## Expressions for the nonlinear transmission performance of multi-mode optical fiber |

Optics Express, Vol. 21, Issue 19, pp. 22834-22846 (2013)

http://dx.doi.org/10.1364/OE.21.022834

Acrobat PDF (682 KB)

### Abstract

We develop an analytical theory which allows us to identify the information spectral density limits of multimode optical fiber transmission systems. Our approach takes into account the Kerr-effect induced interactions of the propagating spatial modes and derives closed-form expressions for the spectral density of the corresponding nonlinear distortion. Experimental characterization results have confirmed the accuracy of the proposed models. Application of our theory in different FMF transmission scenarios has predicted a ~10% variation in total system throughput due to changes associated with inter-mode nonlinear interactions, in agreement with an observed 3dB increase in nonlinear noise power spectral density for a graded index four LP mode fiber.

© 2013 OSA

## 1. Introduction

1. P. J. Winzer, “Optical Networking Beyond WDM,” IEEE Photon. J. **4**(2), 647–651 (2012). [CrossRef]

4. F. Poletti, N. V. Wheeler, M. N. Petrovich, N. Baddela, E. N. Fokoua, J. R. Hayes, D. R. Gray, Z. Li, R. Slavík, and D. J. Richardson, “Towards high-capacity fibre optic communications at the speed of light in vacuum,” Nat. Photonics **7**(4), 279–284 (2013). [CrossRef]

5. V. A. J. M. Sleiffer, Y. Jung, V. Veljanovski, R. G. H. van Uden, M. Kuschnerov, H. Chen, B. Inan, L. G. Nielsen, Y. Sun, D. J. Richardson, S. U. Alam, F. Poletti, J. K. Sahu, A. Dhar, A. M. J. Koonen, B. Corbett, R. Winfield, A. D. Ellis, and H. de Waardt, “73.7 Tb/s (96 x 3 x 256-Gb/s) mode-division-multiplexed DP-16QAM transmission with inline MM-EDFA,” Opt. Express **20**(26), B428–B438 (2012). [CrossRef] [PubMed]

6. Y. Yang, Y. Yan, N. Ahmed, Y. Jeng-Yuan, L. Zhang, Y. Ren, H. Huang, K. M. Birnbaum, B. I. Erkmen, S. Dolinar, M. Tur, and A. E. Willner, “Mode properties and propagation effects of optical orbital angular momentum (OAM) modes in a ring fiber,” IEEE Photon. J. **4**(2), 535–543 (2012). [CrossRef]

7. P. J. Winzer and G. J. Foschini, “MIMO capacities and outage probabilities in spatially multiplexed optical transport systems,” Opt. Express **19**(17), 16680–16696 (2011). [CrossRef] [PubMed]

8. R. Pini, R. Salimbeni, A. F. M. Y. Haider, M. Matera, and C. Lin, “Continuously tunable multiple-order stimulated four-photon mixing in a multimode silica fiber,” Opt. Lett. **9**(3), 79–81 (1984). [CrossRef] [PubMed]

12. R. Essiambre, M. A. Mestre, R. Ryf, A. H. Gnauck, R. W. Tkach, A. R. Chraplyvy, Y. Sun, X. Jiang, and R. Lingle, “Experimental investigation of inter-modal four-wave mixing in few-mode fibers,” IEEE Photon. Technol. Lett. **25**(6), 539–542 (2013). [CrossRef]

13. F. Ferreira, S. Jansen, P. Monteiro, and H. Silva, “Nonlinear semi-analytical model for simulation of few-mode fiber transmission,” IEEE Photon. Technol. Lett. **24**(4), 240–242 (2012). [CrossRef]

16. D. Rafique, S. Sygletos, and A. D. Ellis, “Impact of power allocation strategies in long-haul few-mode fiber transmission systems,” Opt. Express **21**(9), 10801–10809 (2013). [CrossRef] [PubMed]

17. P. P. Mitra and J. B. Stark, “Nonlinear limits to the information capacity of optical fibre communications,” Nature **411**(6841), 1027–1030 (2001). [CrossRef] [PubMed]

19. X. Chen and W. Shieh, “Closed-form expressions for nonlinear transmission performance of densely spaced coherent optical OFDM systems,” Opt. Express **18**(18), 19039–19054 (2010). [CrossRef] [PubMed]

19. X. Chen and W. Shieh, “Closed-form expressions for nonlinear transmission performance of densely spaced coherent optical OFDM systems,” Opt. Express **18**(18), 19039–19054 (2010). [CrossRef] [PubMed]

21. G. Bosco, P. Poggiolini, A. Carena, V. Curri, and F. Forghieri, “Analytical results on channel capacity in uncompensated optical links with coherent detection,” Opt. Express **19**(26), B440–B449 (2011). [CrossRef] [PubMed]

22. P. Poggiolini, A. Carena, V. Curri, G. Bosco, and F. Forghieri, “Analytical Modeling of Non-Linear Propagation in Uncompensated Optical Transmission Links,” IEEE Photon. Technol. Lett. **23**(11), 742–744 (2011). [CrossRef]

23. R. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightwave Technol. **28**(4), 662–701 (2010). [CrossRef]

24. D. Rafique and A. D. Ellis, “Impact of signal-ASE four-wave mixing on the effectiveness of digital back-propagation in 112 Gb/s PM-QPSK systems,” Opt. Express **19**(4), 3449–3454 (2011). [CrossRef] [PubMed]

25. T. Tanimura, M. Nölle, J. K. Fischer, and C. Schubert, “Analytical results on back propagation nonlinear compensator with coherent detection,” Opt. Express **20**(27), 28779–28785 (2012). [CrossRef] [PubMed]

26. G. Rademacher, S. Warm, and K. Petermann, “Analytical description of cross modal nonlinear interaction in mode multiplexed multi-mode fibers,” IEEE Photon. Technol. Lett. **24**(21), 1929–1932 (2012). [CrossRef]

19. X. Chen and W. Shieh, “Closed-form expressions for nonlinear transmission performance of densely spaced coherent optical OFDM systems,” Opt. Express **18**(18), 19039–19054 (2010). [CrossRef] [PubMed]

## 2. Analysis

5. V. A. J. M. Sleiffer, Y. Jung, V. Veljanovski, R. G. H. van Uden, M. Kuschnerov, H. Chen, B. Inan, L. G. Nielsen, Y. Sun, D. J. Richardson, S. U. Alam, F. Poletti, J. K. Sahu, A. Dhar, A. M. J. Koonen, B. Corbett, R. Winfield, A. D. Ellis, and H. de Waardt, “73.7 Tb/s (96 x 3 x 256-Gb/s) mode-division-multiplexed DP-16QAM transmission with inline MM-EDFA,” Opt. Express **20**(26), B428–B438 (2012). [CrossRef] [PubMed]

28. N. Bai, E. Ip, Y.-K. Huang, E. Mateo, F. Yaman, M. J. Li, S. Bickham, S. Ten, J. Liñares, C. Montero, V. Moreno, X. Prieto, V. Tse, K. Man Chung, A. P. Lau, H. Y. Tam, C. Lu, Y. Luo, G. D. Peng, G. Li, and T. Wang, “Mode-division multiplexed transmission with inline few-mode fiber amplifier,” Opt. Express **20**(3), 2668–2680 (2012). [CrossRef] [PubMed]

31. R. Ryf, N. K. Fontaine, M. A. Mestre, S. Randel, X. Palou, C. Bolle, A. H. Gnauck, S. Chandrasekhar, X. Liu, B. Guan, R.-J. Essiambre, P. J. Winzer, S. G. Leon-Saval, J. Bland-Hawthorn, R. Delbue, P. Pupalaikis, A. Sureka, Y. Sun, L. Grüner-Nielsen, R. V. Jensen, and R. Lingle, “12 x 12 MIMO Transmission over 130-km Few-Mode Fiber;” in Proceedings of Frontiers in Optics Conference, 2012 OSA Technical Digest Series (Optical Society of America, 2012), paper FW6C.4.

26. G. Rademacher, S. Warm, and K. Petermann, “Analytical description of cross modal nonlinear interaction in mode multiplexed multi-mode fibers,” IEEE Photon. Technol. Lett. **24**(21), 1929–1932 (2012). [CrossRef]

31. R. Ryf, N. K. Fontaine, M. A. Mestre, S. Randel, X. Palou, C. Bolle, A. H. Gnauck, S. Chandrasekhar, X. Liu, B. Guan, R.-J. Essiambre, P. J. Winzer, S. G. Leon-Saval, J. Bland-Hawthorn, R. Delbue, P. Pupalaikis, A. Sureka, Y. Sun, L. Grüner-Nielsen, R. V. Jensen, and R. Lingle, “12 x 12 MIMO Transmission over 130-km Few-Mode Fiber;” in Proceedings of Frontiers in Optics Conference, 2012 OSA Technical Digest Series (Optical Society of America, 2012), paper FW6C.4.

**18**(18), 19039–19054 (2010). [CrossRef] [PubMed]

*single-mode*transmission system with

*N*identical spans and no dispersion compensation, it has been shown that in the limit of weak nonlinearity the generated nonlinear signal field

_{s}*E*at frequency

_{n}*ω*in mode

_{0}*n*is given by [34, 35

35. D. A. Cleland, A. D. Ellis, and C. H. F. Sturrock, “Precise modelling of four wave mixing products over 400km of step index fibre,” Electron. Lett. **28**(12), 1171–1172 (1992). [CrossRef]

*n*is the nonlinear refractive index,

_{2}*c*the speed of light

*A*the effective area for the interaction between fields at frequencies

_{ijkn}*f*,

_{i,p}*f*,

_{j,q}*f*, and

_{k,r}*f*

_{n}_{,s}where the first subscript denotes the mode, and the second the frequency within that mode.,

*α*represents the loss coefficient,

*L*the span length and

*Δβ*the group velocity mismatch appropriate to the interaction. Generalizing the nonlinear interaction to a multi-mode fiber is straightforward [33] and simply requires identification of the modes and frequencies associated with each of the four interacting waves denoted

_{ijkn,pqrs}*i,j,k*and

*n*and noting that the effective area and group velocity mismatches are given by; Where

*r*and

*θ*are coordinates specifying the position across the transverse field

*E*and

_{x}(r,θ)*β’*the group delay of the

_{x}*x*mode (were x is replaced by

^{th}*i*,

*j*,

*k*or

*n*) . For the purposes of this paper we assume that all modes have identical chromatic dispersion

*β”*and wavelength independent propagation constants

*β*such that the group velocity mismatch is given by;where β” is the group velocity dispersion parameter,

^{~}_{i}*p, q, r*, and

*s*denote the frequency of the components in the

*i*and

^{th}, j^{th}, k^{th}*n*mode respectively. The FWM efficiency over a multi-span system as a function of the spacing between three channels in the same mode is shown by the brown curve in Fig. 1(b). This efficiency curve shows a strongly velocity matched peak for low frequency spacing and rapidly decaying weakly velocity matched peaks comprising both Maker fringes [36

^{th}36. P. D. Maker and R. W. Terhune, “Study of the optical effects due to an induced polarisation third order in the electric field strength,” Phys. Rev. **137**(3A), A801–A818 (1965). [CrossRef]

35. D. A. Cleland, A. D. Ellis, and C. H. F. Sturrock, “Precise modelling of four wave mixing products over 400km of step index fibre,” Electron. Lett. **28**(12), 1171–1172 (1992). [CrossRef]

26. G. Rademacher, S. Warm, and K. Petermann, “Analytical description of cross modal nonlinear interaction in mode multiplexed multi-mode fibers,” IEEE Photon. Technol. Lett. **24**(21), 1929–1932 (2012). [CrossRef]

*B*the total nonlinear noise generated by FWM between a given combination of modes may be calculated by integrating the product of curves such as those shown in Fig. 1 (Eq. (1)) with the signal power spectral density (PSD) in each mode. A closed form solution for this integral was obtained in the case of single mode fiber (corresponding to the case here where β

^{~}

_{n}+ β

^{~}

_{k}-β

^{~}

_{i}+ β

^{~}

_{j}= 0) for an OFDM superchannel with a rectangular spectrum (shown by the black curve in Fig. 1(a)) [19

**18**(18), 19039–19054 (2010). [CrossRef] [PubMed]

*–B/2*to

*B/2*where

*B*represents the bandwidth of the WDM signal. For a few-mode fiber, in order to obtain a closed form expression, the same integral must be performed, but taking into account the VMO (Eq. (5)), and implicitly adopting the same reasonable assumptions as [19

**18**(18), 19039–19054 (2010). [CrossRef] [PubMed]

**18**(18), 19039–19054 (2010). [CrossRef] [PubMed]

*–B/2-Δf*to

_{ijkn}*B/2-Δf*. On one side of the velocity matching peak the integral is truncated, whilst on the other side it is extended, as is apparent by inspection of Fig. 1(b).

_{ijkn}**18**(18), 19039–19054 (2010). [CrossRef] [PubMed]

*η*of:

_{ijkn}*Δf*represents the VMO,

_{ijkn}*A*the effective area of the interaction and

_{ijkn}*ξ*<1 the impact of mode averaging (weakly coupling regime) [14

_{ijkn}14. S. Mumtaz, R.-J. Essiambre, and G. P. Agrawal, “Nonlinear propagation in multimode and multicore fibers: generalisation of the Manakov equations,” J. Lightwave Technol. **31**(3), 398–406 (2013). [CrossRef]

*ξ*for degenerate modes. This closed form analytical expression allows the FWM power generated in mode

_{ijkn}= 2/3*i*originating from signals propagating in modes

*j,k*and

*l*to be calculated by taking the product of this nonlinear efficiency

*η*and the signal power spectral density, of each mode. Direct comparison of Eq. (6) with [19

_{ijkn}**18**(18), 19039–19054 (2010). [CrossRef] [PubMed]

*h*from Eq. (6) has been dropped since its value is approximately unity for all practical few-mode fibers. To simplify analysis in the following sections, we define a summed nonlinear efficiency for the

_{e}*n*mode,

^{th}*a*given by:such that the total nonlinear power generated in the

_{n}*n*mode is given by

^{th}*a*where

_{n}P_{i}P_{j}P_{k}*P*is the signal power spectral density in the appropriate mode.

_{i,j,k}*η*as a function the effective area and VMO. The contour plots show several important features, including (along the x-axis) the expected gradual decay in efficiency as the effective area is increased and the impact of velocity matching. As the VMO increases gradually from zero the nonlinear efficiency decays slightly as Maker fringes fall outside the WDM signal bandwidth. Eventually, when the VMO approaches half the WDM signal bandwidth, the main lobe (see Fig. 1(b)) falls outside the WDM signal bandwidth, inducing a rapid drop in

_{ijkn}*η*In the example shown this results in a sharp discontinuity at a VMO of ± 2.5 THz (signified by the closely spaced contour lines). The frequency offset at which this rapid decay occurs is determined primarily by the signal bandwidth (5 THz in this case), the reminder of the shape is determined by the phase matching parameter

_{ijkn.}*f*and the signal bandwidth. Scaling (normalized in the figure) depends on a number of other fiber parameters, as detailed in Eq. (6). Of course, for any given fiber, only certain combinations of VMO and effective area are possible. The dots in Fig. 2 represent values of

_{w}*η*for different interactions (combinations of modes i,j,k, n) for two specific fibers The dots are color coded according to the mode degraded by the interaction in question (n), the size of the dots are varied simply to enhance visibility of overlapping, or nearly overlapping, points. For each fiber, there are a number of interactions (each with its own effective area and VMO and illustrated by one of the dots) which fall within the high efficiency region (between ± 2.5 THz), including intra-modal effects and partially degenerate inter-mode effects. A number of other FWM interactions clearly exist which are also strongly velocity matched. Comparing the two fibers in Fig. 2, it is apparent that reducing the differential mode delay increases the number of these additional velocity matched interactions. For the higher DMD fiber there are a number of weakly velocity matched interactions (outside the VMO region bounded at ± 2.5 THz), which will be of increased importance if the signal bandwidth

_{ijkn}*B*is extended allowing strong velocity matching at greater VMO (resulting in a movement of the discontinuity in Fig. 2).

## 3. Experimental verification

31. R. Ryf, N. K. Fontaine, M. A. Mestre, S. Randel, X. Palou, C. Bolle, A. H. Gnauck, S. Chandrasekhar, X. Liu, B. Guan, R.-J. Essiambre, P. J. Winzer, S. G. Leon-Saval, J. Bland-Hawthorn, R. Delbue, P. Pupalaikis, A. Sureka, Y. Sun, L. Grüner-Nielsen, R. V. Jensen, and R. Lingle, “12 x 12 MIMO Transmission over 130-km Few-Mode Fiber;” in Proceedings of Frontiers in Optics Conference, 2012 OSA Technical Digest Series (Optical Society of America, 2012), paper FW6C.4.

_{01}and LP

_{11}modes, (giving a total excess coupling loss of 1.5dB). The output comprised coupling lenses, collimator and a phase plate, had an excess loss of ~5dB. The LP

_{01}and LP

_{11}fiber modes were selected using the appropriate phase plate (only one orientation used for LP

_{01}) and the output spectrum recorded using an optical spectrum analyzer connected via a single mode fiber patch cord. The spectrum was used to determine the output power spectral density (and by implication the relative input power spectral density) and the FWM power generated in the central notch. To cover the full measurement the pump power of the second GFA was reduced to achieve a constant power spectral density. Three overlapping sets of data were obtained at different target power spectral densities.

_{01}and LP

_{11}modes, with the inset to Fig. 4(bottom) showing typical recorded output spectra (zoomed in around the central notch) for a range of ASE widths. The inset shows the FWM signal in the gap around 193.79 THz growing as a function of the overall spectral width of the ASE, whilst the input signal amplitude remains approximately constant (see spectral regions between 193.7 and 193.75THz and between 193.83 and 193.9 THz). The resultant data was normalized (arbitrarily) to the efficiency obtained for a 1GHz ASE bandwidth. The results show the expected logarithmic growth trend, punctuated by discrete steps corresponding to the onset of individual inter mode interactions in excellent agreement with theoretical predictions based on summing Eq. (6) over all mode combinations. The fit shown was achieved for VMOs of 0.8, 0.9 and 1.2 ± 0.1 THz (LP

_{01}) and 0.85, 1.2 and 1.35 ± 0.1 THz (LP

_{11}). The fitting parameters suggest inter-mode interaction strengths of around 15% of the intra-mode strength, representing effective areas around 2.5 times greater than the LP

_{01}intra-mode effective area, which is consistent with the overlap integrals for a typical four mode fiber. Of particular significance to the analysis of communication systems, we observe that for signal bandwidth beyond 2.5 THz, the total FWM signal more than 3dB greater than expected from intra-mode nonlinearity alone (purple dotted line). This would be expected to reduce the total information spectral density of a fiber link by approximately 2b/s/Hz in the limit of a high signal to noise ratio (SNR) [3].

## 4. Capacity limits of few mode fiber systems

*η*[19

_{iiii}**18**(18), 19039–19054 (2010). [CrossRef] [PubMed]

38. S. Kilmurray, T. Fehenberger, P. Bayvel, and R. I. Killey, “Comparison of the nonlinear transmission performance of quasi-Nyquist WDM and reduced guard interval OFDM,” Opt. Express **20**(4), 4198–4205 (2012). [CrossRef] [PubMed]

_{NL}). In addition to the assumptions of [19

**18**(18), 19039–19054 (2010). [CrossRef] [PubMed]

14. S. Mumtaz, R.-J. Essiambre, and G. P. Agrawal, “Nonlinear propagation in multimode and multicore fibers: generalisation of the Manakov equations,” J. Lightwave Technol. **31**(3), 398–406 (2013). [CrossRef]

*no*compensation of nonlinearity (inter or intra channel), (3) as a consequence of (2) contributions to the nonlinear noise from interactions between signal and noise may be neglected, (4) that mode coupling equalizes the launch power for each mode and (5) that mode dependent loss is negligible. Considering these assumptions the achievable

*ISD*can be recalculated for a FMF by considering the various interactions which may be velocity-matched within the signal bandwidth and summing over all modes. However, unlike single mode fiber [23

_{NL}23. R. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightwave Technol. **28**(4), 662–701 (2010). [CrossRef]

*ISD*omitting the correction term which accounts for nonlinearity compensation [25

_{NL}25. T. Tanimura, M. Nölle, J. K. Fischer, and C. Schubert, “Analytical results on back propagation nonlinear compensator with coherent detection,” Opt. Express **20**(27), 28779–28785 (2012). [CrossRef] [PubMed]

*P*and

_{s}*P*are the signal and accumulated noise power spectral densities per mode respectively. Equation (6) and Eq. (8) may then be used to calculate the nonlinear information spectral density of a given multi-mode fiber, where

_{N}*P*is the amplified spontaneous emission power spectral density at the end of the link (including a final amplification stage) and

_{N}*P*the signal power spectral density per mode at the output of each amplifier. For large signal to noise ratios, algebraic manipulation of Eq. (8), using the simplification of Eq. (7) enables the relative ISD

_{S}_{NL}of MxM MIMO operation of a multimode fiber to be expressed as:where

*P*is the optimum power spectral density when the fiber is operated in a single LP mode (assuming zero mode coupling),

_{1}*P*the optimum power spectral density when all M modes are utilized and

_{M}*a*and

_{1}*a*the equivalent total nonlinear efficiencies respectively (calculated using Eq. (7)) The optimum power spectral density may be calculated from the analytical solution torecalling that

_{M}*P*represents the accumulated ASE power spectral density. Equation (9) reveals the expected M-fold increase in capacity but with a small reduction from the re-optimization of the optimum signal power spectral density, reduced slightly by the additional nonlinear terms, readily calculated from Eq. (10), and the relative FWM strength

_{N}*a*. The capacity is also directly influenced by the effective area of the fundamental mode through the first term in Eq. (6). Figure 5(a) shows the familiar non-linear information spectral density curve for different four mode fiber designs. For any given fiber, the difference in performance for each mode is small. The fibers represent examples of high and low DMD FMFs with modest fundamental mode effective areas, and a low DMD fiber with a low fundamental mode effective area. Clearly both the overall effective area and the DMD have a direct influence on the maximum capacity of the system. Each mode exhibits a slightly different nonlinear threshold (Eq. (10) calculates the overall optimum launch condition), suggesting that some advantage may be obtained from individually optimizing the relative launch power for each mode [39

_{l}39. D. Rafique, S. Sygletos, and A. D. Ellis, “Impact of power allocation strategies in long-haul few-mode fiber transmission systems,” Opt. Express **21**(9), 10801–10809 (2013). [CrossRef] [PubMed]

7. P. J. Winzer and G. J. Foschini, “MIMO capacities and outage probabilities in spatially multiplexed optical transport systems,” Opt. Express **19**(17), 16680–16696 (2011). [CrossRef] [PubMed]

*P*and

_{1}*a*with the appropriate values for the reference fiber, for example

_{1}*P*and

_{SMF}*a*

_{SMF}.*a*appear as ratios). Of course, the high signal to noise ratio approximation breaks down for the longest transmission distances, as illustrated by the slight convergence of the curves in Fig. 5 (right) beyond 10,000km. We may thus conclude that in the limit of a typical optical signal to noise ratio, the relative performance of different fiber designs depends only on the fiber design itself and on the WDM bandwidth. We may thus optimize a fiber design using the analytical method described here considering only a typical transmission system. A typical design choice is considered in Fig. 6, where we plot the total ISD, at the optimum power spectral density, for a system using a trench assisted graded index FMF of constant core diameter. To obtain the plot, the refractive index curvature and magnitude were varied and we plot the results for values which realized four mode fibers as a function of their maximum DGD (difference between fastest and slowest modes).The capacity clearly reduces monotonically with decreasing DMD favoring large DMD values for optimum nonlinear performance. Since small DMDs are preferred to minimize digital signal processing overheads, the ability to calculate the capacity and optimize the compromise is paramount.

_{M}40. B. Inan, B. Spinnler, F. Ferreira, D. van den Borne, A. Lobato, S. Adhikari, V. A. Sleiffer, M. Kuschnerov, N. Hanik, and S. L. Jansen, “DSP complexity of mode-division multiplexed receivers,” Opt. Express **20**(10), 10859–10869 (2012). [CrossRef] [PubMed]

## 5. Conclusions

## Acknowledgments

## References and links

1. | P. J. Winzer, “Optical Networking Beyond WDM,” IEEE Photon. J. |

2. | D. J. Richardson, “Applied physics. Filling the light pipe,” Science |

3. | A. D. Ellis, “The MODE-GAP project,” in Proceedings of Frontiers in Optics Conference, 2012 OSA Technical Digest Series (Optical Society of America, 2012), paper, FW1D. |

4. | F. Poletti, N. V. Wheeler, M. N. Petrovich, N. Baddela, E. N. Fokoua, J. R. Hayes, D. R. Gray, Z. Li, R. Slavík, and D. J. Richardson, “Towards high-capacity fibre optic communications at the speed of light in vacuum,” Nat. Photonics |

5. | V. A. J. M. Sleiffer, Y. Jung, V. Veljanovski, R. G. H. van Uden, M. Kuschnerov, H. Chen, B. Inan, L. G. Nielsen, Y. Sun, D. J. Richardson, S. U. Alam, F. Poletti, J. K. Sahu, A. Dhar, A. M. J. Koonen, B. Corbett, R. Winfield, A. D. Ellis, and H. de Waardt, “73.7 Tb/s (96 x 3 x 256-Gb/s) mode-division-multiplexed DP-16QAM transmission with inline MM-EDFA,” Opt. Express |

6. | Y. Yang, Y. Yan, N. Ahmed, Y. Jeng-Yuan, L. Zhang, Y. Ren, H. Huang, K. M. Birnbaum, B. I. Erkmen, S. Dolinar, M. Tur, and A. E. Willner, “Mode properties and propagation effects of optical orbital angular momentum (OAM) modes in a ring fiber,” IEEE Photon. J. |

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

8. | R. Pini, R. Salimbeni, A. F. M. Y. Haider, M. Matera, and C. Lin, “Continuously tunable multiple-order stimulated four-photon mixing in a multimode silica fiber,” Opt. Lett. |

9. | R.-J. Essiambre, R. Ryf, M. A. Mestre, A. H. Gnauck, R. Tkach, A. Chraplyvy, S. Randel, Y. Sun, X. Jiang, and R. Lingle, “Inter-modal nonlinear interactions between well separated channels in spatially-multiplexed fiber transmission,” in Proceedings of European Conference on Optical Communication (ECOC 2012), Amsterdam, paper Tu.1.C.4, (2012). [CrossRef] |

10. | N. Mac Suibhne, R. Watts, S. Sygletos, F. C. Garcia Gunning, L. Grüner-Nielsen, and A. D. Ellis, “Nonlinear Pulse Distortion in Few-Mode Fiber,” in Proceedings of European Conference on Optical Communication (ECOC 2012), paper Th.2.F.5, (2012). [CrossRef] |

11. | R. Essiambre, M. A. Mestre, R. Ryf, A. H. Gnauck, R. W. Tkach, A. R. Chraplyvy, Y. Sun, X. Jiang, and R. Lingle, “Experimental observation of inter-modal cross-phase modulation in few-mode fibers,” IEEE Photon. Technol. Lett. |

12. | R. Essiambre, M. A. Mestre, R. Ryf, A. H. Gnauck, R. W. Tkach, A. R. Chraplyvy, Y. Sun, X. Jiang, and R. Lingle, “Experimental investigation of inter-modal four-wave mixing in few-mode fibers,” IEEE Photon. Technol. Lett. |

13. | F. Ferreira, S. Jansen, P. Monteiro, and H. Silva, “Nonlinear semi-analytical model for simulation of few-mode fiber transmission,” IEEE Photon. Technol. Lett. |

14. | S. Mumtaz, R.-J. Essiambre, and G. P. Agrawal, “Nonlinear propagation in multimode and multicore fibers: generalisation of the Manakov equations,” J. Lightwave Technol. |

15. | A. Mecozzi, C. Antonelli, and M. Shtaif, “Coupled Manakov equations in multimode fibers with strongly coupled groups of modes,” Opt. Express |

16. | D. Rafique, S. Sygletos, and A. D. Ellis, “Impact of power allocation strategies in long-haul few-mode fiber transmission systems,” Opt. Express |

17. | P. P. Mitra and J. B. Stark, “Nonlinear limits to the information capacity of optical fibre communications,” Nature |

18. | A. D. Ellis, J. Zhao, and D. Cotter, “Approaching the Non-linear Shannon Limit,” J. Lightwave Technol. |

19. | X. Chen and W. Shieh, “Closed-form expressions for nonlinear transmission performance of densely spaced coherent optical OFDM systems,” Opt. Express |

20. | A. D. Ellis and N. J. Doran, “Are few mode fibres.a practical solution to the capacity crunch”, in Proceedings of 15th International Conference on Transparent Optical Networks (ICTON 2013), paper TuC2.1 (2013). |

21. | G. Bosco, P. Poggiolini, A. Carena, V. Curri, and F. Forghieri, “Analytical results on channel capacity in uncompensated optical links with coherent detection,” Opt. Express |

22. | P. Poggiolini, A. Carena, V. Curri, G. Bosco, and F. Forghieri, “Analytical Modeling of Non-Linear Propagation in Uncompensated Optical Transmission Links,” IEEE Photon. Technol. Lett. |

23. | R. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightwave Technol. |

24. | D. Rafique and A. D. Ellis, “Impact of signal-ASE four-wave mixing on the effectiveness of digital back-propagation in 112 Gb/s PM-QPSK systems,” Opt. Express |

25. | T. Tanimura, M. Nölle, J. K. Fischer, and C. Schubert, “Analytical results on back propagation nonlinear compensator with coherent detection,” Opt. Express |

26. | G. Rademacher, S. Warm, and K. Petermann, “Analytical description of cross modal nonlinear interaction in mode multiplexed multi-mode fibers,” IEEE Photon. Technol. Lett. |

27. | N. Mac Suibhne, A. D. Ellis, F. C. Garcia Gunning, and S. Sygletos, “Experimental verification of four wave mixing efficiency characteristics in a few mode fibre”, in Proceedings of European Conference on Optical Communications (ECOC 2013), paper P1.14 (2013). |

28. | N. Bai, E. Ip, Y.-K. Huang, E. Mateo, F. Yaman, M. J. Li, S. Bickham, S. Ten, J. Liñares, C. Montero, V. Moreno, X. Prieto, V. Tse, K. Man Chung, A. P. Lau, H. Y. Tam, C. Lu, Y. Luo, G. D. Peng, G. Li, and T. Wang, “Mode-division multiplexed transmission with inline few-mode fiber amplifier,” Opt. Express |

29. | S. Randel, R. Ryf, A. H. Gnauck, M. A. Mestre, C. Schmidt, R. Essiambre, P. J. Winzer, R. Delbue, P. Pupalaikis, A. Sureka, Y. Sun, X. Jiang, and R. L. Lingle, “Mode-Multiplexed 6x20-GBd QPSK Transmission over 1200 km DGD-Compensated Few-Mode Fiber,” in Optical Fiber Communication Conference, 2012 OSA Technical Digest Series (Optical Society of America, 2012), paper PDP5C.5. |

30. | E. Ip, M.-J. Li, Y.-K. Huang, A. Tanaka, E. Mateo, W. Wood, J. Hu, Y. Yano, and K. Koreshkov, “146 x6x19-Gbaud wavelength and mode-division multiplexed transmission over 10x50-km spans of few-mode fiber with a gain-equalised few-mode EDFA,” in Optical Fiber Communication Conference, 2013 OSA Technical Digest Series (Optical Society of America, 2013), paper PDP5A.2. |

31. | R. Ryf, N. K. Fontaine, M. A. Mestre, S. Randel, X. Palou, C. Bolle, A. H. Gnauck, S. Chandrasekhar, X. Liu, B. Guan, R.-J. Essiambre, P. J. Winzer, S. G. Leon-Saval, J. Bland-Hawthorn, R. Delbue, P. Pupalaikis, A. Sureka, Y. Sun, L. Grüner-Nielsen, R. V. Jensen, and R. Lingle, “12 x 12 MIMO Transmission over 130-km Few-Mode Fiber;” in Proceedings of Frontiers in Optics Conference, 2012 OSA Technical Digest Series (Optical Society of America, 2012), paper FW6C.4. |

32. | T. Mori, T. Sakamoto, M. Wada, T. Yamamoto, and F. Yamamoto, “A low DMD four LP mode transmission fiber for wide-band WDM-MIMO system,” in Proceedings of Optical Fiber Communication Conference, 2013 OSA Technical Digest Series (Optical Society of America, 2013), paper OTh3K.1. |

33. | G. P. Agrawal, |

34. | A. D. Ellis and W. A. Stallard, “Four Wave mixing in ultra long transmission systems incorporating linear amplifiers”, in Proceedings of IEE Colloquium on Non-Linear Effects in Fibre Communications, |

35. | D. A. Cleland, A. D. Ellis, and C. H. F. Sturrock, “Precise modelling of four wave mixing products over 400km of step index fibre,” Electron. Lett. |

36. | P. D. Maker and R. W. Terhune, “Study of the optical effects due to an induced polarisation third order in the electric field strength,” Phys. Rev. |

37. | O. V. Sinkin, J.-X. Cai, D. G. Foursa, G. Mohs, and A. N. Pilipetskii, “Impact of broadband four-wave mixing on system characterisation,” in Optical Fiber Communication Conference, 2013 OSA Technical Digest Series (Optical Society of America, 2013), paper OTh3G. |

38. | S. Kilmurray, T. Fehenberger, P. Bayvel, and R. I. Killey, “Comparison of the nonlinear transmission performance of quasi-Nyquist WDM and reduced guard interval OFDM,” Opt. Express |

39. | D. Rafique, S. Sygletos, and A. D. Ellis, “Impact of power allocation strategies in long-haul few-mode fiber transmission systems,” Opt. Express |

40. | B. Inan, B. Spinnler, F. Ferreira, D. van den Borne, A. Lobato, S. Adhikari, V. A. Sleiffer, M. Kuschnerov, N. Hanik, and S. L. Jansen, “DSP complexity of mode-division multiplexed receivers,” Opt. Express |

**OCIS Codes**

(060.0060) Fiber optics and optical communications : Fiber optics and optical communications

(060.4370) Fiber optics and optical communications : Nonlinear optics, fibers

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: July 8, 2013

Revised Manuscript: September 15, 2013

Manuscript Accepted: September 16, 2013

Published: September 20, 2013

**Citation**

A. D. Ellis, N. Mac Suibhne, F. C. Garcia Gunning, and S. Sygletos, "Expressions for the nonlinear transmission performance of multi-mode optical fiber," Opt. Express **21**, 22834-22846 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-19-22834

Sort: Year | Journal | Reset

### References

- P. J. Winzer, “Optical Networking Beyond WDM,” IEEE Photon. J.4(2), 647–651 (2012). [CrossRef]
- D. J. Richardson, “Applied physics. Filling the light pipe,” Science330(6002), 327–328 (2010). [CrossRef] [PubMed]
- A. D. Ellis, “The MODE-GAP project,” in Proceedings of Frontiers in Optics Conference, 2012 OSA Technical Digest Series (Optical Society of America, 2012), paper, FW1D.
- F. Poletti, N. V. Wheeler, M. N. Petrovich, N. Baddela, E. N. Fokoua, J. R. Hayes, D. R. Gray, Z. Li, R. Slavík, and D. J. Richardson, “Towards high-capacity fibre optic communications at the speed of light in vacuum,” Nat. Photonics7(4), 279–284 (2013). [CrossRef]
- V. A. J. M. Sleiffer, Y. Jung, V. Veljanovski, R. G. H. van Uden, M. Kuschnerov, H. Chen, B. Inan, L. G. Nielsen, Y. Sun, D. J. Richardson, S. U. Alam, F. Poletti, J. K. Sahu, A. Dhar, A. M. J. Koonen, B. Corbett, R. Winfield, A. D. Ellis, and H. de Waardt, “73.7 Tb/s (96 x 3 x 256-Gb/s) mode-division-multiplexed DP-16QAM transmission with inline MM-EDFA,” Opt. Express20(26), B428–B438 (2012). [CrossRef] [PubMed]
- Y. Yang, Y. Yan, N. Ahmed, Y. Jeng-Yuan, L. Zhang, Y. Ren, H. Huang, K. M. Birnbaum, B. I. Erkmen, S. Dolinar, M. Tur, and A. E. Willner, “Mode properties and propagation effects of optical orbital angular momentum (OAM) modes in a ring fiber,” IEEE Photon. J.4(2), 535–543 (2012). [CrossRef]
- P. J. Winzer and G. J. Foschini, “MIMO capacities and outage probabilities in spatially multiplexed optical transport systems,” Opt. Express19(17), 16680–16696 (2011). [CrossRef] [PubMed]
- R. Pini, R. Salimbeni, A. F. M. Y. Haider, M. Matera, and C. Lin, “Continuously tunable multiple-order stimulated four-photon mixing in a multimode silica fiber,” Opt. Lett.9(3), 79–81 (1984). [CrossRef] [PubMed]
- R.-J. Essiambre, R. Ryf, M. A. Mestre, A. H. Gnauck, R. Tkach, A. Chraplyvy, S. Randel, Y. Sun, X. Jiang, and R. Lingle, “Inter-modal nonlinear interactions between well separated channels in spatially-multiplexed fiber transmission,” in Proceedings of European Conference on Optical Communication (ECOC 2012), Amsterdam, paper Tu.1.C.4, (2012). [CrossRef]
- N. Mac Suibhne, R. Watts, S. Sygletos, F. C. Garcia Gunning, L. Grüner-Nielsen, and A. D. Ellis, “Nonlinear Pulse Distortion in Few-Mode Fiber,” in Proceedings of European Conference on Optical Communication (ECOC 2012), paper Th.2.F.5, (2012). [CrossRef]
- R. Essiambre, M. A. Mestre, R. Ryf, A. H. Gnauck, R. W. Tkach, A. R. Chraplyvy, Y. Sun, X. Jiang, and R. Lingle, “Experimental observation of inter-modal cross-phase modulation in few-mode fibers,” IEEE Photon. Technol. Lett.25(6), 535–538 (2013). [CrossRef]
- R. Essiambre, M. A. Mestre, R. Ryf, A. H. Gnauck, R. W. Tkach, A. R. Chraplyvy, Y. Sun, X. Jiang, and R. Lingle, “Experimental investigation of inter-modal four-wave mixing in few-mode fibers,” IEEE Photon. Technol. Lett.25(6), 539–542 (2013). [CrossRef]
- F. Ferreira, S. Jansen, P. Monteiro, and H. Silva, “Nonlinear semi-analytical model for simulation of few-mode fiber transmission,” IEEE Photon. Technol. Lett.24(4), 240–242 (2012). [CrossRef]
- S. Mumtaz, R.-J. Essiambre, and G. P. Agrawal, “Nonlinear propagation in multimode and multicore fibers: generalisation of the Manakov equations,” J. Lightwave Technol.31(3), 398–406 (2013). [CrossRef]
- A. Mecozzi, C. Antonelli, and M. Shtaif, “Coupled Manakov equations in multimode fibers with strongly coupled groups of modes,” Opt. Express20(21), 23436–23441 (2012). [CrossRef] [PubMed]
- D. Rafique, S. Sygletos, and A. D. Ellis, “Impact of power allocation strategies in long-haul few-mode fiber transmission systems,” Opt. Express21(9), 10801–10809 (2013). [CrossRef] [PubMed]
- P. P. Mitra and J. B. Stark, “Nonlinear limits to the information capacity of optical fibre communications,” Nature411(6841), 1027–1030 (2001). [CrossRef] [PubMed]
- A. D. Ellis, J. Zhao, and D. Cotter, “Approaching the Non-linear Shannon Limit,” J. Lightwave Technol.28(4), 423–433 (2010). [CrossRef]
- X. Chen and W. Shieh, “Closed-form expressions for nonlinear transmission performance of densely spaced coherent optical OFDM systems,” Opt. Express18(18), 19039–19054 (2010). [CrossRef] [PubMed]
- A. D. Ellis and N. J. Doran, “Are few mode fibres.a practical solution to the capacity crunch”, in Proceedings of 15th International Conference on Transparent Optical Networks (ICTON 2013), paper TuC2.1 (2013).
- G. Bosco, P. Poggiolini, A. Carena, V. Curri, and F. Forghieri, “Analytical results on channel capacity in uncompensated optical links with coherent detection,” Opt. Express19(26), B440–B449 (2011). [CrossRef] [PubMed]
- P. Poggiolini, A. Carena, V. Curri, G. Bosco, and F. Forghieri, “Analytical Modeling of Non-Linear Propagation in Uncompensated Optical Transmission Links,” IEEE Photon. Technol. Lett.23(11), 742–744 (2011). [CrossRef]
- R. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightwave Technol.28(4), 662–701 (2010). [CrossRef]
- D. Rafique and A. D. Ellis, “Impact of signal-ASE four-wave mixing on the effectiveness of digital back-propagation in 112 Gb/s PM-QPSK systems,” Opt. Express19(4), 3449–3454 (2011). [CrossRef] [PubMed]
- T. Tanimura, M. Nölle, J. K. Fischer, and C. Schubert, “Analytical results on back propagation nonlinear compensator with coherent detection,” Opt. Express20(27), 28779–28785 (2012). [CrossRef] [PubMed]
- G. Rademacher, S. Warm, and K. Petermann, “Analytical description of cross modal nonlinear interaction in mode multiplexed multi-mode fibers,” IEEE Photon. Technol. Lett.24(21), 1929–1932 (2012). [CrossRef]
- N. Mac Suibhne, A. D. Ellis, F. C. Garcia Gunning, and S. Sygletos, “Experimental verification of four wave mixing efficiency characteristics in a few mode fibre”, in Proceedings of European Conference on Optical Communications (ECOC 2013), paper P1.14 (2013).
- N. Bai, E. Ip, Y.-K. Huang, E. Mateo, F. Yaman, M. J. Li, S. Bickham, S. Ten, J. Liñares, C. Montero, V. Moreno, X. Prieto, V. Tse, K. Man Chung, A. P. Lau, H. Y. Tam, C. Lu, Y. Luo, G. D. Peng, G. Li, and T. Wang, “Mode-division multiplexed transmission with inline few-mode fiber amplifier,” Opt. Express20(3), 2668–2680 (2012). [CrossRef] [PubMed]
- S. Randel, R. Ryf, A. H. Gnauck, M. A. Mestre, C. Schmidt, R. Essiambre, P. J. Winzer, R. Delbue, P. Pupalaikis, A. Sureka, Y. Sun, X. Jiang, and R. L. Lingle, “Mode-Multiplexed 6x20-GBd QPSK Transmission over 1200 km DGD-Compensated Few-Mode Fiber,” in Optical Fiber Communication Conference, 2012 OSA Technical Digest Series (Optical Society of America, 2012), paper PDP5C.5.
- E. Ip, M.-J. Li, Y.-K. Huang, A. Tanaka, E. Mateo, W. Wood, J. Hu, Y. Yano, and K. Koreshkov, “146 x6x19-Gbaud wavelength and mode-division multiplexed transmission over 10x50-km spans of few-mode fiber with a gain-equalised few-mode EDFA,” in Optical Fiber Communication Conference, 2013 OSA Technical Digest Series (Optical Society of America, 2013), paper PDP5A.2.
- R. Ryf, N. K. Fontaine, M. A. Mestre, S. Randel, X. Palou, C. Bolle, A. H. Gnauck, S. Chandrasekhar, X. Liu, B. Guan, R.-J. Essiambre, P. J. Winzer, S. G. Leon-Saval, J. Bland-Hawthorn, R. Delbue, P. Pupalaikis, A. Sureka, Y. Sun, L. Grüner-Nielsen, R. V. Jensen, and R. Lingle, “12 x 12 MIMO Transmission over 130-km Few-Mode Fiber;” in Proceedings of Frontiers in Optics Conference, 2012 OSA Technical Digest Series (Optical Society of America, 2012), paper FW6C.4.
- T. Mori, T. Sakamoto, M. Wada, T. Yamamoto, and F. Yamamoto, “A low DMD four LP mode transmission fiber for wide-band WDM-MIMO system,” in Proceedings of Optical Fiber Communication Conference, 2013 OSA Technical Digest Series (Optical Society of America, 2013), paper OTh3K.1.
- G. P. Agrawal, Nonlinear fiber optics, (Springer Berlin, 2000).
- A. D. Ellis and W. A. Stallard, “Four Wave mixing in ultra long transmission systems incorporating linear amplifiers”, in Proceedings of IEE Colloquium on Non-Linear Effects in Fibre Communications, 159, 6/1–6/4 (1990).
- D. A. Cleland, A. D. Ellis, and C. H. F. Sturrock, “Precise modelling of four wave mixing products over 400km of step index fibre,” Electron. Lett.28(12), 1171–1172 (1992). [CrossRef]
- P. D. Maker and R. W. Terhune, “Study of the optical effects due to an induced polarisation third order in the electric field strength,” Phys. Rev.137(3A), A801–A818 (1965). [CrossRef]
- O. V. Sinkin, J.-X. Cai, D. G. Foursa, G. Mohs, and A. N. Pilipetskii, “Impact of broadband four-wave mixing on system characterisation,” in Optical Fiber Communication Conference, 2013 OSA Technical Digest Series (Optical Society of America, 2013), paper OTh3G.
- S. Kilmurray, T. Fehenberger, P. Bayvel, and R. I. Killey, “Comparison of the nonlinear transmission performance of quasi-Nyquist WDM and reduced guard interval OFDM,” Opt. Express20(4), 4198–4205 (2012). [CrossRef] [PubMed]
- D. Rafique, S. Sygletos, and A. D. Ellis, “Impact of power allocation strategies in long-haul few-mode fiber transmission systems,” Opt. Express21(9), 10801–10809 (2013). [CrossRef] [PubMed]
- B. Inan, B. Spinnler, F. Ferreira, D. van den Borne, A. Lobato, S. Adhikari, V. A. Sleiffer, M. Kuschnerov, N. Hanik, and S. L. Jansen, “DSP complexity of mode-division multiplexed receivers,” Opt. Express20(10), 10859–10869 (2012). [CrossRef] [PubMed]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.