## 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 Optical Society of America

## 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]. Inter-core crosstalk (XT) is one of the most important properties of uncoupled MCFs, and suppression of the XT has been actively studied [2–10]. Recently, Koshiba *et al.* derived a closed-form expression of the average power-coupling coefficient between cores in the MCF [11]. 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

*A*is the complex amplitude,

*κ*the mode coupling coefficient from Core

_{nm}*n*to Core

*m*,

*β*= 2π

*n*

_{eff}/

*λ*the propagation constant,

*n*

_{eff}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 [*z*_{1}, *z*_{2}] can be expressed as

*z*

_{1},

*z*

_{2}] can be expressed as

*R*is the autocorrelation function (ACF) of

_{ff}*f*(

*z*), Δ

*z*is

*z*

_{2}−

*z*

_{1}, and the correlation length

*l*

_{c}of

*R*is assumed to be adequately shorter than Δ

_{ff}*z*. ${R}_{ff}\left(\zeta \right)$ 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*>>

*l*

_{c}, 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:

*β*is the angular wave number. Note that $\tilde{\nu}$,

*n*

_{eff}, and

*β*have common subscripts, e.g., ${\tilde{\nu}}_{\text{c}}={n}_{\text{eff,c}}/\lambda ={\beta}_{\text{c}}/\left(2\pi \right)$. To describe the PSDs with respect to $\tilde{\nu}$ 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 $\tilde{\nu}$, and the following equation holds between

*f*(

*z*) and the PSDs of

*f*(

*z*):

*β*(the angular wave number in the medium) is defined as:

Figure 1
shows the schematics of perturbations on *β*, or how *β*_{v} can vary. As shown in Figs. 1(a) and 1(b), the bend and the structure fluctuation can induce a slight change in *β*_{v} in one core, which can occur ether in the single-core fiber or in the MCF. In the single-core fiber, by assuming proper *R _{ff}* or

*S*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

_{ff}*β*

_{v}in a core when assuming another core as a reference of the propagation constant. Fini

*et al.*[4] and Hayashi

*et al.*[5] assumed that

*β*

_{v}in the MCF is induced by the macrobend and twist of the MCF as

*x*,

_{n}*y*) and (

_{n}*r*,

_{n}*θ*) are the local Cartesian and polar coordinates of Core

_{n}*n*in a fiber cross-section, respectively,

*θ*= 0 is the radial direction of the macrobend,

_{n}*θ*

_{f}the angle between the

*x*-axis and the radial direction of the macrobend, and

*R*

_{b}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 *β*_{v}, *R _{ff}*, or

*S*that can include the perturbations of both the bend and the structure fluctuation. Therefore, by assuming that

_{ff}*R*includes only the effect of structure fluctuation and does not include that of macrobend and twist, Koshiba

_{ff}*et al.*investigated the effects of correlation length

*l*

_{c}and of the shape of the ACF

*R*on the average crosstalk

_{ff}*μ*[10,11]. They investigated some types of

_{X}*R*, and found that the exponential ACF (EAF)

_{ff}*μ*of the MCFs. The EAF have been introduced to microbending loss analysis [12]. Since the PSD of the EAF is the Lorentzian distribution, the power-coupling coefficient was obtained from Eq. (8) as [10,11]:

_{X}*β*´

_{c,}

*is*

_{nm}*β*´

_{c,}

*−*

_{n}*β*´

_{c,}

*, and*

_{m}*β*´

_{c}is redefined

*β*

_{c}that includes the effects of macrobend and twist:

*μ*estimated using coupled-power equation with the power-coupling coefficient of Eq. (11) may be valid in cases where changes of

_{X}*R*

_{b}and

*θ*are gradual enough compared to

*l*

_{c}, since Δ

*β*´

_{c,}

*—which is variable and includes macrobend and twist— is substituted to Δ*

_{nm}*β*

_{c,}

*—which is constant— in Eq. (8).*

_{nm}By assuming constant *R*_{b} and twist rate, Koshiba *et al.* also analytically derived an average power-coupling coefficient *h̅*, which is averaged over *θ*, as [11]:

*B*can be approximated as

_{nm}*β*

_{c,}

*if*

_{n}D_{nm}*β*

_{c,}

*/*

_{m}*β*

_{c,}

*≈1,*

_{n}*D*is the center-to-center distance between Core

_{nm}*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 Δ*β*´_{c,}* _{nm}* can be written as

*θ*represents the angle between the radial direction of the bend and a line segment from Core

_{nm}*m*to Core

*n*, Δ

*β*

_{b,}

*the difference of*

_{nm}*β*variation between Core

*m*and Core

*n*from the macrobend, and $\Delta {\beta}_{\text{b},nm}^{\text{dev}}$ the peak deviation of Δ

*β*

_{b,}

*.*

_{nm}Let ${p}_{{\theta}_{nm}}\left({\theta}_{nm}\right)$ and ${p}_{{R}_{\text{b}}}\left({R}_{\text{b}}\right)$ be the probability density functions of *θ _{nm}* and of

*R*

_{b}, respectively, along the MCF; by assuming that ${p}_{{\theta}_{nm}}\left({\theta}_{nm}\right)$ and ${p}_{{R}_{\text{b}}}\left({R}_{\text{b}}\right)$ are statistically independent, the twist of the MCF is gradual enough, and average crosstalk is adequately low; the average crosstalk

*μ*from Core

_{X,nm}*m*to Core

*n*can be expressed as

*θ*; therefore, by substituting $\Delta \beta =-\Delta {\beta}_{\text{b},nm}\left({R}_{\text{b}},{\theta}_{nm}\right)=-\Delta {\beta}_{\text{b},nm}^{\text{dev}}\left({R}_{\text{b}}\right)\mathrm{cos}{\theta}_{nm}$ and using Eq. (8) and $\mathrm{sin}\left(\mathrm{arc}\mathrm{cos}x\right)=\sqrt{1-{x}^{2}}$, Eq. (22) can be rewritten as

_{nm}*S*is the Lorentzian distribution as shown in Eq. (11). By using the arcsine distribution:

_{ff}*β*

_{b}, Eq. (23) can be rewritten as

*f*and

*g*with respect to

*x*, and the expression with respect to $\tilde{\nu}$ is also shown for comparison. If we consider the case where PSD

*S*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

_{ff}*S*in Eq. (8).

_{ff}Particularly where |Δ*β*_{c,}* _{nm}*| and the bandwidth of ${S}_{ff}^{\left(\beta \right)}$ are adequately smaller than $\Delta {\beta}_{\text{b},nm}^{\text{dev}}$,

*S*becomes a narrow delta-function-like distribution and the convolution contains only a gradually varying part of ${p}_{\Delta {\beta}_{\text{b}}}\left(\Delta {\beta}_{\text{c},nm}\right)$; therefore, Eq. (25) can be approximated as

_{ff}*β*

_{b}—shown in Eq. (24)— with constant

*R*

_{b}, as shown in [13]. In case of homogeneous MCFs (Δ

*β*

_{c}

*= 0), Eq. (26) is reduced to*

_{,nm}*β*

_{c}< 0.21

*β*

_{c}

*D/R*

_{b}; therefore, Eq. (27) may be also used for estimating the crosstalk of a bent heterogeneous MCF with small Δ

*β*

_{c}.

Figure 2
shows comparisons between *h̅* calculated by using Eq. (25) and *h̅* calculated by using Eqs. (13)–(17). Figures 2(a) and 2(b) show the PSDs normalized with respect to the Lorentzian *S _{ff}* and to the arcsine distribution ${p}_{\Delta {\beta}_{\text{b}}}$, 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

*h̅*calculated by using Eq. (25) and dashed lines represent

*h̅*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

*h̅*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 [3,6], hole- or trench-assisted core structures [9,14,15], and photonic-crystal structures [16–18].

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 ${\overline{h}}_{nm}\left(\Delta {\beta}_{\text{c},nm},{R}_{\text{b}}\right)$ in Eq. (25) shown in Fig. 3 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 Δ*β*_{c} to suppress the phase matching [2,4,19]. As shown as the non-phase-matching regions in Fig. 3, Δ*β*_{c} larger than $\Delta {\beta}_{\text{b}}^{\text{dev}}$ can prevent the bend-induced phase-matching between dissimilar cores, and can suppress the crosstalk [4,5]. 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 *R*_{pk} [5,11]:

*R*

_{b}where Eq. (26) can be infinite, or the maximal

*R*

_{b}where the phase matching due to the macrobend can occur even if there is no structure fluctuation. Some margin from

*R*

_{pk}is needed for avoiding the phase matching induced by the spectral broadening of

*S*, due to the structural fluctuations. In heterogeneous MCFs, it is preferred if the correlation length

_{ff}*l*

_{c}of the structural fluctuation can be elongated, because the spectral broadening of

*S*can be narrowed and the PSD leakage into the non-phase-matching region can be suppressed, as shown in Figs. 2(b) and 3.

_{ff}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].

Since the typical winding radii of fiber spools are around 10 cm and *R*_{pk} less than 10 cm requires very large difference in core structure [5], 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] is an exception, but it has a large difference in optical properties between cores so that *R*_{pk} 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 *R*_{pk} around 5 cm while achieving a similar *A*_{eff} of around 80 µm^{2} at 1550 nm, and other good optical properties [21]. Tu *et al.* also reported that up to two kinds of dissimilar trench-assisted cores can be designed to achieve *R*_{pk} around 5 cm while achieving a similar *A*_{eff} of around 100 µm^{2} [22].

#### 4.2. Utilization of the bend-induced perturbation

The bend can also be utilized for the phase matching suppression [9]. 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 (Δ*β*_{c} = 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 *R*_{pk} 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 |Δ*β*_{c,}* _{nm}*| and the bandwidth of ${S}_{ff}^{\left(\beta \right)}$ are adequately smaller than $\Delta {\beta}_{\text{b},nm}^{\text{dev}}$. 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 *R _{ff}* due to the structure fluctuation has a very short correlation length

*l*

_{c}, the power spectrum spreads broadly over the propagation constant mismatch Δ

*β*

_{c}, and thus the PSD may be suppressed even in case of an unbent homogeneous MCF (Δ

*β*

_{c}= 0, 1/

*R*

_{b}= 0). A homogeneous MCF utilizing the longitudinal structural fluctuations was conceptually proposed by Takenaga

*et al.*as “quasi-homogeneous MCF” in [3,6]. 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, ${l}_{\text{c}}\Delta {\beta}_{\text{b},nm}^{\text{dev}}\simeq {l}_{\text{c}}{\beta}_{\text{c}}D/{R}_{\text{b}}$ 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

*h̅*from Eq. (27) at Δ

*β*

_{c}= 0, respectively.

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

In Sections 2 and 3, *S _{ff}* only includes structure fluctuation, and ${p}_{\Delta {\beta}_{\text{b}}}\left(\Delta {\beta}_{\text{c},nm}\right)$ includes macrobend perturbation that gradually varies in longitudinal direction. Based on the assumption that the macrobend perturbation is gradual enough compared to

*l*

_{c}, we may redefine the

*S*as the PSD of high frequency perturbations other than the macrobend perturbation, and thus

_{ff}*S*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

_{ff}*l*

_{c}of

*S*. The microbend may induce

_{ff}*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

*S*for the structure fluctuation and the microbend could possibly be different from the Lorentzian—that is,

_{ff}*S*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.

_{ff}However, as a first step, we will evaluate the effect of the microbend on the crosstalk by assuming that *S _{ff}* for the structure fluctuation and the microbend, in this paper.

Figure 4
shows the dependences of the average power-coupling coefficient *h̅* on microbend conditions for a heterogeneous MCF-A [23] and for a homogeneous MCF-B [24]. Fiber properties of the MCFs are shown in Table 1
. For MCF-A, *h̅* from the center core to an outer core was obtained. For MCF-B, the average of *h̅*s between the three pairs of the neighboring cores was obtained. The values of *h̅* 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*. *h̅* at *T* = 0 N was measured using a 140-mm-radius bobbin *without* the sandpaper. *h̅* 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 *l*_{c} by using Eqs. (13)–(17) or Eq. (25). Figure 5
shows comparisons of the average power-coupling coefficients *h̅*s obtained from the measurement results and those obtained from Eqs. (13)–(17). Figure 5(a) shows the dependences of *h̅* in MCF-A on *R*_{b}, and on whether the microbend is applied or not—that is, difference of *l*_{c}— at *R*_{b} = 140 mm. When the microbend is not applied, *l*_{c} can be estimated to be around 3 cm for this measurement. When the microbend is applied, *l*_{c} can be estimated to be around 1–4 mm. Figure 5(b) shows the dependences of *h̅* in MCF-A and MCF-B on the propagation constant mismatch Δ*β*_{c} and on whether the microbend is applied or not, at *R*_{b} = 140 mm. We can see that Δ*β*_{c} between dissimilar cores in MCF-A is in the non-phase-matching region and *h̅* between dissimilar cores is increased by the decreasing of *l*_{c}, or by the broadening of the bandwidth of *S _{ff}*. On the other hand, Δ

*β*

_{c}in MCF-B is designed to be zero and in the center of the phase-matching region; therefore,

*h̅*in MCF-B is hardly affected by the decreasing of

*l*

_{c}, at least if

*l*

_{c}is larger than 1 mm. Based on this evaluation,

*l*

_{c}was shortened from around 3 cm to around 1–4 mm by applying the microbend in these experiments. Though this shortening of

*l*

_{c}did not affect

*h̅*in MCF-B at

*R*

_{b}= 140 mm, the shortening of

*l*

_{c}may decrease

*h̅*in the (quasi-)homogeneous MCF if

*R*

_{b}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 *S _{ff}* 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]. 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:

*J*

_{0}(

*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:

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