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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 2 — Jan. 16, 2012
  • pp: 949–959
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Crosstalk in multicore fibers with randomness: gradual drift vs. short-length variations

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


Optics Express, Vol. 20, Issue 2, pp. 949-959 (2012)
http://dx.doi.org/10.1364/OE.20.000949


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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

Multicore fibers (MCF) are a major focus of recent fiber research [1

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.

11

11. 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.

]. They offer one of the few remaining avenues for dramatic scaling of the capacity-per-fiber. At the same time, they face significant technical challenges; the potential advantages and viability of this technology compared to a multiple single-core alternative are unclear. One significant potential obstacle is crosstalk and the limitations on core density implied by a tradeoff with crosstalk, effective area and cutoff. It is important to use the right models in order to understand crosstalk: oversimplified models have given crosstalk estimates orders of magnitude too small in some cases. We have presented statistical models including random fiber perturbations [12

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]

]. These display power-coupling dynamics consistent with empirical measurements [7

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]

] and can be used to understand the impact of fiber layout: For example, a fiber measured on a spool may show significantly different crosstalk than the same fiber cabled and deployed in a realistic telecommunications link.

2. Basic model

The basic coupled-mode model includes diagonal terms D and coupling terms K:

ddza=i(D+K)a
(1)

This is illustrated in Fig. 1
Fig. 1 Coupled-mode model.
in the simplest case of a 2-core coupler. It is useful to use the “interaction picture” formulation, where we switch variables:
a=Pu
(2)
ddzP=iDP.
(3)
Each element of 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

ddzu=iP1KPu
(4)

The solution to Eq. (4) is a transfer matrix 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:
dUm,n=i0Ldz[P1KP]m,n.
(5)
dUm,n=i0Ldzκm,nexp[i0zdζδβn,m(ζ)]
(6)
Here, δβn,m = Dn,n-Dm,m gives the differential phase between the cores and κm,n is the m,n element of K. The notation is slightly different than used previously.

A variety of random perturbations and imperfections can be included in δβn,m and its variation with fiber length. This could include variability in the fiber itself, temperature or strain effects, etc. In [9

9. 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.

], 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:

Mm,n=|dUm,n|2
(7)

In addition, significant length-variation is unavoidable in a realistic fiber link, unless the orientation of the fiber relative to the bend is strictly controlled along with all other perturbations. Orientation drift and bend perturbations lead to intermittent resonant coupling as illustrated in Fig. 2
Fig. 2 The standard bend model includes a bend-induced index tilt proportional to the curvature. Bends and other length-varying perturbations shift the index mismatch between cores, and lead to intermittent resonant coupling.
.

The incremental crosstalk between two cores over length L is |dUm,n|2; the magnitude squared of Eq. (6) gives the double-integral:
|dUm,n|2=0Ldz'0Ldzκm,n*(z')κm,n(z)exp[iz'zdζδβn,m(ζ)]
(8)
This equation is analogous to Eq. (11) of [12

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]

], but appears more compact because, in our current notation, δβn,m includes both the length-varying and non-varying parts. If we neglect length variation of the coupling coefficients themselves, then

|dUm,n|2=|κm,n|20Ldz'0Ldzexp[iz'zdζδβn,m(ζ)]
(9)

The statistical average in the integral is the correlation of the accumulated differential phase between positions 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:

|dUm,n|2L|κm,n|2dzexp[iz'zdζδβn,m(ζ)]
(10)

The separation of length scales is reasonable, at least in some relevant cases: Phase oscillations occur on length scale 2π/δβ, typically centimeters. The correlation length for perturbations is much longer, perhaps on the order of 1 meter. The segment length 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.

Previously [12

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]

] we derived,
|dUm,n(ω)|2=|κm,n|2LSff(δβn,m0)
(11)
Here δβ0 is the non-length-varying component (e.g. due to the intentional index mismatch between two cores) Sff is the power-spectral density (PSD) of f, and f is defined as the accumulated phase of the length varying part of of δβ:

f(z)=exp[i0zdζδβn,mvar(ζ,ω)].
(12)

3. Gradual variations

The above derivation is oversimplified: if the rate of variation were truly zero, we would not be able to assume ergodicity: the statistical average might differ significantly from the actual crosstalk, even over long lengths. Worse, the correlation length would be much larger than 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

Many of the power-spectral density (PSD) calculations previously presented [12

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]

,15

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

] illustrate the regime of gradual variations just discussed. This is illustrated in Fig. 3
Fig. 3 The phase-matching factor of crosstalk is calculated as a power spectral density (solid) and compared to the quasi-static approximation (dashed black) for the simple case where the index perturbation is due to constant bend radius and gradual orientation drift.
, which compares the PSD calculation and quasi-static approximation for a bend perturbation with slow orientation drift. Several bend radii are shown. For simplicity, the curvature in each case is assumed perfectly constant. For the PSD calculations, we used 40m long randomly generated processes, and averaged 16 realizations to smooth out fluctuations. For the quasi-static formula, constant curvature implies b = b0cos(θ) with bend orientation θ uniformly distributed over all angles and
b0=2πγncorea/Rbend.
(16)
Here 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):

pδβ(b)=pθ(θ)|db/dθ|+pθ(θ)|db/dθ|=21/(2π)|b0sin(θ)|
(17)
pδβ(b)=1b0π1(b/b0)2
(18)

5. Variations that are not gradual: spin

Intuitively, we would guess that gradual variations will play an important role in crosstalk for realistic telecom fibers. Preform variations will generally manifest on long lengths of fiber. Changes in random orientation drift (which modulates the bend perturbation) likely occur on something like a 1m scale. Cables are typically designed to protect fibers from abrupt kinks and stresses, and so many other perturbations may similarly vary on a length scale much larger than 1/Δβ. However, short-length variations are certainly realistic: for example, fiber can be intentionally spun so that its orientation varies on a centimeter length scale. The PSD formulation allows us to cover gradual and short-length variations on an equal footing.

The fine structure remains even if there is some variability in both the curvature and spin rate. As long as Rbend and spin rate drift slowly, they seem constant on a length scale shorter than this drift; that is, locally
f(z)exp(iaγncoreΛspinλRbendsin(2πz/Λspin))
(19)
and so the PSD reflects a quasi periodicity with period Λspin. This is illustrated in Fig. 5
Fig. 5 Power spectral density is calculated for the same parameters as Fig. 4, but with much larger variance in the bend radius. Structure imposed by the spin periodicity is still clearly visible.
, 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/R0bend plus a random component with standard deviation 1/R0bend. 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/Λspin.

The highly structured spectrum could potentially have significant consequences for system performance. We can model the fiber as having random preform perturbations δβ0 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 δβ0nm. The highly structured PSD means that each time the PSD is sampled at a different random δβ0nm, 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 δβ0nm values are controlled to index precision ~10−5.

6. Crosstalk reduction: fast spin

The results of Figs. 4 and 5 do not suggest successful crosstalk reduction. For example, comparing the solid (spun) and dashed (un-spun) curves of the same bend radius (colors), we see that spin sometimes increases and sometimes decreases crosstalk. In contrast, Fig. 6
Fig. 6 For a very fast, well-controlled spin (1turn/cm) spin periodicity leaves large gaps in the power spectral density: Crosstalk in this calculation is dramatically reduced for cores with index mismatch in between 0 and 1.55 × 10−4 .
shows an analogous calculation where the spin rate, 100turns/meter, is fast relative to the magnitude of the bend-induced index perturbations. For this fast spin rate, the quasi-periodicity means that the PSD peaks are spaced by 2π/Λspin in δβ units or λ/Λspin 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.

The calculation of Fig. 6 is highly idealized in that only bend perturbations are included. Figure 7
Fig. 7 While fast spin can disrupt bend-mediated phase matching between cores, other perturbations may not share the spin periodicity. Low-crosstalk regimes remain as long as non-bend perturbations are not too large.
repeats the calculation with gradually-varying non-bend perturbations included, and with a random length-varying component to the bend radius itself. It is assumed that the spin modulates the bend perturbation only, and so the non-bend perturbations do not have any periodicity imposed by the spin. Non-bend perturbations are normal-distributed with standard deviations 2 × 10−5 (left) or 4 × 10−5 (right). From the results, we see that the spun (solid) crosstalk falls orders of magnitude below the un-spun (dashed) crosstalk as long as the non-bend perturbations are not too large. The larger the non-bend perturbations are, the more tightly the index mismatch control must be to obtain large crosstalk suppression. Successful crosstalk suppression can then be accomplished by simultaneously: generating spin with very short and well-defined spin period, reducing the non-bend length-variation index perturbations, and arranging for the index mismatch between neighboring cores to fall in-between the peaks of the PSD, which occur at multiples of λ/Λspin. Fast spin (with pitch ~1mm) has been demonstrated experimentally [16

16. 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]

], but clearly adds a fabrication challenge to the difficulty of strict index-profile control; prospects for low-cost manufacturing will need further study.

7. Conclusions

This derivation also illustrates that there can be more to crosstalk than simply the probability of phase-matching. For perturbations that vary on a centimeter length scale, the approximation of gradual variation breaks down. The power spectral density captures the phase matching contribution in the general case, including any effects analogous to quasi-phase matching that arise from short-length variations.

Spin modulates the bend perturbation, and is an example where variations can reasonably have a length scale of a few centimeters or smaller. Our calculations show that spin could contribute significant sensitivity to small changes in core index mismatch, and thus variability to the crosstalk of nearly-identical fibers.

We also describe a regime where a very short-period spin is used to drastically reduce crosstalk, and discuss the fabrication tolerances (control of non-bend perturbations) needed to realize such a regime. This approach would be technically challenging to realize at low cost, but illustrates how improved crosstalk models can lead to novel crosstalk mitigation approaches.

Appendix

In the oversimplified derivation of Eq. (14), we assumed that δβn,m equaled a constant b over the entire segment length L. We can more realistically assume that δβn,m drifts slowly along the length. That is, to evaluate Eq. (10), we say that the random δβn,m has some random value b at the center of the interval ζ=(z+z’)/2, and is drifting at some random rate of drift b’. Then δβn,m(ζ)=b+b’[ζ-(z+z’)/2], and so
z'zdζδβn,m(ζ)=b(zz')+b'(zz')2/2
(20)
Equation (10) becomes

|dUm,n|2L|κm,n|2dbpδβ(b)dzeib(zz')eib'(zz')2/2
(21)

Strictly speaking, the right hand side should be averaged over the statistics of 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.
|dUm,n|2L|κm,n|2dbpδβ(b)2πib'eib2/(2b')
(22)
|dUm,n|22πL|κm,n|2dbpδβ(b)i2b'πeib2/(2b')
(23)
If b’ is small, the function sampling the probability distribution is delta-like:
i2b'πeib2/(2b')δ(b)
(24)
So that, again, we get Eq. (14)
|dUm,n|22πL|κm,n|2pδβ(0)
(25)
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 pδβ(b), and so the actual measured crosstalk may reasonably approximate the statistical average.

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.

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.

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 Optical Fiber Communication Conference, OSA Technical Digest (CD) (Optical Society of America, 2011), paper PDPB7.

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]

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 19(17), 16665–16671 (2011). [CrossRef] [PubMed]

9.

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.

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 19(17), 16715–16721 (2011). [CrossRef] [PubMed]

11.

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.

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]

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. 18(4), 477–488 (1982). [CrossRef]

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

  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. 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.
  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 Optical Fiber Communication Conference, OSA Technical Digest (CD) (Optical Society of America, 2011), paper PDPB7.
  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. Express19(17), 16576–16592 (2011). [CrossRef] [PubMed]
  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. Express19(17), 16665–16671 (2011). [CrossRef] [PubMed]
  9. 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.
  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. Express19(17), 16715–16721 (2011). [CrossRef] [PubMed]
  11. 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.
  12. 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]
  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. Express19(26), B102–B111 (2011). [PubMed]
  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.18(4), 477–488 (1982). [CrossRef]

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