Abstract

We study the response of space-division multiplexed fiber links to an excitation by a short impulse of the optical intensity. We show that, in the presence of full mixing, the intensity impulse response is Gaussian, confirming recently reported experimental observations, and relate its variance to the mean square of the mode dispersion vector of the link τ⃗. The good agreement between our theory and the previously published experiments provides solid foundations to the random coupling model of SDM fiber links, and provides a tool for efficient design of MIMO-DSP receivers.

© 2015 Optical Society of America

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References

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  1. R. Ryf, S. Randel, A.H. Gnauck, C. Bolle, A. Sierra, S. Mumtaz, M. Esmaeelpour, E.C. Burrows, R. Essiambre, P.J. Winzer, D. W. Peckham, A.H. McCurdy, and R. Lingle, “Mode-Division Multiplexing Over 96 km of Few-Mode Fiber Using Coherent 6 × 6 MIMO Processing,” J. Lightwave Technol. 30, 521–531 (2012). doi:
    [Crossref]
  2. R. Ryf, R.-J. Essiambre, A. H. Gnauck, S. Randel, M. A. Mestre, C. Schmidt, P. J. Winzer, R. Delbue, P. Pupalaikis, A. Sureka, T. Hayashi, T. Taru, and T. Sasaki, “SDM Transmission over 4200-km 3-Core Microstructured Fiber,” OFC 2011, PDP5C.2 (2011).
  3. R. Ryf, N.K. Fontaine, B. Guan, R.-J. Essiambre, S. Randel, A. H. Gnauck, S. Chandrasekhar, A. Adamiecki, G. Raybon, B. Ercan, R. P. Scott, S.J. Ben Yoo, T. Hayashi, T. Nagashima, and T. Sasaki, “1705-km Transmission over Coupled-Core Fibre Supporting 6 Spatial Modes,” ECOC 2014, Cannes - France, PD.3.2 (2014).
  4. R. Ryf, N.K. Fontaine, B. Guan, B. Huang, M. Esmaeelpour, S. Randel, A. H. Gnauck, S. Chandrasekhar, A. Adamiecki, G. Raybon, R.W. Tkach, R. Shubochkin, Y. Sun, and R. Lingle, “305-km Combined Wavelength and Mode-Multiplexed Transmission over Conventional Graded-Index Multimode Fibre,” ECOC 2014, Cannes - France, PD.3.5 (2014).
  5. 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, 11718–11733 (2012).
    [Crossref] [PubMed]
  6. 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]
  7. C. Antonelli, A. Mecozzi, and M. Shtaif, “The delay spread in fibers for SDM transmission: dependence on fiber parameters and perturbations,” Opt. Express 23, 2196–2202 (2015).
  8. The coefficient κ2 is equivalent to the mean square differential-group-delay (DGD) per kilometer, whose specification is standard in single-mode fiber systems [6].
  9. K-P. Ho and J.M Kahn, “Statistics of group delays in multi-mode fibers with strong mode coupling,” J. Lightwave Technol. 29, 3119–3128 (2011).
    [Crossref]
  10. C.W. Gardiner, “Stochastic methods for physics, chemistry and natural sciences,” Springer-Verlag, NY, 1983.
  11. 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]

2015 (1)

2012 (2)

2011 (1)

2000 (2)

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]

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]

Adamiecki, A.

R. Ryf, N.K. Fontaine, B. Guan, R.-J. Essiambre, S. Randel, A. H. Gnauck, S. Chandrasekhar, A. Adamiecki, G. Raybon, B. Ercan, R. P. Scott, S.J. Ben Yoo, T. Hayashi, T. Nagashima, and T. Sasaki, “1705-km Transmission over Coupled-Core Fibre Supporting 6 Spatial Modes,” ECOC 2014, Cannes - France, PD.3.2 (2014).

R. Ryf, N.K. Fontaine, B. Guan, B. Huang, M. Esmaeelpour, S. Randel, A. H. Gnauck, S. Chandrasekhar, A. Adamiecki, G. Raybon, R.W. Tkach, R. Shubochkin, Y. Sun, and R. Lingle, “305-km Combined Wavelength and Mode-Multiplexed Transmission over Conventional Graded-Index Multimode Fibre,” ECOC 2014, Cannes - France, PD.3.5 (2014).

Antonelli, C.

Ben Yoo, S.J.

R. Ryf, N.K. Fontaine, B. Guan, R.-J. Essiambre, S. Randel, A. H. Gnauck, S. Chandrasekhar, A. Adamiecki, G. Raybon, B. Ercan, R. P. Scott, S.J. Ben Yoo, T. Hayashi, T. Nagashima, and T. Sasaki, “1705-km Transmission over Coupled-Core Fibre Supporting 6 Spatial Modes,” ECOC 2014, Cannes - France, PD.3.2 (2014).

Bolle, C.

Burrows, E.C.

Chandrasekhar, S.

R. Ryf, N.K. Fontaine, B. Guan, R.-J. Essiambre, S. Randel, A. H. Gnauck, S. Chandrasekhar, A. Adamiecki, G. Raybon, B. Ercan, R. P. Scott, S.J. Ben Yoo, T. Hayashi, T. Nagashima, and T. Sasaki, “1705-km Transmission over Coupled-Core Fibre Supporting 6 Spatial Modes,” ECOC 2014, Cannes - France, PD.3.2 (2014).

R. Ryf, N.K. Fontaine, B. Guan, B. Huang, M. Esmaeelpour, S. Randel, A. H. Gnauck, S. Chandrasekhar, A. Adamiecki, G. Raybon, R.W. Tkach, R. Shubochkin, Y. Sun, and R. Lingle, “305-km Combined Wavelength and Mode-Multiplexed Transmission over Conventional Graded-Index Multimode Fibre,” ECOC 2014, Cannes - France, PD.3.5 (2014).

Delbue, R.

R. Ryf, R.-J. Essiambre, A. H. Gnauck, S. Randel, M. A. Mestre, C. Schmidt, P. J. Winzer, R. Delbue, P. Pupalaikis, A. Sureka, T. Hayashi, T. Taru, and T. Sasaki, “SDM Transmission over 4200-km 3-Core Microstructured Fiber,” OFC 2011, PDP5C.2 (2011).

Ercan, B.

R. Ryf, N.K. Fontaine, B. Guan, R.-J. Essiambre, S. Randel, A. H. Gnauck, S. Chandrasekhar, A. Adamiecki, G. Raybon, B. Ercan, R. P. Scott, S.J. Ben Yoo, T. Hayashi, T. Nagashima, and T. Sasaki, “1705-km Transmission over Coupled-Core Fibre Supporting 6 Spatial Modes,” ECOC 2014, Cannes - France, PD.3.2 (2014).

Esmaeelpour, M.

R. Ryf, S. Randel, A.H. Gnauck, C. Bolle, A. Sierra, S. Mumtaz, M. Esmaeelpour, E.C. Burrows, R. Essiambre, P.J. Winzer, D. W. Peckham, A.H. McCurdy, and R. Lingle, “Mode-Division Multiplexing Over 96 km of Few-Mode Fiber Using Coherent 6 × 6 MIMO Processing,” J. Lightwave Technol. 30, 521–531 (2012). doi:
[Crossref]

R. Ryf, N.K. Fontaine, B. Guan, B. Huang, M. Esmaeelpour, S. Randel, A. H. Gnauck, S. Chandrasekhar, A. Adamiecki, G. Raybon, R.W. Tkach, R. Shubochkin, Y. Sun, and R. Lingle, “305-km Combined Wavelength and Mode-Multiplexed Transmission over Conventional Graded-Index Multimode Fibre,” ECOC 2014, Cannes - France, PD.3.5 (2014).

Essiambre, R.

Essiambre, R.-J.

R. Ryf, R.-J. Essiambre, A. H. Gnauck, S. Randel, M. A. Mestre, C. Schmidt, P. J. Winzer, R. Delbue, P. Pupalaikis, A. Sureka, T. Hayashi, T. Taru, and T. Sasaki, “SDM Transmission over 4200-km 3-Core Microstructured Fiber,” OFC 2011, PDP5C.2 (2011).

R. Ryf, N.K. Fontaine, B. Guan, R.-J. Essiambre, S. Randel, A. H. Gnauck, S. Chandrasekhar, A. Adamiecki, G. Raybon, B. Ercan, R. P. Scott, S.J. Ben Yoo, T. Hayashi, T. Nagashima, and T. Sasaki, “1705-km Transmission over Coupled-Core Fibre Supporting 6 Spatial Modes,” ECOC 2014, Cannes - France, PD.3.2 (2014).

Fontaine, N.K.

R. Ryf, N.K. Fontaine, B. Guan, R.-J. Essiambre, S. Randel, A. H. Gnauck, S. Chandrasekhar, A. Adamiecki, G. Raybon, B. Ercan, R. P. Scott, S.J. Ben Yoo, T. Hayashi, T. Nagashima, and T. Sasaki, “1705-km Transmission over Coupled-Core Fibre Supporting 6 Spatial Modes,” ECOC 2014, Cannes - France, PD.3.2 (2014).

R. Ryf, N.K. Fontaine, B. Guan, B. Huang, M. Esmaeelpour, S. Randel, A. H. Gnauck, S. Chandrasekhar, A. Adamiecki, G. Raybon, R.W. Tkach, R. Shubochkin, Y. Sun, and R. Lingle, “305-km Combined Wavelength and Mode-Multiplexed Transmission over Conventional Graded-Index Multimode Fibre,” ECOC 2014, Cannes - France, PD.3.5 (2014).

Gardiner, C.W.

C.W. Gardiner, “Stochastic methods for physics, chemistry and natural sciences,” Springer-Verlag, NY, 1983.

Gnauck, A. H.

R. Ryf, N.K. Fontaine, B. Guan, B. Huang, M. Esmaeelpour, S. Randel, A. H. Gnauck, S. Chandrasekhar, A. Adamiecki, G. Raybon, R.W. Tkach, R. Shubochkin, Y. Sun, and R. Lingle, “305-km Combined Wavelength and Mode-Multiplexed Transmission over Conventional Graded-Index Multimode Fibre,” ECOC 2014, Cannes - France, PD.3.5 (2014).

R. Ryf, N.K. Fontaine, B. Guan, R.-J. Essiambre, S. Randel, A. H. Gnauck, S. Chandrasekhar, A. Adamiecki, G. Raybon, B. Ercan, R. P. Scott, S.J. Ben Yoo, T. Hayashi, T. Nagashima, and T. Sasaki, “1705-km Transmission over Coupled-Core Fibre Supporting 6 Spatial Modes,” ECOC 2014, Cannes - France, PD.3.2 (2014).

R. Ryf, R.-J. Essiambre, A. H. Gnauck, S. Randel, M. A. Mestre, C. Schmidt, P. J. Winzer, R. Delbue, P. Pupalaikis, A. Sureka, T. Hayashi, T. Taru, and T. Sasaki, “SDM Transmission over 4200-km 3-Core Microstructured Fiber,” OFC 2011, PDP5C.2 (2011).

Gnauck, A.H.

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, B.

R. Ryf, N.K. Fontaine, B. Guan, B. Huang, M. Esmaeelpour, S. Randel, A. H. Gnauck, S. Chandrasekhar, A. Adamiecki, G. Raybon, R.W. Tkach, R. Shubochkin, Y. Sun, and R. Lingle, “305-km Combined Wavelength and Mode-Multiplexed Transmission over Conventional Graded-Index Multimode Fibre,” ECOC 2014, Cannes - France, PD.3.5 (2014).

R. Ryf, N.K. Fontaine, B. Guan, R.-J. Essiambre, S. Randel, A. H. Gnauck, S. Chandrasekhar, A. Adamiecki, G. Raybon, B. Ercan, R. P. Scott, S.J. Ben Yoo, T. Hayashi, T. Nagashima, and T. Sasaki, “1705-km Transmission over Coupled-Core Fibre Supporting 6 Spatial Modes,” ECOC 2014, Cannes - France, PD.3.2 (2014).

Hayashi, T.

R. Ryf, R.-J. Essiambre, A. H. Gnauck, S. Randel, M. A. Mestre, C. Schmidt, P. J. Winzer, R. Delbue, P. Pupalaikis, A. Sureka, T. Hayashi, T. Taru, and T. Sasaki, “SDM Transmission over 4200-km 3-Core Microstructured Fiber,” OFC 2011, PDP5C.2 (2011).

R. Ryf, N.K. Fontaine, B. Guan, R.-J. Essiambre, S. Randel, A. H. Gnauck, S. Chandrasekhar, A. Adamiecki, G. Raybon, B. Ercan, R. P. Scott, S.J. Ben Yoo, T. Hayashi, T. Nagashima, and T. Sasaki, “1705-km Transmission over Coupled-Core Fibre Supporting 6 Spatial Modes,” ECOC 2014, Cannes - France, PD.3.2 (2014).

Ho, K-P.

Huang, B.

R. Ryf, N.K. Fontaine, B. Guan, B. Huang, M. Esmaeelpour, S. Randel, A. H. Gnauck, S. Chandrasekhar, A. Adamiecki, G. Raybon, R.W. Tkach, R. Shubochkin, Y. Sun, and R. Lingle, “305-km Combined Wavelength and Mode-Multiplexed Transmission over Conventional Graded-Index Multimode Fibre,” ECOC 2014, Cannes - France, PD.3.5 (2014).

Kahn, J.M

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]

Lingle, R.

R. Ryf, S. Randel, A.H. Gnauck, C. Bolle, A. Sierra, S. Mumtaz, M. Esmaeelpour, E.C. Burrows, R. Essiambre, P.J. Winzer, D. W. Peckham, A.H. McCurdy, and R. Lingle, “Mode-Division Multiplexing Over 96 km of Few-Mode Fiber Using Coherent 6 × 6 MIMO Processing,” J. Lightwave Technol. 30, 521–531 (2012). doi:
[Crossref]

R. Ryf, N.K. Fontaine, B. Guan, B. Huang, M. Esmaeelpour, S. Randel, A. H. Gnauck, S. Chandrasekhar, A. Adamiecki, G. Raybon, R.W. Tkach, R. Shubochkin, Y. Sun, and R. Lingle, “305-km Combined Wavelength and Mode-Multiplexed Transmission over Conventional Graded-Index Multimode Fibre,” ECOC 2014, Cannes - France, PD.3.5 (2014).

McCurdy, A.H.

Mecozzi, A.

Mestre, M. A.

R. Ryf, R.-J. Essiambre, A. H. Gnauck, S. Randel, M. A. Mestre, C. Schmidt, P. J. Winzer, R. Delbue, P. Pupalaikis, A. Sureka, T. Hayashi, T. Taru, and T. Sasaki, “SDM Transmission over 4200-km 3-Core Microstructured Fiber,” OFC 2011, PDP5C.2 (2011).

Mumtaz, S.

Nagashima, T.

R. Ryf, N.K. Fontaine, B. Guan, R.-J. Essiambre, S. Randel, A. H. Gnauck, S. Chandrasekhar, A. Adamiecki, G. Raybon, B. Ercan, R. P. Scott, S.J. Ben Yoo, T. Hayashi, T. Nagashima, and T. Sasaki, “1705-km Transmission over Coupled-Core Fibre Supporting 6 Spatial Modes,” ECOC 2014, Cannes - France, PD.3.2 (2014).

Peckham, D. W.

Pupalaikis, P.

R. Ryf, R.-J. Essiambre, A. H. Gnauck, S. Randel, M. A. Mestre, C. Schmidt, P. J. Winzer, R. Delbue, P. Pupalaikis, A. Sureka, T. Hayashi, T. Taru, and T. Sasaki, “SDM Transmission over 4200-km 3-Core Microstructured Fiber,” OFC 2011, PDP5C.2 (2011).

Randel, S.

R. Ryf, S. Randel, A.H. Gnauck, C. Bolle, A. Sierra, S. Mumtaz, M. Esmaeelpour, E.C. Burrows, R. Essiambre, P.J. Winzer, D. W. Peckham, A.H. McCurdy, and R. Lingle, “Mode-Division Multiplexing Over 96 km of Few-Mode Fiber Using Coherent 6 × 6 MIMO Processing,” J. Lightwave Technol. 30, 521–531 (2012). doi:
[Crossref]

R. Ryf, R.-J. Essiambre, A. H. Gnauck, S. Randel, M. A. Mestre, C. Schmidt, P. J. Winzer, R. Delbue, P. Pupalaikis, A. Sureka, T. Hayashi, T. Taru, and T. Sasaki, “SDM Transmission over 4200-km 3-Core Microstructured Fiber,” OFC 2011, PDP5C.2 (2011).

R. Ryf, N.K. Fontaine, B. Guan, R.-J. Essiambre, S. Randel, A. H. Gnauck, S. Chandrasekhar, A. Adamiecki, G. Raybon, B. Ercan, R. P. Scott, S.J. Ben Yoo, T. Hayashi, T. Nagashima, and T. Sasaki, “1705-km Transmission over Coupled-Core Fibre Supporting 6 Spatial Modes,” ECOC 2014, Cannes - France, PD.3.2 (2014).

R. Ryf, N.K. Fontaine, B. Guan, B. Huang, M. Esmaeelpour, S. Randel, A. H. Gnauck, S. Chandrasekhar, A. Adamiecki, G. Raybon, R.W. Tkach, R. Shubochkin, Y. Sun, and R. Lingle, “305-km Combined Wavelength and Mode-Multiplexed Transmission over Conventional Graded-Index Multimode Fibre,” ECOC 2014, Cannes - France, PD.3.5 (2014).

Raybon, G.

R. Ryf, N.K. Fontaine, B. Guan, B. Huang, M. Esmaeelpour, S. Randel, A. H. Gnauck, S. Chandrasekhar, A. Adamiecki, G. Raybon, R.W. Tkach, R. Shubochkin, Y. Sun, and R. Lingle, “305-km Combined Wavelength and Mode-Multiplexed Transmission over Conventional Graded-Index Multimode Fibre,” ECOC 2014, Cannes - France, PD.3.5 (2014).

R. Ryf, N.K. Fontaine, B. Guan, R.-J. Essiambre, S. Randel, A. H. Gnauck, S. Chandrasekhar, A. Adamiecki, G. Raybon, B. Ercan, R. P. Scott, S.J. Ben Yoo, T. Hayashi, T. Nagashima, and T. Sasaki, “1705-km Transmission over Coupled-Core Fibre Supporting 6 Spatial Modes,” ECOC 2014, Cannes - France, PD.3.2 (2014).

Ryf, R.

R. Ryf, S. Randel, A.H. Gnauck, C. Bolle, A. Sierra, S. Mumtaz, M. Esmaeelpour, E.C. Burrows, R. Essiambre, P.J. Winzer, D. W. Peckham, A.H. McCurdy, and R. Lingle, “Mode-Division Multiplexing Over 96 km of Few-Mode Fiber Using Coherent 6 × 6 MIMO Processing,” J. Lightwave Technol. 30, 521–531 (2012). doi:
[Crossref]

R. Ryf, R.-J. Essiambre, A. H. Gnauck, S. Randel, M. A. Mestre, C. Schmidt, P. J. Winzer, R. Delbue, P. Pupalaikis, A. Sureka, T. Hayashi, T. Taru, and T. Sasaki, “SDM Transmission over 4200-km 3-Core Microstructured Fiber,” OFC 2011, PDP5C.2 (2011).

R. Ryf, N.K. Fontaine, B. Guan, R.-J. Essiambre, S. Randel, A. H. Gnauck, S. Chandrasekhar, A. Adamiecki, G. Raybon, B. Ercan, R. P. Scott, S.J. Ben Yoo, T. Hayashi, T. Nagashima, and T. Sasaki, “1705-km Transmission over Coupled-Core Fibre Supporting 6 Spatial Modes,” ECOC 2014, Cannes - France, PD.3.2 (2014).

R. Ryf, N.K. Fontaine, B. Guan, B. Huang, M. Esmaeelpour, S. Randel, A. H. Gnauck, S. Chandrasekhar, A. Adamiecki, G. Raybon, R.W. Tkach, R. Shubochkin, Y. Sun, and R. Lingle, “305-km Combined Wavelength and Mode-Multiplexed Transmission over Conventional Graded-Index Multimode Fibre,” ECOC 2014, Cannes - France, PD.3.5 (2014).

Sasaki, T.

R. Ryf, N.K. Fontaine, B. Guan, R.-J. Essiambre, S. Randel, A. H. Gnauck, S. Chandrasekhar, A. Adamiecki, G. Raybon, B. Ercan, R. P. Scott, S.J. Ben Yoo, T. Hayashi, T. Nagashima, and T. Sasaki, “1705-km Transmission over Coupled-Core Fibre Supporting 6 Spatial Modes,” ECOC 2014, Cannes - France, PD.3.2 (2014).

R. Ryf, R.-J. Essiambre, A. H. Gnauck, S. Randel, M. A. Mestre, C. Schmidt, P. J. Winzer, R. Delbue, P. Pupalaikis, A. Sureka, T. Hayashi, T. Taru, and T. Sasaki, “SDM Transmission over 4200-km 3-Core Microstructured Fiber,” OFC 2011, PDP5C.2 (2011).

Schmidt, C.

R. Ryf, R.-J. Essiambre, A. H. Gnauck, S. Randel, M. A. Mestre, C. Schmidt, P. J. Winzer, R. Delbue, P. Pupalaikis, A. Sureka, T. Hayashi, T. Taru, and T. Sasaki, “SDM Transmission over 4200-km 3-Core Microstructured Fiber,” OFC 2011, PDP5C.2 (2011).

Scott, R. P.

R. Ryf, N.K. Fontaine, B. Guan, R.-J. Essiambre, S. Randel, A. H. Gnauck, S. Chandrasekhar, A. Adamiecki, G. Raybon, B. Ercan, R. P. Scott, S.J. Ben Yoo, T. Hayashi, T. Nagashima, and T. Sasaki, “1705-km Transmission over Coupled-Core Fibre Supporting 6 Spatial Modes,” ECOC 2014, Cannes - France, PD.3.2 (2014).

Shtaif, M.

Shubochkin, R.

R. Ryf, N.K. Fontaine, B. Guan, B. Huang, M. Esmaeelpour, S. Randel, A. H. Gnauck, S. Chandrasekhar, A. Adamiecki, G. Raybon, R.W. Tkach, R. Shubochkin, Y. Sun, and R. Lingle, “305-km Combined Wavelength and Mode-Multiplexed Transmission over Conventional Graded-Index Multimode Fibre,” ECOC 2014, Cannes - France, PD.3.5 (2014).

Sierra, A.

Sun, Y.

R. Ryf, N.K. Fontaine, B. Guan, B. Huang, M. Esmaeelpour, S. Randel, A. H. Gnauck, S. Chandrasekhar, A. Adamiecki, G. Raybon, R.W. Tkach, R. Shubochkin, Y. Sun, and R. Lingle, “305-km Combined Wavelength and Mode-Multiplexed Transmission over Conventional Graded-Index Multimode Fibre,” ECOC 2014, Cannes - France, PD.3.5 (2014).

Sureka, A.

R. Ryf, R.-J. Essiambre, A. H. Gnauck, S. Randel, M. A. Mestre, C. Schmidt, P. J. Winzer, R. Delbue, P. Pupalaikis, A. Sureka, T. Hayashi, T. Taru, and T. Sasaki, “SDM Transmission over 4200-km 3-Core Microstructured Fiber,” OFC 2011, PDP5C.2 (2011).

Taru, T.

R. Ryf, R.-J. Essiambre, A. H. Gnauck, S. Randel, M. A. Mestre, C. Schmidt, P. J. Winzer, R. Delbue, P. Pupalaikis, A. Sureka, T. Hayashi, T. Taru, and T. Sasaki, “SDM Transmission over 4200-km 3-Core Microstructured Fiber,” OFC 2011, PDP5C.2 (2011).

Tkach, R.W.

R. Ryf, N.K. Fontaine, B. Guan, B. Huang, M. Esmaeelpour, S. Randel, A. H. Gnauck, S. Chandrasekhar, A. Adamiecki, G. Raybon, R.W. Tkach, R. Shubochkin, Y. Sun, and R. Lingle, “305-km Combined Wavelength and Mode-Multiplexed Transmission over Conventional Graded-Index Multimode Fibre,” ECOC 2014, Cannes - France, PD.3.5 (2014).

Winzer, P. J.

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, 11718–11733 (2012).
[Crossref] [PubMed]

R. Ryf, R.-J. Essiambre, A. H. Gnauck, S. Randel, M. A. Mestre, C. Schmidt, P. J. Winzer, R. Delbue, P. Pupalaikis, A. Sureka, T. Hayashi, T. Taru, and T. Sasaki, “SDM Transmission over 4200-km 3-Core Microstructured Fiber,” OFC 2011, PDP5C.2 (2011).

Winzer, P.J.

J. Lightwave Technol. (2)

Opt. Express (2)

Opt. Lett. (1)

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)

The coefficient κ2 is equivalent to the mean square differential-group-delay (DGD) per kilometer, whose specification is standard in single-mode fiber systems [6].

C.W. Gardiner, “Stochastic methods for physics, chemistry and natural sciences,” Springer-Verlag, NY, 1983.

R. Ryf, R.-J. Essiambre, A. H. Gnauck, S. Randel, M. A. Mestre, C. Schmidt, P. J. Winzer, R. Delbue, P. Pupalaikis, A. Sureka, T. Hayashi, T. Taru, and T. Sasaki, “SDM Transmission over 4200-km 3-Core Microstructured Fiber,” OFC 2011, PDP5C.2 (2011).

R. Ryf, N.K. Fontaine, B. Guan, R.-J. Essiambre, S. Randel, A. H. Gnauck, S. Chandrasekhar, A. Adamiecki, G. Raybon, B. Ercan, R. P. Scott, S.J. Ben Yoo, T. Hayashi, T. Nagashima, and T. Sasaki, “1705-km Transmission over Coupled-Core Fibre Supporting 6 Spatial Modes,” ECOC 2014, Cannes - France, PD.3.2 (2014).

R. Ryf, N.K. Fontaine, B. Guan, B. Huang, M. Esmaeelpour, S. Randel, A. H. Gnauck, S. Chandrasekhar, A. Adamiecki, G. Raybon, R.W. Tkach, R. Shubochkin, Y. Sun, and R. Lingle, “305-km Combined Wavelength and Mode-Multiplexed Transmission over Conventional Graded-Index Multimode Fibre,” ECOC 2014, Cannes - France, PD.3.5 (2014).

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Figures (1)

Fig. 1
Fig. 1 Left panel: The mode-averaged intensity waveform I(t; z) at several points z along the link obtained by transmitting a Nyquist signal with bandwidth B = 20 GHz in the three-core fiber described in [7]. The solid blue curves are the results of simulations, whereas the dashed black curves refer to Eqs. (5) and (6). The red dotted curves correspond to the IIR waveform of Eq. (6) normalized so that their peak value is one. Right top panel: The broadening T2(z) of the propagating intensity waveform I(t; z) versus propagation distance. The squares show simulation results, whereas the dotted red curve is a plot of T2 = κ2z – the temporal mean square width of the IIR waveform. Right bottom panel: Relative deviation of the broadening T2(z) from the theoretical value, ε = [T2(z) − κ2z]/κ2z, versus propagation distance. The solid thin curves show the standard deviation envelope ±〈ε21/2, as given in Eq. (16).

Equations (16)

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H ( t ) = d ω 2 π exp ( i ω t ) U ( L , 0 ; ω ) ψ ˜ 0 ( ω ) ,
I ( t ) = 1 2 N k = 1 2 N [ j = 1 2 N | H j , k ( t ) | 2 ] = 1 2 N Trace [ H ( t ) H ( t ) ] ,
I ˜ ( Ω ) = d ω 2 π ψ ˜ 0 * ( ω ) ψ ˜ 0 ( ω + Ω ) R ˜ ( Ω , ω ) ,
R ˜ ( Ω , ω ; z ) = 1 2 N Trace [ U ( z , 0 ; ω + Ω ) U ( z , 0 ; ω ) ] ,
I ( t ) = d t r ( t t ) I 0 ( t ) ,
r ( t ) = 1 2 π T 2 exp ( t 2 2 T 2 ) ,
d r ˜ ( Ω ; z ) = 1 2 N { Trace [ U ( z , 0 ; ω + Ω ) d U ( z , 0 ; ω ) ] + Trace [ d U ( z , 0 ; ω + Ω ) U ( z , 0 ; ω ) ] + Trace [ d U ( z , 0 ; ω + Ω ) d U ( z , 0 ; ω ) ] } .
d U ( z , 0 ; ω ) = [ i ω β ω Λ 2 N d z ω 2 κ 2 2 d z ] U ( z , 0 ; ω ) ,
M n ( z ) = t n I ( t ; z ) d t = ( i Ω ) n I ˜ ( Ω ; z ) | Ω = 0 ,
M 1 ( z ) = 1 2 N d ω 2 π ψ 0 * ( ω ) Trace { [ i ω ψ ˜ 0 ( ω ) U ( z , 0 ; ω ) ] U ( z , 0 ; ω ) } ,
M 2 ( z ) = 1 2 N d ω 2 π Trace { [ i ω ψ ˜ 0 ( ω ) U ( z , 0 ; ω ) ] [ i ω ψ ˜ 0 ( ω ) U ( z , 0 ; ω ) ] } .
i ω U ( z , 0 ; ω ) = τ ( ω ) Λ 2 N U ( z , 0 ; ω ) ,
T 2 ( z ) = d ω 2 π | ψ ˜ 0 ( ω ) | 2 τ 2 ( ω ) ( 2 N ) 2 ,
ε 2 = 1 T 4 B 2 π B π B d ω 2 π π B π B d ω 2 π τ 2 ( ω ) τ 2 ( ω ) ( 2 N ) 4 1 .
ε 2 = 1 b 2 ( 4 N 2 1 ) b / 2 b / 2 d w b / 2 b / 2 d w f ( w w )
ε 2 16 π 3 ( 4 N 2 1 ) b = 8 π 3 N 4 N 2 1 B MD B ,

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