Abstract

We develop a simplified high-order multi-span Volterra series transfer function (SH-MS-VSTF), basing our derivation on the well-known third-order Volterra series transfer function (VSTF). We notice that when applying an approach based on a recursive method and considering the phased-array factor, the order of the expression for the transfer function grows as 3 raised to the number of considered spans. By imposing a frequency-flat approximation to the higher-order terms that are usually neglected in the commonly used VSTF approach, we are able to reduce the overall expression order to the typical third-order plus a complex correction factor. We carry on performance comparisons between the purposed SH-MS-VSTF, the well-known split-step Fourier method (SSFM), and the third-order VSTF. The SH-MS-VSTF exhibits a uniform improvement of about two orders of magnitude in the normalized mean squared deviation with respect to the other methods. This can be translated in a reduction of the overall number of steps required to fully analyze the transmission link up to 99.75% with respect to the SSFM, and 98.75% with respect to the third-order VSTF, respectively, for the same numerical accuracy.

© 2017 Optical Society of America

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Unveiling nonlinear effects in dense coherent optical WDM systems with Volterra series

Jacklyn D. Reis and António L. Teixeira
Opt. Express 18(8) 8660-8670 (2010)

References

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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  14. G. Shulkind and M. Nazarathy, “Nonlinear digital back propagation compensator for coherent optical OFDM based on factorizing the Volterra series transfer function,” Opt. Express 21(11), 13146–13161 (2013).
    [Crossref]
  15. F. P. Guiomar, J. D. Reis, A. Carena, G. Bosco, A. L. Teixeira, and A. N. Pinto, “Experimental demonstration of a frequency-domain Volterra series nonlinear equalizer in polarization-multiplexed transmission,” Opt. Express 21(1), 276–288 (2013).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  19. Z. Dong, A. Lau, and C. Lu, “OSNR monitoring for QPSK and 16-QAM systems in presence of fiber nonlinearities for digital coherent receivers,” Opt. Express 20(17), 19520–19534 (2012).
    [Crossref] [PubMed]
  20. E. Flood, W.H. Guo, D. Reid, M. Lynch, A. Bradley, L. Barry, and J. Donegan, “Interferometer Based In-Band OSNR Monitoring of Single and Dual Polarisation QPSK Signals,” in Proceedings of 36th European Conference on Optical Communication, (ECOC, 2010), paper Th9C6.
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    [Crossref]

2013 (3)

2012 (4)

2011 (2)

J. Pan and C.-H. Cheng, “Nonlinear electrical compensation for the coherent optical OFDM system,” J. Lightwave Technol. 29(2), 215–221 (2011).
[Crossref]

F. P. Guiomar, J. D. Reis, A. L. Teixeira, and A. N. Pinto, “Digital postcompensation using Volterra series transfer function,” IEEE Photon. Technol. Lett. 23(19), 1412–1414 (2011).
[Crossref]

2010 (1)

2009 (2)

Y. Gao, F. Zgang, L. Dou, Z. Chen, and A. Xu, “Intra-channel nonlinearities mitigation in pseudo-linear coherent QPSK transmission systems via nonlinear electrical equalizer,” Opt. Commun. 282(12), 2421–2425 (2009).
[Crossref]

L. N. Binh, “Linear and nonlinear transfer function of single mode fiber for optical transmission systems,” J. Opt. Soc. Am. A 26(7), 1564–1575 (2009).
[Crossref]

2008 (3)

1997 (1)

K. Peddanarappagari and M. Brandt-Pearce, “Volterra series transfer function of single-mode fibers,” J. Lightwave Technol. 15(12), 2232–2241 (1997).
[Crossref]

1986 (1)

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2013).

Barry, L.

E. Flood, W.H. Guo, D. Reid, M. Lynch, A. Bradley, L. Barry, and J. Donegan, “Interferometer Based In-Band OSNR Monitoring of Single and Dual Polarisation QPSK Signals,” in Proceedings of 36th European Conference on Optical Communication, (ECOC, 2010), paper Th9C6.

Binh, L. N.

Bosco, G.

Bradley, A.

E. Flood, W.H. Guo, D. Reid, M. Lynch, A. Bradley, L. Barry, and J. Donegan, “Interferometer Based In-Band OSNR Monitoring of Single and Dual Polarisation QPSK Signals,” in Proceedings of 36th European Conference on Optical Communication, (ECOC, 2010), paper Th9C6.

Brandt-Pearce, M.

K. Peddanarappagari and M. Brandt-Pearce, “Volterra series transfer function of single-mode fibers,” J. Lightwave Technol. 15(12), 2232–2241 (1997).
[Crossref]

Cai, Y.

Carena, A.

Chen, X.

Chen, Z.

Y. Gao, F. Zgang, L. Dou, Z. Chen, and A. Xu, “Intra-channel nonlinearities mitigation in pseudo-linear coherent QPSK transmission systems via nonlinear electrical equalizer,” Opt. Commun. 282(12), 2421–2425 (2009).
[Crossref]

Cheng, C.-H.

Cho, P.

Cui, K.

Curri, V.

Donegan, J.

E. Flood, W.H. Guo, D. Reid, M. Lynch, A. Bradley, L. Barry, and J. Donegan, “Interferometer Based In-Band OSNR Monitoring of Single and Dual Polarisation QPSK Signals,” in Proceedings of 36th European Conference on Optical Communication, (ECOC, 2010), paper Th9C6.

Dong, Z.

Dou, L.

Y. Gao, F. Zgang, L. Dou, Z. Chen, and A. Xu, “Intra-channel nonlinearities mitigation in pseudo-linear coherent QPSK transmission systems via nonlinear electrical equalizer,” Opt. Commun. 282(12), 2421–2425 (2009).
[Crossref]

Flood, E.

E. Flood, W.H. Guo, D. Reid, M. Lynch, A. Bradley, L. Barry, and J. Donegan, “Interferometer Based In-Band OSNR Monitoring of Single and Dual Polarisation QPSK Signals,” in Proceedings of 36th European Conference on Optical Communication, (ECOC, 2010), paper Th9C6.

Forghieri, F.

Gao, Y.

Y. Gao, F. Zgang, L. Dou, Z. Chen, and A. Xu, “Intra-channel nonlinearities mitigation in pseudo-linear coherent QPSK transmission systems via nonlinear electrical equalizer,” Opt. Commun. 282(12), 2421–2425 (2009).
[Crossref]

Goldfarb, G.

Guiomar, F. P.

Guo, W.H.

E. Flood, W.H. Guo, D. Reid, M. Lynch, A. Bradley, L. Barry, and J. Donegan, “Interferometer Based In-Band OSNR Monitoring of Single and Dual Polarisation QPSK Signals,” in Proceedings of 36th European Conference on Optical Communication, (ECOC, 2010), paper Th9C6.

Hauske, F. N.

Huang, Y.

Ip, E.

Kahn, J. M.

Karagodsky, V.

Khurgin, J.

Kim, I.

Lau, A.

Li, G.

Li, L.

Li, X.

Lin, C.

C. Lin, “Digital nonlinear compensation for next-generation optical communication systems using advanced modulation formats,” Master’s thesis, Friedrich-Alexander-Universität Erlangen-Nürnberg, 2014.

Liu, L.

Lu, C.

Lynch, M.

E. Flood, W.H. Guo, D. Reid, M. Lynch, A. Bradley, L. Barry, and J. Donegan, “Interferometer Based In-Band OSNR Monitoring of Single and Dual Polarisation QPSK Signals,” in Proceedings of 36th European Conference on Optical Communication, (ECOC, 2010), paper Th9C6.

Mateo, E.

Meiman, Y.

Nazarathy, M.

G. Shulkind and M. Nazarathy, “Nonlinear digital back propagation compensator for coherent optical OFDM based on factorizing the Volterra series transfer function,” Opt. Express 21(11), 13146–13161 (2013).
[Crossref]

M. Nazarathy, J. Khurgin, R. Weidenfeld, Y. Meiman, P. Cho, R. Noe, I. Shpantzer, and V. Karagodsky, “Phased-array cancellation of nonlinear FWM in coherent OFDM dispersive multi-span links,” Opt. Express 16(20), 15777–15810 (2008).
[Crossref] [PubMed]

R. Weidenfeld, M. Nazarathy, R. Noe, and I. Shpantzer, “Volterra nonlinear compensation of 100G coherent OFDM with baud-rate ADC, tolerable complexity and lowintra-channel FWM/XPM error propagation,” in Optical Fiber Communication Conference, OSA Technical Digest Series (Optical Society of America, 2010), paper OTuE3.
[Crossref]

Noe, R.

M. Nazarathy, J. Khurgin, R. Weidenfeld, Y. Meiman, P. Cho, R. Noe, I. Shpantzer, and V. Karagodsky, “Phased-array cancellation of nonlinear FWM in coherent OFDM dispersive multi-span links,” Opt. Express 16(20), 15777–15810 (2008).
[Crossref] [PubMed]

R. Weidenfeld, M. Nazarathy, R. Noe, and I. Shpantzer, “Volterra nonlinear compensation of 100G coherent OFDM with baud-rate ADC, tolerable complexity and lowintra-channel FWM/XPM error propagation,” in Optical Fiber Communication Conference, OSA Technical Digest Series (Optical Society of America, 2010), paper OTuE3.
[Crossref]

Pan, J.

Pask, C.

Peddanarappagari, K.

K. Peddanarappagari and M. Brandt-Pearce, “Volterra series transfer function of single-mode fibers,” J. Lightwave Technol. 15(12), 2232–2241 (1997).
[Crossref]

Pinto, A. N.

Poggiolini, P.

Reid, D.

E. Flood, W.H. Guo, D. Reid, M. Lynch, A. Bradley, L. Barry, and J. Donegan, “Interferometer Based In-Band OSNR Monitoring of Single and Dual Polarisation QPSK Signals,” in Proceedings of 36th European Conference on Optical Communication, (ECOC, 2010), paper Th9C6.

Reis, J. D.

Shpantzer, I.

M. Nazarathy, J. Khurgin, R. Weidenfeld, Y. Meiman, P. Cho, R. Noe, I. Shpantzer, and V. Karagodsky, “Phased-array cancellation of nonlinear FWM in coherent OFDM dispersive multi-span links,” Opt. Express 16(20), 15777–15810 (2008).
[Crossref] [PubMed]

R. Weidenfeld, M. Nazarathy, R. Noe, and I. Shpantzer, “Volterra nonlinear compensation of 100G coherent OFDM with baud-rate ADC, tolerable complexity and lowintra-channel FWM/XPM error propagation,” in Optical Fiber Communication Conference, OSA Technical Digest Series (Optical Society of America, 2010), paper OTuE3.
[Crossref]

Shulkind, G.

G. Shulkind and M. Nazarathy, “Nonlinear digital back propagation compensator for coherent optical OFDM based on factorizing the Volterra series transfer function,” Opt. Express 21(11), 13146–13161 (2013).
[Crossref]

Teixeira, A. L.

Vatarescu, A.

Weidenfeld, R.

M. Nazarathy, J. Khurgin, R. Weidenfeld, Y. Meiman, P. Cho, R. Noe, I. Shpantzer, and V. Karagodsky, “Phased-array cancellation of nonlinear FWM in coherent OFDM dispersive multi-span links,” Opt. Express 16(20), 15777–15810 (2008).
[Crossref] [PubMed]

R. Weidenfeld, M. Nazarathy, R. Noe, and I. Shpantzer, “Volterra nonlinear compensation of 100G coherent OFDM with baud-rate ADC, tolerable complexity and lowintra-channel FWM/XPM error propagation,” in Optical Fiber Communication Conference, OSA Technical Digest Series (Optical Society of America, 2010), paper OTuE3.
[Crossref]

Xie, C.

Xiong, Q.

Xu, A.

Y. Gao, F. Zgang, L. Dou, Z. Chen, and A. Xu, “Intra-channel nonlinearities mitigation in pseudo-linear coherent QPSK transmission systems via nonlinear electrical equalizer,” Opt. Commun. 282(12), 2421–2425 (2009).
[Crossref]

Yaman, F.

Zgang, F.

Y. Gao, F. Zgang, L. Dou, Z. Chen, and A. Xu, “Intra-channel nonlinearities mitigation in pseudo-linear coherent QPSK transmission systems via nonlinear electrical equalizer,” Opt. Commun. 282(12), 2421–2425 (2009).
[Crossref]

IEEE Photon. Technol. Lett. (1)

F. P. Guiomar, J. D. Reis, A. L. Teixeira, and A. N. Pinto, “Digital postcompensation using Volterra series transfer function,” IEEE Photon. Technol. Lett. 23(19), 1412–1414 (2011).
[Crossref]

J. Lightwave Technol. (6)

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

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

Opt. Commun. (1)

Y. Gao, F. Zgang, L. Dou, Z. Chen, and A. Xu, “Intra-channel nonlinearities mitigation in pseudo-linear coherent QPSK transmission systems via nonlinear electrical equalizer,” Opt. Commun. 282(12), 2421–2425 (2009).
[Crossref]

Opt. Express (7)

X. Li, X. Chen, G. Goldfarb, E. Mateo, I. Kim, F. Yaman, and 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]

J. D. Reis and A. L. Teixeira, “Unveiling nonlinear effects in dense coherent optical WDM systems with Volterra series,” Opt. Express 18(8), 8660–8670 (2010).
[Crossref] [PubMed]

F. P. Guiomar, J. D. Reis, A. L. Teixeira, and A. N. Pinto, “Mitigation of intra-channel nonlinearities using a frequency-domain Volterra series equalizer,” Opt. Express 20(2), 1360–1369 (2012).
[Crossref] [PubMed]

M. Nazarathy, J. Khurgin, R. Weidenfeld, Y. Meiman, P. Cho, R. Noe, I. Shpantzer, and V. Karagodsky, “Phased-array cancellation of nonlinear FWM in coherent OFDM dispersive multi-span links,” Opt. Express 16(20), 15777–15810 (2008).
[Crossref] [PubMed]

Z. Dong, A. Lau, and C. Lu, “OSNR monitoring for QPSK and 16-QAM systems in presence of fiber nonlinearities for digital coherent receivers,” Opt. Express 20(17), 19520–19534 (2012).
[Crossref] [PubMed]

G. Shulkind and M. Nazarathy, “Nonlinear digital back propagation compensator for coherent optical OFDM based on factorizing the Volterra series transfer function,” Opt. Express 21(11), 13146–13161 (2013).
[Crossref]

F. P. Guiomar, J. D. Reis, A. Carena, G. Bosco, A. L. Teixeira, and A. N. Pinto, “Experimental demonstration of a frequency-domain Volterra series nonlinear equalizer in polarization-multiplexed transmission,” Opt. Express 21(1), 276–288 (2013).
[Crossref] [PubMed]

Other (4)

E. Flood, W.H. Guo, D. Reid, M. Lynch, A. Bradley, L. Barry, and J. Donegan, “Interferometer Based In-Band OSNR Monitoring of Single and Dual Polarisation QPSK Signals,” in Proceedings of 36th European Conference on Optical Communication, (ECOC, 2010), paper Th9C6.

R. Weidenfeld, M. Nazarathy, R. Noe, and I. Shpantzer, “Volterra nonlinear compensation of 100G coherent OFDM with baud-rate ADC, tolerable complexity and lowintra-channel FWM/XPM error propagation,” in Optical Fiber Communication Conference, OSA Technical Digest Series (Optical Society of America, 2010), paper OTuE3.
[Crossref]

C. Lin, “Digital nonlinear compensation for next-generation optical communication systems using advanced modulation formats,” Master’s thesis, Friedrich-Alexander-Universität Erlangen-Nürnberg, 2014.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2013).

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Figures (5)

Fig. 1
Fig. 1 Simulation setup adopted for the validation of the high-order VSTF analytical formulation. a) a single-polarization QPSK signal is propagated over 12 fiber spans with ideal inline amplification providing an optical gain, G, that exactly compensates the fiber loss. b) a reference solution of nonlinear propagation is obtained with the asymmetric split-step Fourier method with a very short step-size of 10 m.
Fig. 2
Fig. 2 Comparison of the NSD dependence on step-size length for the three methods (SSFM, third-order VSTF and simplified high-order multi-span VSTF (SH-MS-VSTF)) for the quasi-linear scenario.
Fig. 3
Fig. 3 Comparison of the NSD dependence on step-size length for the three methods (SSFM, third-order VSTF and simplified high-order multi-span VSTF (SH-MS-VSTF)) for the highly nonlinear scenario.
Fig. 4
Fig. 4 Comparison of the NSD dependence on step-size length for the three methods (SSFM, third-order VSTF and simplified high-order multi-span VSTF (SH-MS-VSTF)) for the standard scenario.
Fig. 5
Fig. 5 Comparison of the NSD dependence on step-size length for the three methods (SSFM, third-order VSTF and simplified high-order multi-span VSTF (SH-MS-VSTF)) for the reference scenario and using 60 spans.

Tables (2)

Tables Icon

Table 1 Propagation parameters.

Tables Icon

Table 2 Values of P0 and γ used in our analysis.

Equations (43)

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

A ( t , z ) z = α 2 A ( t , z ) i β 2 2 2 A ( t , z ) t 2 + i γ | A ( t , z ) | 2 A ( t , z ) ,
A ˜ ( ω , z ) z = A ˜ ( ω , z ) 2 ( α i β 2 ω 2 ) + i γ 4 π 2 A ˜ ( ω 1 , z ) A ˜ * ( ω 2 , z ) A ˜ ( ω ω 1 + ω 2 , z ) d ω 1 d ω 2 .
A ˜ ( ω , z + L span ) H 1 ( ω , L span ) A ˜ ( ω , z ) + A ˜ ( ω 1 , z ) A ˜ * ( ω 2 , z ) A ˜ ( ω ω 1 + ω 2 , z ) H 3 ( ω , ω 1 , ω 2 , L span ) d ω 1 d ω 2
H 1 ( ω , L span ) = exp ( α 2 L span + i β 2 2 ω 2 L span )
H 3 ( ω , ω 1 , ω 2 , L span ) = i γ 4 π 2 H 1 ( ω , L span ) H 3 ( ω , ω 1 , ω 2 , L span ) = i γ 4 π 2 H 1 ( ω , L span ) 1 exp ( α L span + i β 2 ( ω 1 ω ) ( ω 1 ω 2 ) L span ) α i β 2 ( ω 1 ω ) ( ω 1 ω 2 )
A ˜ ( ω , z + L span ) = H 1 ( ω , L span ) A ˜ ( ω , z ) + i γ 4 π 2 H 1 ( ω , L span ) A ˜ ( ω 1 , z ) A ˜ * ( ω 2 , z ) A ˜ ( ω ω 1 + ω 2 , z ) H 3 ( ω , ω 1 , ω 2 , L span ) d ω 1 d ω 2 .
A ˜ LI ( ω , z + L span ) = H 1 ( ω , L span ) A ˜ ( ω , z )
A ˜ NL ( ω , z + L span ) = i γ 4 π 2 H 1 ( ω , L span ) A ˜ ( ω 1 , z ) A ˜ * ( ω 2 , z ) A ˜ ( ω ω 1 + ω 2 , z ) H 3 ( ω , ω 1 , ω 2 , L span ) d ω 1 d ω 2 .
A ˜ ( ω , z + L span ) = A ˜ LI ( ω , z + L span ) + A ˜ NL ( ω , z + L span ) .
A ˜ ( ω , z + 2 L span ) = H 1 ( ω , L span ) exp ( α 2 L span ) A ˜ ( ω , z + L span ) + i γ 4 π 2 H 1 ( ω , L span ) exp ( 3 2 α L span ) A ˜ ( ω 1 , z + L span ) A ˜ * ( ω 2 , z + L span ) A ˜ ( ω ω 1 + ω 2 , z + L span ) H 3 ( ω , ω 1 , ω 2 , L span ) d ω 1 d ω 2 .
A ˜ ( 3 ) ( ω , z + 2 L span ) = H 1 2 ( ω , L span ) exp ( α 2 L span ) A ˜ ( ω , z ) + i γ H 1 2 ( ω , L span ) exp ( α 2 L span ) A ˜ ( ω 1 , z ) A ˜ * ( ω 2 , z ) A ˜ ( ω ω 1 + ω 2 , z ) H 3 ( ω , ω 1 , ω 2 , L span ) d ω 1 d ω 2 .
A ˜ ( 9 ) ( ω , z + 2 L span ) = i γ H 1 ( ω , L span ) exp ( 3 2 α L span ) H 3 ( ω , ω 1 , ω 2 , L span ) [ H 1 ( ω 1 , L span ) ( A ˜ ( ω 1 , z ) + i γ H 3 ( ω 1 , ω 1 , ω 2 , L span ) A ˜ ( ω 1 , z ) A ˜ * ( ω 2 , z ) A ˜ ( ω 1 ω 1 + ω 2 , z ) d ω 1 d ω 2 ) ] [ H 1 ( ω 2 , L span ) ( A ˜ ( ω 2 , z ) + i γ H 3 ( ω 2 , ω 1 , ω 2 , L span ) A ˜ ( ω 1 , z ) A ˜ * ( ω 2 , z ) A ˜ ( ω 2 ω 1 + ω 2 , z ) d ω 1 d ω 2 ) ] * [ H 1 ( ω ω 1 + ω 2 , L span ) ( A ˜ ( ω ω 1 + ω 2 , z ) + i γ H 3 ( ω ω 1 + ω 2 , ω 1 , ω 2 , L span ) A ˜ ( ω 1 , z ) A ˜ * ( ω 2 , z ) A ˜ ( ω ω 1 + ω 2 ω 1 + ω 2 , z ) d ω 1 d ω 2 ) ] d ω 1 d ω 2 .
ω 1 = ω 1 ,
A ˜ ( ω 1 , z ) = A ˜ ( ω 1 , z )
A ˜ ( ω 1 ω 1 + ω 2 , z ) = A ˜ ( ω 2 , z ) .
A ˜ * ( ω 2 , z ) A ˜ ( ω 2 , z ) d ω 2 ,
H 3 ( ω 1 , ω 1 , ω 2 , L span ) = 1 exp ( α L span + i β 2 ( ω 1 ω 1 ) ( ω 1 ω 2 ) L span ) α i β 2 ( ω 1 ω 1 ) ( ω 1 ω 2 )
H 3 XPM = 1 exp ( α L span ) α ,
H 1 ( ω i , L span ) ( A ˜ ( ω i , z ) + i γ H 3 ( ω i , ω 1 , ω 2 , L span ) A ˜ ( ω 1 , z ) A ˜ * ( ω 2 , z ) A ˜ ( ω i ω 1 + ω 2 , z ) d ω 1 d ω 2 ) H 1 ( ω i , L span ) A ˜ ( ω i , z ) ( 1 + i γ P 0 H 3 XPM )
A ˜ ( 9 ) ( ω , z + 2 L span ) i γ H 1 ( ω , L span ) exp ( 3 2 α L span ) H 1 ( ω 1 , L span ) A ˜ ( ω 1 , z ) [ 1 + i γ P 0 H 3 XPM ] H 1 * ( ω 2 , L span ) A ˜ * ( ω 2 , z ) [ 1 i γ P 0 H 3 XPM ] H 1 ( ω ω 1 + ω 2 , L span ) A ˜ ( ω ω 1 + ω 2 , z ) [ 1 + i γ P 0 H 3 XPM ] H 3 ( ω , ω 1 , ω 2 , L span ) d ω 1 d ω 2 .
[ 1 + i γ P 0 H 3 XPM ] 2 [ 1 i γ P 0 H 3 XPM ] = 1 + 2 i γ P 0 H 3 XPM γ 2 P 0 2 ( H 3 XPM ) 2 i γ P 0 H 3 XPM + i γ 3 P 0 3 ( H 3 XPM ) 3 + 2 γ 2 P 0 2 ( H 3 XPM ) 2 = 1 + i γ P 0 H 3 XPM + γ 2 P 0 2 ( H 3 XPM ) 2 + i γ 3 P 0 3 ( H 3 XPM ) 2 .
1 + i γ P 0 H 3 XPM + γ 2 P 0 2 ( H 3 XPM ) 2 + i γ 3 P 0 3 ( H 3 XPM ) 3 1 + i γ P 0 H 3 XPM = 1 + i F ( )
A ˜ ( 9 ) ( ω , z + 2 L span ) i γ H 1 ( ω , L span ) exp ( 3 2 α L span ) A ˜ ( ω 1 , z ) A ˜ * ( ω 2 , z ) A ˜ ( ω ω 1 + ω 2 , z ) [ 1 + i F ( ) ] H 1 ( ω 1 , L span ) H 1 * ( ω 2 , L span ) H 1 ( ω ω 1 + ω 2 , L span ) H 3 ( ω , ω 1 , ω 2 , L span ) d ω 1 d ω 2 .
H 1 ( ω 1 , L span ) H 1 * ( ω 2 , L span ) H 1 ( ω ω 1 + ω 2 , L span ) = exp ( α L span ) H 1 ( ω , L span ) exp ( i β 2 ( ω 1 ω 2 ) ( ω 1 ω ) L span ) .
A ˜ ( 9 ) ( ω , z + 2 L span ) i γ H 1 2 ( ω , L span ) exp ( α 2 L span ) A ˜ ( ω 1 , z ) A ˜ * ( ω 2 , z ) A ˜ ( ω ω 1 + ω 2 , z ) [ 1 + i F ( ) ] H 3 ( ω , ω 1 , ω 2 , L span ) exp ( i β 2 ( ω 1 ω 2 ) ( ω 1 ω ) L span ) d ω 1 d ω 2 .
A ˜ ( ω , z + 2 L span ) = H 1 2 ( ω , L span ) exp ( α 2 L span ) A ˜ ( ω , z ) + i γ H 1 2 ( ω , L span ) exp ( α 2 L span ) A ˜ ( ω 1 , z ) A ˜ * ( ω 2 , z ) A ˜ ( ω ω 1 + ω 2 , z ) H 3 ( ω , ω 1 , ω 2 , L span ) [ exp ( i β 2 ( ω 1 ω 2 ) ( ω 1 ω ) L span ) [ 1 + i F ( ) ] + 1 ] d ω 1 d ω 2 .
exp ( i β 2 ( ω 1 ω 2 ) ( ω 1 ω ) L span ) + 1 1 + i F ( ) ,
exp ( i β 2 ( ω 1 ω 2 ) ( ω 1 ω ) L span ) + 1 1 + F 2 ( ) i F ( ) 1 + F 2 ( ) .
exp ( i β 2 ( ω 1 ω 2 ) ( ω 1 ω ) L span ) + 1 i F ( ) 1 + F 2 ( ) exp ( i β 2 ( ω 1 ω 2 ) ( ω 1 ω ) L span ) + 1 i F ( ) exp ( i β 2 ( ω 1 ω 2 ) ( ω 1 ω ) L span ) + 1 .
A ˜ ( ω , z + 2 L span ) = H 1 2 ( ω , L span ) exp ( α 2 L span ) A ˜ ( ω , z ) + i γ H 1 2 ( ω , L span ) exp ( α 2 L span ) A ˜ ( ω 1 , z ) A ˜ * ( ω 2 , z ) A ˜ ( ω ω 1 + ω 2 , z ) H 3 ( ω , ω 1 , ω 2 , L span ) [ exp ( i β 2 ( ω 1 ω 2 ) ( ω 1 ω ) L span ) + 1 ] [ 1 + i F ( ) ] d ω 1 d ω 2 .
A ˜ LI ( ω , z + n S L span ) = H 1 n S ( ω , L span ) A ˜ ( ω , z ) exp ( α 2 ( n S 1 ) L span )
A ˜ NL ( ω , z + n S L span ) = i γ H 1 ( ω , n S L span ) exp ( 3 α 2 L span ( n S 1 ) ) A ˜ ( ω 1 , z ) A ˜ * ( ω 2 , z ) A ˜ ( ω ω 1 + ω 2 , z ) [ 1 + i F ( ) ] n S 1 [ H 1 ( ω 1 , L span ) H 1 * ( ω 2 , L span ) H 1 ( ω ω 1 + ω 2 , L span ) ] n S 1 H 3 ( ω , ω 1 , ω 2 , L span ) d ω 1 d ω 2 .
A ˜ ( ω , z + n S L span ) = H 1 M S ( ω , L span ) A ˜ ( ω , z ) + i γ H 1 M S ( ω , n S L span ) A ˜ ( ω 1 , z ) A ˜ * ( ω 2 , z ) A ˜ ( ω ω 1 + ω 2 , z ) H 3 M S ( ω , ω 1 , ω 2 , n S L span ) [ 1 + i F ( ) ] n S 1 d ω 1 d ω 2 ,
H 1 M S ( ω , n S L span ) = H 1 n S ( ω , L span ) exp ( α 2 ( n S 1 ) L span )
H 3 M S ( ω , ω 1 , ω 2 , n S L span ) = H 3 ( ω , ω 1 , ω 2 , L span ) n S = 1 n spans exp ( i β 2 ( ω 1 ω 2 ) ( ω 1 ω ) ( n S 1 ) L span )
[ 1 + i F ( ) ] n S 1 1 + i n S F ( ) = 1 + i n S γ P 0 1 exp ( α L span ) α .
A ˜ ( ω , z + n S L span ) H 1 M S ( ω , n S L span ) A ˜ ( ω , z ) + i γ H 1 M S ( ω , n S L span ) A ˜ ( ω 1 , z ) A ˜ * ( ω 2 , z ) A ˜ ( ω ω 1 + ω 2 , z ) H 3 M S ( ω , ω 1 , ω 2 , n S L span ) [ 1 + i n S γ P 0 1 exp ( α L span ) α ] d ω 1 d ω 2 .
N steps = n spans n S .
NSD = | S out S ref | 2 d t | S ref | 2 d t ,
OSNR sim = S N real + N num
1 OSNR sim = N real + N num S = 1 OSNR real + 1 OSNR num = 1 OSNR real + NSD ,
NSD 1 OSNR real ,
1 OSNR sim 1 OSNR real .

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