Abstract

We present a formalism and a technique for analyzing the propagation of optical signals through semiconductor traveling-wave amplifiers. The formalism retains, in particular, the spontaneous recombination term, and the newly developed technique permits analytic, real-time solutions to be obtained, so that both the transient and the steady-state regimes are characterized. This end has been achieved by iteratively applying the method of in order to solve the set of coupled differential equations that result from the formulation of the characteristics physical problem. From the real-time solutions so obtained it is also possible easily to recognize the amplitude and phase-modulation effects that occur not only in the steady state but also in the transient regime.

© 1992 Optical Society of America

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References

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  1. R. Bellman, G. Birnbaum, W. G. Wagner, “Transmission of monochromatic radiation in a two level system,” J. Appl. Phys. 34, 780–782 (1963).
    [CrossRef]
  2. L. M. Frantz, J. S. Nodvik, “Theory of pulse propagation in a laser amplifier,” J. Appl. Phys. 34, 2346–2349 (1963).
    [CrossRef]
  3. M. J. O’Mahony, “Semiconductor laser optical amplifiers for use in future fiber systems,” IEEE. J. Lightwave Technol. 6, 531–534 (1988).
    [CrossRef]
  4. I. W. Marshall, D. M. Spirit, M. J. O’Mahonoy, “Picosecond pulse response of a travelling wave semiconductor laser amplifier,” Electron. Lett. 23,(1987).
  5. G. P. Agrawal, N. A. Olsson, “Self-phase modulation and spectral broadening of optical pulses in semiconductor laser amplifiers,” IEEE J. Quantum Electron. 25, 2297–2306 (1989).
    [CrossRef]
  6. G. P. Agrawal, “Population pulsations and nondegenerate four-wave mixing in semiconductor lasers and amplifiers,” J. Opt. Soc. Am. B 5, 147–158 (1988).
    [CrossRef]
  7. T. E. Darcie, R. M. Jopson, R. W. Tkach, “Intermodulation distortion in optical amplifiers from carrier-density modulation,” Electron. Lett. 23, 1392–1394 (1987).
    [CrossRef]
  8. T. E. Darcie, R. M. Jopson, “Interactions in optical amplifiers for multifrequency lightwave system,” Electron. Lett. 24, 638–640 (1988).
    [CrossRef]
  9. M. J. Adams, J. V. Collins, I. D. Henning, “Analysis of semiconductor laser optical amplifiers,” IEE Proc. J. Optoelectron. 132, 58–63 (1985).
    [CrossRef]
  10. G. F. Carrier, C. E. Pearson, Partial Differential Equations (Academic, New York, 1988).

1989 (1)

G. P. Agrawal, N. A. Olsson, “Self-phase modulation and spectral broadening of optical pulses in semiconductor laser amplifiers,” IEEE J. Quantum Electron. 25, 2297–2306 (1989).
[CrossRef]

1988 (3)

T. E. Darcie, R. M. Jopson, “Interactions in optical amplifiers for multifrequency lightwave system,” Electron. Lett. 24, 638–640 (1988).
[CrossRef]

M. J. O’Mahony, “Semiconductor laser optical amplifiers for use in future fiber systems,” IEEE. J. Lightwave Technol. 6, 531–534 (1988).
[CrossRef]

G. P. Agrawal, “Population pulsations and nondegenerate four-wave mixing in semiconductor lasers and amplifiers,” J. Opt. Soc. Am. B 5, 147–158 (1988).
[CrossRef]

1987 (2)

I. W. Marshall, D. M. Spirit, M. J. O’Mahonoy, “Picosecond pulse response of a travelling wave semiconductor laser amplifier,” Electron. Lett. 23,(1987).

T. E. Darcie, R. M. Jopson, R. W. Tkach, “Intermodulation distortion in optical amplifiers from carrier-density modulation,” Electron. Lett. 23, 1392–1394 (1987).
[CrossRef]

1985 (1)

M. J. Adams, J. V. Collins, I. D. Henning, “Analysis of semiconductor laser optical amplifiers,” IEE Proc. J. Optoelectron. 132, 58–63 (1985).
[CrossRef]

1963 (2)

R. Bellman, G. Birnbaum, W. G. Wagner, “Transmission of monochromatic radiation in a two level system,” J. Appl. Phys. 34, 780–782 (1963).
[CrossRef]

L. M. Frantz, J. S. Nodvik, “Theory of pulse propagation in a laser amplifier,” J. Appl. Phys. 34, 2346–2349 (1963).
[CrossRef]

Adams, M. J.

M. J. Adams, J. V. Collins, I. D. Henning, “Analysis of semiconductor laser optical amplifiers,” IEE Proc. J. Optoelectron. 132, 58–63 (1985).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal, N. A. Olsson, “Self-phase modulation and spectral broadening of optical pulses in semiconductor laser amplifiers,” IEEE J. Quantum Electron. 25, 2297–2306 (1989).
[CrossRef]

G. P. Agrawal, “Population pulsations and nondegenerate four-wave mixing in semiconductor lasers and amplifiers,” J. Opt. Soc. Am. B 5, 147–158 (1988).
[CrossRef]

Bellman, R.

R. Bellman, G. Birnbaum, W. G. Wagner, “Transmission of monochromatic radiation in a two level system,” J. Appl. Phys. 34, 780–782 (1963).
[CrossRef]

Birnbaum, G.

R. Bellman, G. Birnbaum, W. G. Wagner, “Transmission of monochromatic radiation in a two level system,” J. Appl. Phys. 34, 780–782 (1963).
[CrossRef]

Carrier, G. F.

G. F. Carrier, C. E. Pearson, Partial Differential Equations (Academic, New York, 1988).

Collins, J. V.

M. J. Adams, J. V. Collins, I. D. Henning, “Analysis of semiconductor laser optical amplifiers,” IEE Proc. J. Optoelectron. 132, 58–63 (1985).
[CrossRef]

Darcie, T. E.

T. E. Darcie, R. M. Jopson, “Interactions in optical amplifiers for multifrequency lightwave system,” Electron. Lett. 24, 638–640 (1988).
[CrossRef]

T. E. Darcie, R. M. Jopson, R. W. Tkach, “Intermodulation distortion in optical amplifiers from carrier-density modulation,” Electron. Lett. 23, 1392–1394 (1987).
[CrossRef]

Frantz, L. M.

L. M. Frantz, J. S. Nodvik, “Theory of pulse propagation in a laser amplifier,” J. Appl. Phys. 34, 2346–2349 (1963).
[CrossRef]

Henning, I. D.

M. J. Adams, J. V. Collins, I. D. Henning, “Analysis of semiconductor laser optical amplifiers,” IEE Proc. J. Optoelectron. 132, 58–63 (1985).
[CrossRef]

Jopson, R. M.

T. E. Darcie, R. M. Jopson, “Interactions in optical amplifiers for multifrequency lightwave system,” Electron. Lett. 24, 638–640 (1988).
[CrossRef]

T. E. Darcie, R. M. Jopson, R. W. Tkach, “Intermodulation distortion in optical amplifiers from carrier-density modulation,” Electron. Lett. 23, 1392–1394 (1987).
[CrossRef]

Marshall, I. W.

I. W. Marshall, D. M. Spirit, M. J. O’Mahonoy, “Picosecond pulse response of a travelling wave semiconductor laser amplifier,” Electron. Lett. 23,(1987).

Nodvik, J. S.

L. M. Frantz, J. S. Nodvik, “Theory of pulse propagation in a laser amplifier,” J. Appl. Phys. 34, 2346–2349 (1963).
[CrossRef]

O’Mahonoy, M. J.

I. W. Marshall, D. M. Spirit, M. J. O’Mahonoy, “Picosecond pulse response of a travelling wave semiconductor laser amplifier,” Electron. Lett. 23,(1987).

O’Mahony, M. J.

M. J. O’Mahony, “Semiconductor laser optical amplifiers for use in future fiber systems,” IEEE. J. Lightwave Technol. 6, 531–534 (1988).
[CrossRef]

Olsson, N. A.

G. P. Agrawal, N. A. Olsson, “Self-phase modulation and spectral broadening of optical pulses in semiconductor laser amplifiers,” IEEE J. Quantum Electron. 25, 2297–2306 (1989).
[CrossRef]

Pearson, C. E.

G. F. Carrier, C. E. Pearson, Partial Differential Equations (Academic, New York, 1988).

Spirit, D. M.

I. W. Marshall, D. M. Spirit, M. J. O’Mahonoy, “Picosecond pulse response of a travelling wave semiconductor laser amplifier,” Electron. Lett. 23,(1987).

Tkach, R. W.

T. E. Darcie, R. M. Jopson, R. W. Tkach, “Intermodulation distortion in optical amplifiers from carrier-density modulation,” Electron. Lett. 23, 1392–1394 (1987).
[CrossRef]

Wagner, W. G.

R. Bellman, G. Birnbaum, W. G. Wagner, “Transmission of monochromatic radiation in a two level system,” J. Appl. Phys. 34, 780–782 (1963).
[CrossRef]

Electron. Lett. (3)

I. W. Marshall, D. M. Spirit, M. J. O’Mahonoy, “Picosecond pulse response of a travelling wave semiconductor laser amplifier,” Electron. Lett. 23,(1987).

T. E. Darcie, R. M. Jopson, R. W. Tkach, “Intermodulation distortion in optical amplifiers from carrier-density modulation,” Electron. Lett. 23, 1392–1394 (1987).
[CrossRef]

T. E. Darcie, R. M. Jopson, “Interactions in optical amplifiers for multifrequency lightwave system,” Electron. Lett. 24, 638–640 (1988).
[CrossRef]

IEE Proc. J. Optoelectron. (1)

M. J. Adams, J. V. Collins, I. D. Henning, “Analysis of semiconductor laser optical amplifiers,” IEE Proc. J. Optoelectron. 132, 58–63 (1985).
[CrossRef]

IEEE J. Quantum Electron. (1)

G. P. Agrawal, N. A. Olsson, “Self-phase modulation and spectral broadening of optical pulses in semiconductor laser amplifiers,” IEEE J. Quantum Electron. 25, 2297–2306 (1989).
[CrossRef]

IEEE. J. Lightwave Technol. (1)

M. J. O’Mahony, “Semiconductor laser optical amplifiers for use in future fiber systems,” IEEE. J. Lightwave Technol. 6, 531–534 (1988).
[CrossRef]

J. Appl. Phys. (2)

R. Bellman, G. Birnbaum, W. G. Wagner, “Transmission of monochromatic radiation in a two level system,” J. Appl. Phys. 34, 780–782 (1963).
[CrossRef]

L. M. Frantz, J. S. Nodvik, “Theory of pulse propagation in a laser amplifier,” J. Appl. Phys. 34, 2346–2349 (1963).
[CrossRef]

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

Other (1)

G. F. Carrier, C. E. Pearson, Partial Differential Equations (Academic, New York, 1988).

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Figures (9)

Fig. 1
Fig. 1

Schematic of an optical TWA, showing the two input signals of field amplitudes B0 and b0(t), Eq. (1).

Fig. 2
Fig. 2

Variation of excess intensity (pulse) amplification at output, Ap(L, tL), Eq. (27), with time, tL, for three different input signal ratios R, 15, 10, and 7. Results for two values of the antiguidance factor, β = 0 (upper curves) and β = 5 (lower curves) are shown for each R. The cw input signal intensity is 0.05 mW/μm2 (equivalent to B0 = 0.1).

Fig. 3
Fig. 3

Variation of the phase, ϕ(L, tL), Eqs. (25), of the pulse signal for β = 5. The other parameters are the same as for Fig. 2.

Fig. 4
Fig. 4

Spatial variation of the excess intensity (pulse) amplification, Ap, with time t as the parameter, for t equal to 10, 100, 250, 500, 750, and 1000 ps, R = 7, β = 5, and B0 = 0.1.

Fig. 5
Fig. 5

Spatial variation of the space- and time-dependent inversion population density, n(z, t), with time as the parameter. The other quantities are the same as for Fig. 4.

Fig. 6
Fig. 6

Spatial variation of the excess intensity amplification, Ap, with the parameter t in the range of the transit time tL (=2.33 ps). All other quantities are as given in Fig. 4. Note that Ap is zero for z > v1t. Although the curves for different times appear to be superimposed, in fact they are marginally separated because of the correspondingly small change in the inversion population with time. The curves for t > tL show, as expected, a finite Ap throughout the length (0 < z < L = 200 μm) of the device and are similar to those in Fig. 4.

Fig. 7
Fig. 7

Spatial variation of n(z, t) with the parameter t in the range of the transit time (tL). All other quantities are as given in Fig. 4. The interesting feature here is the presence of an extremum, which occurs because of the competing effects of the stimulated and the spontaneous recombination rates. For t sufficiently greater than tL the distribution has monotonic variation over the length of the device, as can be seen from Fig. 5.

Fig. 8
Fig. 8

Variation of the excess intensity (pulse) amplification at the output, Ap(L, tL), Eq. (27), with time tL, for signal input ratio R = 10 and β = 5. The cw input signal intensity is 0.005 mW/μm2 (equivalent to B0 = 0.01). The amplifier is in the unsaturated regime. The change in amplification with time is small for operation in this regime, which may be viewed as the amplified pulse’s having small (amplitude) distortion.

Fig. 9
Fig. 9

Variation of the excess intensity (pulse) amplification at the output, Ap(L, tL), Eq. (27), with time tL, for signal input ratio R = 1000 and β = 5. The cw input signal intensity is 0.05 mW/μm2 (equivalent to B0 = 0.1). The amplifier is in the saturation regime. Note that the excess intensity is the same as in Fig. 8. The results plotted here show a much larger variation in Ap(L, t), which implies a much larger pulse (amplitude) distortion than that in the unsaturated regime operation (Fig. 8).

Equations (69)

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E 0 ( t ) = B 0 exp ( - i ω 0 t ) + b 0 ( t ) exp ( - i ω 1 t ) ,
^ = sc + N ( z , t ) ,
D ( z , t ) = D s ( z , t ) + D N ( z , t ) ,
D N ( z , t ) = 0 N ( z , t ) E ( z , t ) ,
D s ( z , t ) = FT [ D s ( z , ω ) ] ,
D s ( z , ω ) = 0 sc ( ω ) E ( z , t ) .
2 z 2 E ( z , t ) = μ 0 [ 2 t 2 D s ( z , t ) + 2 t 2 D N ( z , t ) ] .
2 t 2 D N ( z , t ) 0 N ( z , t ) 2 t 2 E ( z , t ) .
2 t 2 D s ( z , t ) = 0 2 t 2 - sc ( ω ) E ( z , ω ) exp ( - i t ω ) d ω .
E ( z , t ) = B ( z ) exp [ i ( k 0 s z - ω 0 t ) ] + b ( z , t ) exp [ i ( k 1 s z - ω 1 t ) ] ,
k j s = ( n j ω j / c ) ,             n j = ( sc j ) 1 / 2 ,             j = 0 , 1.
[ 2 i k 0 s d B ( z ) d z + k 0 2 B ( z ) N ( z , t ) ] exp [ - i ( ω 0 t - k 0 s z ) ] + [ 2 i k 1 s b ( z , t ) z + 2 i ( s c 1 ω 1 + ω 1 2 sc p ) b ( z , t ) t + k 1 2 b ( z , t ) N ( z , t ) ] exp [ - i ( ω 1 t - k 1 s z ) ] = 0 ,
k 0 2 = μ 0 0 ω 0 2 ,             k 1 2 = μ 0 0 ω 1 2 , sc p = sc p             at             p = 0 ,             p = ω - ω 1 .
N ( z , t ) = ¯ ( z ) + ˜ ( z , t ) .
d B ( z ) d z + k 0 2 ¯ 2 i k 0 s B ( z ) = 0 ,
b ( z , t ) z + 1 ν 1 b ( z , t ) t + ¯ k 1 2 2 i k 1 s b ( z , t ) = - ˜ 2 i { k 0 2 k 1 s B ( z ) exp [ - i ( Ω t + Δ k z ) ] + k 1 2 k 1 s b ( z , t ) } ,
Ω = ω 0 - ω 1 ,             Δ k = k 1 s - k 0 s ,             ν 1 = k 1 s / ( sc 1 ω 1 ) .
N t = I q V - N τ s - g ( N ) ω 0 E 2 + D 2 N ,
g ( N ) = γ g ^ ( N ) = γ a ( N - N 0 ) .
b ( z , t ) B ( z ) ,
E 2 B ( z ) 2 + 2 Re [ B * ( z ) b ( z , t ) exp ( i Ω t ) ] .
N ( z , t ) = N ¯ ( z ) + n ( z , t ) .
N ¯ ( z ) = N 0 ( I / I 0 ) + P 0 1 + P 0 ,
n t + D 1 n = - N ¯ ( z ) - N 0 P s τ s [ B * ( z ) b ( z , t ) exp ( i Ω t ) + B ( z ) b * ( z , t ) exp ( - i Ω t ) ] ,
I 0 = q V N 0 / τ s , P s = ω 0 / γ a τ s ( saturation intensity ) , P 0 ( z ) = B ( z ) 2 / P s [ ( normalized ) dc optical intensity distribution ] , D 1 = ( 1 + P 0 ) / τ s .
N ( z , t ) = ¯ ( z ) + ˜ ( z , t ) = A [ N ( z , t ) - N 0 ]
¯ ( z ) = A [ N ¯ ( z ) - N 0 ] ,             ˜ ( z , t ) = A n ( z , t ) .
d B ( z ) d z + α 0 B ( z ) = 0 ,
P c = 0 L B ( z ) 2 d z / L .
1 ν 1 b ( z , t ) t + b ( z , t ) z + α 0 b ( z , t ) = γ ( 1 - i β ) 2 a n ( z , t ) { B ( z ) exp [ - i ( Ω t + Δ k z ) ] + b ( z , t ) } ,
n ( z , t ) t + D 1 n ( z , t ) = - ( N c - N 0 ) τ s [ B * ( z ) b ( z , t ) exp ( i Ω t ) + B ( z ) b * ( z , t ) exp ( - i Ω t ) ] .
B ( 0 ) = B 0 ,             b ( 0 , t ) = b 0 f ( t ) , b ( z , 0 ) = 0 ,             n ( z , 0 ) = 0.
B ( z ) = B 0 exp ( - α 0 z ) .
b 1 ( z , t ) = b 0 H ( t - z / v 1 ) exp ( - α 0 z ) , n 1 ( z , t ) = - D 31 exp [ - ( α 0 + α 0 * ) z ] [ 1 - exp ( - D 1 t z ) ] , b 2 ( z , t ) = b 0 exp ( - α 0 z ) [ 1 - α 0 F ( z ) ] , n 2 ( z , t ) = n 1 ( z , t ) + D 2 D 31 exp [ - ( α 0 + α 0 * ) z ] × { exp [ - ( α 0 + α 0 * ) z ] - 1 } × [ 1 - exp ( - D 1 t z ) - D 1 t z exp ( - D 1 t z ) ] ,
b 3 ( z , t ) = b 0 exp ( - α 0 z ) [ 1 + α 0 F 1 ( z , t ) + α 0 2 F 2 ( z , t ) ] ,
t z = t - z / v 1 , D 1 = ( 1 + P c ) / τ s .
E ( z , t ) = [ B ( z ) + b ( z , t ) ] exp ( - i ω 0 t ) = B T ( z , t ) exp ( - i ω 0 t ) ,
B ( z ) = B ( z ) exp [ i ψ ( z ) ] , b ( z , t ) = b ( z , t ) exp [ i ϕ ( z , t ) ] , B T ( z , t ) = B ( z ) + b ( z , t ) = B T ( z , t ) exp [ i θ ( z , t ) ] .
S ( z , t ) = E ( z , t ) E * ( z , t ) = E ( z , t ) 2 ,
p ( z , t ) = S ( z , t ) - B ( z ) 2 .
A p ( z , t ) = p ( z , t ) / p ( 0 , t ) = [ S ( z , t ) - B ( z ) 2 ] / [ B 0 + b 0 2 - B 0 2 ] .
2 E = μ 0 [ 2 D s t 2 + 2 D N t 2 ] ,
2 E z 2 = μ 0 [ 2 D s t 2 + 2 D N t 2 ] .
E ( z , t ) = B ( z ) exp [ i ( k 0 s z - ω 0 t ) ] + b ( z , t ) exp [ i ( k 1 s z - ω 1 t ) ] .
E ( z , ω ) = B ( z ) exp ( i k 0 s z ) δ ( ω - ω 0 ) + b ˜ ( z , p ) exp ( i k 1 s z ) ,
2 D s ( z , t ) t 2 = 0 2 t 2 - sc ( ω ) [ B ( z ) exp ( i k 0 s z ) δ ( ω - ω 0 ) + b ˜ ( z , ω - ω 0 ) exp ( i k 1 s z ) ] exp ( - i t ω ) d ω .
2 D s t 2 = - 0 sc ( ω 0 ) ω 2 B ( z ) exp [ i ( k 0 s z - ω 0 t ) ] - 0 [ ω 1 2 sc 1 b ( z , t ) + i ( 2 sc 1 ω 1 + ω 1 2 sc p ) b ( z , t ) t - ( sc 1 + 2 ω 1 sc p ) 2 b ( z , t ) t 2 ] exp [ i ( k 1 s z - ω 1 t ) ]
2 D N ( z , t ) t 2 = 0 2 [ N ( z , t ) E ( z , t ) ] t 2 0 N ( z , t ) 2 E ( z , t ) t 2 = - 0 N { ω 1 2 b ( z , t ) exp [ i ( k 1 s z - ω 1 t ) ] + ω 0 2 B ( z ) exp [ i ( k 0 s z - ω 0 t ) ] } .
k 0 2 B | k 0 d B d z | | d 2 B d z 2 | , k 1 2 b | k 1 b z | | 2 b z 2 | , ω 1 2 b | ω 1 b t | | 2 b t 2 | .
[ 2 i k 0 s d B d z + k 0 2 N B ( z ) ] exp [ i ( k 0 s z - ω 0 t ) ] + [ 2 i k 1 s b z + 2 i ( sc 1 ω 1 + ω 1 2 sc p ) b t + N k 1 2 b ] × exp [ i ( k 1 s z - ω 1 t ) ] = 0.
N ( z , t ) = ¯ ( z ) + ˜ ( z , t ) .
d B d z + ¯ k 0 2 2 i k 0 s B ( z ) = 0 ,
1 ν 1 b ( z , t ) t + b ( z , t ) z + ¯ k 1 2 2 i k 1 s b ( z , t ) = - ˜ 2 i [ k 0 2 k 1 s B ( z ) exp [ - i ( Ω t + Δ k z ) ] + k 1 2 k 1 s b ( z , t ) ] ,
k 0 2 = μ 0 0 ω 0 2 ,             k 1 2 = μ 0 0 ω 1 2 , sc p = sc p p             at p = 0 ( p = ω - ω 1 ) ,
Ω = ω 0 - ω 1 ,             Δ k = k 1 s - k 0 s , ν 1 = k 1 s sc 1 ω 1 + sc p k 1 s sc 1 ω 1             ( neglecting dispersion ) .
d B ( z ) d z + α 0 B ( z ) = 0 ,
P c = 1 L 0 L B ( z ) 2 d z ,
1 ν 1 b ( z , t ) t + b ( z , t ) z + α 0 b ( z , t ) = γ ( 1 - i β ) 2 a n ( z , t ) { B ( z ) exp [ - i ( Ω t + Δ k z ) ] + b ( z , t ) } ,
n ( z , t ) t + D 1 n ( z , t ) = - ( N c - N 0 ) τ s [ B * ( z ) b ( z , t ) exp ( i Ω t ) + B ( z ) b * ( z , t ) exp ( - i Ω t ) ] ,
α 0 = - g 0 ( 1 - i β ) 2 ( 1 + P c ) , g 0 = γ a N 0 ( I I 0 - 1 ) , I 0 = q V N 0 τ s , D 1 = 1 + P c τ s .
B ( 0 ) = B 0 ,             b ( 0 , t ) = b 0 f ( t ) , b ( z , 0 ) = 0 ,             n ( z , t ) = 0.
B ( z ) = B 0 exp ( - α 0 z ) .
a D t u + b D z u = G ( z , t , u ) ,
d z / d t = b / a ,             d u / d t = G / a ,
f 1 ( z , t , u ) = c 1 ,             f 2 ( z , t , u ) = c 2 .
F ( c 1 , c 2 ) = 0
b 1 ( z , t ) = b 0 H ( t - z / v 1 ) exp ( - α 0 z ) , n 1 ( z , t ) = - D 31 exp ( - α 1 z ) [ 1 - exp ( - D 1 t z ) ] , b 2 ( z , t ) = b 0 exp ( - α 0 z ) [ 1 - α 0 F ( z ) ] , n 2 ( z , t ) = n 1 ( z , t ) + A 2 D 2 D 31 exp ( - α 1 z ) [ exp ( - α 1 z ) - 1 ] ,
b 3 ( z , t ) = b 0 exp ( - α 0 z ) [ 1 + α 0 F 1 ( z , t ) + α 0 2 F 2 ( z , t ) ] ,
F ( z , t ) = 2 D 2 A 1 [ exp ( - α 1 z ) - 1 ] / α 1 , F 1 ( z , t ) = F ( z , t ) + D 4 [ 1 - exp ( - α 1 z ) ] × { A 1 b 0 - A 2 D 2 ( B 0 + b 0 ) × [ exp ( - α 1 z ) - 1 ] / 2 } / α 1 , F 2 ( z , t ) = D 4 [ A 1 2 b 0 D 2 + D 2 b 0 A 1 A 2 { [ 1 - exp ( - α 1 z ) ] / α 1 } 2 , A 1 = 1 - exp ( - D 1 t z ) , A 2 = A 1 - D 1 t z exp ( - D 1 t z ) , D 1 = ( 1 + P c ) / τ s ,             D 2 = [ B 0 2 / ( 1 + P c ) ] ( 1 + 1 / R ) , D 3 = 2 b 0 B 0 / ( 1 + P c ) , D 31 = D 3 ( N c - N 0 ) ,             D 4 = 2 B 0 / ( 1 + P c ) , α 1 = α 0 + α 0 * , t z = t - z / v 1 .

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