Abstract

Coupled-mode and coupled-power theories are described for multi-core fiber design and analysis. First, in order to satisfy the law of power conservation, mode-coupling coefficients are redefined and then, closed-form power-coupling coefficients are derived based on exponential, Gaussian, and triangular autocorrelation functions. Using the coupled-mode and coupled-power theories, impacts of random phase-offsets and correlation lengths on crosstalk in multi-core fibers are investigated for the first time. The simulation results are in good agreement with the measurement results. Furthermore, from the simulation results obtained by both theories, it is confirmed that the reciprocity is satisfied in multi-core fibers.

© 2011 OSA

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    [CrossRef] [PubMed]
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  4. 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]
  5. K. Imamura, Y. Tsuchida, K. Mukasa, R. Sugizaki, K. Saitoh, and M. Koshiba, “Investigation on multi-core fibers with large Aeff and low micro bending loss,” Opt. Express 19(11), 10595–10603 (2011).
    [CrossRef] [PubMed]
  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]
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    [CrossRef]
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    [CrossRef]
  9. W.-P. Huang, “Coupled-mode theory for optical waveguides: an overview,” J. Opt. Soc. Am. A 11(3), 963–983 (1994).
    [CrossRef]
  10. D. Marcuse, “Derivation of coupled power equations,” Bell Syst. Tech. J. 51, 229–237 (1972).
  11. K. Petermann, “Microbending loss in monomode fibers,” Electron. Lett. 12(4), 107–109 (1976).
    [CrossRef]
  12. D. Marcuse, “Microdeformation losses of single-mode fibers,” Appl. Opt. 23(7), 1082–1091 (1984).
    [CrossRef] [PubMed]

2011

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

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]

K. Imamura, Y. Tsuchida, K. Mukasa, R. Sugizaki, K. Saitoh, and M. Koshiba, “Investigation on multi-core fibers with large Aeff and low micro bending loss,” Opt. Express 19(11), 10595–10603 (2011).
[CrossRef] [PubMed]

2010

2002

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]

1994

1988

A. W. Snyder and A. Ankiewicz, “Optical fiber couplers-optimum solution for unequal cores,” J. Lightwave Technol. 6(3), 463–474 (1988).
[CrossRef]

1985

A. Hardy and W. Streifer, “Coupled mode theory of parallel waveguides,” J. Lightwave Technol. 3(5), 1135–1146 (1985).
[CrossRef]

1984

1976

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

1972

D. Marcuse, “Derivation of coupled power equations,” Bell Syst. Tech. J. 51, 229–237 (1972).

Ankiewicz, A.

A. W. Snyder and A. Ankiewicz, “Optical fiber couplers-optimum solution for unequal cores,” J. Lightwave Technol. 6(3), 463–474 (1988).
[CrossRef]

Arakawa, Y.

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]

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

Fini, J. M.

Guan, N.

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

Hardy, A.

A. Hardy and W. Streifer, “Coupled mode theory of parallel waveguides,” J. Lightwave Technol. 3(5), 1135–1146 (1985).
[CrossRef]

Huang, W.-P.

Imamura, K.

Koshiba, M.

K. Imamura, Y. Tsuchida, K. Mukasa, R. Sugizaki, K. Saitoh, and M. Koshiba, “Investigation on multi-core fibers with large Aeff and low micro bending loss,” Opt. Express 19(11), 10595–10603 (2011).
[CrossRef] [PubMed]

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

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]

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]

Marcuse, D.

D. Marcuse, “Microdeformation losses of single-mode fibers,” Appl. Opt. 23(7), 1082–1091 (1984).
[CrossRef] [PubMed]

D. Marcuse, “Derivation of coupled power equations,” Bell Syst. Tech. J. 51, 229–237 (1972).

Matsuo, S.

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]

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

Mukasa, K.

Petermann, K.

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

Saitoh, K.

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

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]

K. Imamura, Y. Tsuchida, K. Mukasa, R. Sugizaki, K. Saitoh, and M. Koshiba, “Investigation on multi-core fibers with large Aeff and low micro bending loss,” Opt. Express 19(11), 10595–10603 (2011).
[CrossRef] [PubMed]

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]

Sasaki, Y.

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]

Snyder, A. W.

A. W. Snyder and A. Ankiewicz, “Optical fiber couplers-optimum solution for unequal cores,” J. Lightwave Technol. 6(3), 463–474 (1988).
[CrossRef]

Streifer, W.

A. Hardy and W. Streifer, “Coupled mode theory of parallel waveguides,” J. Lightwave Technol. 3(5), 1135–1146 (1985).
[CrossRef]

Sugizaki, R.

Takenaga, K.

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]

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

Tanigawa, S.

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

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]

Taunay, T. F.

Tsuchida, Y.

Yan, M. F.

Zhu, B.

Appl. Opt.

Bell Syst. Tech. J.

D. Marcuse, “Derivation of coupled power equations,” Bell Syst. Tech. J. 51, 229–237 (1972).

Electron. Lett.

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

IEEE J. Quantum Electron.

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]

IEICE Electron. Express

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]

IEICE Trans. Commun.

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

J. Lightwave Technol.

A. Hardy and W. Streifer, “Coupled mode theory of parallel waveguides,” J. Lightwave Technol. 3(5), 1135–1146 (1985).
[CrossRef]

A. W. Snyder and A. Ankiewicz, “Optical fiber couplers-optimum solution for unequal cores,” J. Lightwave Technol. 6(3), 463–474 (1988).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Express

Other

T. Hayashi, T. Nagashima, O. Shimakawa, T. Sasaki, and E. Sasaoka, “Crosstalk variation of multi-core fibre due to fibre bend,” in Proceedings of 36th European Conference and Exhibition on Optical Communication (Institute of Electrical and Electronics Engineers, 2010), paper We.8.F.6.

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Figures (11)

Fig. 1
Fig. 1

Schematics of (a) bent multi-core fiber and (b) fiber cross-section.

Fig. 2
Fig. 2

Random phase-offsets applied to all cores at every segment.

Fig. 3
Fig. 3

Cross-section of a quasi-homogeneous 7-core fiber [4].

Fig. 4
Fig. 4

Bending-diameter dependence of crosstalk calculated from coupled-mode theory. Dotted line: ds = 0.01 m, solid line: ds = 0.05 m, dashed line: ds = 0.1 m, dashed and dotted line: ds = 0.5 m, closed circles: measured data [4].

Fig. 5
Fig. 5

Bending-diameter dependence of crosstalk calculated from coupled-power theory with exponential autocorrelation function. Dotted line: dc = 0.01 m, solid line: dc = 0.05 m, dashed line: dc = 0.1 m, dashed and dotted line: dc = 0.5 m, dashed and double-dotted line: Eq. (25) [4], closed circles: measured data [4].

Fig. 6
Fig. 6

Propagation-distance dependence of power-coupling coefficients for bending diameter of 250 mm. Dotted line: dc = 0.01 m, solid line: dc = 0.05 m, dashed line: dc = 0.1 m, dashed and dotted line: dc = 0.5 m.

Fig. 7
Fig. 7

Propagation-distance dependence of power-coupling coefficients for bending diameter of 1,000 mm. Dotted line: dc = 0.01 m, solid line: dc = 0.05 m, dashed line: dc = 0.1 m, dashed and dotted line: dc = 0.5 m.

Fig. 8
Fig. 8

Bending-diameter dependence of crosstalk calculated from coupled-power theory with Gaussian autocorrelation function. Dotted line: dc = 0.01 m, solid line: dc = 0.05 m, dashed line: dc = 0.1 m, dashed and dotted line: dc = 0.5 m, closed circles: measured data [4].

Fig. 9
Fig. 9

Bending-diameter dependence of crosstalk calculated from coupled-power theory with triangular autocorrelation function. Dotted line: dc = 0.01 m, solid line: dc = 0.05 m, dashed line: dc = 0.1 m, dashed and dotted line: dc = 0.5 m, closed circles: measured data [4].

Fig. 10
Fig. 10

Bending-diameter dependence of crosstalk calculated from coupled-mode theory. Left figures: from core 1 to core 4 (top) and from core 4 to core 1 (bottom), middle figures: from core 1 to core 5 (top) and from core 5 to core 1 (bottom), right figures: from core 4 to core 5 (top) and from core 5 to core 4 (bottom). Dotted line: dc = 0.01 m, solid line: dc = 0.05 m, dashed line: dc = 0.1 m, dashed and dotted line: dc = 0.5 m, closed circles: measured data [4].

Fig. 11
Fig. 11

Bending-diameter dependence of crosstalk calculated from coupled-power theory with triangular autocorrelation function. Left figures: from core 1 to core 4 (top) and from core 4 to core 1 (bottom), middle figures: from core 1 to core 5 (top) and from core 5 to core 1 (bottom), right figures: from core 4 to core 5 (top) and from core 5 to core 4 (bottom). Dotted line: dc = 0.01 m, solid line: dc = 0.05 m, dashed line: dc = 0.1 m, dashed and dotted line: dc = 0.5 m, closed circles: measured data [4].

Equations (25)

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d A m dz =j nm κ mn A n (z)exp(jΔ β mn z)f(z)
f(z)=exp[j( ϕ m ϕ n )]δf(z)
ϕ m = 0 z β m Λ R b ( z ) cos θ m ( z )d z
θ m (z)=γz+(m2) π 3 + θ 0
κ nm = κ mn C mn Δ β mn
F mn = ( κ mn C mn Δ β mn /2) 2 κ mn κ nm + (Δ β mn /2) 2
F mn = κ mn 2 κ mn 2 + (Δ β mn /2) 2
Κ mn κ mn C mn Δ β mn 2
Κ nm κ nm C nm Δ β nm 2 = κ nm + C mn Δ β mn 2
Κ mn = κ mn + κ nm 2 = Κ nm
d P m dz = nm h mn (z)[ P n (z) P m (z)]
Δ β mn =Δ β mn + Λ R b ( z ) [ β m cos θ m ( z ) β n cos θ n ( z )]
d A m dz =j nm Κ mn A n (z)exp(jΔ β mn z)δf(z)
P m (z)= [ Κ mn A n (0) 0 z exp(j β mn ξ)δf(ξ) dξ ] [ Κ mn A n * (0) 0 z exp(j β mn η)δ f * (η) dη ]
P m (z)= Κ mn 2 P n (0) 0 z dη 0 z exp[jΔ β mn (ξη)] δf(ξ) f * (η) dξ
P m (z)= Κ mn 2 P n (0) 0 z dη [ 0 z exp(jΔ β mn ζ) δf(η+ζ)δ f * (η) dζ ]
P m (z)=z P n (0) Κ mn 2 2 exp(jΔ β mn ζ)R(ζ) dζ
h mn = P m (z) z P n (0) = Κ mn 2 2 S(Δ β mn )
R(ζ)=exp( | ζ | d c )
R(ζ)=exp[ ( ζ d c ) 2 ]
R(ζ)={ 1 | ζ | d c , | ζ | d c 0, | ζ |> d c
h mn (z)= Κ mn 2 d c 1+ [ Δ β mn (z) d c ] 2
h mn (z)= π Κ mn 2 d c 2 exp{ [ Δ β mn (z) d c 2 ] 2 }
h mn (z)= Κ mn 2 d c sin 2 [ Δ β mn (z) d c /2 ] 2 [ Δ β mn (z) d c /2 ] 2
h mn (z)= 4 Κ mn 2 π 4 Κ mn 2 + [ Δ β mn (z) ] 2

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