Abstract

An analytical model for the cross-phase modulation (XPM) variance in dispersion-managed coherent fiber-optic systems is developed based on the first order perturbation theory. The XPM variance is analytically calculated for arbitrary pulse shapes. For a non-Gaussian pulse, the summation of time-shifted Gaussian pulses is used to fit the target pulse shape, which not only provides a good approximation of the non-Gaussian pulse but also allows explicit derivation of the XPM variance. The analytically estimated XPM variance is found to be in good agreement with numerical simulations.

© 2014 Optical Society of America

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  1. D. Marcuse, A. R. Chraplyvy, R. W. Tkach, “Dependence of cross-phase modulation on channel number in fiber WDM systems,” J. Lightwave Technol. 12(5), 885–890 (1994).
    [CrossRef]
  2. T.-K. Chiang, N. Kagi, M. E. Marhic, L. G. Kazovsky, “Cross-phase modulation in fiber links with multiple optical amplifiers and dispersion compensators,” J. Lightwave Technol. 14(3), 249–260 (1996).
    [CrossRef]
  3. 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(9), 1271–1273 (1998).
    [CrossRef]
  4. M. Shtaif, M. Eiselt, “Analysis of intensity interference caused by cross-phase modulation in dispersive optical fibers,” IEEE Photon. Technol. Lett. 10(7), 979–981 (1998).
    [CrossRef]
  5. R. Hui, K. Demarest, C. Allen, “Cross-phase modulation in multispan WDM optical fiber systems,” J. Lightwave Technol. 17(6), 1018–1026 (1999).
    [CrossRef]
  6. A. V. T. Cartaxo, “Cross-phase modulation in intensity modulation-direct detection WDM systems with multiple optical amplifiers and dispersion compensators,” J. Lightwave Technol. 17(2), 178–190 (1999).
    [CrossRef]
  7. Z. Jiang, C. Fan, “A comprehensive study on XPM- and SRS-induced noise in cascaded IM-DD optical fiber transmission systems,” J. Lightwave Technol. 21(4), 953–960 (2003).
    [CrossRef]
  8. S. Kumar, D. Yang, “Second-order theory for self-phase modulation and cross-phase modulation in optical fibers,” J. Lightwave Technol. 23(6), 2073–2080 (2005).
    [CrossRef]
  9. P. Poggiolini, A. Carena, V. Curri, G. Bosco, F. Forghieri, “Analytical modeling of nonlinear propagation in uncompensated optical transmission links,” IEEE Photon. Technol. Lett. 23(11), 742–744 (2011).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  12. G. Bosco, R. Cigliutti, A. Nespola, A. Carena, V. Curri, F. Forghieri, Y. Yamamoto, T. Sasaki, Y. Jiang, P. Poggiolini, “Experimental investigation of nonlinear interference accumulation in uncompensated links,” IEEE Photon. Technol. Lett. 24(14), 1230–1232 (2012).
    [CrossRef]
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    [CrossRef] [PubMed]
  14. A. Mecozzi, R. Essiambre, “Nonlinear Shannon limit in pseudolinear coherent systems,” J. Lightwave Technol. 30(12), 2011–2024 (2012).
    [CrossRef]
  15. R. Dar, M. Feder, A. Mecozzi, M. Shtaif, “Properties of nonlinear noise in long, dispersion-uncompensated fiber links,” Opt. Express 21(22), 25685–25699 (2013).
    [CrossRef] [PubMed]
  16. P. Johannisson, M. Karlsson, “Perturbation analysis of nonlinear propagation in a strongly dispersive optical communication system,” J. Lightwave Technol. 31(8), 1273–1282 (2013).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  19. R. Dar, M. Shtaif, M. Feder, “New bounds on the capacity of the nonlinear fiber-optic channel,” Opt. Lett. 39(2), 398–401 (2014).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  21. S. F. Boys, “Electronic wave functions. I. A general method of calculation for the stationary states of any molecular system,” Proc. R. Soc. London A Math. Phys. Sci. 200(1063), 542–554 (1950).
    [CrossRef]
  22. A. Szabo and N. S. Ostlund, Modern Quantum Chemistry (McGraw-Hill, 1989), Chap. 2.
  23. X. Li, X. Chen, G. Goldfarb, E. Mateo, I. Kim, F. Yaman, G. Li, “Electronic post-compensation of WDM transmission impairments using coherent detection and digital signal processing,” Opt. Express 16(2), 880–888 (2008).
    [CrossRef] [PubMed]

2014 (2)

2013 (2)

R. Dar, M. Feder, A. Mecozzi, M. Shtaif, “Properties of nonlinear noise in long, dispersion-uncompensated fiber links,” Opt. Express 21(22), 25685–25699 (2013).
[CrossRef] [PubMed]

P. Johannisson, M. Karlsson, “Perturbation analysis of nonlinear propagation in a strongly dispersive optical communication system,” J. Lightwave Technol. 31(8), 1273–1282 (2013).
[CrossRef]

2012 (6)

2011 (1)

P. Poggiolini, A. Carena, V. Curri, G. Bosco, F. Forghieri, “Analytical modeling of nonlinear propagation in uncompensated optical transmission links,” IEEE Photon. Technol. Lett. 23(11), 742–744 (2011).
[CrossRef]

2008 (1)

2005 (1)

2003 (1)

1999 (2)

1998 (2)

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(9), 1271–1273 (1998).
[CrossRef]

M. Shtaif, M. Eiselt, “Analysis of intensity interference caused by cross-phase modulation in dispersive optical fibers,” IEEE Photon. Technol. Lett. 10(7), 979–981 (1998).
[CrossRef]

1996 (1)

T.-K. Chiang, N. Kagi, M. E. Marhic, L. G. Kazovsky, “Cross-phase modulation in fiber links with multiple optical amplifiers and dispersion compensators,” J. Lightwave Technol. 14(3), 249–260 (1996).
[CrossRef]

1994 (1)

D. Marcuse, A. R. Chraplyvy, R. W. Tkach, “Dependence of cross-phase modulation on channel number in fiber WDM systems,” J. Lightwave Technol. 12(5), 885–890 (1994).
[CrossRef]

1950 (1)

S. F. Boys, “Electronic wave functions. I. A general method of calculation for the stationary states of any molecular system,” Proc. R. Soc. London A Math. Phys. Sci. 200(1063), 542–554 (1950).
[CrossRef]

Allen, C.

R. Hui, K. Demarest, C. Allen, “Cross-phase modulation in multispan WDM optical fiber systems,” J. Lightwave Technol. 17(6), 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(9), 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. Lightwave Technol. 30(10), 1524–1539 (2012).
[CrossRef]

G. Bosco, R. Cigliutti, A. Nespola, A. Carena, V. Curri, F. Forghieri, Y. Yamamoto, T. Sasaki, Y. Jiang, P. Poggiolini, “Experimental investigation of nonlinear interference accumulation in uncompensated links,” IEEE Photon. Technol. Lett. 24(14), 1230–1232 (2012).
[CrossRef]

P. Poggiolini, A. Carena, V. Curri, G. Bosco, F. Forghieri, “Analytical modeling of nonlinear propagation in uncompensated optical transmission links,” IEEE Photon. Technol. Lett. 23(11), 742–744 (2011).
[CrossRef]

Boys, S. F.

S. F. Boys, “Electronic wave functions. I. A general method of calculation for the stationary states of any molecular system,” Proc. R. Soc. London A Math. Phys. Sci. 200(1063), 542–554 (1950).
[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. Lightwave Technol. 30(10), 1524–1539 (2012).
[CrossRef]

G. Bosco, R. Cigliutti, A. Nespola, A. Carena, V. Curri, F. Forghieri, Y. Yamamoto, T. Sasaki, Y. Jiang, P. Poggiolini, “Experimental investigation of nonlinear interference accumulation in uncompensated links,” IEEE Photon. Technol. Lett. 24(14), 1230–1232 (2012).
[CrossRef]

P. Poggiolini, A. Carena, V. Curri, G. Bosco, F. Forghieri, “Analytical modeling of nonlinear propagation in uncompensated optical transmission links,” IEEE Photon. Technol. Lett. 23(11), 742–744 (2011).
[CrossRef]

Cartaxo, A. V. T.

Chen, X.

Chiang, T.-K.

T.-K. Chiang, N. Kagi, M. E. Marhic, L. G. Kazovsky, “Cross-phase modulation in fiber links with multiple optical amplifiers and dispersion compensators,” J. Lightwave Technol. 14(3), 249–260 (1996).
[CrossRef]

Chraplyvy, A. R.

D. Marcuse, A. R. Chraplyvy, R. W. Tkach, “Dependence of cross-phase modulation on channel number in fiber WDM systems,” J. Lightwave Technol. 12(5), 885–890 (1994).
[CrossRef]

Cigliutti, R.

G. Bosco, R. Cigliutti, A. Nespola, A. Carena, V. Curri, F. Forghieri, Y. Yamamoto, T. Sasaki, Y. Jiang, P. Poggiolini, “Experimental investigation of nonlinear interference accumulation in uncompensated links,” IEEE Photon. Technol. Lett. 24(14), 1230–1232 (2012).
[CrossRef]

Curri, V.

G. Bosco, R. Cigliutti, A. Nespola, A. Carena, V. Curri, F. Forghieri, Y. Yamamoto, T. Sasaki, Y. Jiang, P. Poggiolini, “Experimental investigation of nonlinear interference accumulation in uncompensated links,” IEEE Photon. Technol. Lett. 24(14), 1230–1232 (2012).
[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. Lightwave Technol. 30(10), 1524–1539 (2012).
[CrossRef]

P. Poggiolini, A. Carena, V. Curri, G. Bosco, F. Forghieri, “Analytical modeling of nonlinear propagation in uncompensated optical transmission links,” IEEE Photon. Technol. Lett. 23(11), 742–744 (2011).
[CrossRef]

Dar, R.

Demarest, K.

R. Hui, K. Demarest, C. Allen, “Cross-phase modulation in multispan WDM optical fiber systems,” J. Lightwave Technol. 17(6), 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(9), 1271–1273 (1998).
[CrossRef]

Derevyanko, S.

Eiselt, M.

M. Shtaif, M. Eiselt, “Analysis of intensity interference caused by cross-phase modulation in dispersive optical fibers,” IEEE Photon. Technol. Lett. 10(7), 979–981 (1998).
[CrossRef]

Essiambre, R.

Fan, C.

Feder, M.

Forghieri, F.

G. Bosco, R. Cigliutti, A. Nespola, A. Carena, V. Curri, F. Forghieri, Y. Yamamoto, T. Sasaki, Y. Jiang, P. Poggiolini, “Experimental investigation of nonlinear interference accumulation in uncompensated links,” IEEE Photon. Technol. Lett. 24(14), 1230–1232 (2012).
[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. Lightwave Technol. 30(10), 1524–1539 (2012).
[CrossRef]

P. Poggiolini, A. Carena, V. Curri, G. Bosco, F. Forghieri, “Analytical modeling of nonlinear propagation in uncompensated optical transmission links,” IEEE Photon. Technol. Lett. 23(11), 742–744 (2011).
[CrossRef]

Goldfarb, G.

Hui, R.

R. Hui, K. Demarest, C. Allen, “Cross-phase modulation in multispan WDM optical fiber systems,” J. Lightwave Technol. 17(6), 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(9), 1271–1273 (1998).
[CrossRef]

Jiang, Y.

G. Bosco, R. Cigliutti, A. Nespola, A. Carena, V. Curri, F. Forghieri, Y. Yamamoto, T. Sasaki, Y. Jiang, P. Poggiolini, “Experimental investigation of nonlinear interference accumulation in uncompensated links,” IEEE Photon. Technol. Lett. 24(14), 1230–1232 (2012).
[CrossRef]

Jiang, Z.

Johannisson, P.

P. Johannisson, M. Karlsson, “Perturbation analysis of nonlinear propagation in a strongly dispersive optical communication system,” J. Lightwave Technol. 31(8), 1273–1282 (2013).
[CrossRef]

Kagi, N.

T.-K. Chiang, N. Kagi, M. E. Marhic, L. G. Kazovsky, “Cross-phase modulation in fiber links with multiple optical amplifiers and dispersion compensators,” J. Lightwave Technol. 14(3), 249–260 (1996).
[CrossRef]

Karlsson, M.

P. Johannisson, M. Karlsson, “Perturbation analysis of nonlinear propagation in a strongly dispersive optical communication system,” J. Lightwave Technol. 31(8), 1273–1282 (2013).
[CrossRef]

Kazovsky, L. G.

T.-K. Chiang, N. Kagi, M. E. Marhic, L. G. Kazovsky, “Cross-phase modulation in fiber links with multiple optical amplifiers and dispersion compensators,” J. Lightwave Technol. 14(3), 249–260 (1996).
[CrossRef]

Kim, I.

Kumar, S.

Li, G.

Li, X.

Liang, X.

Marcuse, D.

D. Marcuse, A. R. Chraplyvy, R. W. Tkach, “Dependence of cross-phase modulation on channel number in fiber WDM systems,” J. Lightwave Technol. 12(5), 885–890 (1994).
[CrossRef]

Marhic, M. E.

T.-K. Chiang, N. Kagi, M. E. Marhic, L. G. Kazovsky, “Cross-phase modulation in fiber links with multiple optical amplifiers and dispersion compensators,” J. Lightwave Technol. 14(3), 249–260 (1996).
[CrossRef]

Mateo, E.

Mecozzi, A.

Nespola, A.

G. Bosco, R. Cigliutti, A. Nespola, A. Carena, V. Curri, F. Forghieri, Y. Yamamoto, T. Sasaki, Y. Jiang, P. Poggiolini, “Experimental investigation of nonlinear interference accumulation in uncompensated links,” IEEE Photon. Technol. Lett. 24(14), 1230–1232 (2012).
[CrossRef]

Poggiolini, P.

G. Bosco, R. Cigliutti, A. Nespola, A. Carena, V. Curri, F. Forghieri, Y. Yamamoto, T. Sasaki, Y. Jiang, P. Poggiolini, “Experimental investigation of nonlinear interference accumulation in uncompensated links,” IEEE Photon. Technol. Lett. 24(14), 1230–1232 (2012).
[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. Lightwave Technol. 30(10), 1524–1539 (2012).
[CrossRef]

P. Poggiolini, “The GN model of non-linear propagation in uncompensated coherent optical systems,” J. Lightwave Technol. 30(24), 3857–3879 (2012).
[CrossRef]

P. Poggiolini, A. Carena, V. Curri, G. Bosco, F. Forghieri, “Analytical modeling of nonlinear propagation in uncompensated optical transmission links,” IEEE Photon. Technol. Lett. 23(11), 742–744 (2011).
[CrossRef]

Sasaki, T.

G. Bosco, R. Cigliutti, A. Nespola, A. Carena, V. Curri, F. Forghieri, Y. Yamamoto, T. Sasaki, Y. Jiang, P. Poggiolini, “Experimental investigation of nonlinear interference accumulation in uncompensated links,” IEEE Photon. Technol. Lett. 24(14), 1230–1232 (2012).
[CrossRef]

Shahi, S. N.

Shtaif, M.

Sorokina, M.

Tkach, R. W.

D. Marcuse, A. R. Chraplyvy, R. W. Tkach, “Dependence of cross-phase modulation on channel number in fiber WDM systems,” J. Lightwave Technol. 12(5), 885–890 (1994).
[CrossRef]

Turitsyn, S.

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(9), 1271–1273 (1998).
[CrossRef]

Yamamoto, Y.

G. Bosco, R. Cigliutti, A. Nespola, A. Carena, V. Curri, F. Forghieri, Y. Yamamoto, T. Sasaki, Y. Jiang, P. Poggiolini, “Experimental investigation of nonlinear interference accumulation in uncompensated links,” IEEE Photon. Technol. Lett. 24(14), 1230–1232 (2012).
[CrossRef]

Yaman, F.

Yang, D.

IEEE Photon. Technol. Lett. (4)

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(9), 1271–1273 (1998).
[CrossRef]

M. Shtaif, M. Eiselt, “Analysis of intensity interference caused by cross-phase modulation in dispersive optical fibers,” IEEE Photon. Technol. Lett. 10(7), 979–981 (1998).
[CrossRef]

P. Poggiolini, A. Carena, V. Curri, G. Bosco, F. Forghieri, “Analytical modeling of nonlinear propagation in uncompensated optical transmission links,” IEEE Photon. Technol. Lett. 23(11), 742–744 (2011).
[CrossRef]

G. Bosco, R. Cigliutti, A. Nespola, A. Carena, V. Curri, F. Forghieri, Y. Yamamoto, T. Sasaki, Y. Jiang, P. Poggiolini, “Experimental investigation of nonlinear interference accumulation in uncompensated links,” IEEE Photon. Technol. Lett. 24(14), 1230–1232 (2012).
[CrossRef]

J. Lightwave Technol. (10)

P. Johannisson, M. Karlsson, “Perturbation analysis of nonlinear propagation in a strongly dispersive optical communication system,” J. Lightwave Technol. 31(8), 1273–1282 (2013).
[CrossRef]

D. Marcuse, A. R. Chraplyvy, R. W. Tkach, “Dependence of cross-phase modulation on channel number in fiber WDM systems,” J. Lightwave Technol. 12(5), 885–890 (1994).
[CrossRef]

T.-K. Chiang, N. Kagi, M. E. Marhic, L. G. Kazovsky, “Cross-phase modulation in fiber links with multiple optical amplifiers and dispersion compensators,” J. Lightwave Technol. 14(3), 249–260 (1996).
[CrossRef]

A. V. T. Cartaxo, “Cross-phase modulation in intensity modulation-direct detection WDM systems with multiple optical amplifiers and dispersion compensators,” J. Lightwave Technol. 17(2), 178–190 (1999).
[CrossRef]

R. Hui, K. Demarest, C. Allen, “Cross-phase modulation in multispan WDM optical fiber systems,” J. Lightwave Technol. 17(6), 1018–1026 (1999).
[CrossRef]

Z. Jiang, C. Fan, “A comprehensive study on XPM- and SRS-induced noise in cascaded IM-DD optical fiber transmission systems,” J. Lightwave Technol. 21(4), 953–960 (2003).
[CrossRef]

S. Kumar, D. Yang, “Second-order theory for self-phase modulation and cross-phase modulation in optical fibers,” J. Lightwave Technol. 23(6), 2073–2080 (2005).
[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. Lightwave Technol. 30(10), 1524–1539 (2012).
[CrossRef]

A. Mecozzi, R. Essiambre, “Nonlinear Shannon limit in pseudolinear coherent systems,” J. Lightwave Technol. 30(12), 2011–2024 (2012).
[CrossRef]

P. Poggiolini, “The GN model of non-linear propagation in uncompensated coherent optical systems,” J. Lightwave Technol. 30(24), 3857–3879 (2012).
[CrossRef]

Opt. Express (4)

Opt. Lett. (2)

Proc. R. Soc. London A Math. Phys. Sci. (1)

S. F. Boys, “Electronic wave functions. I. A general method of calculation for the stationary states of any molecular system,” Proc. R. Soc. London A Math. Phys. Sci. 200(1063), 542–554 (1950).
[CrossRef]

Other (2)

A. Szabo and N. S. Ostlund, Modern Quantum Chemistry (McGraw-Hill, 1989), Chap. 2.

R. Dar, M. Feder, A. Mecozzi, and M. Shtaif, “Time varying ISI model for nonlinear interference noise,” in Optical Fiber Communication Conference, OSA Technical Digest (Optical Society of America, 2014), paper W2A.62.
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Fitting a Nyquist pulse x1(t) and a raised cosine pulse x2(t) using a summation of time-shifted Gaussian functions. (roll-off factor = 0.6, number of Gaussian functions = 6).

Fig. 2
Fig. 2

Schematic of a dispersion-managed WDM fiber-optic transmission system. Tx: transmitter; Rx: receiver; MUX: multiplexer; DMUX: demultiplexer; DCF: dispersion compensating fiber.

Fig. 3
Fig. 3

XPM variance vs. average launch power per channel.

Fig. 4
Fig. 4

Constellation of received signal (Pave = 4 dBm, number of fiber span = 1, Ψres = 0).

Fig. 5
Fig. 5

XPM variance vs. residual dispersion per span (Pave = 0 dBm).

Fig. 6
Fig. 6

XPM variance vs. number of WDM channels (Pave = 0 dBm).

Fig. 7
Fig. 7

XPM variance vs. transmission distance (Pave = 0 dBm).

Tables (2)

Tables Icon

Table 1 Fitting Parameters Optimized by LSM (Nyquist Pulse, x1(t))

Tables Icon

Table 2 Fitting Parameters Optimized by LSM (Raised Cosine Pulse, x2(t))

Equations (75)

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=1,2andl=3k.
u 1 (0,T)= P a 0 g(0,T),
u 2 (0,T)= P n=N N b n g(0,Tn T s )exp(jΩT),
g(0,T)=exp( T 2 2 T 0 2 ),
a 0 or b 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 b n exp( (T τ n ) 2 2 T 1 2 jΩT+jθ(z) ) ,
T 1 = T 0 2 jS(z) ,
τ n =n T s +S(z)Ω,
θ(z)=S(z) Ω 2 /2,
S(z)= 0 z β 2 (s)ds .
j u k (1) z β 2 (z) 2 2 u k (1) T 2 = e w(z) [ | u k (0) | 2 +2 | u l (0) | 2 ] u k (0) ,k=1,2andl=3k.
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+jT) 2 δ(z,s) ]ds,
R= R 1 + R 2 + R 3 ,
C= C 1 R 1 + C 2 R 2 + C 3 R 3 ,
D=jC/R,
δ(z,s)=[ 1jRA(z,s) ]/R,
A(z,s)=2[ S(z)S(s) ].
F(z,T)=2 e w(z) | u 2 (0) | 2 u 1 (0) =2 P 3/2 a 0 η(z) m n b m b 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),XPM ( L tot ,T)=j2 P 3/2 a 0 m n b m b n X mn ( L tot ,T),
X mn ( L tot ,T)= U mn ( L tot , T ) h RX (T T )d T ,
U mn ( L tot ,T)= 0 L tot η (s) δ( L tot ,s)R(s) exp[ (D+jT) 2 δ( L tot ,s) ]ds,
η (s)=η(s)exp( k=1 3 C k 2 R k + C 2 R ).
δ u 1 = γ 0 u 1 (1),XPM .
Var{ δ u 1 }=E{ | δ u 1 | 2 } | E{ δ u 1 } | 2 .
K 1 =E{ | b n | 2 }, K 2 =E{ | b n | 4 }.
σ XPM 2 =4 γ 0 2 P 3 | a 0 | 2 ( ( K 2 K 1 2 ) m | X mm | 2 + K 1 2 m n mn | X mn | 2 ).
h (T)= k=1 K ξ k exp[ ( T μ k T s ) 2 2 ( θ k T s ) 2 ] ,
x 1 (t)=sinc( t T s ) cos(aπt/ T s ) 1 (2at/ T s ) 2 ,
x 2 (t)={ 1,| t |< 1a 2 T s 1 2 [ 1sin( π a T s | t | π 2a ) ], 1a 2 T s | t | 1+a 2 T s 0,| t |> 1+a 2 T s ,
u 1 (1),XPM,NG ( L tot ,T)=j2 P 3/2 a 0 m n b m b n Y mn ( L tot ,T),
Y mn ( L tot ,T)= V mn ( L tot , T ) h RX (T T )d T ,
V mn = 0 L tot k 1 =1 K k 2 =1 K k 3 =1 K η (s) δ( L tot ,s)R(s) exp[ (D+jT) 2 δ( L tot ,s) ]ds,
T 1,k = ( θ k T s ) 2 jS(z) ,
η(z)= e w(z) ξ k 1 ξ k 2 ξ k 3 θ k 1 θ k 2 θ k 3 T s 3 T 1, k 1 T 1, k 2 T 1, k 3 ,
C 1 (z)= τ m, k 1 , C 2 (z)= τ n, k 2 , C 3 (z)= μ k 3 T s ,
R 1 (z)= 1 2 T 1, k 1 2 , R 2 (z)= 1 2 ( T 1, k 2 ) 2 , R 3 (z)= 1 2 T 1, k 3 2 ,
τ n,k =(n+ μ k ) T s +S(z)Ω.
σ XPM 2 (T)= | a 0 | 2 G(T),
G(T)=4 γ 0 2 P 3 ( ( K 2 K 1 2 ) m | Y mm | 2 + K 1 2 m n mn | Y mn | 2 ).
σ XPM_overall 2 = | a 0 | 2 G(0)+E{ | a n | 2 } k=M k0 M G(k T s ) .
discrepancy= | σ XPM 2 (numerical) σ XPM 2 (analytical) | σ XPM 2 (numerical) ×100%.
h (T)= k=1 K ξ k exp[ ( T μ k T s ) 2 2 ( θ k T s ) 2 ] .
e(T)=h(T) h (T).
T=mΔT,
χ= m=1 M e m 2 = m=1 M [ h m h m ] 2 ,
χ ξ k =2 m ( h m h m )exp[ ( mΔT μ k T s ) 2 2 ( θ k T s ) 2 ] ,
χ μ k =2 m ( h m h m ) ξ k ( mΔT μ k T s ) θ k 2 T s exp[ ( mΔT μ k T s ) 2 2 ( θ k T s ) 2 ] ,
χ θ k =2 m ( h m h m ) ξ k ( mΔT μ k T s ) 2 θ k 3 T s 3 exp[ ( mΔT μ k T s ) 2 2 ( θ k T s ) 2 ] .
ξ k (i+1) = ξ k (i) χ ξ k Δξ, μ k (i+1) = μ k (i) χ μ k Δu, θ k (i+1) = θ k (i) χ θ k Δθ,
u 1 (0,T)= P a 0 h (T),
u 2 (0,T)= P n=N N b n h (Tn T s )exp(jΩT),
exp( a x 2 bx ) dx= π a exp( b 2 4a ),
u 1 (0) (z,T)= P a 0 k=1 K ξ k θ k T s T 1,k exp[ (T μ k T s ) 2 2 T 1,k 2 ],
u 2 (0) (z,T)= P n=N N b n k=1 K ξ k θ k T s T 1,k exp[ (T τ n,k ) 2 2 T 1,k 2 jΩT+jθ(z) ],
T 1,k = ( θ k T s ) 2 jS(z) ,
τ n,k =(n+ μ k ) T s +S(z)Ω.
F(z,T)=2 P 3/2 a 0 m n b m b n k 1 =1 K k 2 =1 K k 3 =1 K η(z) exp[ k =1 3 [ T C k (z) ] 2 R k (z) ],
η(z)= e w(z) ξ k 1 ξ k 2 ξ k 3 θ k 1 θ k 2 θ k 3 T s 3 T 1, k 1 T 1, k 2 T 1, k 3 ,
C 1 (z)= τ m, k 1 , C 2 (z)= τ n, k 2 , C 3 (z)= μ k 3 T s ,
R 1 (z)= 1 2 T 1, k 1 2 , R 2 (z)= 1 2 ( T 1, k 2 ) 2 , R 3 (z)= 1 2 T 1, k 3 2 .
u 1 (1),XPM,NG ( L tot ,T)=j2 P 3/2 a 0 m n b m b n Y mn ( L tot ,T),
Y mn ( L tot ,T)= V mn ( L tot , T ) h RX (T T )d T ,
V mn = 0 L tot k 1 =1 K k 2 =1 K k 3 =1 K η (s) δ( L tot ,s)R(s) exp[ (D+jT) 2 δ( L tot ,s) ]ds.

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