Abstract

An analytical model to calculate the variance of cross-phase modulation (XPM) distortion in a wavelength-division multiplexed (WDM) fiber-optic system is developed. The method is based on the first order perturbation technique and it is applicable for both dispersion managed and dispersion uncompensated systems. For dispersion managed systems, it is shown that the variance of XPM distortion scales as Nx where N is the number of spans and x ∈ [1, 2] depending on the amount of inline-dispersion compensation. The analytical model is found to be in good agreement with simulations in most of the cases.

© 2014 Optical Society of America

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References

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  1. R. Hui, Y. Wang, K. Demarest, C. Allen, “Frequency response of cross-phase modulation in multispan WDM optical fiber systems,” IEEE Photon. Technol. Lett. 10, 1271–1273 (1998).
    [CrossRef]
  2. R. Hui, K. Demarest, C. Allen, “Cross-phase modulation in multispan WDM optical fiber systems,” J. Lightw. Technol. 17, 1018–1026 (1999).
    [CrossRef]
  3. M. Eiselt, M. Shtaif, L.D. Garrett, “Contribution of timing jitter and amplitude distortion to XPM system penalty in WDM systems,” IEEE Photon. Technol. Lett. 11, 748–750 (1999).
    [CrossRef]
  4. D. Marcuse, A.R. Chraplyvy, R.W. Tkach, “Dependence of cross-phase modulation on channel number in fiber WDM systems,” J. Lightw. Technol. 12, 885–890 (1994).
    [CrossRef]
  5. A.T. Cartaxo, “Cross-phase modulation in intensity modulation-direct detection WDM systems with multiple optical amplifiers and dispersion compensators,” J. Lightw. Technol. 17, 178–190 (1999).
    [CrossRef]
  6. Z. Jiang, C. Fan, “A comprehensive study on XPM-and SRS-induced noise in cascaded IM-DD optical fiber transmission systems,” J. Lightw. Technol. 21, 953–960 (2003).
    [CrossRef]
  7. S. Kumar, D. Yang., “Second-order theory for self-phase modulation and cross-phase modulation in optical fibers,” J. Lightw. Technol. 23, 2073–2080 (2005).
    [CrossRef]
  8. J. Tang, “The channel capacity of a multispan DWDM system employing dispersive nonlinear optical fibers and an ideal coherent optical receiver,” J. Lightw. Technol. 20, 1095–1101 (2002).
    [CrossRef]
  9. Z. Tao, W. Yan, L. Liu, L. Li, S. Oda, T. Hoshida, J. C. Rasmussen, “Simple Fiber Model for Determination of XPM Effects,” J. Lightw. Technol. 29, 974–986 (2011).
    [CrossRef]
  10. M. Nazarathy, J. Khurgin, R. Weidenfeld, Y. Meiman, P. Cho, R. Noe, I. Shpantzer, V. Karagodsky, “Phased-array cancellation of nonlinear FWM in coherent OFDM dispersive multi-span links,” Opt. Express 16, 15777–15810 (2008).
    [CrossRef] [PubMed]
  11. A. Carena, V. Curri, G. Bosco, P. Poggiolini, F. Forghieri, “Modeling of the impact of nonlinear propagation effects in uncompensated optical coherent transmission links,” J. Lightw. Technol. 30, 1524–1539 (2012).
    [CrossRef]
  12. P. Poggiolini, A. Carena, V. Curri, G. Bosco, F. Forghieri, “Analytical modeling of non-linear propagation in uncompensated optical transmission links,” IEEE Photon. Technol. Lett. 23, 742–744 (2011).
    [CrossRef]
  13. A. Mecozzi, Rene Essiambre, “Nonlinear Shannon limit in pseudolinear coherent systems,” J. Lightw. Technol. 30, 2011–2024 (2012).
    [CrossRef]
  14. G. P. Agrawal, Nonlinear Fiber Optics, 4 (Academic Press, 2007).
  15. S. Kumar, S. N. Shahi, D. Yang., “Analytical modeling of a single channel nonlinear fiber optic system based on QPSK,” Opt. Express 20, 27740–27755 (2012).
    [CrossRef] [PubMed]
  16. D. Yang, S. Kumar, “Analysis of intrachannel impairments for coherent systems based on phase-shift keying,” J. Lightw. Technol. 27, 2916–2923 (2009).

2012 (3)

A. Carena, V. Curri, G. Bosco, P. Poggiolini, F. Forghieri, “Modeling of the impact of nonlinear propagation effects in uncompensated optical coherent transmission links,” J. Lightw. Technol. 30, 1524–1539 (2012).
[CrossRef]

A. Mecozzi, Rene Essiambre, “Nonlinear Shannon limit in pseudolinear coherent systems,” J. Lightw. Technol. 30, 2011–2024 (2012).
[CrossRef]

S. Kumar, S. N. Shahi, D. Yang., “Analytical modeling of a single channel nonlinear fiber optic system based on QPSK,” Opt. Express 20, 27740–27755 (2012).
[CrossRef] [PubMed]

2011 (2)

P. Poggiolini, A. Carena, V. Curri, G. Bosco, F. Forghieri, “Analytical modeling of non-linear propagation in uncompensated optical transmission links,” IEEE Photon. Technol. Lett. 23, 742–744 (2011).
[CrossRef]

Z. Tao, W. Yan, L. Liu, L. Li, S. Oda, T. Hoshida, J. C. Rasmussen, “Simple Fiber Model for Determination of XPM Effects,” J. Lightw. Technol. 29, 974–986 (2011).
[CrossRef]

2009 (1)

D. Yang, S. Kumar, “Analysis of intrachannel impairments for coherent systems based on phase-shift keying,” J. Lightw. Technol. 27, 2916–2923 (2009).

2008 (1)

2005 (1)

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

2003 (1)

Z. Jiang, C. Fan, “A comprehensive study on XPM-and SRS-induced noise in cascaded IM-DD optical fiber transmission systems,” J. Lightw. Technol. 21, 953–960 (2003).
[CrossRef]

2002 (1)

J. Tang, “The channel capacity of a multispan DWDM system employing dispersive nonlinear optical fibers and an ideal coherent optical receiver,” J. Lightw. Technol. 20, 1095–1101 (2002).
[CrossRef]

1999 (3)

R. Hui, K. Demarest, C. Allen, “Cross-phase modulation in multispan WDM optical fiber systems,” J. Lightw. Technol. 17, 1018–1026 (1999).
[CrossRef]

M. Eiselt, M. Shtaif, L.D. Garrett, “Contribution of timing jitter and amplitude distortion to XPM system penalty in WDM systems,” IEEE Photon. Technol. Lett. 11, 748–750 (1999).
[CrossRef]

A.T. Cartaxo, “Cross-phase modulation in intensity modulation-direct detection WDM systems with multiple optical amplifiers and dispersion compensators,” J. Lightw. Technol. 17, 178–190 (1999).
[CrossRef]

1998 (1)

R. Hui, Y. Wang, K. Demarest, C. Allen, “Frequency response of cross-phase modulation in multispan WDM optical fiber systems,” IEEE Photon. Technol. Lett. 10, 1271–1273 (1998).
[CrossRef]

1994 (1)

D. Marcuse, A.R. Chraplyvy, R.W. Tkach, “Dependence of cross-phase modulation on channel number in fiber WDM systems,” J. Lightw. Technol. 12, 885–890 (1994).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics, 4 (Academic Press, 2007).

Allen, C.

R. Hui, K. Demarest, C. Allen, “Cross-phase modulation in multispan WDM optical fiber systems,” J. Lightw. Technol. 17, 1018–1026 (1999).
[CrossRef]

R. Hui, Y. Wang, K. Demarest, C. Allen, “Frequency response of cross-phase modulation in multispan WDM optical fiber systems,” IEEE Photon. Technol. Lett. 10, 1271–1273 (1998).
[CrossRef]

Bosco, G.

A. Carena, V. Curri, G. Bosco, P. Poggiolini, F. Forghieri, “Modeling of the impact of nonlinear propagation effects in uncompensated optical coherent transmission links,” J. Lightw. Technol. 30, 1524–1539 (2012).
[CrossRef]

P. Poggiolini, A. Carena, V. Curri, G. Bosco, F. Forghieri, “Analytical modeling of non-linear propagation in uncompensated optical transmission links,” IEEE Photon. Technol. Lett. 23, 742–744 (2011).
[CrossRef]

Carena, A.

A. Carena, V. Curri, G. Bosco, P. Poggiolini, F. Forghieri, “Modeling of the impact of nonlinear propagation effects in uncompensated optical coherent transmission links,” J. Lightw. Technol. 30, 1524–1539 (2012).
[CrossRef]

P. Poggiolini, A. Carena, V. Curri, G. Bosco, F. Forghieri, “Analytical modeling of non-linear propagation in uncompensated optical transmission links,” IEEE Photon. Technol. Lett. 23, 742–744 (2011).
[CrossRef]

Cartaxo, A.T.

A.T. Cartaxo, “Cross-phase modulation in intensity modulation-direct detection WDM systems with multiple optical amplifiers and dispersion compensators,” J. Lightw. Technol. 17, 178–190 (1999).
[CrossRef]

Cho, P.

Chraplyvy, A.R.

D. Marcuse, A.R. Chraplyvy, R.W. Tkach, “Dependence of cross-phase modulation on channel number in fiber WDM systems,” J. Lightw. Technol. 12, 885–890 (1994).
[CrossRef]

Curri, V.

A. Carena, V. Curri, G. Bosco, P. Poggiolini, F. Forghieri, “Modeling of the impact of nonlinear propagation effects in uncompensated optical coherent transmission links,” J. Lightw. Technol. 30, 1524–1539 (2012).
[CrossRef]

P. Poggiolini, A. Carena, V. Curri, G. Bosco, F. Forghieri, “Analytical modeling of non-linear propagation in uncompensated optical transmission links,” IEEE Photon. Technol. Lett. 23, 742–744 (2011).
[CrossRef]

Demarest, K.

R. Hui, K. Demarest, C. Allen, “Cross-phase modulation in multispan WDM optical fiber systems,” J. Lightw. Technol. 17, 1018–1026 (1999).
[CrossRef]

R. Hui, Y. Wang, K. Demarest, C. Allen, “Frequency response of cross-phase modulation in multispan WDM optical fiber systems,” IEEE Photon. Technol. Lett. 10, 1271–1273 (1998).
[CrossRef]

Eiselt, M.

M. Eiselt, M. Shtaif, L.D. Garrett, “Contribution of timing jitter and amplitude distortion to XPM system penalty in WDM systems,” IEEE Photon. Technol. Lett. 11, 748–750 (1999).
[CrossRef]

Essiambre, Rene

A. Mecozzi, Rene Essiambre, “Nonlinear Shannon limit in pseudolinear coherent systems,” J. Lightw. Technol. 30, 2011–2024 (2012).
[CrossRef]

Fan, C.

Z. Jiang, C. Fan, “A comprehensive study on XPM-and SRS-induced noise in cascaded IM-DD optical fiber transmission systems,” J. Lightw. Technol. 21, 953–960 (2003).
[CrossRef]

Forghieri, F.

A. Carena, V. Curri, G. Bosco, P. Poggiolini, F. Forghieri, “Modeling of the impact of nonlinear propagation effects in uncompensated optical coherent transmission links,” J. Lightw. Technol. 30, 1524–1539 (2012).
[CrossRef]

P. Poggiolini, A. Carena, V. Curri, G. Bosco, F. Forghieri, “Analytical modeling of non-linear propagation in uncompensated optical transmission links,” IEEE Photon. Technol. Lett. 23, 742–744 (2011).
[CrossRef]

Garrett, L.D.

M. Eiselt, M. Shtaif, L.D. Garrett, “Contribution of timing jitter and amplitude distortion to XPM system penalty in WDM systems,” IEEE Photon. Technol. Lett. 11, 748–750 (1999).
[CrossRef]

Hoshida, T.

Z. Tao, W. Yan, L. Liu, L. Li, S. Oda, T. Hoshida, J. C. Rasmussen, “Simple Fiber Model for Determination of XPM Effects,” J. Lightw. Technol. 29, 974–986 (2011).
[CrossRef]

Hui, R.

R. Hui, K. Demarest, C. Allen, “Cross-phase modulation in multispan WDM optical fiber systems,” J. Lightw. Technol. 17, 1018–1026 (1999).
[CrossRef]

R. Hui, Y. Wang, K. Demarest, C. Allen, “Frequency response of cross-phase modulation in multispan WDM optical fiber systems,” IEEE Photon. Technol. Lett. 10, 1271–1273 (1998).
[CrossRef]

Jiang, Z.

Z. Jiang, C. Fan, “A comprehensive study on XPM-and SRS-induced noise in cascaded IM-DD optical fiber transmission systems,” J. Lightw. Technol. 21, 953–960 (2003).
[CrossRef]

Karagodsky, V.

Khurgin, J.

Kumar, S.

S. Kumar, S. N. Shahi, D. Yang., “Analytical modeling of a single channel nonlinear fiber optic system based on QPSK,” Opt. Express 20, 27740–27755 (2012).
[CrossRef] [PubMed]

D. Yang, S. Kumar, “Analysis of intrachannel impairments for coherent systems based on phase-shift keying,” J. Lightw. Technol. 27, 2916–2923 (2009).

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

Li, L.

Z. Tao, W. Yan, L. Liu, L. Li, S. Oda, T. Hoshida, J. C. Rasmussen, “Simple Fiber Model for Determination of XPM Effects,” J. Lightw. Technol. 29, 974–986 (2011).
[CrossRef]

Liu, L.

Z. Tao, W. Yan, L. Liu, L. Li, S. Oda, T. Hoshida, J. C. Rasmussen, “Simple Fiber Model for Determination of XPM Effects,” J. Lightw. Technol. 29, 974–986 (2011).
[CrossRef]

Marcuse, D.

D. Marcuse, A.R. Chraplyvy, R.W. Tkach, “Dependence of cross-phase modulation on channel number in fiber WDM systems,” J. Lightw. Technol. 12, 885–890 (1994).
[CrossRef]

Mecozzi, A.

A. Mecozzi, Rene Essiambre, “Nonlinear Shannon limit in pseudolinear coherent systems,” J. Lightw. Technol. 30, 2011–2024 (2012).
[CrossRef]

Meiman, Y.

Nazarathy, M.

Noe, R.

Oda, S.

Z. Tao, W. Yan, L. Liu, L. Li, S. Oda, T. Hoshida, J. C. Rasmussen, “Simple Fiber Model for Determination of XPM Effects,” J. Lightw. Technol. 29, 974–986 (2011).
[CrossRef]

Poggiolini, P.

A. Carena, V. Curri, G. Bosco, P. Poggiolini, F. Forghieri, “Modeling of the impact of nonlinear propagation effects in uncompensated optical coherent transmission links,” J. Lightw. Technol. 30, 1524–1539 (2012).
[CrossRef]

P. Poggiolini, A. Carena, V. Curri, G. Bosco, F. Forghieri, “Analytical modeling of non-linear propagation in uncompensated optical transmission links,” IEEE Photon. Technol. Lett. 23, 742–744 (2011).
[CrossRef]

Rasmussen, J. C.

Z. Tao, W. Yan, L. Liu, L. Li, S. Oda, T. Hoshida, J. C. Rasmussen, “Simple Fiber Model for Determination of XPM Effects,” J. Lightw. Technol. 29, 974–986 (2011).
[CrossRef]

Shahi, S. N.

Shpantzer, I.

Shtaif, M.

M. Eiselt, M. Shtaif, L.D. Garrett, “Contribution of timing jitter and amplitude distortion to XPM system penalty in WDM systems,” IEEE Photon. Technol. Lett. 11, 748–750 (1999).
[CrossRef]

Tang, J.

J. Tang, “The channel capacity of a multispan DWDM system employing dispersive nonlinear optical fibers and an ideal coherent optical receiver,” J. Lightw. Technol. 20, 1095–1101 (2002).
[CrossRef]

Tao, Z.

Z. Tao, W. Yan, L. Liu, L. Li, S. Oda, T. Hoshida, J. C. Rasmussen, “Simple Fiber Model for Determination of XPM Effects,” J. Lightw. Technol. 29, 974–986 (2011).
[CrossRef]

Tkach, R.W.

D. Marcuse, A.R. Chraplyvy, R.W. Tkach, “Dependence of cross-phase modulation on channel number in fiber WDM systems,” J. Lightw. Technol. 12, 885–890 (1994).
[CrossRef]

Wang, Y.

R. Hui, Y. Wang, K. Demarest, C. Allen, “Frequency response of cross-phase modulation in multispan WDM optical fiber systems,” IEEE Photon. Technol. Lett. 10, 1271–1273 (1998).
[CrossRef]

Weidenfeld, R.

Yan, W.

Z. Tao, W. Yan, L. Liu, L. Li, S. Oda, T. Hoshida, J. C. Rasmussen, “Simple Fiber Model for Determination of XPM Effects,” J. Lightw. Technol. 29, 974–986 (2011).
[CrossRef]

Yang, D.

D. Yang, S. Kumar, “Analysis of intrachannel impairments for coherent systems based on phase-shift keying,” J. Lightw. Technol. 27, 2916–2923 (2009).

Yang., D.

S. Kumar, S. N. Shahi, D. Yang., “Analytical modeling of a single channel nonlinear fiber optic system based on QPSK,” Opt. Express 20, 27740–27755 (2012).
[CrossRef] [PubMed]

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

IEEE Photon. Technol. Lett. (3)

M. Eiselt, M. Shtaif, L.D. Garrett, “Contribution of timing jitter and amplitude distortion to XPM system penalty in WDM systems,” IEEE Photon. Technol. Lett. 11, 748–750 (1999).
[CrossRef]

R. Hui, Y. Wang, K. Demarest, C. Allen, “Frequency response of cross-phase modulation in multispan WDM optical fiber systems,” IEEE Photon. Technol. Lett. 10, 1271–1273 (1998).
[CrossRef]

P. Poggiolini, A. Carena, V. Curri, G. Bosco, F. Forghieri, “Analytical modeling of non-linear propagation in uncompensated optical transmission links,” IEEE Photon. Technol. Lett. 23, 742–744 (2011).
[CrossRef]

J. Lightw. Technol. (10)

A. Mecozzi, Rene Essiambre, “Nonlinear Shannon limit in pseudolinear coherent systems,” J. Lightw. Technol. 30, 2011–2024 (2012).
[CrossRef]

R. Hui, K. Demarest, C. Allen, “Cross-phase modulation in multispan WDM optical fiber systems,” J. Lightw. Technol. 17, 1018–1026 (1999).
[CrossRef]

A. Carena, V. Curri, G. Bosco, P. Poggiolini, F. Forghieri, “Modeling of the impact of nonlinear propagation effects in uncompensated optical coherent transmission links,” J. Lightw. Technol. 30, 1524–1539 (2012).
[CrossRef]

D. Yang, S. Kumar, “Analysis of intrachannel impairments for coherent systems based on phase-shift keying,” J. Lightw. Technol. 27, 2916–2923 (2009).

D. Marcuse, A.R. Chraplyvy, R.W. Tkach, “Dependence of cross-phase modulation on channel number in fiber WDM systems,” J. Lightw. Technol. 12, 885–890 (1994).
[CrossRef]

A.T. Cartaxo, “Cross-phase modulation in intensity modulation-direct detection WDM systems with multiple optical amplifiers and dispersion compensators,” J. Lightw. Technol. 17, 178–190 (1999).
[CrossRef]

Z. Jiang, C. Fan, “A comprehensive study on XPM-and SRS-induced noise in cascaded IM-DD optical fiber transmission systems,” J. Lightw. Technol. 21, 953–960 (2003).
[CrossRef]

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

J. Tang, “The channel capacity of a multispan DWDM system employing dispersive nonlinear optical fibers and an ideal coherent optical receiver,” J. Lightw. Technol. 20, 1095–1101 (2002).
[CrossRef]

Z. Tao, W. Yan, L. Liu, L. Li, S. Oda, T. Hoshida, J. C. Rasmussen, “Simple Fiber Model for Determination of XPM Effects,” J. Lightw. Technol. 29, 974–986 (2011).
[CrossRef]

Opt. Express (2)

Other (1)

G. P. Agrawal, Nonlinear Fiber Optics, 4 (Academic Press, 2007).

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Figures (10)

Fig. 1
Fig. 1

Multispan WDM fiber-optic system.

Fig. 2
Fig. 2

(a) Absolute of mean XPM distortion, and (b) Mean of absolute square of XPM distortion. Following parameters were assumed: Ppeak = 0 dBm, β2 = −10 ps2/km, and no. of spans = 10.

Fig. 3
Fig. 3

Variance of XPM impairment. Parameters are the same as that of Fig. 2.

Fig. 4
Fig. 4

The mean XPM variance versus (a) peak power, and (b) fiber dispersion. The other parameters are same as that of Fig. 2.

Fig. 5
Fig. 5

The mean XPM variance versus (a) no. of spans, and (b) channel spacing. The other parameters are same as that of Fig. 2.

Fig. 6
Fig. 6

The mean XPM variance versus peak power for (a) 2-channel WDM, and (b) 5-channel WDM system. 28 Gbaud WDM system and the standard single-mode fiber (SSMF) with the following parameters is used: β2 = −22 ps2/km, γ = 1.1 W−1km−1 and loss α = 0.2 dB/km The other parameters are same as that of Fig. 2.

Fig. 7
Fig. 7

A typical fiber-optic transmission system with inline-compensation. TF:transmission fiber, DCF:dispersion compensating fiber.

Fig. 8
Fig. 8

The mean XPM variance versus pre-compensation percent. Following parameters were assumed: Ppeak = 0 dBm, length of inline DCF Linline = 4.8 km.

Fig. 9
Fig. 9

The mean XPM variance versus inline-compensation ratio. Pre-compensation ratio = 50%, Ppeak = 0 dBm, and no. of spans = 10.

Fig. 10
Fig. 10

The mean XPM variance versus no. of spans for η = 100% (Dres = 0 ps/nm), 90% (Dres = 61 ps/nm), and 80% (Dres = 123 ps/nm). Pre-compensation ratio = 50%, and Ppeak = 0 dBm.

Equations (67)

Equations on this page are rendered with MathJax. Learn more.

j q z β 2 ( z ) 2 2 q T 2 + γ 0 | q | 2 q = j α ( z ) 2 q ,
q ( z , T ) = exp [ w ( z ) / 2 ] u ( z , T ) ,
j u z β 2 ( z ) 2 2 u T 2 + γ ( z ) | u | 2 u = 0 ,
u = u 1 + u 2 ,
j u k z β 2 ( z ) 2 2 u k T 2 = γ ( z ) [ | u k | 2 + 2 | u l | 2 ] u k , k = 1 , 2 and l = 3 k
u 1 ( 0 , T ) = P a 0 p ( 0 , T ) ,
u 2 ( 0 , T ) = P n a n p ( 0 , T n T s ) exp ( j Ω t ) ,
p ( 0 , T ) = exp ( T 2 / 2 T 0 2 )
a n = x n + j y n 2 ,
u k = u k 0 + γ 0 u k ( 1 ) + γ 0 2 u k ( 2 ) + , k = 1 , 2
j u k ( 0 ) z β 2 2 2 u k ( 0 ) T 2 = 0 .
u 1 ( 0 ) ( z , T ) = P T 0 T 1 a 0 exp [ T 2 2 T 1 2 ] ,
u 2 ( 0 ) ( z , T ) = P T 0 T 1 n a n exp [ ( T τ n ) 2 2 T 1 2 j Ω t + j θ ( z ) ] ,
T 1 = ( T 0 2 j S ( z ) ) 1 / 2 ,
τ n = n T s + S ( z ) Ω ,
θ ( z ) = S ( z ) Ω 2 2 ,
S ( z ) = 0 z β 2 ( s ) d s .
j u k ( 1 ) z β 2 ( z ) 2 2 u k ( 1 ) T 2 = exp [ w ( z ) ] [ | u k ( 0 ) | 2 + 2 | u l ( 0 ) | 2 ] u k ( 0 ) . k = 1 , 2 and l = 3 k .
j f z β 2 ( z ) 2 2 f T 2 = F ( z , T ) ,
F ( z , T ) = η ( z ) exp { k = 1 3 [ T C k ( z ) ] 2 R k ( z ) } ,
f ( z , T ) = j 0 z η ( s ) δ ( z , s ) R ( s ) exp [ k = 1 3 C k 2 R k + C 2 R ] exp [ ( D + j T ) 2 δ ( z , s ) ] d s ,
R = R 1 + R 2 + R 3 ,
C = C 1 R 1 + C 2 R 2 + C 3 R 3 ,
D = j C R ,
δ = 1 j R A ( z , s ) R ,
A ( z , s ) = 2 [ S ( z ) S ( s ) ] .
F ( z , T ) = 2 exp [ w ( z ) ] | u 2 ( 0 ) | 2 u 1 ( 0 ) , = 2 P 3 / 2 a 0 η ( z ) m n a m a n * exp { k = 1 3 [ T C k ( z ) ] 2 R k ( z ) } ,
η ( z ) = T 0 3 exp [ w ( z ) ] T 1 ( z ) | T 1 ( z ) | 2 ,
C 1 ( z ) = τ m ( z ) , C 2 ( z ) = τ n ( z ) , C 3 ( z ) = 0 ,
R 1 = R 3 = 1 2 T 1 2 , R 2 = 1 2 ( T 1 * ) 2 .
u 1 ( 1 ) , X P M ( z , T ) = j 2 P 3 / 2 a 0 m n a m a n * X m n ( z , T )
X m n ( z , T ) = 0 z η ( s ) δ ( z , s ) R ( s ) exp [ ( D + j T ) 2 δ ( z , s ) ] d s ,
η ( s ) = η ( s ) exp ( k = 1 3 C k 2 R k + C 2 R ) ,
u 1 ( 1 ) , S P M ( z , T ) = j P 3 / 2 a 0 | a 0 | 2 0 z η ( s ) δ ( z , s ) R ( s ) exp ( T 2 δ ( z , s ) ) d s ,
u 1 ( 1 ) = u 1 ( 1 ) , S P M + u 1 ( 1 ) , X P M .
Var { δ u 1 } = E { | δ u 1 | 2 } | E { δ u 1 } | 2 ,
Var { δ u 1 } = Var { γ 0 u 1 ( 1 ) , X P M } .
E { δ u 1 } = E { γ 0 u 1 ( 1 ) , X P M } = j 2 γ 0 P 3 / 2 a 0 m n E { a m a n * } X m n ( z , T ) .
E { a m a n * } = K 1 δ m n ,
K 1 = E { | a m | 2 } ,
E { δ u 1 } = j 2 γ 0 P 3 / 2 a 0 K 1 m X m m ( z , T ) .
| E { δ u 1 } | 2 = 4 γ 0 2 P 3 | a 0 | 2 K 1 2 m m X m m ( z , T ) X m m * ( z , T ) .
E { | δ u 1 | 2 } = E { | γ 0 u 1 ( 1 ) , X P M | 2 }
= 4 γ 0 2 P 3 | a 0 | 2 m n m n E { a m a n * a m * a n } X m n X m n * .
E { a m a n * a m * a n } = E { | a m | 2 | a m | 2 } { K 2 m = m K 1 2 m m
K 2 = E { | a m | 4 } .
E { | δ u 1 | 2 } = 4 γ 0 2 P 3 | a 0 | 2 ( K 1 2 m m m m X m m X m m * + K 2 m | X m m | 2 ) .
E { a m a n * a m * a n } = K 1 2 δ m m δ n n .
E { | δ u 1 | 2 } = 4 γ 0 2 P 3 | a 0 | 2 K 1 2 m n m n | X m n | 2 .
E { a m a n * a m * a n } = K 1 E { a m * a n } = 0 ,
E { a m * a n } = 0 m n .
E { | δ u 1 | 2 } = 4 γ 0 2 P 3 | a 0 | 2 ( K 1 2 m m m m X m m X m m * + K 2 m | X m m | 2 + K 1 2 m m m m | X m m | 2 ) .
Var { δ u 1 } = 4 γ 0 2 P 3 | a 0 | 2 ( K 2 m | X m m | 2 + K 1 2 m m m m | X m m | 2 K 1 2 m | X m m | 2 ) , = γ 0 2 P 3 | a 0 | 2 ( ( K 2 K 1 ) 2 m | X m m | 2 + K 1 2 m m m m | X m m | 2 ) .
Var { δ u 1 } = γ 0 2 P 3 m m m m | X m m | 2 .
σ X P M 2 = Var { δ u 1 } ¯ = 1 T s 0 T s Var { δ u 1 } d t .
Pre-compensation ratio = L pre L pre + L post × 100 ,
η = Accumulated dispersion of inline DCF per span Accumulated dispersion of TF per span × 100 .
d f ˜ ( z , ω ) d z j ω 2 β 2 ( z ) 2 f ˜ ( z , ω ) = j F ˜ ( z , ω ) ,
F ˜ ( z , ω ) = η ( z ) exp ( k = 1 3 C k 2 R k ) exp [ R T 2 + T ( 2 C + i ω ) ] d T = π η R exp [ ω 2 4 R D ω ] ,
R = R 1 + R 2 + R 3 , C = C 1 R 1 + C 2 R 2 + C 3 R 3 ,
D = i C R ,
η ( z ) = η ( z ) exp ( k = 1 3 C k 2 R k + C 2 R ) .
f ˜ ( z , ω ) = j 0 z F ˜ ( s , ω ) exp [ j ω 2 A ( z , s ) / 4 ] d s ,
A ( z , s ) = 2 [ S ( z ) S ( s ) ] .
f ( z , T ) = j π 2 π 0 z η ( s ) R ( s ) exp [ 4 ω 2 δ ω ( D + j T ) ] d ω d s ,
δ ( z , s ) = 1 R ( s ) j A ( z , s ) .
f ( z , T ) = j 0 z η ( s ) δ ( z , s ) R ( s ) exp [ ( D ( s ) + j T ) 2 δ ( z , s ) ] d s .

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