Abstract

Mode-dependent loss (MDL) is a major factor limiting the achievable information rate in multiple-input multiple-output space-division multiplexed systems. In this paper we show that its impact on system performance, which we quantify in terms of the capacity reduction relative to a reference MDL-free system, may depend strongly on the operation of the inline optical amplifiers. This dependency is particularly strong in low mode-count systems. In addition, we discuss ways in which the signal-to-noise ratio of the MDL-free reference system can be defined and quantify the differences in the predicted capacity loss. Finally, we stress the importance of correctly accounting for the effect of MDL on the accumulation of amplification noise.

© 2015 Optical Society of America

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References

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  1. P. J. Winzer and G. J. Foschini, “MIMO capacities and outage probabilities in spatially multiplexed optical transport systems,” Opt. Express 19, 16680–16696 (2011).
    [Crossref] [PubMed]
  2. K-P. Ho and J. M. Kahn, “Mode-dependent loss and gain: statistics and effect on mode-division multiplexing,” Opt. Express 19, 16612–16635 (2011).
    [Crossref] [PubMed]
  3. S. Warm and K. Petermann, “Splice loss requirements in multi-mode fiber mode-division-multiplex transmission links,” Opt. Express 21, 519–532 (2013).
    [Crossref] [PubMed]
  4. K-P. Ho and J. M. Kahn, “Frequency Diversity in Mode-Division Multiplexing Systems,” IEEE J. Lightwave Technol. 29, 3719–3726 (2011).
    [Crossref]
  5. A. Andrusier, M. Shtaif, C. Antonelli, and A. Mecozzi, “Assessing the effects of mode-dependent loss in space-division multiplexed systems,” IEEE J. Lightwave Technol. 32, 1317–1322 (2014).
    [Crossref]
  6. A. Juarez, E. Krune, S. Warm, C. A. Bunge, and K. Petermann, “Modeling of mode coupling in multimode fibers with respect to bandwidth and loss,” IEEE J. Lightwave Technol. 32, 1549–1558 (2014).
    [Crossref]
  7. A. Andrusier, M. Shtaif, C. Antonelli, and A. Mecozzi, “Characterization of mode-dependent loss in SDM systems,” Proc. OFC, Paper Th1J.2 (2014).
  8. A. Lobato, F. Ferreira, M. Kuschnerov, D. van den Borne, S. L. Jansen, A. Napoli, B. Spinnler, and B. Lankl, “Impact of mode coupling on the mode-dependent loss tolerance in few-mode fiber transmission,” Opt. Express 20, 29776–29783 (2012).
    [Crossref]
  9. K. Guan, P. J. Winzer, and M. Shtaif, “On the BER performance of MIMO-SDM systems with finite constellation inputs,” IEEE Photon. Technol. Letters 23, 1223–1226 (2014).
  10. A. Mecozzi and M. Shtaif, “Signal-to-noise-ratio degradation caused by polarization-dependent loss and the effect of dynamic gain equalization,” IEEE J. Lightwave Technol. 22, 1856– 1871 (2004).
    [Crossref]
  11. M. Shtaif, “Performance degradation in coherent polarization multiplexed systems as a result of polarization dependent loss,” Opt. Express 16, 13918–13932 (2008).
    [Crossref] [PubMed]
  12. J. M. Wiesenfeld, L. D. Garrett, M. Shtaif, M. H. Eiselt, and R. W. Tkach, “Effects of DGE channel bandwidth on nonlinear ULH systems,” OFC 2005, Paper OWA2.
  13. R. N. Mahalati, D. Askarov, and J. M. Kahn, “Adaptive modal gain equalization techniques in multi-mode erbium-doped fiber amplifiers,” J. of Lightwave Technol. 32, 2133 –2143 (2014).
    [Crossref]
  14. A. Lobato, F. Ferreira, B. Inan, and S. Adhikari, “Maximum-likelihood detection in few-mode fiber transmission with mode-dependent loss,” IEEE Photon. Technol. Lett. 25, 1095–1098 (2013).
    [Crossref]
  15. C. Antonelli, A. Mecozzi, M. Shtaif, and P. J. Winzer, “Stokes-space analysis of modal dispersion in fibers with multiple mode transmission,” Opt. Express 20, 11718–11783 (2012).
    [Crossref] [PubMed]
  16. K-P. Ho and J.M Kahn, “Statistics of group delays in multi-mode fibers with strong mode coupling,” IEEE J. Lightwave Technol. 29, 3119–3128 (2011).
    [Crossref]
  17. Q. Hu and W. Shieh, “Autocorrelation function of channel matrix in few-mode fibers with strong mode coupling,” Opt. Express 21, 22153–22165 (2013).
    [Crossref] [PubMed]
  18. The limit of large SNR implies that the capacity C = log2 [det (I + S0TT†Q−1)], where T is the channel transfer matrix and Q is the noise coherency matrix, can be approximated as log2 [det (S0TT†Q−1)] [5]. In this limit the MDL-induced capacity loss is independent of the SNR parameter S0.
  19. J.P. Gordon and H. Kogelnik, “PMD fundamentals: polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. USA 97, 4541–4550 (2000).
    [Crossref] [PubMed]
  20. L. E. Nelson, C. Antonelli, A. Mecozzi, M. Birk, P. Magill, A. Schex, and L. Rapp, “Statistics of polarization dependent loss in an installed long-haul WDM system,” Opt. Express 19, 6790–6796 (2011)
    [Crossref] [PubMed]
  21. P. J. Winzer and G. J. Foschini, “Optical MIMO-SDM system capacities,” OFC 2014, Paper Th1J.1.
  22. C.W. Gardiner, Handbook of Stochastic Methods: for Physics, Chemistry and Natural Sciences (Springer-Verlag, 1983).
    [Crossref]

2014 (4)

A. Andrusier, M. Shtaif, C. Antonelli, and A. Mecozzi, “Assessing the effects of mode-dependent loss in space-division multiplexed systems,” IEEE J. Lightwave Technol. 32, 1317–1322 (2014).
[Crossref]

A. Juarez, E. Krune, S. Warm, C. A. Bunge, and K. Petermann, “Modeling of mode coupling in multimode fibers with respect to bandwidth and loss,” IEEE J. Lightwave Technol. 32, 1549–1558 (2014).
[Crossref]

K. Guan, P. J. Winzer, and M. Shtaif, “On the BER performance of MIMO-SDM systems with finite constellation inputs,” IEEE Photon. Technol. Letters 23, 1223–1226 (2014).

R. N. Mahalati, D. Askarov, and J. M. Kahn, “Adaptive modal gain equalization techniques in multi-mode erbium-doped fiber amplifiers,” J. of Lightwave Technol. 32, 2133 –2143 (2014).
[Crossref]

2013 (3)

2012 (2)

2011 (5)

2008 (1)

2004 (1)

A. Mecozzi and M. Shtaif, “Signal-to-noise-ratio degradation caused by polarization-dependent loss and the effect of dynamic gain equalization,” IEEE J. Lightwave Technol. 22, 1856– 1871 (2004).
[Crossref]

2000 (1)

J.P. Gordon and H. Kogelnik, “PMD fundamentals: polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. USA 97, 4541–4550 (2000).
[Crossref] [PubMed]

Adhikari, S.

A. Lobato, F. Ferreira, B. Inan, and S. Adhikari, “Maximum-likelihood detection in few-mode fiber transmission with mode-dependent loss,” IEEE Photon. Technol. Lett. 25, 1095–1098 (2013).
[Crossref]

Andrusier, A.

A. Andrusier, M. Shtaif, C. Antonelli, and A. Mecozzi, “Assessing the effects of mode-dependent loss in space-division multiplexed systems,” IEEE J. Lightwave Technol. 32, 1317–1322 (2014).
[Crossref]

A. Andrusier, M. Shtaif, C. Antonelli, and A. Mecozzi, “Characterization of mode-dependent loss in SDM systems,” Proc. OFC, Paper Th1J.2 (2014).

Antonelli, C.

A. Andrusier, M. Shtaif, C. Antonelli, and A. Mecozzi, “Assessing the effects of mode-dependent loss in space-division multiplexed systems,” IEEE J. Lightwave Technol. 32, 1317–1322 (2014).
[Crossref]

C. Antonelli, A. Mecozzi, M. Shtaif, and P. J. Winzer, “Stokes-space analysis of modal dispersion in fibers with multiple mode transmission,” Opt. Express 20, 11718–11783 (2012).
[Crossref] [PubMed]

L. E. Nelson, C. Antonelli, A. Mecozzi, M. Birk, P. Magill, A. Schex, and L. Rapp, “Statistics of polarization dependent loss in an installed long-haul WDM system,” Opt. Express 19, 6790–6796 (2011)
[Crossref] [PubMed]

A. Andrusier, M. Shtaif, C. Antonelli, and A. Mecozzi, “Characterization of mode-dependent loss in SDM systems,” Proc. OFC, Paper Th1J.2 (2014).

Askarov, D.

R. N. Mahalati, D. Askarov, and J. M. Kahn, “Adaptive modal gain equalization techniques in multi-mode erbium-doped fiber amplifiers,” J. of Lightwave Technol. 32, 2133 –2143 (2014).
[Crossref]

Birk, M.

Bunge, C. A.

A. Juarez, E. Krune, S. Warm, C. A. Bunge, and K. Petermann, “Modeling of mode coupling in multimode fibers with respect to bandwidth and loss,” IEEE J. Lightwave Technol. 32, 1549–1558 (2014).
[Crossref]

Eiselt, M. H.

J. M. Wiesenfeld, L. D. Garrett, M. Shtaif, M. H. Eiselt, and R. W. Tkach, “Effects of DGE channel bandwidth on nonlinear ULH systems,” OFC 2005, Paper OWA2.

Ferreira, F.

A. Lobato, F. Ferreira, B. Inan, and S. Adhikari, “Maximum-likelihood detection in few-mode fiber transmission with mode-dependent loss,” IEEE Photon. Technol. Lett. 25, 1095–1098 (2013).
[Crossref]

A. Lobato, F. Ferreira, M. Kuschnerov, D. van den Borne, S. L. Jansen, A. Napoli, B. Spinnler, and B. Lankl, “Impact of mode coupling on the mode-dependent loss tolerance in few-mode fiber transmission,” Opt. Express 20, 29776–29783 (2012).
[Crossref]

Foschini, G. J.

Gardiner, C.W.

C.W. Gardiner, Handbook of Stochastic Methods: for Physics, Chemistry and Natural Sciences (Springer-Verlag, 1983).
[Crossref]

Garrett, L. D.

J. M. Wiesenfeld, L. D. Garrett, M. Shtaif, M. H. Eiselt, and R. W. Tkach, “Effects of DGE channel bandwidth on nonlinear ULH systems,” OFC 2005, Paper OWA2.

Gordon, J.P.

J.P. Gordon and H. Kogelnik, “PMD fundamentals: polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. USA 97, 4541–4550 (2000).
[Crossref] [PubMed]

Guan, K.

K. Guan, P. J. Winzer, and M. Shtaif, “On the BER performance of MIMO-SDM systems with finite constellation inputs,” IEEE Photon. Technol. Letters 23, 1223–1226 (2014).

Ho, K-P.

K-P. Ho and J. M. Kahn, “Mode-dependent loss and gain: statistics and effect on mode-division multiplexing,” Opt. Express 19, 16612–16635 (2011).
[Crossref] [PubMed]

K-P. Ho and J. M. Kahn, “Frequency Diversity in Mode-Division Multiplexing Systems,” IEEE J. Lightwave Technol. 29, 3719–3726 (2011).
[Crossref]

K-P. Ho and J.M Kahn, “Statistics of group delays in multi-mode fibers with strong mode coupling,” IEEE J. Lightwave Technol. 29, 3119–3128 (2011).
[Crossref]

Hu, Q.

Inan, B.

A. Lobato, F. Ferreira, B. Inan, and S. Adhikari, “Maximum-likelihood detection in few-mode fiber transmission with mode-dependent loss,” IEEE Photon. Technol. Lett. 25, 1095–1098 (2013).
[Crossref]

Jansen, S. L.

Juarez, A.

A. Juarez, E. Krune, S. Warm, C. A. Bunge, and K. Petermann, “Modeling of mode coupling in multimode fibers with respect to bandwidth and loss,” IEEE J. Lightwave Technol. 32, 1549–1558 (2014).
[Crossref]

Kahn, J. M.

R. N. Mahalati, D. Askarov, and J. M. Kahn, “Adaptive modal gain equalization techniques in multi-mode erbium-doped fiber amplifiers,” J. of Lightwave Technol. 32, 2133 –2143 (2014).
[Crossref]

K-P. Ho and J. M. Kahn, “Frequency Diversity in Mode-Division Multiplexing Systems,” IEEE J. Lightwave Technol. 29, 3719–3726 (2011).
[Crossref]

K-P. Ho and J. M. Kahn, “Mode-dependent loss and gain: statistics and effect on mode-division multiplexing,” Opt. Express 19, 16612–16635 (2011).
[Crossref] [PubMed]

Kahn, J.M

K-P. Ho and J.M Kahn, “Statistics of group delays in multi-mode fibers with strong mode coupling,” IEEE J. Lightwave Technol. 29, 3119–3128 (2011).
[Crossref]

Kogelnik, H.

J.P. Gordon and H. Kogelnik, “PMD fundamentals: polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. USA 97, 4541–4550 (2000).
[Crossref] [PubMed]

Krune, E.

A. Juarez, E. Krune, S. Warm, C. A. Bunge, and K. Petermann, “Modeling of mode coupling in multimode fibers with respect to bandwidth and loss,” IEEE J. Lightwave Technol. 32, 1549–1558 (2014).
[Crossref]

Kuschnerov, M.

Lankl, B.

Lobato, A.

A. Lobato, F. Ferreira, B. Inan, and S. Adhikari, “Maximum-likelihood detection in few-mode fiber transmission with mode-dependent loss,” IEEE Photon. Technol. Lett. 25, 1095–1098 (2013).
[Crossref]

A. Lobato, F. Ferreira, M. Kuschnerov, D. van den Borne, S. L. Jansen, A. Napoli, B. Spinnler, and B. Lankl, “Impact of mode coupling on the mode-dependent loss tolerance in few-mode fiber transmission,” Opt. Express 20, 29776–29783 (2012).
[Crossref]

Magill, P.

Mahalati, R. N.

R. N. Mahalati, D. Askarov, and J. M. Kahn, “Adaptive modal gain equalization techniques in multi-mode erbium-doped fiber amplifiers,” J. of Lightwave Technol. 32, 2133 –2143 (2014).
[Crossref]

Mecozzi, A.

A. Andrusier, M. Shtaif, C. Antonelli, and A. Mecozzi, “Assessing the effects of mode-dependent loss in space-division multiplexed systems,” IEEE J. Lightwave Technol. 32, 1317–1322 (2014).
[Crossref]

C. Antonelli, A. Mecozzi, M. Shtaif, and P. J. Winzer, “Stokes-space analysis of modal dispersion in fibers with multiple mode transmission,” Opt. Express 20, 11718–11783 (2012).
[Crossref] [PubMed]

L. E. Nelson, C. Antonelli, A. Mecozzi, M. Birk, P. Magill, A. Schex, and L. Rapp, “Statistics of polarization dependent loss in an installed long-haul WDM system,” Opt. Express 19, 6790–6796 (2011)
[Crossref] [PubMed]

A. Mecozzi and M. Shtaif, “Signal-to-noise-ratio degradation caused by polarization-dependent loss and the effect of dynamic gain equalization,” IEEE J. Lightwave Technol. 22, 1856– 1871 (2004).
[Crossref]

A. Andrusier, M. Shtaif, C. Antonelli, and A. Mecozzi, “Characterization of mode-dependent loss in SDM systems,” Proc. OFC, Paper Th1J.2 (2014).

Napoli, A.

Nelson, L. E.

Petermann, K.

A. Juarez, E. Krune, S. Warm, C. A. Bunge, and K. Petermann, “Modeling of mode coupling in multimode fibers with respect to bandwidth and loss,” IEEE J. Lightwave Technol. 32, 1549–1558 (2014).
[Crossref]

S. Warm and K. Petermann, “Splice loss requirements in multi-mode fiber mode-division-multiplex transmission links,” Opt. Express 21, 519–532 (2013).
[Crossref] [PubMed]

Rapp, L.

Schex, A.

Shieh, W.

Shtaif, M.

K. Guan, P. J. Winzer, and M. Shtaif, “On the BER performance of MIMO-SDM systems with finite constellation inputs,” IEEE Photon. Technol. Letters 23, 1223–1226 (2014).

A. Andrusier, M. Shtaif, C. Antonelli, and A. Mecozzi, “Assessing the effects of mode-dependent loss in space-division multiplexed systems,” IEEE J. Lightwave Technol. 32, 1317–1322 (2014).
[Crossref]

C. Antonelli, A. Mecozzi, M. Shtaif, and P. J. Winzer, “Stokes-space analysis of modal dispersion in fibers with multiple mode transmission,” Opt. Express 20, 11718–11783 (2012).
[Crossref] [PubMed]

M. Shtaif, “Performance degradation in coherent polarization multiplexed systems as a result of polarization dependent loss,” Opt. Express 16, 13918–13932 (2008).
[Crossref] [PubMed]

A. Mecozzi and M. Shtaif, “Signal-to-noise-ratio degradation caused by polarization-dependent loss and the effect of dynamic gain equalization,” IEEE J. Lightwave Technol. 22, 1856– 1871 (2004).
[Crossref]

A. Andrusier, M. Shtaif, C. Antonelli, and A. Mecozzi, “Characterization of mode-dependent loss in SDM systems,” Proc. OFC, Paper Th1J.2 (2014).

J. M. Wiesenfeld, L. D. Garrett, M. Shtaif, M. H. Eiselt, and R. W. Tkach, “Effects of DGE channel bandwidth on nonlinear ULH systems,” OFC 2005, Paper OWA2.

Spinnler, B.

Tkach, R. W.

J. M. Wiesenfeld, L. D. Garrett, M. Shtaif, M. H. Eiselt, and R. W. Tkach, “Effects of DGE channel bandwidth on nonlinear ULH systems,” OFC 2005, Paper OWA2.

van den Borne, D.

Warm, S.

A. Juarez, E. Krune, S. Warm, C. A. Bunge, and K. Petermann, “Modeling of mode coupling in multimode fibers with respect to bandwidth and loss,” IEEE J. Lightwave Technol. 32, 1549–1558 (2014).
[Crossref]

S. Warm and K. Petermann, “Splice loss requirements in multi-mode fiber mode-division-multiplex transmission links,” Opt. Express 21, 519–532 (2013).
[Crossref] [PubMed]

Wiesenfeld, J. M.

J. M. Wiesenfeld, L. D. Garrett, M. Shtaif, M. H. Eiselt, and R. W. Tkach, “Effects of DGE channel bandwidth on nonlinear ULH systems,” OFC 2005, Paper OWA2.

Winzer, P. J.

K. Guan, P. J. Winzer, and M. Shtaif, “On the BER performance of MIMO-SDM systems with finite constellation inputs,” IEEE Photon. Technol. Letters 23, 1223–1226 (2014).

C. Antonelli, A. Mecozzi, M. Shtaif, and P. J. Winzer, “Stokes-space analysis of modal dispersion in fibers with multiple mode transmission,” Opt. Express 20, 11718–11783 (2012).
[Crossref] [PubMed]

P. J. Winzer and G. J. Foschini, “MIMO capacities and outage probabilities in spatially multiplexed optical transport systems,” Opt. Express 19, 16680–16696 (2011).
[Crossref] [PubMed]

P. J. Winzer and G. J. Foschini, “Optical MIMO-SDM system capacities,” OFC 2014, Paper Th1J.1.

IEEE J. Lightwave Technol. (5)

K-P. Ho and J. M. Kahn, “Frequency Diversity in Mode-Division Multiplexing Systems,” IEEE J. Lightwave Technol. 29, 3719–3726 (2011).
[Crossref]

A. Andrusier, M. Shtaif, C. Antonelli, and A. Mecozzi, “Assessing the effects of mode-dependent loss in space-division multiplexed systems,” IEEE J. Lightwave Technol. 32, 1317–1322 (2014).
[Crossref]

A. Juarez, E. Krune, S. Warm, C. A. Bunge, and K. Petermann, “Modeling of mode coupling in multimode fibers with respect to bandwidth and loss,” IEEE J. Lightwave Technol. 32, 1549–1558 (2014).
[Crossref]

A. Mecozzi and M. Shtaif, “Signal-to-noise-ratio degradation caused by polarization-dependent loss and the effect of dynamic gain equalization,” IEEE J. Lightwave Technol. 22, 1856– 1871 (2004).
[Crossref]

K-P. Ho and J.M Kahn, “Statistics of group delays in multi-mode fibers with strong mode coupling,” IEEE J. Lightwave Technol. 29, 3119–3128 (2011).
[Crossref]

IEEE Photon. Technol. Lett. (1)

A. Lobato, F. Ferreira, B. Inan, and S. Adhikari, “Maximum-likelihood detection in few-mode fiber transmission with mode-dependent loss,” IEEE Photon. Technol. Lett. 25, 1095–1098 (2013).
[Crossref]

IEEE Photon. Technol. Letters (1)

K. Guan, P. J. Winzer, and M. Shtaif, “On the BER performance of MIMO-SDM systems with finite constellation inputs,” IEEE Photon. Technol. Letters 23, 1223–1226 (2014).

J. of Lightwave Technol. (1)

R. N. Mahalati, D. Askarov, and J. M. Kahn, “Adaptive modal gain equalization techniques in multi-mode erbium-doped fiber amplifiers,” J. of Lightwave Technol. 32, 2133 –2143 (2014).
[Crossref]

Opt. Express (8)

M. Shtaif, “Performance degradation in coherent polarization multiplexed systems as a result of polarization dependent loss,” Opt. Express 16, 13918–13932 (2008).
[Crossref] [PubMed]

L. E. Nelson, C. Antonelli, A. Mecozzi, M. Birk, P. Magill, A. Schex, and L. Rapp, “Statistics of polarization dependent loss in an installed long-haul WDM system,” Opt. Express 19, 6790–6796 (2011)
[Crossref] [PubMed]

K-P. Ho and J. M. Kahn, “Mode-dependent loss and gain: statistics and effect on mode-division multiplexing,” Opt. Express 19, 16612–16635 (2011).
[Crossref] [PubMed]

P. J. Winzer and G. J. Foschini, “MIMO capacities and outage probabilities in spatially multiplexed optical transport systems,” Opt. Express 19, 16680–16696 (2011).
[Crossref] [PubMed]

C. Antonelli, A. Mecozzi, M. Shtaif, and P. J. Winzer, “Stokes-space analysis of modal dispersion in fibers with multiple mode transmission,” Opt. Express 20, 11718–11783 (2012).
[Crossref] [PubMed]

A. Lobato, F. Ferreira, M. Kuschnerov, D. van den Borne, S. L. Jansen, A. Napoli, B. Spinnler, and B. Lankl, “Impact of mode coupling on the mode-dependent loss tolerance in few-mode fiber transmission,” Opt. Express 20, 29776–29783 (2012).
[Crossref]

S. Warm and K. Petermann, “Splice loss requirements in multi-mode fiber mode-division-multiplex transmission links,” Opt. Express 21, 519–532 (2013).
[Crossref] [PubMed]

Q. Hu and W. Shieh, “Autocorrelation function of channel matrix in few-mode fibers with strong mode coupling,” Opt. Express 21, 22153–22165 (2013).
[Crossref] [PubMed]

Proc. Natl. Acad. Sci. USA (1)

J.P. Gordon and H. Kogelnik, “PMD fundamentals: polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. USA 97, 4541–4550 (2000).
[Crossref] [PubMed]

Other (5)

P. J. Winzer and G. J. Foschini, “Optical MIMO-SDM system capacities,” OFC 2014, Paper Th1J.1.

C.W. Gardiner, Handbook of Stochastic Methods: for Physics, Chemistry and Natural Sciences (Springer-Verlag, 1983).
[Crossref]

The limit of large SNR implies that the capacity C = log2 [det (I + S0TT†Q−1)], where T is the channel transfer matrix and Q is the noise coherency matrix, can be approximated as log2 [det (S0TT†Q−1)] [5]. In this limit the MDL-induced capacity loss is independent of the SNR parameter S0.

J. M. Wiesenfeld, L. D. Garrett, M. Shtaif, M. H. Eiselt, and R. W. Tkach, “Effects of DGE channel bandwidth on nonlinear ULH systems,” OFC 2005, Paper OWA2.

A. Andrusier, M. Shtaif, C. Antonelli, and A. Mecozzi, “Characterization of mode-dependent loss in SDM systems,” Proc. OFC, Paper Th1J.2 (2014).

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Figures (7)

Fig. 1
Fig. 1 Amplification schemes considered in the analysis. In Scheme 1 each amplifier restores the mode-averaged signal power over the entire band of frequencies. In Scheme 2 the mode-averaged signal power is restored separately for each sub-band. In Scheme 3 each amplifier restores the mode-averaged power of a sub-band to the value that it had at the input of the span. Scheme 4 operates in the same way as Scheme 3, except that it restores the mode-averaged signal power across the entire band of frequencies.
Fig. 2
Fig. 2 The left panel shows the PDF of the MDL-induced capacity loss per mode for a system consisting of 20 amplified spans where N = 10 spatial modes are transmitted and for a link MDL of 24.6 dB (corresponding to about 5.5 dB per span). The right panel shows the corresponding complementary cumulative distribution function (CCDF), namely the probability of the capacity loss per mode exceeding the value on the abscissa.
Fig. 3
Fig. 3 The left panel shows the average capacity loss per mode as a function of the average link MDL in the limit of large SNR for amplification schemes 1, 2, and 3 of Fig. 1. The right panel shows the corresponding values of the outage capacity loss per mode. The results are shown for the cases of N = 3 and N = 10 spatial modes.
Fig. 4
Fig. 4 The top left panel shows the average capacity loss per mode as a function of the average link MDL for SNR values of S0 = 10, 20, 30, and 40 dB. The top right panel shows the corresponding values of the average capacity ratio. The bottom left and right panels show the outage capacity loss per mode and the outage capacity ratio, respectively, versus the average link MDL for the same SNR values as in the top panels.
Fig. 5
Fig. 5 The left panel shows the ratio between the second and the third terms of Eq. (5) in the form of a scatter plot versus the average link MDL. See text for details. The right panel shows the ratio between Sref and S0. Amplification scheme 1 has been assumed in both plots, but the same behavior has been observed for all amplification schemes.
Fig. 6
Fig. 6 The left panel shows by thick curve the PDF of the capacity loss per mode for the same system as in Fig. 2, which is obtained by assuming amplification scheme 1 and by setting the reference SNR to S ref = S 0 γ 0 γ 0 . The thin curve is taken from Fig. 2, and is shown for comparison. The center panel shows the corresponding CCDFs. The right panel shows the outage capacity loss in the two cases for increasing values of the average link MDL.
Fig. 7
Fig. 7 The left panel shows by thick curve the PDF of the capacity loss per mode for the same system as in Fig. 2, which is obtained for amplification scheme 1 by assuming that spatially isotropic noise is loaded at the receiver, rather than at the inline amplifier sites as it should be. The thin curve is the corresponding PDF taken from Fig. 2. The right panel shows the ratio between the average capacity losses per mode obtained for the two noise models versus the average link MDL.

Tables (1)

Tables Icon

Table 1 Analytic description of the amplification schemes of Fig. 1.

Equations (41)

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= C ref C 2 N ,
= Γ 2 3 log ( 2 ) ,
σ 1 2 = 2 20 ( 4 N 2 1 ) .
σ 2 2 = [ 22 15 ( 5 α 0 L 6 ) α 0 2 L 2 ] σ 1 2 ,
C = 2 N log 2 ( S 0 ) + 2 N log 2 ( γ 0 γ 0 ) Δ C MA + log 2 { det [ ( I + Γ Λ ) ( I + Γ Λ ) 1 ] } Δ C MD ,
C = 2 N log 2 ( S 0 ) + 2 N ln ( 2 ) ( Δ γ 0 Δ γ 0 Γ 2 Γ 2 2 ) ,
C = log 2 [ det ( I + S 0 Q 1 / 2 T T Q 1 / 2 ) ] ,
T T = γ 0 ( I + Γ Λ ) , Q = γ 0 ( I + Γ Λ ) ,
C 2 N log 2 ( S 0 γ 0 γ 0 ) + Γ 2 Γ 2 2 ln ( 2 ) log 2 ( S 0 ) 1 2 log 2 ( γ 0 2 γ 2 γ 0 2 γ 2 ) ,
C 2 N = log 2 ( S 0 ) 1 2 log ( 2 ) ( γ 0 2 γ 2 γ 0 2 γ 2 1 ) .
= log 2 ( S ref ) C 2 N = log 2 ( γ 0 γ 0 ) + 1 2 log ( 2 ) ( γ 0 2 γ 2 γ 0 2 γ 2 1 ) .
d γ 0 d z = ( α 0 + α g ) γ 0 α γ , d γ d z = ( α 0 + α g ) γ γ 0 α .
d γ 0 = ( α 0 + α σ α 2 2 g ) γ 0 d z d W γ ,
d γ = ( α 0 + α σ α 2 2 g ) γ d z γ 0 d W ,
d γ 0 = ( α 0 g ) γ 0 d z d W γ , d γ = ( α 0 g ) γ d z γ 0 d W .
d ( γ 0 2 γ 2 ) d z = 2 ( α 0 g ) ( γ 0 2 γ 2 ) σ α 2 γ 0 2 .
γ 0 2 γ 2 = exp { 2 I ( 0 , z ) σ α 2 z } ,
d γ 0 = ( α 0 g ) γ 0 d z d W γ + g α 0 L d z , d γ = ( α 0 g ) γ d z γ 0 d W ,
γ 0 2 γ 2 = 2 α 0 L 0 L d z g ( z ) γ 0 ( z ) exp { 2 I ( z , z ) σ 2 ( z z ) } .
= log 2 ( γ 0 γ 0 ) + 1 2 ln ( 2 ) [ 2 α 0 L 0 L d z g ( z ) γ 0 ( z ) exp { 2 I ( 0 , z ) + σ 2 z } 1 ] ,
γ 0 = 0 L exp { I ( z , z ) } [ d W z γ ( z ) + g ( z ) α 0 L d z ] .
γ = 0 L exp { I ( z , z ) } γ 0 ( z ) d W z ,
γ = 1 L 0 z z d W z , γ 0 = z L + 1 L 0 z d W z 0 z z d W z 1 L 0 z I ( z , z ) d z 1 α 0 L I ( 0 , z ) ,
ln ( 2 ) = ln ( γ 0 γ 0 ) + σ α 2 L 3 + 1 L 2 0 L ( L z ) 0 z z d W z d W z + 1 L 0 L I ( 0 , z ) d z I ( 0 , L ) α 0 L ,
γ = 0 z d W z , γ 0 = 1 I ( 0 , z ) + 0 z 0 z d W z d W z .
ln ( 2 ) = σ α 2 L 3 + 1 L 2 0 L ( L z ) 0 z z d W z d W z .
= σ α 2 L 3 ln ( 2 ) .
σ 1 2 = [ 1 L 2 ln ( 2 ) 0 L ( L z ) 0 z z d W z d W z ] 2 .
[ 0 L ( L z ) 0 z [ z + f ( z ) ] d W z d W z ] 2 = σ α 4 4 N 2 1 0 L ( L z ) 2 [ z + f ( z ) ] 3 f 3 ( z ) 3 d z
σ 1 2 = σ α 4 L 2 ln 2 ( 2 ) 180 ( 4 N 2 1 ) .
d W z 1 , i d W z 2 , j = D 1 σ α 2 δ i , j δ ( z 1 z 2 ) d z 1 d z 2
d W z 1 d W z 2 d W z 3 d W z 4 = i , j d W z 1 , i d W z 2 , i d W z 3 , j d W z 4 , j = σ α 4 δ ( z 1 z 2 ) δ ( z 3 z 4 ) d z 1 d z 2 d z 3 d z 4 + D 1 σ α 4 δ ( z 1 z 3 ) δ ( z 2 z 4 ) d z 1 d z 2 d z 3 d z 4 + D 1 σ α 4 δ ( z 1 z 4 ) δ ( z 2 z 3 ) d z 1 d z 2 d z 3 d z 4 .
I ( 0 , z ) = 0 z d W z 0 z d W z .
ln ( 2 ) = σ 2 L 3 + 1 L 2 0 L ( L z ) 0 z [ z + L L α 0 ( L z ) ] d W z d W z
σ 2 2 = 22 σ α 4 L 2 ln 2 ( 2 ) 180 ( 4 N 2 1 ) 5 α 0 L 6 ln 2 ( 2 ) 12 ( 4 N 2 1 ) σ α 4 α 0 2
C ref = 2 N log 2 ( S 0 γ 0 γ 0 )
= Γ 2 Γ 2 2 ln ( 2 ) 1 2 ln ( 2 ) ( γ 2 γ 2 ) .
= 1 2 ln ( 2 ) ( 0 L 0 L d W z d W z 1 L 2 0 L 0 L z z d W z d W z ) ,
[ 0 L 0 L d W z d W z ] [ 0 L 0 L z z d W z d W z ] , [ 0 L 0 L d W z d W z ] 2 , [ 0 L 0 L z z d W z d W z ] 2
σ 1 2 = 55 σ α 4 L 2 180 ln 2 ( 2 ) ( 4 N 2 1 ) = 55 σ 1 2 .
RX = log 2 ( 1 γ 0 ) + 1 2 ln ( 2 ) ( 1 γ 0 2 γ 2 1 ) 1 2 ln ( 2 ) [ 2 I ( 0 , L ) + σ α 2 L ] ,

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