Abstract

High spectral efficiency modulation format based unrepeatered transmission systems using distributed Raman amplifier (DRA) have attracted much attention recently. To enhance the reach and optimize system performance, careful design of DRA is required based on the analysis of various types of impairments and their balance. In this paper, we study various pump RIN induced distortions on high spectral efficiency modulation formats. The vector theory of both 1st and higher-order stimulated Raman scattering (SRS) effect using Jones-matrix formalism is presented. The pump RIN will induce three types of distortion on high spectral efficiency signals: intensity noise stemming from SRS, phase noise stemming from cross phase modulation (XPM), and polarization crosstalk stemming from cross polarization modulation (XPolM). An analytical model for the statistical property of relative phase noise (RPN) in higher order DRA without dealing with complex vector theory is derived. The impact of pump RIN induced impairments are analyzed in polarization-multiplexed (PM)-QPSK and PM-16QAM-based unrepeatered systems simulations using 1st, 2nd and 3rd-order forward pumped Raman amplifier. It is shown that at realistic RIN levels, negligible impairments will be induced to PM-QPSK signals in 1st and 2nd order DRA, while non-negligible impairments will occur in 3rd order case. PM-16QAM signals suffer more penalties compared to PM-QPSK with the same on-off gain where both 2nd and 3rd order DRA will cause non-negligible performance degradations. We also investigate the performance of digital signal processing (DSP) algorithms to mitigate such impairments.

© 2015 Optical Society of America

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References

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  1. X. Zhou, L. E. Nelson, R. Isaac, P. Magill, B. Zhu, D. W. Peckham, P. Borel, and K. Carlson, “1200km Transmission of 50GHz spaced, 5×504-Gb/s PDM-32-64 hybrid QAM using Electrical and Optical Spectral Shaping,” in Tech. Digest of the Conference on Optical Fiber Communication (2012), paper OM2A.2.
  2. X. Zhou, L. E. Nelson, R. Isaac, P. D. Magill, B. Zhu, P. Borel, K. Carlson, and D. W. Peckham, “12000km Transmission of 100GHz spaced, 8×495-Gb/s PDM Time-Domain Hybrid QPSK-8QAM Signals,” in Tech. Digest of the Conference on Optical Fiber Communication (2013), paper OTu2B.4.
  3. L. E. Nelson, X. Zhou, B. Zhu, M. F. Yan, P. W. Wisk, and P. D. Magill, “All-Raman-amplified, 73 nm seamless band transmission of 9 Tb/s over 6000 km of fiber,” IEEE Photonics Technol. Lett. 26(3), 242–245 (2014).
    [Crossref]
  4. R.-J. Essiambre, P. Winzer, J. Bromage, and C. H. Kim, “Design of bidirectionally pumped fiber amplifiers generating double Rayleigh backscattering,” IEEE Photonics Technol. Lett. 14(7), 914–916 (2002).
    [Crossref]
  5. C. R. S. Fludger, V. Handerek, and R. J. Mears, “Pump to signal RIN transfer in Raman fiber amplifiers,” J. Lightwave Technol. 19(8), 1140–1148 (2001).
    [Crossref]
  6. Q. Lin and G. P. Agrawal, “Vector theory of stimulated Raman scattering and its application to fiber-based Raman amplifiers,” J. Opt. Soc. Am. B 20(8), 1616–1631 (2003).
    [Crossref]
  7. S. Sergeyev, S. Popov, and A. T. Friberg, “Modeling polarization-dependent gain in fiber Raman amplifiers with randomly varying birefringence,” Opt. Commun. 262(1), 114–119 (2006).
    [Crossref]
  8. E. S. Son, J. H. Lee, and Y. C. Chung, “Statistics of polarization-dependent gain in fiber Raman amplifiers,” J. Lightwave Technol. 23(3), 1219–1226 (2005).
    [Crossref]
  9. J. Bromage, “Raman amplification for fiber communication systems,” J. Lightwave Technol. 22(1), 79–93 (2004).
    [Crossref]
  10. M. D. Mermelstein, K. Brar, and C. Headley, “RIN transfer measurement and moldeling in dual-order Raman fiber amplifiers,” J. Lightwave Technol. 21(6), 1518–1523 (2003).
    [Crossref]
  11. S. Jiang and P. Gallion, “Theoretical analysis on the PMD-assisted pump-to-signal noise transfer in distributed fiber Raman amplifiers,” J. Lightwave Technol. 25(10), 3185–3192 (2007).
    [Crossref]
  12. J. Cheng, M. Tang, S. Fu, P. P. Shum, and D. Liu, “Relative phase noise induced impairment in M-ary phase-shift-keying coherent optical communication system using distributed fiber Raman amplifier,” Opt. Lett. 38(7), 1055–1057 (2013).
    [Crossref] [PubMed]
  13. J. Cheng, M. Tang, S. Fu, P. P. Shum, D. Liu, M. Xiang, Z. Feng, and D. Yu, “Relative phase noise estimation and mitigation in Raman amplified coherent optical communication system,” Opt. Express 22(2), 1257–1266 (2014).
    [Crossref] [PubMed]
  14. J. Cheng, M. Tang, S. Fu, P. P. Shum, D. Yu, L. Wang, and D. Liu, “Relative phase noise-induced phase error and system impairment in pump depletion/nondepletion regime,” J. Lightwave Technol. 32(12), 2277–2286 (2014).
    [Crossref]
  15. J. Wu, J. Cheng, M. Tang, L. Deng, F. Songnian, P. P. Shum, and D. Liu, “Relative phase noise induced impairment in CO-OFDM optical communication system with distributed fiber Raman amplifier,” Opt. Lett. 39(10), 2841–2844 (2014).
    [Crossref] [PubMed]
  16. T. J. Xia, D. L. Peterson, G. A. Wellbrock, D. Chang, P. Perrier, H. Fevrier, S. Ten, C. Tower, and G. Mills, “557-km Unrepeatered 100G Transmission with Commercial Raman DWDM System, Enhanced ROPA, and Cabled Large Aeff Ultra-Low Loss Fiber in OSP Environment,” in Tech. Digest of the Conference on Optical Fiber Communication (2014), paper Th5A.7.
    [Crossref]
  17. B. Zhu, P. Borel, K. Carlson, X. Jiang, D. W. Peckham, and R. J. Lingle, “Unrepeatered transmission of 3.2-Tb/s (32×120-Gb/s) over 445-km fiber link,” IEEE Photon. Technol. Lett. 25(19), 1863–1866 (2013).
    [Crossref]
  18. H. Bissessur, P. Bousselet, D. A. Mongardien, and I. Brylski, “Ultra-long 10 Gb/s Unrepeatered WDM Transmission up to 601 km,” in Tech. Digest of the Conference on Optical Fiber Communication (2010), paper OTuD.6.
    [Crossref]
  19. M. Winter, C.-A. Bunge, D. Setti, and K. Petermann, “A statistical treatment of cross-polarization modulation in DWDM systems,” J. Lightwave Technol. 27(17), 3739–3751 (2009).
    [Crossref]
  20. G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic, 2000).
  21. Q. Lin and G. P. Agrawal, “Effects of polarization-mode dispersion on cross-phase modulation in dispersion-managed wavelength-division-multiplexed systems,” J. Lightwave Technol. 22(4), 977–987 (2004).
    [Crossref]
  22. D. Chang, H. D. Pedro, P. Perrier, H. Fevrier, S. Ten, C. Towery, I. Davis, and S. Makovejs, “150 x 120 Gb/s Unrepeatered Transmission over 333.6 km and 389.6 km (with ROPA) G.652 Fiber,” in Proc. European Conference on Optical Communications (2014), paper Tu.1.5.4.
  23. H. Bissessur, C. Bastide, S. Dubost, S. Etinne, and D. Mongardien, “8 Tb/s unrepeatered transmission of real-time processed 200 Gb/s PDM 16-QAM over 363 km,” in Proc. European Conference on Optical Communications (2014), paper Tu.1.5.3.
    [Crossref]
  24. J. D. Downie, J. Hurley, I. Roudas, D. Pikula, and J. A. Garza-Alanis, “Unrepeatered 256 Gb/s PM-16QAM transmission over up to 304 km with simple system configurations,” Opt. Express 22(9), 10256–10261 (2014).
    [Crossref] [PubMed]
  25. A. Bononi, P. Serena, N. Rossi, and D. Sperti, “Which is the Dominant Nonlinearity in Long-haul PDM-QPSKCoherent Transmissions,” in Proc. European Conference on Optical Communications (2010), paper Th.10.E.1.
  26. L. Li, Z. Tao, L. Liu, W. Yan, S. Oda, T. Hoshida, and J. C. Rasmussen, “Nonlinear Polarization Crosstalk Canceller for Dual-Polarization Digital Coherent Receivers,” in Tech. Digest of the Conference on Optical Fiber Communication (2010), paper OWE3.
    [Crossref]

2014 (5)

2013 (2)

B. Zhu, P. Borel, K. Carlson, X. Jiang, D. W. Peckham, and R. J. Lingle, “Unrepeatered transmission of 3.2-Tb/s (32×120-Gb/s) over 445-km fiber link,” IEEE Photon. Technol. Lett. 25(19), 1863–1866 (2013).
[Crossref]

J. Cheng, M. Tang, S. Fu, P. P. Shum, and D. Liu, “Relative phase noise induced impairment in M-ary phase-shift-keying coherent optical communication system using distributed fiber Raman amplifier,” Opt. Lett. 38(7), 1055–1057 (2013).
[Crossref] [PubMed]

2009 (1)

2007 (1)

2006 (1)

S. Sergeyev, S. Popov, and A. T. Friberg, “Modeling polarization-dependent gain in fiber Raman amplifiers with randomly varying birefringence,” Opt. Commun. 262(1), 114–119 (2006).
[Crossref]

2005 (1)

2004 (2)

2003 (2)

2002 (1)

R.-J. Essiambre, P. Winzer, J. Bromage, and C. H. Kim, “Design of bidirectionally pumped fiber amplifiers generating double Rayleigh backscattering,” IEEE Photonics Technol. Lett. 14(7), 914–916 (2002).
[Crossref]

2001 (1)

Agrawal, G. P.

Borel, P.

B. Zhu, P. Borel, K. Carlson, X. Jiang, D. W. Peckham, and R. J. Lingle, “Unrepeatered transmission of 3.2-Tb/s (32×120-Gb/s) over 445-km fiber link,” IEEE Photon. Technol. Lett. 25(19), 1863–1866 (2013).
[Crossref]

Brar, K.

Bromage, J.

J. Bromage, “Raman amplification for fiber communication systems,” J. Lightwave Technol. 22(1), 79–93 (2004).
[Crossref]

R.-J. Essiambre, P. Winzer, J. Bromage, and C. H. Kim, “Design of bidirectionally pumped fiber amplifiers generating double Rayleigh backscattering,” IEEE Photonics Technol. Lett. 14(7), 914–916 (2002).
[Crossref]

Bunge, C.-A.

Carlson, K.

B. Zhu, P. Borel, K. Carlson, X. Jiang, D. W. Peckham, and R. J. Lingle, “Unrepeatered transmission of 3.2-Tb/s (32×120-Gb/s) over 445-km fiber link,” IEEE Photon. Technol. Lett. 25(19), 1863–1866 (2013).
[Crossref]

Cheng, J.

Chung, Y. C.

Deng, L.

Downie, J. D.

Essiambre, R.-J.

R.-J. Essiambre, P. Winzer, J. Bromage, and C. H. Kim, “Design of bidirectionally pumped fiber amplifiers generating double Rayleigh backscattering,” IEEE Photonics Technol. Lett. 14(7), 914–916 (2002).
[Crossref]

Feng, Z.

Fludger, C. R. S.

Friberg, A. T.

S. Sergeyev, S. Popov, and A. T. Friberg, “Modeling polarization-dependent gain in fiber Raman amplifiers with randomly varying birefringence,” Opt. Commun. 262(1), 114–119 (2006).
[Crossref]

Fu, S.

Gallion, P.

Garza-Alanis, J. A.

Handerek, V.

Headley, C.

Hurley, J.

Jiang, S.

Jiang, X.

B. Zhu, P. Borel, K. Carlson, X. Jiang, D. W. Peckham, and R. J. Lingle, “Unrepeatered transmission of 3.2-Tb/s (32×120-Gb/s) over 445-km fiber link,” IEEE Photon. Technol. Lett. 25(19), 1863–1866 (2013).
[Crossref]

Kim, C. H.

R.-J. Essiambre, P. Winzer, J. Bromage, and C. H. Kim, “Design of bidirectionally pumped fiber amplifiers generating double Rayleigh backscattering,” IEEE Photonics Technol. Lett. 14(7), 914–916 (2002).
[Crossref]

Lee, J. H.

Lin, Q.

Lingle, R. J.

B. Zhu, P. Borel, K. Carlson, X. Jiang, D. W. Peckham, and R. J. Lingle, “Unrepeatered transmission of 3.2-Tb/s (32×120-Gb/s) over 445-km fiber link,” IEEE Photon. Technol. Lett. 25(19), 1863–1866 (2013).
[Crossref]

Liu, D.

Magill, P. D.

L. E. Nelson, X. Zhou, B. Zhu, M. F. Yan, P. W. Wisk, and P. D. Magill, “All-Raman-amplified, 73 nm seamless band transmission of 9 Tb/s over 6000 km of fiber,” IEEE Photonics Technol. Lett. 26(3), 242–245 (2014).
[Crossref]

Mears, R. J.

Mermelstein, M. D.

Nelson, L. E.

L. E. Nelson, X. Zhou, B. Zhu, M. F. Yan, P. W. Wisk, and P. D. Magill, “All-Raman-amplified, 73 nm seamless band transmission of 9 Tb/s over 6000 km of fiber,” IEEE Photonics Technol. Lett. 26(3), 242–245 (2014).
[Crossref]

Peckham, D. W.

B. Zhu, P. Borel, K. Carlson, X. Jiang, D. W. Peckham, and R. J. Lingle, “Unrepeatered transmission of 3.2-Tb/s (32×120-Gb/s) over 445-km fiber link,” IEEE Photon. Technol. Lett. 25(19), 1863–1866 (2013).
[Crossref]

Petermann, K.

Pikula, D.

Popov, S.

S. Sergeyev, S. Popov, and A. T. Friberg, “Modeling polarization-dependent gain in fiber Raman amplifiers with randomly varying birefringence,” Opt. Commun. 262(1), 114–119 (2006).
[Crossref]

Roudas, I.

Sergeyev, S.

S. Sergeyev, S. Popov, and A. T. Friberg, “Modeling polarization-dependent gain in fiber Raman amplifiers with randomly varying birefringence,” Opt. Commun. 262(1), 114–119 (2006).
[Crossref]

Setti, D.

Shum, P. P.

Son, E. S.

Songnian, F.

Tang, M.

Wang, L.

Winter, M.

Winzer, P.

R.-J. Essiambre, P. Winzer, J. Bromage, and C. H. Kim, “Design of bidirectionally pumped fiber amplifiers generating double Rayleigh backscattering,” IEEE Photonics Technol. Lett. 14(7), 914–916 (2002).
[Crossref]

Wisk, P. W.

L. E. Nelson, X. Zhou, B. Zhu, M. F. Yan, P. W. Wisk, and P. D. Magill, “All-Raman-amplified, 73 nm seamless band transmission of 9 Tb/s over 6000 km of fiber,” IEEE Photonics Technol. Lett. 26(3), 242–245 (2014).
[Crossref]

Wu, J.

Xiang, M.

Yan, M. F.

L. E. Nelson, X. Zhou, B. Zhu, M. F. Yan, P. W. Wisk, and P. D. Magill, “All-Raman-amplified, 73 nm seamless band transmission of 9 Tb/s over 6000 km of fiber,” IEEE Photonics Technol. Lett. 26(3), 242–245 (2014).
[Crossref]

Yu, D.

Zhou, X.

L. E. Nelson, X. Zhou, B. Zhu, M. F. Yan, P. W. Wisk, and P. D. Magill, “All-Raman-amplified, 73 nm seamless band transmission of 9 Tb/s over 6000 km of fiber,” IEEE Photonics Technol. Lett. 26(3), 242–245 (2014).
[Crossref]

Zhu, B.

L. E. Nelson, X. Zhou, B. Zhu, M. F. Yan, P. W. Wisk, and P. D. Magill, “All-Raman-amplified, 73 nm seamless band transmission of 9 Tb/s over 6000 km of fiber,” IEEE Photonics Technol. Lett. 26(3), 242–245 (2014).
[Crossref]

B. Zhu, P. Borel, K. Carlson, X. Jiang, D. W. Peckham, and R. J. Lingle, “Unrepeatered transmission of 3.2-Tb/s (32×120-Gb/s) over 445-km fiber link,” IEEE Photon. Technol. Lett. 25(19), 1863–1866 (2013).
[Crossref]

IEEE Photon. Technol. Lett. (1)

B. Zhu, P. Borel, K. Carlson, X. Jiang, D. W. Peckham, and R. J. Lingle, “Unrepeatered transmission of 3.2-Tb/s (32×120-Gb/s) over 445-km fiber link,” IEEE Photon. Technol. Lett. 25(19), 1863–1866 (2013).
[Crossref]

IEEE Photonics Technol. Lett. (2)

L. E. Nelson, X. Zhou, B. Zhu, M. F. Yan, P. W. Wisk, and P. D. Magill, “All-Raman-amplified, 73 nm seamless band transmission of 9 Tb/s over 6000 km of fiber,” IEEE Photonics Technol. Lett. 26(3), 242–245 (2014).
[Crossref]

R.-J. Essiambre, P. Winzer, J. Bromage, and C. H. Kim, “Design of bidirectionally pumped fiber amplifiers generating double Rayleigh backscattering,” IEEE Photonics Technol. Lett. 14(7), 914–916 (2002).
[Crossref]

J. Lightwave Technol. (8)

C. R. S. Fludger, V. Handerek, and R. J. Mears, “Pump to signal RIN transfer in Raman fiber amplifiers,” J. Lightwave Technol. 19(8), 1140–1148 (2001).
[Crossref]

M. D. Mermelstein, K. Brar, and C. Headley, “RIN transfer measurement and moldeling in dual-order Raman fiber amplifiers,” J. Lightwave Technol. 21(6), 1518–1523 (2003).
[Crossref]

J. Bromage, “Raman amplification for fiber communication systems,” J. Lightwave Technol. 22(1), 79–93 (2004).
[Crossref]

Q. Lin and G. P. Agrawal, “Effects of polarization-mode dispersion on cross-phase modulation in dispersion-managed wavelength-division-multiplexed systems,” J. Lightwave Technol. 22(4), 977–987 (2004).
[Crossref]

E. S. Son, J. H. Lee, and Y. C. Chung, “Statistics of polarization-dependent gain in fiber Raman amplifiers,” J. Lightwave Technol. 23(3), 1219–1226 (2005).
[Crossref]

S. Jiang and P. Gallion, “Theoretical analysis on the PMD-assisted pump-to-signal noise transfer in distributed fiber Raman amplifiers,” J. Lightwave Technol. 25(10), 3185–3192 (2007).
[Crossref]

M. Winter, C.-A. Bunge, D. Setti, and K. Petermann, “A statistical treatment of cross-polarization modulation in DWDM systems,” J. Lightwave Technol. 27(17), 3739–3751 (2009).
[Crossref]

J. Cheng, M. Tang, S. Fu, P. P. Shum, D. Yu, L. Wang, and D. Liu, “Relative phase noise-induced phase error and system impairment in pump depletion/nondepletion regime,” J. Lightwave Technol. 32(12), 2277–2286 (2014).
[Crossref]

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

Opt. Commun. (1)

S. Sergeyev, S. Popov, and A. T. Friberg, “Modeling polarization-dependent gain in fiber Raman amplifiers with randomly varying birefringence,” Opt. Commun. 262(1), 114–119 (2006).
[Crossref]

Opt. Express (2)

Opt. Lett. (2)

Other (9)

X. Zhou, L. E. Nelson, R. Isaac, P. Magill, B. Zhu, D. W. Peckham, P. Borel, and K. Carlson, “1200km Transmission of 50GHz spaced, 5×504-Gb/s PDM-32-64 hybrid QAM using Electrical and Optical Spectral Shaping,” in Tech. Digest of the Conference on Optical Fiber Communication (2012), paper OM2A.2.

X. Zhou, L. E. Nelson, R. Isaac, P. D. Magill, B. Zhu, P. Borel, K. Carlson, and D. W. Peckham, “12000km Transmission of 100GHz spaced, 8×495-Gb/s PDM Time-Domain Hybrid QPSK-8QAM Signals,” in Tech. Digest of the Conference on Optical Fiber Communication (2013), paper OTu2B.4.

T. J. Xia, D. L. Peterson, G. A. Wellbrock, D. Chang, P. Perrier, H. Fevrier, S. Ten, C. Tower, and G. Mills, “557-km Unrepeatered 100G Transmission with Commercial Raman DWDM System, Enhanced ROPA, and Cabled Large Aeff Ultra-Low Loss Fiber in OSP Environment,” in Tech. Digest of the Conference on Optical Fiber Communication (2014), paper Th5A.7.
[Crossref]

H. Bissessur, P. Bousselet, D. A. Mongardien, and I. Brylski, “Ultra-long 10 Gb/s Unrepeatered WDM Transmission up to 601 km,” in Tech. Digest of the Conference on Optical Fiber Communication (2010), paper OTuD.6.
[Crossref]

G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic, 2000).

D. Chang, H. D. Pedro, P. Perrier, H. Fevrier, S. Ten, C. Towery, I. Davis, and S. Makovejs, “150 x 120 Gb/s Unrepeatered Transmission over 333.6 km and 389.6 km (with ROPA) G.652 Fiber,” in Proc. European Conference on Optical Communications (2014), paper Tu.1.5.4.

H. Bissessur, C. Bastide, S. Dubost, S. Etinne, and D. Mongardien, “8 Tb/s unrepeatered transmission of real-time processed 200 Gb/s PDM 16-QAM over 363 km,” in Proc. European Conference on Optical Communications (2014), paper Tu.1.5.3.
[Crossref]

A. Bononi, P. Serena, N. Rossi, and D. Sperti, “Which is the Dominant Nonlinearity in Long-haul PDM-QPSKCoherent Transmissions,” in Proc. European Conference on Optical Communications (2010), paper Th.10.E.1.

L. Li, Z. Tao, L. Liu, W. Yan, S. Oda, T. Hoshida, and J. C. Rasmussen, “Nonlinear Polarization Crosstalk Canceller for Dual-Polarization Digital Coherent Receivers,” in Tech. Digest of the Conference on Optical Fiber Communication (2010), paper OWE3.
[Crossref]

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Figures (9)

Fig. 1
Fig. 1

ACF of RPN for (a) 1st order, (b) 2nd order, and (c) 3rd order DRA, where the RIN of 1st order pump in 1st order DRA, 2nd order pump in 2nd order DRA and 3rd order pump in 3rd order DRA is at −110 dB/Hz, and the RIN of the intermediate pumps in higher order DRA is at −140 dB/Hz.

Fig. 2
Fig. 2

ACF of RPolN for (a) 1st order, (b) 2nd order, and (c) 3rd order DRA, where the RIN of 1st order pump in 1st order DRA, 2nd order pump in 2nd order DRA and 3rd order pump in 3rd order DRA is at −110 dB/Hz, and the RIN of the intermediate pumps in higher order DRA is at −140 dB/Hz.

Fig. 3
Fig. 3

(a) simulation set-up, and (b) DSP algorithm block diagram.

Fig. 4
Fig. 4

Power evolution for (a) 1st order DRA, (b) 2nd order DRA and (c) 3rd order DRA, (d) the calculated variance of RPN in 1st, 2nd and 3rd order DRA.

Fig. 5
Fig. 5

The SNR penalty for PM-QPSK at BER = 10−3 for 1st, 2nd, and 3rd order DRA.

Fig. 6
Fig. 6

SNR penalties for PM-16QAM at BER = 10−3 for 1st, 2nd, and 3rd order DRA.

Fig. 7
Fig. 7

The constellation diagram of x-polarization QPSK signal with (a) RIN only, (b) RIN and RPN, (c) RIN, RPN and RPolN. The 2nd order forward pump with −105 dB/Hz is used.

Fig. 8
Fig. 8

SNR penalty for PM-16QAM in (a) 1st order, (b) 2nd order, and (c) 3rd order DRA.

Fig. 9
Fig. 9

The SNR penalty with and without using mitigation algorithms for (a) PM-QPSK and (b) PM-16QAM.

Tables (1)

Tables Icon

Table 1 Fiber parameters

Equations (31)

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

σ 1 =( 1 0 0 -1 ) σ 2 =( 0 1 1 0 ) σ 3 =( 0 -i i 0 )
d| A p dz = α p 2 | A p i 2 B p σ| A p + i γ p 3 [ 2 A p | A p + κ a κ b | A p * A p * | ]| A p + 2i γ p 3 [ ( 1+ δ a ) A s | A s +( 1+ δ b )| A s A s |+( κ a κ b + δ b )| A s * A s * | ]| A p 1 2 [ g 2 A s | A s + g 1 | A s A s |+ g 2 | A s * A s * | ]| A p
d| A s dz = α s 2 | A s i 2 B s σ| A s + i γ s 3 [ 2 A s | A s + κ a κ b | A s * A s * | ]| A s + 2i γ p 3 [ ( 1+ δ a ) A p | A p +( 1+ δ b )| A p A p |+( κ a κ b + δ b )| A p * A p * | ]| A s + 1 2 [ g 2 A p | A p + g 1 | A p A p |+ g 2 | A p * A p * | ]| A s
| A p z = α p 2 | A p i 2 B p σ| A p β 1p | A p t + i γ p 3 [ 2 A p | A p +| A p * A p * | ]| A p
| A s z = α s 2 | A s i 2 B s σ| A s β 1s | A s t i β 2s 2 2 | A s t 2 + i2 γ s 3 [ A p | A p +| A p A p |+| A p * A p * | ]| A s + g R | A p A p || A s
| A p z =δ β 1 | A p τ α p 2 | A p + i8 γ p 9 P p | A p
| A s z = α s 2 | A s i β 2s 2 2 | A s τ 2 i 2 Ωbσ| A s +( i4 γ s 3 + 1 2 g R ) P p | A s +( i4 γ s 9 + 1 2 g R ) P p P ^ p σ| A s
b( z 1 )b( z 2 ) = I ¯ 3 D p 2 exp( | z 1 z 2 | / L C ) 2 L C
| A p2 z =δ β 1 p2,s | A p2 τ α p2 2 | A p2 + i8 γ p2 9 P p2 | A p2 +( i4 γ p2 3 1 2 g R p2,p1 ) P p1 | A p2 +( i4 γ p2 9 1 2 g R p2,p1 ) P p1 P ^ p1 σ| A p2
| A p1 z =δ β 1 p1,s | A p1 τ α p1 2 | A p1 i 2 Ω p2,p1 bσ| A p1 + i8 γ p1 9 P p1 | A p1 +( i4 γ p1 3 + 1 2 g R p2,p1 ) P p2 | A p1 +( i4 γ p1 9 + 1 2 g R p2,p1 ) P p2 P ^ p2 σ| A p1
| A s z = α s 2 | A s i β 2s 2 2 | A s τ 2 i 2 Ω p2,s bσ| A s +( i4 γ s 3 + 1 2 g R p2,s ) P p2 | A s +( i4 γ s 9 + 1 2 g R p2,s ) P p2 P ^ p2 σ| A s +( i4 γ s 3 + 1 2 g R p1,s ) P p1 | A s +( i4 γ s 9 + 1 2 g R p1,s ) P p1 P ^ p1 σ| A s
| A p3 z =δ β 1 p3,s | A p3 τ α p3 2 | A p3 + i8 γ p3 9 P p3 | A p3 +( i4 γ p3 3 1 2 g R p3,p2 ) P p2 | A p3 +( i4 γ p3 9 1 2 g R p3,p2 ) P p2 P ^ p2 σ| A p3 +( i4 γ p3 3 1 2 g R p3,p1 ) P p1 | A p3 +( i4 γ p3 9 1 2 g R p3,p1 ) P p1 P ^ p1 σ| A p3
| A p2 z =δ β 1 p2,s | A p2 τ α p2 2 | A p2 i 2 Ω p3,p2 bσ| A p2 + i8 γ p2 9 P p2 | A p2 +( i4 γ p2 3 + 1 2 g R p3,p2 ) P p3 | A p2 +( i4 γ p2 9 + 1 2 g R p3,p2 ) P p3 P ^ p3 σ| A p2 +( i4 γ p2 3 1 2 g R p2,p1 ) P p1 | A p2 +( i4 γ p2 9 1 2 g R p2,p1 ) P p1 P ^ p1 σ| A p2
| A p1 z =δ β 1 p1,s | A p1 τ α p1 2 | A p1 i 2 Ω p3,p1 bσ| A p1 + i8 γ p1 9 P p1 | A p1 +( i4 γ p1 3 + 1 2 g R p3,p1 ) P p3 | A p1 +( i4 γ p1 9 + 1 2 g R p3,p1 ) P p3 P ^ p3 σ| A p1 +( i4 γ p1 3 + 1 2 g R p2,p1 ) P p2 | A p1 +( i4 γ p1 9 + 1 2 g R p2,p1 ) P p2 P ^ p2 σ| A p1
| A s z = α s 2 | A s i β 2s 2 2 | A s τ 2 i 2 Ω p3,s bσ| A s +( i4 γ s 3 + 1 2 g R p3,s ) P p3 | A s +( i4 γ s 9 + 1 2 g R p3,s ) P p3 P ^ p3 σ| A s +( i4 γ s 3 + 1 2 g R p2,s ) P p2 | A s +( i4 γ s 9 + 1 2 g R p2,s ) P p2 P ^ p2 σ| A s +( i4 γ s 3 + 1 2 g R p1,s ) P p1 | A s +( i4 γ s 9 + 1 2 g R p1,s ) P p1 P ^ p1 σ| A s
θ( L )= 4 γ s 3 0 L ( P p3 + P p2 + P p1 )dz
P k ( z,t )= P ¯ k ( z )[ 1+ m k ( z )exp( iΩt ) ]
Δθ( L,Ω )= 4 γ s 3 0 L [ P ¯ p3 ( z ) m p3 ( z )+ P ¯ p2 ( z ) m p2 ( z )+ P ¯ p1 ( z ) m p1 ( z ) ]exp( iΩt )dz
PS D RPN ( L,Ω )= | Δθ( L,Ω ) | 2
σ RPN 2 = ν 1 ν 2 PS D RPN ( L,Ω ) dΩ
P p3 z =δ β 1 p3,s P p3 τ α p3 P p3 g R p3,p2 P p2 P p3 g R p3,p1 P p1 P p3
P p2 z =δ β 1 p2,s P p2 τ α p2 P p2 + g R p3,p2 P p3 P p2 g R p2,p1 P p1 P p2
P p1 z =δ β 1 p1,s P p1 τ α p1 P p1 + g R p3,p1 P p3 P p1 + g R p2,p1 P p2 P p1
d P ¯ p3 dz = α p3 P ¯ p3 g R p3,p2 P ¯ p2 P ¯ p3 g R p3,p1 P ¯ p1 P ¯ p3
d P ¯ p2 dz = α p2 P ¯ p2 + g R p3,p2 P ¯ p3 P ¯ p2 g R p2,p1 P ¯ p1 P ¯ p2
d P ¯ p1 dz = α p1 P ¯ p1 + g R p3,p1 P ¯ p3 P ¯ p1 + g R p2,p1 P ¯ p2 P ¯ p1
d m p3 dz +iΩδ β 1 p3,s m p3 = g R p3,p2 P ¯ p2 m p2 g R p3,p1 P ¯ p1 m p1
d m p2 dz +iΩδ β 1 p2,s m p2 = g R p3,p2 P ¯ p3 m p3 g R p2,p1 P ¯ p1 m p1
d m p1 dz +iΩδ β 1 p1,s m p1 = g R p3,p1 P ¯ p3 m p3 + g R p2,p1 P ¯ p2 m p2
[ r x r y ]=[ 1 w xy 2 w xy w yx 1 w yx 2 ][ s x s y ]
w xy = r x - s x s y , w yx = r y - s y s x

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