## Abstract

We analyze the impact of Differential Mode Delay (DMD) Management on the nonlinear impairments in mode-division multiplexed transmission systems. It is found out that DMD Management can lead to a degraded performance, due to enhanced intermodal nonlinear interaction. This can be attributed to an increased correlation of co-propagating channels, similar to the effects that show up in dispersion managed single-mode systems.

© 2015 Optical Society of America

## 1. Introduction

Mode Division Multiplexing (MDM) through Multi-mode fibers (MMF) is a promising approach to overcome the upcoming capacity crunch in long haul optical fiber networks [1, 2].

MMF can be easily designed to support more than hundred fiber modes, roughly corresponding to an increase of the per-fiber capacity of more than hundred times, as the potential (linear) capacity scales with the number of fiber modes. One of the main challenges in designing MDM systems with multi-mode fibers is the group-delay difference (Differential Mode Delay, DMD) between the eigenmodes that are guided by the fiber. Inevitable coupling between the fiber modes, due to manufacturing tolerances, splices and connectors or fiber bends require the usage of Multiple-Input-Multiple-Output (MIMO) receivers to equalize for mode coupling. The memory length and therefore the complexity of such receivers increases with the group-delay difference between the fastest and the slowest mode of the fiber [3].

The first approach to meet this challenge is to use fibers with low DMD by designing the index profile accordingly. This method can be adequate for short fiber links, while it requires MIMO receiver structures with very large memory requirements when using it in long-haul systems. DMD management has been proposed to drastically reduce the receiver complexity by alternating two different fibers where the fastest mode of the first fiber section is the slowest mode of the second section [4].

Nonlinear effects in few-mode fiber systems have recently started to gain interest, since the ultimate limitation of the data throughput in such systems might be determined by Kerr-effect based nonlinearities. Nonlinear Kerr-effect based interaction between signals that travel in different fiber modes have been of particular interest since they show features that are very different from commonly used single-mode systems [5, 6].

DMD management leads to a correlation of signals that travel in different fiber modes, similar to the effect of periodic dispersion compensation in single mode systems. Consequently, an additional signal impairment due to increased intermodal nonlinear interaction might appear. This paper numerically analyzes this effect with an exemplary few-mode fiber transmission system.

## 2. DMD management

Figure 1 shows the concept of DMD management for a few-mode fiber link that supports an LP_{01} and two degenerated LP_{11} modes. The fibers are designed with a power law refractive-index profile. By varying the profile exponent around the ideal value of 2 for the parabolic profile [7], it is possible to design two fibers where the group-velocity of the LP_{01} mode in the first fiber equals the group-velocity of the LP_{11a} mode and vice versa

Figure 2 illustrates schematically the advantage of DMD management. The delay spread between the fastest and the slowest mode that relates directly to the MIMO receiver complexity increases linearly for a non DMD managed system when no linear mode coupling is considered (Fig. 2(a)). When considering unitary coupling between all modes with a large number of coupling points, the delay spread increases with the square root of the transmission distance [8, 9]. In a case with non-unitary coupling, as it is introduces e.g. by splices, the delay spread is between the two graphs of Fig. 2(a), as indicated by the marked area [9, 10], depending on the strength and the nature of the coupling effect. Figure 2(b) shows the delay spread for a link with DMD management. Without linear coupling, the delay spread is periodically perfectly compensated. With unitary coupling, the delay spread follows the same square-root dependence as without DMD management. With splice related coupling effects, the delay spread is indicated by the marked area, between the square-root dependance and full compensation. In [10], it was shown that it is favorable to build transmission links where the two different fibers are alternated very often in order to drastically reduce the MIMO complexity. On the other hand, periodic DMD management introduces a periodic signal correlation, similar to periodic chromatic dispersion management in single-mode fiber systems that can potentially lead to an increased nonlinear signal degradation.

## 3. Simulation setup

The considered transmission link is shown in Fig. 3. Each transmitter generates one 28 GBaud QPSK signal (no WDM) that is passed through a root-raised cosine filter with a roll-off factor of 0.2. The signals are ideally multiplexed on each of the fiber modes, where, for calculation simplicity, only one polarization is excited. The link consists of 10 spans. Each span is composed of 2 to 40 DMD-compensated segments that sum up to a total of 80 km distance. Each segment consists of two fibers that ideally compensate the entire group-delay spread, according to Fig. 1.

The group-delay difference between the LP_{01} and the LP_{11} modes is arbitrarily set to 24*ps/km*. ach mode’s chromatic dispersion is equal at 16*ps*/(*nm* · *km*). The effective area of the LP_{01} and the LP_{11} modes are 45*μm*^{2} and 60*μm*^{2}, respectively. The intermodal effective area between the LP_{01} and the LP_{11} modes is 90*μm*^{2}. A nonlinear material coefficient of *n*_{2} = 2.6 · 10^{−20}*m*^{2}/*W* is considered.

All losses, including mode-dependent losses, are ideally compensated by a noise-free amplifier at the end of each span. A static frequency domain equalizer compensates for the accumulated chromatic dispersion. Noise loading is performed before an adaptive frequency domain equalizer that serves as a MIMO receiver. The impact of nonlinear interaction is evaluated by assessing the required OSNR (ROSNR) for a certain Bit Error Rate (BER)

Along the transmission link, splices occur about every 2 km (+− 100 m), independent of the DMD management scheme, while this distance is slightly varied in order to reduce the build-up of numerical artifacts. Splices are characterized by a lateral misalignment that leads to coupling and loss [11, 12]. The lateral misalignment of all splice-points within one link follows a Gaussian distribution with zero mean.

## 4. Nonlinear Schrödinger equation for multi-mode fibers

The propagation of a pulse *A _{i}* that propagates in the

*i*th mode of a multi-mode fiber is governed by a set of Nonlinear Schrödinger Equations (NLSE), if linear mode coupling inside one segment of fiber is only considered between the two polarizations of each fiber mode, following the approaches presented in [13]:

*β*

_{0i},

*β*

_{1i}and

*β*

_{2i}are the first three Taylor coefficients of the

*i*th mode’s propagation constants, where

*β*

_{1i}is the group-delay parameter and

*β*

_{2i}represents the dispersion parameter.

*α*is the attenuation coefficient that is chosen to be equal for all modes. The summation in equ. (1) needs to be performed over all modes that propagate in the fiber. The nonlinear parameter can be obtained as

*γ*=

_{ij}*n*

_{2}

*ω*

_{0}

*f*with

_{ij}/c*n*

_{2}being the nonlinear index coefficient and

*f*represents the inverse of the commonly used effective core area, while

_{ii}*f*is the inverse of the intermodal effective area of modes

_{ij}*i*and

*j*. In this work, the Multi-mode NLSE is solved with the Split-Step Fourier method.

## 5. Simulation results: DMD management with no linear mode-coupling

In order to assess the impact of the DMD management on the nonlinear-induced performance penalty, we first investigate a transmission link with no linear mode coupling. This includes the assumption that DMD-compensated segments are ideally connected and the faster mode from one segment can be transferred crosstalk-free to the slower mode of the following segment.

Figure 4 shows the required OSNR (ROSNR) to reach a Bit-Error Rate of 10^{−3} as a function of number of DMD-compensated segments, while keeping the length of one span (80 km) constant. Figure 4(a) represents the signal in the LP_{01} mode, while each graph represents a different input power per mode. The ROSNR increases from about 14.8 dB to 15.7 dB when increasing the number of DMD-compensated segments from 2 to 40 at 0 dBm input power. The effect is weaker for lower input powers, but still present, indicating that nonlinear impairments are the cause of the signal degradation. Figure 4(b) shows the same graphs for the signal that propagates in the LP_{11a} mode. For 0 dBm input power, the ROSNR increases from about 14.3 to 14.6 dB when increasing the number of DMD-compensated segments. The ROSNR is in general smaller for the LP_{11a} mode compared to the LP_{01} mode as the intramodal effective area of the LP_{11a} mode is larger, leading to lower nonlinear signal distortion.

The negative effect of DMD management on the transmission performance can be explained by the increased correlation between the signals that travel in different fiber modes when using DMD management with short segment lengths. This leads to enhanced intermodal interaction between modes that travel at different group-delay and is similar to the transmission penalties that can be observed in periodically dispersion compensated single-mode transmission systems.

The LP_{01} mode has intermodal nonlinear interaction with two modes that have a different group-delay (LP_{11a} and LP_{11b} mode); hence the nonlinear interaction with this two modes is increased by the DMD management. The LP_{11a} mode, on the other hand, has intermodal nonlinear interaction with one mode that has a different group-delay (LP_{01} mode) and one mode with equal group-delay (LP_{11b} mode). The intermodal interaction with the LP_{11b} mode is always maximal due to the equal group-delays. Consequently, DMD management can only increase the nonlinear interaction with the LP_{01} mode and not with the LP_{11b} mode. Therefore, DMD management has a larger impact on the nonlinear impairments in the LP_{01} mode compared to the LP_{11a} mode as it can be seen by comparing Fig. 4(a) and Fig. 4(b).

## 6. Simulation results: DMD management with splice-induced linear mode-coupling

Finally, we analyze the impact of splices on the transmission performance. Splices are placed about every 2 km along the fiber link. The lateral misalignment of all splices follow a Gaussian distribution that leads to an overall splice loss per span of 0.5 dB. Figure 5 shows the ROSNR as a function of the number of filter tabs for different numbers of segments per span for the the signal that was launched into (a) the LP_{01} and (b) into the LP_{11a} mode. Each graph represents the average of 10 link realizations with the same number of segments but different splice random number seeds. It can be seen that a larger number of DMD-compensated segments leads to drastically decreased filter tab requirement. This is accompanied by the trade-off that the minimum reachable ROSNR value is higher a larger number of DMD-compensated segments.

When comparing Fig. 5 with Fig. 4, it can be seen that furthermore the performance of the signal in the LP_{01} is slightly improved by the splice-induced coupling, while the signal in the LP_{11a} mode shows a degradation. This can be attributed to the mode-dependent loss as well as the special amplification scheme that was used in this simulation. The interplay of MDL and the used amplification enables a situation where the signal of the LP_{01} mode has a disproportionate amount of the overall power inside the fiber that leads to the performance shift.

## 7. Conclusion

We analyzed the impact of Differential Mode Delay (DMD) management on the performance of Mode-Division Multiplexed transmission links with few-mode fibers. DMD management leads to an increased OSNR requirement as a result of enhanced intermodal nonlinear interaction. This trend is existent when considering transmission links without linear mode coupling as well as when introducing coupling as it appears e.g. due to lateral misalignment at splice points. Nevertheless, DMD management drastically reduces the required memory length in the MIMO-receiver at the cost of a moderately increased OSNR requirement.

## Acknowledgments

This work was funded by the German Research Foundation ( DFG project PE 319/29-1).

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