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 = [x₁, ⋯ , x_M]^T and y⃗ = [y₁, ⋯ , y_M]^T are the complex amplitudes of the eigenmodes of a MMF supporting in total 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 N₀ per eigenmode is added at the receiver. E₀ 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_ΣS = M/trace(H̱′H̱′^†). H̱′^† is the conjugate transpose of H̱′. The mode averaged signal to noise ratio (SNR) at the receiver is SNR = E₀/N₀ and is set to 20 dB for all investigations. The transmission characteristic of the MMF link at laser emission frequency f₀ is described by the M × M transmission matrix

 

Fig. 1 Spliced linear MIMO MDM transmission system.

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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_x,y = {0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7μm} 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_x,y. The used GI-MMF parameters for fibers guiding M = {3, 10, 28, 55} modes are shown in Table 1. Throughout this work only one polarization is considered.

Table 1. Multimode fiber parameters at wavelength λ = 1.55μm

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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]:

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_x,y, number of splice points K and number of guided modes M:

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_x,y 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 σ_x,y for MMFs with equally excited modes. As expected, splice losses decrease with an increased core radius.

 

Fig. 2 Mean splice loss $\overline{{\alpha }_S}$ as a function of fiber misalignment for fibers with M = {3, 10, 28, 55} modes

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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.

 

Fig. 3 Overall splice MDL σ_S,MDL (left) and mean splice MDL difference ${\overline{\text{MDL}}}_{S\mathrm{\Delta }}$ (right) as a function of mean splice loss $\overline{{\alpha }_S}$ for fibers with M = {3, 10, 28, 55} modes. The circle-marked values represent a fiber misalignment of σ_x,y = 0.4μm.

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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 · log₁₀(x)).

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.

 

Fig. 4 Overall link MDL σ_Σ,MDL as a function of accumulated splice MDL $\sqrt{K}\cdot {\sigma }_{S,\text{MDL}}$ for fibers with M = {3, 10, 28, 55} modes and splice MDL values σ_S,MDL = 0.05dB (left) and σ_S,MDL = 0.2dB (right). The red reference curve corresponds to i.i.d coupling segments as described in Eq. (11) while the black lines indicate a $\sqrt{K}$ (solid) and a K^x (dashed) dependance.

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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:

 

Fig. 5 Overall link MDL σ_Σ,MDL (left) and mean link MDL difference ${\overline{\text{MDL}}}_{\mathrm{\Sigma }\mathrm{\Delta }}$ (right) after K = 128 splice points. The circle-marked values represent a fiber misalignment of σ_x,y = 0.4μm.

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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:

 

Fig. 6 AWGN model of a spliced MDM transmission system with mode permutation every K_per splices.

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

 

Fig. 7 Overall link MDL σ_Σ,MDL as a function of accumulated splice MDL $\sqrt{K}{\sigma }_{S,\text{MDL}}$ for a link ensemble without mode permutator and a link ensemble with mode permutation every K_per = {16, 8, 4, 2, 1} splices. The black reference curve corresponds to i.i.d coupling segments as described in Eq. (11). The fiber misalignment is σ_x,y = 0.4μm. Left) 3-mode MMF with σ_S,MDL = 0.12dB, right) 55 mode MMF with σ_S,MDL = 0.08dB.

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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 H̱H̱^†[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.

 

Fig. 8 Eigenvalue distribution of H̱H̱^† (top left), cumulative distribution function (cdf) of the relative capacity C_r (bottom left) and overall distribution of MDL (right) for fibers guiding M = 3 (dashed) and M = 28 (solid) modes without mode permutation. For both fibers the overall link MDL is σ_Σ,MDL = 8dB and the number of splices is K = 256.

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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_per = 1) does not significantly change the MDL distribution compared to mode permutation every K_per = 16 splices. The different capacity distributions for both MMFs still show a significant lower outage capacity C_r for the 3 mode fiber.

 

Fig. 9 Eigenvalue distribution of H̱H̱^† (top left), cumulative distribution function (cdf) of the relative capacity C_r (bottom left) and overall distribution of MDL (right) for fibers guiding M = 3 (dashed) and M = 28 (solid) modes and mode permutation every K_per = 16 splices. For both fibers the overall link MDL is σ_Σ,MDL = 4dB and the number of splices is K = 256.

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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_r of two fibers with M = 3 (Fig. 10 left) and M = 55 modes (Fig. 10 right). The relative capacity C_r of the 3 mode fiber for a given link MDL σ_Σ,MDL is always smaller than the capacity C_r of the 55 mode fiber, regardless if mode permutating devices are used or not (cf. section 3.2). For example, a link MDL σ_Σ,MDL ≈ 5dB (without mode permutation) leads to a relative capacity C_r ≈ 68% for a 3 mode fiber and to C_r ≈ 88% for a 55 mode fiber. It is also notable that mode permutation reduce the relative capacity for a given link MDL σ_Σ,MDL, as it changes the distribution of the MDL values (see also Fig. 9).

 

Fig. 10 Relative capacity C_r at outage 10⁻⁴ as a function of overall link MDL σ_Σ,MDL for K={8, 16, 32, 64, 128, 256, 512, 1024} splice points for each fibers with M = 3 (left) and M = 55 modes (right) and varying splice losses.

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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.

 

Fig. 11 Distribution of the relative capacity C_r for a MMF with M = 28 modes after K = {64, 128, 256} splice points and a mean splice loss $\overline{{\alpha }_S}$ of 0.05dB. Gray shaded graphs represent a single MDM link realization, while color shaded graphs represent an ensemble of different link realizations. Gray and color shaded graphs are differently scaled.

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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_r 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 C_r ≥ 90%. Thus, a splice every ≈ 3km would limit the linear transmission distance with a reasonable capacity C_r = 90% to about 300 km for both fibers. A reduced capacity of C_r = 80% would double the transmission distances.

 

Fig. 12 Relative capacity C_r at outage 10⁻⁴ for a MMF with M = 3, M = 10, M = 28 and M = 55 eigenmodes.

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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_r 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 K. Figure 13 right shows the relative capacity C_r 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. 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_r > 90%.

 

Fig. 13 Relative capacity C_r at outage 10⁻⁴ as a function of (left) mean splice loss $\overline{{\alpha }_S}\times K$ number of splices and (right) mean link loss $\overline{{\alpha }_{\mathrm{\Sigma }S}}$ for fibers with M = {3, 10, 28, 55} modes, different fiber splice losses and different number of splices K.

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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]:

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_r < 50% to C_r = 90% with one mode permutator every K_per = 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 C_r = 90%. Therefore, a mode permutation every K_per = 4 splices increases the number of tolerable splices from 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.

 

Fig. 14 Relative capacity C_r at outage probability 10⁻⁴ after 512 splices as a function of splice loss $\overline{{\alpha }_S}$ for a link ensemble without mode permutation (no perm.) and a link ensemble with a mode permutator every K_per = {16, 8, 4, 2, 1} splices. Left) a MMF with 3 modes and right) a MMF with 55 modes.

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

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_per = 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 C_r > 90%.