## Multi-core fiber design and analysis: coupled-mode theory and coupled-power theory |

Optics Express, Vol. 19, Issue 26, pp. B102-B111 (2011)

http://dx.doi.org/10.1364/OE.19.00B102

Acrobat PDF (1904 KB)

### 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

## 1. Introduction

1. J. M. Fini, B. Zhu, T. F. Taunay, and M. F. Yan, “Statistics of crosstalk in bent multicore fibers,” Opt. Express **18**(14), 15122–15129 (2010). [CrossRef] [PubMed]

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]

1. J. M. Fini, B. Zhu, T. F. Taunay, and M. F. Yan, “Statistics of crosstalk in bent multicore fibers,” Opt. Express **18**(14), 15122–15129 (2010). [CrossRef] [PubMed]

1. J. M. Fini, B. Zhu, T. F. Taunay, and M. F. Yan, “Statistics of crosstalk in bent multicore fibers,” Opt. Express **18**(14), 15122–15129 (2010). [CrossRef] [PubMed]

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]

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]

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]

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]

## 2. Coupled-mode theory

### 2.1 Redefinition of mode-coupling coefficients

**18**(14), 15122–15129 (2010). [CrossRef] [PubMed]

*A*is the mode amplitude in core

_{m}*m*,

*z*is the propagation direction,

*κ*is the mode-coupling coefficient (MCC) from core

_{mn}*n*to core

*m*, Δ

*β*=

_{mn}*β*−

_{m}*β*= −Δ

_{n}*β*is the propagation- constant difference with

_{nm}*β*and

_{m}*β*being the propagation constants of modes in core

_{n}*m*and core

*n*, respectively, and

*f*is the phase function describing bending and twisting effects. The phase function is separated into two parts. One is the deterministic part, exp[

*j*(

*ϕ*−

_{m}*ϕ*)], and the other is the random part,

_{n}*δf*,with

*ϕ*being the phase in core

_{m}*m*caused by bend and/or twist. For regular 7-core fibers as shown in Fig. 1(b),

*ϕ*

_{1}= 0 and

*ϕ*

_{2}to

*ϕ*

_{7}are expressed as [2]where

*Λ*is the core pitch,

*R*

_{b}is the bending radius, and

*θ*,with

_{m}*γ*and

*θ*

_{0}being the twist pitch and the twist offset, respectively. The phase in core

*m*,

*ϕ*, for arbitrarily located cores is also reported in [1

_{m}**18**(14), 15122–15129 (2010). [CrossRef] [PubMed]

*κ*=

_{mn}*κ*. However, for non-identical cores, they are not symmetric and therefore, when using conventional CMEs, total power is not conserved. In this case, using the cross-power term,

_{nm}*C*=

_{mn}*C*, the relation between

_{nm}*κ*and

_{mn}*κ*is rewritten as [7

_{nm}7. A. Hardy and W. Streifer, “Coupled mode theory of parallel waveguides,” J. Lightwave Technol. **3**(5), 1135–1146 (1985). [CrossRef]

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

*n*to core

*m*is also rewritten as [7

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

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

*κ*=

_{mn}*κ*,

_{nm}*C*= 0), the maximum power-conversion efficiency is reduced to

_{mn}*K*and

_{mn}*K*, can be written as average of usual MCCs,

_{nm}*κ*and

_{mn}*κ*,

_{nm}9. W.-P. Huang, “Coupled-mode theory for optical waveguides: an overview,” J. Opt. Soc. Am. A **11**(3), 963–983 (1994). [CrossRef]

*δf*, the total link is divided into finite segments of arbitrary but equal length,

*d*

_{s}, as shown in Fig. 2 , and then, random phase-offsets generated by using uniform random numbers, exp(

*jϕ*

_{rnd}), are applied to all cores at every segment. The segment length used in CMT is thought to be a stochastic parameter corresponding to the correlation length used in CPT.

### 2.2 Simulation results

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]

**8**(6), 385–390 (2011). [CrossRef]

**8**(6), 385–390 (2011). [CrossRef]

**8**(6), 385–390 (2011). [CrossRef]

**18**(14), 15122–15129 (2010). [CrossRef] [PubMed]

*d*

_{s}= 0.05 m agree well with the measurement results [4

**8**(6), 385–390 (2011). [CrossRef]

## 3. Coupled-power theory

### 3.1 Derivation of power-coupling coefficients

*P*is the average power in core

_{m}*m*and

*h*is the power-coupling coefficient (PCC). As PCCs should be symmetric, the starting point for deriving PCCs is CMEs with redefined MCCs,

_{mn}*K*=

_{mn}*K*. In order to obtain longitudinally varying, in other words, local PCCs, considering Eq. (3) and defining the local propagation-constant difference at

_{nm}*z*=

*z*’,

*δf*, is assumed to be stationary and the ensemble average is equal to zero,

*z*sufficiently close to

*z*= 0,

**18**(14), 15122–15129 (2010). [CrossRef] [PubMed]

*ζ*=

*ξ*−

*η*, we obtain

*δf*, is a stationary random process and therefore, it has an autocorrelation function,

*R*(0) = 1. Noting that the first integral in Eq. (16) yields the fiber length,

*z*, and that the autocorrelation function contributes only over the order of the correlation length, Eq. (16) is rewritten as

*d*

_{c}is the correlation length. The corresponding PCCs are, for EAF,for GAF,and for TAF,where

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]

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]

### 3.2 Simulation results

**8**(6), 385–390 (2011). [CrossRef]

**8**(6), 385–390 (2011). [CrossRef]

## 4. Conclusion

## Acknowledgments

## References and links

1. | J. M. Fini, B. Zhu, T. F. Taunay, and M. F. Yan, “Statistics of crosstalk in bent multicore fibers,” Opt. Express |

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

3. | 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. |

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 |

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 |

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. |

7. | A. Hardy and W. Streifer, “Coupled mode theory of parallel waveguides,” J. Lightwave Technol. |

8. | A. W. Snyder and A. Ankiewicz, “Optical fiber couplers-optimum solution for unequal cores,” J. Lightwave Technol. |

9. | W.-P. Huang, “Coupled-mode theory for optical waveguides: an overview,” J. Opt. Soc. Am. A |

10. | D. Marcuse, “Derivation of coupled power equations,” Bell Syst. Tech. J. |

11. | K. Petermann, “Microbending loss in monomode fibers,” Electron. Lett. |

12. | D. Marcuse, “Microdeformation losses of single-mode fibers,” Appl. Opt. |

**OCIS Codes**

(060.2270) Fiber optics and optical communications : Fiber characterization

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

**ToC Category:**

Fibers, Fiber Devices, and Amplifiers

**History**

Original Manuscript: September 28, 2011

Revised Manuscript: October 12, 2011

Manuscript Accepted: October 21, 2011

Published: November 16, 2011

**Virtual Issues**

European Conference on Optical Communication 2011 (2011) *Optics Express*

**Citation**

Masanori Koshiba, Kunimasa Saitoh, Katsuhiro Takenaga, and Shoichiro Matsuo, "Multi-core fiber design and analysis: coupled-mode theory and coupled-power theory," Opt. Express **19**, B102-B111 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-26-B102

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### References

- J. M. Fini, B. Zhu, T. F. Taunay, and M. F. Yan, “Statistics of crosstalk in bent multicore fibers,” Opt. Express18(14), 15122–15129 (2010). [CrossRef] [PubMed]
- 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.
- 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. Express8(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. Express19(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]
- 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]
- W.-P. Huang, “Coupled-mode theory for optical waveguides: an overview,” J. Opt. Soc. Am. A11(3), 963–983 (1994). [CrossRef]
- D. Marcuse, “Derivation of coupled power equations,” Bell Syst. Tech. J.51, 229–237 (1972).
- K. Petermann, “Microbending loss in monomode fibers,” Electron. Lett.12(4), 107–109 (1976). [CrossRef]
- D. Marcuse, “Microdeformation losses of single-mode fibers,” Appl. Opt.23(7), 1082–1091 (1984). [CrossRef] [PubMed]

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