## Abstract

We investigate numerically the influence of fiber splices and fiber connectors to the statistics of mode dependent loss (MDL) and multiple-input multiple-output (MIMO) outage capacity in mode multiplexed multi-mode fiber links. Our results indicate required splice losses much lower than currently feasible to achieve a reasonable outage capacity in long-haul transmission systems. Splice losses as low as 0.03dB may effectively lead to an outage of MIMO channels after only a few hundred kilometers transmission length. In a first approximation, the relative capacity solely depends on the accumulated splice loss and should be less than ≈ 2dB to ensure a relative capacity of 90%. We also show that discrete mode permutation (mixing) within the transmission line may effectively increase the maximum transmission distance by a factor of 5 for conventional splice losses.

© 2012 Optical Society of America

## 1. Introduction

Mode division multiplex (MDM) is a promising approach to increase the transmission capacity of optical transmission fibers. In principle, MDM may be realized with any fiber supporting more than one mode, where each mode is utilized to transmit a different data stream. Thus the fiber capacity will be increased by the number of guided modes. Using this definition, polarization multiplexing (PolMux) is a MDM system with two (polarization) modes. In order to increase the fiber capacity further, multi-mode fibers (MMF) [1–4] or multi-core fibers (MCF) [5], guiding more than two modes might be used. Ideally, the linear capacity of these MDM systems is the linear capacity of a single mode fiber times the number of guided modes *M*. Considering a MMF based MDM approach, inevitable modal cross-talk due to (linear) mode coupling makes it necessary to use multiple-input multiple-output (MIMO) approaches (well known from wireless communications [6]) to recover the original data signal at the receiver. While modal cross-talk might be mitigated by MIMO receivers, mode dependent loss (MDL) reduces irreversible the overall MDM capacity and can effectively reduce the number of MIMO channels, as it was shown in [7, 8].

First approaches to analyze the MDL and MIMO capacity statistics in the strong-coupling regime are based on the assumption of statistically independent mode coupling, modeled by random matrices [7, 8]. In [9], a more physical description of mode coupling due to statistical bends is used to analyze the system capacity. In this contribution, we numerically analyze a MDM system with realistic coupling statistics from fiber splices and fiber connectors in gradient-index MMFs (GI-MMF) guiding different number of modes [10]. Any fiber nonlinearities which also influence the system capacity [11] are neglected in this work.

## 2. Spliced MMF MDM model

Neglecting fiber nonlinearities, a multiple-input multiple-output MDM transmission system may be described by
$\overrightarrow{y}=\sqrt{{E}_{0}}\underset{\_}{H}\cdot \overrightarrow{x}+\overrightarrow{n}$, with the transfer matrix
$\underset{\_}{H}=\sqrt{{\alpha}_{\mathrm{\Sigma}S}}{\underset{\_}{H}}^{\prime}$ as illustrated in Fig. 1. Here, *x⃗* = [*x*_{1}, ⋯ , *x _{M}*]

*and*

^{T}*y⃗*= [

*y*

_{1}, ⋯ ,

*y*]

_{M}*are the complex amplitudes of the eigenmodes of a MMF supporting in total*

^{T}*M*modes at the mode-selective transmitter and at the coherent mode-selective receiver, respectively. Assuming the same noise power in all modes, complex gaussian noise

*n⃗*with power spectral density

*N*

_{0}per eigenmode is added at the receiver.

*E*

_{0}is the average energy transmitted per symbol and per fiber eigenmode. Using this definition the total power increases with the number of fiber eigenmodes

*M*. The link loss (here only splice loss) will be compensated by

*α*

_{Σ}

*=*

_{S}*M*/trace(

*H*̱′

*H*̱′

^{†}).

*H*̱′

^{†}is the conjugate transpose of

*H*̱′. The mode averaged signal to noise ratio (SNR) at the receiver is SNR =

*E*

_{0}/

*N*

_{0}and is set to 20 dB for all investigations. The transmission characteristic of the MMF link at laser emission frequency

*f*

_{0}is described by the

*M*×

*M*transmission matrix

*K*is the number of splices (independent fiber sections) and

*M*×

*M*diagonal phase matrix with random phase entries ${\varphi}_{m}^{\left(k\right)}=[0,2\pi ]$ that accounts for modal noise due to random changes of the phase constants of the fiber eigenmodes induced by strain, temperature variation, etc. [12, 13]. Modal coupling due to fiber misalignment at splice points is considered by a

*M*×

*M*coupling matrix

*C*̱

*, with entries*

_{k}*e⃗*direction is approximated by Laguerre-Gaussian modes, as done in [15, 16]:

_{x}*l*denotes the circumferential order and

*q*the radial order, ${\text{L}}_{q}^{\left(l\right)}\left(x\right)$ represents the generalized Laguerre polynomials of order

*l*and degree

*q*and $w=\sqrt{a/\left({k}_{0}NA\right)}$ is the spot size of the fundamental mode with core radius

*a*, numerical aperture

*NA*and the free space wavenumber

*k*

_{0}.

*C*has to be chosen to fulfill the orthogonal relation of the fiber eigenmodes (4), i.e. ∬

_{l,q}*·*

_{A}E_{i}*E*d

_{j}*A*=

*δ*.

_{i,j}To obtain insight in the influence of fiber splices to the capacity of MDM MMF transmission links, it is reasonable to assume independent normal (gaussian) distributed fiber misalignments in *x* and *y* direction with zero mean and standard variation *σ _{x,y}* at splice points along a transmission link. For each MMF link with a given number of splices

*K*= {8, 16, 32, 64, 128, 258, 512, 1024} and a given standard deviation of fiber misalignment

*σ*= {0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7

_{x,y}*μ*m} we numerically simulated an ensemble with 100,000 realizations with random phase differences ${\varphi}_{m}^{\left(k\right)}$ of the fiber eigenmodes and random fiber misalignments defined by the standard deviation

*σ*. The used GI-MMF parameters for fibers guiding

_{x,y}*M*= {3, 10, 28, 55} modes are shown in Table 1. Throughout this work only one polarization is considered.

Assuming that the channel matrix
$\underset{\_}{H}=\sqrt{{\alpha}_{\mathrm{\Sigma}S}}{\underset{\_}{H}}^{\prime}$ is known at the receiver, but unknown at the transmitter, the *M* mode MDM MIMO capacity at a single fequency for equal launch power per mode, normalized to the capacity of *M* times the capacity of a SMF systems may be expressed by [17]:

*λ*represent the eigenvalues of

_{i}*H*̱

*H*̱

^{†}. The maximum relative capacity is

*C*= 1 and represents either a fiber bundle or a MDM transmission system without mode dependent loss (i.e.

_{r}*λ*= 1 for all MIMO channels). Due to modal noise and varying coupling statistics the capacity is not a fixed value for a given SNR, but rather changes over time or for different link realizations. Therefore, it is reasonable to define the (relative) outage capacity

_{i}*C*as the maximal (relative) capacity which can be obtained with a probability of with

_{o}*p*(

_{c}*C*) the probability density function of the capacity for a link ensemble. For clarity reasons, we omit the subscript for the outage capacity

*C*and use

_{o}*C*as the relative outage capacity. In case of frequency dependence of the MDM system, the outage capacity may even approach the average channel capacity due to frequency diversity [18], which is not considered in this work.

_{r}As it is obvious from Eq. (5) the capacity strongly depends on the MIMO system eigenvalues *λ _{i}*, also called mode dependent loss (MDL). It should be pointed out that MDL is not equivalent to the loss difference of MMF eigenmodes. Different definitions can be found in the literature for MDL. In this paper, following the definitions in [7],

*σ*

_{Σ,MDL}is the overall standard deviation of all eigenvalues of a link ensemble defined by fiber misalignment

*σ*, number of splice points

_{x,y}*K*and number of guided modes

*M*:

*j*-th link realization and std(

*x*) the standard deviation. It is also common to define the MDL difference for a given link realization [8]:

*H*̱ for

*K*= 1 splice. Splice MDL values are denoted as

*σ*

_{S,MDL}and ${\overline{\text{MDL}}}_{S\mathrm{\Delta}}$.

## 3. Mode dependent loss (MDL)

Because it is more convenient to characterize splices by their splice losses rather than by their spatial misalignment *σ _{x,y}*, any further results which depend on the fiber misalignment

*σ*will be presented with respect to the mean fiber splice loss $\overline{{\alpha}_{S}}$. Figure 2 shows the ensemble average splice loss $\overline{{\alpha}_{S}}$ for different fiber misalignments

_{x,y}*σ*for MMFs with equally excited modes. As expected, splice losses decrease with an increased core radius.

_{x,y}Considering two fibers with different numerical apertures but the same number of guided modes *M*, the mode coupling characteristics are the same for equal splice losses. Thus, the MDL and the system capacity as a function of splice loss
$\overline{{\alpha}_{S}}$ are independent of the fiber’s numerical aperture and the following results will be also valid for numerical aperture different as in Table 1.

Before the MDL statistics of the transmission link can be analyzed, Fig. 3 shows the MDL values of the considered fiber splices. Due to the fact that the tolerable fiber misalignment for a given splice loss
$\overline{{\alpha}_{S}}$ increases with the number of supported modes *M*, fibers supporting many modes show a much higher overall splice MDL *σ*_{S,MDL} than fibers supporting only few modes.

The overall *link* MDL *σ*_{Σ,MDL}, as a key parameter for the system capacity, might be analytically estimated for the following cases (considering all MDL values in units of decibel, i.e. 10 · log_{10}(*x*)).

- In case of
*cross-talk free*MDL elements, the transfer matrix of the*k*-th element and the total link may be expressed by a diagonal matrix, where the elements define MDL. Assuming*K*concatenated, independent and statistically identical*cross-talk free*random MDL elements with per segment MDL*σ*_{S,MDL}, the overall link MDL standard deviation is - K.P. Ho and J. Kahn [7] have shown that in case of
*K*independent and identically distributed (i.i.d.) random*coupling*segments with MDL*σ*_{S,MDL}, the standard deviation of the overall link MDL slightly increases over the link of*cross talk free*MDL elements:

For the analyzed MDM systems with fiber splices and/or fiber connectors, none of these assumptions seems to hold and the overall link MDL *σ*_{Σ,MDL} has to be estimated numerically. Figure 4 compares the numerically estimated overall link MDL *σ*_{Σ,MDL} with the predictions from Eqs. (9) and (11) for splices with MDL values *σ*_{S,MDL} = 0.05dB and *σ*_{S,MDL} = 0.2dB.

Due to the highly correlated coupling of the splice points the overall link MDL *σ*_{Σ,MDL} increases tremendously over the estimated standard deviation by Eq. (11). The link MDL *σ*_{Σ,MDL} as a function of splice numbers *K* can be approximated by:

*σ*

_{S,MDL}≈ 0.05dB...0.2dB and fibers with

*M*= {3, 10, 28, 55} modes. The exponent

*x*and therefore the coupling correlation between different splice points decreases only slightly with an increased fiber splice loss $\overline{{\alpha}_{S}}$ (i.e. increased splice MDL

*σ*

_{S,MDL}). Reducing the number of guided modes leads to a minor decrease of the exponent

*x*. With this in mind, it is clear that the overall link MDL standard deviation

*σ*

_{Σ,MDL}significantly differs for MMFs with the same mean splice loss $\overline{{\alpha}_{S}}$ but different number of modes (Fig. 5), as the splice MDL

*σ*

_{S,MDL}depends not only on the splice loss $\overline{{\alpha}_{S}}$ but also on the mode number

*M*.

Later on in section 3.2, we show the relation between the overall link MDL *σ*_{Σ,MDL} and the MIMO capacity and explain, why the MDL alone is not a sufficient criterion to estimate the system MIMO capacity. But before, we discuss the ability of discrete mode permutation (or mode mixing) to reduce the overall link MDL value.

#### 3.1. Mode permutation

A possible approach to reduce the correlation of the modal coupling at different splice points and thus the overall link MDL *σ*_{Σ,MDL} is to add uncorrelated coupling to the fiber link. This could be done effectively by a mode mixing device, which randomly permutes the fiber eigen-modes every *K _{per}* splices as sketched in Fig. 6[19]. We consider an ideal mode permutator without mode dependent loss that is described by a random permutation matrix. For example, a coupling matrix for a 3 mode fiber might look like:

The authors are aware that it is rather unrealistic to practically realize this kind of mode mixing device, and that more realistic mode scramblers might introduce not negligible MDL and further cross-talk. Figure 7 shows the influence of mode permutation every *K _{per}* = {16, 8, 4, 2, 1} fiber splices to the link MDL. Even mode mixing after only

*K*= 16 splices is able to reduce the overall link MDL significantly. In this case, the link MDL can be approximated by

_{per}Adding a mode permutator after each fiber splice (*K _{per}* = 1) leads to completely uncorrelated modal coupling, which results in an overall link MDL

*σ*

_{Σ,MDL}similar as described in [7].

#### 3.2. MDL and eigenvalue distribution

In principle, it is possible to calculate the ergodic or outage capacity for arbitrary MIMO systems with the knowledge of the distribution of the eigenvalues *λ _{i}* of

*H*̱

*H*̱

^{†}[7]. In case of independent and identically distributed (i.i.d.) random coupling matrices, we know from [7] that the overall MDL distribution can be described by a Maxwellian distribution for a 2 mode system a or semi-circle distribution for a system with an infinitive number of modes (e.g. 55 modes). We also know from [8] that the relative outage capacity in i.i.d. coupling systems only slightly depends on the number of guided modes

*M*or the number of coupling sections

*K*but mainly on the link MDL. However, in this more realistic system model, based on coupling effects at splices, the outage capacity and the MDL distribution have to be estimated numerically for each fiber as they do not relate to any common probability distribution and as they depend on the number of guided modes.

Figure 8 (top left) shows as an example the eigenvalue distributions for each eigenvalue of two fibers with *M* = 3 and *M* = 28 modes and an overall link MDL *σ*_{Σ,MDL} = 8dB after *K* = 256 splices. In case of the 28 mode fiber, a clustering of the eigenvalues between 2 dB and 3 dB with an increased slope of the eigenvalue distribution can be observed. This leads to a high peak for high MDL values in the overall link MDL distribution (Fig. 8 right) and a more flat MDL distribution with long tails at low MDL values. The 3 mode fiber shows a similar overall MDL distribution, but with an even more pronounced peak at high MDL values. The resulting relative capacities *C _{r}* for these systems are plotted in Fig. 8 bottom left and show that the same MDL (here,

*σ*

_{Σ,MDL}= 8dB) for different number of guided modes leads to different outage capacities and capacity distributions with a significantly lower outage capacity for the 3 mode fiber.

The influence of mode permutation every *K _{per}* = 16 splices to the overall MDL and capacity distribution after

*K*= 256 splices is shown in Fig. 9. Due to the de-correlation of the coupling process at the splice points, a clustering of the eigenvalues, as seen in Fig. 8, cannot be observed any more in Fig. 9 top left. This leads to overall MDL distributions (Fig. 9 right) more similar to the expected distributions for i.i.d. coupling matrices. Increasing the number of mode permutators further to one per splice (

*K*= 1) does not significantly change the MDL distribution compared to mode permutation every

_{per}*K*= 16 splices. The different capacity distributions for both MMFs still show a significant lower outage capacity

_{per}*C*for the 3 mode fiber.

_{r}## 4. Outage capacity

#### 4.1. Capacity as a function of MDL

As it was shown in the previous section, the relative capacity *C _{r}* depends not only on the overall link MDL

*σ*

_{Σ,MDL}, but also on the number of guided modes

*M*. Figure 10 compares the relative capacity

*C*of two fibers with

_{r}*M*= 3 (Fig. 10 left) and

*M*= 55 modes (Fig. 10 right). The relative capacity

*C*of the 3 mode fiber for a given link MDL

_{r}*σ*

_{Σ,MDL}is always smaller than the capacity

*C*of the 55 mode fiber, regardless if mode permutating devices are used or not (cf. section 3.2). For example, a link MDL

_{r}*σ*

_{Σ,MDL}≈ 5dB (without mode permutation) leads to a relative capacity

*C*≈ 68% for a 3 mode fiber and to

_{r}*C*≈ 88% for a 55 mode fiber. It is also notable that mode permutation reduce the relative capacity for a given link MDL

_{r}*σ*

_{Σ,MDL}, as it changes the distribution of the MDL values (see also Fig. 9).

#### 4.2. Capacity as a Function of Splice Loss

All the previously performed simulations considered the statistical influence of modal noise and fiber splices to the MDL and therefore to the system capacity. It is reasonable to model the splice points of the MDM transmission system in a statistical manner if we are interested in the overall performance of such a system. For example, the expected mean losses of fiber splices are known and we are looking for the capacity of a link not yet deployed. But if we are only interested in the capacity statistics of one specific link, all splices of the link are fixed and the only reason for capacity fluctuations is modal noise. Figure 11 shows as an example the capacity distribution of a 28 mode MMF after *K* = {64, 128, 256} splice points with a mean splice loss
$\overline{{\alpha}_{S}}=0.05\hspace{0.17em}\text{dB}$. The ensemble of different links (color shaded graphs) lead to a relative broad capacity distribution, while the capacity distribution resulting only from modal noise (gray shaded graphs) (i.e. a single link realization) is much narrower. Thus, modal noise leads to capacity fluctuations, but the main impact in our scenario is the statistical modeling of the fiber splices. All the following results (as well as the MDL results in previous sections) are based on ensembles of different link realizations rather than on a single link realization.

Figure 12 shows the relative capacity *C _{r}* at an outage probability of 10

^{−4}as a contour plot for fibers with

*M*= {3, 10, 28, 55} modes. As expected, the capacity

*C*depends on the number of guided modes and decreases with the number of fiber splices and the mean splice loss $\overline{{\alpha}_{S}}$. Considering a mean splice loss $\overline{{\alpha}_{S}}=0.05\hspace{0.17em}\text{dB}$ for the 3 mode MMF (i.e. realistic splice loss for single mode fibers) or a realistic splice loss $\overline{{\alpha}_{S}}=0.03\hspace{0.17em}\text{dB}$ for the 55 mode fiber up to 100 splices might by tolerated for a relative capacity

_{r}*C*≥ 90%. Thus, a splice every ≈ 3km would limit the linear transmission distance with a reasonable capacity

_{r}*C*= 90% to about 300 km for both fibers. A reduced capacity of

_{r}*C*= 80% would double the transmission distances.

_{r}Figure 13 left shows the summary of Fig. 12, as it shows the relative capacity *C _{r}* as a function of the product splice loss
$\overline{{\alpha}_{S}}\times K$ number of splices. It can be seen that the relative capacity

*C*can be approximated only by the product of mean splice loss $\overline{{\alpha}_{S}}\times K$ number of splices, regardless of the number of fiber modes or the individual splice loss $\overline{{\alpha}_{S}}$ and number of splices

_{r}*K*. Figure 13 right shows the relative capacity

*C*as a function of the mean link loss $\overline{{\alpha}_{\mathrm{\Sigma}S}}$, which differs for a MMF from the product of mean splice loss $\overline{{\alpha}_{S}}\times K$ number of splices, because of the changing modal power distribution in a MMF along a spliced fiber link. For a MMF with many modes, e.g.

_{r}*M*= {28, 55}, the relative capacity can be solely estimated by the mean link loss $\overline{{\alpha}_{\mathrm{\Sigma}S}}$, as it becomes independent of the number of splices and the splice loss. But for the capacity estimation regardless the number of modes

*M*, the product splice loss $\overline{{\alpha}_{S}}\times K$ number of splices is the more suitable parameter. Links with $\overline{{\alpha}_{S}}\times K\le 2\hspace{0.17em}\text{dB}$ achieve a relative capacity of

*C*> 90%.

_{r}Figure 13 clearly shows that in MDM systems the origin of losses have an important influence to the system performance. As an example, considering a link with 10 spans and 19 dB loss per span (mode independent fiber losses only), 0 dBm signal launch power per mode and an EDFA noise figure of 5 dB, the achievable SNR* _{a}* is [20]:

*R*= 25GBaud equals the OSNR the SNR. The linear capacity for a

_{s}*M*mode system, neglecting fiber splices is:

*to SNR*

_{a}_{a,s}= 23dB. This results to a none MIMO (i.e. no MDL) capacity of

*M*· 4.59bit/s/Hz but to a reduced MDM MIMO capacity of approximately 0.7 ·

*M*· 4.59bit/s/Hz =

*M*· 3.21bit/s/Hz (cf. Fig. 13 for $\overline{{\alpha}_{s}}\times K=9\hspace{0.17em}\text{dB}$). Therefore, splice losses degrade the system performance much stronger than mode independent fiber losses.

#### Mode permutation

Figure 14 shows the influence of mode permutators to the outage capacity after *K* = 512 splices for a 3 mode and a 55 mode fiber. As expected from the MDL statistics in the previous section, the relative outage capacity increases with the number of mode permutators. Considering a splice loss
$\overline{{\alpha}_{S}}=0.05\hspace{0.17em}\text{dB}$ for the 3 mode fiber, the relative capacity *C _{r}* increases from

*C*< 50% to

_{r}*C*= 90% with one mode permutator every

_{r}*K*= 4 splices. The same number of mode permutators are required in case of the 55 mode fiber with an assumed splice loss $\overline{{\alpha}_{S}}=0.03\hspace{0.17em}\text{dB}$ to achieve

_{per}*C*= 90%. Therefore, a mode permutation every

_{r}*K*= 4 splices increases the number of tolerable splices from

_{per}*K*≈ 100 to

*K*≈ 500. In contrast to MMF links without mode permutation, the link loss (or accumulated splice loss) does not solely describe the relative capacity of these systems. Therefore, graphs relating the link loss and the relative capacity of MMF links with additional mode permutation are omitted.

Beside fiber splices, in real fibers also mode dependent fiber losses, bending losses, modal cross talk due to fiber imperfections, etc. will influence the system capacity. Assuming un-realistically that these additional coupling effects introduce uncorrelated random coupling no additional MDL, the splice loss requirements (even for systems without mode permutation) will not fall below the values for one mode permutator per splice (*K _{per}* = 1). On the contrary, additional MDL due to mode dependent fiber losses, bending losses, etc. might lead to higher requirements for the splice losses. Therefore, the results for one mode permutator per splice (

*K*= 1) can be interpreted as the minimum splice loss requirements in MMF MDM transmission systems.

_{per}## 5. Conclusion

We have analyzed the influence of fiber splices/connectors to the outage capacity and MDL of a MDM transmission system with gradient-index multi-mode fibers guiding different number of modes. Our results show that MDM long-haul transmission systems (e.g. more than 1.500 km with one splice every ≈ 3km) with a reasonable relative capacity of *C _{r}* ≥ 90% require splice losses much lower as nowadays achievable (i.e. < 0.006 dB ⋯ 0.01 dB depending on the number of modes). With conventional splice losses, the transmission distance is restricted to 300 km (i.e. 100 splice points). By using mode permutators every

*K*= 4 splices, it is possible to increase the transmission distance by a factor of 5. For systems without additional mode permutation, the relative capacity mainly depends on the mean link loss or the product splice loss $\overline{{\alpha}_{S}}\times K$ number of splices. MDM links with $\overline{{\alpha}_{S}}\times K\le 2\text{dB}$ will achieve a relative capacity of

_{per}*C*> 90%.

_{r}## Acknowledgments

This work was partly funded by Deutsche Forschungsgemeinschaft (DFG).

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