## Mode coupling dynamics and communication strategies for multi-core fiber systems |

Optics Express, Vol. 20, Issue 4, pp. 4548-4563 (2012)

http://dx.doi.org/10.1364/OE.20.004548

Acrobat PDF (1087 KB)

### Abstract

The propagation dynamics of 7-core multi-core fibers (MCFs) with identical and three-types of cores are analytically derived based on the coupled-mode theory. The mode coupling dynamics can be aperiodic with transmission distance for MCF with identical cores. For MCFs with heterogeneous cores, it is found that even though signals from different core groups will not couple with each other, the coupling within their own group is significantly affected by the presence of other core groups. Joint signal processing techniques to mitigate mode coupling induced-cross-talks such as least mean square (LMS) algorithm and maximum likelihood (ML) detection are investigated and corresponding transmission performance are determined for coherent as well as intensity modulated formats. It is shown that aperiodic mode coupling in intensity modulated systems induces cross-talks that are difficult to eliminate through signal processing. The analytical insights may help in optimizing MCF designs and corresponding signal processing techniques for future high capacity MCF transmission systems.

© 2012 OSA

## 1. Introduction

3. F. Yaman, N. Bai, B. Zhu, T. Wang, and G. Li, “Long distance transmission in few-mode fibers,” Opt. Express **18**(12), 13250–13257 (2010). [CrossRef] [PubMed]

4. E. Ip, A. P. T. Lau, D. J. F. Barros, and J. M. Kahn, “Coherent detection in optical fiber systems,” Opt. Express **16**(2), 753–791 (2008). [CrossRef] [PubMed]

5. M. Koshiba, K. Saitoh, and Y. Kokubun, “Heterogeneous multicore fibres: proposal and design principle,” IEICE Electron. Express **6**(2), 98–103 (2009). [CrossRef]

11. S. Randel, R. Ryf, A. Sierra, P. J. Winzer, A. H. Gnauck, C. A. Bolle, R.-J. Essiambre, D. W. Peckham, A. McCurdy, and R. Lingle Jr., “6×56-Gb/s mode-division multiplexed transmission over 33-km few-mode fiber enabled by 6×6 MIMO equalization,” Opt. Express **19**(17), 16697–16707 (2011). [CrossRef] [PubMed]

6. C. P. Tsekrekos, A. Martinez, F. M. Huijskens, and A. M. J. Koonen, “Design considerations for transparent mode group diversity multiplexing,” IEEE Photon. Technol. Lett. **18**(22), 2359–2361 (2006). [CrossRef]

19. P. Glas, M. Naumann, A. Schirrmacher, and Th. Pertsch, “The multicore fiber—a novel design for a diode pumped fiber laser,” Opt. Commun. **151**(1-3), 187–195 (1998). [CrossRef]

20. N. N. Elkin, A. P. Napartovich, V. N. Troshchieva, and D. V. Vysotsky, “Mode competition in multi-core fiber amplifier,” Opt. Commun. **277**(2), 390–396 (2007). [CrossRef]

22. B. Zhu, T. F. Taunay, M. Fishteyn, X. Liu, S. Chandrasekhar, M. F. Yan, J. M. Fini, E. M. Monberg, and F. V. Dimarcello, “112-Tb/s space-division multiplexed DWDM transmission with 14-b/s/Hz aggregate spectral efficiency over a 76.8-km seven-core fiber,” Opt. Express **19**(17), 16665–16671 (2011). [CrossRef] [PubMed]

5. M. Koshiba, K. Saitoh, and Y. Kokubun, “Heterogeneous multicore fibres: proposal and design principle,” IEICE Electron. Express **6**(2), 98–103 (2009). [CrossRef]

28. N. Kishi, E. Yamashita, and K. Atsuki, “Modal and coupling-field analysis of optical fibers with circularly distributed multiple cores and a central core,” J. Lightwave Technol. **4**(8), 991–996 (1986). [CrossRef]

## 2. Coupled-mode formalism

*n*non-identical cores (labeled core 1, 2, …,

*n*) arbitrarily embedded near the center of the cladding. The

*p*

^{th}core is identified by its radius and refractive index

*a*and

_{p}*n*

_{1}

*respectively while the cladding has a refractive index of*

_{p}*n*

_{2}. The cores are spatially positioned such that the field from each core is relatively well isolated and perturbation methods with conventional coupled-mode theory can be used to analyze the mode coupling dynamics [29

29. A. W. Snyder, “Coupled-mode theory for optical fibers,” J. Opt. Soc. Am. **62**(11), 1267–1277 (1972). [CrossRef]

_{01}fundamental mode and we denote the amplitude of the LP

_{01}mode of the

*p*

^{th}core as

*A*(

_{p}*z*). The simultaneous mode coupling between all the cores of a MCF is governed by a set of coupled-mode equations [29

29. A. W. Snyder, “Coupled-mode theory for optical fibers,” J. Opt. Soc. Am. **62**(11), 1267–1277 (1972). [CrossRef]

31. H. A. Haus and L. Molter-Orr, “Coupled multiple waveguide systems,” IEEE J. Quantum Electron. **19**(5), 840–844 (1983). [CrossRef]

**A**(

*z*) = [

*A*

_{1}(

*z*)

*A*

_{2}(

*z*) …

*A*(

_{n}*z*)]

^{T}is a column vector and T denotes the transpose,

*z*is the direction of propagation, and

**C**is a

*n*×

*n*matrix with elements

*c*given bywhere

_{pq}*β*represents the propagation constant for the LP

_{p}_{01}mode of core

*p*. The coupling coefficient

*C*is a measure of the spatial overlapping of the mode fields of core

_{pq}*p*and

*q*over the cross-sectional area of core

*q*. Using the addition theorem [32] to express the mode field of core

*p*in terms of the local coordinate system of core

*q*and with the help of the eigenvalue equation for a step-index optical fiber,

*C*can be obtained analytically as

_{pq}*J*,

_{l}*I*, and

_{l}*K*are the Bessel function of the first kind and the modified Bessel functions of the first and second kinds of order

_{l}*l*respectively and

*d*is the distance between the centers of core

_{pq}*p*and

*q*. The normalized fiber parameters

*U*,

_{p}*V*, and

_{p}*W*are defined as

_{p}*U*=

_{p}*a*[(2π

_{p}*n*

_{1}

*/*

_{p}*λ*)

^{2}−

*β*

_{p}^{2}]

^{1/2},

*V*=

_{p}*a*(2π/

_{p}*λ*)(

*n*

_{1}

_{p}^{2}−

*n*

_{2}

^{2})

^{1/2}, and

*W*=

_{p}*a*[

_{p}*β*

_{p}^{2}− (2π

*n*

_{2}/

*λ*)

^{2}]

^{1/2}with

*λ*being the free-space wavelength. The index contrast Δ

*= (*

_{p}*n*

_{1}

_{p}^{2}–

*n*

_{2}

^{2})/(2

*n*

_{1}

_{p}^{2}) can be approximated as (

*n*

_{1}

*–*

_{p}*n*

_{2})/

*n*

_{1}

*which corresponds to the relative core-cladding index difference. The electric field of the individual core, denoted as*

_{p}*E*(

_{p}*z*), can be expressed in terms of its mode amplitude

*A*(

_{p}*z*) as

**C**are

*z*-dependent. Using Eq. (4), we can translate Eq. (1) to an eigenvalue problemwhere

**E**(

*z*) = [

*E*

_{1}(

*z*)

*E*

_{2}(

*z*) …

*E*(

_{n}*z*)]

^{T}and

**R**contains

*z*-independent elements

*r*given by

_{pq}**E**(

*z*) = exp(−

**R**

*z*)

**E**(0), the solution to Eq. (5) is obtained asandwhere δ

*is the Kronecker delta function,*

_{pq}*γ*is an eigenvalue of

_{p}**R**, and

**v**

*is the corresponding eigenvector. Equation (7) is a generalized solution for an*

_{p}*n*-core MCF that describes the power exchange between the modes of individual cores as the light propagates.

*n*

_{2}= 1.4440 (pure silica at 1.550 μm) and identical core radii

*a*= 4.5 μm for all

_{p}*p*= 1, 2, …, 7. Of course, the analytical derivations can be well applied to other 7-core MCFs reported in recent experiments with various geometry and index profiles [33, 34].

## 3. Analysis of mode coupling dynamics in multi-core fibers

### 3.1 Homogeneous 7-core MCF

*= Δ for all*

_{p}*p*, i.e. a homogeneous 7-core MCF as shown in Fig. 1 . The coupled-mode equations are given by Eq. (1) with the matrix

**C**into

**R**using Eq. (6) yields a sequence of eigenvalues in increasing mode order given byand

*C*is given by Eq. (3). When light is launched into core 1, i.e.

_{pq}*A*

_{1}(0) = 1 and

*A*(0) = 0 for

_{p}*p*≠ 1, analytical solutions for the mode amplitudes at distance

*z*are obtained asandwhile the normalized mode powers |

*A*(

_{p}*z*)|

^{2}can be expressed asandwhich are periodic in

*z*. The minima of |

*A*

_{1}(

*z*)|

^{2}are located at the maxima of |

*A*(

_{p}*z*)|

^{2}. According to Eq. (13), the normalized mode power drops to its first minimum at a distance

**ℒ**

_{c1 }given bywhich is shorter than that of a 2-core model by a factor of

*A*

_{2}(0) = 1 and

*A*(0) = 0 for

_{p}*p*≠ 2, the mode amplitudes for various cores are given by and

28. N. Kishi, E. Yamashita, and K. Atsuki, “Modal and coupling-field analysis of optical fibers with circularly distributed multiple cores and a central core,” J. Lightwave Technol. **4**(8), 991–996 (1986). [CrossRef]

29. A. W. Snyder, “Coupled-mode theory for optical fibers,” J. Opt. Soc. Am. **62**(11), 1267–1277 (1972). [CrossRef]

37. 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]

**ℒ**

_{cp}, defined as the propagation length at which the normalized mode power in the

*p*

^{th}launching core drops from one to its first minimum. This definition is general as it incorporates both aperiodic and periodic coupling dynamics. The generalized coupling length

**ℒ**

_{c2}can be calculated numerically from Eq. (18) by setting its first derivative

*C*

_{12}) and 1/7, 0.6622/

*C*

_{12}and 0.1681, 1.024/

*C*

_{12}and 0.1927, and 1.550/

*C*

_{12}and 0.6187 for core 1, 3, 4, and 5, respectively. The generalized coupling length serves as a measure for evaluating the coupling strength of composite waveguides such as MCFs. The dependency of the generalized coupling lengths

**ℒ**

_{c1}and

**ℒ**

_{c2}on the relative index difference Δ is shown in Fig. 3 for different pitch values Λ. The normalized mode powers at

**ℒ**

_{c1}and

**ℒ**

_{c2}are obtained from Eqs. (16) and (23), respectively. As shown in Fig. 3,

**ℒ**

_{c1}is smaller than

**ℒ**

_{c2}and both of them increase with Δ or Λ.

### 3.2 Heterogeneous 7-core MCF with three types of cores

_{2}= Δ

_{4}= Δ

_{6}, Δ

_{3}= Δ

_{5}= Δ

_{7}, and Δ

_{1}≠ Δ

_{2}≠ Δ

_{3}as shown in Fig. 4 , which is a configuration that has attracted most of the attention to date [5

5. M. Koshiba, K. Saitoh, and Y. Kokubun, “Heterogeneous multicore fibres: proposal and design principle,” IEICE Electron. Express **6**(2), 98–103 (2009). [CrossRef]

**C**is given by

*β*=

_{pq}*β*−

_{p}*β*. The eigenvalues can be calculated explicitly aswhere

_{q}**ℒ**

_{c3}, and the modal power |

*A*

_{3}(

**ℒ**

_{c3})|

^{2}can be obtained from Eqs. (31), (32), and (33) respectively by switching Δ

_{2}with Δ

_{3},

*A*

_{2}(

*z*) with

*A*

_{3}(

*z*),

*A*

_{4}(

*z*) with

*A*

_{5}(

*z*),

*A*

_{6}(

*z*) with

*A*

_{7}(

*z*), and

**ℒ**

_{c2}with

**ℒ**

_{c3}. Light launching into core 1 simply does not couple out i.e. |

*A*

_{1}(

*z*)|

^{2}= 1 and |

*A*(

_{p}*z*)|

^{2}= 0 for

*p*≠ 1. The propagation dynamics of a heterogeneous 7-core MCF with Δ

_{1}= 0.370%, Δ

_{2}= Δ

_{4}= Δ

_{6}= 0.325%, and Δ

_{3}= Δ

_{5}= Δ

_{7}= 0.360% are shown in Fig. 5 for the case when light is launched into core 2 along with that of a homogeneous 3-core MCF (assuming cores 1, 3, 5, and 7 are absent) for comparison. The generalized coupling lengths for the two cases are 9.38 and 1.14 km respectively, indicating the strong effect of cores 1, 3, 5, and 7 on coupling dynamics of core 2, 4, and 6.

**ℒ**

_{c2}and

**ℒ**

_{c3}on Δ

_{2}is shown in Fig. 6 for various combinations of Δ

_{1}and Δ

_{3}. From the figure, the generalized coupling lengths can increase substantially as the eigenvalues in Eq. (32) become close to each other, which can be identified from the peaks in Fig. 6. This can be understood as the case when the propagation constants of different eigenmodes of the composite MCF structure become equal. On the other hand, the dips in Fig. 6 correspond to cases where Δ

_{1}= Δ

_{2}, Δ

_{1}= Δ

_{3}, and/or Δ

_{2}= Δ

_{3}for which the generalized coupling lengths drop because of enhanced coupling between core groups.

## 4. Communication strategies and transmission performance for 7-core MCFs in the presence of mode coupling

*z*is given by

**y**= [

*y*

_{1}

*y*

_{2}

*y*

_{3}

*y*

_{4}

*y*

_{5}

*y*

_{6}

*y*

_{7}]

^{T}can be expressed aswhere

**υ**= [

*υ*

_{1}

*υ*

_{2}

*υ*

_{3}

*υ*

_{4}

*υ*

_{5}

*υ*

_{6}

*υ*

_{7}]

^{T}are independent identically distributed (i.i.d) additive white Gaussian noise (AWGN) that collectively model shot noise and thermal noise from the photo receivers. In principle, one can equalize the channel and obtain a joint estimate of the transmitted signal

**T**is unitary,

**W**

^{H}**υ**are i.i.d. AWGN with the same covariance matrix as

**υ**and hence the effects of cross-talks or mode coupling can be perfectly compensated. In practice however, the coupling dynamics will be distorted by bending and other fabrication imperfections such that

**W**

*may not be known. In this case, adaptive signal processing techniques such as the least mean squares (LMS) algorithm will most likely be used to ‘learn’ the channel and mitigate cross-talks. In particular,*

^{H}*N*sets of training data

*n +*1

^{th}update of

**W**is given by [38]where

*μ*is the step size of the LMS update,

**y**

^{(}

^{n}^{)}is the

*n*

^{th}received vector and

*e*(

*n*) is the error vector. For a signal-to-noise ratio (SNR) of 20 dB, the convergence behavior of the LMS algorithm for a homogeneous 7-core fiber using quadrature phase-shift-keying (QPSK) modulation format are shown in Fig. 7 with different step sizes. The propagation distance is

*z*=

**ℒ**

_{c}and the initial state is set to be

**W**

^{(1)}=

**I**. It can be seen that when the step size is large enough, convergence can be achieved with around 100 to 200 symbols. Further simulation results suggest that the speed of convergence stays more or less the same for

*z*> 0.5

**ℒ**

_{c}.

**E**(0) i.e.

**E**(0) ∈ {0, 1}

^{7}, the output signal is given bywhere

**t**

*is the*

_{p}*p*

^{th}row of

**T**. Unfortunately, due to the magnitude squared operation by the photo-diode, linear equalization techniques cannot fully equalize or eliminate the cross-talks. Figure 8 shows the convergence behavior of the LMS algorithm for an OOK system with transmission distance

*z*=

**ℒ**

_{c}. The SNR is 20dB and the initial state of the equalization filter is set to be

**W**

^{(1)}=

**I**. It is obvious from the convergence behavior that only a small amount of cross-talk can be mitigated by the LMS algorithm and subsequent transmission performance using linear equalization is far from that of a cross-talk-free system. Further simulation results with larger number of LMS iterations and smaller step sizes do not help reduce the cross-talks.

*z*and

**T**and such constellations are irregular in general. In this case, joint Maximum-likelihood detection on the received signals

**y**can be described as

*z*. From the figure, it can be seen that the BER lower bounds are aperiodic with

*z*due to the aperiodic coupling dynamics of homogeneous 7-core MCF described in Section 3A. Starting from

*z*and approximately stays constant for

*z*>

**ℒ**

_{c1}and the performance does not approach back to that at

## 5. Conclusions

## Appendix

_{2}= Δ

_{4}and Δ

_{3}= Δ

_{5}. The eigenvalues of this 4-core MCF are analytically given bywhere

*ξ*=

*C*

_{23}

*C*

_{32}/(

*α*

_{22}−

*α*

_{33})

^{2}<< 1 i.e. when the non-identical cores are well-isolated,

*α*

_{22}−

*α*

_{33})[1 + 2

*ξ*] for

*α*

_{22}>

*α*

_{33}and

*α*

_{22}−

*α*

_{33})[1 + 2

*ξ*] for

*α*

_{22}<

*α*

_{33}. In this case, Eq. (46) can be written as

*ξ*and

*ξ*is small, the coupling dynamics within cores 2 and 4 depend on

*ξ*and

*C*

_{24}/(

*α*

_{22}−

*α*

_{33}) which is typically a small value as well. In this case,

*ξ*will not be negligible if such two terms are of the same order of magnitude. Equation (49) can be re-written asyieldingand

*jγ*is the propagation constant for the

_{p}*p*

^{th}eigenmode, one can deduce that power transfer characteristics for composite waveguides depend on the differences between the propagation constants of the eigenmodes not the propagation constants of individual core modes. The use of eigenmodes for analyzing signal transmission in composite fiber structures has also been reported recently [39

39. C. Xia, N. Bai, I. Ozdur, X. Zhou, and G. Li, “Supermodes for optical transmission,” Opt. Express **19**(17), 16653–16664 (2011). [CrossRef] [PubMed]

*ξ*is so small such that

*ξ*<<

*|C*

_{24}/(

*α*

_{22}−

*α*

_{33})| << 1, |

*A*

_{2}(

*z*)|

^{2}≅ cos

^{2}(

*C*

_{24}

*z*) i.e. the coupling dynamics truly approach to that of the corresponding 2-core system.

## Acknowledgments

## References and links

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**OCIS Codes**

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

(060.2310) Fiber optics and optical communications : Fiber optics

(060.2400) Fiber optics and optical communications : Fiber properties

(060.2430) Fiber optics and optical communications : Fibers, single-mode

(060.4005) Fiber optics and optical communications : Microstructured fibers

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: October 10, 2011

Revised Manuscript: December 20, 2011

Manuscript Accepted: February 1, 2012

Published: February 9, 2012

**Citation**

Florence Y. M. Chan, Alan Pak Tao Lau, and Hwa-Yaw Tam, "Mode coupling dynamics and communication strategies for multi-core fiber systems," Opt. Express **20**, 4548-4563 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-4-4548

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