Abstract

We study random coupling induced crosstalk between groups of degenerate modes in spatially multiplexed optical transmission. Our analysis shows that the average crosstalk is primarily determined by the wavenumber mismatch, by the correlation length of the random perturbations, and by the coherence length of the degenerate modes, whereas the effect of a deterministic group velocity difference is negligible. The standard deviation of the crosstalk is shown to be comparable to its average value, implying that crosstalk measurements are inherently noisy.

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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, “Statistics of group delays in multi-mode fibers with strong mode coupling,” J. Light-wave Technol. 29, 3119–3128 (2011).
    [Crossref]
  3. 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 96 km of few-mode fiber using coherent 6 × 6 MIMO processing,” J. Lightwave Technol. 30, 521–531 (2012).
    [Crossref]
  4. C. Koebele, M. Salsi, D. Sperti, P. Tran, P. Brindel, H. Mardoyan, S. Bigo, A. Boutin, F. Verluise, P. Sillard, M. Astruc, L. Provost, F. Cerou, and G. Charlet, “Two mode transmission at 2x100Gb/s, over 40km-long prototype few-mode fiber, using LCOS-based programmable mode multiplexer, and demultiplexer.” Opt. Express 19, 16593–16600 (2011).
    [Crossref] [PubMed]
  5. D. Gloge, “Weakly guiding fibers,” Appl. Opt. 10, 2252–2258 (1971).
    [Crossref] [PubMed]
  6. R. Ryf, R. J. Essiambre, S. Randel, M. A. Mestre, C. Schmidt, and P. J. Winzer, “Impulse response analysis of coupled-core 3-core fibers,” Proceedings of ECOC 2012, Paper Mo.1.F (2012).
  7. 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]
  8. The deterministic coupling between members of an LP mode group [9] is not included in B as its effect is assumed to be masked by the presence of the random perturbations which are represented by the second term in the square brackets of Eq. (1). This is consistent with the fact that strong random coupling between the constituent pseudo-modes is observed in experiments.
  9. H. Kogelnik and P. J. Winzer, “Modal birefringence in weakly guiding fibers,” J. Lightwave Technol. 30, 2240–2243 (2012).
    [Crossref]
  10. 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]
  11. Strictly speaking the matrices Ul and Uq are independent up to a scalar phase factor that is negligible as compared to the phase difference due to the deterministic wavenumber mismatch.
  12. The quantity that we refer to as the coherence length of the field describes the propagation distance along which the field decorrelates due to the fiber perturbations. It is not related to the coherence length of the light source.
  13. 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]
  14. J. Vuong, P. Ramantanis, A. Seck, D. Bendimerad, and Y. Frignac, “Understanding discrete linear mode coupling in few-mode fiber transmission systems,” Proceedings of ECOC 2011, Paper Tu.5.B.2 (2011).
  15. C. W. Gardiner, Stochastic methods for physics, chemistry and natural sciences (Springer-Verlag, NY, 1983).

2012 (3)

2011 (3)

2001 (1)

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]

1971 (1)

Antonelli, C.

Astruc, M.

Bendimerad, D.

J. Vuong, P. Ramantanis, A. Seck, D. Bendimerad, and Y. Frignac, “Understanding discrete linear mode coupling in few-mode fiber transmission systems,” Proceedings of ECOC 2011, Paper Tu.5.B.2 (2011).

Bigo, S.

Bolle, C.

Boutin, A.

Brindel, P.

Burrows, E. C.

Cerou, F.

Charlet, G.

Esmaeelpour, M.

Essiambre, R. J.

Foschini, G. J.

Frignac, Y.

J. Vuong, P. Ramantanis, A. Seck, D. Bendimerad, and Y. Frignac, “Understanding discrete linear mode coupling in few-mode fiber transmission systems,” Proceedings of ECOC 2011, Paper Tu.5.B.2 (2011).

Galtarossa, A.

Gardiner, C. W.

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

Gloge, D.

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]

Ho, K. P.

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

Kahn, J. M.

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

Koebele, C.

Kogelnik, H.

H. Kogelnik and P. J. Winzer, “Modal birefringence in weakly guiding fibers,” J. Lightwave Technol. 30, 2240–2243 (2012).
[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]

Lingle, R.

Mardoyan, H.

McCurdy, A. H.

Mecozzi, A.

Mestre, M. A.

R. Ryf, R. J. Essiambre, S. Randel, M. A. Mestre, C. Schmidt, and P. J. Winzer, “Impulse response analysis of coupled-core 3-core fibers,” Proceedings of ECOC 2012, Paper Mo.1.F (2012).

Mumtaz, S.

Palmieri, L.

Peckham, D. W.

Provost, L.

Ramantanis, P.

J. Vuong, P. Ramantanis, A. Seck, D. Bendimerad, and Y. Frignac, “Understanding discrete linear mode coupling in few-mode fiber transmission systems,” Proceedings of ECOC 2011, Paper Tu.5.B.2 (2011).

Randel, S.

Ryf, R.

Salsi, M.

Schiano, M.

Schmidt, C.

R. Ryf, R. J. Essiambre, S. Randel, M. A. Mestre, C. Schmidt, and P. J. Winzer, “Impulse response analysis of coupled-core 3-core fibers,” Proceedings of ECOC 2012, Paper Mo.1.F (2012).

Seck, A.

J. Vuong, P. Ramantanis, A. Seck, D. Bendimerad, and Y. Frignac, “Understanding discrete linear mode coupling in few-mode fiber transmission systems,” Proceedings of ECOC 2011, Paper Tu.5.B.2 (2011).

Shtaif, M.

Sierra, A.

Sillard, P.

Sperti, D.

Tambosso, T.

Tran, P.

Verluise, F.

Vuong, J.

J. Vuong, P. Ramantanis, A. Seck, D. Bendimerad, and Y. Frignac, “Understanding discrete linear mode coupling in few-mode fiber transmission systems,” Proceedings of ECOC 2011, Paper Tu.5.B.2 (2011).

Winzer, P. J.

Appl. Opt. (1)

J. Light-wave Technol. (1)

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

J. Lightwave Technol. (2)

Opt. Express (3)

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 (6)

Strictly speaking the matrices Ul and Uq are independent up to a scalar phase factor that is negligible as compared to the phase difference due to the deterministic wavenumber mismatch.

The quantity that we refer to as the coherence length of the field describes the propagation distance along which the field decorrelates due to the fiber perturbations. It is not related to the coherence length of the light source.

R. Ryf, R. J. Essiambre, S. Randel, M. A. Mestre, C. Schmidt, and P. J. Winzer, “Impulse response analysis of coupled-core 3-core fibers,” Proceedings of ECOC 2012, Paper Mo.1.F (2012).

J. Vuong, P. Ramantanis, A. Seck, D. Bendimerad, and Y. Frignac, “Understanding discrete linear mode coupling in few-mode fiber transmission systems,” Proceedings of ECOC 2011, Paper Tu.5.B.2 (2011).

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

The deterministic coupling between members of an LP mode group [9] is not included in B as its effect is assumed to be masked by the presence of the random perturbations which are represented by the second term in the square brackets of Eq. (1). This is consistent with the fact that strong random coupling between the constituent pseudo-modes is observed in experiments.

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Figures (1)

Fig. 1
Fig. 1

(a) Average energy coupled from non-degenerate group q to a two-fold degenerate group l versus propagation distance normalized to Lc. The solid black line represents the complete expression given by Eq. (7), whereas the thin red and the dotted green lines were obtained in simulation with different parameters (see text). The dashed black line shows the simplified crosstalk expression given by Eq. (8). The shaded area marks one standard deviation from the mean as given by Eq. (9). (b) Standard deviation of the crosstalk σulq as obtained from Eq. (9) (thick-black curve) and from numerical simulations (thin red). (c) Crosstalk probability density function (circles) and chi-squared distribution fit (solid line).

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Equations (9)

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d ℰ → d z = i [ B + 1 2 N b → ˜ ( ω , z ) ⋅ Λ → ( 2 N ) ] ℰ → ,
d e → l d z = i 2 N b l l e → l + i 2 N ∑ j ≠ l M e i β j l z b l j e → n ,
e → l ( z ) = U l ( z , 0 ) e → l ( 0 ) + i 2 N ∑ j ≠ l M ∫ 0 z U l ( z , z ′ ) b l j ( z ′ ) e → j ( z ′ ) e i β j l z ′ d z ′ ,
e → l ( z ) = i 2 N ∫ 0 z e i β q l z ′ U l ( z , z ′ ) b l q ( z ′ ) U q ( z ′ , 0 ) e → q ( 0 ) d z ′ .
u l q = ∫ − ∞ + ∞ P ˜ ( ω ) | e → l ( z , ω ) | 2 d ω 2 π ,
〈 | e → l ( z ) | 2 〉 = g l N Re { ∫ 0 z e − i β q l ξ e − ( 1 / L l + 1 / L q ) ξ ( z − ξ ) f ( ξ ) d ξ } .
〈 u l q 〉 = 2 n 0 g l N Re { ∫ − ∞ + ∞ P ˜ ( ω ) e − K z + K z − 1 K 2 } d ω 2 π ,
〈 u l q 〉 = n 0 β l q , 0 2 2 g l N z L eff ( 1 + ε l q ) ,
σ u l q = 〈 u l q 〉 2 g l .

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