Abstract

We theoretically study optical transmission characteristics of wavelength-division multiplexed (WDM) and polarization-multiplexed (POLMUX) signals using high-order optical quadrature-amplitude-modulation (QAM) formats up to 256. First, we conduct intensive computer simulations on bit-error rates (BERs) in WDM POLMUX QAM transmission systems and find maximum transmission distances under the influence of nonlinear impairments. Next, to elucidate the physics behind such nonlinear transmission characteristics, we calculate the distribution of constellation points for QAM signals as functions of the the launched power, the transmission distance, and the symbol rate. These results lead to a closed-form formula for BER of any QAM formats. From such formula, we derive simple laws that determine the maximum transmission distance and the optimum power as functions of the QAM order and the symbol rate. These laws can well explain the simulation results.

© 2011 OSA

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  1. A. H. Gnauck, P. J. Winzer, S. Chandrasekhar, X. Liu, B. Zhu, and D. W. Peckham, “10×224-Gb/s WDM transmission of 28-Gbaud PDM 16-QAM on a 50-GHz grid transmission over 1,200 km of fiber,” in 2010 OSA Technical Digest of Optical Fiber Communication Conference (Optical Society of America, 2010), PDPB8.
  2. A. Sano, T. Kobayashi, K. Ishihara, H. Masuda, S. Yamamoto, K. Mori, E. Yamazaki, E. Yoshida, Y. Miyamoto, T. Yamada, and H. Yamazaki, “240-Gb/s polarization-multiplexed 64-QAM modulation and blind detection using PLC-LN hybrid integrated modulator and digital coherent receiver,” in Proceedings of European Conference on Optical Communication (Sept.2009), PD2.2.
  3. M. Nakazawa, S. Okamoto, T. Omiya, K. Kasai, and M. Yoshida, “256 QAM (64 Gbit/s) coherent optical transmission over 160 km with an optical bandwidth of 5.4 GHz,” in 2010 OSA Technical Digest of Optical Fiber Communication Conference (Optical Society of America, 2010), OMJ5.
  4. D.-S. Ly-Gagnon, S. Tsukamoto, K. Katoh, and K. Kikuchi, “Coherent detection of optical quadrature phase-shift keying signals with carrier phase estimation,” J. Lightwave Technol. 24, 12–21 (2006).
    [CrossRef]
  5. H. Sun, K.-T. Wu, and K. Roberts, “Real-time measurements of a 40 Gb/s coherent system,” Opt. Express 16, 873–879 (2008).
    [CrossRef] [PubMed]
  6. K. Kikuchi, “Ultra long-haul optical transmission characteristics of wavelength-division multiplexed dual-polarisation 16-quadrature-amplitude-modulation signals,” Electron. Lett. 46, 433–434 (2010).
    [CrossRef]
  7. S. G. Evangelides, L. F. Mollenauer, J. P. Gordon, and N. S. Bergano, “Polarization multiplexing with solitons,” J. Lightwave Technol. 10, 28–35 (1992).
    [CrossRef]
  8. T. Kato, M. Hirano, M. Onishi, and M. Nishimura, “Ultra-low nonlinearity low-loss pure silica core fibre for long-haul WDM transmission,” Electron. Lett. 35, 1615–1617 (1999).
    [CrossRef]
  9. M. Seimetz, “Laser linewidth limitations for optical systems with high order modulation employing feed forward digital carrier phase estimation,” in 2008 OSA Technical Digest of Optical Fiber Communication Conference (Optical Society of America, 2008), OTuM2.
  10. J. P. Gordon and L. F. Mollenauer, “Phase noise in photonic communications systems using linear amplifiers,” Opt. Lett. 15, 1351–1353 (1990).
    [CrossRef] [PubMed]
  11. E. Ip and J. M. Kahn, “Compensation of dispersion and nonlinear impairments using digital backpropagation,” J. Lightwave Technol. 26, 3416–3425 (2008).
    [CrossRef]
  12. G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 1989), Chap. 5.
  13. P. Poggiolini, A. Carena, V. Curri, G. Bosco, and F. Forghieri, “Analytical modeling of nonlinear propagation in uncompensated optical transmission links,” IEEE Photon. Technol. Lett. 23, 742–744 (2011).
    [CrossRef]
  14. X. Chen and W. Shieh, “Closed-form expressions for nonlinear transmission performance of densely spaced coherent OFDM systems,” Opt. Express 18, 19039–19054 (2010).
    [CrossRef] [PubMed]

2011

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

2010

K. Kikuchi, “Ultra long-haul optical transmission characteristics of wavelength-division multiplexed dual-polarisation 16-quadrature-amplitude-modulation signals,” Electron. Lett. 46, 433–434 (2010).
[CrossRef]

A. H. Gnauck, P. J. Winzer, S. Chandrasekhar, X. Liu, B. Zhu, and D. W. Peckham, “10×224-Gb/s WDM transmission of 28-Gbaud PDM 16-QAM on a 50-GHz grid transmission over 1,200 km of fiber,” in 2010 OSA Technical Digest of Optical Fiber Communication Conference (Optical Society of America, 2010), PDPB8.

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

2008

2006

1999

T. Kato, M. Hirano, M. Onishi, and M. Nishimura, “Ultra-low nonlinearity low-loss pure silica core fibre for long-haul WDM transmission,” Electron. Lett. 35, 1615–1617 (1999).
[CrossRef]

1992

S. G. Evangelides, L. F. Mollenauer, J. P. Gordon, and N. S. Bergano, “Polarization multiplexing with solitons,” J. Lightwave Technol. 10, 28–35 (1992).
[CrossRef]

1990

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 1989), Chap. 5.

Bergano, N. S.

S. G. Evangelides, L. F. Mollenauer, J. P. Gordon, and N. S. Bergano, “Polarization multiplexing with solitons,” J. Lightwave Technol. 10, 28–35 (1992).
[CrossRef]

Bosco, G.

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

Carena, A.

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

Chandrasekhar, S.

A. H. Gnauck, P. J. Winzer, S. Chandrasekhar, X. Liu, B. Zhu, and D. W. Peckham, “10×224-Gb/s WDM transmission of 28-Gbaud PDM 16-QAM on a 50-GHz grid transmission over 1,200 km of fiber,” in 2010 OSA Technical Digest of Optical Fiber Communication Conference (Optical Society of America, 2010), PDPB8.

Chen, X.

Curri, V.

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

Evangelides, S. G.

S. G. Evangelides, L. F. Mollenauer, J. P. Gordon, and N. S. Bergano, “Polarization multiplexing with solitons,” J. Lightwave Technol. 10, 28–35 (1992).
[CrossRef]

Forghieri, F.

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

Gnauck, A. H.

A. H. Gnauck, P. J. Winzer, S. Chandrasekhar, X. Liu, B. Zhu, and D. W. Peckham, “10×224-Gb/s WDM transmission of 28-Gbaud PDM 16-QAM on a 50-GHz grid transmission over 1,200 km of fiber,” in 2010 OSA Technical Digest of Optical Fiber Communication Conference (Optical Society of America, 2010), PDPB8.

Gordon, J. P.

S. G. Evangelides, L. F. Mollenauer, J. P. Gordon, and N. S. Bergano, “Polarization multiplexing with solitons,” J. Lightwave Technol. 10, 28–35 (1992).
[CrossRef]

J. P. Gordon and L. F. Mollenauer, “Phase noise in photonic communications systems using linear amplifiers,” Opt. Lett. 15, 1351–1353 (1990).
[CrossRef] [PubMed]

Hirano, M.

T. Kato, M. Hirano, M. Onishi, and M. Nishimura, “Ultra-low nonlinearity low-loss pure silica core fibre for long-haul WDM transmission,” Electron. Lett. 35, 1615–1617 (1999).
[CrossRef]

Ip, E.

Ishihara, K.

A. Sano, T. Kobayashi, K. Ishihara, H. Masuda, S. Yamamoto, K. Mori, E. Yamazaki, E. Yoshida, Y. Miyamoto, T. Yamada, and H. Yamazaki, “240-Gb/s polarization-multiplexed 64-QAM modulation and blind detection using PLC-LN hybrid integrated modulator and digital coherent receiver,” in Proceedings of European Conference on Optical Communication (Sept.2009), PD2.2.

Kahn, J. M.

Kasai, K.

M. Nakazawa, S. Okamoto, T. Omiya, K. Kasai, and M. Yoshida, “256 QAM (64 Gbit/s) coherent optical transmission over 160 km with an optical bandwidth of 5.4 GHz,” in 2010 OSA Technical Digest of Optical Fiber Communication Conference (Optical Society of America, 2010), OMJ5.

Kato, T.

T. Kato, M. Hirano, M. Onishi, and M. Nishimura, “Ultra-low nonlinearity low-loss pure silica core fibre for long-haul WDM transmission,” Electron. Lett. 35, 1615–1617 (1999).
[CrossRef]

Katoh, K.

Kikuchi, K.

K. Kikuchi, “Ultra long-haul optical transmission characteristics of wavelength-division multiplexed dual-polarisation 16-quadrature-amplitude-modulation signals,” Electron. Lett. 46, 433–434 (2010).
[CrossRef]

D.-S. Ly-Gagnon, S. Tsukamoto, K. Katoh, and K. Kikuchi, “Coherent detection of optical quadrature phase-shift keying signals with carrier phase estimation,” J. Lightwave Technol. 24, 12–21 (2006).
[CrossRef]

Kobayashi, T.

A. Sano, T. Kobayashi, K. Ishihara, H. Masuda, S. Yamamoto, K. Mori, E. Yamazaki, E. Yoshida, Y. Miyamoto, T. Yamada, and H. Yamazaki, “240-Gb/s polarization-multiplexed 64-QAM modulation and blind detection using PLC-LN hybrid integrated modulator and digital coherent receiver,” in Proceedings of European Conference on Optical Communication (Sept.2009), PD2.2.

Liu, X.

A. H. Gnauck, P. J. Winzer, S. Chandrasekhar, X. Liu, B. Zhu, and D. W. Peckham, “10×224-Gb/s WDM transmission of 28-Gbaud PDM 16-QAM on a 50-GHz grid transmission over 1,200 km of fiber,” in 2010 OSA Technical Digest of Optical Fiber Communication Conference (Optical Society of America, 2010), PDPB8.

Ly-Gagnon, D.-S.

Masuda, H.

A. Sano, T. Kobayashi, K. Ishihara, H. Masuda, S. Yamamoto, K. Mori, E. Yamazaki, E. Yoshida, Y. Miyamoto, T. Yamada, and H. Yamazaki, “240-Gb/s polarization-multiplexed 64-QAM modulation and blind detection using PLC-LN hybrid integrated modulator and digital coherent receiver,” in Proceedings of European Conference on Optical Communication (Sept.2009), PD2.2.

Miyamoto, Y.

A. Sano, T. Kobayashi, K. Ishihara, H. Masuda, S. Yamamoto, K. Mori, E. Yamazaki, E. Yoshida, Y. Miyamoto, T. Yamada, and H. Yamazaki, “240-Gb/s polarization-multiplexed 64-QAM modulation and blind detection using PLC-LN hybrid integrated modulator and digital coherent receiver,” in Proceedings of European Conference on Optical Communication (Sept.2009), PD2.2.

Mollenauer, L. F.

S. G. Evangelides, L. F. Mollenauer, J. P. Gordon, and N. S. Bergano, “Polarization multiplexing with solitons,” J. Lightwave Technol. 10, 28–35 (1992).
[CrossRef]

J. P. Gordon and L. F. Mollenauer, “Phase noise in photonic communications systems using linear amplifiers,” Opt. Lett. 15, 1351–1353 (1990).
[CrossRef] [PubMed]

Mori, K.

A. Sano, T. Kobayashi, K. Ishihara, H. Masuda, S. Yamamoto, K. Mori, E. Yamazaki, E. Yoshida, Y. Miyamoto, T. Yamada, and H. Yamazaki, “240-Gb/s polarization-multiplexed 64-QAM modulation and blind detection using PLC-LN hybrid integrated modulator and digital coherent receiver,” in Proceedings of European Conference on Optical Communication (Sept.2009), PD2.2.

Nakazawa, M.

M. Nakazawa, S. Okamoto, T. Omiya, K. Kasai, and M. Yoshida, “256 QAM (64 Gbit/s) coherent optical transmission over 160 km with an optical bandwidth of 5.4 GHz,” in 2010 OSA Technical Digest of Optical Fiber Communication Conference (Optical Society of America, 2010), OMJ5.

Nishimura, M.

T. Kato, M. Hirano, M. Onishi, and M. Nishimura, “Ultra-low nonlinearity low-loss pure silica core fibre for long-haul WDM transmission,” Electron. Lett. 35, 1615–1617 (1999).
[CrossRef]

Okamoto, S.

M. Nakazawa, S. Okamoto, T. Omiya, K. Kasai, and M. Yoshida, “256 QAM (64 Gbit/s) coherent optical transmission over 160 km with an optical bandwidth of 5.4 GHz,” in 2010 OSA Technical Digest of Optical Fiber Communication Conference (Optical Society of America, 2010), OMJ5.

Omiya, T.

M. Nakazawa, S. Okamoto, T. Omiya, K. Kasai, and M. Yoshida, “256 QAM (64 Gbit/s) coherent optical transmission over 160 km with an optical bandwidth of 5.4 GHz,” in 2010 OSA Technical Digest of Optical Fiber Communication Conference (Optical Society of America, 2010), OMJ5.

Onishi, M.

T. Kato, M. Hirano, M. Onishi, and M. Nishimura, “Ultra-low nonlinearity low-loss pure silica core fibre for long-haul WDM transmission,” Electron. Lett. 35, 1615–1617 (1999).
[CrossRef]

Peckham, D. W.

A. H. Gnauck, P. J. Winzer, S. Chandrasekhar, X. Liu, B. Zhu, and D. W. Peckham, “10×224-Gb/s WDM transmission of 28-Gbaud PDM 16-QAM on a 50-GHz grid transmission over 1,200 km of fiber,” in 2010 OSA Technical Digest of Optical Fiber Communication Conference (Optical Society of America, 2010), PDPB8.

Poggiolini, P.

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

Roberts, K.

Sano, A.

A. Sano, T. Kobayashi, K. Ishihara, H. Masuda, S. Yamamoto, K. Mori, E. Yamazaki, E. Yoshida, Y. Miyamoto, T. Yamada, and H. Yamazaki, “240-Gb/s polarization-multiplexed 64-QAM modulation and blind detection using PLC-LN hybrid integrated modulator and digital coherent receiver,” in Proceedings of European Conference on Optical Communication (Sept.2009), PD2.2.

Seimetz, M.

M. Seimetz, “Laser linewidth limitations for optical systems with high order modulation employing feed forward digital carrier phase estimation,” in 2008 OSA Technical Digest of Optical Fiber Communication Conference (Optical Society of America, 2008), OTuM2.

Shieh, W.

Sun, H.

Tsukamoto, S.

Winzer, P. J.

A. H. Gnauck, P. J. Winzer, S. Chandrasekhar, X. Liu, B. Zhu, and D. W. Peckham, “10×224-Gb/s WDM transmission of 28-Gbaud PDM 16-QAM on a 50-GHz grid transmission over 1,200 km of fiber,” in 2010 OSA Technical Digest of Optical Fiber Communication Conference (Optical Society of America, 2010), PDPB8.

Wu, K.-T.

Yamada, T.

A. Sano, T. Kobayashi, K. Ishihara, H. Masuda, S. Yamamoto, K. Mori, E. Yamazaki, E. Yoshida, Y. Miyamoto, T. Yamada, and H. Yamazaki, “240-Gb/s polarization-multiplexed 64-QAM modulation and blind detection using PLC-LN hybrid integrated modulator and digital coherent receiver,” in Proceedings of European Conference on Optical Communication (Sept.2009), PD2.2.

Yamamoto, S.

A. Sano, T. Kobayashi, K. Ishihara, H. Masuda, S. Yamamoto, K. Mori, E. Yamazaki, E. Yoshida, Y. Miyamoto, T. Yamada, and H. Yamazaki, “240-Gb/s polarization-multiplexed 64-QAM modulation and blind detection using PLC-LN hybrid integrated modulator and digital coherent receiver,” in Proceedings of European Conference on Optical Communication (Sept.2009), PD2.2.

Yamazaki, E.

A. Sano, T. Kobayashi, K. Ishihara, H. Masuda, S. Yamamoto, K. Mori, E. Yamazaki, E. Yoshida, Y. Miyamoto, T. Yamada, and H. Yamazaki, “240-Gb/s polarization-multiplexed 64-QAM modulation and blind detection using PLC-LN hybrid integrated modulator and digital coherent receiver,” in Proceedings of European Conference on Optical Communication (Sept.2009), PD2.2.

Yamazaki, H.

A. Sano, T. Kobayashi, K. Ishihara, H. Masuda, S. Yamamoto, K. Mori, E. Yamazaki, E. Yoshida, Y. Miyamoto, T. Yamada, and H. Yamazaki, “240-Gb/s polarization-multiplexed 64-QAM modulation and blind detection using PLC-LN hybrid integrated modulator and digital coherent receiver,” in Proceedings of European Conference on Optical Communication (Sept.2009), PD2.2.

Yoshida, E.

A. Sano, T. Kobayashi, K. Ishihara, H. Masuda, S. Yamamoto, K. Mori, E. Yamazaki, E. Yoshida, Y. Miyamoto, T. Yamada, and H. Yamazaki, “240-Gb/s polarization-multiplexed 64-QAM modulation and blind detection using PLC-LN hybrid integrated modulator and digital coherent receiver,” in Proceedings of European Conference on Optical Communication (Sept.2009), PD2.2.

Yoshida, M.

M. Nakazawa, S. Okamoto, T. Omiya, K. Kasai, and M. Yoshida, “256 QAM (64 Gbit/s) coherent optical transmission over 160 km with an optical bandwidth of 5.4 GHz,” in 2010 OSA Technical Digest of Optical Fiber Communication Conference (Optical Society of America, 2010), OMJ5.

Zhu, B.

A. H. Gnauck, P. J. Winzer, S. Chandrasekhar, X. Liu, B. Zhu, and D. W. Peckham, “10×224-Gb/s WDM transmission of 28-Gbaud PDM 16-QAM on a 50-GHz grid transmission over 1,200 km of fiber,” in 2010 OSA Technical Digest of Optical Fiber Communication Conference (Optical Society of America, 2010), PDPB8.

2010 OSA Technical Digest of Optical Fiber Communication Conference

A. H. Gnauck, P. J. Winzer, S. Chandrasekhar, X. Liu, B. Zhu, and D. W. Peckham, “10×224-Gb/s WDM transmission of 28-Gbaud PDM 16-QAM on a 50-GHz grid transmission over 1,200 km of fiber,” in 2010 OSA Technical Digest of Optical Fiber Communication Conference (Optical Society of America, 2010), PDPB8.

Electron. Lett.

K. Kikuchi, “Ultra long-haul optical transmission characteristics of wavelength-division multiplexed dual-polarisation 16-quadrature-amplitude-modulation signals,” Electron. Lett. 46, 433–434 (2010).
[CrossRef]

T. Kato, M. Hirano, M. Onishi, and M. Nishimura, “Ultra-low nonlinearity low-loss pure silica core fibre for long-haul WDM transmission,” Electron. Lett. 35, 1615–1617 (1999).
[CrossRef]

IEEE Photon. Technol. Lett.

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

J. Lightwave Technol.

Opt. Express

Opt. Lett.

Other

M. Seimetz, “Laser linewidth limitations for optical systems with high order modulation employing feed forward digital carrier phase estimation,” in 2008 OSA Technical Digest of Optical Fiber Communication Conference (Optical Society of America, 2008), OTuM2.

G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 1989), Chap. 5.

A. Sano, T. Kobayashi, K. Ishihara, H. Masuda, S. Yamamoto, K. Mori, E. Yamazaki, E. Yoshida, Y. Miyamoto, T. Yamada, and H. Yamazaki, “240-Gb/s polarization-multiplexed 64-QAM modulation and blind detection using PLC-LN hybrid integrated modulator and digital coherent receiver,” in Proceedings of European Conference on Optical Communication (Sept.2009), PD2.2.

M. Nakazawa, S. Okamoto, T. Omiya, K. Kasai, and M. Yoshida, “256 QAM (64 Gbit/s) coherent optical transmission over 160 km with an optical bandwidth of 5.4 GHz,” in 2010 OSA Technical Digest of Optical Fiber Communication Conference (Optical Society of America, 2010), OMJ5.

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Figures (11)

Fig. 1
Fig. 1

Simulation model of QAM transmission and digital coherent detection.

Fig. 2
Fig. 2

BER characteristics of QAM transmission systems. (a): 4QAM, (b): 16QAM, (c): 64QAM, and (d): 256QAM. Green, red, black, and blue curves represent BERs when we transmit a single channel, co-polarized three WDM channels, co-polarized five WDM channels, and POLMUX five WDM channels, respectively. The symbol rate is 12.5 Gsymbol/s, and the WDM channel spacing is 25 GHz. Numbers of spans n corresponding to (a)–(d) are 160, 37, 10, and 3, respectively.

Fig. 3
Fig. 3

Maximum number of spans n as a function of the order of QAM m. The solid line shows the relation of nm −1.

Fig. 4
Fig. 4

Maximum number of spans n for 4, 16, 64, and 256QAM as functions of the symbol rate. POLMUX five WDM channels are transmitted, and the WDM channel spacing is twice as large as the symbol rate. Solid lines represent the slope of nB −2/3.

Fig. 5
Fig. 5

Optimum power levels Popt for 4QAM transmission as a function of the symbol rate. POLMUX five WDM channels are transmitted through 160 spans, and the WDM channel spacing is twice as large as the symbol rate. The solid line represents the slope of Popt B 1/3.

Fig. 6
Fig. 6

Distributions of constellation points of a 4QAM signal. The complex amplitude is normalized such that the mean square of its absolute value is unity. σ nor 2 stands for the variance of the distribution.

Fig. 7
Fig. 7

Distributions of the real part of the received complex amplitude. (a): Pave =−14 dBm, (b): Pave = −9 dBm, and (c): Pave =−4 dBm. Black curves show simulation results, and red ones are Gaussian fits. Five-channel WDM and POLMUX signals are transmitted. The symbol rate is 12.5 Gsymbol/s, the WDM channel spacing 25 GHz, and the number of spans 160.

Fig. 8
Fig. 8

Variance σ nor 2 of the Gaussian distribution of the constellation points as a function of the launched power. The blue curve is obtained when γ = 0 and ASE is included, whereas the red curve is obtained when ASE is neglected and fiber nonlinearity is included. The black curve is calculated when both of ASE and fiber nonlinearity are taken into account.

Fig. 9
Fig. 9

Variance σ nor 2 of the Gaussian distribution of the constellation points as a function of the number of spans. The launched power Pave is −7 dBm. The red curve and the black curve are obtained when symbol rates are 100 Gsymbol/s and 12.5 Gsymbol/s, respectively. Blue lines are linear fits to these curves.

Fig. 10
Fig. 10

Variance σ nor 2 of the Gaussian distribution of the constellation points as a function of the symbol rate. The launched power Pave is −7 dBm, and the number of spans 100.

Fig. 11
Fig. 11

BER curves calculated from the simple BER formula Eq. (8). The symbol rate is 12.5 Gsymbol/s. The black, red, blue, and green curves are BER curves for 4QAM, 16QAM, 64QAM, and 256QAM, respectively. The symbol rate is 12.5 Gsymbol/s and the WDM channel spacing is 25 GHz. Five-channel WDM and POLMUX signals are transmitted, and numbers of spans are the same as those used in Sec. 3.

Equations (15)

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

E x z = α 2 E x i 2 β 2 2 E x T 2 + 8 i 9 γ ( | E x | 2 + | E y | 2 ) E x ,
E y z = α 2 E y i 2 β 2 2 E y T 2 + 8 i 9 γ ( | E x | 2 + | E y | 2 ) E y ,
σ nor 2 = C p ( γ P ave ) 2 ( Ln ) ,
BER D e erfc ( δ 2 2 σ 2 ) ,
P ave ( m ) = 1 3 ( m 1 ) δ 2 .
σ 2 = h f n ( G 1 ) n s p B ,
σ n 2 = 2 P ave σ nor 2 = 2 γ 2 L n C p P ave 3 ,
BER D e erfc ( 1 ( m 1 ) n B C P ave + ( m 1 ) n P ave 2 C n ) ,
C = 3 2 h f ( G 1 ) n s p ,
C n = 3 4 L γ 2 C p .
P ave = ( C n 2 C ) 1 / 3 B 1 / 3 ,
BER min = D e erfc ( C n 1 / 3 ( 2 C ) 2 / 3 3 ( m 1 ) n B 2 / 3 ) .
( m 1 ) n B 2 / 3 = C ,
n B 2 / 3 ( m 1 ) 1 B 2 / 3 m 1 .
P o p t B 1 / 3 .

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