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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 5 — Mar. 11, 2013
  • pp: 5401–5412
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Physical interpretation of intercore crosstalk in multicore fiber: effects of macrobend, structure fluctuation, and microbend

Tetsuya Hayashi, Takashi Sasaki, Eisuke Sasaoka, Kunimasa Saitoh, and Masanori Koshiba  »View Author Affiliations


Optics Express, Vol. 21, Issue 5, pp. 5401-5412 (2013)
http://dx.doi.org/10.1364/OE.21.005401


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Abstract

We have derived an intuitively interpretable expression of the average power-coupling coefficient for estimating the inter-core crosstalk of the multicore fiber. Based on the derived expression, we discuss how the structure fluctuation and macrobend can affect the crosstalk, and organize previously reported methods for crosstalk suppression. We also discuss how the microbending can affect the crosstalk in homogeneous and heterogeneous MCFs, based on the derived expression and previously reported measurement results.

© 2013 OSA

1. Introduction

Spatial division multiplexing using a multi-core fiber (MCF) is a strong candidate technology to overcome the capacity limit of single-core fiber transmission systems [1

1. T. Morioka, “New generation optical infrastructure technologies: EXAT initiative towards 2020 and beyond,” in OptoElectron. Commun. Conf. (OECC) (2009), paper FT4. [CrossRef]

]. Inter-core crosstalk (XT) is one of the most important properties of uncoupled MCFs, and suppression of the XT has been actively studied [2

2. M. Koshiba, K. Saitoh, and Y. Kokubun, “Heterogeneous multi-core fibers: proposal and design principle,” IEICE Electron. Express 6(2), 98–103 (2009). [CrossRef]

10

10. 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). [CrossRef] [PubMed]

]. Recently, Koshiba et al. derived a closed-form expression of the average power-coupling coefficient between cores in the MCF [11

11. M. Koshiba, K. Saitoh, K. Takenaga, and S. Matsuo, “Analytical expression of average power-coupling coefficients for estimating intercore crosstalk in multicore fibers,” IEEE Photon. J. 4(5), 1987–1995 (2012). [CrossRef]

]. The closed-form expression is powerful, and easy to estimate the average crosstalk; however, it is difficult to interpret physical meaning of the expression intuitively.

In this paper, we derive another expression of the average power-coupling coefficient, whose physical meaning is easily interpreted. Based on the derived expression, we discuss how the structure fluctuation and macrobend can affect the crosstalk, and explain previously reported methods for crosstalk suppression. We also discuss how the microbending can affect the crosstalk in homogeneous and heterogeneous MCFs, based on the derived expression and previously reported measurement results.

2. Review and clarification of previous derivation of power-coupling coefficient

In this section, we review the derivation of the power-coupling coefficient between cores from the longitudinally perturbed coupled-mode equation. We also clarify an ambiguous point of the definition of power spectral density in the derivation.

Since the propagation constants of cores in the MCF are perturbed by bend, twist, structure fluctuation, and so on, the coupling between cores in the MCF can be described using the coupled power equation. The power-coupling coefficient can be derived from the coupled mode equation with longitudinally perturbed propagation constants. The coupled mode equation can be expressed as
dAndz=jκnmexp[j0z(βmβn)dz]Am=jκnmexp[j(βc,mβc,n)zj0z(βv,mβv,n)dz]Am,
(1)
where A is the complex amplitude, κnm the mode coupling coefficient from Core n to Core m, β = 2πneff/λ the propagation constant, neff the effective refractive index, and λ the wavelength. Subscripts c and v of β represent constant and variable perturbed parts of β, respectively

Based on Eq. (1), in case of low crosstalk, the crosstalk in amplitude within the fiber segment [z1, z2] can be expressed as
Δxnm=ΔAnAmjκnmz1z2exp[j(βc,mβc,n)z]f(z)dz,
(2)
by using
f(z)exp[j0z[βv,m(z)βv,n(z)]dz].
(3)
Accordingly, the average crosstalk increase in power within the segment [z1, z2] can be expressed as
ΔXnm=|Δxnm|2=ΔxnmΔxnm*κnm2z1z2z1z2exp[j(βc,mβc,n)(zz)]f(z)f*(z)dzdzκnm2z1z2z1zz2zexp[jΔβc,nmζ]f(z+ζ)f*(z)dζdzκnm2z1z2dzRff(ζ)exp(jΔβc,nmζ)dζκnm2ΔzRff(ζ)exp(jΔβc,nmζ)dζ,
(4)
where Rff is the autocorrelation function (ACF) of f(z), Δz is z2z1, and the correlation length lc of Rff is assumed to be adequately shorter than Δz. Rff(ζ) can be understood as the correlation between the coupled and non-coupled lights that are propagated for the length of ζ after the coupling. For example, where ζ >> lc, the coupled and non-coupled lights becomes incoherent even if the lights are very coherent. Based on the Wiener–Khinchin theorem, the power spectrum density (PSD) is the Fourier transform of the ACF:
Sff(ν˜)(ν˜)=Rff(ζ)exp(j2πν˜ζ)dζ,
(5)
where ν˜=neff/λ=β/(2π) represents the wave number (or spatial frequency) in the medium—whereas the propagation constant β is the angular wave number. Note that ν˜, neff, and β have common subscripts, e.g., ν˜c=neff,c/λ=βc/(2π). To describe the PSDs with respect to ν˜ and β with common expressions, we would like to define the PSD with respect to β, whose total power is equivalent to Eq. (5). From the Parseval’s theorem, the average power of f(z), or expected value of |f(z)|2, is equivalent to the integral of the PSD over whole ν˜, and the following equation holds between f(z) and the PSDs of f(z):
Sff(ν˜)(ν˜)dν˜=Sff(ν˜)(ν˜)dν˜dβdβ=E[|f(z)|2]=1,
(6)
where E[·] represents the expected value. Therefore, in this paper, the PSD Sff(β)(β) with the scale of the propagation constant β (the angular wave number in the medium) is defined as:
Sff(β)(β)Sff(ν˜)(ν˜)dν˜dβ=12πSff(ν˜)(ν˜)=12πRff(ζ)exp(jβζ)dζ.
(7)
From Eqs. (4)(7), the power-coupling coefficient can be expressed as

hnm=ΔXnmΔzκnm2Sff(ν˜)(Δneff,c,nmλ)=κnm2[2πSff(β)(Δβc,nm)].
(8)

Figure 1
Fig. 1 Schematics of perturbations on the propagation constant. (a) a slight change of the propagation constant in a core due to bend, (b) a slight change of the propagation constant in a core due to structure fluctuation, and (c) a bend-induced change of the propagation constant in a core when assuming another core as a reference of the propagation constant.
shows the schematics of perturbations on β, or how βv can vary. As shown in Figs. 1(a) and 1(b), the bend and the structure fluctuation can induce a slight change in βv in one core, which can occur ether in the single-core fiber or in the MCF. In the single-core fiber, by assuming proper Rff or Sff for the perturbations shown in Figs. 1(a) and 1(b), Eq. (8) is utilized for analyzing the power coupling between modes in the multi-mode fiber, microbend loss—power coupling from the core modes to the cladding modes, and so on. In the MCF, as shown in Fig. 1(c), the bend can induce relatively large βv in a core when assuming another core as a reference of the propagation constant. Fini et al. [4

4. 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]

] and Hayashi et al. [5

5. T. Hayashi, T. Nagashima, O. Shimakawa, T. Sasaki, and E. Sasaoka, “Crosstalk variation of multi-core fibre due to fibre bend,” in Eur. Conf. Opt. Commun. (ECOC) (2010), paper We.8.F.6. [CrossRef]

] assumed that βv in the MCF is induced by the macrobend and twist of the MCF as
βv,n=βc,nxncosθf(z)ynsinθf(z)Rb(z)=βc,nrncosθn(z)Rb(z),
(9)
and investigated the crosstalk characteristics of the MCFs. Here, (xn, yn) and (rn, θn) are the local Cartesian and polar coordinates of Core n in a fiber cross-section, respectively, θn = 0 is the radial direction of the macrobend, θf the angle between the x-axis and the radial direction of the macrobend, and Rb the macrobend radius of the MCF—that is, the distance between the center of the macrobend and the origin of the local coordinates.

However, it is not easy to assume a proper βv, Rff, or Sff that can include the perturbations of both the bend and the structure fluctuation. Therefore, by assuming that Rff includes only the effect of structure fluctuation and does not include that of macrobend and twist, Koshiba et al. investigated the effects of correlation length lc and of the shape of the ACF Rff on the average crosstalk μX [10

10. 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). [CrossRef] [PubMed]

,11

11. M. Koshiba, K. Saitoh, K. Takenaga, and S. Matsuo, “Analytical expression of average power-coupling coefficients for estimating intercore crosstalk in multicore fibers,” IEEE Photon. J. 4(5), 1987–1995 (2012). [CrossRef]

]. They investigated some types of Rff, and found that the exponential ACF (EAF)
Rff(ζ)=exp(|ζ|/lc)
(10)
is proper for estimating actual μX of the MCFs. The EAF have been introduced to microbending loss analysis [12

12. K. Petermann, “Microbending loss in monomode fibers,” Electron. Lett. 12(4), 107–109 (1976). [CrossRef]

]. Since the PSD of the EAF is the Lorentzian distribution, the power-coupling coefficient was obtained from Eq. (8) as [10

10. 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). [CrossRef] [PubMed]

,11

11. M. Koshiba, K. Saitoh, K. Takenaga, and S. Matsuo, “Analytical expression of average power-coupling coefficients for estimating intercore crosstalk in multicore fibers,” IEEE Photon. J. 4(5), 1987–1995 (2012). [CrossRef]

]:
hnm(z)=κnm21π1/(2πlc)1/(2πlc)2+[Δneff,c,nm(z)/λ]2=κnm22π1π1/lc1/lc2+[Δβc,nm(z)]2=κnm22lc1+[Δβc,nm(z)lc]2,
(11)
where Δβ´c,nm is β´c,nβ´c,m, and β´c is redefined βc that includes the effects of macrobend and twist:
βc,n=βc,n(1+xncosθf(z)ynsinθf(z)Rb(z))=βc,n(1+rncosθn(z)Rb(z)).
(12)
Average crosstalk μX estimated using coupled-power equation with the power-coupling coefficient of Eq. (11) may be valid in cases where changes of Rb and θ are gradual enough compared to lc, since Δβ´c,nm—which is variable and includes macrobend and twist— is substituted to Δβc,nm—which is constant— in Eq. (8).

By assuming constant Rb and twist rate, Koshiba et al. also analytically derived an average power-coupling coefficient , which is averaged over θ, as [11

11. M. Koshiba, K. Saitoh, K. Takenaga, and S. Matsuo, “Analytical expression of average power-coupling coefficients for estimating intercore crosstalk in multicore fibers,” IEEE Photon. J. 4(5), 1987–1995 (2012). [CrossRef]

]:
h¯nm=κnm22lc[1/a(b+ac)+1/c(b+ac)],
(13)
a=1+(Δβc,nmlcBnmlc/Rb)21+lc(Δβc,nmlcβc,nDnmlc/Rb)2,
(14)
b=1+(Δβc,nmlc)2(Bnmlc/Rb)21+(Δβc,nmlc)2(βc,nDnmlc/Rb)2,
(15)
c=1+(Δβc,nmlc+Bnmlc/Rb)21+(Δβc,nmlc+βc,nDnmlc/Rb)2,
(16)
Bnm=(βc,nxnβc,mxm)2+(βc,nynβc,mym)2,
(17)
where Bnm can be approximated as βc,nDnm if βc,m/βc,n ≈1, Dnm is the center-to-center distance between Core m and Core n. They also reported that Eqs. (13)(17) agreed well with measurement results. However, it is difficult to interpret physical meaning of Eqs. (13)(17) intuitively.

3. Derivation of an intuitive expression of average power-coupling coefficient

To understand the physical meaning of the average power-coupling coefficient, we derive another expression of the average power-coupling coefficient in this section. For simplicity, the center of Core m is taken as the origin of the local coordinate, and accordingly Δβ´c,nm can be written as
Δβc,nm(Rb,θnm)=Δβc,nm+Δβb,nm(Rb,θnm),
(18)
Δβb,nm(Rb,θnm)=Δβb,nmdev(Rb)cosθnm,
(19)
Δβb,nmdev(Rb)=βc,nDnmRb,
(20)
where θnm represents the angle between the radial direction of the bend and a line segment from Core m to Core n, Δβb,nm the difference of β variation between Core m and Core n from the macrobend, and Δβb,nmdev the peak deviation of Δβb,nm.

Let pθnm(θnm) and pRb(Rb) be the probability density functions of θnm and of Rb, respectively, along the MCF; by assuming that pθnm(θnm) and pRb(Rb) are statistically independent, the twist of the MCF is gradual enough, and average crosstalk is adequately low; the average crosstalk μX,nm from Core m to Core n can be expressed as
μX,nm(L)0Lhnm(z)dzL[1L0Lhnm(z)dz]LE[hnm]L0pRb(Rb)h¯nm(Rb)dRb,
(21)
where the average power-coupling coefficient is
h¯nm(Rb)=02πpθnm(θnm)hnm(Rb,θnm)dθnm.
(22)
By assuming that the twist of the MCF is random enough and the MCF is adequately long, pθnm(θnm) can be assumed to be constant ( = 1/(2π)) over all θnm; therefore, by substituting Δβ=Δβb,nm(Rb,θnm)=Δβb,nmdev(Rb)cosθnm and using Eq. (8) and sin(arccosx)=1x2, Eq. (22) can be rewritten as
h¯nm(Rb)=02π12πhnm(Rb,θnm)dθnm=02π12πκnm22πSff(β)[Δβc,nm(Rb,θnm)]dθnm=20πκnm2Sff(β)[Δβc,nm+Δβb,nmdev(Rb)cosθnm]dθnm=2πΔβb,nmdevΔβb,nmdevκnm2π[Δβb,nmdev(Rb)]2Δβ2Sff(β)(Δβc,nmΔβ)d(Δβ).
(23)
where Sff is the Lorentzian distribution as shown in Eq. (11). By using the arcsine distribution:
pΔβb(Δβc,nm)={1π(Δβb,nmdev)2Δβc,nm2,|Δβc,nm|Δβb,nmdev,0,otherwise,
(24)
which is the probability distribution of Δβb, Eq. (23) can be rewritten as
h¯nm(Δβc,nm,Rb)=κnm22π(pΔβbSff(β))Δβ(Δβc,nm)=κnm2(pΔν˜bSff(ν˜))Δν˜(Δν˜c,nm),
(25)
where the expression of (fg)x denotes the convolution of f and g with respect to x, and the expression with respect to ν˜ is also shown for comparison. If we consider the case where PSD Sff in Eq. (8) includes both the effects of the structure fluctuation and the macrobend, the convolution term in Eq. (25) may be understood as the PSD Sff in Eq. (8).

Particularly where |Δβc,nm| and the bandwidth of Sff(β) are adequately smaller than Δβb,nmdev, Sff becomes a narrow delta-function-like distribution and the convolution contains only a gradually varying part of pΔβb(Δβc,nm); therefore, Eq. (25) can be approximated as
h¯nm(Δβc,nm,Rb)κnm2[2πpΔβb(Δβc,nm)]=κnm2pΔν˜b(Δν˜c,nm)κnm22(βc,nDnmRb)2Δβc,nm2=κnm2λπ(neff,c,nDnmRb)2Δneff,c,nm2,
(26)
which is also obtained from Eq. (8) by approximating the PSD Sff(β) as the probability distribution of Δβb—shown in Eq. (24)— with constant Rb, as shown in [13

13. J. M. Fini, B. Zhu, T. F. Taunay, M. F. Yan, and K. S. Abedin, “Crosstalk in multicore fibers with randomness: gradual drift vs. short-length variations,” Opt. Express 20(2), 949–959 (2012). [CrossRef] [PubMed]

]. In case of homogeneous MCFs (Δβc,nm = 0), Eq. (26) is reduced to
h¯nm(Rb)κnm22Rbβc,nDnm=κnm2λRbπneff,c,nDnm,
(27)
which coincides with Eq. (19) in [9

9. 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 Eq. (25) in [11

11. M. Koshiba, K. Saitoh, K. Takenaga, and S. Matsuo, “Analytical expression of average power-coupling coefficients for estimating intercore crosstalk in multicore fibers,” IEEE Photon. J. 4(5), 1987–1995 (2012). [CrossRef]

]. The difference between Eq. (26) and Eq. (27) is less than 0.1 dB when Δβc < 0.21βcD/Rb; therefore, Eq. (27) may be also used for estimating the crosstalk of a bent heterogeneous MCF with small Δβc.

Figure 2
Fig. 2 Comparisons between calculated by using Eq. (25) and calculated by using Eqs. (13)(17). (a) normalized with respect to the Lorentzian, (b) normalized with respect to the arcsine distribution. Solid lines: calculated by using Eq. (25), dashed lines: calculated by using Eqs. (13)(17). The solid lines and the dashed lines are overlapped.
shows comparisons between calculated by using Eq. (25) and calculated by using Eqs. (13)(17). Figures 2(a) and 2(b) show the PSDs normalized with respect to the Lorentzian Sff and to the arcsine distribution pΔβb, respectively. The Lorentzian and arcsine distributions represent the spectra of the perturbations induced by the structure fluctuation and by the macrobend, respectively. Solid lines represent calculated by using Eq. (25) and dashed lines represent calculated by using Eqs. (13)(17); however, the solid lines and the dashed lines are overlapped, and we can only see the solid lines. Accordingly, it was clearly confirmed that Eq. (25) is equivalent to the expression of with Eqs. (13)(17), and it can be also said that the set of Eqs. (13)(17) is a closed-form solution of the convolution of the Lorentzian and the arcsine distribution.

4. Crosstalk suppression methods related to macrobend and structure fluctuation

Based on the above derivations, it can be understood that the crosstalk is proportional to the power of the mode-coupling coefficient and to the PSD of the perturbations. Of course, the suppression of the mode-coupling coefficient is important and various ways were proposed for confining power into cores such as high-index and small-diameter core structure [3

3. K. Takenaga, S. Tanigawa, N. Guan, S. Matsuo, K. Saitoh, and M. Koshiba, “Reduction of crosstalk by quasi-homogeneous solid multi-core fiber,” in Opt. Fiber Commun. Conf. (OFC) (2010), paper OWK7.

,6

6. K. Takenaga, Y. Arakawa, S. Tanigawa, N. Guan, S. Matsuo, K. Saitoh, and M. Koshiba, “An investigation on crosstalk in multi-core fibers by introducing random fluctuation along longitudinal direction,” IEICE Trans. Commun. E94-B(2), 409–416 (2011). [CrossRef]

], hole- or trench-assisted core structures [9

9. 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]

,14

14. K. Saitoh, T. Matsui, T. Sakamoto, M. Koshiba, and S. Tomita, “Multi-core hole-assisted fibers for high core density space division multiplexing,” in OptoElectron. Commun. Conf. (OECC) (2010), paper 7C2–1.

,15

15. K. Takenaga, Y. Arakawa, S. Tanigawa, N. Guan, S. Matsuo, K. Saitoh, and M. Koshiba, “Reduction of crosstalk by trench-assisted multi-core fiber,” in Opt. Fiber Commun. Conf. (OFC) (2011), paper OWJ4.

], and photonic-crystal structures [16

16. D. M. Taylor, C. R. Bennett, T. J. Shepherd, L. F. Michaille, M. D. Nielsen, and H. R. Simonsen, “Demonstration of multi-core photonic crystal fibre in an optical interconnect,” Electron. Lett. 42(6), 331–332 (2006). [CrossRef]

18

18. K. Imamura, K. Mukasa, Y. Mimura, and T. Yagi, “Multi-core holey fibers for the long-distance (>100 km) ultra large capacity transmission,” in Opt. Fiber Commun. Conf. (OFC) (2009), paper OTuC3.

].

The PSD can be intuitively explained as the amount of the phase matching. Accordingly, how to suppress the PSD can be understood as how to suppress the phase matching. In this section, the methods for suppressing the phase matching are described.

The phase matching suppression methods can be categorized into some types according to how to utilize what kind of the perturbations. Here, three types of suppression methods are explained in the following subsections. A schematic example of h¯nm(Δβc,nm,Rb) in Eq. (25) shown in Fig. 3
Fig. 3 A schematic example of the average power-coupling coefficient , as a function of the propagation constant mismatch Δβc and the curvature 1/Rb, in case that twist of an MCF is gradual and random enough. (a) a 3-dimensional plot, (b) a contour map of log(). Thick solid lines in (b) are the thresholds between the phase-matching region and the non-phase-matching region.
will help with understanding, along with Fig. 2.

4.1 Utilization of the propagation constant mismatch

One is the method utilizing the propagation constant mismatch Δβc to suppress the phase matching [2

2. M. Koshiba, K. Saitoh, and Y. Kokubun, “Heterogeneous multi-core fibers: proposal and design principle,” IEICE Electron. Express 6(2), 98–103 (2009). [CrossRef]

,4

4. 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]

,19

19. G. Le Noane, D. Boscher, P. Grosso, J. C. Bizeul, and C. Botton, “Ultra high density cables using a new concept of bunched multicore monomode fibers: A key for the future FTTH networks,” in Int. Wire Cable Symp. (IWCS) (1994), 203–210.

]. As shown as the non-phase-matching regions in Fig. 3, Δβc larger than Δβbdev can prevent the bend-induced phase-matching between dissimilar cores, and can suppress the crosstalk [4

4. 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]

,5

5. T. Hayashi, T. Nagashima, O. Shimakawa, T. Sasaki, and E. Sasaoka, “Crosstalk variation of multi-core fibre due to fibre bend,” in Eur. Conf. Opt. Commun. (ECOC) (2010), paper We.8.F.6. [CrossRef]

]. In other words, for suppressing the crosstalk, the bending radius of the MCF has to be managed to be adequately larger than the critical bending radius Rpk [5

5. T. Hayashi, T. Nagashima, O. Shimakawa, T. Sasaki, and E. Sasaoka, “Crosstalk variation of multi-core fibre due to fibre bend,” in Eur. Conf. Opt. Commun. (ECOC) (2010), paper We.8.F.6. [CrossRef]

,11

11. M. Koshiba, K. Saitoh, K. Takenaga, and S. Matsuo, “Analytical expression of average power-coupling coefficients for estimating intercore crosstalk in multicore fibers,” IEEE Photon. J. 4(5), 1987–1995 (2012). [CrossRef]

]:
Rpk=Bnm|Δβc,nm|Dnmβc,n|Δβc,nm|=Dnmneff,c,n|Δneff,c,nm|,
(28)
that is, Rb where Eq. (26) can be infinite, or the maximal Rb where the phase matching due to the macrobend can occur even if there is no structure fluctuation. Some margin from Rpk is needed for avoiding the phase matching induced by the spectral broadening of Sff, due to the structural fluctuations. In heterogeneous MCFs, it is preferred if the correlation length lc of the structural fluctuation can be elongated, because the spectral broadening of Sff can be narrowed and the PSD leakage into the non-phase-matching region can be suppressed, as shown in Figs. 2(b) and 3.

If most part of an MCF is deployed in gentle-bend conditions, a slight difference in propagation constants or effective indices may be enough for the phase matching suppression [4

4. 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]

].

Since the typical winding radii of fiber spools are around 10 cm and Rpk less than 10 cm requires very large difference in core structure [5

5. T. Hayashi, T. Nagashima, O. Shimakawa, T. Sasaki, and E. Sasaoka, “Crosstalk variation of multi-core fibre due to fibre bend,” in Eur. Conf. Opt. Commun. (ECOC) (2010), paper We.8.F.6. [CrossRef]

], most of crosstalk measurements and transmission experiments reported in various papers are considered to have been conducted in the phase-matching region. A hexagonal MCF with three kinds of cores reported in [20

20. J. Sakaguchi, Y. Awaji, N. Wada, T. Hayashi, T. Nagashima, T. Kobayashi, and M. Watanabe, “Propagation characteristics of seven-core fiber for spatial and wavelength division multiplexed 10-Gbit/s channels,” in Opt. Fiber Commun. Conf. (OFC) (2011), paper OWJ2.

] is an exception, but it has a large difference in optical properties between cores so that Rpk can be smaller than the bobbin radius of 140 mm.

Recently, Saitoh et al. reported that up to two kinds of dissimilar step-index cores can be designed to achieve Rpk around 5 cm while achieving a similar Aeff of around 80 µm2 at 1550 nm, and other good optical properties [21

21. K. Saitoh, M. Koshiba, K. Takenaga, and S. Matsuo, “Low-crosstalk multi-core fibers for long-haul transmission,” Proc. SPIE 8284, 82840I, 82840I-8 (2012). [CrossRef]

]. Tu et al. also reported that up to two kinds of dissimilar trench-assisted cores can be designed to achieve Rpk around 5 cm while achieving a similar Aeff of around 100 µm2 [22

22. J. Tu, K. Saitoh, M. Koshiba, K. Takenaga, and S. Matsuo, “Design and analysis of large-effective-area heterogeneous trench-assisted multi-core fiber,” Opt. Express 20(14), 15157–15170 (2012). [CrossRef] [PubMed]

].

4.2. Utilization of the bend-induced perturbation

The bend can also be utilized for the phase matching suppression [9

9. 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]

]. As shown in Figs. 2(a) and 3, enlargement of the bend-induced perturbation—caused by the increase of the curvature or the decrease of the bending radius— can spread the PSD and suppress the crosstalk even in case of homogeneous MCFs (Δβc = 0). Identical core structure is rather desirable for suppressing the PSD. The PSD changes gradually with the bend radius, and there is no drastic PSD increase like that around Rpk in case of heterogeneous MCFs, since the PSD is suppressed in the phase-matching region. As shown in Eq. (27), the average crosstalk of a homogeneous MCF is proportional to the average bending radius, where |Δβc,nm| and the bandwidth of Sff(β) are adequately smaller than Δβb,nmdev. Therefore, if the average bending radius of the MCF is managed to be smaller than a certain value, or if the MCF is deployed in bend-challenged conditions, low crosstalk can be achieved with identical cores.

4.3. Utilization of the longitudinal structural fluctuation

As shown in Figs. 2(b) and 3(a), the power spectrum of the perturbations is broadened by the longitudinal structural fluctuations. If the Rff due to the structure fluctuation has a very short correlation length lc, the power spectrum spreads broadly over the propagation constant mismatch Δβc, and thus the PSD may be suppressed even in case of an unbent homogeneous MCF (Δβc = 0, 1/Rb = 0). A homogeneous MCF utilizing the longitudinal structural fluctuations was conceptually proposed by Takenaga et al. as “quasi-homogeneous MCF” in [3

3. K. Takenaga, S. Tanigawa, N. Guan, S. Matsuo, K. Saitoh, and M. Koshiba, “Reduction of crosstalk by quasi-homogeneous solid multi-core fiber,” in Opt. Fiber Commun. Conf. (OFC) (2010), paper OWK7.

,6

6. K. Takenaga, Y. Arakawa, S. Tanigawa, N. Guan, S. Matsuo, K. Saitoh, and M. Koshiba, “An investigation on crosstalk in multi-core fibers by introducing random fluctuation along longitudinal direction,” IEICE Trans. Commun. E94-B(2), 409–416 (2011). [CrossRef]

]. To the author’s knowledge, the crosstalk suppression by the structural fluctuation has not been actually observed yet, because the bend-induced perturbations are much larger than the fluctuation induced perturbations in the measurement conditions. However, the structural fluctuation may work when the MCF is cabled and installed in very-gently-bent conditions. For example, lcΔβb,nmdevlcβcD/Rb of 1.3, 1/3, 1.0 × 10−1, 1.0 × 10−2, and 1.0 × 10−3 correspond to 1-dB, 5-dB, 10-dB, 20-dB, and 30-dB decreases in from Eq. (27) at Δβc = 0, respectively.

5. Applicability of the average power-coupling coefficient for estimating microbend-affected crosstalk

In Sections 2 and 3, Sff only includes structure fluctuation, and pΔβb(Δβc,nm) includes macrobend perturbation that gradually varies in longitudinal direction. Based on the assumption that the macrobend perturbation is gradual enough compared to lc, we may redefine the Sff as the PSD of high frequency perturbations other than the macrobend perturbation, and thus Sff may include not only the effect of structure fluctuation but also the effect of microbend, in Eqs. (13)(17) and Eq. (25). In this case, the increase of the microbend can be understood as the decrease of the correlation length lc of Sff. The microbend may induce β perturbations both within a core (Fig. 1(a)) and between cores (Fig. 1(c)); and may be induced by various ways such as winding on a sandpaper-coated bobbin, winding on a wire mesh bobbin, and actual cabling. Therefore, the shape of Sff for the structure fluctuation and the microbend could possibly be different from the Lorentzian—that is, Sff only for the structure fluctuation— and depend on how the microbend is induced. Thus, we need to investigate further details of the effect of the microbend on the crosstalk through experiment.

However, as a first step, we will evaluate the effect of the microbend on the crosstalk by assuming that Sff for the structure fluctuation and the microbend, in this paper.

Figure 4
Fig. 4 Dependences of the microbend on the average crosstalk (average power-coupling coefficient) for MCF-A (heterogeneous) and MCF-B (homogeneous), measured by wavelength averaging [26] with 100-m fiber [23,24] at λ = 1550 nm.
shows the dependences of the average power-coupling coefficient on microbend conditions for a heterogeneous MCF-A [23

23. T. Hayashi, T. Sasaki, and E. Sasaoka, “Microbending-induced crosstalk increase in heterogeneous multi-core fiber,” in Eur. Conf. Opt. Commun. (ECOC) (2011), paper Mo.1.LeCervin.3.

] and for a homogeneous MCF-B [24

24. T. Hayashi, T. Sasaki, and E. Sasaoka, “Multi-core fibers and their crosstalk characteristics,” in IEEE Photonics Society Summer Topical Meeting Series (2012), paper TuC4.1.

]. Fiber properties of the MCFs are shown in Table 1

Table 1. Characteristics of the Evaluated MCFs

table-icon
View This Table
. For MCF-A, from the center core to an outer core was obtained. For MCF-B, the average of s between the three pairs of the neighboring cores was obtained. The values of were obtained from values of measured average crosstalk and fiber length by using coupled power equation. The microbend was applied by winding the MCFs on a 140-mm-radius bobbin with sandpaper (grade P240) at winding tension T. at T = 0 N was measured using a 140-mm-radius bobbin without the sandpaper. of the heterogeneous MCF was increased by the microbend, but that of the homogeneous MCF was varied only slightly.

These results may be well explained as the shortening of lc by using Eqs. (13)(17) or Eq. (25). Figure 5
Fig. 5 Comparisons of the average power-coupling coefficient s obtained from the measurements and from Eqs. (13)(17). (a) The dependences of in MCF-A on the bending radius Rb and on the microbend. (b) The dependence of in MCF-A and MCF-B on the propagation constant mismatch Δβc and on the microbend at Rb = 140 mm. Closed-marks: measured without the microbend, open-marks: measured with the microbend, triangulars: measured by averaging the crosstalk by rewinding 2-m fiber 10 times [5], circles: measured by wavelength averaging [26] with 100-m fiber [23,24]. Solid lines: calculated at lc = 3 cm, dashed lines: at lc = 4 mm, dotted lines: at lc = 3 mm, dashed-dotted lines: at lc = 2 mm, dashed-two dotted lines: at lc = 1 mm,
shows comparisons of the average power-coupling coefficients s obtained from the measurement results and those obtained from Eqs. (13)(17). Figure 5(a) shows the dependences of in MCF-A on Rb, and on whether the microbend is applied or not—that is, difference of lc— at Rb = 140 mm. When the microbend is not applied, lc can be estimated to be around 3 cm for this measurement. When the microbend is applied, lc can be estimated to be around 1–4 mm. Figure 5(b) shows the dependences of in MCF-A and MCF-B on the propagation constant mismatch Δβc and on whether the microbend is applied or not, at Rb = 140 mm. We can see that Δβc between dissimilar cores in MCF-A is in the non-phase-matching region and between dissimilar cores is increased by the decreasing of lc, or by the broadening of the bandwidth of Sff. On the other hand, Δβc in MCF-B is designed to be zero and in the center of the phase-matching region; therefore, in MCF-B is hardly affected by the decreasing of lc, at least if lc is larger than 1 mm. Based on this evaluation, lc was shortened from around 3 cm to around 1–4 mm by applying the microbend in these experiments. Though this shortening of lc did not affect in MCF-B at Rb = 140 mm, the shortening of lc may decrease in the (quasi-)homogeneous MCF if Rb is adequately large. Thus, we may consider that the microbend is possible to be utilized for suppressing the crosstalk in a very straight homogeneous MCF.

We evaluated the effect of the microbend on the crosstalk by assuming Sff for the structure fluctuation and the microbend as the Lorentzian. However, further investigation of the effect of the microbend on the crosstalk is necessary for elucidating the detailed characteristics of the effect, as we mentioned above.

6. Conclusions

We derived an intuitively interpretable expression of the average power-coupling coefficient as the convolution of the arcsine distribution—the spectrum of the perturbation induced by the macrobend— and the Lorentzian distribution—the spectrum of the perturbation induced by the structure fluctuation. The derived expression was confirmed to be equivalent to the closed-form expression derived in [11

11. M. Koshiba, K. Saitoh, K. Takenaga, and S. Matsuo, “Analytical expression of average power-coupling coefficients for estimating intercore crosstalk in multicore fibers,” IEEE Photon. J. 4(5), 1987–1995 (2012). [CrossRef]

]. Based on the newly derived expression, we showed how the structure fluctuation and macrobend can affect the crosstalk, and organized previously reported methods for crosstalk suppression. We also discussed how the microbend can affect the crosstalk in homogeneous and heterogeneous MCFs. Though the average power-coupling coefficient affected by the microbend may be the convolution of the arcsine distribution and the spectrum of the high frequency perturbation, including the effects of the structure fluctuation and the microbend, we evaluated previously reported measurement results based on the derived expression—the convolution of the arcsine distribution and the Lorentzian distribution, as a first step. The crosstalk increase due to the microbend in the non-phase-matching region of the heterogeneous MCF and the crosstalk insensitivity to the microbend in the homogeneous MCF can be explained by the shortening of the correlation length of the high frequency perturbation. Further investigation on the effect of the microbend on the crosstalk will be reported in the future.

Appendix: Calculation of Eq. (25) for Figs. 2 and 3

Based on the following relationships of the Fourier transform:
J0(Δβbζ)FouriertransformpΔν˜b(Δν˜c)=2πpΔβb(Δβc),
(29)
Rff(ζ)FouriertransformSff(ν˜)(Δν˜c)=2πSff(β)(Δβc),
(30)
fgFouriertransform(FG)Δν˜=12π(FG)Δβ,
(31)
where J0(x) is the Bessel function of the first kind of order zero, and functions F and G represents the Fourier transform of functions f and g respectively; we calculated Eq. (25) numerically by performing the fast Fourier transform (FFT) on the following relations:
J0(Δβbζ)Rff(ζ)Fouriertransform(pΔν˜bSff(ν˜))Δν˜b(Δν˜c)=2π(pΔβbSff(β))Δβb(Δβc).
(32)
We calculated adequately broad bandwidths of the PSD so that the aliasing noise caused by the FFT can be suppressed in the plotted ranges.

References and links

1.

T. Morioka, “New generation optical infrastructure technologies: EXAT initiative towards 2020 and beyond,” in OptoElectron. Commun. Conf. (OECC) (2009), paper FT4. [CrossRef]

2.

M. Koshiba, K. Saitoh, and Y. Kokubun, “Heterogeneous multi-core fibers: proposal and design principle,” IEICE Electron. Express 6(2), 98–103 (2009). [CrossRef]

3.

K. Takenaga, S. Tanigawa, N. Guan, S. Matsuo, K. Saitoh, and M. Koshiba, “Reduction of crosstalk by quasi-homogeneous solid multi-core fiber,” in Opt. Fiber Commun. Conf. (OFC) (2010), paper OWK7.

4.

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]

5.

T. Hayashi, T. Nagashima, O. Shimakawa, T. Sasaki, and E. Sasaoka, “Crosstalk variation of multi-core fibre due to fibre bend,” in Eur. Conf. Opt. Commun. (ECOC) (2010), paper We.8.F.6. [CrossRef]

6.

K. Takenaga, Y. Arakawa, S. Tanigawa, N. Guan, S. Matsuo, K. Saitoh, and M. Koshiba, “An investigation on crosstalk in multi-core fibers by introducing random fluctuation along longitudinal direction,” IEICE Trans. Commun. E94-B(2), 409–416 (2011). [CrossRef]

7.

T. Hayashi, T. Taru, O. Shimakawa, T. Sasaki, and E. Sasaoka, “Low-crosstalk and low-loss multi-core fiber utilizing fiber bend,” in Opt. Fiber Commun. Conf. (OFC) (2011), paper OWJ3.

8.

T. Hayashi, T. Taru, O. Shimakawa, T. Sasaki, and E. Sasaoka, “Ultra-low-crosstalk multi-core fiber feasible to ultra-long-haul transmission,” in Opt. Fiber Commun. Conf. (OFC) (2011), paper PDPC2.

9.

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]

10.

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). [CrossRef] [PubMed]

11.

M. Koshiba, K. Saitoh, K. Takenaga, and S. Matsuo, “Analytical expression of average power-coupling coefficients for estimating intercore crosstalk in multicore fibers,” IEEE Photon. J. 4(5), 1987–1995 (2012). [CrossRef]

12.

K. Petermann, “Microbending loss in monomode fibers,” Electron. Lett. 12(4), 107–109 (1976). [CrossRef]

13.

J. M. Fini, B. Zhu, T. F. Taunay, M. F. Yan, and K. S. Abedin, “Crosstalk in multicore fibers with randomness: gradual drift vs. short-length variations,” Opt. Express 20(2), 949–959 (2012). [CrossRef] [PubMed]

14.

K. Saitoh, T. Matsui, T. Sakamoto, M. Koshiba, and S. Tomita, “Multi-core hole-assisted fibers for high core density space division multiplexing,” in OptoElectron. Commun. Conf. (OECC) (2010), paper 7C2–1.

15.

K. Takenaga, Y. Arakawa, S. Tanigawa, N. Guan, S. Matsuo, K. Saitoh, and M. Koshiba, “Reduction of crosstalk by trench-assisted multi-core fiber,” in Opt. Fiber Commun. Conf. (OFC) (2011), paper OWJ4.

16.

D. M. Taylor, C. R. Bennett, T. J. Shepherd, L. F. Michaille, M. D. Nielsen, and H. R. Simonsen, “Demonstration of multi-core photonic crystal fibre in an optical interconnect,” Electron. Lett. 42(6), 331–332 (2006). [CrossRef]

17.

K. Imamura, K. Mukasa, R. Sugizaki, Y. Mimura, and T. Yagi, “Multi-core holey fibers for ultra large capacity wide-band transmission,” in Eur. Conf. Opt. Commun. (ECOC) (2008), paper P.1.17. [CrossRef]

18.

K. Imamura, K. Mukasa, Y. Mimura, and T. Yagi, “Multi-core holey fibers for the long-distance (>100 km) ultra large capacity transmission,” in Opt. Fiber Commun. Conf. (OFC) (2009), paper OTuC3.

19.

G. Le Noane, D. Boscher, P. Grosso, J. C. Bizeul, and C. Botton, “Ultra high density cables using a new concept of bunched multicore monomode fibers: A key for the future FTTH networks,” in Int. Wire Cable Symp. (IWCS) (1994), 203–210.

20.

J. Sakaguchi, Y. Awaji, N. Wada, T. Hayashi, T. Nagashima, T. Kobayashi, and M. Watanabe, “Propagation characteristics of seven-core fiber for spatial and wavelength division multiplexed 10-Gbit/s channels,” in Opt. Fiber Commun. Conf. (OFC) (2011), paper OWJ2.

21.

K. Saitoh, M. Koshiba, K. Takenaga, and S. Matsuo, “Low-crosstalk multi-core fibers for long-haul transmission,” Proc. SPIE 8284, 82840I, 82840I-8 (2012). [CrossRef]

22.

J. Tu, K. Saitoh, M. Koshiba, K. Takenaga, and S. Matsuo, “Design and analysis of large-effective-area heterogeneous trench-assisted multi-core fiber,” Opt. Express 20(14), 15157–15170 (2012). [CrossRef] [PubMed]

23.

T. Hayashi, T. Sasaki, and E. Sasaoka, “Microbending-induced crosstalk increase in heterogeneous multi-core fiber,” in Eur. Conf. Opt. Commun. (ECOC) (2011), paper Mo.1.LeCervin.3.

24.

T. Hayashi, T. Sasaki, and E. Sasaoka, “Multi-core fibers and their crosstalk characteristics,” in IEEE Photonics Society Summer Topical Meeting Series (2012), paper TuC4.1.

25.

W.-P. Huang, “Coupled-mode theory for optical waveguides: an overview,” J. Opt. Soc. Am. A 11(3), 963–983 (1994). [CrossRef]

26.

T. Hayashi, T. Taru, O. Shimakawa, T. Sasaki, and E. Sasaoka, “Characterization of crosstalk in ultra-low-crosstalk multi-core fiber,” J. Lightwave Technol. 30(4), 583–589 (2012). [CrossRef]

OCIS Codes
(060.0060) Fiber optics and optical communications : Fiber optics and optical communications
(060.2280) Fiber optics and optical communications : Fiber design and fabrication
(060.2400) Fiber optics and optical communications : Fiber properties
(060.4005) Fiber optics and optical communications : Microstructured fibers

ToC Category:
Fiber Optics and Optical Communications

History
Original Manuscript: December 27, 2012
Revised Manuscript: February 18, 2013
Manuscript Accepted: February 19, 2013
Published: February 26, 2013

Citation
Tetsuya Hayashi, Takashi Sasaki, Eisuke Sasaoka, Kunimasa Saitoh, and Masanori Koshiba, "Physical interpretation of intercore crosstalk in multicore fiber: effects of macrobend, structure fluctuation, and microbend," Opt. Express 21, 5401-5412 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-5-5401


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References

  1. T. Morioka, “New generation optical infrastructure technologies: EXAT initiative towards 2020 and beyond,” in OptoElectron. Commun. Conf. (OECC) (2009), paper FT4. [CrossRef]
  2. M. Koshiba, K. Saitoh, and Y. Kokubun, “Heterogeneous multi-core fibers: proposal and design principle,” IEICE Electron. Express6(2), 98–103 (2009). [CrossRef]
  3. K. Takenaga, S. Tanigawa, N. Guan, S. Matsuo, K. Saitoh, and M. Koshiba, “Reduction of crosstalk by quasi-homogeneous solid multi-core fiber,” in Opt. Fiber Commun. Conf. (OFC) (2010), paper OWK7.
  4. 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]
  5. T. Hayashi, T. Nagashima, O. Shimakawa, T. Sasaki, and E. Sasaoka, “Crosstalk variation of multi-core fibre due to fibre bend,” in Eur. Conf. Opt. Commun. (ECOC) (2010), paper We.8.F.6. [CrossRef]
  6. K. Takenaga, Y. Arakawa, S. Tanigawa, N. Guan, S. Matsuo, K. Saitoh, and M. Koshiba, “An investigation on crosstalk in multi-core fibers by introducing random fluctuation along longitudinal direction,” IEICE Trans. Commun.E94-B(2), 409–416 (2011). [CrossRef]
  7. T. Hayashi, T. Taru, O. Shimakawa, T. Sasaki, and E. Sasaoka, “Low-crosstalk and low-loss multi-core fiber utilizing fiber bend,” in Opt. Fiber Commun. Conf. (OFC) (2011), paper OWJ3.
  8. T. Hayashi, T. Taru, O. Shimakawa, T. Sasaki, and E. Sasaoka, “Ultra-low-crosstalk multi-core fiber feasible to ultra-long-haul transmission,” in Opt. Fiber Commun. Conf. (OFC) (2011), paper PDPC2.
  9. 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]
  10. 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). [CrossRef] [PubMed]
  11. M. Koshiba, K. Saitoh, K. Takenaga, and S. Matsuo, “Analytical expression of average power-coupling coefficients for estimating intercore crosstalk in multicore fibers,” IEEE Photon. J.4(5), 1987–1995 (2012). [CrossRef]
  12. K. Petermann, “Microbending loss in monomode fibers,” Electron. Lett.12(4), 107–109 (1976). [CrossRef]
  13. J. M. Fini, B. Zhu, T. F. Taunay, M. F. Yan, and K. S. Abedin, “Crosstalk in multicore fibers with randomness: gradual drift vs. short-length variations,” Opt. Express20(2), 949–959 (2012). [CrossRef] [PubMed]
  14. K. Saitoh, T. Matsui, T. Sakamoto, M. Koshiba, and S. Tomita, “Multi-core hole-assisted fibers for high core density space division multiplexing,” in OptoElectron. Commun. Conf. (OECC) (2010), paper 7C2–1.
  15. K. Takenaga, Y. Arakawa, S. Tanigawa, N. Guan, S. Matsuo, K. Saitoh, and M. Koshiba, “Reduction of crosstalk by trench-assisted multi-core fiber,” in Opt. Fiber Commun. Conf. (OFC) (2011), paper OWJ4.
  16. D. M. Taylor, C. R. Bennett, T. J. Shepherd, L. F. Michaille, M. D. Nielsen, and H. R. Simonsen, “Demonstration of multi-core photonic crystal fibre in an optical interconnect,” Electron. Lett.42(6), 331–332 (2006). [CrossRef]
  17. K. Imamura, K. Mukasa, R. Sugizaki, Y. Mimura, and T. Yagi, “Multi-core holey fibers for ultra large capacity wide-band transmission,” in Eur. Conf. Opt. Commun. (ECOC) (2008), paper P.1.17. [CrossRef]
  18. K. Imamura, K. Mukasa, Y. Mimura, and T. Yagi, “Multi-core holey fibers for the long-distance (>100 km) ultra large capacity transmission,” in Opt. Fiber Commun. Conf. (OFC) (2009), paper OTuC3.
  19. G. Le Noane, D. Boscher, P. Grosso, J. C. Bizeul, and C. Botton, “Ultra high density cables using a new concept of bunched multicore monomode fibers: A key for the future FTTH networks,” in Int. Wire Cable Symp. (IWCS) (1994), 203–210.
  20. J. Sakaguchi, Y. Awaji, N. Wada, T. Hayashi, T. Nagashima, T. Kobayashi, and M. Watanabe, “Propagation characteristics of seven-core fiber for spatial and wavelength division multiplexed 10-Gbit/s channels,” in Opt. Fiber Commun. Conf. (OFC) (2011), paper OWJ2.
  21. K. Saitoh, M. Koshiba, K. Takenaga, and S. Matsuo, “Low-crosstalk multi-core fibers for long-haul transmission,” Proc. SPIE8284, 82840I, 82840I-8 (2012). [CrossRef]
  22. J. Tu, K. Saitoh, M. Koshiba, K. Takenaga, and S. Matsuo, “Design and analysis of large-effective-area heterogeneous trench-assisted multi-core fiber,” Opt. Express20(14), 15157–15170 (2012). [CrossRef] [PubMed]
  23. T. Hayashi, T. Sasaki, and E. Sasaoka, “Microbending-induced crosstalk increase in heterogeneous multi-core fiber,” in Eur. Conf. Opt. Commun. (ECOC) (2011), paper Mo.1.LeCervin.3.
  24. T. Hayashi, T. Sasaki, and E. Sasaoka, “Multi-core fibers and their crosstalk characteristics,” in IEEE Photonics Society Summer Topical Meeting Series (2012), paper TuC4.1.
  25. W.-P. Huang, “Coupled-mode theory for optical waveguides: an overview,” J. Opt. Soc. Am. A11(3), 963–983 (1994). [CrossRef]
  26. T. Hayashi, T. Taru, O. Shimakawa, T. Sasaki, and E. Sasaoka, “Characterization of crosstalk in ultra-low-crosstalk multi-core fiber,” J. Lightwave Technol.30(4), 583–589 (2012). [CrossRef]

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