## Abstract

We designed and fabricated a multi-core fiber (MCF) in which seven identical trench-assisted pure-silica cores were arranged hexagonally. To design MCF, the relation among the crosstalk, fiber parameters, and fiber bend was derived using a new approximation model based on the coupled-mode theory with the equivalent index model. The mean values of the statistical distributions of the crosstalk were observed to be extremely low and estimated to be less than −30 dB even after 10,000-km propagation because of the trench-assisted cores and utilization of the fiber bend. The attenuation of each core was very low for MCFs (0.175–0.181 dB/km at 1550 nm) because of the pure-silica cores. Both the crosstalk and attenuation values are the lowest achieved in MCFs.

©2011 Optical Society of America

## 1. Introduction

Exponentially increasing demand for transmission capacity is the driving force for research in high-capacity optical transmission systems. Recently, ultra-high-capacity systems have achieved capacities up to 100 Tb/s per fiber by employing time-, wavelength-, polarization-division multiplexing, and multilevel modulations [1–3]. However, transmission capacity is rapidly approaching its fundamental limit [4]. Mode- and spatial-division multiplexing are attractive for capacity enlargement [5]. Mode-division multiplexing has been demonstrated in a one wavelength channel over simple multimode fibers using multiple-input multiple-output signal processing [6,7]. Recently, spatial-division multiplexing has been demonstrated in ultra-high-capacity transmissions [8,9] without any added signal processing for demultiplexing spatial channels because of low inter-core crosstalk of multi-core fibers (MCFs).

In the last few years, crosstalk suppression has been a primary concern in MCF research for high-capacity long-distance transmissions [10–17], after the proposal of all-solid heterogeneous MCFs utilizing the phase mismatch between the fiber cores for reducing the crosstalk [10,11]. The measured crosstalk in fabricated heterogeneous MCF showed discrepancies in calculations based on conventional coupled-mode equations [12]. The crosstalk and its length dependence were calculated using the coupled-power theory by assuming random fluctuations on the fiber structure along the longitudinal direction [13]. However, the fluctuations of fabricated MCFs were difficult to measure and were estimated from the measured crosstalk. Recently, theoretical [14,15] and experimental [15] reports showed that crosstalk significantly varies on the basis of the bending radius of the MCF and is a stochastic value. Calculations based on the coupled-mode theory considering fiber bend are in good agreement with the experimental measurements. We designed and fabricated an ultra-low-crosstalk homogeneous MCF by utilizing the fiber bend and employing trench-assisted cores [16] and characterized the crosstalk of the MCF statistically [17].

This paper reports the design and fabrication of an ultra-low crosstalk and low-loss MCF, based on [15–17], including a detailed description of an approximation model for estimating crosstalk and measured bending radius dependences of the crosstalk of the fabricated MCF, which have not been previously reported. The equivalent index model was introduced into the coupled-mode theory to consider the fiber bend effects. Bending-radius dependence and stochasticity of the crosstalk are briefly described. We developed an approximation model for estimating the longitudinal evolution and probability distribution of the crosstalk and derived a relation between the mean crosstalk and fiber parameters. We designed and fabricated an ultra-low crosstalk MCF based on the approximation model. To date, the lowest value for the mean crosstalk increase per fiber length was achieved and the lowest value for MCF attenuation was observed owing to the pure-silica cores.

## 2. Design of multi-core fiber with low crosstalk

#### 2.1 Coupled-mode equation with equivalent index model

Figure 1
shows the oscillatory power conversion between the coupled waveguides based on the coupled-mode theory [18]. The power conversion efficiency, i.e., peak of the normalized power of coupled light, and the cycle length of the oscillation are dependent on the mode-coupling coefficient *κ* and the difference in propagation constants *β* of each waveguide. The power conversion efficiency between waveguides *n* and *m* is expressed as

*F*decreases when the propagation constants are different, as shown in Fig. 1(b). Because

*β*denotes 2π

*n*

_{eff}/λ in which the wavelength λ ~10

^{−6}and

*n*

_{eff}represents the effective refractive index of the waveguide, the power conversion efficiency

*F*considerably decreases by slight differences in the effective indices between the waveguides when

*κ*is small.

Heterogeneous MCFs were proposed in which the effective indices of the neighboring cores are different based on this property of the coupled-mode theory [10,11]. However, the measured crosstalk of fabricated heterogeneous MCFs was reported to be ~40 dB larger than the power conversion efficiency [12]. It was also reported that the measured crosstalk of fabricated *homogeneous* MCFs did not oscillate as shown in Fig. 1(a), but accumulated linearly along the fiber length [13].

Considering that the effects of the fiber bend on the crosstalk may play an important role in such discrepancies, we introduced the equivalent index model [19] to the coupled-mode theory. A bent fiber can be represented as a corresponding straight fiber, which has an equivalent refractive-index profile:

*r*,

*θ*) represents the local polar coordinates from a determined origin in a cross section of the fiber,

*θ*is the angle from a radial direction of the bend,

*R*is the bending radius of the fiber, and

*n*(

*r*,

*θ*) is the intrinsic refractive index. Here,

*r*cos

*θ*represents the displacement in the radial axis of the bend from the origin. Accordingly, the relative index difference between the equivalent index and the intrinsic index can be written as

*r*cos

*θ*, the bending radius

*R*, and the relative index difference Δ

_{eq}. Core pitches or core-to-core distances of MCFs are generally more than 30 µm for suppressing the crosstalk [11–17]. Accordingly, the relative index difference Δ

_{eq}easily exceeds 0.005%; the core Δ difference of 0.005% between the neighboring cores was reported to induce adequate phase mismatch for suppressing the crosstalk [11], where core Δ is the relative index difference between the core and cladding.

If we define the center of Core *m* as the origin of the local polar coordinates, the effective index of Core *n* can be expressed as

*n*is

*n*

_{eff}is the effective index,

*D*is the distance from the center of Core

_{nm}*m*to that of Core

*n*,

*θ*is the angle of a line segment connecting Core

_{nm}*m*and Core

*n*in the radial direction,

*λ*is the wavelength, and

*β*is the propagation constant. Because

*β*

_{eq,}

*is a variable, the coupled-mode equation for the MCF can be expressed as*

_{n}*A*represents the slowly varying complex amplitude of the electric field,

*κ*is the mode-coupling coefficient from Core

_{nm}*m*to Core

*n*, and

*z*is the longitudinal axis of the MCF. Here, Eq. (7) can be rewritten as

*A*|

_{n}^{2}simulated using Eqs. (6) and (8), where

*R*is constant and the effective indices of the coupled cores are slightly different. Discrete dominant changes were observed at every phase-matching point where

*δn*

_{eq,}

*equals zero (written as*

_{nm}*δn*in Fig. 3):

_{eq}These changes appear random, because the phase offsets between Core *m* and Core *n* are different for each phase-matching point. The phase offsets can easily fluctuate in practice by slight variations in the bending radius, twist rate of fiber, among others. Therefore, the dominant crosstalk changes are practically stochastic. Figure 4
shows the simulated and measured relationship between the bending radius and the crosstalk of a two-meter-long heterogeneous MCF. The simulation method is described in [15]. Core Δ of each core of the MCF was 0.38%. The crosstalk was measured between two cores of diameters 8.1 µm and 9.4 µm separated by 30 µm. The simulated and measured results are in good agreement. The crosstalk degraded at small bending radii and peaked around a bending radius *R*
_{pk}:

*R*>

*R*

_{pk}. For

*R*<

*R*

_{pk}, the crosstalk gradually decreased from the peak, as

*R*approaches zero. This can be understood as follows: the phases of each core are matched, to be precise, at phase-matching

*regions*where

*δn*

_{eq,}

*is*

_{nm}*nearly*equal to zero, and lengths of the regions shorten when

*R*becomes smaller.

*R*

_{pk}was ~65 mm even with such a large difference in core diameters. For core diameters 8.9 µm and 9.4 µm,

*R*

_{pk}was ~245 mm.

Because of the large dependence of the crosstalk on the bending radius, it should be noted that careful management of the fiber bend is necessary in heterogeneous MCFs. Furthermore, very large differences in core structures are necessary for suppressing the crosstalk when the MCF is wound on a commonly used bobbin whose winding radius ranges from 70 to 150 mm. If the crosstalk is not suppressed for an MCF wound on a bobbin, then the optical properties of each core of the MCF may be affected by the crosstalk and measurements may not be performed precisely. To prevent such a case, large differences in the core structures are needed, and accordingly, the optical properties of each core have to be largely different from one another. To overcome these limitations, we utilize the phase mismatch induced by the bend for suppressing the crosstalk in homogeneous MCF in which the optical properties of each core of the MCF are the same.

## 2.2 Model for longitudinal evolution of crosstalk in MCF

To deal with the stochastic evolution of the crosstalk, we consider the case of a two-core fiber that is bent at a constant radius and twisted continuously at a constant rate. We approximate the dominant crosstalk changes by the following discrete changes:

*A*

_{n}_{,}

*represents the amplitude*

_{N}*A*of Core

*n*after the

*N*-th phase-matching point,

*ϕ*

_{rnd}is the phase offset between Core

*m*and Core

*n*, and Κ

*is the coefficient for the discrete changes caused by the coupling from Core*

_{nm}*m*to Core

*n*. We assume

*ϕ*

_{rnd}is a random number, because it significantly varies with slight variations in the bending radius, twist rate of the fiber, among others. Here, we assume that the crosstalk is adequately low (|

*A*

_{n}_{,}

*| << 1) so that*

_{N}*A*

_{m}_{,}

*can be approximated by*

_{N}*A*

_{m}_{,0}= 1 and the crosstalk by the coupled power |

*A*

_{n}_{,}

*|*

_{N}^{2}. In this case, because ℜ[Κ

*exp(*

_{nm}*jϕ*

_{rnd})] and ℑ[Κ

*exp(*

_{nm}*jϕ*

_{rnd})] have a variance

*σ*

^{2}of |Κ

*|*

_{nm}^{2}/2, ℜ

*A*

_{n}_{,}

*and ℑ*

_{N}*A*

_{n}_{,}

*have Gaussian profiles whose variance*

_{N}*σ*

^{2}is

*N*|Κ

*|*

_{nm}^{2}/2 if

*N*is adequately large due to the statistical independence and the central limit theorem. Furthermore, if the cores have identical structures, an absolute value of Κ

*can be derived as follows (see Appendix):*

_{nm}*γ*represents the twist rate. We assume that

*N*=

*Lγ*/

*π*where

*L*is the length of the fiber. Therefore, the variances

*σ*

_{2}

^{2}of ℜ

*A*

_{n}_{,}

*and ℑ*

_{N}*A*

_{n}_{,}

*can be expressed asDue to the distributions of ℜ*

_{N}*A*

_{n}_{,}

*and ℑ*

_{N}*A*

_{n}_{,}

*, |*

_{N}*A*

_{n}_{,}

*|*

_{N}^{2}/

*σ*

_{2}

^{2}has a chi-square distribution with two degrees of freedom. Accordingly, the probability distribution of

*XT*= |

*A*

_{n}_{,}

*|*

_{N}^{2}, i.e., the crosstalk distribution, can be expressed as

*f*(

_{k}*x*) and

*F*(

_{k}*x*) denote the probability distribution and the cumulative distribution function, respectively, of the chi-square distribution with

*k*degrees of freedom.

So far, our discussions have not considered the polarization coupling in which two polarization modes randomly couple with each other. Consequently, the coupled power can be statistically distributed equally between the two polarization modes. Therefore, the variances *σ*
_{4}
^{2} in ℜ*A _{n}*

_{,}

*and ℑ*

_{N}*A*

_{n}_{,}

*of the two polarization modes can be obtained as*

_{N}*A*

_{n}_{,}

*|*

_{N}^{2}/

*σ*

^{2}has a chi-square distribution with four degrees of freedom. The crosstalk distribution can be expressed as

*XT*, i.e., the mean crosstalk

*XT*, can be obtained as

_{μ}*h*denotes the mean crosstalk increase per unit length. The crosstalk is linearly proportional to

*R*and

*L*at low crosstalk. The

*Q*-quantile of the distribution of

*XT*can be expressed as

*F*

_{k}^{−1}(

*Q*) is the inverse function of the cumulative distribution function of the chi-square distribution with

*k*degrees of freedom. Based on Eq. (21), the shape of the crosstalk distribution on the decibel scale is independent of the mean crosstalk

*XT*. For example, regarding the case of the four degrees of freedom, the 0.9999-quantile of the crosstalk distribution can be obtained by adding ~7.69 dB to

_{μ}*XT*on the decibel scale.

_{μ}The mean crosstalk from multiple cores to one core can be represented as a sum of the mean crosstalk from each of the multiple cores to the single core, because the mean crosstalk is linearly proportional to variance, and the variance can be summed in such cases. In a seven-core fiber whose cores are arranged in a hexagonal lattice, its center core is surrounded by six outer cores so that the mean crosstalk of the center core is the worst and can be expressed as

where Λ denotes the core pitch or the pitch of the lattice, and*XT*

_{coeff}represents the dependence of the mean crosstalk on the bending radius

*R*and fiber length

*L*. The mean crosstalk increase per length

*h*can be also regarded as the power coupling coefficient in the coupled-power theory [13,20]. Prediction of the crosstalk evolution based on the coupled-power equation is considered more rigorous if the crosstalk is large, because it considers the power decrease in the cores from which the power is coupled. Given that Core 1 is the center core of the seven-core fiber and Cores 2–7 are the outer cores, and if the center core is excited with power

*P*

_{1}(0) = 1 but all outer cores are not excited, the power of each core is expressed as follows [13]:

*P*(

_{n,m}*z*) represents the power of Core

*n*when Core

*m*is excited. If each of the outer cores is excited with

*P*(0) = 1 but the center core is not excited, the power of each core can be obtained as follows:

_{n}*XT*> −20 dB as shown in Fig. 5 .

_{μ}Equations (13)–(27) are considered to be applicable not only in the case when the fiber is continuously twisted but also when a sufficiently long fiber is twisted randomly, because they are independent of the twist rate.

#### 2.3 Design of multi-core fiber

For the application for long-distance high-capacity networks, we designed a homogeneous seven-core fiber with the following conditions:

- i. Attenuation of each core should be less than that of the standard single-mode fiber (SSMF),
- ii. Mode field diameter (MFD) of each core should range from 9.5 to 10.5 µm at λ = 1550 nm,
- iii. Cable cutoff wavelength (λ
_{cc}) of each core should be less than 1530 nm, - iv. 0.9999-quantile of crosstalk of the center core should be less than −30 dB after 100-km propagation at λ = 1625 nm and
*R*< 200 mm, - v. Attenuation degradations of the outer cores should be less than 0.001 dB/km.

_{eff}) of ~80 µm

^{2}. A larger A

_{eff}decreases the nonlinearity, but weakens the confinement of power in each core, which makes the crosstalk suppression difficult. In this way, we adhere to condition (ii). The target of λ

_{cc}was determined after ITU-T G.654 so that it is much longer than λ

_{cc}of SSMF. We elongated λ

_{cc}for improving the power confinement, because ultra-high-capacity transmissions have been demonstrated using only C- and L-bands (C + L band) [1–3] because of the limited bandwidths of the amplifiers and the relatively high attenuation of the other bands. The target of the crosstalk was also determined at λ = 1625 nm considering the application for the C + L band, because longer wavelengths weaken the confinement and increase the crosstalk. Other parameters in condition (iv) were determined so that low crosstalk can be achieved when the MCF is wound on a bobbin. The attenuation degradation of the outer cores should be suppressed.

Various approaches, such as MCF with hole-assisted structures [21], can be adopted for suppressing the crosstalk whilst keeping the other optical properties suitable for transmissions. In this study, we employed a trench-assisted profile (see Fig. 6
), which is simple to fabricate and reduces the mode-coupling coefficients while achieving conditions (ii) and (iii). Table 1
lists the designed optical properties of each core at λ = 1550 nm. λ_{cc} and MFD were designed to meet the target conditions and A_{eff} was ~80 µm^{2}. Chromatic dispersion (CD) was moderately higher than that in conventional SMFs. The higher dispersion can induce larger phase mismatches between wavelength channels so that the nonlinear effects in the MCF can be more suppressed [22]. The dispersion slope (D. Slope) was 0.063 ps/nm^{2}/km. We set the core pitch Λ to 45 µm, because condition (iv) is also achieved for Λ > 39.2 µm with the designed core, as shown in Fig. 7
. The attenuation degradation of outer cores of MCFs was reported to be due to microbending loss [12], and/or due to coupling with the fiber coating and macro-bending loss [23], induced by a high refractive index of the coating. We considered that the latter factor is the cause of the attenuation degradation, because the microbending loss dependence on fiber diameter is induced by the change of fiber rigidity rather than that of cladding thickness [24]. Figure 8
shows the relationships between the cladding diameter and the attenuation degradation for *R* = 140 mm and λ = 1625 nm, which were simulated with the designed structure in case that the simulated outer core is positioned at the outermost side of the bend. To meet condition (v), the cladding diameter was designed to be 150 µm so that the attenuation degradation of Cores 2–7 can be less than 0.001 dB/km. Calculation results in this section were obtained using a full-vector finite-element method [25].

## 3. Fabrication

#### 3.1 Fabrication and measured properties of the designed MCF

We fabricated an MCF with pure-silica cores for reducing the attenuation of each core. The cross section of the MCF in Fig. 9
shows the seven trench-assisted cores and a marker for core identification. The actual core pitch, cladding diameter, and coating diameter were 45 µm, 150 µm, and 256 µm, respectively. The attenuation spectra are shown in Fig. 10
. Very low MCF attenuation (0.175–0.181 dB/km at λ = 1550 nm, 0.192–0.202 dB/km over the C + L band) was observed for each core, and distinctive degradations for the outer cores were not observed. It was confirmed that each core was fabricated as we designed, from good agreement between the designed and measured values of the optical properties shown in Table 2
. Macrobending losses were observed to be very low due to the trench-assisted structure and long λ_{cc}. Polarization mode dispersion (PMD) was also measured for the C + L band. One of the values of PMD was observed to exceed 0.2 ps/√km, but we consider that PMD can be decreased by improving the fabrication process.

#### 3.2 Crosstalk characteristics of the fabricated MCF

We obtained the mean crosstalk of the fabricated MCF from the statistical crosstalk distribution measured using the wavelength sweeping method [17]. The MCF was 17.4 km long and wound on a bobbin whose winding radius was 140 mm. The crosstalk distribution was obtained at λ = 1550 nm and λ = 1625nm. An example of the measured distribution of the crosstalk from Core 1 to Core 5 at λ = 1625 nm is shown in Fig. 11 . The measured distribution fitted well to Eq. (17).

The values of the mean crosstalk between the neighboring cores of the MCF obtained by fitting the measured crosstalk distributions with Eq. (17) are plotted in Fig. 12(a) . The average and maximum values of the measured mean crosstalk were −79.5 dB and −77.6 dB, respectively, at λ = 1550 nm, and −69.8 dB and −67.7 dB, respectively, at λ = 1625 nm. The values between non-neighboring cores were too low to measure. The mean crosstalk of the center core, which was coupled from six outer cores, was −72.3 dB at λ = 1550 nm and −62.1 dB at λ = 1625 nm from the sums of the measured values. The measured values [plotted in Fig. 12(a)] are in good agreement with the calculated ones [plotted in Fig. 12(b)]. Variations in the measured values are considered to be due to variations in core pitches and index profiles between the cores.

Bending radius dependences of the mean crosstalk were also observed among combinations of Cores 1, 4, and 5. Figure 13 shows the results for λ = 1625 nm. The number of radii where the mean crosstalk was measured was two, and when the origin of the coordinates are included, the mean crosstalk was observed to be linearly proportional to the bending radius, as predicted in Eq. (19). Slight difference of the mean crosstalk of opposite directions in the each graph of Fig. 13 is considered to be because of measurement error due to the extremely low crosstalk.

From the results, the theoretical approximation model described in Section 2.2 was validated, and the crosstalk of the MCF was confirmed to be extremely low.

## 4. Estimation of the crosstalk of the fabricated MCF after long-distance propagation

Based on Eq. (22), the dependence of the mean crosstalk of the fabricated MCF on propagation length *L* [km] and bending radius *R* [mm] can be estimated. The crosstalk coefficients *XT*
_{coeff} for the crosstalk of the center core were 2.41 × 10^{−11} /km/mm at λ = 1550 nm and 2.56 × 10^{−10} /km/mm at λ = 1625 nm. The relationship between *L*, *R*, and the mean crosstalk are shown in Fig. 14
. The mean crosstalk after 10,000-km propagation at λ = 1625 nm can be less than −30 dB if *R* is less than 391 mm. The MCF can be applied to such a bending radius by appropriately cabling the MCF, e.g., by cabling the MCFs into helical slots in a cable. The 0.9999-quantile of the crosstalk distribution (~*XT _{μ}* + 7.7 [dB]) after 100-km propagation was estimated to be less than −30 dB.

## 5. Conclusion

In this paper, we reported the design and fabrication of an ultra-low crosstalk and low-loss MCF, based on [15–17], including a detailed description of an approximation model for estimating crosstalk and measured bending radius dependences of the crosstalk of the fabricated MCF. We introduced the equivalent index model to the coupled-mode equations for the crosstalk in MCFs and found that the crosstalk is significantly affected by the fiber bend and is a stochastic value. The stochasticity of the crosstalk is induced by fluctuations of equivalent propagation constants in each MCF core, because of the slight perturbations of bends and twists in the MCF. To deal with such a random longitudinal evolution of the crosstalk, we developed an approximation model and derived the statistical distribution of the crosstalk and a relationship between the fiber parameters and the mean value of the crosstalk distribution, i.e., the mean crosstalk. Considering a homogenous MCF, the mean crosstalk was linearly proportional to the bending radius and fiber length. We designed a trench-assisted seven-core MCF based on the above theoretical scheme. We fabricated the MCF with pure-silica cores, and achieved, to the best of our knowledge, the lowest attenuation for the MCF, 0.175–0.181 dB/km at λ = 1550 nm, and 0.192–0.202 dB/km over the C + L band. We obtained the statistical distributions of the crosstalk of the fabricated MCF, and the measurement results were in good agreement with the developed model. The mean crosstalk between the neighboring cores was found to be less than −77.6 dB at λ = 1550 nm and less than −67.7 dB at λ = 1625 nm when the MCF was wound on a 140-mm-radius bobbin. The mean crosstalk from six outer cores to one center core was −72.3 dB at λ = 1550 nm and −62.1 dB at λ = 1625 nm calculated from sums of the measured values. From the measurement results and the validated model, the mean crosstalk from the six outer cores to the center core at λ = 1625 nm even after the 10,000-km propagation was estimated to be less than −30 dB in a practical applicable bending radius range of the MCF.

## Appendix: Derivation of Eq. (12)

Assuming that *A _{n}*

_{,0}= 0 and

*A*

_{m}_{,0}= 1, Eq. (11) can be rewritten as

Here, *ϕ*
_{rnd}(1) does not affect the absolute value of *A _{n}*

_{,1}; thus, we assume

*ϕ*

_{rnd}(1) = 0. Accordingly, Κ

*can be expressed as*

_{nm}*A _{n}*

_{,1}is the amplitude of Core

*n*after the first phase-matching point and can be derived from Eqs. (6) and (8). Eq. (6) for Core

*m*and Core

*n*can be rewritten as

where *A*(z) represents the amplitude evolution along the longitudinal axis *z* of the fiber. Here, we assume that the fiber is twisted continuously at a constant rate *γ*; hence, Eq. (8) can be rewritten as

If phase-matching points exist, then the first and the second points are at z = *π*/(2*γ*), and z = 3*π*/(2*γ*), respectively, in this case. Therefore, *A _{n}*

_{,1}can be expressed as

*A*(

_{n}*π*/

*γ*) and Eq. (29) can be rewritten as

Assuming that *A _{m}*(

*z*) ≅

*A*(0) = 1 when the crosstalk is adequately low, Eq. (32) can be derived as follows:

_{m}using the following equation (obtained by substituting *t*=exp(*jθ*) to Eq. (11.2) in [26]):

where *J _{ν}*(

*x*) denotes the Bessel function of the first kind of order

*ν*and

*ν*is an integer. Assuming that

*β*=

_{n}*β*=

_{m}*β*, we obtain

Eq. (35) can be rewritten as

using the following equation (Eqs. (11.137) and (11.138) in [26]):

Assuming that *β* ~ 10^{7}, *D _{nm}* ~ 4 × 10

^{−5},

*γ*< ~1, and

*R < ~*1, Eq. (37) holds for small

*ν*. Eq. (37) does not hold for large

*ν*, but the Bessel function in the second term of the right-hand side of Eq. (35) is divided by

*ν*so that Eq. (36) holds. The summation term in the imaginary part in the square brackets of Eq. (36) can be rewritten as

The summation term of the cosine on the right-hand side of Eq. (38) gives odd *ν* values and equals zero, and the remaining terms on the right-hand side give even values for *ν*. Thus, it can be rewritten as

using the following equation (see p. 888 in [26]):

Using Eqs. (38) and (39), Eq. (36) can be rewritten as

Thus, |Κ* _{nm}*| can be derived as Eq. (12). As shown in Fig. 15
, Eq. (12) was also validated by comparing values of |Κ

*| obtained using Eq. (12) and those simulated numerically using Eqs. (30)–(32) for two identical step-index cores for all combinations of core Δ (0.35, 0.4%),*

_{nm}*D*(35, 40 µm),

_{nm}*R*(60, 120, 180, 240, 300 mm), and

*γ*(0.02

*π*, 0.2

*π*, 2

*π*, 20

*π*radians).

## Acknowledgments

This research is supported by the National Institute of Information and Communications Technology (NICT), Japan under “Research on Innovative Optical Fiber Technology”.

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