Abstract

The effect of nonlinear phase noise in dispersion-managed optical transmission systems is studied. The variance of the nonlinear phase noise in systems based on differential phase-shift keying (DPSK) in the presence of dispersion is examined analytically, and a semianalytical expression to calculate the error probability including intrachannel four-wave mixing, linear phase noise, and nonlinear phase noise for systems based on DPSK is derived. In addition, for the on–off keying (OOK) format, the formula for the error probability including amplified spontaneous emission noise and intrachannel nonlinear effects has been given. On the basis of the semianalytical expressions, we have compared the error probability of systems based on DPSK and OOK, and the results show that, to reach a given bit error rate of 109 for a specific long-haul system, the difference between the signal-to-noise ratio required by the DPSK format and that required by the OOK format is around 6  dB for the launch power of 0 dBm, and the difference becomes larger as the launch power increases.

© 2006 Optical Society of America

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References

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  1. X. Wei and X. Liu, "Analysis of intrachannel four-wave mixing in differential phase-shift keying transmission with large dispersion," Opt. Lett. 28, 2300-2302 (2003).
    [CrossRef] [PubMed]
  2. K. P. Ho, "Error probability of DPSK signals with intrachannel four-wave-mixing in highly dispersive transmission systems," IEEE Photon. Technol. Lett. 17, 789-791 (2005).
    [CrossRef]
  3. J. P. Gordon and L. F. Mollenauer, "Phase noise in photonic communications systems using linear amplifiers," Opt. Lett. 15, 1352-1353 (1990).
    [CrossRef]
  4. A. G. Green, P. P. Mitra, and L. G. L. Wegener, "Effect of chromatic dispersion on nonlinear phase noise," Opt. Lett. 28, 2455-2457 (2003).
    [CrossRef] [PubMed]
  5. S. Kumar, "Impact of dispersion on nonlinear phase noise in optical transmission systems," Opt. Lett. 30, 3278-3280 (2005).
    [CrossRef]
  6. A. Mecozzi, "Limits to long-haul coherent transmission set by the Kerr nonlinearity and noise of the in-line amplifiers," J. Lightwave Technol. 12, 1993-2000 (1994).
    [CrossRef]
  7. A. Mecozzi, "Long-distance transmission at zero dispersion combined effect of the Kerr nonlinearity and the noise of the in-line amplifiers," J. Opt. Soc. Am. B 11, 462-469 (1994).
    [CrossRef]
  8. K. P. Ho and H. C. Wang, "Comparison of nonlinear phase noise and intrachannel four-wave mixing for RZ-DPSK signals in dispersive transmission systems," IEEE Photon. Technol. Lett. 17, 1426-1428 (2005).
    [CrossRef]
  9. A. Gnauck and P. Winzer, "Optical phase-shift-keyed transmission," J. Lightwave Technol. 23, 115-130 (2005).
    [CrossRef]
  10. P. C. Jain, "Error probabilities in binary angle modulation," IEEE Trans. Inf. Theory IT-20, 36-42 (1974).
    [CrossRef]
  11. D. Middleton, An Introduction to Statistical Communication Theory (McGraw-Hill, 1960).
  12. K. P. Ho, "Impact of nonlinear phase noise to DPSK signals: a comparison of different models," IEEE Photon. Technol. Lett. 16, 1403-1405 (2004).
    [CrossRef]
  13. P. C. Jain and N. M. Blachman, "Detection of a PSK signal transmitted through a hard-limited channel," IEEE Trans. Inf. Theory IT-19, 623-630 (1973).
    [CrossRef]
  14. K. P. Ho, "Asymptotic probability density of nonlinear phase noise," Opt. Lett. 28, 1350-1352 (2003).
    [CrossRef] [PubMed]
  15. A. Mecozzi, "Probability density functions of the nonlinear phase noise," Opt. Lett. 29, 673-675 (2004).
    [CrossRef] [PubMed]
  16. S. Kumar and D. Yang, "Second-order theory for self-phase modulation and cross-phase phase modulation in optical fibers," J. Lightwave Technol. 23, 2073-2080 (2005).
    [CrossRef]
  17. P. Humblet and M. Arizoglu, "On the bit error rate of lightwave systems with optical amplifiers," J. Lightwave Technol. 9, 1576-1582 (1991).
    [CrossRef]
  18. J. G. Proakis, Digital Communications, 4th ed. (McGraw-Hill, 2000).
  19. M. Manna and E. Golovchenko, "FWM resonances in dispersion slop-matched and nonzero-dispersion fiber maps," IEEE Photon. Technol. Lett. 14, 929-931 (2002).
    [CrossRef]
  20. S. Kumar, J. C. Mauro, S. Raghavan, and D. Q. Chowdhury, "Intrachannel nonlinear penalties in dispersion-managed transmission systems," IEEE J. Sel. Top. Quantum Electron. 8, 626-631 (2002).
    [CrossRef]
  21. G. P. Agrawal, Nonlinear Fiber Optics (Academic, 1995), pp. 67-68.

2005

K. P. Ho, "Error probability of DPSK signals with intrachannel four-wave-mixing in highly dispersive transmission systems," IEEE Photon. Technol. Lett. 17, 789-791 (2005).
[CrossRef]

K. P. Ho and H. C. Wang, "Comparison of nonlinear phase noise and intrachannel four-wave mixing for RZ-DPSK signals in dispersive transmission systems," IEEE Photon. Technol. Lett. 17, 1426-1428 (2005).
[CrossRef]

A. Gnauck and P. Winzer, "Optical phase-shift-keyed transmission," J. Lightwave Technol. 23, 115-130 (2005).
[CrossRef]

S. Kumar and D. Yang, "Second-order theory for self-phase modulation and cross-phase phase modulation in optical fibers," J. Lightwave Technol. 23, 2073-2080 (2005).
[CrossRef]

S. Kumar, "Impact of dispersion on nonlinear phase noise in optical transmission systems," Opt. Lett. 30, 3278-3280 (2005).
[CrossRef]

2004

A. Mecozzi, "Probability density functions of the nonlinear phase noise," Opt. Lett. 29, 673-675 (2004).
[CrossRef] [PubMed]

K. P. Ho, "Impact of nonlinear phase noise to DPSK signals: a comparison of different models," IEEE Photon. Technol. Lett. 16, 1403-1405 (2004).
[CrossRef]

2003

2002

M. Manna and E. Golovchenko, "FWM resonances in dispersion slop-matched and nonzero-dispersion fiber maps," IEEE Photon. Technol. Lett. 14, 929-931 (2002).
[CrossRef]

S. Kumar, J. C. Mauro, S. Raghavan, and D. Q. Chowdhury, "Intrachannel nonlinear penalties in dispersion-managed transmission systems," IEEE J. Sel. Top. Quantum Electron. 8, 626-631 (2002).
[CrossRef]

1994

A. Mecozzi, "Long-distance transmission at zero dispersion combined effect of the Kerr nonlinearity and the noise of the in-line amplifiers," J. Opt. Soc. Am. B 11, 462-469 (1994).
[CrossRef]

A. Mecozzi, "Limits to long-haul coherent transmission set by the Kerr nonlinearity and noise of the in-line amplifiers," J. Lightwave Technol. 12, 1993-2000 (1994).
[CrossRef]

1991

P. Humblet and M. Arizoglu, "On the bit error rate of lightwave systems with optical amplifiers," J. Lightwave Technol. 9, 1576-1582 (1991).
[CrossRef]

1990

J. P. Gordon and L. F. Mollenauer, "Phase noise in photonic communications systems using linear amplifiers," Opt. Lett. 15, 1352-1353 (1990).
[CrossRef]

1974

P. C. Jain, "Error probabilities in binary angle modulation," IEEE Trans. Inf. Theory IT-20, 36-42 (1974).
[CrossRef]

1973

P. C. Jain and N. M. Blachman, "Detection of a PSK signal transmitted through a hard-limited channel," IEEE Trans. Inf. Theory IT-19, 623-630 (1973).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 1995), pp. 67-68.

Arizoglu, M.

P. Humblet and M. Arizoglu, "On the bit error rate of lightwave systems with optical amplifiers," J. Lightwave Technol. 9, 1576-1582 (1991).
[CrossRef]

Blachman, N. M.

P. C. Jain and N. M. Blachman, "Detection of a PSK signal transmitted through a hard-limited channel," IEEE Trans. Inf. Theory IT-19, 623-630 (1973).
[CrossRef]

Chowdhury, D. Q.

S. Kumar, J. C. Mauro, S. Raghavan, and D. Q. Chowdhury, "Intrachannel nonlinear penalties in dispersion-managed transmission systems," IEEE J. Sel. Top. Quantum Electron. 8, 626-631 (2002).
[CrossRef]

Gnauck, A.

Golovchenko, E.

M. Manna and E. Golovchenko, "FWM resonances in dispersion slop-matched and nonzero-dispersion fiber maps," IEEE Photon. Technol. Lett. 14, 929-931 (2002).
[CrossRef]

Gordon, J. P.

J. P. Gordon and L. F. Mollenauer, "Phase noise in photonic communications systems using linear amplifiers," Opt. Lett. 15, 1352-1353 (1990).
[CrossRef]

Green, A. G.

Ho, K. P.

K. P. Ho and H. C. Wang, "Comparison of nonlinear phase noise and intrachannel four-wave mixing for RZ-DPSK signals in dispersive transmission systems," IEEE Photon. Technol. Lett. 17, 1426-1428 (2005).
[CrossRef]

K. P. Ho, "Error probability of DPSK signals with intrachannel four-wave-mixing in highly dispersive transmission systems," IEEE Photon. Technol. Lett. 17, 789-791 (2005).
[CrossRef]

K. P. Ho, "Impact of nonlinear phase noise to DPSK signals: a comparison of different models," IEEE Photon. Technol. Lett. 16, 1403-1405 (2004).
[CrossRef]

K. P. Ho, "Asymptotic probability density of nonlinear phase noise," Opt. Lett. 28, 1350-1352 (2003).
[CrossRef] [PubMed]

Humblet, P.

P. Humblet and M. Arizoglu, "On the bit error rate of lightwave systems with optical amplifiers," J. Lightwave Technol. 9, 1576-1582 (1991).
[CrossRef]

Jain, P. C.

P. C. Jain, "Error probabilities in binary angle modulation," IEEE Trans. Inf. Theory IT-20, 36-42 (1974).
[CrossRef]

P. C. Jain and N. M. Blachman, "Detection of a PSK signal transmitted through a hard-limited channel," IEEE Trans. Inf. Theory IT-19, 623-630 (1973).
[CrossRef]

Kumar, S.

Liu, X.

Manna, M.

M. Manna and E. Golovchenko, "FWM resonances in dispersion slop-matched and nonzero-dispersion fiber maps," IEEE Photon. Technol. Lett. 14, 929-931 (2002).
[CrossRef]

Mauro, J. C.

S. Kumar, J. C. Mauro, S. Raghavan, and D. Q. Chowdhury, "Intrachannel nonlinear penalties in dispersion-managed transmission systems," IEEE J. Sel. Top. Quantum Electron. 8, 626-631 (2002).
[CrossRef]

Mecozzi, A.

Middleton, D.

D. Middleton, An Introduction to Statistical Communication Theory (McGraw-Hill, 1960).

Mitra, P. P.

Mollenauer, L. F.

J. P. Gordon and L. F. Mollenauer, "Phase noise in photonic communications systems using linear amplifiers," Opt. Lett. 15, 1352-1353 (1990).
[CrossRef]

Proakis, J. G.

J. G. Proakis, Digital Communications, 4th ed. (McGraw-Hill, 2000).

Raghavan, S.

S. Kumar, J. C. Mauro, S. Raghavan, and D. Q. Chowdhury, "Intrachannel nonlinear penalties in dispersion-managed transmission systems," IEEE J. Sel. Top. Quantum Electron. 8, 626-631 (2002).
[CrossRef]

Wang, H. C.

K. P. Ho and H. C. Wang, "Comparison of nonlinear phase noise and intrachannel four-wave mixing for RZ-DPSK signals in dispersive transmission systems," IEEE Photon. Technol. Lett. 17, 1426-1428 (2005).
[CrossRef]

Wegener, L. G. L.

Wei, X.

Winzer, P.

Yang, D.

IEEE J. Sel. Top. Quantum Electron.

S. Kumar, J. C. Mauro, S. Raghavan, and D. Q. Chowdhury, "Intrachannel nonlinear penalties in dispersion-managed transmission systems," IEEE J. Sel. Top. Quantum Electron. 8, 626-631 (2002).
[CrossRef]

IEEE Photon. Technol. Lett.

M. Manna and E. Golovchenko, "FWM resonances in dispersion slop-matched and nonzero-dispersion fiber maps," IEEE Photon. Technol. Lett. 14, 929-931 (2002).
[CrossRef]

K. P. Ho, "Error probability of DPSK signals with intrachannel four-wave-mixing in highly dispersive transmission systems," IEEE Photon. Technol. Lett. 17, 789-791 (2005).
[CrossRef]

K. P. Ho and H. C. Wang, "Comparison of nonlinear phase noise and intrachannel four-wave mixing for RZ-DPSK signals in dispersive transmission systems," IEEE Photon. Technol. Lett. 17, 1426-1428 (2005).
[CrossRef]

K. P. Ho, "Impact of nonlinear phase noise to DPSK signals: a comparison of different models," IEEE Photon. Technol. Lett. 16, 1403-1405 (2004).
[CrossRef]

IEEE Trans. Inf. Theory

P. C. Jain and N. M. Blachman, "Detection of a PSK signal transmitted through a hard-limited channel," IEEE Trans. Inf. Theory IT-19, 623-630 (1973).
[CrossRef]

P. C. Jain, "Error probabilities in binary angle modulation," IEEE Trans. Inf. Theory IT-20, 36-42 (1974).
[CrossRef]

J. Lightwave Technol.

P. Humblet and M. Arizoglu, "On the bit error rate of lightwave systems with optical amplifiers," J. Lightwave Technol. 9, 1576-1582 (1991).
[CrossRef]

A. Mecozzi, "Limits to long-haul coherent transmission set by the Kerr nonlinearity and noise of the in-line amplifiers," J. Lightwave Technol. 12, 1993-2000 (1994).
[CrossRef]

A. Gnauck and P. Winzer, "Optical phase-shift-keyed transmission," J. Lightwave Technol. 23, 115-130 (2005).
[CrossRef]

S. Kumar and D. Yang, "Second-order theory for self-phase modulation and cross-phase phase modulation in optical fibers," J. Lightwave Technol. 23, 2073-2080 (2005).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Lett.

Other

D. Middleton, An Introduction to Statistical Communication Theory (McGraw-Hill, 1960).

J. G. Proakis, Digital Communications, 4th ed. (McGraw-Hill, 2000).

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 1995), pp. 67-68.

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Figures (11)

Fig. 1
Fig. 1

Schematic of a dispersion-managed optical transmission system.

Fig. 2
Fig. 2

Variance of phase noise for a single pulse. The solid and dotted curves show the numerical and analytical results, respectively. The dashed curve shows the variance of linear phase noise by letting the nonlinear coefficient be zero in the simulation.

Fig. 3
Fig. 3

Variance of phase noise including the effects of SPM and IXPM. The solid and dotted curves show the numerical and analytical results of the phase noise, respectively. The dashed curve shows the linear phase noise only.

Fig. 4
Fig. 4

Variance of phase noise due to IFWM alone and due to the nonlinear phase noise including the effects of SPM and IXPM for the resonant dispersion map.

Fig. 5
Fig. 5

Variance of phase noise including the effects of SPM and IXPM for a nonresonant dispersion map. The solid and dotted curves show the numerical and analytical results of the phase noise, respectively. The dashed curve shows the linear phase noise obtained by setting the nonlinear coefficient to zero in the simulation.

Fig. 6
Fig. 6

Variance of the phase noise due to IFWM and the nonlinear phase noise for a nonresonant dispersion-managed system [ D = D 1 , D 2 = D 1 + 2.5 ( ps / nm ) / km ] .

Fig. 7
Fig. 7

Dependence of the BER on precompensation length for systems based on DPSK ( SNR = 15   dB ) .

Fig. 8
Fig. 8

Dependence of the BER on precompensation length for systems based on OOK ( SNR = 15   dB ) .

Fig. 9
Fig. 9

Dependence of the BER on the SNR for systems based on DPSK and OOK in the absence of nonlinearity.

Fig. 10
Fig. 10

(Color online) Dependence of the SNR on the launch power to reach the given BER of 10 9 . The solid curve with x's and the solid curve with diamonds are the required SNR for nonresonant dispersion-managed systems based on DPSK and OOK under different launch powers, respectively. The dotted curve with x's and the dotted curve with diamonds are the required SNR for resonant dispersion-managed systems based on DPSK and OOK under different launch powers.

Fig. 11
Fig. 11

(Color online) Required SNR to reach the given BERs of 10 6 , 10 7 , 10 8 , 10 9 , and 10 10 for different launch powers for various systems examined in our paper (nonresonant DPSK, resonant DPSK, nonresonant OOK, and resonant OOK).

Equations (232)

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10 9
6   dB
10 9
3   dBm
3   dB
0   dBm
6   dB
s ( t ) = | u ( t ) + u ( t T b ) | 2 | u ( t ) u ( t T b ) | 2 , = 4 A ( t ) A ( t T b ) cos ( Δϕ ) ,
u ( t ) = A ( t ) exp [ j ϕ ( t ) ]
Δ ϕ = ϕ ( t ) ϕ ( t T b )
T b
u ( t )
s = R cos ( Δ ϕ ) ,
Δ ϕ
( pdf )
Δ ϕ
Pe = - 0 q s ( r ) d r ,
q s ( r )
Ψ s ( ν )
C ( x ) = x q s ( r ) d r = 1 2 1 π 0 Im [ Ψ s ( ν ) exp ( j ν x ) ] d ν ν ,
Ψ s ( ν ) = E [ exp ( j ν s ) ] = E [ exp ( j ν R cos Δ ϕ ) ]
Im { }
E { }
Pe = C ( 0 ) = 1 2 1 π 0 Im [ Ψ s ( ν ) ] d ν ν = 1 2 1 π 0 E [ sin ( ν R cos Δ ϕ ) ] d ν ν = 1 2 2 π n = 0 ( 1 ) n E 0 J 2 n + 1 ( ν R ) ν d ν × cos [ ( 2 n + 1 ) Δ ϕ ] = 1 2 2 π n = 0 ( 1 ) n ( 2 n + 1 ) E { cos [ ( 2 n + 1 ) Δ ϕ ] } .
Δ ϕ = ϕ ( t ) ϕ ( t T b ) = [ ϕ L ( t ) + ϕ NL ( t ) + ϕ IFWM ( t ) ] [ ϕ L ( t T b ) + ϕ NL ( t T b ) + ϕ IFWM ( t T b ) ] ,
ϕ L ( t )
ASE , ϕ NL ( t )
ϕ IFWM ( t )
Δ ϕ = Δ ϕ L + NL + Δ ϕ IFWM ,
Δ ϕ L + NL = ϕ L ( t ) + ϕ NL ( t ) ϕ L ( t T b ) ϕ NL ( t T b ) ,
Δ ϕ IFWM = ϕ IFWM ( t ) ϕ IFWM ( t T b ) .
Δ ϕ L + NL
E { sin [ ( 2 n + 1 ) Δ ϕ L + NL ] } = 0
Pe = 1 2 2 π n = 0 ( 1 ) n ( 2 n + 1 ) E { cos [ ( 2 n + 1 ) Δ ϕ L + NL ] } × E { cos [ ( 2 n + 1 ) Δ ϕ IFWM ] } .
ϕ L ( t )
ϕ L ( t T b )
ϕ NL ( t )
ϕ NL ( t T b )
ϕ L ( t )
ϕ L ( t T b )
p ( ϕ L ) = 1 2 π + 1 π m = 1 C m cos ( m ϕ L ) ,
C m = π ρ s 2 exp ( ρ s / 2 )
[ I ( m 1 ) / 2 ( ρ s 2 )
+ I ( m + 1 ) / 2 ( ρ s 2 ) ] ,
z i
z i
E z i
C z i
T z i
u k ( 0 ) ( z i , t ) = E 0 π T z i exp [ ( t k T b ) 2 ( 1 + j C z i ) 2 T z i       2 + j θ z i ] ,
E 0 = P 0 π T 0
T 0
T b
T z i = T 0 4 + S z i 2 T 0 ,
C z i = S z i T 0 2 , E z i = E 0 ,
θ z i = 1 2 tan 1 ( S z i T 0 2 ) , S z i = 0 z i β 2 ( s ) d s .
u k
u k = u k ( 0 ) + γ u k ( 1 ) + γ 2 u k ( 2 ) + ,
u k
u k ( m )
j u k ( 1 ) z β 2 ( z ) 2 2 u k ( 1 ) t 2 = exp ( 0 z α ( s ) d s ) × [ | u k ( 0 ) | 2 + 2 l = N / 2 N / 2 | u l ( 0 ) | 2 ] u k ( 0 ) , l k , l ,   k = N / 2 , , N / 2 ,
N + 1
β 2 ( z )
α ( s )
z i
u 0 ( L ) = E z i π T ( L ) exp { j [ θ z i + θ ( L ) ] } [ 1 + j γ E z i h ( z i ) ] ,
h ( z i ) = T z i 2 j S ( L ) ( 1 + j C z i ) π × l = N / 2 N / 2 b l z i L exp [ w ( y ) a l ( y ) ] T ( y ) T z i 2 j S ( y ) ( 1 + j C z i ) × d y δ ( L , y ) R ( y ) ,
R ( y ) = 1 T 2 ( y ) + 1 + j C ( y ) 2 T 2 ( y ) ,
δ ( L , y ) = 1 2 j [ S ( L ) S ( y ) ] R ( y ) R ( y ) ,
b l = { 1 if l = 0 2 otherwise ,
a l ( y ) = { 1 + j C ( y ) 3 + j C ( y ) 1 T 2 ( y ) + 4 [ 3 + j C ( y ) ] δ ( L z i , y ) } × ( l T b ) 2 ,
θ ( L ) = 1 2 tan 1 [ S ( L ) T z i 2 + C z i S ( L ) ] ,
S ( y ) = z i y β 2 ( s ) d s , w ( y ) = z i y α ( s ) d s ,
T ( y ) = [ T z i 2 + C z i S ( y ) ] 2 + S 2 ( y ) T z i ,
C ( y ) = S ( y ) + C z i T z i 2 + C z i 2 S ( y ) T z i 2 .
ϕ NL , z i = tan 1 { γ E z i Re [ h ( z i ) ] 1 γ E z i Im [ h ( z i ) ] }
γ Re [ h ( z i ) ] { 1 + γ E z i Im [ h ( z i ) ] } E z i .
z i
δ E z i
z i
δθ NL , z i 2 = γ 2 Re [ h ( z i ) ] 2 { 1 + 2 γ E z i Im [ h ( z i ) ] } 2 δ E z i 2 .
δ E z i 2 = 2 ρ E z i ,
ρ = n sp h f ( G 1 ) .
σ NL 2
Δ ϕ L + NL
p ( Δ ϕ L + NL ) = 1 2 π + 1 π m = 1 C m 2 exp [ - ( 2 n + 1 ) 2 σ NL 2 ] × cos ( m Δ ϕ L + NL ) .
cos [ ( 2 n + 1 ) Δ ϕ L + NL ]
E { cos [ ( 2 n + 1 ) Δ ϕ L + NL ] } = 0 2 π cos [ ( 2 n + 1 ) Δ ϕ L + NL ] p ( Δ ϕ L + NL ) ϕ L + NL = π 4 ρ s e ρ s [ I n ( ρ s 2 ) + I n + 1 ( ρ s 2 ) ] 2 exp [ ( 2 n + 1 ) 2 σ NL 2 ] .
Pe = 1 2 ρ s e ρ s 2 n = 0 ( 1 ) n ( 2 n + 1 ) [ I n ( ρ s 2 ) + I n + 1 ( ρ s 2 ) ] 2 × exp [ ( 2 n + 1 ) 2 σ NL 2 ] E { cos [ ( 2 n + 1 ) Δ ϕ IFWM ] } .
cos [ ( 2 n + 1 ) Δ ϕ IFWM ]
ϕ IFWM ( t )
ϕ IFWM ( t T b )
Δ ϕ IFWM = ϕ IFWM ( t ) ϕ IFWM ( t T b )
Pe { b n } = 1 2 ρ s e ρ s 2 n = 0 ( 1 ) n ( 2 n + 1 ) [ I n ( ρ s 2 ) + I n + 1 ( ρ s 2 ) ] 2 × exp [ ( 2 n + 1 ) 2 σ NL 2 ] cos [ ( 2 n + 1 ) Δ ϕ IFWM ] .
Pe ALL = E { 1 2 ρ s e ρ s 2 n = 0 ( 1 ) n ( 2 n + 1 ) [ I n ( ρ s 2 ) + I n + 1 ( ρ s 2 ) ] 2 × exp [ ( 2 n + 1 ) 2 σ NL 2 ] × cos [ ( 2 n + 1 ) Δ ϕ IFWM ] } ,
S = | u ( t ) + n ( t ) | 2 ,
u ( t )
n ( t )
f p ( x ) = 1 N 0 ( x P ) 1 / 2 exp ( x + P N 0 ) I 0 ( 2 x P N 0 ) , x > 0 ,
N 0 / 2
I 0
p ( 0 | 1 ) = 0 ζ f P 0 ( x ) d x = 1 Q 1 ( 2 P 0 N 0 , 2 ζ N 0 ) ,
p ( 1 | 0 ) = ζ f 0 ( x ) d x = Q 1 ( 0 , 2 ζ N 0 ) = exp ( ζ N 0 ) ,
P 0
Q 1 ( , )
Q 1 ( a , b ) = exp [ ( a + b ) / 2 ] k = 0 ( a b ) k / 2 I k ( a b ) .
Pe = [ p ( 0 | 1 ) + p ( 1 | 0 ) ] / 2
s = | u ( t ) + δ u + n ( t ) | 2 ,
δ u 1
u ( t ) + δ u 1
P = | u + δ u 1 | 2 = P 0 | 1 + δ u 1 P 0 | 2 .
{ b n }
p { b n } ( 0 | 1 ) = 1 Q 1 ( 2 P 0 | 1 + δ u 1 P 0 | 2 N 0 , 2 ζ N 0 ) .
p NL ( 0 | 1 ) = E { 1 Q 1 ( 2 P 0 | 1 + δ u 1 P 0 | 2 N 0 , 2 ζ N 0 ) } .
p NL ( 1 | 0 ) = E { Q 1 ( 2 | δ u 0 | 2 N 0 , 2 ζ N 0 ) } ,
δ u 0
Pe NL = [ p NL ( 0 | 1 ) + p NL ( 1 | 0 ) ] / 2 .
Pe NL
40   km
D 1
γ 1 = 2.5 W 1 km 1
α 1 = 0.2 dB / km
D 2
γ 2 = 10.0 W 1
km 1
α 2 = 0.2 dB / km
n sp = 1
D pre
D post
γ 2 = 10 W 1
km 1
α pre = α post = 0.5 dB / km
1 .55   μm
24   THz
46   GHz
40   Gbits / s
80   km
D 1 + D 2 = 0
8.33   ps
2   mW
17   pulses
8 .33   ps
25   ps
D 1
D 2
D 1
D 2
n sp = 0
n sp = 1
| D 1 | =
| D 2 |
σ IFWM 2 = ϕ 2 ϕ 2
t = 0
n sp = 0
D 1
D 2
D 1
D 2
100 ps / nm
D = D 1
D 2 = | D 1 |
40   Gbits / s
8 .33   ps
D 1
D 2
γ 1 = 2.5 W 1 km 1
γ 2 = 10 W 1 km 1
α 1 = α 2
D pre = D post
α pre = α post
800   km
E { cos [ ( 2 n + 1 ) Δ ϕ IFWM }
Δ φ IFWM
24   THz
Δ ϕ IFWM = ϕ IFWM 0 ϕ IFWM 1 ,
ϕ IFWM j = sin 1 [ Im ( u j ) / | u j | ] , j = 0 , 1 ,
u 0
u 1
10   km
5 .5   km
550 ps / nm
5.5   km
1   mW
p NL ( 0 | 1 )
8.0   km
10 9
10 9
10 9
10 9
3   dBm
0   dBm
6   dB
33%
D 1 = 17 ( ps / nm ) / km
D 2 = 17 ( ps / nm ) / km
10 9
10 9
3   dB
u k ( 0 , t ) = P 0 exp [ ( t k T b ) 2 2 T 0 2 ] = E 0 T eff exp [ ( t k T b ) 2 2 T 0 2 ] ,
T b
T 0
P 0
T eff
π T 0
E 0
E 0 = P 0 T eff
z i
u k ( 0 ) ( z i , t ) = E 0 π T z i exp [ ( t k T b ) 2 ( 1 + j C z i ) 2 T z i 2 + j θ z i ] .
z i
u k ( 0 ) ( z , t ) = E 0 π T ( z ) exp { j [ θ z i + θ ( z ) ] } × exp { ( t k T b ) 2 [ 1 + j C ( z ) ] 2 T ( z ) 2 } ,
T ( z ) = [ T z i 2 + C z i S ( z ) ] + S 2 ( z ) T z i ,
C ( z ) = S ( z ) + C z i T z i 2 + C z i 2 S ( z ) T z i 2 ,
θ ( z ) = 1 2 tan 1 [ S ( z ) T z i 2 + C z i S ( z ) ] .
l T b ( l = N / 2 , , N / 2 )
F ( z , t ) = l = N / 2 N / 2 { 2 exp [ w ( z ) ] | u l ( 0 ) | 2 u 0 ( 0 ) } = 2 l = N / 2 N / 2 ( exp [ w ( z ) ] E 0 π T ( z ) × exp [ ( t l T b ) 2 2 T ( z ) 2 ] E 0 π T ( z ) × exp { j [ θ z i + θ ( z ) ] } exp { t 2 [ 1 + j C ( z ) ] 2 T ( z ) 2 } ) = η ( z ) l = N / 2 N / 2 exp { m = 1 2 [ t D m , l ( z ) ] 2 R m ( z ) } ,
η ( z ) = 2 exp [ w ( z ) ] E 0 π T ( z ) E 0 π T ( z )
× exp { j [ θ z i + θ ( z ) ] } ,
{ R 1 ( z ) = 1 T ( z ) 2 R 2 ( z ) = [ 1 + j C ( z ) ] 2 T ( z ) 2     D 1 , l = l T b , D 2 , l = 0 .
u 0 ( 1 ) ,IXPM ( z , t ) = j z i z η ( y ) exp [ R ¯ τ 2 ( t D ¯ ) 2 δ ] δ ( z , y ) R ( y ) d y ,
{ R ¯ = R 1 R 2 R 1 + R 2 R = R 1 + R 2 D ¯ = τ R 2 R 1 + R 2 δ ( z , y ) = 1 2 j R [ S ( z ) S ( y ) ] R .
u 0 ( 1 ) ,SPM ( z , t ) = j z i z η ( y ) δ ( z , y ) R ( y ) exp { t 2 δ } d y ,
η ( z ) = exp [ w ( z ) ] E 0 π T ( z ) E 0 π T ( z ) × exp { j [ θ z i + θ ( z ) ] } .
u 0 ( L , t ) = u 0 ( 0 ) ( L , t ) + γ u 0 ( 1 ) ,SPM ( L , t ) + γ u 0 ( 1 ) ,IXPM ( L , t ) .
[ D = D 1 , D 2 = D 1 + 2.5 ( ps / nm ) / km ]
( SNR = 15   dB )
( SNR = 15   dB )
10 9
10 6
10 7
10 8
10 9
10 10

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