## Statistics of crosstalk in bent multicore fibers |

Optics Express, Vol. 18, Issue 14, pp. 15122-15129 (2010)

http://dx.doi.org/10.1364/OE.18.015122

Acrobat PDF (1201 KB)

### Abstract

A statistical theory for crosstalk in multicore fibers is derived from coupled-mode equations including bend-induced perturbations. Bends are shown to play a crucial role in crosstalk, explaining large disagreement between experiments and previous calculations. The average crosstalk of a fiber segment is related to the statistics of the bend radius and orientation, including spinning along the fiber length. This framework allows efficient and accurate estimates of cross-talk for realistic telecommunications links.

© 2010 OSA

## 1. Introduction

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. **17**(5), 807–810 (1999). [CrossRef]

3. 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]

## 2. Realistic link models based on coupled-mode equations

*T*, each of which incorporates the statistics of the random perturbations in a portion of the fiber. If

*v(z)*is a vector of mode coefficients at position z along the fiber, then

*C*, bend perturbations by

*B*, and the unperturbed mode effective index (including manufacturing variation, etc.) is given by

*A*,

*C*is non-diagonal, and typically has elements <<10

^{−6}(in effective index units) for telecommunications fibers of interest. Generally

*A*and

*C*will be slowly varying, and so we assume for simplicity that they are constant over a segment length ΔL = z

_{p + 1}-z

_{p}. A realistic model cannot neglect

*A*, since even accidental variation in effective index [2

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. **17**(5), 807–810 (1999). [CrossRef]

^{−5}-10

^{−4}. In addition,

*A*may include larger intentional index skew. The conformal mapping model gives a bend-induced index perturbation of the fiber index profile [9

9. D. Marcuse, “Influence of curvature on the losses of doubly clad fibers,” Appl. Opt. **21**(23), 4208–4213 (1982). [CrossRef] [PubMed]

10. K. Petermann and R. Kuhne, “Upper and lower limits for the microbending loss in arbitrary single-mode fibers,” J. Lightwave Technol. **4**(1), 2–7 (1986). [CrossRef]

11. J. W. Nicholson, J. M. Fini, A. D. Yablon, P. S. Westbrook, K. Feder, and C. Headley, “Demonstration of bend-induced nonlinearities in large-mode-area fibers,” Opt. Lett. **32**(17), 2562–2564 (2007). [CrossRef] [PubMed]

*m*is well localized to core center position [x

_{m},y

_{m}], and so the bend perturbation is simply

^{−6}-10

^{−3}. The orientation θ and radius R

_{bend}of the fiber curvature have a random component. Below we assumed a correlation length of order 1m in the random variations.

## 3. The importance of bends

*a*= 42μm apart in a regular triangular arrangement (following [5]). This leads to approximately the same coupling coefficient

*C*=

_{n,m}*c*≈4.6 × 10

_{0}^{−9}at 1550 nm for all nearest-neighbor cores. Other elements of

*C*were small and were neglected in the simulation. The outer six cores have Δn

_{m}

^{eff}= ± Δn

_{skew}, alternating in sign, so that all nearest-neighbors are skewed by at least Δn

_{skew}(values given in the figure legend). This can be accomplished, for example, by slightly varying the core diameters. Comparing the straight-fiber result (b) to the equivalent results assuming a 1m radius of curvature (c), we see that even gentle bends have a huge impact. Further experiments will be needed to show whether the mechanisms included in the coupled-mode model [Eq. (2)] are sufficient for predictive modeling of real links, but there is little question that the perturbation [Eq. (3)] must be included in some form in any realistic model. Ignoring bends gives a grossly unrealistic sense of the effectiveness of this strategy, in particular for small skew values. Introducing bends also causes crosstalk to accumulate with length in the calculation, as it does in real fibers [unlike in Fig. 2(b), where the crosstalk varies sinusoidally]. Using the improved model with bends, we can identify the correct skew required for low crosstalk in a realistic system.

*a/R*

_{bend}and are therefore much larger than

*c*unless the fiber is held extremely straight, R

_{0}_{bend}>1km. The point of skewing the effective index of cores is to prevent phase-matched coupling between them; however, if the bend perturbation is large enough, it can bring skewed cores into phase-matched resonance. The result is to remove much of the benefit of the skew strategy unless the skew is larger than γn

_{core}

*a/R*

_{bend}. For R

_{bend}= 1m, γn

_{core}

*a/R*

_{bend}≈5 × 10

^{−5}explaining why the largest skew (7 × 10

^{−5}) in Fig. 2 is sufficient to prevent phase-matched coupling while the smaller skews are not.

## 4. Average crosstalk for a bent multicore fiber

*z*], so that the bend statistics can be considered stationary within each segment. The statistics are certainly not stationary for the entire link (statistics in a bend-challenged portion are drastically different from those in a bend-managed portion of the same fiber), but the assumption of stationary statistics over an individual segment (e.g. ~10m) is quite reasonable.

_{p},z_{p + 1}*A*+

*B*is diagonal (and thus integrable), a natural approach is to write the interaction-picture evolution of the re-phased transfer matrix

*U*,

*C*is small, a first-order approximation is valid over a small enough segment,

*A*is approximately constant over the interval, and

*P*(z

_{p}) = 1for n≠m, where we have introduced the propagation-constant mismatch,

*f*, the accumulated bend phase,

*f*is a stationary random process, it has an autocorrelation function

*R*,

_{ff}*f*has a finite correlation length, then

*R*becomes small beyond this length. If the correlation length is much smaller than Δ

_{ff}*L*, then we can neglect boundary terms and identify the FT of the autocorrelation as the Power Spectral Density (PSD) [12]

*regular*multicore fibers, nearest-neighbors all have the same spacing

*a*, and so for nearest neighbor cores n,m with displacement angle θ

_{m,n},

*regular*multicore fiber, but are simply evaluated at different Δβ.

*L*=

*z*

_{2}-

*z*

_{1}to satisfy three approximations: 1. Δ

*L*>> correlation length of

*f*, 2. Δ

*L*is small enough that higher order perturbations can be neglected within the interval, and 3. Δ

*L*<< length scale of variation in

*A*and

*C*. Thus these results may not rigorously apply when the coupling is too large, since the first two conditions may be contradictory.

*C*

_{nm}, index mismatches Δβ

_{nm}, and bend statistics essentially impact the crosstalk independently of one another, once the bend statistics are averaged. There is no need to do simulations varying the combinations of these three quantities to see how they interact. Further, simulations of a total link can be done using a coarse-grained concatenation model, confident that the fine-grained variations in bend radius have been appropriately averaged.

## 5. Low crosstalk fiber design

_{nm}= Δβ

_{nm}/(2π/λ) for λ = 1550 nm]. The parameters of the above examples were used (Δ

*L*= 40m,

*a*= 42μm,

*c*≈4.6 × 10

_{0}^{−9})—but the results apply to other values of

*a*and

*c*using scaling rules: Cross Talk ∝Δ

_{0}*L|c*|

_{0}^{2}and

*R*

_{bend}scales along with core spacing. In this example, the curvature has constant and random components,with the standard deviation of the curvature 20% of its nominal value:

*K*with maximum spin rate 2π/(0.5m), gulp period 5m. The random orientation drift rate

_{0}*g*has standard deviation 2.5rad/m. Both random components

_{2}*g*and

_{1}*g*were generated by filtering white noise with a raised-cosine finite-impulse-response filter of length 0.5m. Two random orientation and curvature functions with these statistics are shown in Fig. 4(a) for illustration. Once the random processes are defined, standard methods allow efficient calculation of the power spectral density; for each pseudo-random realization of

_{2}*f*, a standard algorithm (p. 581 of [12], with raised-cosine window) was applied. To further reduce the noise in the final result of Fig. 4, the PSD estimates for 16 independent realizations were averaged. The smooth appearance of these curves indicates that the statistical average has been approximated—if needed, many more realizations could be averaged without significant computational burden.

^{−4}, then we simply read off the cross-talk at this value on the x-axis. This amount of skew successfully reduces cross-talk by orders of magnitude in a link with typical bend radius of around 1m (pink curve), but gives no improvement over an un-skewed design if the bends are significantly tighter than 1m radius (green or black curves). Thus a single plot derived from efficient Fast Fourier Transform operations captures the essence of an ensemble of individual brute-force simulations (such as Fig. 2). By separating the influence of bending from length and raw coupling strength (

*c*), Eq. (12) allows us calculate a family of bend conditions such as in Fig. 4, and apply these to all possible fiber designs (all designs reducing to parameters

_{0}*c*, Δ

_{0}*L*, index mismatch, and

*a*) simultaneously, without redoing the calculation for each fiber. Further, this analysis is appropriate in the most relevant case for real applications: where the fiber layout has significant randomness.

_{b0}of order 10cm. Using scaling rules to adjust the calculation of Fig. 4 to the parameters of the measurement in [5] (2km length and 40μm core spacing), the calculated average cross-talk is 20-27dB, in rough agreement with the 17-20dB crosstalk measured in [5]. Better agreement may be anticipated using a measured index profile for the core (rather than the step-index description used here), accurate spool characteristics, etc.

*a*= 42μm,

*c*≈4.6 × 10

_{0}^{−9}), with a small intentional skew Δn

_{skew}= 1 × 10

^{−5}and random perturbations to each outer core index uniformly distributed between ± 1 × 10

^{−5}. In this example, the nominal bend radius was

*R*

_{b0}= 4m,

*g*had standard deviation 0.02/m,

_{1}*K*was a constant spin of 1 turn/10m, and g

_{0}_{2}had standard deviation ~.16rad/m. For simplicity, in both calculations we assumed that dissimilar cores result in effective index skewing but cause negligible variation in the coupling values between the nearest-neighbor core (

*C*

_{n,m}=

*c*or 0). The deviations between the brute-force and statistical results are initially large, as expected: the analytical result is only valid for lengths much greater than the correlation length of the random inputs. The asymptotic agreement is very good and is reached in reasonable lengths (much shorter than a total communications link), validating this approach.

_{0}## 6. Conclusions

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

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

7. | J. M. Fini, B. Zhu, T. F. Taunay, and M. F. Yan, “Bends in the design of low-crosstalk multicore fiber communications links,” to be published in the 15th OptoElectronics and Communications Conference 2010. |

8. | S. Kumar, U. H. Manyam, and V. Srikant, “Optical fibers having cores with different propagation constants, and methods of manufacturing same,” US Patent 6611648 (2003). |

9. | D. Marcuse, “Influence of curvature on the losses of doubly clad fibers,” Appl. Opt. |

10. | K. Petermann and R. Kuhne, “Upper and lower limits for the microbending loss in arbitrary single-mode fibers,” J. Lightwave Technol. |

11. | J. W. Nicholson, J. M. Fini, A. D. Yablon, P. S. Westbrook, K. Feder, and C. Headley, “Demonstration of bend-induced nonlinearities in large-mode-area fibers,” Opt. Lett. |

12. | K. S. Shanmugan, and A. M. Breipohl, |

**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: April 16, 2010

Revised Manuscript: June 7, 2010

Manuscript Accepted: June 23, 2010

Published: June 30, 2010

**Citation**

John M. Fini, Benyuan Zhu, Thierry F. Taunay, and Man F. Yan, "Statistics of crosstalk in bent multicore fibers," Opt. Express **18**, 15122-15129 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-14-15122

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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).
- 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, “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.
- J. M. Fini, B. Zhu, T. F. Taunay, and M. F. Yan, “Bends in the design of low-crosstalk multicore fiber communications links,” to be published in the 15th OptoElectronics and Communications Conference 2010.
- S. Kumar, U. H. Manyam, and V. Srikant, “Optical fibers having cores with different propagation constants, and methods of manufacturing same,” US Patent 6611648 (2003).
- D. Marcuse, “Influence of curvature on the losses of doubly clad fibers,” Appl. Opt. 21(23), 4208–4213 (1982). [CrossRef] [PubMed]
- K. Petermann and R. Kuhne, “Upper and lower limits for the microbending loss in arbitrary single-mode fibers,” J. Lightwave Technol. 4(1), 2–7 (1986). [CrossRef]
- J. W. Nicholson, J. M. Fini, A. D. Yablon, P. S. Westbrook, K. Feder, and C. Headley, “Demonstration of bend-induced nonlinearities in large-mode-area fibers,” Opt. Lett. 32(17), 2562–2564 (2007). [CrossRef] [PubMed]
- K. S. Shanmugan, and A. M. Breipohl, Random Signals (John Wiley and Sons, 1988).

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