## 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 Optical Society of America

## 1. Introduction

Multicore fibers [1,2] offer large improvements in cost and compactness for some systems, including some short-length interconnects where fiber congestion is a problem. At the same time, demand is quickly driving core telecommunications links towards fundamental limits of capacity per fiber, even assuming advanced modulation formats [3]. Thus, multicore fiber research now sees a resurgence of interest as one of the few remaining avenues for further scaling this capacity. While it is clear that a multi-core fiber can carry a lot of capacity, this technology will only be able to compete broadly with standard multi-fiber solutions if several technical hurdles are overcome.

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.

Strategies for achieving low-cross-talk with low nonlinearity are thus of great interest [4–8]. We have recently shown that bend-induced index perturbations are crucial in low-cross-talk design [6]. There, we looked at simple, deterministic arrangements of bend radius and orientation. In this paper, we turn to the more interesting case where bend radius and orientation are random processes. We derive an expression for the cross-talk in a randomly-bent segment averaged over the statistics of the bend. Our results give simple design rules for low-crosstalk fibers, and agree with measurements much better than previous calculations.

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

Realistic links are made up of segments with very different bend characteristics. They may include short bend-challenged lengths and much longer lengths with far more controlled layout, as illustrated in Fig. 1
. Like PMD, cross-talk can be thought of as a concatenation of transfer matrices *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

In this paper, we derive properties of the transfer matrix $T({z}_{p+1},{z}_{p})$ 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*,

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 = z_{p + 1}-z_{p}. A realistic model cannot neglect *A*, since even accidental variation in effective index [2] 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]. It is well known, and is a proven, standard method in modeling macrobending loss, microbending loss [10], and bend-induced mode field distortion [11]. For fibers of interest here, basis mode *m* is well localized to core center position [x_{m},y_{m}], and so the bend perturbation is simply

Here γ≈1 can include stress corrections. The magnitude of these elements can range from ~10^{−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

Bend perturbations are crucial to modeling crosstalk, despite several models without bends in the literature. For example, Fig. 2
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], and has generated interest more recently [5–8]. 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]). 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.

This should not be surprising—bend-induced index perturbations are of order *a/R*
_{bend} and are therefore much larger than *c _{0}* unless the fiber is held extremely straight, R

_{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.

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 . 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

We now start the analysis of cross-talk by dividing the total link into appropriate segments [*z _{p},z_{p + 1}*], 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.

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*,

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

If *A* is approximately constant over the interval, and *P*(z_{p}) = 1

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

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

Since *f* is a stationary random process, it has an autocorrelation function *R _{ff}*,

If *f* has a finite correlation length, then *R _{ff}* becomes small beyond this length. If the correlation length is much smaller than Δ

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

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 θ_{m,n},

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 Δ*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.

The analytical result vastly improves our understanding of the design, because it means that the coupling coefficients *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

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)
shows the average cross-talk between two cores as a function of their index mismatch [Δn_{nm} = Δβ_{nm}/(2π/λ) for λ = 1550 nm]. The parameters of the above examples were used (Δ*L* = 40m, *a* = 42μm, *c _{0}*≈4.6 × 10

^{−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,

The bend orientation

has a deterministic gulp*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.

Useful results can be read easily from this type of figure: For example, we might evaluate a low-crosstalk fiber design using intentional effective index skew for nearest-neighbor cores [4]. If the skew gives typical index mismatch of around 10^{−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 _{0}*), 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

*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.

Although bends were not considered or described in detail in [5], measurements there provide an interesting initial comparison with these calculations: for a laboratory measurement, one might guess that the fiber is spooled with R_{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.

The formulas were tested by comparing several brute-force transfer matrix propagations (solid) with the analytical results (dashed), for example, as in Fig. 5
. 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, *c _{0}*≈4.6 × 10

^{−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

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.

In this paper we perform the statistical averaging of multicore fiber propagation over the bend statistics of a short fiber segment, perhaps ~10m in a link of several km. The results highlight the importance of bending in low-crosstalk design, showing that a range of realistic bend radii result in very different cross-talk predictions. Our analysis shows that the degree of skew proposed in previous analyses would be insufficient to reduce crosstalk significantly, but that large reductions in crosstalk can be achieved using our elegant design approach. Further, the results show that crosstalk of a fiber in a typical laboratory arrangement (coiled to radius ~10cm) may be drastically different from crosstalk of the same fiber in actual deployment, where bend radius is likely much larger.

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.

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