Abstract

Modal dispersion (MD) in a multimode fiber may be considered as a generalized form of polarization mode dispersion (PMD) in single mode fibers. Using this analogy, we extend the formalism developed for PMD to characterize MD in fibers with multiple spatial modes. We introduce a MD vector defined in a D-dimensional extended Stokes space whose square length is the sum of the square group delays of the generalized principal states. For strong mode coupling, the MD vector undertakes a D-dimensional isotropic random walk, so that the distribution of its length is a chi distribution with D degrees of freedom. We also characterize the largest differential group delay, that is the difference between the delays of the fastest and the slowest principal states, and show that it too is very well approximated by a chi distribution, although in general with a smaller number of degrees of freedom. Finally, we study the spectral properties of MD in terms of the frequency autocorrelation functions of the MD vector, of the square modulus of the MD vector, and of the largest differential group delay. The analytical results are supported by extensive numerical simulations.

© 2012 OSA

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  1. A. R. Chraplyvy, “The coming capacity crunch,” European Conference on Optical Communication 2009 (ECOC09), plenary talk (2009).
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    [CrossRef]
  3. Peter J. Winzer and Gerard J. Foschini, “MIMO capacities and outage probabilities in spatially multiplexed optical transport systems,” Opt. Express 19, 16680–16696 (2011).
    [CrossRef] [PubMed]
  4. S. Chandrasekhar, A. H. Gnauck, X. Liu, P. J. Winzer, Y. Pan, E. C. Burrows, B. Zhu, T. F. Taunay, M. Fishteyn, M. F. Yan, J. M. Fini, E. M. Monberg, and F. V. Dimarcello, “WDM/SDM transmission of 10 × 128-Gb/s PDM-QPSK over 2688-km 7-core fiber with a per-fiber net aggregate spectral-efficiency distance product of 40,320 km×b/s/Hz,” Proceedings of European Conference on Optical Communication 2011 (ECOC11), Paper Th.13.C4 (2011).
  5. S. Randel, R. Ryf, A. Sierra, P.J. Winzer, A.H. Gnauck, C.A. Bolle, R-J. Essiambre, D.W. Peckham, A. McCurdy, and R. Lingle “6×56-Gb/s mode-division multiplexed transmission over 33-km few-mode fiber enabled by 6×6 MIMO equalization,” Opt”. Express 19, 16697–16707 (2011).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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  16. D. Cassioli, M. Z. Win, and A. F. Molisch, “The ultra-wide bandwidth indoor channel: from statistical model to simulations,” IEEE J. Sel. Areas Commun. 20, 1247–1257 (2002).
    [CrossRef]
  17. In the literature the shape parameter KN is typically referred to as the number of degrees of freedom of the chi distribution. We refrain from this terminology so as to avoid confusion with the number of degrees of freedom Deff representing the dimension of the manyfold spanned by the MD vector in the generalized Stokes space.
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    [CrossRef]
  19. M. Shtaif and A. Mecozzi, “Study of the frequency autocorrelation of the differential group delay in fibers with polarization mode dispersion,” Opt. Lett. 25, 707–709 (2000).
    [CrossRef]
  20. R. Ryf, S. Randel, A. H. Gnauck, C. Bolle, A. Sierra, S. Mumtaz, M. Esmaeelpour, E. C. Burrows, R-J. Essiambre, P. J. Winzer, D. W. Peckham, A. H. McCurdy, and R. Lingle, “Mode-division multiplexing over 96km of few-mode fiber using coherent 6 × 6 MIMO processing,” J. Lightwave Technol. 30, 521–531 (2012).
    [CrossRef]
  21. M. Gell-Mann, “Symmetries of baryons and mesons,” Phys. Rev. 125, 1067–1084 (1962).
    [CrossRef]
  22. A. Galtarossa, L. Palmieri, M. Schiano, and T. Tambosso, “Measurement of birefringence correlation length in long, single-mode fibers,” Opt. Lett. 26, 962–964 (2001).
    [CrossRef]
  23. F. Curti, B. Daino, G. De Marchis, and F. Matera, “Statistical treatment of the evolution of the principal states of polarization in single-mode fibers,” J. Lightwave Technol. 8, 1162–1166 (1990).
    [CrossRef]
  24. The more common terminology for the vector operation defined in Eq. (28) is Lie bracket. We prefer the term generalized cross-product for emphasizing the analogy with the case of PMD.

2012 (2)

2011 (4)

2010 (1)

R. W. Tkach, “Scaling optical communications for the next decade and beyond,” Bell Labs Tech. J. 14, 3–9 (2010).
[CrossRef]

2006 (1)

2005 (1)

2002 (1)

D. Cassioli, M. Z. Win, and A. F. Molisch, “The ultra-wide bandwidth indoor channel: from statistical model to simulations,” IEEE J. Sel. Areas Commun. 20, 1247–1257 (2002).
[CrossRef]

2001 (2)

2000 (2)

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]

M. Shtaif and A. Mecozzi, “Study of the frequency autocorrelation of the differential group delay in fibers with polarization mode dispersion,” Opt. Lett. 25, 707–709 (2000).
[CrossRef]

1999 (1)

1995 (1)

N. Gisin, “Statistics of polarization dependent losses,” Opt. Commun. 114, 399–405 (1995).
[CrossRef]

1990 (1)

F. Curti, B. Daino, G. De Marchis, and F. Matera, “Statistical treatment of the evolution of the principal states of polarization in single-mode fibers,” J. Lightwave Technol. 8, 1162–1166 (1990).
[CrossRef]

1962 (1)

M. Gell-Mann, “Symmetries of baryons and mesons,” Phys. Rev. 125, 1067–1084 (1962).
[CrossRef]

Astruc, M.

Bai, N.

E. Ip, N. Bai, Y.K. Huang, E. Mateo, F. Yaman, S. Bickham, H.Y. Tam, C. Lu, M.J. Li, S. Ten, A.P. Tao Lau, V. Tse, G.D. Peng, C. Montero, X. Prieto, and G. Li, “88 × 3 × 112-Gb/s WDM transmission over 50 km of three-mode fiber with inline few-mode fiber amplifier,” Proceedings of European Conference on Optical Communication 2011 (ECOC11), Paper Th.13.C2 (2011).

Bickham, S.

E. Ip, N. Bai, Y.K. Huang, E. Mateo, F. Yaman, S. Bickham, H.Y. Tam, C. Lu, M.J. Li, S. Ten, A.P. Tao Lau, V. Tse, G.D. Peng, C. Montero, X. Prieto, and G. Li, “88 × 3 × 112-Gb/s WDM transmission over 50 km of three-mode fiber with inline few-mode fiber amplifier,” Proceedings of European Conference on Optical Communication 2011 (ECOC11), Paper Th.13.C2 (2011).

Bigo, S.

Bolle, C.

Bolle, C.A.

S. Randel, R. Ryf, A. Sierra, P.J. Winzer, A.H. Gnauck, C.A. Bolle, R-J. Essiambre, D.W. Peckham, A. McCurdy, and R. Lingle “6×56-Gb/s mode-division multiplexed transmission over 33-km few-mode fiber enabled by 6×6 MIMO equalization,” Opt”. Express 19, 16697–16707 (2011).
[CrossRef] [PubMed]

Boutin, A.

Brentel, J.

Brindel, P.

Burrows, E. C.

R. Ryf, S. Randel, A. H. Gnauck, C. Bolle, A. Sierra, S. Mumtaz, M. Esmaeelpour, E. C. Burrows, R-J. Essiambre, P. J. Winzer, D. W. Peckham, A. H. McCurdy, and R. Lingle, “Mode-division multiplexing over 96km of few-mode fiber using coherent 6 × 6 MIMO processing,” J. Lightwave Technol. 30, 521–531 (2012).
[CrossRef]

S. Chandrasekhar, A. H. Gnauck, X. Liu, P. J. Winzer, Y. Pan, E. C. Burrows, B. Zhu, T. F. Taunay, M. Fishteyn, M. F. Yan, J. M. Fini, E. M. Monberg, and F. V. Dimarcello, “WDM/SDM transmission of 10 × 128-Gb/s PDM-QPSK over 2688-km 7-core fiber with a per-fiber net aggregate spectral-efficiency distance product of 40,320 km×b/s/Hz,” Proceedings of European Conference on Optical Communication 2011 (ECOC11), Paper Th.13.C4 (2011).

Cassioli, D.

D. Cassioli, M. Z. Win, and A. F. Molisch, “The ultra-wide bandwidth indoor channel: from statistical model to simulations,” IEEE J. Sel. Areas Commun. 20, 1247–1257 (2002).
[CrossRef]

Chandrasekhar, S.

S. Chandrasekhar, A. H. Gnauck, X. Liu, P. J. Winzer, Y. Pan, E. C. Burrows, B. Zhu, T. F. Taunay, M. Fishteyn, M. F. Yan, J. M. Fini, E. M. Monberg, and F. V. Dimarcello, “WDM/SDM transmission of 10 × 128-Gb/s PDM-QPSK over 2688-km 7-core fiber with a per-fiber net aggregate spectral-efficiency distance product of 40,320 km×b/s/Hz,” Proceedings of European Conference on Optical Communication 2011 (ECOC11), Paper Th.13.C4 (2011).

Charlet, G.

Chraplyvy, A. R.

A. R. Chraplyvy, “The coming capacity crunch,” European Conference on Optical Communication 2009 (ECOC09), plenary talk (2009).

Curti, F.

F. Curti, B. Daino, G. De Marchis, and F. Matera, “Statistical treatment of the evolution of the principal states of polarization in single-mode fibers,” J. Lightwave Technol. 8, 1162–1166 (1990).
[CrossRef]

Daino, B.

F. Curti, B. Daino, G. De Marchis, and F. Matera, “Statistical treatment of the evolution of the principal states of polarization in single-mode fibers,” J. Lightwave Technol. 8, 1162–1166 (1990).
[CrossRef]

De Marchis, G.

F. Curti, B. Daino, G. De Marchis, and F. Matera, “Statistical treatment of the evolution of the principal states of polarization in single-mode fibers,” J. Lightwave Technol. 8, 1162–1166 (1990).
[CrossRef]

Dimarcello, F. V.

S. Chandrasekhar, A. H. Gnauck, X. Liu, P. J. Winzer, Y. Pan, E. C. Burrows, B. Zhu, T. F. Taunay, M. Fishteyn, M. F. Yan, J. M. Fini, E. M. Monberg, and F. V. Dimarcello, “WDM/SDM transmission of 10 × 128-Gb/s PDM-QPSK over 2688-km 7-core fiber with a per-fiber net aggregate spectral-efficiency distance product of 40,320 km×b/s/Hz,” Proceedings of European Conference on Optical Communication 2011 (ECOC11), Paper Th.13.C4 (2011).

Esmaeelpour, M.

Essiambre, R-J.

R. Ryf, S. Randel, A. H. Gnauck, C. Bolle, A. Sierra, S. Mumtaz, M. Esmaeelpour, E. C. Burrows, R-J. Essiambre, P. J. Winzer, D. W. Peckham, A. H. McCurdy, and R. Lingle, “Mode-division multiplexing over 96km of few-mode fiber using coherent 6 × 6 MIMO processing,” J. Lightwave Technol. 30, 521–531 (2012).
[CrossRef]

S. Randel, R. Ryf, A. Sierra, P.J. Winzer, A.H. Gnauck, C.A. Bolle, R-J. Essiambre, D.W. Peckham, A. McCurdy, and R. Lingle “6×56-Gb/s mode-division multiplexed transmission over 33-km few-mode fiber enabled by 6×6 MIMO equalization,” Opt”. Express 19, 16697–16707 (2011).
[CrossRef] [PubMed]

S. Randel, M. Magarini, R. Ryf, R-J. Essiambre, A. H. Gnauck, P. J. Winzer, T. Hayashi, T. Taru, and T. Sasaki “MIMO-based signal processing of spatially multiplexed 112-Gb/s PDM-QPSK signals using strongly-coupled 3-core fiber,” Proceedings of European Conference on Optical Communication 2011 (ECOC11), Paper Tu.5.B1 (2011).

Fan, S.

Fini, J. M.

S. Chandrasekhar, A. H. Gnauck, X. Liu, P. J. Winzer, Y. Pan, E. C. Burrows, B. Zhu, T. F. Taunay, M. Fishteyn, M. F. Yan, J. M. Fini, E. M. Monberg, and F. V. Dimarcello, “WDM/SDM transmission of 10 × 128-Gb/s PDM-QPSK over 2688-km 7-core fiber with a per-fiber net aggregate spectral-efficiency distance product of 40,320 km×b/s/Hz,” Proceedings of European Conference on Optical Communication 2011 (ECOC11), Paper Th.13.C4 (2011).

Fishteyn, M.

S. Chandrasekhar, A. H. Gnauck, X. Liu, P. J. Winzer, Y. Pan, E. C. Burrows, B. Zhu, T. F. Taunay, M. Fishteyn, M. F. Yan, J. M. Fini, E. M. Monberg, and F. V. Dimarcello, “WDM/SDM transmission of 10 × 128-Gb/s PDM-QPSK over 2688-km 7-core fiber with a per-fiber net aggregate spectral-efficiency distance product of 40,320 km×b/s/Hz,” Proceedings of European Conference on Optical Communication 2011 (ECOC11), Paper Th.13.C4 (2011).

Foschini, Gerard J.

Galtarossa, A.

Gell-Mann, M.

M. Gell-Mann, “Symmetries of baryons and mesons,” Phys. Rev. 125, 1067–1084 (1962).
[CrossRef]

Gisin, N.

N. Gisin, “Statistics of polarization dependent losses,” Opt. Commun. 114, 399–405 (1995).
[CrossRef]

Gnauck, A. H.

R. Ryf, S. Randel, A. H. Gnauck, C. Bolle, A. Sierra, S. Mumtaz, M. Esmaeelpour, E. C. Burrows, R-J. Essiambre, P. J. Winzer, D. W. Peckham, A. H. McCurdy, and R. Lingle, “Mode-division multiplexing over 96km of few-mode fiber using coherent 6 × 6 MIMO processing,” J. Lightwave Technol. 30, 521–531 (2012).
[CrossRef]

S. Randel, M. Magarini, R. Ryf, R-J. Essiambre, A. H. Gnauck, P. J. Winzer, T. Hayashi, T. Taru, and T. Sasaki “MIMO-based signal processing of spatially multiplexed 112-Gb/s PDM-QPSK signals using strongly-coupled 3-core fiber,” Proceedings of European Conference on Optical Communication 2011 (ECOC11), Paper Tu.5.B1 (2011).

S. Chandrasekhar, A. H. Gnauck, X. Liu, P. J. Winzer, Y. Pan, E. C. Burrows, B. Zhu, T. F. Taunay, M. Fishteyn, M. F. Yan, J. M. Fini, E. M. Monberg, and F. V. Dimarcello, “WDM/SDM transmission of 10 × 128-Gb/s PDM-QPSK over 2688-km 7-core fiber with a per-fiber net aggregate spectral-efficiency distance product of 40,320 km×b/s/Hz,” Proceedings of European Conference on Optical Communication 2011 (ECOC11), Paper Th.13.C4 (2011).

Gnauck, A.H.

S. Randel, R. Ryf, A. Sierra, P.J. Winzer, A.H. Gnauck, C.A. Bolle, R-J. Essiambre, D.W. Peckham, A. McCurdy, and R. Lingle “6×56-Gb/s mode-division multiplexed transmission over 33-km few-mode fiber enabled by 6×6 MIMO equalization,” Opt”. Express 19, 16697–16707 (2011).
[CrossRef] [PubMed]

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]

Hayashi, T.

S. Randel, M. Magarini, R. Ryf, R-J. Essiambre, A. H. Gnauck, P. J. Winzer, T. Hayashi, T. Taru, and T. Sasaki “MIMO-based signal processing of spatially multiplexed 112-Gb/s PDM-QPSK signals using strongly-coupled 3-core fiber,” Proceedings of European Conference on Optical Communication 2011 (ECOC11), Paper Tu.5.B1 (2011).

Ho, K-P.

Huang, Y.K.

E. Ip, N. Bai, Y.K. Huang, E. Mateo, F. Yaman, S. Bickham, H.Y. Tam, C. Lu, M.J. Li, S. Ten, A.P. Tao Lau, V. Tse, G.D. Peng, C. Montero, X. Prieto, and G. Li, “88 × 3 × 112-Gb/s WDM transmission over 50 km of three-mode fiber with inline few-mode fiber amplifier,” Proceedings of European Conference on Optical Communication 2011 (ECOC11), Paper Th.13.C2 (2011).

Ip, E.

E. Ip, N. Bai, Y.K. Huang, E. Mateo, F. Yaman, S. Bickham, H.Y. Tam, C. Lu, M.J. Li, S. Ten, A.P. Tao Lau, V. Tse, G.D. Peng, C. Montero, X. Prieto, and G. Li, “88 × 3 × 112-Gb/s WDM transmission over 50 km of three-mode fiber with inline few-mode fiber amplifier,” Proceedings of European Conference on Optical Communication 2011 (ECOC11), Paper Th.13.C2 (2011).

James, D. F. V.

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A 64, 052312 (2001).
[CrossRef]

Kahn, J. M

Kahn, J. M.

Karlsson, M.

Koebele, C.

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]

Kwiat, P. G.

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A 64, 052312 (2001).
[CrossRef]

Li, G.

E. Ip, N. Bai, Y.K. Huang, E. Mateo, F. Yaman, S. Bickham, H.Y. Tam, C. Lu, M.J. Li, S. Ten, A.P. Tao Lau, V. Tse, G.D. Peng, C. Montero, X. Prieto, and G. Li, “88 × 3 × 112-Gb/s WDM transmission over 50 km of three-mode fiber with inline few-mode fiber amplifier,” Proceedings of European Conference on Optical Communication 2011 (ECOC11), Paper Th.13.C2 (2011).

Li, M.J.

E. Ip, N. Bai, Y.K. Huang, E. Mateo, F. Yaman, S. Bickham, H.Y. Tam, C. Lu, M.J. Li, S. Ten, A.P. Tao Lau, V. Tse, G.D. Peng, C. Montero, X. Prieto, and G. Li, “88 × 3 × 112-Gb/s WDM transmission over 50 km of three-mode fiber with inline few-mode fiber amplifier,” Proceedings of European Conference on Optical Communication 2011 (ECOC11), Paper Th.13.C2 (2011).

Lingle, R.

R. Ryf, S. Randel, A. H. Gnauck, C. Bolle, A. Sierra, S. Mumtaz, M. Esmaeelpour, E. C. Burrows, R-J. Essiambre, P. J. Winzer, D. W. Peckham, A. H. McCurdy, and R. Lingle, “Mode-division multiplexing over 96km of few-mode fiber using coherent 6 × 6 MIMO processing,” J. Lightwave Technol. 30, 521–531 (2012).
[CrossRef]

S. Randel, R. Ryf, A. Sierra, P.J. Winzer, A.H. Gnauck, C.A. Bolle, R-J. Essiambre, D.W. Peckham, A. McCurdy, and R. Lingle “6×56-Gb/s mode-division multiplexed transmission over 33-km few-mode fiber enabled by 6×6 MIMO equalization,” Opt”. Express 19, 16697–16707 (2011).
[CrossRef] [PubMed]

Liu, X.

S. Chandrasekhar, A. H. Gnauck, X. Liu, P. J. Winzer, Y. Pan, E. C. Burrows, B. Zhu, T. F. Taunay, M. Fishteyn, M. F. Yan, J. M. Fini, E. M. Monberg, and F. V. Dimarcello, “WDM/SDM transmission of 10 × 128-Gb/s PDM-QPSK over 2688-km 7-core fiber with a per-fiber net aggregate spectral-efficiency distance product of 40,320 km×b/s/Hz,” Proceedings of European Conference on Optical Communication 2011 (ECOC11), Paper Th.13.C4 (2011).

Lu, C.

E. Ip, N. Bai, Y.K. Huang, E. Mateo, F. Yaman, S. Bickham, H.Y. Tam, C. Lu, M.J. Li, S. Ten, A.P. Tao Lau, V. Tse, G.D. Peng, C. Montero, X. Prieto, and G. Li, “88 × 3 × 112-Gb/s WDM transmission over 50 km of three-mode fiber with inline few-mode fiber amplifier,” Proceedings of European Conference on Optical Communication 2011 (ECOC11), Paper Th.13.C2 (2011).

Magarini, M.

S. Randel, M. Magarini, R. Ryf, R-J. Essiambre, A. H. Gnauck, P. J. Winzer, T. Hayashi, T. Taru, and T. Sasaki “MIMO-based signal processing of spatially multiplexed 112-Gb/s PDM-QPSK signals using strongly-coupled 3-core fiber,” Proceedings of European Conference on Optical Communication 2011 (ECOC11), Paper Tu.5.B1 (2011).

Mardoyan, H.

Mateo, E.

E. Ip, N. Bai, Y.K. Huang, E. Mateo, F. Yaman, S. Bickham, H.Y. Tam, C. Lu, M.J. Li, S. Ten, A.P. Tao Lau, V. Tse, G.D. Peng, C. Montero, X. Prieto, and G. Li, “88 × 3 × 112-Gb/s WDM transmission over 50 km of three-mode fiber with inline few-mode fiber amplifier,” Proceedings of European Conference on Optical Communication 2011 (ECOC11), Paper Th.13.C2 (2011).

Matera, F.

F. Curti, B. Daino, G. De Marchis, and F. Matera, “Statistical treatment of the evolution of the principal states of polarization in single-mode fibers,” J. Lightwave Technol. 8, 1162–1166 (1990).
[CrossRef]

McCurdy, A.

S. Randel, R. Ryf, A. Sierra, P.J. Winzer, A.H. Gnauck, C.A. Bolle, R-J. Essiambre, D.W. Peckham, A. McCurdy, and R. Lingle “6×56-Gb/s mode-division multiplexed transmission over 33-km few-mode fiber enabled by 6×6 MIMO equalization,” Opt”. Express 19, 16697–16707 (2011).
[CrossRef] [PubMed]

McCurdy, A. H.

Mecozzi, A.

Molisch, A. F.

D. Cassioli, M. Z. Win, and A. F. Molisch, “The ultra-wide bandwidth indoor channel: from statistical model to simulations,” IEEE J. Sel. Areas Commun. 20, 1247–1257 (2002).
[CrossRef]

Monberg, E. M.

S. Chandrasekhar, A. H. Gnauck, X. Liu, P. J. Winzer, Y. Pan, E. C. Burrows, B. Zhu, T. F. Taunay, M. Fishteyn, M. F. Yan, J. M. Fini, E. M. Monberg, and F. V. Dimarcello, “WDM/SDM transmission of 10 × 128-Gb/s PDM-QPSK over 2688-km 7-core fiber with a per-fiber net aggregate spectral-efficiency distance product of 40,320 km×b/s/Hz,” Proceedings of European Conference on Optical Communication 2011 (ECOC11), Paper Th.13.C4 (2011).

Montero, C.

E. Ip, N. Bai, Y.K. Huang, E. Mateo, F. Yaman, S. Bickham, H.Y. Tam, C. Lu, M.J. Li, S. Ten, A.P. Tao Lau, V. Tse, G.D. Peng, C. Montero, X. Prieto, and G. Li, “88 × 3 × 112-Gb/s WDM transmission over 50 km of three-mode fiber with inline few-mode fiber amplifier,” Proceedings of European Conference on Optical Communication 2011 (ECOC11), Paper Th.13.C2 (2011).

Mumtaz, S.

Munro, W. J.

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A 64, 052312 (2001).
[CrossRef]

Nazarathy, M.

Palmieri, L.

Pan, Y.

S. Chandrasekhar, A. H. Gnauck, X. Liu, P. J. Winzer, Y. Pan, E. C. Burrows, B. Zhu, T. F. Taunay, M. Fishteyn, M. F. Yan, J. M. Fini, E. M. Monberg, and F. V. Dimarcello, “WDM/SDM transmission of 10 × 128-Gb/s PDM-QPSK over 2688-km 7-core fiber with a per-fiber net aggregate spectral-efficiency distance product of 40,320 km×b/s/Hz,” Proceedings of European Conference on Optical Communication 2011 (ECOC11), Paper Th.13.C4 (2011).

Peckham, D. W.

Peckham, D.W.

S. Randel, R. Ryf, A. Sierra, P.J. Winzer, A.H. Gnauck, C.A. Bolle, R-J. Essiambre, D.W. Peckham, A. McCurdy, and R. Lingle “6×56-Gb/s mode-division multiplexed transmission over 33-km few-mode fiber enabled by 6×6 MIMO equalization,” Opt”. Express 19, 16697–16707 (2011).
[CrossRef] [PubMed]

Peng, G.D.

E. Ip, N. Bai, Y.K. Huang, E. Mateo, F. Yaman, S. Bickham, H.Y. Tam, C. Lu, M.J. Li, S. Ten, A.P. Tao Lau, V. Tse, G.D. Peng, C. Montero, X. Prieto, and G. Li, “88 × 3 × 112-Gb/s WDM transmission over 50 km of three-mode fiber with inline few-mode fiber amplifier,” Proceedings of European Conference on Optical Communication 2011 (ECOC11), Paper Th.13.C2 (2011).

Prieto, X.

E. Ip, N. Bai, Y.K. Huang, E. Mateo, F. Yaman, S. Bickham, H.Y. Tam, C. Lu, M.J. Li, S. Ten, A.P. Tao Lau, V. Tse, G.D. Peng, C. Montero, X. Prieto, and G. Li, “88 × 3 × 112-Gb/s WDM transmission over 50 km of three-mode fiber with inline few-mode fiber amplifier,” Proceedings of European Conference on Optical Communication 2011 (ECOC11), Paper Th.13.C2 (2011).

Provost, L.

Randel, S.

R. Ryf, S. Randel, A. H. Gnauck, C. Bolle, A. Sierra, S. Mumtaz, M. Esmaeelpour, E. C. Burrows, R-J. Essiambre, P. J. Winzer, D. W. Peckham, A. H. McCurdy, and R. Lingle, “Mode-division multiplexing over 96km of few-mode fiber using coherent 6 × 6 MIMO processing,” J. Lightwave Technol. 30, 521–531 (2012).
[CrossRef]

S. Randel, R. Ryf, A. Sierra, P.J. Winzer, A.H. Gnauck, C.A. Bolle, R-J. Essiambre, D.W. Peckham, A. McCurdy, and R. Lingle “6×56-Gb/s mode-division multiplexed transmission over 33-km few-mode fiber enabled by 6×6 MIMO equalization,” Opt”. Express 19, 16697–16707 (2011).
[CrossRef] [PubMed]

S. Randel, M. Magarini, R. Ryf, R-J. Essiambre, A. H. Gnauck, P. J. Winzer, T. Hayashi, T. Taru, and T. Sasaki “MIMO-based signal processing of spatially multiplexed 112-Gb/s PDM-QPSK signals using strongly-coupled 3-core fiber,” Proceedings of European Conference on Optical Communication 2011 (ECOC11), Paper Tu.5.B1 (2011).

Ryf, R.

R. Ryf, S. Randel, A. H. Gnauck, C. Bolle, A. Sierra, S. Mumtaz, M. Esmaeelpour, E. C. Burrows, R-J. Essiambre, P. J. Winzer, D. W. Peckham, A. H. McCurdy, and R. Lingle, “Mode-division multiplexing over 96km of few-mode fiber using coherent 6 × 6 MIMO processing,” J. Lightwave Technol. 30, 521–531 (2012).
[CrossRef]

S. Randel, R. Ryf, A. Sierra, P.J. Winzer, A.H. Gnauck, C.A. Bolle, R-J. Essiambre, D.W. Peckham, A. McCurdy, and R. Lingle “6×56-Gb/s mode-division multiplexed transmission over 33-km few-mode fiber enabled by 6×6 MIMO equalization,” Opt”. Express 19, 16697–16707 (2011).
[CrossRef] [PubMed]

S. Randel, M. Magarini, R. Ryf, R-J. Essiambre, A. H. Gnauck, P. J. Winzer, T. Hayashi, T. Taru, and T. Sasaki “MIMO-based signal processing of spatially multiplexed 112-Gb/s PDM-QPSK signals using strongly-coupled 3-core fiber,” Proceedings of European Conference on Optical Communication 2011 (ECOC11), Paper Tu.5.B1 (2011).

Salsi, M.

Sasaki, T.

S. Randel, M. Magarini, R. Ryf, R-J. Essiambre, A. H. Gnauck, P. J. Winzer, T. Hayashi, T. Taru, and T. Sasaki “MIMO-based signal processing of spatially multiplexed 112-Gb/s PDM-QPSK signals using strongly-coupled 3-core fiber,” Proceedings of European Conference on Optical Communication 2011 (ECOC11), Paper Tu.5.B1 (2011).

Schiano, M.

Shtaif, M.

Sierra, A.

R. Ryf, S. Randel, A. H. Gnauck, C. Bolle, A. Sierra, S. Mumtaz, M. Esmaeelpour, E. C. Burrows, R-J. Essiambre, P. J. Winzer, D. W. Peckham, A. H. McCurdy, and R. Lingle, “Mode-division multiplexing over 96km of few-mode fiber using coherent 6 × 6 MIMO processing,” J. Lightwave Technol. 30, 521–531 (2012).
[CrossRef]

S. Randel, R. Ryf, A. Sierra, P.J. Winzer, A.H. Gnauck, C.A. Bolle, R-J. Essiambre, D.W. Peckham, A. McCurdy, and R. Lingle “6×56-Gb/s mode-division multiplexed transmission over 33-km few-mode fiber enabled by 6×6 MIMO equalization,” Opt”. Express 19, 16697–16707 (2011).
[CrossRef] [PubMed]

Sillard, P.

Simony, E.

Sperti, D.

Tam, H.Y.

E. Ip, N. Bai, Y.K. Huang, E. Mateo, F. Yaman, S. Bickham, H.Y. Tam, C. Lu, M.J. Li, S. Ten, A.P. Tao Lau, V. Tse, G.D. Peng, C. Montero, X. Prieto, and G. Li, “88 × 3 × 112-Gb/s WDM transmission over 50 km of three-mode fiber with inline few-mode fiber amplifier,” Proceedings of European Conference on Optical Communication 2011 (ECOC11), Paper Th.13.C2 (2011).

Tambosso, T.

Tao Lau, A.P.

E. Ip, N. Bai, Y.K. Huang, E. Mateo, F. Yaman, S. Bickham, H.Y. Tam, C. Lu, M.J. Li, S. Ten, A.P. Tao Lau, V. Tse, G.D. Peng, C. Montero, X. Prieto, and G. Li, “88 × 3 × 112-Gb/s WDM transmission over 50 km of three-mode fiber with inline few-mode fiber amplifier,” Proceedings of European Conference on Optical Communication 2011 (ECOC11), Paper Th.13.C2 (2011).

Taru, T.

S. Randel, M. Magarini, R. Ryf, R-J. Essiambre, A. H. Gnauck, P. J. Winzer, T. Hayashi, T. Taru, and T. Sasaki “MIMO-based signal processing of spatially multiplexed 112-Gb/s PDM-QPSK signals using strongly-coupled 3-core fiber,” Proceedings of European Conference on Optical Communication 2011 (ECOC11), Paper Tu.5.B1 (2011).

Taunay, T. F.

S. Chandrasekhar, A. H. Gnauck, X. Liu, P. J. Winzer, Y. Pan, E. C. Burrows, B. Zhu, T. F. Taunay, M. Fishteyn, M. F. Yan, J. M. Fini, E. M. Monberg, and F. V. Dimarcello, “WDM/SDM transmission of 10 × 128-Gb/s PDM-QPSK over 2688-km 7-core fiber with a per-fiber net aggregate spectral-efficiency distance product of 40,320 km×b/s/Hz,” Proceedings of European Conference on Optical Communication 2011 (ECOC11), Paper Th.13.C4 (2011).

Ten, S.

E. Ip, N. Bai, Y.K. Huang, E. Mateo, F. Yaman, S. Bickham, H.Y. Tam, C. Lu, M.J. Li, S. Ten, A.P. Tao Lau, V. Tse, G.D. Peng, C. Montero, X. Prieto, and G. Li, “88 × 3 × 112-Gb/s WDM transmission over 50 km of three-mode fiber with inline few-mode fiber amplifier,” Proceedings of European Conference on Optical Communication 2011 (ECOC11), Paper Th.13.C2 (2011).

Tkach, R. W.

R. W. Tkach, “Scaling optical communications for the next decade and beyond,” Bell Labs Tech. J. 14, 3–9 (2010).
[CrossRef]

Tran, P.

Tse, V.

E. Ip, N. Bai, Y.K. Huang, E. Mateo, F. Yaman, S. Bickham, H.Y. Tam, C. Lu, M.J. Li, S. Ten, A.P. Tao Lau, V. Tse, G.D. Peng, C. Montero, X. Prieto, and G. Li, “88 × 3 × 112-Gb/s WDM transmission over 50 km of three-mode fiber with inline few-mode fiber amplifier,” Proceedings of European Conference on Optical Communication 2011 (ECOC11), Paper Th.13.C2 (2011).

Verluise, F.

White, A. G.

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A 64, 052312 (2001).
[CrossRef]

Win, M. Z.

D. Cassioli, M. Z. Win, and A. F. Molisch, “The ultra-wide bandwidth indoor channel: from statistical model to simulations,” IEEE J. Sel. Areas Commun. 20, 1247–1257 (2002).
[CrossRef]

Winzer, P. J.

R. Ryf, S. Randel, A. H. Gnauck, C. Bolle, A. Sierra, S. Mumtaz, M. Esmaeelpour, E. C. Burrows, R-J. Essiambre, P. J. Winzer, D. W. Peckham, A. H. McCurdy, and R. Lingle, “Mode-division multiplexing over 96km of few-mode fiber using coherent 6 × 6 MIMO processing,” J. Lightwave Technol. 30, 521–531 (2012).
[CrossRef]

S. Randel, M. Magarini, R. Ryf, R-J. Essiambre, A. H. Gnauck, P. J. Winzer, T. Hayashi, T. Taru, and T. Sasaki “MIMO-based signal processing of spatially multiplexed 112-Gb/s PDM-QPSK signals using strongly-coupled 3-core fiber,” Proceedings of European Conference on Optical Communication 2011 (ECOC11), Paper Tu.5.B1 (2011).

S. Chandrasekhar, A. H. Gnauck, X. Liu, P. J. Winzer, Y. Pan, E. C. Burrows, B. Zhu, T. F. Taunay, M. Fishteyn, M. F. Yan, J. M. Fini, E. M. Monberg, and F. V. Dimarcello, “WDM/SDM transmission of 10 × 128-Gb/s PDM-QPSK over 2688-km 7-core fiber with a per-fiber net aggregate spectral-efficiency distance product of 40,320 km×b/s/Hz,” Proceedings of European Conference on Optical Communication 2011 (ECOC11), Paper Th.13.C4 (2011).

Winzer, P.J.

S. Randel, R. Ryf, A. Sierra, P.J. Winzer, A.H. Gnauck, C.A. Bolle, R-J. Essiambre, D.W. Peckham, A. McCurdy, and R. Lingle “6×56-Gb/s mode-division multiplexed transmission over 33-km few-mode fiber enabled by 6×6 MIMO equalization,” Opt”. Express 19, 16697–16707 (2011).
[CrossRef] [PubMed]

Winzer, Peter J.

Yaman, F.

E. Ip, N. Bai, Y.K. Huang, E. Mateo, F. Yaman, S. Bickham, H.Y. Tam, C. Lu, M.J. Li, S. Ten, A.P. Tao Lau, V. Tse, G.D. Peng, C. Montero, X. Prieto, and G. Li, “88 × 3 × 112-Gb/s WDM transmission over 50 km of three-mode fiber with inline few-mode fiber amplifier,” Proceedings of European Conference on Optical Communication 2011 (ECOC11), Paper Th.13.C2 (2011).

Yan, M. F.

S. Chandrasekhar, A. H. Gnauck, X. Liu, P. J. Winzer, Y. Pan, E. C. Burrows, B. Zhu, T. F. Taunay, M. Fishteyn, M. F. Yan, J. M. Fini, E. M. Monberg, and F. V. Dimarcello, “WDM/SDM transmission of 10 × 128-Gb/s PDM-QPSK over 2688-km 7-core fiber with a per-fiber net aggregate spectral-efficiency distance product of 40,320 km×b/s/Hz,” Proceedings of European Conference on Optical Communication 2011 (ECOC11), Paper Th.13.C4 (2011).

Zhu, B.

S. Chandrasekhar, A. H. Gnauck, X. Liu, P. J. Winzer, Y. Pan, E. C. Burrows, B. Zhu, T. F. Taunay, M. Fishteyn, M. F. Yan, J. M. Fini, E. M. Monberg, and F. V. Dimarcello, “WDM/SDM transmission of 10 × 128-Gb/s PDM-QPSK over 2688-km 7-core fiber with a per-fiber net aggregate spectral-efficiency distance product of 40,320 km×b/s/Hz,” Proceedings of European Conference on Optical Communication 2011 (ECOC11), Paper Th.13.C4 (2011).

Bell Labs Tech. J. (1)

R. W. Tkach, “Scaling optical communications for the next decade and beyond,” Bell Labs Tech. J. 14, 3–9 (2010).
[CrossRef]

IEEE J. Sel. Areas Commun. (1)

D. Cassioli, M. Z. Win, and A. F. Molisch, “The ultra-wide bandwidth indoor channel: from statistical model to simulations,” IEEE J. Sel. Areas Commun. 20, 1247–1257 (2002).
[CrossRef]

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N. Gisin, “Statistics of polarization dependent losses,” Opt. Commun. 114, 399–405 (1995).
[CrossRef]

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Opt. Lett. (5)

Opt”. Express (1)

S. Randel, R. Ryf, A. Sierra, P.J. Winzer, A.H. Gnauck, C.A. Bolle, R-J. Essiambre, D.W. Peckham, A. McCurdy, and R. Lingle “6×56-Gb/s mode-division multiplexed transmission over 33-km few-mode fiber enabled by 6×6 MIMO equalization,” Opt”. Express 19, 16697–16707 (2011).
[CrossRef] [PubMed]

Phys. Rev. (1)

M. Gell-Mann, “Symmetries of baryons and mesons,” Phys. Rev. 125, 1067–1084 (1962).
[CrossRef]

Phys. Rev. A (1)

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A 64, 052312 (2001).
[CrossRef]

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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 (6)

A. R. Chraplyvy, “The coming capacity crunch,” European Conference on Optical Communication 2009 (ECOC09), plenary talk (2009).

E. Ip, N. Bai, Y.K. Huang, E. Mateo, F. Yaman, S. Bickham, H.Y. Tam, C. Lu, M.J. Li, S. Ten, A.P. Tao Lau, V. Tse, G.D. Peng, C. Montero, X. Prieto, and G. Li, “88 × 3 × 112-Gb/s WDM transmission over 50 km of three-mode fiber with inline few-mode fiber amplifier,” Proceedings of European Conference on Optical Communication 2011 (ECOC11), Paper Th.13.C2 (2011).

S. Chandrasekhar, A. H. Gnauck, X. Liu, P. J. Winzer, Y. Pan, E. C. Burrows, B. Zhu, T. F. Taunay, M. Fishteyn, M. F. Yan, J. M. Fini, E. M. Monberg, and F. V. Dimarcello, “WDM/SDM transmission of 10 × 128-Gb/s PDM-QPSK over 2688-km 7-core fiber with a per-fiber net aggregate spectral-efficiency distance product of 40,320 km×b/s/Hz,” Proceedings of European Conference on Optical Communication 2011 (ECOC11), Paper Th.13.C4 (2011).

S. Randel, M. Magarini, R. Ryf, R-J. Essiambre, A. H. Gnauck, P. J. Winzer, T. Hayashi, T. Taru, and T. Sasaki “MIMO-based signal processing of spatially multiplexed 112-Gb/s PDM-QPSK signals using strongly-coupled 3-core fiber,” Proceedings of European Conference on Optical Communication 2011 (ECOC11), Paper Tu.5.B1 (2011).

In the literature the shape parameter KN is typically referred to as the number of degrees of freedom of the chi distribution. We refrain from this terminology so as to avoid confusion with the number of degrees of freedom Deff representing the dimension of the manyfold spanned by the MD vector in the generalized Stokes space.

The more common terminology for the vector operation defined in Eq. (28) is Lie bracket. We prefer the term generalized cross-product for emphasizing the analogy with the case of PMD.

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Figures (4)

Fig. 1
Fig. 1

Probability density function of τ for N = 2, normalized to its root-mean square value. The coupling magnitude increases from no coupling in (a) to full coupling in (d). A similar behavior has been verified for a number of larger values of N.

Fig. 2
Fig. 2

Top panel. The solid line represents the shape parameter KN of the chi PDF that provides the best approximation of the distribution of the largest DGD T as a function of N; the dashed line shows the mean square value of T normalized to 〈τ2〉/N2. Bottom panels. Probability density function of T for (A) N = 3, (B) N = 50 and (C) N = 100. Dots are the results of Monte-Carlo simulations, solid lines are the plot of chi PDFs with parameters taken from the top panel.

Fig. 3
Fig. 3

Top panels: normalized autocorrelation function of τ⃗ according to theory (solid curve) and simulations (stars). Bottom panels: the normalized autocorrelation function of τ2 according to theory (solid curve) and simulations (stars) as well as the autocorrelation function of T (circles). Dot-dashed curves represent the re-scaled ACF of τ2 Eq. (5), modified by replacing D with KN as from Fig. 2.

Fig. 4
Fig. 4

(a) Probability density function of the length of the MD vector τ⃗LP11 that represents the four LP11 modes (subspace dimension DLP11 = 15) and of the length of the MD vector τ⃗LP01 that represents the two LP01 modes (subspace dimension DLP11 = 3). The degree of coupling between the two groups of modes is discussed in the text. Figures (b) and (c) show the ACFs of the the two MD vectors. Symbols refer to numerical simulations, solid lines to the theory.

Equations (33)

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τ 2 = 2 N i = 1 2 N t i 2 .
K N 3 + 10.39 × ( N 1 ) 1.36
T 2 ( N 1 ) 2 + 24.7 × ( N 1 ) + 16.14 0.2532 × ( N 1 ) 2 + 7.401 × ( N 1 ) + 16.14 τ 2 / N 2 ,
R τ ( ω ) = τ ( Ω + ω ) τ ( Ω ) = D ω 2 [ 1 exp ( ω 2 τ 2 D ) ] ,
R τ 2 ( ω ) = τ 2 ( Ω + ω ) τ 2 ( Ω ) = τ 2 2 + 4 τ 2 ω 2 4 D ω 4 [ 1 exp ( ω 2 τ 2 D ) ] .
| ψ ( z ) = U ( z , z 0 ) | ψ ( z 0 ) ,
d U ( z , z 0 ) d z = i B ( z ) U ( z , z 0 )
d U ( z , z 0 ) d ω = i T ( z , z 0 ) U ( z , z 0 ) ,
1 2 N Trace { Λ i Λ j } = δ i , j ,
H = 1 2 N ( η 0 1 + η Λ ) ,
η Λ = j = 1 4 N 2 1 η j Λ j .
η i = trace { Λ i H } and η 0 = trace { H } .
| ψ ψ | = 1 2 N ( 1 + ψ Λ ) .
ψ j = trace { Λ j | ψ ψ | } = ψ | Λ j | ψ ,
| ψ | ϕ | 2 = 1 2 N ( 1 + ψ ϕ ) ,
1 2 N trace { ( A Λ ) ( B Λ ) } = A B ,
S ψ i = θ i ,
2 N | S | 2 = θ i 2 .
d U ( z , z 0 ) d z = i 2 N ( β 0 1 + β Λ ) U ( z , z 0 )
d U ( z , z 0 ) d ω = i 2 N ( τ 0 1 + τ Λ ) U ( z , z 0 ) ,
τ Λ = 2 N i d U d ω U = 2 N i d U 2 d ω U 2 + U 2 ( 2 N i d U 1 d ω U 1 ) U 2 = τ 2 Λ + U 2 ( τ 1 Λ ) U 2 ,
| τ 1 ( R ) | 2 = 1 2 N trace { [ U 2 ( τ 1 Λ ) U 2 ] 2 } = | τ 1 | 2 ,
z ( τ Λ ) = β ω Λ + i 2 N [ ( β Λ ) ( τ Λ ) ( τ Λ ) ( β Λ ) ] ,
τ ( L ) = 0 L R ( L , z ) β ω d z
U ( L , z ) ( β ω Λ ) U ( L , z ) = ( R ( L , z ) β ω ) Λ .
τ 2 = 2 N i = 1 2 N t i 2 ,
f i , j , k = i ( 2 N ) 2 trace { Λ k ( Λ i Λ j Λ j Λ i ) } .
A × β = i , j , k f i , j , k A i B j e k
τ z = β ω + β × τ ,
S z = β × S , S ω = τ × S ,
d τ = d W + ω d W × τ ,
Λ 1 = σ 1 = ( 1 0 0 1 ) , Λ 2 = σ 2 = ( 0 1 1 0 ) , Λ 3 = σ 3 = ( 0 i i 0 ) .
Λ 1 = 2 ( 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 ) , Λ 2 = 2 ( 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 ) , Λ 3 = 2 ( 0 i 0 0 i 0 0 0 0 0 0 0 0 0 0 0 ) , Λ 4 = 2 ( 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 ) , Λ 5 = 2 ( 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 ) , Λ 6 = 2 ( 0 0 0 0 0 0 0 0 0 0 0 i 0 0 i 0 ) , Λ 7 = 2 ( 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 ) , Λ 8 = 2 ( 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 ) , Λ 9 = 2 ( 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 ) , Λ 10 = 2 ( 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 ) , Λ 11 = 2 ( 0 0 i 0 0 0 0 0 i 0 0 0 0 0 0 0 ) , Λ 12 = 2 ( 0 0 0 i 0 0 0 0 0 0 0 0 i 0 0 0 ) , Λ 13 = 2 ( 0 0 0 0 0 0 i 0 0 i 0 0 0 0 0 0 ) , Λ 14 = 2 ( 0 0 0 0 0 0 0 i 0 0 0 0 0 i 0 0 ) , Λ 15 = ( 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ) .

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