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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 14 — Jul. 5, 2010
  • pp: 15122–15129
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Statistics of crosstalk in bent multicore fibers

John M. Fini, Benyuan Zhu, Thierry F. Taunay, and Man F. Yan  »View Author Affiliations


Optics Express, Vol. 18, Issue 14, pp. 15122-15129 (2010)
http://dx.doi.org/10.1364/OE.18.015122


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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

When comparing the cost and performance of a multi-core to a multi-fiber solution, cross-talk is an obvious potential disadvantage of multi-core. Multi-core fibers can be engineered to have low cross-talk, but only subject to tradeoffs with other important parameters: effective area, density of cores, and cutoff. Nonlinearity and density of cores put important limits on the ultimate capacity scaling, and also determine the amount of signal processing (and thus electronic power consumption) that is necessary to recover transmitted information.

2. Realistic link models based on coupled-mode equations

v(L)=T(L,0)v(0)=T(L,zN1)...T(z2,z1)T(z1,0)v(0)
(1)

In this paper, we derive properties of the transfer matrix T(zp+1,zp) for a relatively short segment based on a coupled-mode equation. In general, this equation could include a variety of effects, but it is crucial that we include at least three terms: the coupling is given by C, bend perturbations by B, and the unperturbed mode effective index (including manufacturing variation, etc.) is given by A,

ddzT=i2πλ(A+B+C)T.
(2)

Of these, only 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 = zp + 1-zp. 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]

] can easily be ~10−5-10−4. 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]

]. It is well known, and is a proven, standard method in modeling macrobending loss, microbending loss [10

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]

], and bend-induced mode field distortion [11

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]

]. For fibers of interest here, basis mode m is well localized to core center position [xm,ym], and so the bend perturbation is simply

Bm,m=γncoreRbend(xmcosθ+ymsinθ).
(3)

Here γ≈1 can include stress corrections. The magnitude of these elements can range from ~10−6-10−3. The orientation θ and radius Rbend 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

Bend perturbations are crucial to modeling crosstalk, despite several models without bends in the literature. For example, Fig. 2
Fig. 2 The six outer cores have alternating positive and negative skew (Δnm eff = ± Δnskew) relative to the center core in the fiber design (a). Transfer-matrix simulation results show the fraction of power coupled from the center core for an over-simplified model with no bends (b), and for an improved model including a 1m bend radius (c). Skew in effective index is given by the legend, and illustrated in the inset.
shows solutions of the coupled mode model applied to a multicore design with skewed effective index between adjacent cores. The idea of intentionally skewing neighboring cores was discussed many years ago [4

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).

], and has generated interest more recently [5

5. 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.

8

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).

]. For this example, we used seven step-index cores with nominal contrast Δ = 0.0031 and diameter 10.4μm, spaced a = 42μm apart in a regular triangular arrangement (following [5

5. 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.

]). This leads to approximately the same coupling coefficient Cn,m = c0≈4.6 × 10−9 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 Δnm eff = ± Δnskew, alternating in sign, so that all nearest-neighbors are skewed by at least Δnskew (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.

This should not be surprising—bend-induced index perturbations are of order a/R bend and are therefore much larger than c0 unless the fiber is held extremely straight, Rbend>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 γncore a/R bend. For Rbend = 1m, γncore 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.

Since the bend-perturbation depends on the orientation of the bend θ with respect to the core geometry, and since this orientation typically cannot be held constant along the length, phase-matched resonances will be associated with certain positions where the bend-perturbation cancels the unperturbed index mismatch. Thus intermittent resonant coupling results from bend orientation drift, depicted schematically in Fig. 3
Fig. 3 Intermittent resonant coupling is illustrated schematically. Two neighboring cores (blue and red) have an effective index mismatch caused by slight core size difference. Lower plots show bent-fiber equivalent index profile (black) and mode effective index for the two cores (blue and red lines). As bend orientation drifts, the bend perturbation to this mismatch oscillates. If the bend is tight enough, each pair of cores sees index matched resonance twice for each 2π of orientation drift. Coupling between cores is frustrated at other orientations (grey arrows), but is efficient near the index-matched orientation.
. In the figure, two cores (labeled blue and red) are highlighted. They have an unperturbed index mismatch, and a bend-induced relative index shift that oscillates as the orientation of the bend drifts (relative to the fiber cross-section). Twice with each full turn of the orientation drift, the index mismatch is canceled by the bend perturbation, resulting in resonant coupling between these cores. This basic mechanism applies generally whenever the bend perturbation is large enough—whether the fabricated mismatches are accidental or designed, and whether they are due to variations in core index or core radius. In Fig. 2(c), we assumed the fiber orientation drift was slow and regular, making a full twist every 20m, crudely representing an unintentional spin or twist introduced during draw, cabling, etc. Discrete jumps corresponding to these intermittent phase-matching events are discernable in Fig. 2, and are even more pronounced in other simulations we have done.

4. Average crosstalk for a bent multicore fiber

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

ddzPi2πλ(A+B)P
(4)
T=PU
(5)
ddzU=i2πλP1CPU
(6)

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

Un,m(zp+1,zp)δn,m+i2πλCnmzpzp+1dzPn,n1Pm,m
(7)

If A is approximately constant over the interval, and P(zp) = 1
Un,m(zp+1,zp)i2πλCnmzpzp+1dzexp(iΔβm,n(zzp))exp(zpzi2πλ(Bm,mBn,n)),
(8)
for n≠m, where we have introduced the propagation-constant mismatch,

Δβm,n=2πλ(Am,mAn,n).
(9)

This begins to resemble the Fourier transform (FT) of f, the accumulated bend phase,

f(z)exp(zpzi2πλ(Bm,mBn,n)).
(10)

The crosstalk is the average fraction of power coupled between cores n and m

|Un,m|2|2πCnmλ|2zpzp+1dzzpzp+1dzexp(iΔβm,n(zz))f(z)f*(z),
(11)

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

|Un,m|2|2πCnmλ|2zpzp+1dzzpzp+1dzexp(iΔβm,n(zz))Rff(zz).
(12)

If f has a finite correlation length, then Rff becomes small beyond this length. If the correlation length is much smaller than ΔL, then we can neglect boundary terms and identify the FT of the autocorrelation as the Power Spectral Density (PSD) [12

12. K. S. Shanmugan, and A. M. Breipohl, Random Signals (John Wiley and Sons, 1988).

]

|Un,m|2|2πCnmλ|2(zp+1zp)Sff(Δβm,n)
(13)

In the case of regular multicore fibers, nearest-neighbors all have the same spacing a, and so for nearest neighbor cores n,m with displacement angle θm,n,

f(z)exp(z1zi2πλγncoreRbendacos(θθm,n))
(14)

The autocorrelation function and PSD are the same for all nearest-neighbor pairs in a regular multicore fiber, but are simply evaluated at different Δβ.

This derivation relies on choosing the interval ΔL = z 2-z 1 to satisfy three approximations: 1. ΔL>> correlation length of f, 2. ΔL is small enough that higher order perturbations can be neglected within the interval, and 3. ΔL << 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.

5. Low crosstalk fiber design

We have calculated cross-talk using the above statistical formulas for many sets of fiber bend statistics. The calculations are extremely fast, enabling unhindered exploration of the parameter space. Figure 4(b)
Fig. 4 Calculations illustrating the statistical model: Two curvature (a) and bend orientation functions (b) were randomly generated from the same statistics and plotted versus length for Rb0 = 0.5m. Broad oscillations common to both orientation curves are the gulp, while differences between the traces are the random orientation drift. Average cross-talk (c) between two neighboring cores [Eq. (13)] is plotted as a function of effective index mismatch for several nominal bend radii.
shows the average cross-talk between two cores as a function of their index mismatch [Δnnm = Δβnm/(2π/λ) for λ = 1550 nm]. The parameters of the above examples were used (ΔL = 40m, a = 42μm, c0≈4.6 × 10−9)—but the results apply to other values of a and c0 using scaling rules: Cross Talk ∝ΔL|c0|2 and R bend scales along with core spacing. In this example, the curvature has constant and random components,
1/Rb=1/Rb0+g1
(15)
with the standard deviation of the curvature 20% of its nominal value:

Rb0|g1|21/2=0.2.
(16)

The bend orientation
dθ/dz=K0(z)+g2(z)
(17)
has a deterministic gulp K0 with maximum spin rate 2π/(0.5m), gulp period 5m. The random orientation drift rate g2 has standard deviation 2.5rad/m. Both random components g1 and g2 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 f, a standard algorithm (p. 581 of [12

12. K. S. Shanmugan, and A. M. Breipohl, Random Signals (John Wiley and Sons, 1988).

], 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.

Although bends were not considered or described in detail in [5

5. 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.

], measurements there provide an interesting initial comparison with these calculations: for a laboratory measurement, one might guess that the fiber is spooled with Rb0 of order 10cm. Using scaling rules to adjust the calculation of Fig. 4 to the parameters of the measurement in [5

5. 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.

] (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

5. 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.

]. 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.

The formulas were tested by comparing several brute-force transfer matrix propagations (solid) with the analytical results (dashed), for example, as in Fig. 5
Fig. 5 The statistical average cross-talk expression, Eq. (12) (dashed), is validated by comparison with brute-force transfer-matrix propagation (solid). Asymptotic agreement is reached for each of the neighboring-core pairs, represented by different colors.
. Each color represents a different outer core coupled to the center core (except for black, which is the total coupling from the center core). Several examples were tested—the results shown use similar fiber parameters as above (a = 42μm, c0≈4.6 × 10−9), with a small intentional skew Δnskew = 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, g1 had standard deviation 0.02/m, K0 was a constant spin of 1 turn/10m, and g2 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 = c0 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.

6. Conclusions

Bends and fabrication variations qualitatively change the cross-talk of a multicore fiber, and are crucial to any realistic crosstalk simulations. By understanding this bend impact, we can choose low-crosstalk designs suited to realistic systems, for example skewing the cores enough to avoid intermittent resonant phase-matching.

These results offer a framework for correct design of multicore fibers, cables, and layout practices. This framework enables efficient long-distance concatenation models where the small-scale variation has already been correctly averaged away. It also calls for more careful consideration of the actual bends that would be present in a realistic multicore fiber link. The framework can now be applied to the wide range of applications and installations conditions under consideration for this emerging fiber type.

References and links

1.

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.

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]

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 Optical Fiber Communication Conference, OSA Technical Digest (CD) (Optical Society of America, 2010), paper OWK6.

6.

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.

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. 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]

12.

K. S. Shanmugan, and A. M. Breipohl, Random Signals (John Wiley and Sons, 1988).

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

  1. 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.
  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]
  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 Optical Fiber Communication Conference, OSA Technical Digest (CD) (Optical Society of America, 2010), paper OWK6.
  6. 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.
  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. 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]
  12. K. S. Shanmugan, and A. M. Breipohl, Random Signals (John Wiley and Sons, 1988).

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