## Crosstalk in multicore fibers with randomness: gradual drift vs. short-length variations |

Optics Express, Vol. 20, Issue 2, pp. 949-959 (2012)

http://dx.doi.org/10.1364/OE.20.000949

Acrobat PDF (889 KB)

### Abstract

Random perturbations play an important role in the crosstalk of multicore fibers, and can be captured by statistical coupled-mode calculations. In this approach, phase matching contributes a multiplicative factor to the average crosstalk, depending on the perturbation statistics and any intentional heterogeneity of neighboring cores. The impact of perturbations is shown to be qualitatively different depending on whether they are gradually varying, or have short-length (centimeter-scale) variations. This insight implies a novel crosstalk suppression strategy: fast modulation of a bend perturbation by spinning the fiber can disrupt the bend-induced phase matching.

© 2012 OSA

## 1. Introduction

12. 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]

7. 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]

## 2. Basic model

*a*is an amplitude of a local mode that accumulates an explicit phase according to the corresponding element of

*D*, while in the representation

*u*, these phases are implicit. We obtain a modified evolution equation

12. 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]

13. H. Haus, W. Huang, S. Kawakami, and N. Whitaker, “Coupled-mode theory of optical waveguides,” J. Lightwave Technol. **5**(1), 16–23 (1987). [CrossRef]

14. 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). [PubMed]

*u(z = L) = Uu(z = 0)*. If the coupling is not too large, then we can use a simple 1st-order perturbative expression for the transfer matrix

*U = 1 + dU*, with: Here,

*δβ*gives the differential phase between the cores and

_{n,m}= D_{n,n}-D_{m,m}*κ*is the

_{m,n}*m,n*element of

*K*. The notation is slightly different than used previously.

_{n,m}and its variation with fiber length. This could include variability in the fiber itself, temperature or strain effects, etc. In [9], we showed that including random variations is essential to a realistic model: even if other variations could be eliminated with perfect fabrication, bend perturbations are sufficiently large to make the deterministic coupled-mode model grossly unrealistic. The power transfer matrix

*M*is extracted from the coupled-mode model by taking the average over the random perturbations:

*|dU*; the magnitude squared of Eq. (6) gives the double-integral:This equation is analogous to Eq. (11) of [12

_{m,n}|^{2}12. 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]

*δβ*includes both the length-varying and non-varying parts. If we neglect length variation of the coupling coefficients themselves, then

_{n,m}*z*and

*z’*. This correlation is clearly 1 when

*z = z’*. It may be approximately sinusoidal in

*z-z’*over short lengths, but if there is sufficient disorder (e. g. sufficient orientation drift of the fiber), the integrand will be zero for

*|z’-z|*greater than some correlation length. If we choose the increment

*L*of fiber length much larger than this correlation length, then the bounds of the second integral can be ignored. Now assuming the variations are statistically stationary:

*L*can be chosen somewhat arbitrarily; if we choose a length of order 100m-1km, then the segment is much larger than the correlation length, as required.

**18**(14), 15122–15129 (2010). [CrossRef] [PubMed]

^{0}is the non-length-varying component (e.g. due to the intentional index mismatch between two cores)

*S*is the power-spectral density (PSD) of

_{ff}*f*, and

*f*is defined as the accumulated phase of the length varying part of of δβ:

## 3. Gradual variations

*b*, in the integrals. The statistical average could be calculated directly using the probability distribution p

_{δβ}:If we naively evaluate this expression, the integral in

*z*gives

*2πδ(b)*, and so,That is, in the quasi-static limit, the integral in Eq. (10) approaches the probability density measuring how often phase matching (

*δβ = 0*) is achieved during the process of gradual intermittent phase matching illustrated in Fig. 2. This reinforces the intuition of previous detailed calculations: if there is an intentional index mismatch between cores, this will suppress crosstalk only insofar as random perturbations cannot cancel the deterministic component. The probability density

*p*has units of

_{δβ}(0)*1/δβ*. If there is no systematic index mismatch,

*p*will typically have a peak value at

_{δβ}(b)*b = 0*proportional to one over the spread

*Δβ*in

*δβ*(or the corresponding spread in random index perturbations

*Δn*):This inverse-proportionality to the bend perturbation gives an intuitive explanation for the previous numerical predictions [12

_{eff}= Δβλ/2π**18**(14), 15122–15129 (2010). [CrossRef] [PubMed]

7. 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]

*L*, violating the assumptions leading to Eq. (10). However, the final result is correct: in the Appendix, we show that same expression results if we allow δβ to drift at a small but finite rate along the fiber length.

## 4. Gradual variations: numerical examples

**18**(14), 15122–15129 (2010). [CrossRef] [PubMed]

*b = b*cos(θ) with bend orientation θ uniformly distributed over all angles andHere

_{0}*a*is the core spacing, and γ includes a stress correction to the purely geometrical (γ = 1) bend perturbation model. The probability density function for

*b*is then derived with the usual change-of-variable formulas (recognizing that each

*b*corresponds to two angles, θ and -θ):

## 5. Variations that are not gradual: spin

*R*and spin rate drift slowly, they seem constant on a length scale shorter than this drift; that is, locallyand so the PSD reflects a quasi periodicity with period Λ

_{bend}_{spin}. This is illustrated in Fig. 5 , where the same calculation is repeated, but where there is now considerable random variation in the bend radius: the curvature is now equal to the nominal value (indicated in the legend) 1/R

^{0}

_{bend}plus a random component with standard deviation 1/R

^{0}

_{bend}. The tail of the PSD is now much larger, since the random curvature variation can lead to large bend perturbations. The PSD curves are generally smoother, since there is more disorder in the system. However, the dominant peaks clearly still show the structure of the quasi-periodicity, falling at multiples of λ/Λ

_{spin}.

^{0}that are constant with length (on a ~1km scale), in addition to the bend perturbations. The bend perturbations may have identical statistics for the different core pairs, and for each kilometer of fiber, but each core pair and each spliced fiber has a different random δβ

^{0}

_{nm}. The highly structured PSD means that each time the PSD is sampled at a different random δβ

^{0}

_{nm}, the total crosstalk will be very different, and there is thus a high variability of crosstalk from between core pairs, or from one fiber to a spliced (nearly identical fiber). This is true even if the fabrication tolerances are quite good: if the δβ

^{0}

_{nm}values are controlled to index precision ~10

^{−5}.

## 6. Crosstalk reduction: fast spin

_{spin}in δβ units or λ/Λ

_{spin}on the index mismatch axis. For 1550nm wavelength this is 1.55x10

^{−4}. The calculated crosstalk is suppressed by orders of magnitude by the spin for cores with index mismatch far from these peaks. For example, cores with index mismatch of (0.8 ± 0.5) × 10

^{−4}show large calculated crosstalk suppression.

## 7. Conclusions

**18**(14), 15122–15129 (2010). [CrossRef] [PubMed]

7. 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]

## Appendix

*δβ*equaled a constant

_{n,m}*b*over the entire segment length

*L*. We can more realistically assume that

*δβ*drifts slowly along the length. That is, to evaluate Eq. (10), we say that the random

_{n,m}*δβ*has some random value

_{n,m}*b*at the center of the interval

*ζ=(z+z’)/2*, and is drifting at some random rate of drift

*b’.*Then

*δβ*, and soEquation (10) becomes

_{n,m}(ζ)=b+b’[ζ-(z+z’)/2]*b’*, but we can treat

*b’*as a deterministic value in anticipation that it will soon drop out. The integral is now of a well-known form: the Fourier transform of a Gaussian. If

*b’*is small, the function sampling the probability distribution is delta-like:So that, again, we get Eq. (14)Note that the actual value of

*b’*falls out, as long as it is small enough so that the sampling function can resolve the probability distribution. In this model with constant

*b’*,

*L*is chosen not much larger than the correlation length, and so a single segment will not be ergodic. However, a concatenation of many of these segments will sample many different values of

*b*in proportion to

*p*, and so the actual measured crosstalk may reasonably approximate the statistical average.

_{δβ}(b)## References and links

1. | S. Iano, T. Sato, S. Sentsui, T. Kuroha, and Y. Nishimura, “Multicore optical fiber,” in |

2. | B. Rosinski, J. W. D. Chi, P. Grosso, and J. Le Bihan, “Multichannel transmission of a multicore fiber coupled with vertical-cavity surface-emitting lasers,” J. Lightwave Technol. |

3. | R. J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightwave Technol. |

4. | G. Le Noane, P. Grosso, and I. Hardy, “Small, high precision, multicore optical guides and process for the production of said guides,” US Patent 5519801 (1996). |

5. | J. Sakaguchi, Y. Awaji, N. Wada, A. Kanno, T. Kawanishi, T. Hayashi, T. Taru, T. Kobayashi, and M. Watanabe, “109-Tb/s (7x97x172-Gb/s SDM/WDM/PDM) QPSK transmission through 16.8-km homogeneous multi-core fiber,” in |

6. | B. Zhu, T. Taunay, M. Fishteyn, X. Liu, S. Chandrasekhar, M. Yan, J. Fini, E. Monberg, and F. Dimarcello, “Space-, Wavelength-, Polarization-Division Multiplexed Transmission of 56-Tb/s over a 76.8-km Seven-Core Fiber,” in |

7. | 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 |

8. | B. Zhu, T. F. Taunay, M. Fishteyn, X. Liu, S. Chandrasekhar, M. F. Yan, J. M. Fini, E. M. Monberg, and F. V. Dimarcello, “112-Tb/s space-division multiplexed DWDM transmission with 14-b/s/Hz aggregate spectral efficiency over a 76.8-km seven-core fiber,” Opt. Express |

9. | J. M. Fini, B. Zhu, T. F. Taunay, and M. F. Yan, “Low cross-talk design of multi-core fibers,” in |

10. | K. S. Abedin, T. F. Taunay, M. Fishteyn, M. F. Yan, B. Zhu, J. M. Fini, E. M. Monberg, F. V. Dimarcello, and P. W. Wisk, “Amplification and noise properties of an erbium-doped multicore fiber amplifier,” Opt. Express |

11. | K. Imamura, K. Mukasa, and T. Yagi, “Investigation on Multi-Core Fibers with Large Aeff and Low Micro Bending Loss,” in |

12. | J. M. Fini, B. Zhu, T. F. Taunay, and M. F. Yan, “Statistics of crosstalk in bent multicore fibers,” Opt. Express |

13. | H. Haus, W. Huang, S. Kawakami, and N. Whitaker, “Coupled-mode theory of optical waveguides,” J. Lightwave Technol. |

14. | M. Koshiba, K. Saitoh, K. Takenaga, and S. Matsuo, “Multi-core fiber design and analysis: coupled-mode theory and coupled-power theory,” Opt. Express |

15. | J. M. Fini, “Crosstalk in multi-core optical fibres,” in Proceedings of ECOC, paper Mo.1.LeCervin.4 (2011). |

16. | D. Payne, A. Barlow, and J. Hansen, “Development of low-and high-birefringence optical fibers,” IEEE J. Quantum Electron. |

**OCIS Codes**

(060.2280) Fiber optics and optical communications : Fiber design and fabrication

(060.2330) Fiber optics and optical communications : Fiber optics communications

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: October 28, 2011

Revised Manuscript: December 13, 2011

Manuscript Accepted: December 14, 2011

Published: January 4, 2012

**Citation**

John M. Fini, Benyuan Zhu, Thierry F. Taunay, Man F. Yan, and Kazi S. Abedin, "Crosstalk in multicore fibers with randomness: gradual drift vs. short-length variations," Opt. Express **20**, 949-959 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-2-949

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### References

- S. Iano, T. Sato, S. Sentsui, T. Kuroha, and Y. Nishimura, “Multicore optical fiber,” in Optical Fiber Communication, 1979 OSA Technical Digest Series (Optical Society of America, 1979), paper WB1.
- B. Rosinski, J. W. D. Chi, P. Grosso, and J. Le Bihan, “Multichannel transmission of a multicore fiber coupled with vertical-cavity surface-emitting lasers,” J. Lightwave Technol.17(5), 807–810 (1999). [CrossRef]
- R. J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightwave Technol.28(4), 662–701 (2010). [CrossRef]
- G. Le Noane, P. Grosso, and I. Hardy, “Small, high precision, multicore optical guides and process for the production of said guides,” US Patent 5519801 (1996).
- J. Sakaguchi, Y. Awaji, N. Wada, A. Kanno, T. Kawanishi, T. Hayashi, T. Taru, T. Kobayashi, and M. Watanabe, “109-Tb/s (7x97x172-Gb/s SDM/WDM/PDM) QPSK transmission through 16.8-km homogeneous multi-core fiber,” in Optical Fiber Communication Conference, OSA Technical Digest (CD) (Optical Society of America, 2011), paper PDPB6.
- B. Zhu, T. Taunay, M. Fishteyn, X. Liu, S. Chandrasekhar, M. Yan, J. Fini, E. Monberg, and F. Dimarcello, “Space-, Wavelength-, Polarization-Division Multiplexed Transmission of 56-Tb/s over a 76.8-km Seven-Core Fiber,” in Optical Fiber Communication Conference, OSA Technical Digest (CD) (Optical Society of America, 2011), paper PDPB7.
- 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. Express19(17), 16576–16592 (2011). [CrossRef] [PubMed]
- B. Zhu, T. F. Taunay, M. Fishteyn, X. Liu, S. Chandrasekhar, M. F. Yan, J. M. Fini, E. M. Monberg, and F. V. Dimarcello, “112-Tb/s space-division multiplexed DWDM transmission with 14-b/s/Hz aggregate spectral efficiency over a 76.8-km seven-core fiber,” Opt. Express19(17), 16665–16671 (2011). [CrossRef] [PubMed]
- J. M. Fini, B. Zhu, T. F. Taunay, and M. F. Yan, “Low cross-talk design of multi-core fibers,” in Conference on Lasers and Electro-Optics/International Quantum Electronics Conference, OSA Technical Digest (CD) (Optical Society of America, 2010), paper CTuAA3.
- K. S. Abedin, T. F. Taunay, M. Fishteyn, M. F. Yan, B. Zhu, J. M. Fini, E. M. Monberg, F. V. Dimarcello, and P. W. Wisk, “Amplification and noise properties of an erbium-doped multicore fiber amplifier,” Opt. Express19(17), 16715–16721 (2011). [CrossRef] [PubMed]
- K. Imamura, K. Mukasa, and T. Yagi, “Investigation on Multi-Core Fibers with Large Aeff and Low Micro Bending Loss,” in Optical Fiber Communication Conference, OSA Technical Digest (CD) (Optical Society of America, 2010), paper OWK6.
- J. M. Fini, B. Zhu, T. F. Taunay, and M. F. Yan, “Statistics of crosstalk in bent multicore fibers,” Opt. Express18(14), 15122–15129 (2010). [CrossRef] [PubMed]
- H. Haus, W. Huang, S. Kawakami, and N. Whitaker, “Coupled-mode theory of optical waveguides,” J. Lightwave Technol.5(1), 16–23 (1987). [CrossRef]
- M. Koshiba, K. Saitoh, K. Takenaga, and S. Matsuo, “Multi-core fiber design and analysis: coupled-mode theory and coupled-power theory,” Opt. Express19(26), B102–B111 (2011). [PubMed]
- J. M. Fini, “Crosstalk in multi-core optical fibres,” in Proceedings of ECOC, paper Mo.1.LeCervin.4 (2011).
- D. Payne, A. Barlow, and J. Hansen, “Development of low-and high-birefringence optical fibers,” IEEE J. Quantum Electron.18(4), 477–488 (1982). [CrossRef]

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