Abstract

With ideal nonlinearity compensation using digital back propagation (DBP), the transmission performance of an optical fiber channel has been considered to be limited by nondeterministic nonlinear signal-ASE interaction. In this paper, we conduct theoretical and numerical study on nonlinearity compensation using DBP in the presence of polarization-mode dispersion (PMD). Analytical expressions of transmission performance with DBP are derived and substantiated by numerical simulations for polarization-division-multiplexed systems under the influence of PMD effects. We find that nondeterministic distributed PMD impairs the effectiveness of DBP-based nonlinearity compensation much more than nonlinear signal-ASE interaction, and is therefore the fundamental limitation to single-mode fiber channel capacity.

© 2012 OSA

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References

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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  6. W. Shieh and Y. Tang, “Ultrahigh-speed signal transmission over nonlinear and dispersive fiber optic channel: the multicarrier advantage,” IEEE Photon. J.2(3), 276–283 (2010).
    [CrossRef]
  7. X. Liu, F. Buchali, and R. W. Tkach, “Improving the nonlinear tolerance of polarization-division-multiplexed CO-OFDM in long-haul fiber transmission,” J. Lightwave Technol.27(16), 3632–3640 (2009).
    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  18. R. J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightwave Technol.28(4), 662–701 (2010).
    [CrossRef]
  19. D. Rafique and A. D. Ellis, “Impact of signal-ASE four-wave mixing on the effectiveness of digital back-propagation in 112 Gb/s PM-QPSK systems,” Opt. Express19(4), 3449–3454 (2011).
    [CrossRef] [PubMed]
  20. G. Gao, X. Chen, W. Shieh, “Limitation of fiber nonlinearity compensation using digital back propagation,” OFC’2012, paper OMA3.
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    [CrossRef] [PubMed]
  22. X. Chen and W. Shieh, “Closed-form expressions for nonlinear transmission performance of densely spaced coherent optical OFDM systems,” Opt. Express18(18), 19039–19054 (2010).
    [CrossRef] [PubMed]
  23. W. Shieh and X. Chen, “Information spectral efficiency and launch power density limits due to fiber nonlinearity for coherent optical OFDM systems,” IEEE Photon. J.3(2), 158–173 (2011).
    [CrossRef]
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    [CrossRef]
  27. D. Marcuse, C. R. Menyuk, and P. K. A. Wai, “Application of the Manakov-PMD equation to studies of signal propagation in optical fibers with randomly varying birefringence,” J. Lightwave Technol.15(9), 1735–1746 (1997).
    [CrossRef]
  28. P. P. Mitra and J. B. Stark, “Nonlinear limits to the information capacity of optical fiber communications,” Nature411(6841), 1027–1030 (2001).
    [CrossRef] [PubMed]
  29. A. Vannucci and A. Bononi, “Statistical characterization of the Jones Matrix of long fibers affected by polarization mode dispersion (PMD),” J. Lightwave Technol.20(5), 811–821 (2002).
    [CrossRef]
  30. A. Bononi and A. Vannucci, “Statistics of the Jones matrix of fibers affected by polarization mode dispersion,” Opt. Lett.26(10), 675–677 (2001).
    [CrossRef] [PubMed]

2011

2010

2009

2008

2007

2006

2005

2002

A. Vannucci and A. Bononi, “Statistical characterization of the Jones Matrix of long fibers affected by polarization mode dispersion (PMD),” J. Lightwave Technol.20(5), 811–821 (2002).
[CrossRef]

2001

P. P. Mitra and J. B. Stark, “Nonlinear limits to the information capacity of optical fiber communications,” Nature411(6841), 1027–1030 (2001).
[CrossRef] [PubMed]

A. Bononi and A. Vannucci, “Statistics of the Jones matrix of fibers affected by polarization mode dispersion,” Opt. Lett.26(10), 675–677 (2001).
[CrossRef] [PubMed]

1997

D. Marcuse, C. R. Menyuk, and P. K. A. Wai, “Application of the Manakov-PMD equation to studies of signal propagation in optical fibers with randomly varying birefringence,” J. Lightwave Technol.15(9), 1735–1746 (1997).
[CrossRef]

1996

P. K. A. Wai and C. R. Menyuk, “Polarization mode dispersion, decorrelation, and diffusion in optical fibers with randomly varying birefringence,” J. Lightwave Technol.14(2), 148–157 (1996).
[CrossRef]

1992

Adamiecki, A.

Athaudage, C.

W. Shieh and C. Athaudage, “coherent optical orthogonal frequency division multiplexing,” Electron. Lett.42(10), 587–589 (2006).
[CrossRef]

Barros, D. J. F.

Bayvel, P.

Bononi, A.

A. Vannucci and A. Bononi, “Statistical characterization of the Jones Matrix of long fibers affected by polarization mode dispersion (PMD),” J. Lightwave Technol.20(5), 811–821 (2002).
[CrossRef]

A. Bononi and A. Vannucci, “Statistics of the Jones matrix of fibers affected by polarization mode dispersion,” Opt. Lett.26(10), 675–677 (2001).
[CrossRef] [PubMed]

Buchali, F.

Calabrò, S.

Chandrasekhar, S.

Chen, X.

W. Shieh and X. Chen, “Information spectral efficiency and launch power density limits due to fiber nonlinearity for coherent optical OFDM systems,” IEEE Photon. J.3(2), 158–173 (2011).
[CrossRef]

X. Chen and W. Shieh, “Closed-form expressions for nonlinear transmission performance of densely spaced coherent optical OFDM systems,” Opt. Express18(18), 19039–19054 (2010).
[CrossRef] [PubMed]

Cho, P.

Chowdhury, A.

de Waardt, H.

Doerr, C. R.

Du, L. B.

Ellis, A. D.

Essiambre, R. J.

Foschini, G. J.

Gavioli, G.

Goebel, B.

Inoue, K.

Ip, E.

Jansen, S. L.

Kahn, J. M.

Karagodsky, V.

Khoe, G.-D.

Khurgin, J.

Killey, R. I.

Kramer, G.

Krummrich, P. M.

Lau, A. P. T.

Leuthold, J.

Li, G.

Liu, X.

Lowery, A. J.

Marcuse, D.

D. Marcuse, C. R. Menyuk, and P. K. A. Wai, “Application of the Manakov-PMD equation to studies of signal propagation in optical fibers with randomly varying birefringence,” J. Lightwave Technol.15(9), 1735–1746 (1997).
[CrossRef]

Mateo, E. F.

Meiman, Y.

Menyuk, C. R.

D. Marcuse, C. R. Menyuk, and P. K. A. Wai, “Application of the Manakov-PMD equation to studies of signal propagation in optical fibers with randomly varying birefringence,” J. Lightwave Technol.15(9), 1735–1746 (1997).
[CrossRef]

P. K. A. Wai and C. R. Menyuk, “Polarization mode dispersion, decorrelation, and diffusion in optical fibers with randomly varying birefringence,” J. Lightwave Technol.14(2), 148–157 (1996).
[CrossRef]

Mitra, P. P.

P. P. Mitra and J. B. Stark, “Nonlinear limits to the information capacity of optical fiber communications,” Nature411(6841), 1027–1030 (2001).
[CrossRef] [PubMed]

Nazarathy, M.

Noe, R.

Rafique, D.

Raybon, G.

Savory, S. J.

Shieh, W.

W. Shieh and X. Chen, “Information spectral efficiency and launch power density limits due to fiber nonlinearity for coherent optical OFDM systems,” IEEE Photon. J.3(2), 158–173 (2011).
[CrossRef]

W. Shieh and Y. Tang, “Ultrahigh-speed signal transmission over nonlinear and dispersive fiber optic channel: the multicarrier advantage,” IEEE Photon. J.2(3), 276–283 (2010).
[CrossRef]

X. Chen and W. Shieh, “Closed-form expressions for nonlinear transmission performance of densely spaced coherent optical OFDM systems,” Opt. Express18(18), 19039–19054 (2010).
[CrossRef] [PubMed]

W. Shieh and C. Athaudage, “coherent optical orthogonal frequency division multiplexing,” Electron. Lett.42(10), 587–589 (2006).
[CrossRef]

Shpantzer, I.

Sinsky, J. H.

Sohler, W.

Spinnler, B.

Stark, J. B.

P. P. Mitra and J. B. Stark, “Nonlinear limits to the information capacity of optical fiber communications,” Nature411(6841), 1027–1030 (2001).
[CrossRef] [PubMed]

Suche, H.

Tang, Y.

W. Shieh and Y. Tang, “Ultrahigh-speed signal transmission over nonlinear and dispersive fiber optic channel: the multicarrier advantage,” IEEE Photon. J.2(3), 276–283 (2010).
[CrossRef]

Tkach, R. W.

van den Borne, D.

Vannucci, A.

A. Vannucci and A. Bononi, “Statistical characterization of the Jones Matrix of long fibers affected by polarization mode dispersion (PMD),” J. Lightwave Technol.20(5), 811–821 (2002).
[CrossRef]

A. Bononi and A. Vannucci, “Statistics of the Jones matrix of fibers affected by polarization mode dispersion,” Opt. Lett.26(10), 675–677 (2001).
[CrossRef] [PubMed]

Wai, P. K. A.

D. Marcuse, C. R. Menyuk, and P. K. A. Wai, “Application of the Manakov-PMD equation to studies of signal propagation in optical fibers with randomly varying birefringence,” J. Lightwave Technol.15(9), 1735–1746 (1997).
[CrossRef]

P. K. A. Wai and C. R. Menyuk, “Polarization mode dispersion, decorrelation, and diffusion in optical fibers with randomly varying birefringence,” J. Lightwave Technol.14(2), 148–157 (1996).
[CrossRef]

Weidenfeld, R.

Winzer, P. J.

Yaman, F.

E. F. Mateo, F. Yaman, and G. Li, “Efficient compensation of inter-channel nonlinear effects via digital backward propagation in WDM optical transmission,” Opt. Express18(14), 15144–15154 (2010).
[CrossRef] [PubMed]

F. Yaman and G. Li, “Nonlinear impairment compensation for polarization-division multiplexed WDM transmission using digital backward propagation,” IEEE Photon. J.2(5), 816–832 (2010).
[CrossRef]

Zhou, X.

Zhu, L.

Appl. Opt.

Electron. Lett.

W. Shieh and C. Athaudage, “coherent optical orthogonal frequency division multiplexing,” Electron. Lett.42(10), 587–589 (2006).
[CrossRef]

IEEE Photon. J.

W. Shieh and Y. Tang, “Ultrahigh-speed signal transmission over nonlinear and dispersive fiber optic channel: the multicarrier advantage,” IEEE Photon. J.2(3), 276–283 (2010).
[CrossRef]

F. Yaman and G. Li, “Nonlinear impairment compensation for polarization-division multiplexed WDM transmission using digital backward propagation,” IEEE Photon. J.2(5), 816–832 (2010).
[CrossRef]

W. Shieh and X. Chen, “Information spectral efficiency and launch power density limits due to fiber nonlinearity for coherent optical OFDM systems,” IEEE Photon. J.3(2), 158–173 (2011).
[CrossRef]

J. Lightwave Technol.

P. K. A. Wai and C. R. Menyuk, “Polarization mode dispersion, decorrelation, and diffusion in optical fibers with randomly varying birefringence,” J. Lightwave Technol.14(2), 148–157 (1996).
[CrossRef]

D. Marcuse, C. R. Menyuk, and P. K. A. Wai, “Application of the Manakov-PMD equation to studies of signal propagation in optical fibers with randomly varying birefringence,” J. Lightwave Technol.15(9), 1735–1746 (1997).
[CrossRef]

A. Vannucci and A. Bononi, “Statistical characterization of the Jones Matrix of long fibers affected by polarization mode dispersion (PMD),” J. Lightwave Technol.20(5), 811–821 (2002).
[CrossRef]

A. Chowdhury, G. Raybon, R. J. Essiambre, J. H. Sinsky, A. Adamiecki, J. Leuthold, C. R. Doerr, and S. Chandrasekhar, “Compensation of intrachannel nonlinearities in 40-Gb/s pseudolinear systems using optical-phase conjugation,” J. Lightwave Technol.23(1), 172–177 (2005).
[CrossRef]

S. L. Jansen, D. van den Borne, B. Spinnler, S. Calabrò, H. Suche, P. M. Krummrich, W. Sohler, G.-D. Khoe, and H. de Waardt, “Optical phase conjugation for ultra-long-haul phase-shift-keyed transmission,” J. Lightwave Technol.24(1), 54–64 (2006).
[CrossRef]

X. Liu, F. Buchali, and R. W. Tkach, “Improving the nonlinear tolerance of polarization-division-multiplexed CO-OFDM in long-haul fiber transmission,” J. Lightwave Technol.27(16), 3632–3640 (2009).
[CrossRef]

R. J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightwave Technol.28(4), 662–701 (2010).
[CrossRef]

E. Ip, “Nonlinear compensation using backpropagation for polarization-multiplexed transmission,” J. Lightwave Technol.28(6), 939–951 (2010).
[CrossRef]

E. Ip and J. M. Kahn, “Compensation of dispersion and nonlinear impairments using digital backpropagation,” J. Lightwave Technol.26(20), 3416–3425 (2008).
[CrossRef]

Nature

P. P. Mitra and J. B. Stark, “Nonlinear limits to the information capacity of optical fiber communications,” Nature411(6841), 1027–1030 (2001).
[CrossRef] [PubMed]

Opt. Express

S. J. Savory, G. Gavioli, R. I. Killey, and P. Bayvel, “Electronic compensation of chromatic dispersion using a digital coherent receiver,” Opt. Express15(5), 2120–2126 (2007).
[CrossRef] [PubMed]

E. Ip, A. P. T. Lau, D. J. F. Barros, and J. M. Kahn, “Coherent detection in optical fiber systems,” Opt. Express16(2), 753–791 (2008).
[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. Express16(20), 15777–15810 (2008).
[CrossRef] [PubMed]

E. F. Mateo, L. Zhu, and G. Li, “Impact of XPM and FWM on the digital implementation of impairment compensation for WDM transmission using backward propagation,” Opt. Express16(20), 16124–16137 (2008).
[CrossRef] [PubMed]

E. F. Mateo, F. Yaman, and G. Li, “Efficient compensation of inter-channel nonlinear effects via digital backward propagation in WDM optical transmission,” Opt. Express18(14), 15144–15154 (2010).
[CrossRef] [PubMed]

L. B. Du and A. J. Lowery, “Improved single channel backpropagation for intra-channel fiber nonlinearity compensation in long-haul optical communication systems,” Opt. Express18(16), 17075–17088 (2010).
[CrossRef] [PubMed]

X. Chen and W. Shieh, “Closed-form expressions for nonlinear transmission performance of densely spaced coherent optical OFDM systems,” Opt. Express18(18), 19039–19054 (2010).
[CrossRef] [PubMed]

E. F. Mateo, X. Zhou, and G. Li, “Improved digital backward propagation for the compensation of inter-channel nonlinear effects in polarization-multiplexed WDM systems,” Opt. Express19(2), 570–583 (2011).
[CrossRef] [PubMed]

D. Rafique and A. D. Ellis, “Impact of signal-ASE four-wave mixing on the effectiveness of digital back-propagation in 112 Gb/s PM-QPSK systems,” Opt. Express19(4), 3449–3454 (2011).
[CrossRef] [PubMed]

Opt. Lett.

Other

A. Milton and I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Abramowitz and Stegun, eds. (Dover, 1964).

G. Gao, X. Chen, W. Shieh, “Limitation of fiber nonlinearity compensation using digital back propagation,” OFC’2012, paper OMA3.

L. B. Du, B. Schmidt, and A. Lowery, “Efficient digital backpropagation for PDM-CO-OFDM optical transmission systems,” OFC’ 2010, paper OTuE2.

E. Ip and J. M. Kahn, “Nonlinear impairment compensation using backpropagation,” in Optical Fiber, New Developments, C. Lethien, Ed., In-Tech, Vienna Austria, December (2009).

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Figures (6)

Fig. 1
Fig. 1

Illustration of nonlinear signal-ASE interaction for transmission systems using DBP. SSMF: standard single-mode fiber, EDFA: erbium doped fiber amplifier.

Fig. 2
Fig. 2

Illustration of PMD influence on nonlinear four-wave mixing generation. Sub.: Subcarrier.

Fig. 3
Fig. 3

Simulation setup for polarization-division-multiplexed CO-OFDM system with digital back propagation. LOs: local oscillators, PBC/S: polarization beam combiner/splitter.

Fig. 4
Fig. 4

Statistical distribution of Q factor for (a) launch power of 15 dBm and PMD of 0.05 ps/sqrt(km) (b) launch power of 15 dBm and PMD of 0.1 ps/sqrt(km), (c) launch power of 18 dBm and PMD of 0.05 ps/sqrt(km), and (d) launch power of 18dBm and PMD of 0.1 ps/sqrt(km). Results are obtained over 10 spans of 100 km SSMF links. For each combination of launch power and PMD coefficient, 100 cases are simulated.

Fig. 5
Fig. 5

SNR versus launch power under different nonlinear interactions. Linear: linear transmission regime without nonlinear effects. N-ASE: nonlinear signal-ASE interaction impaired DBP systems. PMD: distributed PMD impaired DBP systems. The unit of PMD parameter is ps / km . Avg: average SNR obtained from 100 PMD realizations. NBP: no back-propagation nonlinearity compensation. Open symbols are for simulation results and dashed lines for theoretical results. Results are obtained over 10 spans of 100 km SSMF links.

Fig. 6
Fig. 6

System performance over 5-THz C band for (a) SNR versus launch power density for 40 spans 100 km SSMF transmissions and (b) spectral efficiency versus number of spans. Linear: linear transmission regime without nonlinear effects. N-ASE: nonlinear signal-ASE limited regimes. PMD: nonlinearity compensation influenced by PMD, the unit of PMD parameter is ps/ km ; NBP: no back-propagation nonlinearity compensation. For PMD limited regimes, mean system performance has been assumed.

Tables (1)

Tables Icon

Table 1 Fiber Span Parameters used in Simulation

Equations (27)

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

i c g ' z +i α 2 c g ' 1 2 p ( z ) σ ω g c g ' + 1 2 β 2 ω g 2 c g ' +γ[ ( c k '+ c i ' ) c j ' +( c k '+ c j ' ) c i ' ]=0
c g ' =γ[ ( c k + c i ) c j +( c k + c j ) c i ] e αL / 2i β g L 1 e αLjΔ β ijk L jΔ β ijk +α 1exp(jMΔ β ˜ ijk ) 1exp(jΔ β ˜ ijk )
P NL,M = 1 2 k=N/2 N/2 j=N/2 N/2 P g,M = 1 2 k=N/2 N/2 j=N/2 N/2 3 2 P i P j P k γ 2 e αL η 1 η 2 η 1 = | 1 e αL e jΔ β ijk L jΔ β ijk +α | 2 1 β 2 2 ( 2π ) 4 Δ f 4 j 2 ( kj ) 2 + α 2 η 2 = | 1exp( jMΔ β ˜ ijk ) 1exp( jΔ β ˜ ijk ) | 2 = sin 2 ( Mj( kj )Δ f 2 ( 2π ) 2 | β 2 | 2 L 2 ζ 2 /2 ) sin 2 ( j( kj )Δ f 2 ( 2π ) 2 | β 2 | 2 L 2 ζ 2 /2 )
I NL,M = P NL,M Δf = 3 γ 2 I 3 4 β 2 2 ( 2π ) 4 B/2 f 1 B/2 f 1 B/2 B/2 η 1 ( f, f 1 ) η 2 ( f, f 1 ) dfd f 1 = 3 γ 2 I 3 Mln( B/ B 0 ) h e 8πα| β 2 | h e = 2( M1+ e αLζM M e αLζ ) e αLζ M ( e αLζ 1 ) 2 +1
c g,M ' =γ[ ( s k + s i ) s j +( s k + s j ) s i ( c k + c i ) c j ( c k + c j ) c i ] e αL /2 e j β g L                     1 e αL e jΔ β ijk L jΔ β ijk +α 1exp(jMΔ β ˜ ijk ) 1exp(jΔ β ˜ ijk )
P g,M ' = | c g,M ' | 2 = 9 2 P 2 P n γ 2 η 1 η 2
I noise,M = 9 γ 2 M I 2 I n ln( B/B 0 ) h e,M 8πα| β 2 | = 9 γ 2 M I 2 n 0 ln( B/B 0 ) h e,M 4πα| β 2 |
B 0 = α 2 π 2 B| β 2 | ,      h e,M = 2( M1+ e αLζM M e αLζ ) e αLζ M ( e αLζ 1 ) 2 +1
I NASE = M=1 N s I noise,M = M=1 N s 9M γ 2 I 2 n 0 ln( B/B 0 ) h e,M 4πα| β 2 |
I NASE = M=1 N s I noise,M = 9 γ 2 I 2 n 0 ln( B/B 0 ) N s ( N s +1 ) 8πα| β 2 |
I NASE = I 2 I 0,NASE ,  I 0,NASE = 4πα| β 2 | M=1 N s 9M γ 2 n 0 ln( B/B 0 ) h e,M , SNR= I 2 n 0 +( I 2 / I 0,NASE ) I NASE opt = 2 n 0 I 0,NASE , SNR NASE max = I 0,NASE / 8 n 0 ,  S NASE max =2 log 2 ( 1+ I 0,NASE / 8 n 0 )
c g,M ' =γ[ ( c k p+ c i p ) c j p +( c k p+ c j p ) c i p ] e αL / 2i β g L 1 e αLjΔ β ijk L jΔ β ijk +α
c g,M ' =γ[ ( c k p+ c i p ) c j p +( c k p+ c j p ) c i p ( c k + c i ) c j ( c k + c j ) c i ] e αL /2 e j β g L 1 e αL e jΔ β ijk L jΔ β ijk +α
E{ | ( c k p+ c i p ) c j p +( c k p+ c j p ) c i p ( c k + c i ) c j ( c k + c j ) c i | 2 } = 1 4 P 3 [ 129exp( Δ τ m 2 ¯ Δ ϖ 2 8 )3exp( 7 Δ τ m 2 ¯ Δ ϖ 2 24 ) ] =3 P 3 [ 1 3 4 exp( 3 π 3 LM D p 2 f 1 2 16 ) 1 4 exp( 7 π 3 LM D p 2 f 1 2 16 ) ]
P g,M ' =E{ | c g,M ' | 2 }= γ 2 ( Δ β ijk ) 2 + α 2 E{ | ( c k p+ c i p ) c j p +( c k p+ c j p ) c i p ( c k + c i ) c j ( c k + c j ) c i | 2 } = 3 P 3 γ 2 ( Δ β ijk ) 2 + α 2 [ 1 3 4 exp( 3 π 3 LM D p 2 f 1 2 16 ) 1 4 exp( 7LM D p 2 f 1 2 16 ) ]= P g,M 0 R PMD R PMD =2[ 1 3 4 exp( 3 π 3 LM D p 2 f 1 2 16 ) 1 4 exp( 7 π 3 LM D p 2 f 1 2 16 ) ]
I PMD M = 3 γ 2 β 2 2 ( 2π ) 4 B 0 /2 B/2 0 R PMD 1 ( f 1 f ) 2 + f W 4 dfd f 1    = 3 γ 2 I 3 4πα β 2 B 0 /2 B/2 1 f 1 [ 1 3 4 exp( 3 π 3 LM D p 2 f 1 2 16 ) 1 4 exp( 7LM D p 2 f 1 2 16 ) ] d f 1    = 3 γ 2 I 3 4πα β 2 ln( B/ B 0 ) 9 γ 2 I 3 32πα β 2 [ Ei( 3 π 3 B 2 LM D p 2 64 )Ei( 3 π 3 B 0 LM D p 2 64 ) ]              3 γ 2 I 3 32πα β 2 [ Ei( 7 π 3 B 2 LM D p 2 64 )Ei( 7 π 3 B 2 LM D p 2 64 ) ]
I PMD = M=1/2 N s 1/2 I PMD M = 3 γ 2 I 3 32πα β 2 { 8 N s ln( B/ B 0 ) M=1/2 N s 1/2 [ 3 E 1 ( M )+ E 2 ( M ) ] } E 1 ( M )=Ei( 3 π 3 B 2 LM D p 2 64 )Ei( 3 π 3 B 0 LM D p 2 64 ) E 2 ( M )=Ei( 7 π 3 B 2 LM D p 2 64 )Ei( 7 π 3 B 0 LM D p 2 64 )
  I PMD = ( I/ I 0,PMD ) 2 I,  SNR= I 2 n 0 + ( I/ I 0,PMD ) 2 I ,  I PMD opt = ( n 0 I 0,PMD 2 ) 1/3  SNR PMD max = 1 3 ( I 0,PMD n 0 ) 2/3 ,  S PMD max =2 log 2 ( 1+ 1 3 ( I 0,PMD n 0 ) 2/3 ) I 0,PMD = 1 γ 32πα| β 2 | 24 N s ln( B/ B 0 ) M=1/2 N s 1/2 [ 9 E 1 ( M )+3 E 2 ( M ) ]   
I NL = I NASE + I PMD ,    SNR =  I 2 n 0 + I NL
| U x | 2 = | 2( c x k* + n x k* )( c x i + n x i )( s x j + n x j )+( c y k* + n y k* )( c y i + n y i )( c x j + n x j )                   +( c y k* + n y k* )( c y j + n y j )( c x i + n x i )( 2 c x k* c x i c x j + c y k* c y i c x j + c y k* c y j c x i ) | 2           = 9 4 P 2 P n + 9 4 P P n 2 + 9 4 P n 3 9 4 P 2 P n
| ( s k + s i ) s j +( s k + s j ) s i ( c k + c i ) c j ( c k + c j ) c i | 2 = | U x | 2 + | U y | 2 = 9 2 P 2 P n
c k(j)x p = a k(j) c k(j)x + b k(j) c k(j)y ,       c k(j)y p = b k(j) * c k(j)x + a k(j) * c k(j)y   
E{ | U x | 2 }=E{ | A | 2 }+E{ | B | 2 }2Re[ E{ A * B } ]
c ix p* c ix = c ix p c ix * = | c ix | 2 ,   c iy p* c iy = c iy p c iy * = | c iy | 2
E{ | A | 2 }=E( | B | 2 )= 4 | c kx | 2 | c ix | 2 | c jx | 2 + | c ky | 2 | c iy | 2 | c jx | 2 + | c ky | 2 | c jy | 2 | c ix | 2 = 3 4 P 3 E{ A * B }=E[ ( 2 c kx p* c ix p c jx p + c ky p* c iy p c jx p + c ky p* c jy p c ix p ) * ( 2 c kx p* c ix p c jx p + c ky p* c iy p c jx p + c ky p* c jy p c ix p ) ]   = 1 8 P 3 E{ 5 a k * a j +2 b k * b j +2 b j * b k + a k a j }
E{ | U x | 2 }= P 3 8 [ 129exp( Δ τ 2 ¯ Δ ϖ 2 8 )3exp( 7 Δ τ 2 ¯ Δ ϖ 2 24 ) ]
E{ | U | 2 }=E{ | U x | 2 + | U y | 2 }=2E{ | U x | 2 }          = 1 4 P 3 [ 129exp( Δ τ 2 ¯ Δ ϖ 2 8 )3exp( 7 Δ τ 2 ¯ Δ ϖ 2 24 ) ]

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