Abstract

A theoretical analysis is presented of two-wave mixing in photorefractive materials in which the complex amplitude of a weak signal beam has slowly varying but otherwise arbitrary spatial dependence and arbitrary time variation. The set of coupled partial differential equations that couple the dynamics of the optical fields to the photorefractive medium is solved analytically to give the complex amplitude of the transmitted signal in the undepleted pump regime. The results are compared with known results for the transient two-wave mixing of plane waves.

© 1993 Optical Society of America

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  1. P. Gunter, J. P. Huignard, eds., Topics in Applied Physics (Springer-Verlag, New York, 1988), Vol. 61.
    [CrossRef]
  2. P. Gunter, J. P. Huignard, eds., Topics in Applied Physics (Springer-Verlag, New York, 1989), Vol. 62.
  3. N. V. Kukhtarev, V. B. Markov, S. G. Odulov, “Transient energy transfer during hologram formation in LiNbO3 in external electric field,” Opt. Commun. 23, 338–343 (1977).
    [CrossRef]
  4. J. M. Heaton, L. Solymar, “Transient energy transfer during hologram formation in photorefractive crystals,” Opt. Acta 32, 397–408 (1985).
    [CrossRef]
  5. M. Cronin-Golomb, “Analytic solution for photorefractive two-beam coupling with time-varying signals,” in Photorefractive Materials, Effects, and Devices, Vol. 17 of 1987 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1987), pp. 142–145.
  6. M. Horowitz, D. Kligler, B. Fischer, “Time-dependent behavior of photorefractive two- and four-wave mixing,” J. Opt. Soc. Am. B 8, 2204–2217 (1991).
    [CrossRef]
  7. G. Hamel de Montchenault, J. P. Huignard, “Two wave mixing with time modulated signal in Bi12SiO20 theory and application to homodyne front detection,” J. Appl. Phys. 63, 624–627 (1988).
    [CrossRef]
  8. F. Davidson, L. Boutsikaris, “Homodyne detection using photorefractive materials as beam splitters,” Opt. Eng. 29, 369–377 (1990).
    [CrossRef]
  9. A. L. Smirl, G. C. Valley, K. M. Bohnert, T. F. Bogess, “Picosecond photorefractive and free-carrier transient energy transfer in GaAs at 1 μm,” IEEE J. Quantum Electron. 24, 289–303 (1988).
    [CrossRef]
  10. X. S. Yao, V. Dominic, J. Feinberg, “Theory of beam coupling and pulse shaping of mode-locked laser pulses in a photorefractive crystal,” J. Opt. Soc. Am. B 7, 2347–2355 (1990).
    [CrossRef]
  11. G. C. Papen, B. E. A. Saleh, J. A. Tataronis, “Analysis of transient phase conjugation in photorefractive media,” J. Opt. Soc. Am. B 5, 1763–1774 (1988).
    [CrossRef]
  12. G. C. Papen, B. E. A. Saleh, J. A. Tataronis, “Impulse response function of photorefractive phase conjugation,” Opt. Lett. 14, 287–289 (1989).
    [CrossRef] [PubMed]
  13. F. Vachss, P. Yeh, “Image degradation mechanisms in photorefractive amplifiers,” J. Opt. Soc. Am. B 6, 1834–1844 (1989).
    [CrossRef]
  14. L. Boutsikaris, F. Davidson, “Photorefractive two wave mixing of optical beams with arbitrary intensity profiles,” in Conference on Lasers and Electro-Optics, Vol. 10 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), p. 184.
  15. W. Krolikowski, M. Cronin-Golomb, “Photorefractive wave mixing with finite beams,” in OSA Annual Meeting, Vol. 17 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), p. 21.
  16. P. Refregier, L. Solymar, H. Rajbenbach, J. P. Huignard, “Two-beam coupling in photorefractive Bi12SiO20 with moving grating: theory and experiments,” J. Appl. Phys. 58, 45–57 (1985).
    [CrossRef]
  17. N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. Soskin, V. L. Vinetskii, “Holographic storage in electrooptic crystals. I. Steady state,” Ferroelectrics 22, 949–960 (1979).
    [CrossRef]
  18. E. Zauderer, Partial Differential Equations of Applied Mathematics (Wiley, New York, 1983).

1991 (1)

1990 (2)

F. Davidson, L. Boutsikaris, “Homodyne detection using photorefractive materials as beam splitters,” Opt. Eng. 29, 369–377 (1990).
[CrossRef]

X. S. Yao, V. Dominic, J. Feinberg, “Theory of beam coupling and pulse shaping of mode-locked laser pulses in a photorefractive crystal,” J. Opt. Soc. Am. B 7, 2347–2355 (1990).
[CrossRef]

1989 (2)

1988 (3)

G. Hamel de Montchenault, J. P. Huignard, “Two wave mixing with time modulated signal in Bi12SiO20 theory and application to homodyne front detection,” J. Appl. Phys. 63, 624–627 (1988).
[CrossRef]

G. C. Papen, B. E. A. Saleh, J. A. Tataronis, “Analysis of transient phase conjugation in photorefractive media,” J. Opt. Soc. Am. B 5, 1763–1774 (1988).
[CrossRef]

A. L. Smirl, G. C. Valley, K. M. Bohnert, T. F. Bogess, “Picosecond photorefractive and free-carrier transient energy transfer in GaAs at 1 μm,” IEEE J. Quantum Electron. 24, 289–303 (1988).
[CrossRef]

1985 (2)

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

P. Refregier, L. Solymar, H. Rajbenbach, J. P. Huignard, “Two-beam coupling in photorefractive Bi12SiO20 with moving grating: theory and experiments,” J. Appl. Phys. 58, 45–57 (1985).
[CrossRef]

1979 (1)

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

1977 (1)

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, “Transient energy transfer during hologram formation in LiNbO3 in external electric field,” Opt. Commun. 23, 338–343 (1977).
[CrossRef]

Bogess, T. F.

A. L. Smirl, G. C. Valley, K. M. Bohnert, T. F. Bogess, “Picosecond photorefractive and free-carrier transient energy transfer in GaAs at 1 μm,” IEEE J. Quantum Electron. 24, 289–303 (1988).
[CrossRef]

Bohnert, K. M.

A. L. Smirl, G. C. Valley, K. M. Bohnert, T. F. Bogess, “Picosecond photorefractive and free-carrier transient energy transfer in GaAs at 1 μm,” IEEE J. Quantum Electron. 24, 289–303 (1988).
[CrossRef]

Boutsikaris, L.

F. Davidson, L. Boutsikaris, “Homodyne detection using photorefractive materials as beam splitters,” Opt. Eng. 29, 369–377 (1990).
[CrossRef]

L. Boutsikaris, F. Davidson, “Photorefractive two wave mixing of optical beams with arbitrary intensity profiles,” in Conference on Lasers and Electro-Optics, Vol. 10 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), p. 184.

Cronin-Golomb, M.

W. Krolikowski, M. Cronin-Golomb, “Photorefractive wave mixing with finite beams,” in OSA Annual Meeting, Vol. 17 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), p. 21.

M. Cronin-Golomb, “Analytic solution for photorefractive two-beam coupling with time-varying signals,” in Photorefractive Materials, Effects, and Devices, Vol. 17 of 1987 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1987), pp. 142–145.

Davidson, F.

F. Davidson, L. Boutsikaris, “Homodyne detection using photorefractive materials as beam splitters,” Opt. Eng. 29, 369–377 (1990).
[CrossRef]

L. Boutsikaris, F. Davidson, “Photorefractive two wave mixing of optical beams with arbitrary intensity profiles,” in Conference on Lasers and Electro-Optics, Vol. 10 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), p. 184.

Dominic, V.

Feinberg, J.

Fischer, B.

Hamel de Montchenault, G.

G. Hamel de Montchenault, J. P. Huignard, “Two wave mixing with time modulated signal in Bi12SiO20 theory and application to homodyne front detection,” J. Appl. Phys. 63, 624–627 (1988).
[CrossRef]

Heaton, J. M.

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

Horowitz, M.

Huignard, J. P.

G. Hamel de Montchenault, J. P. Huignard, “Two wave mixing with time modulated signal in Bi12SiO20 theory and application to homodyne front detection,” J. Appl. Phys. 63, 624–627 (1988).
[CrossRef]

P. Refregier, L. Solymar, H. Rajbenbach, J. P. Huignard, “Two-beam coupling in photorefractive Bi12SiO20 with moving grating: theory and experiments,” J. Appl. Phys. 58, 45–57 (1985).
[CrossRef]

Kligler, D.

Krolikowski, W.

W. Krolikowski, M. Cronin-Golomb, “Photorefractive wave mixing with finite beams,” in OSA Annual Meeting, Vol. 17 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), p. 21.

Kukhtarev, N. V.

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

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, “Transient energy transfer during hologram formation in LiNbO3 in external electric field,” Opt. Commun. 23, 338–343 (1977).
[CrossRef]

Markov, V. B.

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

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, “Transient energy transfer during hologram formation in LiNbO3 in external electric field,” Opt. Commun. 23, 338–343 (1977).
[CrossRef]

Odulov, S. G.

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

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, “Transient energy transfer during hologram formation in LiNbO3 in external electric field,” Opt. Commun. 23, 338–343 (1977).
[CrossRef]

Papen, G. C.

Rajbenbach, H.

P. Refregier, L. Solymar, H. Rajbenbach, J. P. Huignard, “Two-beam coupling in photorefractive Bi12SiO20 with moving grating: theory and experiments,” J. Appl. Phys. 58, 45–57 (1985).
[CrossRef]

Refregier, P.

P. Refregier, L. Solymar, H. Rajbenbach, J. P. Huignard, “Two-beam coupling in photorefractive Bi12SiO20 with moving grating: theory and experiments,” J. Appl. Phys. 58, 45–57 (1985).
[CrossRef]

Saleh, B. E. A.

Smirl, A. L.

A. L. Smirl, G. C. Valley, K. M. Bohnert, T. F. Bogess, “Picosecond photorefractive and free-carrier transient energy transfer in GaAs at 1 μm,” IEEE J. Quantum Electron. 24, 289–303 (1988).
[CrossRef]

Solymar, L.

P. Refregier, L. Solymar, H. Rajbenbach, J. P. Huignard, “Two-beam coupling in photorefractive Bi12SiO20 with moving grating: theory and experiments,” J. Appl. Phys. 58, 45–57 (1985).
[CrossRef]

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

Soskin, M.

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

Tataronis, J. A.

Vachss, F.

Valley, G. C.

A. L. Smirl, G. C. Valley, K. M. Bohnert, T. F. Bogess, “Picosecond photorefractive and free-carrier transient energy transfer in GaAs at 1 μm,” IEEE J. Quantum Electron. 24, 289–303 (1988).
[CrossRef]

Vinetskii, V. L.

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

Yao, X. S.

Yeh, P.

Zauderer, E.

E. Zauderer, Partial Differential Equations of Applied Mathematics (Wiley, New York, 1983).

Ferroelectrics (1)

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

IEEE J. Quantum Electron. (1)

A. L. Smirl, G. C. Valley, K. M. Bohnert, T. F. Bogess, “Picosecond photorefractive and free-carrier transient energy transfer in GaAs at 1 μm,” IEEE J. Quantum Electron. 24, 289–303 (1988).
[CrossRef]

J. Appl. Phys. (2)

G. Hamel de Montchenault, J. P. Huignard, “Two wave mixing with time modulated signal in Bi12SiO20 theory and application to homodyne front detection,” J. Appl. Phys. 63, 624–627 (1988).
[CrossRef]

P. Refregier, L. Solymar, H. Rajbenbach, J. P. Huignard, “Two-beam coupling in photorefractive Bi12SiO20 with moving grating: theory and experiments,” J. Appl. Phys. 58, 45–57 (1985).
[CrossRef]

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

Opt. Acta (1)

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

Opt. Commun. (1)

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, “Transient energy transfer during hologram formation in LiNbO3 in external electric field,” Opt. Commun. 23, 338–343 (1977).
[CrossRef]

Opt. Eng. (1)

F. Davidson, L. Boutsikaris, “Homodyne detection using photorefractive materials as beam splitters,” Opt. Eng. 29, 369–377 (1990).
[CrossRef]

Opt. Lett. (1)

Other (6)

E. Zauderer, Partial Differential Equations of Applied Mathematics (Wiley, New York, 1983).

L. Boutsikaris, F. Davidson, “Photorefractive two wave mixing of optical beams with arbitrary intensity profiles,” in Conference on Lasers and Electro-Optics, Vol. 10 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), p. 184.

W. Krolikowski, M. Cronin-Golomb, “Photorefractive wave mixing with finite beams,” in OSA Annual Meeting, Vol. 17 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), p. 21.

M. Cronin-Golomb, “Analytic solution for photorefractive two-beam coupling with time-varying signals,” in Photorefractive Materials, Effects, and Devices, Vol. 17 of 1987 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1987), pp. 142–145.

P. Gunter, J. P. Huignard, eds., Topics in Applied Physics (Springer-Verlag, New York, 1988), Vol. 61.
[CrossRef]

P. Gunter, J. P. Huignard, eds., Topics in Applied Physics (Springer-Verlag, New York, 1989), Vol. 62.

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Figures (13)

Fig. 1
Fig. 1

Geometry of two-wave mixing: As(r, t) and Ap are the complex amplitudes of the signal and pump beam, respectively; k1 and k2 are the corresponding wave vectors.

Fig. 2
Fig. 2

Amplitude of the transmitted signal field as a function of the normalized time tg at the entrance (z = 0) and at the output (z = d) of the crystal. The profile along x = y = 0 is shown; Γd = 6, a = 0.5.

Fig. 3
Fig. 3

Same as in Fig. 2 but Γd = 3.

Fig. 4
Fig. 4

Same as in Fig. 2 but Γd = −3.

Fig. 5
Fig. 5

Same as in Fig. 3 but a = 4.

Fig. 6
Fig. 6

Spatial and temporal profile of the amplitude of the signal field (at y = 0) at the entrance of the crystal; a = 0.5.

Fig. 7
Fig. 7

Spatial and temporal profile of the amplitude of the transmitted signal at the output of the crystal (z = d); Γd = 6, d = 1.5, θ1 = 15°, a = 0.5.

Fig. 8
Fig. 8

Same as in Fig. 7 but Γd = −3.

Fig. 9
Fig. 9

Amplitude of the transmitted signal field as a function of the normalized time tg at the entrance (z = 0) and at the output (z = d) of the crystal. The profile along x = y = 0 is shown; Γd = 9.

Fig. 10
Fig. 10

Same as in Fig. 9 but Γd = −9.

Fig. 11
Fig. 11

Spatial and temporal profile of the amplitude of the signal field (at y = 0) at the entrance of the crystal.

Fig. 12
Fig. 12

Spatial and temporal profile of the amplitude of the transmitted signal at the output of the crystal (z = d); Γd = 9, d = 2, θ1 = 20°.

Fig. 13
Fig. 13

Same as in Fig. 12, but Γd = −9.

Equations (45)

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E ( r , t ) = A s ( r , t ) exp j ( k 1 · r w t ) + A p exp j ( k 2 · r w t ) + c . c ,
t E sc ( r , t ) + g E sc ( r , t ) = h A s * ( r , t ) A p ,
g = 1 τ g + j w g ,
h = E sc S g I p .
2 E ( r , t ) = 1 c 2 [ n s 2 + 2 n s Δ n ( r , t ) ] 2 t 2 E ( r , t ) ,
Δ n ( r , t ) = 1 4 n s 3 r eff ( E sc ( r , t ) + c . c )
k 1 · A s * ( r , t ) + w υ 2 t A s * ( r , t ) = j w C υ r eff E sc ( r , t ) A p * ,
cos θ 1 z A s * ( r , t ) + sin θ 1 x A s * ( r , t ) + 1 υ t A s * ( r , t ) = j C r eff E sc ( r , t ) A p * .
cos θ 1 z A s * ( r , Ω ) + sin θ 1 x A s * ( r , Ω ) + j G ( Ω ) A s * ( r , Ω ) = 0 ,
G ( Ω ) = Ω υ + C r eff h g + j Ω | A p | 2 .
x = τ , z = 0 , A s * ( y , Ω ) = f ( τ , y , Ω ) .
d x / d s = sin θ 1 , d z / d s = cos θ 1 , d A s * ( y , Ω ) / d s = j G ( Ω ) A s * ( y , Ω ) .
x = sin θ 1 s + τ , z = cos θ 1 s , A s * ( s , τ , y , Ω ) = f ( s = 0 , τ , y , Ω ) exp [ j G ( Ω ) s ] .
A s * ( r , Ω ) = A s * ( x z tan θ 1 , y , 0 , Ω ) × exp [ j G ( Ω ) z / cos θ 1 ] .
A s * ( r , Ω ) = A s * ( x z tan θ 1 , y , 0 , Ω ) exp ( j τ c Ω ) ,
I s ( r , t ) = I s ( x z tan θ 1 , y , 0 , t τ c ) ,
G ( Ω ) z cos θ 1 = τ c τ g w N + l C r eff h I p τ g 1 + j ( δ 1 + w N ) ,
G ( Ω ) z cos θ 1 = γ l 1 + j δ 1 1 + j ( δ 1 + w N ) ,
G ( Ω ) z cos θ 1 = γ l Δ j ( w N j Δ ) ,
A s * ( r , Ω ) = A s * ( x z tan θ 1 , y , 0 , Ω ) exp ( D j ( w N j Δ ) ) ,
exp ( D j ( w N j Δ ) ) 1 τ g exp ( Δ t τ g ) × { δ ( t τ g ) ( D τ g t ) 1 / 2 J 1 [ 2 ( D t τ g ) 1 / 2 ] u ( t τ g ) } .
A s * ( r , t ) = A s * ( x z tan θ 1 , y , 0 , t ) 1 τ g exp ( Δ t τ g ) δ ( t τ g ) A s * ( x z tan θ 1 , y , 0 , t ) 1 τ g exp ( Δ t τ g ) ( D τ g t ) 1 / 2 J 1 [ 2 ( D t τ g ) 1 / 2 ] u ( t τ g ) .
0 t A s * ( x z tan θ 1 , y , 0 , T ) 1 τ g exp ( Δ ( t T ) τ g ) × ( D τ g t T ) 1 / 2 J 1 { 2 [ D ( t T ) τ g ] 1 / 2 } d T .
0 t A s * ( x z tan θ 1 , y , 0 , T ) exp [ Δ ( t T ) τ g ] × ξ η [ J 0 ( ξ ) ξ ] d T = 0 t A s * ( x z tan θ 1 , y , 0 , T ) × exp [ Δ ( t T ) τ g ] J 0 [ 2 ( D η / τ g ) 1 / 2 ] η | η = t T d T .
A s * ( r , t ) = A s * ( x z tan θ 1 , y , 0 , t ) + 0 t A s * ( x z tan θ 1 , y , 0 , T ) exp [ Δ ( t T ) τ g ] × η J 0 [ 2 ( j γ z Δ η τ g cos θ 1 ) 1 / 2 ] | η = t T d T .
A s * ( x , y , z , t ) = A s * ( x z tan θ 1 , y , 0 , 0 ) K ( z , t ) + 0 t Δ A s * ( x z tan θ 1 , y , 0 , T ) τ g K ( z , t T ) d T + 0 t A s * ( x z tan θ 1 , y , 0 , T ) T K ( z , t T ) d T ,
lim t 0 t K ( z , t T ) d T = 0 K ( z , T ) d T = 0 exp ( α T ) J 0 ( β T ) d T ,
lim t 0 t K ( z , t T ) d T = 1 α exp ( β 2 / 4 α ) = τ g Δ exp ( j γ z / cos θ 1 ) .
j γ z cos θ 1 = ( C r eff E sc S ) z cos θ 1 = C r eff | E sc S | z cos θ 1 = Γ z / 2 ,
Γ = w n s 3 r eff | E sc S | 2 c cos θ 1 = | k 1 | s r eff | E sc S | 2 cos θ 1 .
lim t A s * ( r , t ) = A s * ( x z tan θ 1 , y , 0 ) exp ( Γ z / 2 ) ,
lim t I s ( r , t ) = I s ( x z tan θ 1 , y , 0 ) exp ( Γ z ) .
E sc S = | E sc S | exp ( j ϕ ) ,
| E sc S | = E q [ E 0 2 + E D 2 E 0 2 + ( E D + E q ) 2 ] 1 / 2 ,
tan ϕ = E D E 0 ( 1 + E D E q + E 0 2 E D E q ) ,
τ g = τ d i ( 1 + τ R / τ D ) 2 + ( τ R / τ E ) 2 ( 1 + τ R τ d i / τ D τ I ) ( 1 + τ R / τ D ) + ( τ R / τ E ) 2 ( τ d i / τ I ) ,
w g = 1 τ d i ( τ R / τ E ) ( τ d i / τ I 1 ) ( 1 + τ R / τ D ) 2 + ( τ R / τ E ) 2 ,
τ d i = γ R N A e μ s N D I p ,
τ E = 1 k g μ E 0 ,
τ D = e μ K B T k g 2 ,
τ R = 1 γ R N A ,
τ I = N A s N D I p ,
A s * ( z , t ) = A s * ( 0 , 0 ) K ( z , t ) + 0 t A s * ( 0 , T ) τ K ( z , t T ) d T + 0 t A s * ( 0 , T ) T K ( z , t T ) d T ,
0 t A s * ( 0 , T ) T K ( z , t T ) d T = A s * ( 0 , t ) K ( z , 0 ) A s * ( 0 , 0 ) K ( z , t ) + 0 t A s * ( 0 , T ) K ( z , η ) η | η= t T d T .
A s * ( z , t ) = A s * ( 0 , t ) + 0 t A s * ( 0 , t ) exp ( t T τ ) η × J 0 [ 2 ( γ z η τ ) 1 / 2 ] | η = t T d T .

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