Abstract

In this paper, we evaluate experimentally and model theoretically the intra- and inter-core crosstalk between the polarized core modes in single-mode multi-core fiber media including temporal and longitudinal birefringent effects. Specifically, extensive experimental results on a four-core fiber indicate that the temporal fluctuation of fiber birefringence modifies the intra- and inter-core crosstalk behavior in both linear and nonlinear optical power regimes. To gain theoretical insight into the experimental results, we introduce an accurate multi-core fiber model based on local modes and perturbation theory, which is derived from the Maxwell equations including both longitudinal and temporal birefringent effects. Numerical calculations based on the developed theory are found to be in good agreement with the experimental data.

© 2016 Optical Society of America

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References

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  1. B. J. Puttnam, R. S. Luis, W. Klaus, J. Sakaguchi, J.-M. Delgado Mendinueta, Y. Awaji, N. Wada, Y. Tamura, T. Hayashi, M. Hirano, and J. Marciante, “2.15 Pb/s transmission using a 22 core homogeneous single-mode multi-core fiber and wideband optical comb,” in Eur. Conf. Opt. Commun. (ECOC, 2015), paper PDP 3.1.
    [Crossref]
  2. T. Mizuno, H. Takara, A. Sano, and Y. Miyamoto, “Dense space-division multiplexing transmission systems using multi-core and multi-mode fiber,” J. Lightwave Technol. 34(2), 582–592 (2016).
    [Crossref]
  3. M. Morant, A. Macho, and R. Llorente, “On the suitability of multicore fiber for LTE-advanced MIMO optical fronthaul systems,” J. Lightwave Technol. 34(2), 676–682 (2016).
    [Crossref]
  4. T. Hayashi, T. Sasaki, E. Sasaoka, K. Saitoh, and M. Koshiba, “Physical interpretation of intercore crosstalk in multicore fiber: effects of macrobend, structure fluctuation, and microbend,” Opt. Express 21(5), 5401–5412 (2013).
    [Crossref] [PubMed]
  5. J. M. Fini, B. Zhu, T. F. Taunay, M. F. Yan, and K. S. Abedin, “Statistical models of multicore fiber crosstalk including time delays,” J. Lightwave Technol. 30(12), 2003–2010 (2012).
    [Crossref]
  6. R. S. Luís, B. J. Puttnam, A. V. T. Cartaxo, W. Klaus, J. M. D. Mendinueta, Y. Awaji, N. Wada, T. Nakanishi, T. Hayashi, and T. Sasaki, “Time and modulation frequency dependence of crosstalk in homogeneous multi-core fibers,” J. Lightwave Technol. 34(2), 441–447 (2016).
    [Crossref]
  7. T. Hayashi, T. Taru, O. Shimakawa, T. Sasaki, and E. Sasaoka, “Characterization of crosstalk in ultra-low crosstalk multi-core fiber,” J. Lightwave Technol. 30(4), 583–589 (2012).
    [Crossref]
  8. J. M. Fini, B. Zhu, T. F. Taunay, and M. F. Yan, “Statistics of crosstalk in bent multicore fibers,” Opt. Express 18(14), 15122–15129 (2010).
    [Crossref] [PubMed]
  9. M. Koshiba, K. Saitoh, K. Takenaga, and S. Matsuo, “Multi-core fiber design and analysis: coupled-mode theory and coupled-power theory,” Opt. Express 19(26), B102–B111 (2011).
    [Crossref] [PubMed]
  10. T. Hayashi, T. Taru, O. Shimakawa, T. Sasaki, and E. Sasaoka, “Design and fabrication of ultra-low crosstalk and low-loss multi-core fiber,” Opt. Express 19(17), 16576–16592 (2011).
    [Crossref] [PubMed]
  11. M. Koshiba, K. Saitoh, K. Takenaga, and S. Matsuo, “Analytical expression of average power-coupling coefficients for estimating intercore crosstalk in multicore fibers,” IEEE Photonics J. 4(5), 1987–1995 (2012).
    [Crossref]
  12. A. Macho, M. Morant, and R. Llorente, “Experimental evaluation of nonlinear crosstalk in multi-core fiber,” Opt. Express 23(14), 18712–18720 (2015).
    [Crossref] [PubMed]
  13. A. Macho, M. Morant, and R. Llorente, “Unified model of linear and nonlinear crosstalk in multi-core fiber,” J. Lightwave Technol. 34(13), 3035–3046 (2016).
    [Crossref]
  14. A. Mecozzi, C. Antonelli, and M. Shtaif, “Coupled Manakov equations in multimode fibers with strongly coupled groups of modes,” Opt. Express 20(21), 23436–23441 (2012).
    [Crossref] [PubMed]
  15. A. Mecozzi, C. Antonelli, and M. Shtaif, “Nonlinear propagation in multi-mode fibers in the strong coupling regime,” Opt. Express 20(11), 11673–11678 (2012).
    [Crossref] [PubMed]
  16. S. Mumtaz, R. J. Essiambre, and G. P. Agrawal, “Nonlinear propagation in multimode and multicore fibers: generalization of the manakov equations,” J. Lightwave Technol. 31(3), 398–406 (2013).
    [Crossref]
  17. S. Mumtaz, R.-J. Essiambre, and G. P. Agrawal, “Birefringence effects in space-division multiplexed fiber transmission systems: generalization of Manakov equation,” in IEEE Photonics Society Summer Topical Meeting Series (IEEE, 2012), paper MC3.5.
  18. L. Palmieri and A. Galtarossa, “Coupling effects among degenerate modes in multimode optical fibers,” IEEE Photonics J. 6(6), 0600408 (2014).
    [Crossref]
  19. C. Antonelli, A. Mecozzi, and M. Shtaif, “The delay spread in fibers for SDM transmission: dependence on fiber parameters and perturbations,” Opt. Express 23(3), 2196–2202 (2015).
    [Crossref] [PubMed]
  20. D. Marcuse, “Coupled-mode theory for anisotropic optical waveguides,” Bell Syst. Tech. J. 54(6), 985–995 (1975).
    [Crossref]
  21. D. Wong, “Thermal stability of intrinsic stress birefringence in optical fibers,” J. Lightwave Technol. 8(11), 1757–1761 (1990).
    [Crossref]
  22. D. Marcuse, Theory of Dielectric Optical Waveguides (Elsevier, 1974), Ch. 3.
  23. D. Gloge, “Weakly guiding fibers,” Appl. Opt. 10(10), 2252–2258 (1971).
    [Crossref] [PubMed]
  24. G. P. Agrawal, Nonlinear Fiber Optics, 5th ed. (Elsevier, 2013).
  25. R. W. Boyd, Nonlinear Optics, 3rd ed. (Elsevier, 2008), Chap. 1.
  26. A. M. Weiner, Ultrafast Optics, 1st ed. (John Wiley and Sons, 2009), Chap. 6.
  27. A. V. T. Cartaxo, R. S. Luís, B. J. Puttnam, T. Hayashi, Y. Awaji, and N. Wada, “Dispersion impact on the crosstalk amplitude response of homogeneous multi-core fibers,” IEEE Photonics Technol. Lett. 28(17), 1858–1861 (2016).
    [Crossref]
  28. S. Huard, Polarization of Light (John Wiley & Sons, 1997), Chap. 2.
  29. P. Drexler and F. Pavel, “Optical fiber birefringence effects-sources, utilization and methods of suppression,” in Recent Progress in Optical Fiber Research, Yasin, M., Harun, S., and Arof, H. (eds.), (InTech, 2011), Chap. 7.
  30. M. J. Weber, Handbook of Optical Materials (CRC University, 2003).
  31. K. Iizuka, Elements of Photonics Volume I (Wiley-Interscience, 2002), Chap. 6.
  32. C. D. Poole and D. L. Favin, “Polarization-mode dispersion measurements based on transmission spectra through a polarizer,” J. Lightwave Technol. 12(6), 917–929 (1994).
    [Crossref]
  33. R. Hui and M. O’Sullivan, Fiber Optic Measurement Techniques (Elsevier, 2009), Chap. 4.
  34. M. Karlsson, J. Brentel, and P. A. Andrekson, “Long-term measurement of PMD and polarization drift in installed fibers,” J. Lightwave Technol. 18(7), 941–951 (2000).
    [Crossref]
  35. M. Brodsky, N. J. Frigo, M. Boroditsky, and M. Tur, “Polarization mode dispersion of installed fibers,” J. Lightwave Technol. 24(12), 4584–4599 (2006).
    [Crossref]

2016 (5)

2015 (2)

2014 (1)

L. Palmieri and A. Galtarossa, “Coupling effects among degenerate modes in multimode optical fibers,” IEEE Photonics J. 6(6), 0600408 (2014).
[Crossref]

2013 (2)

2012 (5)

2011 (2)

2010 (1)

2006 (1)

2000 (1)

1994 (1)

C. D. Poole and D. L. Favin, “Polarization-mode dispersion measurements based on transmission spectra through a polarizer,” J. Lightwave Technol. 12(6), 917–929 (1994).
[Crossref]

1990 (1)

D. Wong, “Thermal stability of intrinsic stress birefringence in optical fibers,” J. Lightwave Technol. 8(11), 1757–1761 (1990).
[Crossref]

1975 (1)

D. Marcuse, “Coupled-mode theory for anisotropic optical waveguides,” Bell Syst. Tech. J. 54(6), 985–995 (1975).
[Crossref]

1971 (1)

Abedin, K. S.

Agrawal, G. P.

Andrekson, P. A.

Antonelli, C.

Awaji, Y.

A. V. T. Cartaxo, R. S. Luís, B. J. Puttnam, T. Hayashi, Y. Awaji, and N. Wada, “Dispersion impact on the crosstalk amplitude response of homogeneous multi-core fibers,” IEEE Photonics Technol. Lett. 28(17), 1858–1861 (2016).
[Crossref]

R. S. Luís, B. J. Puttnam, A. V. T. Cartaxo, W. Klaus, J. M. D. Mendinueta, Y. Awaji, N. Wada, T. Nakanishi, T. Hayashi, and T. Sasaki, “Time and modulation frequency dependence of crosstalk in homogeneous multi-core fibers,” J. Lightwave Technol. 34(2), 441–447 (2016).
[Crossref]

Boroditsky, M.

Brentel, J.

Brodsky, M.

Cartaxo, A. V. T.

A. V. T. Cartaxo, R. S. Luís, B. J. Puttnam, T. Hayashi, Y. Awaji, and N. Wada, “Dispersion impact on the crosstalk amplitude response of homogeneous multi-core fibers,” IEEE Photonics Technol. Lett. 28(17), 1858–1861 (2016).
[Crossref]

R. S. Luís, B. J. Puttnam, A. V. T. Cartaxo, W. Klaus, J. M. D. Mendinueta, Y. Awaji, N. Wada, T. Nakanishi, T. Hayashi, and T. Sasaki, “Time and modulation frequency dependence of crosstalk in homogeneous multi-core fibers,” J. Lightwave Technol. 34(2), 441–447 (2016).
[Crossref]

Essiambre, R. J.

Favin, D. L.

C. D. Poole and D. L. Favin, “Polarization-mode dispersion measurements based on transmission spectra through a polarizer,” J. Lightwave Technol. 12(6), 917–929 (1994).
[Crossref]

Fini, J. M.

Frigo, N. J.

Galtarossa, A.

L. Palmieri and A. Galtarossa, “Coupling effects among degenerate modes in multimode optical fibers,” IEEE Photonics J. 6(6), 0600408 (2014).
[Crossref]

Gloge, D.

Hayashi, T.

Karlsson, M.

Klaus, W.

Koshiba, M.

Llorente, R.

Luís, R. S.

R. S. Luís, B. J. Puttnam, A. V. T. Cartaxo, W. Klaus, J. M. D. Mendinueta, Y. Awaji, N. Wada, T. Nakanishi, T. Hayashi, and T. Sasaki, “Time and modulation frequency dependence of crosstalk in homogeneous multi-core fibers,” J. Lightwave Technol. 34(2), 441–447 (2016).
[Crossref]

A. V. T. Cartaxo, R. S. Luís, B. J. Puttnam, T. Hayashi, Y. Awaji, and N. Wada, “Dispersion impact on the crosstalk amplitude response of homogeneous multi-core fibers,” IEEE Photonics Technol. Lett. 28(17), 1858–1861 (2016).
[Crossref]

Macho, A.

Marcuse, D.

D. Marcuse, “Coupled-mode theory for anisotropic optical waveguides,” Bell Syst. Tech. J. 54(6), 985–995 (1975).
[Crossref]

Matsuo, S.

M. Koshiba, K. Saitoh, K. Takenaga, and S. Matsuo, “Analytical expression of average power-coupling coefficients for estimating intercore crosstalk in multicore fibers,” IEEE Photonics J. 4(5), 1987–1995 (2012).
[Crossref]

M. Koshiba, K. Saitoh, K. Takenaga, and S. Matsuo, “Multi-core fiber design and analysis: coupled-mode theory and coupled-power theory,” Opt. Express 19(26), B102–B111 (2011).
[Crossref] [PubMed]

Mecozzi, A.

Mendinueta, J. M. D.

Miyamoto, Y.

Mizuno, T.

Morant, M.

Mumtaz, S.

Nakanishi, T.

Palmieri, L.

L. Palmieri and A. Galtarossa, “Coupling effects among degenerate modes in multimode optical fibers,” IEEE Photonics J. 6(6), 0600408 (2014).
[Crossref]

Poole, C. D.

C. D. Poole and D. L. Favin, “Polarization-mode dispersion measurements based on transmission spectra through a polarizer,” J. Lightwave Technol. 12(6), 917–929 (1994).
[Crossref]

Puttnam, B. J.

A. V. T. Cartaxo, R. S. Luís, B. J. Puttnam, T. Hayashi, Y. Awaji, and N. Wada, “Dispersion impact on the crosstalk amplitude response of homogeneous multi-core fibers,” IEEE Photonics Technol. Lett. 28(17), 1858–1861 (2016).
[Crossref]

R. S. Luís, B. J. Puttnam, A. V. T. Cartaxo, W. Klaus, J. M. D. Mendinueta, Y. Awaji, N. Wada, T. Nakanishi, T. Hayashi, and T. Sasaki, “Time and modulation frequency dependence of crosstalk in homogeneous multi-core fibers,” J. Lightwave Technol. 34(2), 441–447 (2016).
[Crossref]

Saitoh, K.

Sano, A.

Sasaki, T.

Sasaoka, E.

Shimakawa, O.

Shtaif, M.

Takara, H.

Takenaga, K.

M. Koshiba, K. Saitoh, K. Takenaga, and S. Matsuo, “Analytical expression of average power-coupling coefficients for estimating intercore crosstalk in multicore fibers,” IEEE Photonics J. 4(5), 1987–1995 (2012).
[Crossref]

M. Koshiba, K. Saitoh, K. Takenaga, and S. Matsuo, “Multi-core fiber design and analysis: coupled-mode theory and coupled-power theory,” Opt. Express 19(26), B102–B111 (2011).
[Crossref] [PubMed]

Taru, T.

Taunay, T. F.

Tur, M.

Wada, N.

A. V. T. Cartaxo, R. S. Luís, B. J. Puttnam, T. Hayashi, Y. Awaji, and N. Wada, “Dispersion impact on the crosstalk amplitude response of homogeneous multi-core fibers,” IEEE Photonics Technol. Lett. 28(17), 1858–1861 (2016).
[Crossref]

R. S. Luís, B. J. Puttnam, A. V. T. Cartaxo, W. Klaus, J. M. D. Mendinueta, Y. Awaji, N. Wada, T. Nakanishi, T. Hayashi, and T. Sasaki, “Time and modulation frequency dependence of crosstalk in homogeneous multi-core fibers,” J. Lightwave Technol. 34(2), 441–447 (2016).
[Crossref]

Wong, D.

D. Wong, “Thermal stability of intrinsic stress birefringence in optical fibers,” J. Lightwave Technol. 8(11), 1757–1761 (1990).
[Crossref]

Yan, M. F.

Zhu, B.

Appl. Opt. (1)

Bell Syst. Tech. J. (1)

D. Marcuse, “Coupled-mode theory for anisotropic optical waveguides,” Bell Syst. Tech. J. 54(6), 985–995 (1975).
[Crossref]

IEEE Photonics J. (2)

L. Palmieri and A. Galtarossa, “Coupling effects among degenerate modes in multimode optical fibers,” IEEE Photonics J. 6(6), 0600408 (2014).
[Crossref]

M. Koshiba, K. Saitoh, K. Takenaga, and S. Matsuo, “Analytical expression of average power-coupling coefficients for estimating intercore crosstalk in multicore fibers,” IEEE Photonics J. 4(5), 1987–1995 (2012).
[Crossref]

IEEE Photonics Technol. Lett. (1)

A. V. T. Cartaxo, R. S. Luís, B. J. Puttnam, T. Hayashi, Y. Awaji, and N. Wada, “Dispersion impact on the crosstalk amplitude response of homogeneous multi-core fibers,” IEEE Photonics Technol. Lett. 28(17), 1858–1861 (2016).
[Crossref]

J. Lightwave Technol. (11)

C. D. Poole and D. L. Favin, “Polarization-mode dispersion measurements based on transmission spectra through a polarizer,” J. Lightwave Technol. 12(6), 917–929 (1994).
[Crossref]

M. Karlsson, J. Brentel, and P. A. Andrekson, “Long-term measurement of PMD and polarization drift in installed fibers,” J. Lightwave Technol. 18(7), 941–951 (2000).
[Crossref]

M. Brodsky, N. J. Frigo, M. Boroditsky, and M. Tur, “Polarization mode dispersion of installed fibers,” J. Lightwave Technol. 24(12), 4584–4599 (2006).
[Crossref]

T. Mizuno, H. Takara, A. Sano, and Y. Miyamoto, “Dense space-division multiplexing transmission systems using multi-core and multi-mode fiber,” J. Lightwave Technol. 34(2), 582–592 (2016).
[Crossref]

M. Morant, A. Macho, and R. Llorente, “On the suitability of multicore fiber for LTE-advanced MIMO optical fronthaul systems,” J. Lightwave Technol. 34(2), 676–682 (2016).
[Crossref]

J. M. Fini, B. Zhu, T. F. Taunay, M. F. Yan, and K. S. Abedin, “Statistical models of multicore fiber crosstalk including time delays,” J. Lightwave Technol. 30(12), 2003–2010 (2012).
[Crossref]

R. S. Luís, B. J. Puttnam, A. V. T. Cartaxo, W. Klaus, J. M. D. Mendinueta, Y. Awaji, N. Wada, T. Nakanishi, T. Hayashi, and T. Sasaki, “Time and modulation frequency dependence of crosstalk in homogeneous multi-core fibers,” J. Lightwave Technol. 34(2), 441–447 (2016).
[Crossref]

T. Hayashi, T. Taru, O. Shimakawa, T. Sasaki, and E. Sasaoka, “Characterization of crosstalk in ultra-low crosstalk multi-core fiber,” J. Lightwave Technol. 30(4), 583–589 (2012).
[Crossref]

D. Wong, “Thermal stability of intrinsic stress birefringence in optical fibers,” J. Lightwave Technol. 8(11), 1757–1761 (1990).
[Crossref]

A. Macho, M. Morant, and R. Llorente, “Unified model of linear and nonlinear crosstalk in multi-core fiber,” J. Lightwave Technol. 34(13), 3035–3046 (2016).
[Crossref]

S. Mumtaz, R. J. Essiambre, and G. P. Agrawal, “Nonlinear propagation in multimode and multicore fibers: generalization of the manakov equations,” J. Lightwave Technol. 31(3), 398–406 (2013).
[Crossref]

Opt. Express (8)

A. Mecozzi, C. Antonelli, and M. Shtaif, “Coupled Manakov equations in multimode fibers with strongly coupled groups of modes,” Opt. Express 20(21), 23436–23441 (2012).
[Crossref] [PubMed]

A. Mecozzi, C. Antonelli, and M. Shtaif, “Nonlinear propagation in multi-mode fibers in the strong coupling regime,” Opt. Express 20(11), 11673–11678 (2012).
[Crossref] [PubMed]

C. Antonelli, A. Mecozzi, and M. Shtaif, “The delay spread in fibers for SDM transmission: dependence on fiber parameters and perturbations,” Opt. Express 23(3), 2196–2202 (2015).
[Crossref] [PubMed]

J. M. Fini, B. Zhu, T. F. Taunay, and M. F. Yan, “Statistics of crosstalk in bent multicore fibers,” Opt. Express 18(14), 15122–15129 (2010).
[Crossref] [PubMed]

M. Koshiba, K. Saitoh, K. Takenaga, and S. Matsuo, “Multi-core fiber design and analysis: coupled-mode theory and coupled-power theory,” Opt. Express 19(26), B102–B111 (2011).
[Crossref] [PubMed]

T. Hayashi, T. Taru, O. Shimakawa, T. Sasaki, and E. Sasaoka, “Design and fabrication of ultra-low crosstalk and low-loss multi-core fiber,” Opt. Express 19(17), 16576–16592 (2011).
[Crossref] [PubMed]

T. Hayashi, T. Sasaki, E. Sasaoka, K. Saitoh, and M. Koshiba, “Physical interpretation of intercore crosstalk in multicore fiber: effects of macrobend, structure fluctuation, and microbend,” Opt. Express 21(5), 5401–5412 (2013).
[Crossref] [PubMed]

A. Macho, M. Morant, and R. Llorente, “Experimental evaluation of nonlinear crosstalk in multi-core fiber,” Opt. Express 23(14), 18712–18720 (2015).
[Crossref] [PubMed]

Other (11)

D. Marcuse, Theory of Dielectric Optical Waveguides (Elsevier, 1974), Ch. 3.

S. Mumtaz, R.-J. Essiambre, and G. P. Agrawal, “Birefringence effects in space-division multiplexed fiber transmission systems: generalization of Manakov equation,” in IEEE Photonics Society Summer Topical Meeting Series (IEEE, 2012), paper MC3.5.

R. Hui and M. O’Sullivan, Fiber Optic Measurement Techniques (Elsevier, 2009), Chap. 4.

B. J. Puttnam, R. S. Luis, W. Klaus, J. Sakaguchi, J.-M. Delgado Mendinueta, Y. Awaji, N. Wada, Y. Tamura, T. Hayashi, M. Hirano, and J. Marciante, “2.15 Pb/s transmission using a 22 core homogeneous single-mode multi-core fiber and wideband optical comb,” in Eur. Conf. Opt. Commun. (ECOC, 2015), paper PDP 3.1.
[Crossref]

S. Huard, Polarization of Light (John Wiley & Sons, 1997), Chap. 2.

P. Drexler and F. Pavel, “Optical fiber birefringence effects-sources, utilization and methods of suppression,” in Recent Progress in Optical Fiber Research, Yasin, M., Harun, S., and Arof, H. (eds.), (InTech, 2011), Chap. 7.

M. J. Weber, Handbook of Optical Materials (CRC University, 2003).

K. Iizuka, Elements of Photonics Volume I (Wiley-Interscience, 2002), Chap. 6.

G. P. Agrawal, Nonlinear Fiber Optics, 5th ed. (Elsevier, 2013).

R. W. Boyd, Nonlinear Optics, 3rd ed. (Elsevier, 2008), Chap. 1.

A. M. Weiner, Ultrafast Optics, 1st ed. (John Wiley and Sons, 2009), Chap. 6.

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Figures (11)

Fig. 1
Fig. 1

Multi-core fiber comprising different birefringent segments in cores a and b with longitudinal and temporal varying fluctuations in the refractive index tensor.

Fig. 2
Fig. 2

Comparison of the linear mode-coupling coefficients (MCCs) max,ay and ηax,by with the linear MCC kax,bx using the spatial average ratios given by Eqs. (16) for different constant twist rate values fT and core pitch ratio dab/R0: (a) <max,ay/kax,bx> and (b) <ηax,by/kax,bx>. (Legend shown on Fig. 2(b) also applies to Fig. 2(a)).

Fig. 3
Fig. 3

Multi-core fiber simulation model including longitudinal perturbations and temporal birefringence effects.

Fig. 4
Fig. 4

Schematic cross-sections of multi-core fibers simulated: (a) lowly- and (b) highly-birefringent cores.

Fig. 5
Fig. 5

Simulation results of the crosstalk behavior between the polarized core modes in a 150 m two-core fiber with lowly-birefringent cores, considering the temporal fluctuation of the linear birefringence of each core with a twist rate value of fT = 0.05 and 1 turns/m: (a) Simulated linear birefringence evolution in cores a and b throughout a 10-day period; (b) intra-core crosstalk evolution; and (c) direct and cross inter-core crosstalk evolution. (iC-XT: intra-core crosstalk, IC-XT: inter-core crosstalk).

Fig. 6
Fig. 6

Simulation results of the crosstalk behavior between the polarized core modes in a 150 m two-core fiber with highly-birefringent cores considering the temporal fluctuation of the linear birefringence of each core with a twist rate value of fT = 0.05 and 1 turns/m: (a) Simulated linear birefringence evolution in cores a and b throughout a 10-day period; (b) intra-core crosstalk evolution; and (c) direct and cross inter-core crosstalk evolution. (iC-XT: intra-core crosstalk, IC-XT: inter-core crosstalk).

Fig. 7
Fig. 7

Multi-parameter simulation of the crosstalk between polarized core modes varying the bending radius (1/RB) and the twist rate (fT) in a 2 m homogenous two-core fiber with lowly-birefringent cores. (a) intra-core crosstalk mean ay-ax, (b) direct inter-core crosstalk mean bx-ax, and (c) cross inter-core crosstalk mean by-ax.

Fig. 8
Fig. 8

Numerical results of the nonlinear crosstalk evolution considering a theoretical range of the power launch level in the ax polarized core mode in a homogeneous 5 m lowly-birefringent two-core fiber: (a) nonlinear intra-core crosstalk mean between ay-ax, and (b) nonlinear direct and cross inter-core crosstalk mean between bx-ax and by-ax. (iC-XT: intra-core crosstalk, DIC-XT: direct inter-core crosstalk, XIC-XT: cross inter-core crosstalk).

Fig. 9
Fig. 9

Experimental set-up for intra- and inter-core crosstalk evaluation between the polarized core modes of cores 1 and 3 of a four-core fiber (4CF), considering multi-core fiber temporal birefringence fluctuation and both linear and nonlinear power regimes.

Fig. 10
Fig. 10

Experimental results of the temporal linear birefringence fluctuation over different days and months of a 150 m 4CF, and corresponding intra- and inter-core crosstalk mean between cores 1 and 3. A power launch level of PL = 0 dBm was used in linear regime (NL: nonlinear regime with PL = 6 dBm). (a) Linear birefringence evolution of cores 1 and 3, (b) intra-core crosstalk 1x-1y and 3x-3y, and (c) direct- and cross-inter-core crosstalk 1i-3j with i,j = x,y

Fig. 11
Fig. 11

Experimental measurement and simulated results of the intra-core crosstalk (iC-XT) mean evolution with the average value of the linear birefringence. Solid points: measured data of cores 1, 2, 3, 4 in a 150 m homogeneous four-core fiber. Dashed line: simulation results using the coupled-local mode equation, Eq. (17).

Tables (1)

Tables Icon

Table 1 Number of segments and linear birefringence average values considered in the analysis

Equations (36)

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ε ( r , t )= i=x,y 1 2 [ E i, ω 0 ( r ;t )exp( j ω 0 t )+c.c. ]  u ^ i ,
P ( 1 ) ( r , t )= i=x,y 1 2 [ P i, ω 0 ( 1 ) ( r ;t )exp( j ω 0 t )+c.c. ]  u ^ i ,
P ( 3 ) ( r , t ) i=x,y 1 2 [ P i, ω 0 ( 3 ) ( r ;t )exp( j ω 0 t )+c.c. ]  u ^ i ,
P i, ω 0 ( 1 ) ( r ;t )= ε 0 χ ij ( 1 ) ( z;t ) E j, ω 0 ( r ;t ),
P i, ω 0 ( 3 ) ( r ;t )= 3 4 ε 0 χ ijkl ( 3 ) ( ω 0 ; ω 0 , ω 0 , ω 0 ) E j, ω 0 ( r ;t ) E k, ω 0 ( r ;t ) E l, ω 0 ( r ;t ),
χ ijkl ( 3 ) ( ω 0 ; ω 0 , ω 0 , ω 0 )= 1 3 χ xxxx ( 3 ) ( ω 0 ; ω 0 , ω 0 , ω 0 )( δ ij δ kl + δ ik δ jl + δ il δ jk ),
Δ E x, ω 0 ( r ;t )+ k 0 2 [ ε r,x ( r ;t ) E x, ω 0 ( r ;t )+σ( r ;t ) E y, ω 0 ( r ;t ) ] + k 0 2 γ[ ( | E x, ω 0 ( r ;t ) | 2 + 2 3 | E y, ω 0 ( r ;t ) | 2 ) E x, ω 0 ( r ;t )+ 1 3 E y, ω 0 2 ( r ;t ) E x, ω 0 ( r ;t ) ]=0,
Δ E y, ω 0 ( r ;t )+ k 0 2 [ ε r,y ( r ;t ) E y, ω 0 +σ( r ;t ) E x, ω 0 ] + k 0 2 γ[ ( | E y, ω 0 ( r ;t ) | 2 + 2 3 | E x, ω 0 ( r ;t ) | 2 ) E y, ω 0 ( r ;t )+ 1 3 E x, ω 0 2 ( r ;t ) E y, ω 0 ( r ;t ) ]=0,
ε r,i ( r ;t ):= ε r,ci ( r ;t )+Δ ε r,ai ( r ;t )+Δ ε r,bi ( r ;t )= ={ r core a: ε r,ai ( z;t )= ε r,ci ( z;t )+Δ ε r,ai ( z;t )= n ai 2 ( z;t ) r cladding: ε r,ci ( z;t )= n ci 2 ( z;t ) r core b: ε r,bi ( z;t )= ε r,ci ( z;t )+Δ ε r,bi ( z;t )= n bi 2 ( z;t )
σ( r ;t ):={ r core a: σ a ( z;t )= χ a,xy ( 1 ) ( z;t ) r cladding: σ c ( z;t )= χ c,xy ( 1 ) ( z;t ) r core b: σ b ( z;t )= χ b,xy ( 1 ) ( z;t )
E i, ω 0 ( r ;t ) m=a,b E mi, ω 0 ( r ;t )= m=a,b A mi ( z;t ) F mi ( x,y;z,t )exp[ j Φ mi ( z;t ) ] ,
Φ mi ( z;t ):= ϕ mi ( z;t )j 1 2 αz,
ϕ mi ( z;t ):= 0 z β mi ( eq ) ( δ;t )dδ = β mi z+ 0 z β mi ( B ) ( δ;t )+ β mi ( S ) ( δ;t )dδ .
Δ[ F mi ( x,y;z,t )exp( j Φ mi ( z;t ) ) ]+ k 0 2 ε r,mi ( r ;t )[ F mi ( x,y;z,t )exp( j Φ mi ( z;t ) ) ]=0;
Δ T F ai ( x,y;z,t )+ k 0 2 ε r,i ( r ;t ) F ai ( x,y;z,t )= =[ k 0 2 Δ ε r,bi ( r ;t )+j z β ai ( eq ) ( z;t )+ ( β ai ( eq ) ( z;t ) ) 2 jα β ai ( eq ) ( z;t ) ] F ai ( x,y;z,t ),
j z A ax ( z;t )= c ax ( z;t ) A ax ( z;t )+ m ax,ay ( z;t )exp( jΔ ϕ ay,ax ( z;t ) ) A ay ( z;t ) +exp( jΔ ϕ bx,ax ( z;t ) )[ k ax,bx ( z;t )j χ ax,bx ( z;t ) z ] A bx ( z;t ) + η ax,by ( z;t )exp( jΔ ϕ by,ax ( z;t ) ) A by ( z;t ) +exp( αz )[ q ax ( z;t ) | A ax ( z;t ) | 2 + g ax ( z;t ) | A ay ( z;t ) | 2 ] A ax ( z;t ) + 1 2 exp( αz ) g ax ( z;t )exp( j2Δ ϕ ay,ax ( z;t ) ) A ax * ( z;t ) A ay 2 ( z;t ),
χ ax,bx ( z;t ):= β bx ( eq ) ( z;t ) β ax ( eq ) ( z;t ) N ax ( z;t ) F ax ( x,y;z,t ) F bx ( x,y;z,t )d S ,
c ax ( z;t ):= k 0 2 2 β ax ( eq ) ( z;t ) N ax ( z;t ) Δ ε r,bx ( z;t ) F ax 2 ( x,y;z,t )d S ,
m ax,ay ( z;t ):= k 0 2 2 β ax ( eq ) ( z;t ) N ax ( z;t ) σ( r ;t ) F ax ( x,y;z,t ) F ay ( x,y;z,t )d S ,
k ax,bx ( z;t ):= k 0 2 2 β ax ( eq ) ( z;t ) N ax ( z;t ) Δ ε r,ax ( z;t ) F ax ( x,y;z,t ) F bx ( x,y;z,t )d S ,
η ax,by ( z;t ):= k 0 2 2 β ax ( eq ) ( z;t ) N ax ( z;t ) σ( r ;t ) F ax ( x,y;z,t ) F by ( x,y;z,t )d S k 0 2 2 β ax ( eq ) ( z;t ) N ax ( z;t ) [ σ b ( z;t ) F ax ( x,y;z,t ) F by ( x,y;z,t )d S b + σ a ( z;t ) F ax ( x,y;z,t ) F by ( x,y;z,t )d S a ],
σ a( b ) ( z )π R 0 f T ( z ) n a( b ) 4 | p 11 p 12 |,
n a( b ) = ( n ax( bx ) + n ay( by ) ) /2 ,
q ax ( z;t ):= k 0 2 2 β ax ( eq ) ( z;t ) N ax ( z;t ) γ F ax 4 ( x,y;z,t )d S ,
g ax ( z;t ):= k 0 2 3 β ax ( eq ) ( z;t ) N ax ( z;t ) γ F ax 2 ( x,y;z,t ) F ay 2 ( x,y;z,t )d S ,
m ax,ay ( z ) k ax,bx ( z ) σ a 2( n a 2 n c 2 ) u a [ J 0 2 ( u a )+ J 1 2 ( u a ) ] J 0 ( u a ) J 1 ( u a ) K 0 ( w b ) K 0 ( w b d ab / R 0 ) ,
η ax,by ( z ) k ax,bx ( z ) 1 ( n a 2 n c 2 ) [ σ b u a u b J 0 ( u a ) J 0 ( u b ) J 1 ( u b ) J 1 ( u a ) K 0 ( w b ) K 0 ( w a ) K 0 ( w a d ab / R 0 ) K 0 ( w b d ab / R 0 ) + σ a ] σ b + σ a n a 2 n c 2 ,
j( z + α 2 ) A ax ( z;t )= m ax,ay ( z;t )exp( jΔ ϕ ay,ax ( z;t ) ) A ay ( z;t ) + δ=b N k ax,δx ( z;t )exp( jΔ ϕ δx,ax ( z;t ) ) A δx ( z;t ) +( q ax ( z;t ) | A ax ( z;t ) | 2 + g ax ( z;t ) | A ay ( z;t ) | 2 ) A ax ( z;t ) + 1 2 g ax ( z;t )exp( j2Δ ϕ ay,ax ( z;t ) ) A ax * ( z;t ) A ay 2 ( z;t ).
ϕ mi ( z;t )= z 1 z 2 β mi ( eq ) ( ξ;t )dξ β mi ( eq ) ( z;t )( z 2 z 1 )= k 0 n mi ( eq ) ( z;t )( z 2 z 1 ).
n mi,j,s-l ( eq ) ( z;t )= n mi,j,s-l ( t )[ 1+ d m R B,l cos( 2π f T,l z+ θ 0 + θ m ) ],
δ n m,l 0.011 n m 3 d c 2 / R B,l 2 ,
n mi,j,s-l ( t )~ n m [ 1± 1 2 N( μ( t )=Δ n m,j , σ 2 =δ n m,l ) ].
F mi ( r;z,t )={ J 0 ( u mi ( z;t ) R 0 r ); r R 0 K 0 ( w mi ( z;t ) R 0 r ); r> R 0 ,
u mi ( z;t )= ( 1+ 2 ) V mi ( z;t ) / [ 1+ ( 4+ V mi 4 ( z;t ) ) 1/4 ] ,
w mi ( z;t )= V mi 2 ( z;t ) u mi 2 ( z;t ) ,
V mi ( z;t )= k 0 R 0 ( n mi ( eq ) ( z;t ) ) 2 n c 2 ,

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