## Design and analysis of large-effective-area heterogeneous trench-assisted multi-core fiber |

Optics Express, Vol. 20, Issue 14, pp. 15157-15170 (2012)

http://dx.doi.org/10.1364/OE.20.015157

Acrobat PDF (1955 KB)

### Abstract

Based on the overlap integral of electromagnetic fields in neighboring cores, a calculating method is proposed for obtaining the coupling coefficient between two adjacent trench-assisted non-identical cores. And a kind of heterogeneous trench-assisted multi-core fiber (Hetero-TA-MCF) with 12 cores is proposed to achieve large effective area (*A*_{eff}) and high density of cores. As bending radius becomes larger than 50 mm, the crosstalk value at 1550-nm wavelength of the Hetero-TA-MCF is about −42 dB after 100-km propagation and the *A*_{eff} of this Hetero-TA-MCF can reach 100 µm^{2}.

© 2012 OSA

## 1. Introduction

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

*A*

_{eff}) comparing to MCF with step-index profile also has been proposed [5]. It has been proved that the crosstalk between the identical cores will become larger and larger as bending radius (

*R*) increases [6

6. S. Matsuo, K. Takenaga, Y. Arakawa, Y. Sasaki, S. Tanigawa, K. Saitoh, and M. Koshiba, “Crosstalk behavior of cores in multi-core fiber under bent condition,” IEICE Electron. Express **8**(6), 385–390 (2011). [CrossRef]

*R*reaching a threshold value which was called

*R*

_{pk}in [7]. Moreover, if the cores have slight differences in their core radii and core refractive indices, the maximum power transferred between the cores goes down drastically [4

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

*A*

_{eff}as much as possible. Additionally, when we analyze the crosstalk characteristics between a couple of non-identical cores with trench regions, we should know the coupling coefficient (

*κ*) between them. However, to the best of our knowledge, an analytical method which can be used to figure out the value of

*κ*between two trench-assisted non-identical cores has not been reported yet. Therefore, before proposing a new type of Hetero-TA-MCF and analyzing the characteristics of it, we will introduce an effective method for calculating the

*κ*between two trench-assisted non-identical cores by considering the overlap integral of electromagnetic fields in the neighboring cores. Then we will compare such method with finite element method (FEM) [8

8. K. Saitoh and M. Koshiba, “Full-vectorial imaginary-distance beam propagation method based on a finite element scheme: Application to photonic crystal fibers,” IEEE J. Quantum Electron. **38**(7), 927–933 (2002). [CrossRef]

*κ*, an optimal design condition for the Hetero-TA-MCF will be given.

## 2. Calculation of *κ* between trench-assisted non-identical cores

### 2.1 Analysis of refractive-index distribution in the coupled region

*N*

_{1}(

*r, θ*) and

*N*

_{2}(

*r, θ*) represent the refractive-index distribution of each core with trench structure, and

*n*(

*r, θ*) means the refractive-index distribution outside the cores, which are shown in Fig. 1(a), Fig. 1(b), and Fig. 1(c), respectively. The expression of coupling coefficient between two waveguides is expressed as [9]where

*ω*is an angular frequency of the sinusoidally varying electromagnetic fields,

*ε*

_{0}is the permittivity of the medium, and

**means the outward-directed unit vector. The pair of**

*u*_{z}*p*and

*q*is either (1, 2) or (2, 1).

**and**

*E***represent the electric and magnetic fields respectively.**

*H**N*

^{2}

*−N*

_{2}

^{2}is zero except the region inside core 1, so the difference of the refractive-index distribution inside the core 1 is

*n*

_{1}

^{2}

*−n*

_{cl}

^{2}, while according to Fig. 2(b),

*N*

^{2}

*−N*

_{1}

^{2}is zero except the region inside core 2, so the difference of the refractive-index distribution inside the core 2 is

*n*

_{2}

^{2}

*−n*

_{cl}

^{2}. Furthermore, the denominator of

*κ*equals 4

_{pq}*P*, where

*P*means the total power flow [9]. So the expression of

*κ*can be rewritten aswhere

_{pq}**and**

*E*_{p}**represent the amplitude of electric field distribution of core**

*E*_{q}*p*inside the range of core

*p*, and the amplitude of electric field distribution of core

*q*inside the range of core

*p*, respectively.

*a*

_{1}

*denotes the radius of core*

_{-p}*p*.

### 2.2 Derivation of the expression of κ between trench-assisted non-identical cores

*m*(

*m*= 1, 2), cladding and trench along the transversal direction as follows: where

*β*is the propagation constant and

*k*is the wavenumber in a vacuum. The normalized frequency (

*V*

_{1}

*,*

_{-m}*V*

_{2}

*), the normalized transverse wave number in core*

_{-m}*m*(

*U*

_{1}

*), that in cladding (*

_{-m}*W*

_{1}

*) and that in trench (*

_{-m}*W*

_{2}

*) can be expressed as follows: where*

_{-m}*a*

_{1}

*is the radius of core*

_{-m}*m*and Eq. (11) is the approximation which is in error by less than 0.2 percent for 1.5 ≤

*V*

_{1}

*≤ 2.5 [10].*

_{-m}*J*

_{0}(

*ξ*) and the 0th-order Neumann function

_{m}r_{m}*N*

_{0}(

*ξ*) [9], respectively. However,

_{m}r_{m}*N*

_{0}(

*ξ*) diverges infinitely at

_{m}r_{m}*r*= 0. Therefore

*J*

_{0}(

*ξ*) is the proper solution for the field in the core. In the cladding region of fiber, the solutions for Eq. (5) of TM modes are the modified Bessel function of the first kind

_{m}r_{m}*I*

_{0}(

*σ*) and modified Bessel functions of the second kind

_{m}r_{m}*K*

_{0}(

*σ*), respectively. However,

_{m}r_{m}*I*

_{0}(

*σ*) diverges infinitely at

_{m}r_{m}*r*= ∞. Therefore

*K*

_{0}(

*σ*) is the proper solution for the field in the cladding. In the same principle,

_{m}r_{m}*K*

_{0}(

*σ*) is the proper solution for the field in the trench. Nevertheless, in hybrid modes, the solutions for Eq. (5) are given by the product of the

_{m}r_{m}*n*th-order Bessel functions and cos(

*nθ*+

*ψ*). Thus, the z-components of the electric field can be obtained as

*nθ*+

*ψ*), where

*n*is an integer and

*ψ*denotes the phase. As shown in Fig. 3 , core

*m*, inner cladding between core

*m*and trench

*m*, trench

*m*, outer cladding outside trench

*m*, trench

*m’*and inner cladding inside trench

*m’*are abbreviated as Co

*m*, IC

*m*, Tr

*m*, OC, Tr

*m’*, and IC

*m’*.

*D*is the core pitch,

*a*

_{1}

*is the radius of core*

_{-m}*m*,

*a*

_{2}

*is the distance from the center of core*

_{-m}*m*to the inner circumference of trench

*m*, and

*a*

_{3}

*is the distance from the center of core*

_{-m}*m*to the outer circumference of trench

*m*.

*R*,

*R*

_{1}and

*R*

_{2}denote the distance between the center of core

*m*to the objective point in IC

*m’*, the distance from the center of core

*m*to the outer circumference of trench

*m’*and the distance from the center of core

*m*to the inner circumference of trench

*m’*, respectively.

*K*(

_{n}*z*) in Eq. (29) is large, it can be approximated as

*κ*can be expressed as

_{pq}### 2.3 Comparison with finite element method (FEM)

*XT*) of step-index MCF (S-MCF) and trench-assisted MCF (TA-MCF) at 1550-nm wavelength. The blue solid line relates the simulation result of S-MCF. On the other hand, the red solid line represents the result of TA-MCF which is calculated by using the above-mentioned method and the green solid line represents the result of TA-MCF that is obtained by using the FEM [8

8. K. Saitoh and M. Koshiba, “Full-vectorial imaginary-distance beam propagation method based on a finite element scheme: Application to photonic crystal fibers,” IEEE J. Quantum Electron. **38**(7), 927–933 (2002). [CrossRef]

## 3. Model of Hetero-TA-MCF with high density of cores

### 3.1 Design of Hetero-TA-MCF

*n*

_{cl}and

*n*

_{tr}(Δ

_{2}) was assumed to be −0.7%, which is the structural parameter used in the fabricated TA-MCFs [5]. And we define the distance between the outer circumferences of adjacent trenches (

*D*

_{tr}) as a value that is not smaller than 2 µm for making sure the trench not overlap to each other.

*A*

_{eff}to reach 110 µm

^{2}or 80 µm

^{2}, the outer cladding thickness (

*OCT*) needs to be at least 40 µm [12

12. K. Takenaga, Y. Arakawa, Y. Sasaki, S. Tanigawa, S. Matsuo, K. Saitoh, and M. Koshiba, “A large effective area multi-core fiber with an optimized cladding thickness,” Opt. Express **19**(26), B543–B550 (2011). [CrossRef] [PubMed]

13. S. Matsuo, K. Takenaga, Y. Arakawa, Y. Sasaki, S. Taniagwa, K. Saitoh, and M. Koshiba, “Large-effective-area ten-core fiber with cladding diameter of about 200 μm,” Opt. Lett. **36**(23), 4626–4628 (2011). [CrossRef] [PubMed]

*CD*) should not be larger than 200 µm [14

14. T. Matsui, K. Nakajima, and C. Fukai, “Applicability of photonic crystal fiber with uniform air-hole structure to high-speed and wide-band transmission over conventional telecommunication bands,” J. Lightwave Technol. **27**(23), 5410–5416 (2009). [CrossRef]

*A*

_{eff}and

*OCT*, the

*OCT*should be at least 37 µm when the

*A*

_{eff}equals 100 µm

^{2}. Thus, if we set the maximum of the

*CD*to be around 200 µm, the limit value of Λ should be 33 or so when the

*A*

_{eff}equals 100 µm

^{2}. In this case, the core number of 12 here is the limit value for this ring layout due to the required value of

*CD*.

### 3.2 Crosstalk characteristics

*R*which was proposed as

*R*

_{pk}[7]. The crosstalk is degraded at

*R*<

*R*

_{pk}due to the phase-matching between non-identical cores. In this phase-matching region, the bend perturbations are crucial. In the non-phase-matching region of

*R*>

*R*

_{pk}, on the other hand, the crosstalk is dominated by the statistical properties [15

15. 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]

*n*

_{eff}) between cores will be required for pushing the value of

*R*

_{pk}toward sufficiently small range. Figure 8 shows the required Δ

*n*

_{eff}value between two non-identical cores as function of Λ and

*R*

_{pk}. If we try to shift

*R*

_{pk}to

*R*of smaller than 5 cm with Λ being about 33 µm, Δ

*n*

_{eff}should be around 0.001 or larger than it between non-identical cores.

*n*

_{eff}) of the fundamental mode at 1550-nm wavelength as function of core radius and core Δ

_{1}in four conditions — (a)

*r*

_{2}/

*r*

_{1}= 2.0,

*W*/

*r*

_{1}= 1.0, (b)

*r*

_{2}/

*r*

_{1}= 2.0,

*W*/

*r*

_{1}= 1.1, (c)

*r*

_{2}/

*r*

_{1}= 2.0,

*W*/

*r*

_{1}= 1.2, (d)

*r*

_{2}/

*r*

_{1}= 2.0,

*W*/

*r*

_{1}= 1.3, where

*n*

_{eff}was simulated by FEM. The black solid lines and the black dashed lines represent the values of

*n*

_{eff}and

*A*

_{eff}, respectively. The couple of white solid lines correspond to the upper limit of bending loss (

*BL*) of the higher-order mode (HOM) at 1530 nm when

*R*equals 140 mm and the lower limit of bending loss (

*BL*) of the fundamental mode (FM) at 1625 nm when

*R*equals 30 mm. To define the single-mode operation, the bending loss of LP

_{11}-like HOM should be > 1 dB/m at

*R*= 140 mm [14

14. T. Matsui, K. Nakajima, and C. Fukai, “Applicability of photonic crystal fiber with uniform air-hole structure to high-speed and wide-band transmission over conventional telecommunication bands,” J. Lightwave Technol. **27**(23), 5410–5416 (2009). [CrossRef]

*R*= 30 mm, which is described in ITU-T recommendations G.655 and G.656. Therefore, in order to guarantee the transmission with single-mode operation and low bending loss from C-band to L-band, we only research the field that is surrounded by the couple of white solid lines.

*r*

_{2}/

*r*

_{1}to be 2.0 and change

*W*/

*r*

_{1}from 1.0 to 1.3 to investigate how the width of trench region influences the crosstalk between the neighboring cores. In the conditions (a), (b), (c), and (d), we do not set the target value of

*A*

_{eff}to be 110 µm

^{2}, because the core radii of cores are too large to ensure the required value of

*CD*. So we select four pairs of cores with

*A*

_{eff}equaling 100 µm

^{2}for the simulation in conditions (a), (b), (c), and (d). In Fig. 9(a), one of the cores has a radius of 5.34 µm and Δ

_{1}= 0.304% and another core has a radius of 4.86 µm and Δ

_{1}= 0.242%. In Fig. 9(b), one of the cores has a radius of 5.27 µm and Δ

_{1}= 0.293% and another core has a radius of 4.83 µm and Δ

_{1}= 0.239%. In Fig. 9(c), one of the cores has a radius of 5.18 µm and Δ

_{1}= 0.280% and another core has a radius of 4.77 µm and Δ

_{1}= 0.234%. In Fig. 9(d), one of the cores has a radius of 5.03 µm and Δ

_{1}= 0.263% and another core has a radius of 4.75 µm and Δ

_{1}= 0.240%. Furthermore, in order to make sure the trench not overlap to each other, we defined

*D*

_{tr}to be not smaller than 2 µm. Under this requirement, we set each required Λ. The optical properties of the cores in each condition are summarized in Table 2 .

*RCMF*) was used to compare the core density of fibers [12

12. K. Takenaga, Y. Arakawa, Y. Sasaki, S. Tanigawa, S. Matsuo, K. Saitoh, and M. Koshiba, “A large effective area multi-core fiber with an optimized cladding thickness,” Opt. Express **19**(26), B543–B550 (2011). [CrossRef] [PubMed]

*CMF*) is defined aswhere

*N*

_{core}is a number of core and

*CD*means a cladding diameter. And

*RCMF*is a ratio between

*CMF*of a MCF and a standard single core single mode fiber with

*A*

_{eff}= 80 µm

^{2}at 1.55 µm and

*CD*= 125 µm.

*κ*which was proposed above, the crosstalk can be calculated by using the novel power coupling coefficient (PCC) definition [15

15. 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]

*d*

_{c}) and

*d*

_{c}of 0.05-m was proved to agree well with the measurement in [6

6. S. Matsuo, K. Takenaga, Y. Arakawa, Y. Sasaki, S. Tanigawa, K. Saitoh, and M. Koshiba, “Crosstalk behavior of cores in multi-core fiber under bent condition,” IEICE Electron. Express **8**(6), 385–390 (2011). [CrossRef]

*d*

_{c}to be 0.05 m in this simulation of crosstalk, and the twisting rate is defined to be 5 turns per 100 m.

*n*

_{eff}between the adjacent cores decreases and

*CD*should be larger. Therefore, if we want the

*R*

_{pk}of the designed fiber smaller than 50 mm, the Δ

*n*

_{eff}should be around 0.001. In this case,

*r*

_{2}/

*r*

_{1}= 2.0 and

*W*/

*r*

_{1}= 1.0 is the optimal design condition for the Hetero-TA-MCF and under this condition, as

*R*becomes larger than 50 mm, the worst-case crosstalk is around −42 dB after 100-km propagation and it is insensitive to the bending radius.

*r*

_{1}= 5.34 µm, Δ

_{1}= 0.304%, and Λ = 33 µm. For condition (b),

*r*

_{1}= 5.27 µm, Δ

_{1}= 0.293%, and Λ = 34 µm. For condition (c),

*r*

_{1}= 5.18 µm, Δ

_{1}= 0.280%, and Λ = 34 µm. And for condition (d),

*r*

_{1}= 5.03 µm, Δ

_{1}= 0.263%, and Λ = 35 µm. We can find obviously that the crosstalk of the Homo-TA-MCF become larger and larger as increasing the bending radius. Therefore, we can see the merit of Hetero-TA-MCF clearly from this comparison.

## 4. Conclusion

*κ*value between two adjacent trench-assisted non-identical cores. And we designed a sort of Hetero-TA-12-core fiber with

*A*

_{eff}of 100-µm

^{2}and showed that

*RCMF*can reach 5.8. Based on the calculating method of

*κ*, crosstalk characteristics of the Hetero-TA-MCF were analyzed. The parameters that Λ = 33 µm,

*A*

_{eff}= 100 µm

^{2},

*r*

_{2}/

*r*

_{1}= 2.0, and

*W*/

*r*

_{1}= 1.0 were proved to be the best design condition for this model. As

*R*becomes larger than 50 mm, the worst-case crosstalk at 1550-nm wavelength is about −42 dB after 100-km propagation, which also demonstrates that the Hetero-TA-MCF with high density of cores is insensitive to the bending radius.

## Acknowledgments

## References and links

1. | T. Morioka, “New generation optical infrastructure technologies: “EXACT initiative” towards 2020 and beyond,” in |

2. | K. Takenaga, S. Tanigawa, N. Guan, S. Matsuo, K. Saitoh, and M. Koshiba, “Reduction of crosstalk by quasi-homogeneous solid multi-core fiber,” in |

3. | 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. E |

4. | M. Koshiba, K. Saitoh, and Y. Kokubun, “Heterogeneous multi-core fibers: proposal and design principle,” IEICE Electron. Express |

5. | 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 |

6. | S. Matsuo, K. Takenaga, Y. Arakawa, Y. Sasaki, S. Tanigawa, K. Saitoh, and M. Koshiba, “Crosstalk behavior of cores in multi-core fiber under bent condition,” IEICE Electron. Express |

7. | T. Hayashi, T. Nagashima, O. Shimakawa, T. Sasaki, and E. Sasaoka, “Crosstalk variation of multi-core fiber due to fiber bend,” in |

8. | K. Saitoh and M. Koshiba, “Full-vectorial imaginary-distance beam propagation method based on a finite element scheme: Application to photonic crystal fibers,” IEEE J. Quantum Electron. |

9. | K. Okamoto, |

10. | H. D. Rudolph and E. G. Neuman, “Approximations for the eigenvalues of the fundamental mode of a step index glass fiber waveguide,” Nachrichtentech. Elektron. |

11. | A. W. Snyder and J. D. Love, |

12. | K. Takenaga, Y. Arakawa, Y. Sasaki, S. Tanigawa, S. Matsuo, K. Saitoh, and M. Koshiba, “A large effective area multi-core fiber with an optimized cladding thickness,” Opt. Express |

13. | S. Matsuo, K. Takenaga, Y. Arakawa, Y. Sasaki, S. Taniagwa, K. Saitoh, and M. Koshiba, “Large-effective-area ten-core fiber with cladding diameter of about 200 μm,” Opt. Lett. |

14. | T. Matsui, K. Nakajima, and C. Fukai, “Applicability of photonic crystal fiber with uniform air-hole structure to high-speed and wide-band transmission over conventional telecommunication bands,” J. Lightwave Technol. |

15. | M. Koshiba, K. Saitoh, K. Takenaga, and S. Matsuo, “Multi-core fiber design and analysis: coupled-mode theory and coupled-power theory,” Opt. Express |

**OCIS Codes**

(060.2270) Fiber optics and optical communications : Fiber characterization

(060.2280) Fiber optics and optical communications : Fiber design and fabrication

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: April 30, 2012

Revised Manuscript: June 9, 2012

Manuscript Accepted: June 11, 2012

Published: June 21, 2012

**Citation**

Jiajing Tu, Kunimasa Saitoh, Masanori Koshiba, Katsuhiro Takenaga, and Shoichiro Matsuo, "Design and analysis of large-effective-area heterogeneous trench-assisted multi-core fiber," Opt. Express **20**, 15157-15170 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-14-15157

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### References

- T. Morioka, “New generation optical infrastructure technologies: “EXACT initiative” towards 2020 and beyond,” in Proceedings of 14th OptoElectronics and Communications Conference (Institute of Electrical and Electronics Engineers, 2009), paper FT4.
- K. Takenaga, S. Tanigawa, N. Guan, S. Matsuo, K. Saitoh, and M. Koshiba, “Reduction of crosstalk by quasi-homogeneous solid multi-core fiber,” in Optical Fiber Communication Conference, OSA Technical Digest (CD) (Optical Society of America, 2010), paper OWK7.
- 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, 409–416 (2011).
- M. Koshiba, K. Saitoh, and Y. Kokubun, “Heterogeneous multi-core fibers: proposal and design principle,” IEICE Electron. Express6(2), 98–103 (2009). [CrossRef]
- 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 Optical Fiber Communication Conference, OSA Technical Digest (CD) (Optical Society of America, 2011), paper OWJ4.
- S. Matsuo, K. Takenaga, Y. Arakawa, Y. Sasaki, S. Tanigawa, K. Saitoh, and M. Koshiba, “Crosstalk behavior of cores in multi-core fiber under bent condition,” IEICE Electron. Express8(6), 385–390 (2011). [CrossRef]
- T. Hayashi, T. Nagashima, O. Shimakawa, T. Sasaki, and E. Sasaoka, “Crosstalk variation of multi-core fiber due to fiber bend,” in Proceedings of 36th European Conference and Exhibition on Optical Communication (Institute of Electrical and Electronics Engineers, 2010), paper We.8.F.6.
- K. Saitoh and M. Koshiba, “Full-vectorial imaginary-distance beam propagation method based on a finite element scheme: Application to photonic crystal fibers,” IEEE J. Quantum Electron.38(7), 927–933 (2002). [CrossRef]
- K. Okamoto, Fundamentals of Optical Waveguides (Corona Publishing, 1992), Chap. 4.
- H. D. Rudolph and E. G. Neuman, “Approximations for the eigenvalues of the fundamental mode of a step index glass fiber waveguide,” Nachrichtentech. Elektron.29, 328–329 (1976).
- A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, 1983), Chap. 37.
- K. Takenaga, Y. Arakawa, Y. Sasaki, S. Tanigawa, S. Matsuo, K. Saitoh, and M. Koshiba, “A large effective area multi-core fiber with an optimized cladding thickness,” Opt. Express19(26), B543–B550 (2011). [CrossRef] [PubMed]
- S. Matsuo, K. Takenaga, Y. Arakawa, Y. Sasaki, S. Taniagwa, K. Saitoh, and M. Koshiba, “Large-effective-area ten-core fiber with cladding diameter of about 200 μm,” Opt. Lett.36(23), 4626–4628 (2011). [CrossRef] [PubMed]
- T. Matsui, K. Nakajima, and C. Fukai, “Applicability of photonic crystal fiber with uniform air-hole structure to high-speed and wide-band transmission over conventional telecommunication bands,” J. Lightwave Technol.27(23), 5410–5416 (2009). [CrossRef]
- 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]

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