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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 19 — Sep. 23, 2013
  • pp: 22834–22846
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Expressions for the nonlinear transmission performance of multi-mode optical fiber

A. D. Ellis, N. Mac Suibhne, F. C. Garcia Gunning, and S. Sygletos  »View Author Affiliations


Optics Express, Vol. 21, Issue 19, pp. 22834-22846 (2013)
http://dx.doi.org/10.1364/OE.21.022834


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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

2. Analysis

For a periodically amplified single-mode transmission system with Ns identical spans and no dispersion compensation, it has been shown that in the limit of weak nonlinearity the generated nonlinear signal field En at frequency ω0 in mode n is given by [34

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, 159, 6/1–6/4 (1990).

, 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]

];
En=ω0n2cAijknEiEjEk*1eαLejΔβijknLjΔβijkn+αsin(NSΔβijknL/2)sin(ΔβijknL/2)
(1)
where n2 is the nonlinear refractive index, c the speed of light Aijkn the effective area for the interaction between fields at frequencies fi,p,fj,q, fk,r, and fn,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 Δβijkn,pqrs the group velocity mismatch appropriate to the interaction. Generalizing the nonlinear interaction to a multi-mode fiber is straightforward [33

33. G. P. Agrawal, Nonlinear fiber optics, (Springer Berlin, 2000).

] and simply requires identification of the modes and frequencies associated with each of the four interacting waves denoted i,j,k and n and noting that the effective area and group velocity mismatches are given by;
1Aijkn=Ei(r,θ)Ej(r,θ)Ek(r,θ)En(r,θ)drdθEi(r,θ)drdθ.Ej(r,θ)drdθ.Ek(r,θ)drdθ.En(r,θ)drdθ
(2)
Δβijkn=β'n+β'kβ'iβ'j
(3)
Where r and θ are coordinates specifying the position across the transverse field Ex(r,θ) and β’x the group delay of the xth mode (were x is replaced by 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 constantsβ~i such that the group velocity mismatch is given by;
Δβijkn,pqrs=βn~+βk~βi~βj~4π2β''(fi,sfk,r)(fj,qfk,r)
(4)
where β” is the group velocity dispersion parameter, p, q, r, and s denote the frequency of the components in the ith, jth, kth and nth 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

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]

] and quasi velocity matching peaks [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]

] for wider frequency spacing. For inter-mode interactions additional walk-off due to the differential mode delay is present, significantly reducing velocity-matching and hence FWM efficiency at low frequency spacing. However, by balancing the walk-off from these two effects, efficient velocity-matching may be restored. FWM efficiency is maximized at a velocity-matched frequency offset (VMO) where the CD and DMD cancel exactly. The frequency offset (see Fig. 1) may be readily calculated from Eq. (4) to be [26

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]

]
Δfijkn=βn~+βk~βi~βj~2πβ''
(5)
For an optical super-channel with a total bandwidth 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~i + β~j = 0) for an OFDM superchannel with a rectangular spectrum (shown by the black curve in Fig. 1(a)) [19

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]

]. To account for all terms the integral was performed from –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

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]

]. Simple arithmetic shows that the impact of the VMO corresponds exactly to shifting the bounds of the integral performed in [19

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]

] to –B/2-Δfijknto B/2-Δfijkn. 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).

The implications of Eq. (6) are illustrated in Fig. 2
Fig. 2 Relative nonlinear noise power coefficient (in dB relative to SMF) for a step index 12-mode fiber (four LP modes) with a maximum differential mode delay of 880ps/km, 0.2 dB/km loss and assumed chromatic dispersion of 20 ps2/km (left) and with a maximum DMD of 110ps/km (right). Contours show 10log10ijkn)as a function of effective area and VMO for a WDM bandwidth of 5THz. Colored dots represent the calculated values of these parameters for nonlinear noise generated in the LP01(red), LP02(green), LP11(blue) and LP21 (purple) modes for different inter-mode interactions. Fiber parameters calculated using a commercial mode solver.
for two different FMF with high (left) and low (right) DMDs. Contours in Fig. 2 represent potential values of ηijkn 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 ηijkn. 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 fw 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 ηijkn 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 B is extended allowing strong velocity matching at greater VMO (resulting in a movement of the discontinuity in Fig. 2).

To calculate the total nonlinear noise influencing a given mode, all of the interactions influencing that mode must be summed (Eq. (7)). This corresponds to summing the contributions from each of the colored points associated with that mode in Fig. 2 (or the equivalent figure for fiber in question). This is illustrated in Fig. 3
Fig. 3 Nonlinear noise power normalized to LP01 mode at 100GHz bandwidth as a function of WDM signal bandwidth for the high (left) and low (right) DMD fibers of Fig. 2. LP01 (red), LP02 (green), LP11 (blue) and LP21 (purple).
which shows the nonlinear noise power generated at the center of the WDM versus the WDM bandwidth for these two four LP mode fibers. A number of discontinuities are apparent, in addition to the logarithmically increasing background expected for a SMF [37

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.

], with larger number of discontinuities observed for the fiber with the lowest DMD giving a significant increase in the overall nonlinear power spectral density. These discontinuities correspond to the signal bandwidth becoming sufficiently large to result in an additional strongly velocity matched contribution to the total nonlinear noise (see the green curve in Fig. 1(b), showing a WDM bandwidth just beyond such a discontinuity). In terms of Fig. 2, as the WDM bandwidth is increased, the discontinuity (at ± B/2) moves to higher VMO, allowing strong contributions from different points.

3. Experimental verification

We confirmed the predictions of Eq. (6) using the following simple measurement (Fig. 4
Fig. 4 (top) Experimental configuration used to measure few mode fibers (lenses and positioning stages at the ends of the few mode fiber are omitted for clarity). (bottom) Comparison of theoretical (solid lines) and experimental (dots) results for the LP01(red) and LP11 (blue) mode showing the normalized nonlinear power spectral density at the center of the band as a function of the bandwidth of an amplified spontaneous emission source with a 50GHz frequency notch at the center. Theoretical predictions are based on typical measured DMD from the same fiber draw as the fiber used in this experiment [17], also shows theoretical prediction neglecting the inter mode nonlinearity (dotted lines)and typical output spectra (inset for signal widths between 1.5 (red) and 3 (dark blue) THz).
(top) with a 30km low DMD FMF similar to the fiber used in [31

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.

]. A gain-flattened erbium doped fiber amplifier (GFA) was used as a source of amplified spontaneous emission (ASE), which was shaped into a rectangular spectral blocks, representing highly dispersed WDM signals, using a wavelength selective switch (WSS). These blocks were amplified by a 38dBm GFA and the WSS was re-adjusted to ensure a uniform power spectral density. The central 50GHz section of the ASE spectrum was attenuated using the WSS in order to allow monitoring of the FWM signal. The FMF launch was adjusted by offsetting the position of the single mode fiber to maximize nonlinear mixing between the LP01 and LP11 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 LP01 and LP11fiber modes were selected using the appropriate phase plate (only one orientation used for LP01) 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.

4. Capacity limits of few mode fiber systems

Figure 5 (right) illustrates the performance of each fiber type as a function of system length, plotting the total capacity instead of the information spectral density per mode using Eqs. (6) and (8). Whereas the actual performance is dependent on the optical SNR, the difference in ISD between fiber types appears to be largely independent of the system length for all practical scenarios, only noticeably varying for transmission distances around 20,000km in this example. This may be understood by inspecting Eq. (9), where we may observe that the penalty arising from inter-mode nonlinear effects contains no length dependence (all length dependent terms such as aM 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
Fig. 6 Predicted maximum total information spectral density (right axis) and information spectral density per mode (left axis) for trench assisted fiber designs with a core radius of 10.4 µm as index curvature is varied resulting in differences in DMD, plotted as a function of resultant DMD between LP01 and LP21 for positive (red filled circles) and negative values (blue open circles). Inset shows a similar fiber with a core radius of 30 µm (filled blue circles). Solid red line shows a logarithmic fit over points falling within 0.1 and 8 ns/km DMD as a guide to the eye. Both plots are for a 3,200km system with 80km amplifier spacing (4.8dB noise figure), 100 channels (50GHz spacing), 0.2dB loss and dispersion (β”) of 20ps2/km.
, 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.

The inset to Fig. 6 shows a different fiber design, with an increased core diameter, again with the refractive index curvature varied. As expected the overall ISD scales with the increased core area. Our results suggest that effective area dominates over the effect of reduced differential mode delay, although effects such as bend loss and mode dependent loss have been neglected in this preliminary study. This direct dependence of effective area is illustrated in Fig. 7
Fig. 7 Predicted information spectral density per mode for step index fiber designs supporting four LP modes, plotted as a function of resultant LP01 effective area. System parameters correspond to a 3,200km system with 80km amplifier spacing (4.8dB noise figure), 100 channels (50GHz spacing), 0.2dB loss and dispersion (β”) of 20ps2/km.
, which shows the variation in predicted ISD for various step-index fibers as the core radius and refractive index are simultaneously varied such that four LP modes are supported for each fiber.

Combining analytical result of the potential ISD predicted by the approach of this paper with calculations of digital signal processing complexity [40

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]

] should enable fibers to be designed to which optimize the tradeoff between signal processing complexity (proportional to DGD) and nonlinearity without the need for extensive numerical simulations. Since our preliminary results suggest that the system reach, amplifier noise figure etc. do not influence the relative performance of different fiber designs, such an optimization should be possible without detailed knowledge of the precise system configuration.

5. Conclusions

In this paper we have proposed closed form expressions to calculate the nonlinear information spectral density of a multi-mode fiber system. We have shown that the approach accurately predicts experimental measurements. These experimental results suggest that for signal bandwidths above 2.5 THz inter-mode FWM is likely to induce an increase in nonlinear noise of around 3dB and a corresponding reduction in overall fiber information spectral density of a few b/s/Hz (around 10%). We have presented a selection of results which show that the maximum potential ISD decreases with decreasing DMD and that for practical cases the decrease in ISD does not depend on the system design.

Acknowledgments

This work was partly funded by the European Communities 7th Framework Programme FP/2007-2013 (Grant 258033-MODE-GAP), EPSRC (Grant EP/J017582/1-UNLOC), Science Foundation Ireland (Grants 06/IN/I969 and 10/CE/I1853) and The Royal Society (Grant WM120035-TEST).

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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]

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]

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


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