## Random coupling between groups of degenerate fiber modes in mode multiplexed transmission |

Optics Express, Vol. 21, Issue 8, pp. 9484-9490 (2013)

http://dx.doi.org/10.1364/OE.21.009484

Acrobat PDF (941 KB)

### Abstract

We study random coupling induced crosstalk between groups of degenerate modes in spatially multiplexed optical transmission. Our analysis shows that the average crosstalk is primarily determined by the wavenumber mismatch, by the correlation length of the random perturbations, and by the coherence length of the degenerate modes, whereas the effect of a deterministic group velocity difference is negligible. The standard deviation of the crosstalk is shown to be comparable to its average value, implying that crosstalk measurements are inherently noisy.

© 2013 OSA

## 1. Introduction

1. P. J. Winzer and G. J. Foschini, “MIMO capacities and outage probabilities in spatially multiplexed optical transport systems,” Opt. Express **19**, 16680–16696 (2011) [CrossRef] [PubMed] .

2. K. P. Ho and J. M. Kahn, “Statistics of group delays in multi-mode fibers with strong mode coupling,” J. Light-wave Technol. **29**, 3119–3128 (2011) [CrossRef] .

3. R. Ryf, S. Randel, A. H. Gnauck, C. Bolle, A. Sierra, S. Mumtaz, M. Esmaeelpour, E. C. Burrows, R. J. Essiambre, P. J. Winzer, D. W. Peckham, A. H. McCurdy, and R. Lingle, “Mode-division multiplexing over 96 km of few-mode fiber using coherent 6 × 6 MIMO processing,” J. Lightwave Technol. **30**, 521–531 (2012) [CrossRef] .

4. C. Koebele, M. Salsi, D. Sperti, P. Tran, P. Brindel, H. Mardoyan, S. Bigo, A. Boutin, F. Verluise, P. Sillard, M. Astruc, L. Provost, F. Cerou, and G. Charlet, “Two mode transmission at 2x100Gb/s, over 40km-long prototype few-mode fiber, using LCOS-based programmable mode multiplexer, and demultiplexer.” Opt. Express **19**, 16593–16600 (2011) [CrossRef] [PubMed] .

5. D. Gloge, “Weakly guiding fibers,” Appl. Opt. **10**, 2252–2258 (1971) [CrossRef] [PubMed] .

## 2. Analysis

7. C. Antonelli, A. Mecozzi, M. Shtaif, and P. J. Winzer, “Stokes-space analysis of modal dispersion in fibers with multiple mode transmission,” Opt. Express **20**, 11718–11733 (2012) [CrossRef] [PubMed] .

*N*orthogonal spatial modes is described by the equation where 2

*N*is the total number of propagation modes (with the factor of 2 accounting for polarizations), and

*E⃗*is a 2

*N*-dimensional column vector whose components are the complex envelopes of the electric field in the individual modes. The overall number of modes is

*g*is the degeneracy of the

_{j}*j*-th group of modes and

*M*is the number of groups. For example, in a weakly-guiding step index fiber supporting the first two LP mode groups, we have

*N*= 3,

*M*= 2,

*g*

_{1}= 1, and

*g*

_{2}= 2. The 2

*N*× 2

*N*matrix

**B**describes propagation in the absence of perturbations, where the modes are uncoupled. Its only nonzero elements are on the main diagonal and they equal the wavenumbers of the various modes [8

8. The deterministic coupling between members of an LP mode group [9] is not included in **B** as its effect is assumed to be masked by the presence of the random perturbations which are represented by the second term in the square brackets of Eq. (1). This is consistent with the fact that strong random coupling between the constituent pseudo-modes is observed in experiments.

*N*

^{2}− 1 real-valued components and it generalizes the familiar birefringence vector used in the modeling of polarization effects in single mode fibers [10

10. J. P. Gordon and H. Kogelnik, “PMD fundamentals: polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. USA **97**, 4541–4550 (2000) [CrossRef] [PubMed] .

7. C. Antonelli, A. Mecozzi, M. Shtaif, and P. J. Winzer, “Stokes-space analysis of modal dispersion in fibers with multiple mode transmission,” Opt. Express **20**, 11718–11733 (2012) [CrossRef] [PubMed] .

*E⃗*. The term Λ⃗

^{(2N)}is the generalized Pauli matrix vector [7

7. C. Antonelli, A. Mecozzi, M. Shtaif, and P. J. Winzer, “Stokes-space analysis of modal dispersion in fibers with multiple mode transmission,” Opt. Express **20**, 11718–11733 (2012) [CrossRef] [PubMed] .

*N*× 2

*N*traceless Hermitian matrices Λ

*, which reduce to the standard Pauli matrices [10*

_{i}10. J. P. Gordon and H. Kogelnik, “PMD fundamentals: polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. USA **97**, 4541–4550 (2000) [CrossRef] [PubMed] .

*N*= 1). These matrices constitute a basis for the space of 2

*N*× 2

*N*traceless Hermitian matrices. The scalar product

*E⃗*into vectors

*E⃗*of smaller dimension 2

_{j}*g*, whose components are the field envelopes of the degenerate members of the

_{j}*j*-th group. In this representation, the first 2

*g*

_{1}terms of the main diagonal of the matrix

**B**are equal to

*β*

_{1}(the wavenumber of the modes in the first group), the following 2

*g*

_{2}terms are equal to

*β*

_{2}(the wavenumber of the second group) and so on.

**b**

*whose dimension is 2*

_{lj}*g*× 2

_{j}*g*, which can be demonstrated to have the property

_{l}*e⃗*(

_{l}*z*) = exp(−

*iβ*)

_{l}z*E⃗*(

_{l}*z*), Eq. (1) can be rewritten as a set of coupled equations where we defined

*β*=

_{jl}*β*−

_{j}*β*. A formal solution of Eq. (2) is given by where

_{l}**U**

*(*

_{l}*z*, 0) is a unitary matrix satisfying d

**U**

*/d*

_{l}*z*=

*i*

**b**

_{ll}**U**

*/2*

_{l}*g*, and where

_{l}*l*-th group of modes from

*z*′ to

*z*in the absence of random coupling with other groups.

### 2.1. Coupling between groups of modes

*E⃗*(

_{l}*z*) when the input signal is injected only into group

*q*, with

*q*≠

*l*. Since we expect the coupling between different groups to be small due to the large wavenumber difference, we extract

*E⃗*(

_{l}*z*) via a first-order perturbation analysis, which yields The interpretation of Eq. (4) is rather intuitive. The electric field generated in group

*l*at position

*z*′ is proportional to the product of the random coupling matrix

**b**

*(*

_{lq}*z*′) by the field

*e⃗*(

_{q}*z*′) =

**U**

*(*

_{q}*z*′, 0)

*e⃗*(0). The matrix

_{q}**U**

*(*

_{l}*z*,

*z*′) accounts for the propagation of the field in group

*l*from

*z*′ to

*z*and the term exp(

*iβ*′) accounts for the mode-independent phase that it accumulates. In what follows we will assume that the input vector is normalized, so that |

_{ql}z*e⃗*(0)| = 1.

_{q}*q*-th and the

*l*-th groups of modes. When the delay spread in the fiber is dominated by the deterministic walk-off between the groups, the impulse response measured after a distance

*z*will be limited to a time interval whose duration is |

*zβ*

_{ql,1}|, where

*β*

_{ql,1}is the frequency derivative of

*β*at the carrier frequency, namely at

_{ql}*ω*= 0. This picture is in agreement, in terms of the general shape of the impulse response, with the experimental results reported in [3

3. R. Ryf, S. Randel, A. H. Gnauck, C. Bolle, A. Sierra, S. Mumtaz, M. Esmaeelpour, E. C. Burrows, R. J. Essiambre, P. J. Winzer, D. W. Peckham, A. H. McCurdy, and R. Lingle, “Mode-division multiplexing over 96 km of few-mode fiber using coherent 6 × 6 MIMO processing,” J. Lightwave Technol. **30**, 521–531 (2012) [CrossRef] .

*β*≃

_{ql}*β*

_{ql,0}+

*β*

_{ql,1}

*ω*and by neglecting the frequency dependence of the perturbation vector

**20**, 11718–11733 (2012) [CrossRef] [PubMed] .

*l*-th group when a signal is transmitted in the

*q*-th group. This quantity is given by where

*P̃*(

*ω*) is the input signal spectrum, normalized so that

*u*over fiber realizations.

_{lq}### 2.2. Statistics of mode coupling

*u*〉. This involves calculating first 〈|

_{lq}*e⃗*(

_{l}*z*,

*ω*)|

^{2}〉, which in turn involves calculating

*e⃗*(

_{l}*z*,

*ω*)|

^{2}from Eq. (4). Since the statistics of mode coupling should be stationary with respect to frequency within the telecom bandwidth, we may set the offset from the carrier frequency to zero, namely

*ω*= 0, in

**b**

*and*

_{ll}**b**

*(that determine*

_{qq}**U**

*and*

_{l}**U**

*) are statistically independent of each other [11] and of the coupling blocks*

_{q}**b**

*, the averaging of the inner term*

_{lm}**I**

*, and we may write 〈*

_{l}**U**

*(*

_{l}*z*, 0)〉 = exp(−

*z/L*)

_{l}**I**

*, where we assumed exponential decorrelation of the electric field vector. The physical meaning of the correlation length*

_{l}*L*is seen when expressing the longitudinal autocorrelation function of the field vector as

_{l}*L*as the coherence length of the electric field [12].

_{l}**U**

*. To accomplish this task, we refer the reader to the detailed construction of the Λ⃗*

_{q}^{(2}

^{N}^{)}matrix vector, which is presented in the appendix of [7

**20**, 11718–11733 (2012) [CrossRef] [PubMed] .

**b**

*represent off-diagonal blocks of the matrix*

_{lq}**b**

*are proportional to different components of the vector*

_{lq}**b**

*and between the real and imaginary parts of each element. Denoting by*

_{lq}*f*(

*z*) the autocorrelation function of the components of

*b*(

_{j}*z*′)

*b*(

_{k}*z*″)〉 =

*f*(

*z*′ −

*z*″)

*δ*, we obtain

_{jk}**U**

*, with the result: To gain further insight we assume a particular functional form for the autocorrelation function of the mode coupling perturbations,*

_{l}*f*(

*z*) =

*n*

_{0}exp(−

*z/L*), where

_{c}*n*

_{0}quantifies the strength of the perturbations and

*L*is their correlation length. This is a plausible choice, which is customary in PMD studies [13

_{c}13. A. Galtarossa, L. Palmieri, M. Schiano, and T. Tambosso, “Measurement of birefringence correlation length in long, single-mode fibers,” Opt. Lett. **26**, 962–964 (2001) [CrossRef] .

*K*= 1/

*L*

_{eff}+

*iβ*, with

_{lq}*L*

_{eff}= (1/

*L*+ 1/

_{c}*L*+ 1/

_{l}*L*)

_{q}^{−1}. Note that for large propagation distances, the fractional expression inside the integral in Eq. (7) can be approximated by

*z/K*, which shows that the degree of coupling can be reduced by increasing the wavenumber mismatch. Expanding Eq. (7) to fourth order in

*β*≃

_{lq}*β*

_{lq,0}+

*ωβ*

_{lq,1}, yields where

*ε*is a correction term which turns out to be negligibly small for realistic fiber parameters. For example, assuming

_{lq}*ω*

_{rms}/2

*π*= 20 GHz, we obtain

*β*

_{lq,1}= 20 ps/km [3

3. R. Ryf, S. Randel, A. H. Gnauck, C. Bolle, A. Sierra, S. Mumtaz, M. Esmaeelpour, E. C. Burrows, R. J. Essiambre, P. J. Winzer, D. W. Peckham, A. H. McCurdy, and R. Lingle, “Mode-division multiplexing over 96 km of few-mode fiber using coherent 6 × 6 MIMO processing,” J. Lightwave Technol. **30**, 521–531 (2012) [CrossRef] .

*β*

_{lq,1}= 4.35 ns/km [4

4. C. Koebele, M. Salsi, D. Sperti, P. Tran, P. Brindel, H. Mardoyan, S. Bigo, A. Boutin, F. Verluise, P. Sillard, M. Astruc, L. Provost, F. Cerou, and G. Charlet, “Two mode transmission at 2x100Gb/s, over 40km-long prototype few-mode fiber, using LCOS-based programmable mode multiplexer, and demultiplexer.” Opt. Express **19**, 16593–16600 (2011) [CrossRef] [PubMed] .

*β*

_{lq,0}is by orders of magnitude larger than 1 m

^{−1}. This means that with a large enough wavenumber mismatch the group velocity difference is expected to have a negligible effect on system performance [14].

*σ*

_{ulq}. Inspection of Eq. (4) suggests that, if the propagation distance is much longer than the correlation length of

**b**

*, then the output field is the sum of many independent, random contributions, and hence by the central-limit theorem*

_{lq}*e⃗*(

_{l}*z*) becomes a Gaussian vector with 2

*g*independent

_{l}*complex*components. The crosstalk

*u*is proportional to the frequency integral of |

_{lq}*e⃗*(

_{l}*z*)|

^{2}and its statistical properties are affected by the frequency-dependence of the coupling. In the special case where fiber characterization is performed with a continuous-wave (CW) signal,

*u*can be approximated by a chi-squared distributed variable with 4

_{lq}*g*degrees of freedom and standard deviation given by With the characteristic values of

_{l}*g*in the majority of relevant fiber structures (in the case of few mode fibers in the weakly guiding approximation, the largest value of

_{l}*g*is 2), this implies that measurements of crosstalk are necessarily noisy, a fact that needs to be properly taken into account in practical considerations.

_{l}## 3. Results

*L*is much shorter than in reality and equal to only fifteen beat lengths (

_{c}*L*=

_{l}*L*=

_{q}*L*/10. We assumed degeneracy factors of

_{c}*g*= 1 and

_{q}*g*= 2, for the two groups of modes and the perturbation vector

_{l}*z*. It can be shown that their power spectral density (which is constant) is equal to

*L*and with a variance

_{c}*n*

_{0}which was chosen such that a coupling of ∼ 5 × 10

^{−2}is predicted by Eq. (8) at

*L*= 10

*L*.

_{c}*u*〉 as a function of propagation distance, assuming transmission of a CW signal. The thick solid lines (black) represent the full expression Eq. (7) and the thin lines (red) represent the results of Monte Carlo simulations with 50,000 fiber realizations. The dashed lines represent the simplified expression Eq. (8). The excellent agreement between the analytical solutions and the simulations is evident. The small deviation observed when the crosstalk level rises towards 5% represents the saturation of the first order analysis that we used. We note that the accuracy of the analytical results depends only on the overall level of coupling and not on the exact parameter combinations. To demonstrate this, the dotted green curve in Fig. 1(a), which overlaps with the red curve, was calculated with different parameters;

_{lq}*L*=

_{l}*L*=

_{q}*L*/5. The shaded area surrounding the curves marks one standard deviation as given by Eq. (9). Figure 1(b) shows the standard deviation

_{c}*σ*as a function of propagation distance. Once again, the thick black and the thin red lines represent the analytical expression Eq. (9) and the numerical results, respectively. In Fig. 1(c) we plot the crosstalk probability density function for the displayed values of the propagation distance. Symbols represent Monte Carlo simulations, solid lines are the plot of a chi-squared distribution with 4

_{ulq}*g*

_{2}= 8 degrees of freedom and mean value given by Eq. (7).

## 4. Conclusions

## Acknowledgment

## References and links

1. | P. J. Winzer and G. J. Foschini, “MIMO capacities and outage probabilities in spatially multiplexed optical transport systems,” Opt. Express |

2. | K. P. Ho and J. M. Kahn, “Statistics of group delays in multi-mode fibers with strong mode coupling,” J. Light-wave Technol. |

3. | R. Ryf, S. Randel, A. H. Gnauck, C. Bolle, A. Sierra, S. Mumtaz, M. Esmaeelpour, E. C. Burrows, R. J. Essiambre, P. J. Winzer, D. W. Peckham, A. H. McCurdy, and R. Lingle, “Mode-division multiplexing over 96 km of few-mode fiber using coherent 6 × 6 MIMO processing,” J. Lightwave Technol. |

4. | C. Koebele, M. Salsi, D. Sperti, P. Tran, P. Brindel, H. Mardoyan, S. Bigo, A. Boutin, F. Verluise, P. Sillard, M. Astruc, L. Provost, F. Cerou, and G. Charlet, “Two mode transmission at 2x100Gb/s, over 40km-long prototype few-mode fiber, using LCOS-based programmable mode multiplexer, and demultiplexer.” Opt. Express |

5. | D. Gloge, “Weakly guiding fibers,” Appl. Opt. |

6. | R. Ryf, R. J. Essiambre, S. Randel, M. A. Mestre, C. Schmidt, and P. J. Winzer, “Impulse response analysis of coupled-core 3-core fibers,” Proceedings of ECOC 2012, Paper Mo.1.F (2012). |

7. | C. Antonelli, A. Mecozzi, M. Shtaif, and P. J. Winzer, “Stokes-space analysis of modal dispersion in fibers with multiple mode transmission,” Opt. Express |

8. | The deterministic coupling between members of an LP mode group [9] is not included in |

9. | H. Kogelnik and P. J. Winzer, “Modal birefringence in weakly guiding fibers,” J. Lightwave Technol. |

10. | J. P. Gordon and H. Kogelnik, “PMD fundamentals: polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. USA |

11. | Strictly speaking the matrices U are independent up to a scalar phase factor that is negligible as compared to the phase difference due to the deterministic wavenumber mismatch._{q} |

12. | The quantity that we refer to as the |

13. | A. Galtarossa, L. Palmieri, M. Schiano, and T. Tambosso, “Measurement of birefringence correlation length in long, single-mode fibers,” Opt. Lett. |

14. | J. Vuong, P. Ramantanis, A. Seck, D. Bendimerad, and Y. Frignac, “Understanding discrete linear mode coupling in few-mode fiber transmission systems,” Proceedings of ECOC 2011, Paper Tu.5.B.2 (2011). |

15. | C. W. Gardiner, |

**OCIS Codes**

(060.2330) Fiber optics and optical communications : Fiber optics communications

(060.4230) Fiber optics and optical communications : Multiplexing

(060.4510) Fiber optics and optical communications : Optical communications

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: January 4, 2013

Revised Manuscript: February 18, 2013

Manuscript Accepted: March 4, 2013

Published: April 9, 2013

**Citation**

Cristian Antonelli, Antonio Mecozzi, Mark Shtaif, and Peter J. Winzer, "Random coupling between groups of degenerate fiber modes in mode multiplexed transmission," Opt. Express **21**, 9484-9490 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-8-9484

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### References

- P. J. Winzer and G. J. Foschini, “MIMO capacities and outage probabilities in spatially multiplexed optical transport systems,” Opt. Express19, 16680–16696 (2011). [CrossRef] [PubMed]
- K. P. Ho and J. M. Kahn, “Statistics of group delays in multi-mode fibers with strong mode coupling,” J. Light-wave Technol.29, 3119–3128 (2011). [CrossRef]
- R. Ryf, S. Randel, A. H. Gnauck, C. Bolle, A. Sierra, S. Mumtaz, M. Esmaeelpour, E. C. Burrows, R. J. Essiambre, P. J. Winzer, D. W. Peckham, A. H. McCurdy, and R. Lingle, “Mode-division multiplexing over 96 km of few-mode fiber using coherent 6 × 6 MIMO processing,” J. Lightwave Technol.30, 521–531 (2012). [CrossRef]
- C. Koebele, M. Salsi, D. Sperti, P. Tran, P. Brindel, H. Mardoyan, S. Bigo, A. Boutin, F. Verluise, P. Sillard, M. Astruc, L. Provost, F. Cerou, and G. Charlet, “Two mode transmission at 2x100Gb/s, over 40km-long prototype few-mode fiber, using LCOS-based programmable mode multiplexer, and demultiplexer.” Opt. Express19, 16593–16600 (2011). [CrossRef] [PubMed]
- D. Gloge, “Weakly guiding fibers,” Appl. Opt.10, 2252–2258 (1971). [CrossRef] [PubMed]
- R. Ryf, R. J. Essiambre, S. Randel, M. A. Mestre, C. Schmidt, and P. J. Winzer, “Impulse response analysis of coupled-core 3-core fibers,” Proceedings of ECOC 2012, Paper Mo.1.F (2012).
- C. Antonelli, A. Mecozzi, M. Shtaif, and P. J. Winzer, “Stokes-space analysis of modal dispersion in fibers with multiple mode transmission,” Opt. Express20, 11718–11733 (2012). [CrossRef] [PubMed]
- The deterministic coupling between members of an LP mode group [9] is not included in B as its effect is assumed to be masked by the presence of the random perturbations which are represented by the second term in the square brackets of Eq. (1). This is consistent with the fact that strong random coupling between the constituent pseudo-modes is observed in experiments.
- H. Kogelnik and P. J. Winzer, “Modal birefringence in weakly guiding fibers,” J. Lightwave Technol.30, 2240–2243 (2012). [CrossRef]
- J. P. Gordon and H. Kogelnik, “PMD fundamentals: polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. USA97, 4541–4550 (2000). [CrossRef] [PubMed]
- Strictly speaking the matrices Ul and Uq are independent up to a scalar phase factor that is negligible as compared to the phase difference due to the deterministic wavenumber mismatch.
- The quantity that we refer to as the coherence length of the field describes the propagation distance along which the field decorrelates due to the fiber perturbations. It is not related to the coherence length of the light source.
- A. Galtarossa, L. Palmieri, M. Schiano, and T. Tambosso, “Measurement of birefringence correlation length in long, single-mode fibers,” Opt. Lett.26, 962–964 (2001). [CrossRef]
- J. Vuong, P. Ramantanis, A. Seck, D. Bendimerad, and Y. Frignac, “Understanding discrete linear mode coupling in few-mode fiber transmission systems,” Proceedings of ECOC 2011, Paper Tu.5.B.2 (2011).
- C. W. Gardiner, Stochastic methods for physics, chemistry and natural sciences (Springer-Verlag, NY, 1983).

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