Abstract

The quantum theory of noise in a chain of gain–loss elements is reviewed. A new derivation is given of the characteristic function of the photon-number distribution at the output of a long amplifier chain. The results of the quantum theory are compared with those of the semiclassical theory, and it is shown that in most practical cases the semiclassical theory gives an excellent approximation of the full quantum result. It is shown that transparency sets a fundamental limit on the maximum distance over which a signal can be transmitted and hence on the maximum spacing between regenerative stations. An analytic estimate of this limit is given.

© 2000 Optical Society of America

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References

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  1. J.-B. Thomine, G. Aubin, and F. Pirio, “Future trends for high capacity optically amplified submarine systems,” presented at the IEE Colloquium on Transoceanic Cable Communications-TAT 12 and 13 Herald a New Era, March 1996.
  2. L. D. Garrett, R. S. Vodhanel, S. H. Patel, R. W. Tkach, and A. R. Chraplyvy, “Dispersion management in optical networks,” presented at the 1998 International Conference on Communication Technology, Beijing, China, October 22–24, 1998.
  3. N. S. Bergano, “WDM long haul transmission systems,” in Optical Fiber Communication Conference (OFC), Vol. 2 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), p. 30.
  4. W. S. Wong, H. A. Haus, L. A. Jiang, P. B. Hansen, and M. Margalit, “Photon statistics of amplified spontaneous emission noise in a 10 Gbit/s optically preamplified direct detection receiver,” Opt. Lett. 23, 1832–1834 (1998).
    [CrossRef]
  5. H. A. Haus, “The noise figure of optical amplifiers,” IEEE Photon. Technol. Lett. 11, 1602–1604 (1998).
    [CrossRef]
  6. E. Desurvire, “Comments on: The noise figure of optical amplifiers, by H. A. Haus,” IEEE Photon. Technol. Lett. (to be published).
  7. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, 1995).
  8. H. A. Haus and J. A. Mullen, “Quantum noise in linear amplifiers,” Phys. Rev. 128, 2407–2413 (1962).
    [CrossRef]
  9. T. Li and M. C. Teich, “Performance of a lightwave system incorporating a cascade of erbium-doped fiber amplifiers,” Opt. Commun. 91, 41–45 (1992).
    [CrossRef]
  10. E. Desurvire, “A three dimensional quantum-vacuum noise/signal beamsplitter model for nonideal linear optical amplifiers,” Opt. Fiber Technol. Mater. Devices Syst. 5, 82–91 (1999).
    [CrossRef]
  11. A. Mecozzi, “On the optimization of the gain distribution of transmission lines with unequal amplifier spacing,” IEEE Photon. Technol. Lett. 11, 1033–1035 (1998).
    [CrossRef]
  12. T. Shepherd and E. Jakeman, “Statistical analysis of an incoherently coupled, steady-state optical amplifier,” J. Opt. Soc. Am. B 4, 1860–1869 (1987).
    [CrossRef]
  13. A. Mecozzi, F. De Pasquale, and L. Peliti, “Unified approach to stochastic representation in reaction kinetics,” Nuovo Cimento B 100, 733–743 (1987).
  14. D. Marcuse, “Derivation of analytical expressions for the bit-error probability in light-wave systems with optical amplifiers,” J. Lightwave Technol. 8, 1816–1823 (1990).
    [CrossRef]
  15. L. F. Mollenauer, J. P. Gordon, and S. G. Evangelides, “The sliding frequency-guiding filters: an improved form of soliton jitter control,” Opt. Lett. 17, 1575–1577 (1992).
    [CrossRef] [PubMed]
  16. J. C. Livas, “High sensitivity optically preamplified 10 Gb/s receivers,” in Optical Fiber Communication, Vol. 5 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), postdeadline paper PD-4.

1999 (1)

E. Desurvire, “A three dimensional quantum-vacuum noise/signal beamsplitter model for nonideal linear optical amplifiers,” Opt. Fiber Technol. Mater. Devices Syst. 5, 82–91 (1999).
[CrossRef]

1998 (3)

A. Mecozzi, “On the optimization of the gain distribution of transmission lines with unequal amplifier spacing,” IEEE Photon. Technol. Lett. 11, 1033–1035 (1998).
[CrossRef]

W. S. Wong, H. A. Haus, L. A. Jiang, P. B. Hansen, and M. Margalit, “Photon statistics of amplified spontaneous emission noise in a 10 Gbit/s optically preamplified direct detection receiver,” Opt. Lett. 23, 1832–1834 (1998).
[CrossRef]

H. A. Haus, “The noise figure of optical amplifiers,” IEEE Photon. Technol. Lett. 11, 1602–1604 (1998).
[CrossRef]

1992 (2)

T. Li and M. C. Teich, “Performance of a lightwave system incorporating a cascade of erbium-doped fiber amplifiers,” Opt. Commun. 91, 41–45 (1992).
[CrossRef]

L. F. Mollenauer, J. P. Gordon, and S. G. Evangelides, “The sliding frequency-guiding filters: an improved form of soliton jitter control,” Opt. Lett. 17, 1575–1577 (1992).
[CrossRef] [PubMed]

1990 (1)

D. Marcuse, “Derivation of analytical expressions for the bit-error probability in light-wave systems with optical amplifiers,” J. Lightwave Technol. 8, 1816–1823 (1990).
[CrossRef]

1987 (2)

T. Shepherd and E. Jakeman, “Statistical analysis of an incoherently coupled, steady-state optical amplifier,” J. Opt. Soc. Am. B 4, 1860–1869 (1987).
[CrossRef]

A. Mecozzi, F. De Pasquale, and L. Peliti, “Unified approach to stochastic representation in reaction kinetics,” Nuovo Cimento B 100, 733–743 (1987).

1962 (1)

H. A. Haus and J. A. Mullen, “Quantum noise in linear amplifiers,” Phys. Rev. 128, 2407–2413 (1962).
[CrossRef]

De Pasquale, F.

A. Mecozzi, F. De Pasquale, and L. Peliti, “Unified approach to stochastic representation in reaction kinetics,” Nuovo Cimento B 100, 733–743 (1987).

Desurvire, E.

E. Desurvire, “A three dimensional quantum-vacuum noise/signal beamsplitter model for nonideal linear optical amplifiers,” Opt. Fiber Technol. Mater. Devices Syst. 5, 82–91 (1999).
[CrossRef]

Evangelides, S. G.

Gordon, J. P.

Hansen, P. B.

Haus, H. A.

W. S. Wong, H. A. Haus, L. A. Jiang, P. B. Hansen, and M. Margalit, “Photon statistics of amplified spontaneous emission noise in a 10 Gbit/s optically preamplified direct detection receiver,” Opt. Lett. 23, 1832–1834 (1998).
[CrossRef]

H. A. Haus, “The noise figure of optical amplifiers,” IEEE Photon. Technol. Lett. 11, 1602–1604 (1998).
[CrossRef]

H. A. Haus and J. A. Mullen, “Quantum noise in linear amplifiers,” Phys. Rev. 128, 2407–2413 (1962).
[CrossRef]

Jakeman, E.

Jiang, L. A.

Li, T.

T. Li and M. C. Teich, “Performance of a lightwave system incorporating a cascade of erbium-doped fiber amplifiers,” Opt. Commun. 91, 41–45 (1992).
[CrossRef]

Marcuse, D.

D. Marcuse, “Derivation of analytical expressions for the bit-error probability in light-wave systems with optical amplifiers,” J. Lightwave Technol. 8, 1816–1823 (1990).
[CrossRef]

Margalit, M.

Mecozzi, A.

A. Mecozzi, “On the optimization of the gain distribution of transmission lines with unequal amplifier spacing,” IEEE Photon. Technol. Lett. 11, 1033–1035 (1998).
[CrossRef]

A. Mecozzi, F. De Pasquale, and L. Peliti, “Unified approach to stochastic representation in reaction kinetics,” Nuovo Cimento B 100, 733–743 (1987).

Mollenauer, L. F.

Mullen, J. A.

H. A. Haus and J. A. Mullen, “Quantum noise in linear amplifiers,” Phys. Rev. 128, 2407–2413 (1962).
[CrossRef]

Peliti, L.

A. Mecozzi, F. De Pasquale, and L. Peliti, “Unified approach to stochastic representation in reaction kinetics,” Nuovo Cimento B 100, 733–743 (1987).

Shepherd, T.

Teich, M. C.

T. Li and M. C. Teich, “Performance of a lightwave system incorporating a cascade of erbium-doped fiber amplifiers,” Opt. Commun. 91, 41–45 (1992).
[CrossRef]

Wong, W. S.

IEEE Photon. Technol. Lett. (2)

H. A. Haus, “The noise figure of optical amplifiers,” IEEE Photon. Technol. Lett. 11, 1602–1604 (1998).
[CrossRef]

A. Mecozzi, “On the optimization of the gain distribution of transmission lines with unequal amplifier spacing,” IEEE Photon. Technol. Lett. 11, 1033–1035 (1998).
[CrossRef]

J. Lightwave Technol. (1)

D. Marcuse, “Derivation of analytical expressions for the bit-error probability in light-wave systems with optical amplifiers,” J. Lightwave Technol. 8, 1816–1823 (1990).
[CrossRef]

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

Nuovo Cimento B (1)

A. Mecozzi, F. De Pasquale, and L. Peliti, “Unified approach to stochastic representation in reaction kinetics,” Nuovo Cimento B 100, 733–743 (1987).

Opt. Commun. (1)

T. Li and M. C. Teich, “Performance of a lightwave system incorporating a cascade of erbium-doped fiber amplifiers,” Opt. Commun. 91, 41–45 (1992).
[CrossRef]

Opt. Fiber Technol. Mater. Devices Syst. (1)

E. Desurvire, “A three dimensional quantum-vacuum noise/signal beamsplitter model for nonideal linear optical amplifiers,” Opt. Fiber Technol. Mater. Devices Syst. 5, 82–91 (1999).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. (1)

H. A. Haus and J. A. Mullen, “Quantum noise in linear amplifiers,” Phys. Rev. 128, 2407–2413 (1962).
[CrossRef]

Other (6)

E. Desurvire, “Comments on: The noise figure of optical amplifiers, by H. A. Haus,” IEEE Photon. Technol. Lett. (to be published).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, 1995).

J.-B. Thomine, G. Aubin, and F. Pirio, “Future trends for high capacity optically amplified submarine systems,” presented at the IEE Colloquium on Transoceanic Cable Communications-TAT 12 and 13 Herald a New Era, March 1996.

L. D. Garrett, R. S. Vodhanel, S. H. Patel, R. W. Tkach, and A. R. Chraplyvy, “Dispersion management in optical networks,” presented at the 1998 International Conference on Communication Technology, Beijing, China, October 22–24, 1998.

N. S. Bergano, “WDM long haul transmission systems,” in Optical Fiber Communication Conference (OFC), Vol. 2 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), p. 30.

J. C. Livas, “High sensitivity optically preamplified 10 Gb/s receivers,” in Optical Fiber Communication, Vol. 5 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), postdeadline paper PD-4.

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Figures (3)

Fig. 1
Fig. 1

Probability distribution of the chain output for M = 8 and values of n ASE of (a) 3, (b) 10, and (c) 100. The parameter related to the average number of output photons is, for the distributions with higher average photon numbers representing 1’s, n out = 304.074 , 1013.58, 10135.8, respectively, and 0 for those distributions with smaller average numbers that correspond to a transmitted logical 0. Dashed curves, semiclassical approximations; solid curves, the result of the full quantum theory.

Fig. 2
Fig. 2

Ratio n out / n ASE , corresponding to BER’s of (a) 10 - 6 , (b) 10 - 9 , and (c) 10 - 12 , to the number of modes M. Solid curves, the complete semiclassical theory; dashed curves, the Gaussian approximation.

Fig. 3
Fig. 3

Receiver sensitivity obtained with the quantum theory versus preamplifier gain G dB (in decibels). Solid curve, M = 1 ; dashed curve, M = 2 .

Tables (1)

Tables Icon

Table 1 Parameters for Validity of the Semiclassical Approximation for a Logical 0 and for a Logicala

Equations (97)

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a ˆ 1 = Γ 1 / 2 a ˆ 0 + ( 1 - Γ ) 1 / 2 c ˆ 0 ,
a ˆ 2 = G 1 / 2 a ˆ 1 + [ ( n sp - 1 ) ( G - 1 ) ] 1 / 2 c ˆ 1 + [ n sp ( G - 1 ) ] 1 / 2 c ˆ 2 ,
a ˆ 2 = ( G Γ ) 1 / 2 a ˆ 0 + [ G ( 1 - Γ ) ] 1 / 2 c ˆ 0 + [ ( n sp - 1 ) ( G - 1 ) ] 1 / 2 c ˆ 1 + [ n sp ( G - 1 ) ] 1 / 2 c ˆ 2 ,
a ˆ 2 = ( G Γ ) 1 / 2 a ˆ 0 + [ G ( 1 - Γ ) + ( n sp - 1 ) ( G - 1 ) ] 1 / 2 c ˆ 1 + [ n sp ( G - 1 ) ] 1 / 2 c ˆ 2 ,
c ˆ 1 = [ G ( 1 - Γ ) + ( n sp - 1 ) ( G - 1 ) ] - 1 / 2 × { [ G ( 1 - Γ ) ] 1 / 2 c ˆ 0 + [ ( n sp - 1 ) ( G - 1 ) ] 1 / 2 c ˆ 1 } .
a ˆ N = G net 1 / 2 a ˆ 0 + f 1 1 / 2 b ˆ 1 + n ASE 1 / 2 b ˆ 2 ,
G net = g ( N ,   1 ) ,
f 1 = k = 1 N g ( N ,   k + 1 ) [ G k ( 1 - Γ k ) + ( n k , sp - 1 ) ( G k - 1 ) ] ,
n ASE = k = 1 N g ( N ,   k + 1 ) n k , sp ( G k - 1 ) ,
g ( k ,   h ) = j = h k   Γ j G j ,
g ( N ,   N + 1 ) = 1 .
f 1 - n ASE = k = 1 N   g ( N ,   k + 1 ) ( 1 - Γ k G k ) = k = 2 N + 1   g ( N ,   k ) - k = 1 N   g ( N ,   k ) = 1 - g ( N ,   1 )
f 1 = n ASE - ( G net - 1 ) .
a ˆ N = G net 1 / 2 a ˆ 0 + [ n ASE - ( G net - 1 ) ] 1 / 2 b ˆ 1 + n ASE 1 / 2 b ˆ 2 .
[ a ˆ N ,   a ˆ N ] = 1 .
n ˆ N = a ˆ N a ˆ N ,
n ˆ 0 = a ˆ 0 a ˆ 0 ,
G n N ( z ) = z n ˆ N = exp ( - λ n ˆ N ) , λ = - ln   z ,
G n N ( z ) = 1 1 - n ASE ( z - 1 ) G n 0 1 + G net ( z - 1 ) 1 - n ASE ( z - 1 ) .
G n N , tot ( z ) = 1 [ 1 - n ASE ( z - 1 ) ] M × G n 0 1 + G net ( z - 1 ) 1 - n ASE ( z - 1 ) M ,
G n 0 ( z ) = exp [ n ˆ 0 ( z - 1 ) ] .
G n N , tot ( z ) = 1 [ 1 - n ASE ( z - 1 ) ] M × exp - n out n ASE 1 - 1 1 - n ASE ( z - 1 ) ,
n out = G net   M n ˆ 0
G n N , tot ( z ) = z n ˆ N = n = 1   z n P ( n ) ,
P 1 ( n ) = ( n ASE ) n ( 1 + n ASE ) n + M × exp - n out 1 + n ASE L n M - 1 - n out n ASE ( 1 + n ASE ) ,
L n M - 1 ( 0 ) = ( n + M - 1 ) ! n ! ( M - 1 ) !
P 0 ( n ) = ( n + M - 1 ) ! ( n ASE ) n n ! ( M - 1 ) ! ( 1 + n ASE ) n + M exp - n out 1 + n ASE .
G n N , tot ( e - i ω )
= 1 [ 1 - n ASE ( e - i ω - 1 ) ] M × exp - n out n ASE 1 - 1 1 - n ASE ( e - i ω - 1 ) .
n ˆ = G n ( z ) z z = 1 ,
n ˜ ( n ˆ - 1 ) = 2 G n ( z ) z 2 z = 1 ,
n ˆ N = G tot n 0 + n ASE ,
Δ n ˆ N 2 = n ˆ N 2 - n ˆ N 2
= G tot n ˆ 0 + 2 G tot n ˆ 0 n ASE + n ASE ( n ASE + 1 ) × G tot 2 ( Δ n ˆ 0 2 - n ˆ 0 ) .
n ˆ N = G tot M n ˆ 0 + Mn ASE ,
Δ n ˆ N 2 = G tot M n ˆ 0 + 2 G tot   M n ˆ 0 n ASE + Mn ASE ( n ASE + 1 ) + G tot 2   M ( Δ n ˆ 0 2 - n ˆ 0 ) .
n ˆ N = n out + Mn ASE ,
Δ n ˆ N 2 = n out + 2 n out n ASE + Mn ASE ( n ASE + 1 ) ,
x = k = 1 2 M   y k 2 ,
g n N , tot ( ω ) = exp ( i ω x ) = k = 1 2 M - d y k   exp ( iy k 2 ω ) f k ( y k ) ,
g n N , tot ( ω ) = 1 ( 1 - in ASE ω ) M × exp - n out n ASE 1 - 1 1 - in ASE ω .
f ( x ) = 1 n ASE exp - n out n ASE - x n ASE × x n ASE n ASE n out ( M - 1 ) / 2 I M - 1 2 xn out n ASE .
f 0 ( x ) = 1 n ASE ( M - 1 ) ! x n ASE M - 1   exp - x n ASE .
f ( x ) = 1 n ASE ϕ x n ASE ;   n out n ASE ,   M ,
f 0 ( x ) = 1 n ASE ϕ 0 x n ASE ;   M .
P ( n ) f ( x ) | x = n .
h ( ω ,   Ω ) = 1 - Ω < ω < Ω 0 otherwise
g n N , tot ( ω ) h ( ω ,   Ω ) G n N , tot ( e - i ω ) h ( ω ,   Ω ) .
G n N , tot ( e - i ω )
1 ( 1 - n ASE ω 2 / 2 + n ASE i ω ) M × exp - n out n ASE 1 - 1 1 - n ASE ω 2 / 2 + n ASE i ω .
h ( x ,   Ω ) = -   d ω 2 π h ( ω ,   Ω ) = sin ( Ω x ) π x ,
Δ n ˆ N , tot 2 = Mn ASE ( n ASE + 1 ) Mn ASE 2 ,
Δ n ˆ N , tot 2 2 n out   n ASE .
M n ASE π 2 n ASE s 0 = n ASE   / ( π 2 / M ) 1
2 n out   n ASE π 2 n ASE s 1 = n out   / π 1 .
P err = 1 2 0 n th   d xf 1 ( x ) + n th d xf 0 ( x ) = 1 2 0 n th / n ASE   d x ϕ 1 x ;   n out n ASE ,   M + n th / n ASE   d x ϕ 0 ( x ;   M ) ,
f 1 ( n ¯ th ) = f 0 ( n ¯ th ) ϕ 1 n ¯ th n ASE ;   n out n ASE ,   M = ϕ 0 n ¯ th n ASE ;   M .
P err ( Q ) exp ( - Q 2 / 2 ) 2 π Q ,
s = n out Mn ASE
Q = s M 2 s + 1 + 1 .
s = 2 Q 2 M + 2 Q M n out n ASE = 2 ( Q 2 + M Q ) .
n out n ASE = 72 + 12 M .
n out n ASE 72 + 12 M - 8.08   exp ( - M / 200.4 ) × [ 1 - 0.0899   exp ( - M / 4.412 ) ] .
n out n ASE 45.32 + 9.52 M - 1.76   exp ( - M / 28.30 ) - 3.69 ,
n out n ASE 98 + 14 M - 16.45 × exp ( - M / 75.67 ) [ 1 - 0.486   exp ( - M / 42.07 ) ] ,
n ASE = n sp k ( G k - 1 ) ,
n out = I out T ω 0 = 2 I ¯ T ω 0 ,
2 I ¯ T n sp ω 0   k ( G k - 1 )
= 72 + 12 M - 8.08   exp ( - M / 200.4 )
× [ 1 - 0.0899   exp ( - M / 4.412 ) ] .
k ( G k - 1 ) = 3974 .
n out n ASE = n in G n sp ( G - 1 ) .
n sen = n in 2 = n sp ( G - 1 ) G × { 36 + 6 M - 4.04   exp ( - M / 200.4 ) × [ 1 - 0.0899   exp ( - M / 4.412 ) ] } .
P err ( n th ) = 1 2 n = 0 n th P 1 ( n ) + n = n th + 1   P 0 ( n ) .
Δ P err ( n th ) = P err ( n th + 1 ) - P err ( n th ) = ½ [ P 1 ( n th + 1 ) - P 0 ( n th + 1 ) ] .
P 1 ( n th + 1 ) - P 0 ( n th + 1 ) > 0
P 1 ( x ) - P 0 ( x ) = 0
L z α ( x ) = Γ ( z + α + 1 ) Γ ( z + 1 ) Γ ( α + 1 )   1 F 1 ( - z ,   α + 1 ;   x ) ,
a ˆ N = G net ( N ) 1 / 2 a ˆ 0 + f 1 ( N ) 1 / 2 b ˆ 1 ( N ) + n ASE ( N ) 1 / 2 b ˆ 2 ( N ) .
a ˆ N + 1 = ( G N + 1 Γ N + 1 ) 1 / 2 a ˆ N + [ G N + 1 ( 1 - Γ N + 1 ) + ( n sp , N + 1 - 1 ) ( G N + 1 - 1 ) ] 1 / 2 c ˆ 1 + [ n sp , N + 1 ( G N + 1 - 1 ) ] 1 / 2 c ˆ 2 .
b ˆ 1 ( N + 1 ) = f 1 ( N + 1 ) - 1 / 2 { [ G N + 1 Γ N + 1 f 1 ( N ) ] 1 / 2 b ˆ 1 ( N ) + [ G N + 1 ( 1 - Γ N + 1 ) + ( n sp , N + 1 - 1 )
× ( G N + 1 - 1 ) ] 1 / 2 c ˆ 1 } ,
b ˆ 2 ( N + 1 ) = n ASE ( N + 1 ) - 1 / 2 × { [ G N + 1 Γ N + 1 n ASE ( N ) ] 1 / 2 b ˆ 2 ( N ) + [ n sp , N + 1 ( G N + 1 - 1 ) ] 1 / 2 c ˆ 2 } .
f 1 ( N + 1 ) = G N + 1 Γ N + 1 f 1 ( N ) + G N + 1 ( 1 - Γ N + 1 ) + ( n sp , N + 1 - 1 ) ( G N + 1 - 1 ) ,
n ASE ( N + 1 ) = G N + 1 Γ N + 1 n ASE ( N ) + n sp , N + 1 ( G N + 1 - 1 ) .
a ˆ N + 1 = G net ( N + 1 ) 1 / 2 a ˆ 0 + f 1 ( N + 1 ) 1 / 2 b ˆ 1 ( N + 1 ) + n ASE ( N + 1 ) 1 / 2 b ˆ 2 ( N + 1 ) .
exp ( - λ a ˆ a ˆ ) = exp [ ( e - λ - 1 ) a ˆ a ˆ ] n . o . ,
G n N ( z ) = m   1 m ! ( e - λ - 1 ) m a ˆ N m a ˆ N m .
G n N ( z ) = m   1 m ! ( e - λ - 1 ) m × ( G net 1 / 2 a ˆ 0 + n ASE 1 / 2 b ˆ 2 ) m × ( G net 1 / 2 a ˆ 0 + n ASE 1 / 2 b ˆ 2 ) m .
I =   d 2 β 2 π | β 2 β 2 | ,
G n N ( z ) =   d 2 β 2 π | β 2 | 0 b 2 | 2 m   1 m ! ( e - λ - 1 ) m × ( G net 1 / 2 b ˆ 0 + n ASE 1 / 2 β 2 ) m × ( G net 1 / 2 a ˆ 0 + n ASE 1 / 2 β 2 * ) m
G n N ( z ) =   d 2 β 2 π | β 2 | 0 b 2 | 2 × exp { ( e - λ - 1 ) ( G net 1 / 2 a ˆ 0 + n ASE 1 / 2 β 2 ) × ( G net 1 / 2 a ˆ 0 + n ASE 1 / 2 β 2 * ) } n . o .
β 2 | 0 b 2 = exp ( - | β 2 | 2 / 2 ) .
G n N ( z ) = 1 1 - n ASE ( e - λ - 1 ) × exp G net ( e - λ - 1 ) 1 - n ASE ( e - λ - 1 ) a ˆ 0 a ˆ 0 n . o . .
λ = - ln 1 + G net ( e - λ - 1 ) 1 - n ASE ( e - λ - 1 )
G net ( e - λ - 1 ) 1 - n ASE ( e - λ - 1 ) = e - λ - 1
G n N ( z ) = 1 1 - n ASE ( e - λ - 1 ) exp [ ( e - λ - 1 ) a ˆ 0 a ˆ 0 ] n . o = 1 1 - n ASE ( e - λ - 1 ) exp ( - λ n ˆ 0 ) = 1 1 - n ASE ( z - 1 ) 1 + G net ( z - 1 ) 1 - n ASE ( z - 1 ) n ˆ 0 = 1 1 - n ASE ( z - 1 ) G n 0 1 + G net ( z - 1 ) 1 - n ASE ( z - 1 ) ,

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