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 (Aeff) 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 Aeff of this Hetero-TA-MCF can reach 100 µm2.

© 2012 OSA

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References

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  1. 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.
  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 Optical Fiber Communication Conference, OSA Technical Digest (CD) (Optical Society of America, 2010), paper OWK7.
  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. E94-B, 409–416 (2011).
  4. M. Koshiba, K. Saitoh, and Y. Kokubun, “Heterogeneous multi-core fibers: proposal and design principle,” IEICE Electron. Express6(2), 98–103 (2009).
    [CrossRef]
  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 Optical Fiber Communication Conference, OSA Technical Digest (CD) (Optical Society of America, 2011), paper OWJ4.
  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. Express8(6), 385–390 (2011).
    [CrossRef]
  7. 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.
  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]
  9. K. Okamoto, Fundamentals of Optical Waveguides (Corona Publishing, 1992), Chap. 4.
  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.29, 328–329 (1976).
  11. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, 1983), Chap. 37.
  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. Express19(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]
  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]
  15. 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]

2011

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).

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]

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]

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]

2009

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, and Y. Kokubun, “Heterogeneous multi-core fibers: proposal and design principle,” IEICE Electron. Express6(2), 98–103 (2009).
[CrossRef]

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]

1976

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).

Arakawa, Y.

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).

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]

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]

Fukai, C.

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]

Guan, N.

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).

Kokubun, Y.

M. Koshiba, K. Saitoh, and Y. Kokubun, “Heterogeneous multi-core fibers: proposal and design principle,” IEICE Electron. Express6(2), 98–103 (2009).
[CrossRef]

Koshiba, M.

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).

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]

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]

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]

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]

M. Koshiba, K. Saitoh, and Y. Kokubun, “Heterogeneous multi-core fibers: proposal and design principle,” IEICE Electron. Express6(2), 98–103 (2009).
[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]

Matsui, T.

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]

Matsuo, S.

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]

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. 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]

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, 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]

Nakajima, K.

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]

Neuman, E. G.

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).

Rudolph, H. D.

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).

Saitoh, 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. Express8(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 multi-core fibers by introducing random fluctuation along longitudinal direction,” IEICE Trans. Commun. E94-B, 409–416 (2011).

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]

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]

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]

M. Koshiba, K. Saitoh, and Y. Kokubun, “Heterogeneous multi-core fibers: proposal and design principle,” IEICE Electron. Express6(2), 98–103 (2009).
[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]

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. Express8(6), 385–390 (2011).
[CrossRef]

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]

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]

Takenaga, K.

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]

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]

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).

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]

Taniagwa, S.

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]

Tanigawa, S.

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]

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).

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]

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. Express8(6), 385–390 (2011).
[CrossRef]

M. Koshiba, K. Saitoh, and Y. Kokubun, “Heterogeneous multi-core fibers: proposal and design principle,” IEICE Electron. Express6(2), 98–103 (2009).
[CrossRef]

IEICE Trans. Commun. E

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).

J. Lightwave Technol.

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]

Nachrichtentech. Elektron.

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).

Opt. Express

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]

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]

Opt. Lett.

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]

Other

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, 1983), Chap. 37.

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.

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.

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. Okamoto, Fundamentals of Optical Waveguides (Corona Publishing, 1992), Chap. 4.

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Figures (11)

Fig. 1
Fig. 1

The profile of refractive index in two trench-assisted non-identical cores and the part outside the cores.

Fig. 2
Fig. 2

Difference of the refractive-index distributions. (a) N2N22. (b) N2N12.

Fig. 3
Fig. 3

The profile of core m with trench structures.

Fig. 4
Fig. 4

Geometries for the calculation of the coupling coefficient.

Fig. 5
Fig. 5

Schematic of a core with index trench and Homo-TA-7-core model.

Fig. 6
Fig. 6

Simulated crosstalk at 1550-nm wavelength as function of length.

Fig. 7
Fig. 7

Schematic of Hetero-TA-12-core model.

Fig. 8
Fig. 8

Required Δneff as function of the Λ and Rpk.

Fig. 9
Fig. 9

Effective index value of the fundamental mode as function of core radius and core Δ1, where (a) r2/r1 = 2.0, W/r1 = 1.0, (b) r2/r1 = 2.0, W/r1 = 1.1, (c) r2/r1 = 2.0, W/r1 = 1.2, and (d) r2/r1 = 2.0, W/r1 = 1.3.

Fig. 10
Fig. 10

Crosstalk of Hetero-TA-12-core fiber at 100-km propagation as function of bending radius.

Fig. 11
Fig. 11

Crosstalk of Homo-TA-12-core fiber at 100-km propagation as function of bending radius.

Tables (2)

Tables Icon

Table 1 Structural Parameters for Calculation

Tables Icon

Table 2 Optical Properties of the Cores in Different Conditions (1550 nm)

Equations (34)

Equations on this page are rendered with MathJax. Learn more.

N 2 (r,θ)= N 1 2 (r,θ)+ N 2 2 (r,θ) n 2 (r,θ),
κ pq = ω ε 0 + + ( N 2 N q 2 ) E p * E q dxdy + + u z ( E p * × H p + E p × H p * )dxdy ,
κ pq = ω ε 0 4P 0 2π 0 a 1p ( n p 2 n cl 2 ) E p * E q rdrdθ ,
E ˜ =E(r,θ) e j(ωtβz) .
2 E z r 2 + 1 r E z r + 1 r 2 2 E z θ 2 +[ k 2 n (r,θ) 2 β 2 ] E z =0.
ξ m = n m 2 k 2 β m 2 ,
σ m = β m 2 n cl 2 k 2 ,
γ m = β m 2 n tr 2 k 2 ,
V 1m = a 1m k n m 2 n cl 2 ,
V 2m = a 1m k n cl 2 n tr 2 ,
W 1m = a 1m σ m =1.1428 V 1m 0.996,
U 1m = a 1m ξ m = V 1m 2 W 1m 2 ,
W 2m = a 1m γ m = V 2m 2 + W 1m 2 ,
E m z ={ A m J n ( ξ m r m )cos(nθ+ψ)(inCo m) B m K n ( σ m r m )cos(nθ+ψ)(inIC m) C m K n ( γ m r m )cos(nθ+ψ)(inTrm) D m K n ( σ m r m )cos(nθ+ψ)(inOC) E m K n ( γ m r m )cos(nθ+ψ)(inTrm') F m K n ( σ m r m )cos(nθ+ψ)(inIC m') ,
{ A m J n ( U 1m )= B m K n ( W 1m ) B m K n ( W 1m a 2m a 1m )= C m K n ( W 2m a 2m a 1m ) C m K n ( W 2m a 3m a 1m )= D m K n ( W 3m a 3m a 1m ) D m K n ( W 3m R 1 a 1m )= E m K n ( W 4m R 1 a 1m ) E m K n ( W 4m R 2 a 1m )= F m K n ( W 5m R 2 a 1m ) .
D m = L m A m ,
F m = Q m A m ,
L m = J n ( U 1m ) K n ( W 1m a 2m a 1m ) K n ( W 2m a 3m a 1m ) K n ( W 1m ) K n ( W 2m a 2m a 1m ) K n ( W 3m a 3m a 1m ) ,
Q m = L m K n ( W 3m R 1 a 1m ) K n ( W 4m R 2 a 1m ) K n ( W 4m R 1 a 1m ) K n ( W 5m R 2 a 1m ) .
A m = U 1m W 1m β m a 1m 2 V 1m J 1 ( U 1m ) 2P π ε 0 n 1m c ,
E p ={ E p x =j A p β p a 1p U 1p J 0 ( U 1p r a 1p )cosψ E p y =j A p β p a 1p U 1p J 0 ( U 1p r a 1p )sinψ E p z = A p J 1 ( U 1p r a 1p )cos(θ+ψ) ,
E q ={ E q x =j Q q A q β q a 1q W 1q K 0 ( W 1q R a 1q )cosψ E q y =j Q q A q β q a 1q W 1q K 0 ( W 1q R a 1q )sinψ E q z = Q q A q K 1 ( W 1q R a 1q )cos(Θ+ψ) .
R= D 2 + r 2 2Drcosθ Drcosθ,
a 3p = D 2 + R 1 2 2D R 1 cos(πΘ) D R 1 cos(πΘ),
r= D 2 + R 2 2DRcos(πΘ) DRcos(πΘ),
R 1 (D a 3p )(Drcosθ) Dr .
R 2 (D a 2p )(Drcosθ) Dr .
E p * E q = Q q A p A q β p β q a 1p a 1q U 1p W 1q J 0 ( U 1p r a 1p ) K 0 ( W 1q R a 1q ) + Q q A p A q J 1 ( U 1p r a 1p ) K 1 ( W 1q R a 1q )cos(θ+ψ)cos(Θ+ψ).
S 1 = 0 2π 0 a 1p ( n p 2 n cl 2 ) E p * E q rdrdθ =( n p 2 n cl 2 ) L q A p A q β p β q a 1p a 1q U 1p W 1q × 0 2π 0 a 1p J 0 ( U 1p r a 1p ) K 0 ( W 1q R a 1q ) K 1 ( P 1 Drcosθ Dr ) K 1 ( Y 1 Drcosθ Dr ) K 1 ( P 2 Drcosθ Dr ) K 1 ( Y 2 Drcosθ Dr ) rdrdθ ,
K n (z) π 2z exp(z).
S 1 =( n p 2 n cl 2 ) L q A p A q β p β q a 1p a 1q U 1p W 1q π a 1q 2 W 1q D exp( W 1q D a 1q ) × 0 a 1p J 0 ( U 1p r a 1p )exp[( P 2 P 1 + Y 2 Y 1 ) D Dr ]rdr 0 2π exp[( W 1q a 1q P 2 P 1 + Y 2 Y 1 Dr )rcosθ]dθ .
I 0 (z)= 1 π 0 π exp(zcosθ) dθ,
κ pq = k( n p 2 n cl 2 ) W 1p U 1q L q π a 1q 2 W 1q D exp( W 1q D a 1q ) n p n q a 1p a 1q V 1p V 1q J 1 ( U 1p ) J 1 ( U 1q ) × 0 a 1p J 0 ( U 1p r a 1p ) I 0 [( W 1q a 1q P 2 P 1 + Y 2 Y 1 Dr )r]exp[( P 2 P 1 + Y 2 Y 1 ) D Dr ]rdr.
CMF= N core A eff π (CD/2) 2 ,

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