. Mater. 3, 601–617 (1974).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Acta (1)

J. M. Heaton and L. Solymar, “Transient energy transfer during hologram formation in photorefractive crystals,” Opt. Acta 32, 397–408 (1985).
[CrossRef]

Opt. Commun. (2)

L. Solymar and J. M. Heaton, “Transient energy transfer in photorefractive materials; an analytic solution,” Opt. Commun. 51, 76–78 (1984).
[CrossRef]

N. Kukhtarev, V. Markov, and S. Odulov, “Transient energy transfer during hologram formation in LiNbO3in external electric field,” Opt. Commun. 23, 338–343 (1977).
[CrossRef]

Opt. Eng. (2)

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

A. M. Glass, “The photorefractive effect,” Opt. Eng. 17, 470–479 (1978).
[CrossRef]

Sov. Phys. JETP (2)

N. B. Baranova and B. Y. Zel’dovich, “Dislocations of the wave-front surface and zeros of the amplitude,” Sov. Phys. JETP,  62, 925–929 (1982).

V I. Belinicher, V K. Malinovski, and B. I. Sturman, “Photo-galvanic effect in a crystal with polar axis,” Sov. Phys. JETP,  46, 362–366 (1977).

Sov. Phys. Tech. Phys. (1)

N. V. Kukhtarev, V. B. Markov, and S. G. Odulov, “Nonstationary energy exchange during interactions between two light beams in electrooptical crystals,” Sov. Phys. Tech. Phys. 25, 1109–1114 (1980).

Sov. Phys. Usp. (1)

V I. Belinicher and B. I. Sturman, “The photogalvanic effect in media lacking a center of symmetry,” Sov. Phys. Usp. 23, 199–223 (1980).
[CrossRef]

Sov. Tech. Phys. Lett. (1)

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

Other (2)

C. Kittel and H. Kroemer, Thermal Physics (Freeman, San Francisco, Calif, 1980).

M. Braun, Differential Equations and Their Applications (Springer-Verlag, Berlin, 1983).

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Figures (22)

Fig. 1
Fig. 1

Experimental setup used to simultaneously investigate beam coupling and diffraction efficiencies during recording.

Fig. 2
Fig. 2

Dynamic energy transfer in LiNbO3 for E0 = 10 kV/cm. a, Intensity IS for a beam ratio IR/IS = 10:1. b, Intensity IR for a beam ratio IR/IS = 1:10.

Fig. 3
Fig. 3

Absolute intensities recorded versus recording time for a unity recording modulation depth.

Fig. 4
Fig. 4

Theoretical fit to the experimentally measured intensity coupling and diffraction efficiency, m = 1.

Fig. 5
Fig. 5

Effect of the grating phase on the intensities. The solid curve shows the grating phase at the crystal output, which corresponds to the left-hand vertical axis. The dashed curve shows the relative intensity of one of the recording beams, which corresponds to the right-hand vertical axis. The dotted line indicates the phase equal to π.

Fig. 6
Fig. 6

Variation of the grating phase with thickness and recording time. The thickness axis ranges from 0 to 2.5 mm, and the time axis is from 0 to 400 s.

Fig. 7
Fig. 7

Relative intensities recorded versus recording time for a beam ratio IR/IS of 10:1.

Fig. 8
Fig. 8

Theoretical transmitted intensity for the entire recording range; beam ratio IR/IS = 1:1.

Fig. 9
Fig. 9

Theoretical transmitted intensity for the entire recording range; beam ratio IR/IS = 10:1.

Fig. 10
Fig. 10

Recording phase for a photogalvanic field of 10 kV/cm. The thickness axis ranges from 0 to 2.5 mm, while the time axis is from 0 to 20τdi.

Fig. 11
Fig. 11

Recording phase for a photogalvanic field of 15 kV/cm. The thickness axis ranges from 0 to 2.5 mm, while the time axis is from 0 to 20τdi.

Fig. 12
Fig. 12

Recording phase for a photogalvanic field of 15 kV/cm, with the 2π phase jump in the thickness dimension removed. The thickness axis ranges from 0 to 2.5 mm, while the time axis is from 0 to 20τdi.

Fig. 13
Fig. 13

a, Optical phase and b, intensity contours during recording.

Fig. 14
Fig. 14

Recording wave S versus crystal thickness with a zero at point z0. a, Intensity IS; b, wave S in phase space.

Fig. 15
Fig. 15

Space-charge field magnitude through the crystal depth for a photogalvanic field of 15 kV/cm, IR/IS = 10:1. Again the thickness is from 0 to 2.5 mm, while the time is from 0 to 20τdi.

Fig. 16
Fig. 16

Space-charge field phase through the crystal depth for a photogalvanic field of 15 kV/cm, IR/IS = 10:1. The thickness is from 0 to 2.5 mm, while the time is from 0 to 20τdi.

Fig. 17
Fig. 17

Recording phase through the crystal depth for a photogalvanic field of 62 kV/cm, IR/IS = 1:1. The thickness is plotted from 0 to 2.5 mm, while the time is plotted from 0 to 40τdi.

Fig. 18
Fig. 18

Contours of the two recording intensities during recording.

Fig. 19
Fig. 19

Minima of the two recording beams. a, Contours corresponding to IS = 0. b, Contours corresponding to IR = 0.

Fig. 20
Fig. 20

Recording intensity IS versus thickness and time. The thickness is shown from 0 to 2.5 mm, while time ranges from 0 to 40τdi.

Fig. 21
Fig. 21

Phase of the recorded space-charge field for a photogalvanic field of 62 kV/cm. The thickness is shown from 0 to 2.5 mm, while time ranges from 0 to 40τdi.

Fig. 22
Fig. 22

a, Zeros of the space-charge field during recording. b, Zeros of the space charge field during recording, with the intensity zeros indicated by the dotted curves.

Equations (42)

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n t N D + t = 1 e j x ,
N D + t = s I ( N D N D + ) γ R n N D + ,
j = e μ n E k B T μ n x + p I ,
E x = e ( n + N A N D + ) ,
2 δ E t 2 + δ E t ( 1 τ + + 1 τ di + 1 τ D + i τ E ) + δ E ( 1 τ + τ di + 1 τ D τ I + i τ E τ I ) = 2 e K g s R * S ( N D N A n 0 ) ( i τ D 1 τ E ) 2 p ( t ( R * S ) + R * S τ R ) .
τ + = ( s I 0 + 2 γ R n 0 + γ R N A ) 1 ,
τ di = / ( e μ n 0 ) ,
τ D = e / ( μ k B T K g 2 ) ,
τ E = ( K g μ E 0 ) 1 ,
τ I = ( s I 0 + γ R n 0 ) 1 .
τ R = ( γ R N A ) 1 .
T 1 = 1 τ + + 1 τ di + 1 τ D + i τ E ,
T 2 = 1 τ + τ di + 1 τ D τ I + i τ E τ I .
2 δ E t 2 + δ E t ( T 1 ) + δ E ( T 2 ) = e K g s R * S ( N D N A n 0 ) × ( i τ D 1 τ E ) p [ t ( R * S ) + R * S τ R ] .
R * S t + R * S τ R R * S τ R ,
Δ n = ½ n e 3 r 33 δ E ,
δ E t ( T 1 ) + δ E ( T 2 ) = σ R * S ,
R z = i Γ δ E * S ,
S z = i Γ δ E R ,
σ = 2 e K g s ( N D N A n 0 ) ( i τ D 1 τ E ) 2 p ( 1 τ R ) ,
Γ = π n e 3 r 33 2 λ cos θ ,
δ E ( m Δ t ) = σ T 2 R * S [ 1 exp ( T 2 Δ t T 1 ) ] + δ E [ ( m 1 ) Δ t ] exp ( T 2 Δ t T 1 ) ,
n e = 2.259 , γ R = 10 13 m 3 / S , T = 293 ° K , N A = 6 × 10 16 cm 3 , s = 29.0 , μ = 0.8 × 10 4 m 2 / ( Vs ) , r 33 = 33 pm / V .
ϕ g = ϕ E ( ϕ S ϕ R ) ,
N D + = N D 0 + + δ N D + ,
E = E 0 + δ E ,
j = j 0 + δ j ,
I = I 0 + δ I ,
δ I = m I 0 exp [ i K g x + ϕ ( z , t ) ] .
δ n t δ N D + t = μ E 0 δ n x μ n 0 δ E x + k B T μ e 2 ( δ n ) x 2 p e δ I z ,
δ N D + t = s I 0 N D + + s δ I ( N D N A n 0 ) γ R n 0 δ N D + γ R n 0 δ n γ R N A δ n ,
δ E x = e ( δ n δ N D + ) .
e 2 δ E t x = μ E 0 δ n x μ n 0 δ E x + k B T μ e 2 ( δ n ) x 2 p e δ I x .
x δ I = i K g δ I ,
x δ n = i K g δ n ,
x δ E = i K g δ E ,
i K g e δ E t i K g μ E 0 δ n i K g μ n 0 δ E K g 2 k B T μ e δ n i K g p e δ I .
δ N D + = ( i K g / e ) δ E + δ n
δ n t + δ n τ + = ( i K g s I 0 e + i K g γ R n 0 e ) δ E + i K g e δ E t + s δ I ( N D N A n 0 ) ,
τ + ( s I 0 + 2 γ R n 0 + γ R N A ) 1 .
i K g e ( 2 δ E t 2 + 1 τ + δ E t ) + i K g μ n 0 ( δ E τ + + δ E t ) = i K g μ E 0 ( δ n t + δ n τ + ) k B T μ K g 2 e ( δ n t + δ n τ + ) i K g p e ( δ I t + δ I τ + ) .
2 δ E t 2 + δ E t ( T 1 ) + δ E ( T 2 ) = e K g s δ I ( N D N A n 0 ) ( i τ D 1 τ E ) p ( δ I t + δ I τ R ) ,

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