## Abstract

Mode-dependent loss (MDL) is a major factor limiting the achievable information rate in multiple-input multiple-output space-division multiplexed systems. In this paper we show that its impact on system performance, which we quantify in terms of the capacity reduction relative to a reference MDL-free system, may depend strongly on the operation of the inline optical amplifiers. This dependency is particularly strong in low mode-count systems. In addition, we discuss ways in which the signal-to-noise ratio of the MDL-free reference system can be defined and quantify the differences in the predicted capacity loss. Finally, we stress the importance of correctly accounting for the effect of MDL on the accumulation of amplification noise.

© 2015 Optical Society of America

## 1. Introduction

One of the most relevant concerns in the context of space-division multiplexed (SDM) transmission in optical fibers is related to the possible consequences of mode-dependent loss (MDL). This topic has been considered in a number of studies [1–9] and it extends earlier work which was done on MDL in single mode fibers (where the phenomenon is generally referred to as polarization dependent loss – PDL) [10, 11]. Similarly to the case of PDL, the main expected sources of MDL are inline optical amplifiers, mode multiplexers and demultiplexers, imperfect splices between fiber sections [3], and microscopic fiber bends [6]. Yet, in addition to early PDL studies, where the preferred metric for quantifying the system impact of PDL was the deterioration of the worst channel’s signal-to noise-ratio, or of the average bit-error-ratio (BER) [11], in modern *coherent* systems it has also become customary to quantify the impact of MDL in terms of its effect on the system information capacity [1–5]. This metric turns out to be very sensitive to the assumptions that are made on how the system is modeled and operated.

In this paper we argue that the capacity reduction caused by MDL varies significantly with system aspects that have received little attention in previous MDL studies. We show that without explicitly relating to the operation mode of the inline optical amplifiers, the effect of MDL on system capacity cannot be determined with acceptable reliability, most critically in the case of low mode-count systems. Therefore, in a meaningful analysis of the capacity impact of MDL, it is important to specify how the mode averaged signal power is manipulated at the amplification sites. Up to now, different authors have made different assumptions regarding amplification [1–5]. The most common amplification scheme is that in which an inline amplifier resets the *total* signal power (over the entire WDM band) to its value at the link input. Ideally, in this regime all frequency components of the propagating signal undergo the same amplification. Another possible scenario is one where the inline amplifier resets the mode-averaged signal power to a nominal value on a per-frequency basis. This requires the amplifier to be equipped with an adaptive frequency equalization mechanism [10], similar to a dynamic gain equalizing filter (DGEF) in conventional single-mode systems [12]. In spite of the seemingly minor difference between the two approaches, it turns out that they produce different distributions for the MDL-induced capacity loss, which is particularly important in the case of low mode-count transmission. We note that we only consider the effects of frequency equalization, whereas modal equalization, where the gain of each mode can be controlled individually [13] is not addressed in this paper.

Subsequently, we discuss the importance of clearly specifying the MDL-free system relative to which the capacity loss is evaluated. One possibility is to set the SNR of the MDL-free reference system to the ratio between the time averages of the total signal and noise powers (in all modes) measured in the MDL-impaired system. In some cases, when the MDL dynamics is too slow or when the accessible frequency range is not sufficient to allow effective averaging, the reference system SNR ends up depending on the specific MDL realization. While both definitions can be legitimate, they yield different results in terms of the MDL-induced capacity reduction, as we discuss in this paper.

Finally, we quantify the error that is incurred when the effects of MDL on the accumulated amplification noise are ignored by assuming that spatially isotropic noise is added to the transmitted signal at the receiver [2, 3].

The paper is organized as follows. In Sec. 2 we illustrate the importance of the amplification strategy when considering the MDL-induced capacity reduction. In Sec. 3 we discuss the use of the capacity loss per mode relative to a more commonly used capacity ratio when considering the impact of MDL. In Sec. 4 we discuss the importance of precisely defining the MDL-free reference system, relative to which the penalties due to MDL are evaluated, and introduce possible definitions for such reference systems. Section 5 is devoted to demonstrating the importance of accounting for the effect of MDL on the accumulated amplification noise. Finally, in Sec. 6 we provide a description of the numerical procedure that was used to obtain the simulation results that are displayed throughout this paper. The derivation of all the reported analytical results is provided in the appendix.

## 2. Optical amplifier types for SDM and their impact on performance

We consider an SDM link consisting of multiple amplified spans where modes propagate in the strong coupling regime [15, 16]. Each span is modeled as consisting of a large number of statistically independent and identically distributed MDL elements, which, in the absence of any other knowledge, are assumed to be isotropic in the generalized multidimensional Stokes space [15]. The physical meaning of such isotropy is that the propagation modes are equivalent from the standpoint of MDL statistics. The gain and the amplification noise introduced by the amplifiers are assumed to be mode-independent. We note that this is not a limiting assumption, as an amplifier suffering from mode-dependent gain and noise is equivalent to an amplifier having perfectly isotropic gain and noise preceded or followed by an MDL element. For a consistent analysis of capacity, we need to account for the fact that MDL is frequency dependent, with the correlation bandwidth being inversely proportional to the link’s modal dispersion (MD) [15,17]. While the specific MD value may vary from link to link and is still lacking sufficient experimental characterization, the correlation bandwidth is certainly much narrower than the overall WDM band. In what follows we divide the WDM band into narrow sub-bands with frequency independent MDL. In other words, each sub-band represents an independent realization of a frequency-flat MIMO-SDM channel. For each sub-band the information capacity can be readily determined [1,5], and the overall capacity follows from summing the capacities over the individual sub-bands. Note that the division into sub-bands is a standard way of evaluating the capacity on a frequency-varying channel.

We consider two possible schemes in which the inline amplifiers can operate, as illustrated in Fig. 1. In Scheme 1 the amplifiers restore the mode-averaged signal power over the entire band of frequencies. In scheme 2 the mode-averaged signal power is restored separately for each sub-band, which makes Scheme 2 a frequency-resolved power equalization scheme. In order to reconcile the present study with that of [1], we also consider a third scheme which assumes equalization on a per sub-band basis, but applied separately on each span. Namely, each amplifier restores the mode-averaged power of each sub-band to the value that it had at the input of the span. For the sake of completeness, we also consider a fourth scheme, which operates similarly to Scheme 3, except that it restores the signal power across the entire band of frequencies to its value at the beginning of the span. If we denote by **M*** _{k}* the transfer matrix of the

*k*-th span (including the

*k*-th amplifier), and by

**T**

*=*

_{k}**M**

_{k}**M**

_{k−1}...

**M**

_{1}the transfer matrix of the link between the transmitter and the end of the

*k*-th span, the described schemes can be summarized as shown in Table 1, where 2

*N*is the total number of transmitted modes, with the factor of 2 accounting for polarization. The angled brackets denote ensemble averaging, which, under the assumption of frequency ergodicity, is equivalent to frequency averaging over all sub-bands making up the full WDM band. This equivalence requires that the WDM band is sufficiently large relative to the spectral correlation of MDL. Although the accuracy of this assumption might be questionable in the very beginning of the link (where MD may still be small, and hence the correlation bandwidth may be large), it becomes more and more justified when the accumulated MD exceeds the inverse WDM bandwidth. In the limit of strong mode coupling within the individual spans, as we show in the appendix, Scheme 4 is rigorously equivalent to Scheme 1, in spite of the fact that the implementations of the two schemes are very different. For this reason we do not refer to Scheme 4 explicitly in what follows.

Throughout the paper we use the formalism introduced in [5] where all the system consequences of MDL are captured by four quantities; the MDL vector Γ⃗, the noise coherency vector Γ⃗′, the mode averaged signal loss *γ*_{0}, and the mode averaged noise power *γ′*_{0}. However, unlike in [5], we define *γ′*_{0} as a unitless coefficient which similarly to *γ*_{0} is unity in the absence of MDL. Thus the overall mode-averaged noise power (which was denoted by *γ′*_{0} in [5]) is now given by *P _{n}γ′*

_{0}, where

*P*is the sum of the noise powers generated per mode by the amplifiers. While the analytical results that we present here and derive in Sec. 7 are obtained for the case of distributed amplification, we show in what follows that they also hold in realistic system scenarios assuming lumped optical amplifiers. In the present analysis we quantify the impact of MDL on system capacity by means of the capacity loss per mode

_{n}*ℓ*, which is defined as

*C*is the actual system capacity and where

*C*

_{ref}is the capacity of the MDL-free system that is assumed as a reference. The difference between this metric and the frequently encountered capacity ratio

*C/C*

_{ref}will be discussed in Sec. 3. Note that the choice of the reference system is important in assessing the impact of MDL. We assume that the reference system operates at an SNR of ${S}_{\text{ref}}={S}_{0}\frac{\u3008{\gamma}_{0}\u3009}{\u3008{\gamma}_{0}^{\prime}\u3009}$, which is the value that one would obtain by measuring the ratio between the aggregate signal and noise powers averaged over time or frequency. Also, we assume that no channel state information is available at the transmitter, which hence excites the 2

*N*modes with the same average power. The capacity of the reference system is in this case

*C*

_{ref}= 2

*N*log

_{2}(1 +

*S*

_{ref}). First, we argue that the average capacity loss due to MDL is the same for all the considered amplification schemes and it is given by where Γ is the length of the MDL vector Γ⃗. The difference between the schemes is in the

*fluctuations*of the capacity loss, which is characterized by the variance of

*ℓ*, and consequently determines the system’s outage behavior. As shown in the appendix, in the case of Scheme 1, where the amplifiers are set to reproduce the total mode-averaged output power in the WDM band, the variance of the MDL-induced capacity reduction is given by For Scheme 2, where the amplifiers equalize the mode-averaged signal power separately for the individual sub-bands, the variance is given by

*L*is the overall fiber length and

*α*

_{0}is the mode-averaged fiber loss coefficient. For long fiber links, Eq. (4) reduces to ${\sigma}_{2}^{2}\simeq 22{\sigma}_{1}^{2}$. An equivalent expression for Scheme 3 could not be derived, and hence its performance will only be analyzed numerically in what follows.

An important consequence of Eqs. (3) and (4) is that in the regime of large mode counts the fluctuations of the capacity loss per mode become negligible, as can be seen from the fact that their standard deviation is proportional to ∼ 〈*ℓ*〉/2*N*. This observation suggests that in such a regime only a small margin needs to be allocated to prevent MDL-induced outages. Of course, as will be discussed later, the situation is significantly different in systems with low mode counts.

The different behaviors of the three schemes are compared in Fig. 2, where we plot the probability density function (PDF) and the complementary cumulative distribution function (CCDF) of *ℓ*, in the left and right panels, respectively, in the limit of large SNR [18]. These plots were produced for a system consisting of 20×100 km amplified spans of fiber supporting *N* = 10 spatial modes (this mode count is consistent with the situation of a weakly guiding fiber with normalized frequency *V* ≃ 6, where LP_{01}, LP_{11}, LP_{21}, LP_{02}, LP_{31}, and LP_{12} modes are guided). The mean MDL of the link was 24.6 dB (corresponding to
$24.6/\sqrt{20}\simeq 5.5\hspace{0.17em}\text{dB}$ per span), and the details on how the simulations were conducted are provided in Section 6. As predicted by the theory, the average capacity loss per mode 〈*ℓ*〉 is the same for the three schemes, but the widths of the distributions are notably different, with *σ*_{2}/*σ*_{1} ≃ 5.3 and *σ*_{3}/*σ*_{1} ≃ 1.9. The theoretically predicted ratio given by Eq. (4) is *σ*_{2}/*σ*_{1} = 4.7. Given the large MDL value used in the simulation, the accuracy of the analytical result (derived in the limit of small to moderate MDL values) is quite impressive. The right panel of Fig. 2 can be used to extract the outage capacity loss per mode in the case where the MDL is approximately constant within the bandwidth of a WDM channel. For example, when setting the outage probability to the value of 10^{−4}, the outage capacity loss per mode *ℓ*_{out} for scheme 2 is 0.2 bits larger than in the case of scheme 1.

These points are further addressed in Fig. 3, where we plot the average and the outage values of the capacity loss per mode as a function of the mean MDL of the link, for the two cases of *N* = 3 and *N* = 10. The difference in the outage capacities between the amplification schemes is more evident in the case *N* = 3, and in general it decreases when the number of modes increases. In fact, the difference between the outage capacity values of any two schemes *a* and *b*, denoted by *ℓ*_{out,}* _{a}* and

*ℓ*

_{out,}

*, is proportional to the difference between the corresponding standard deviations of the capacity loss per mode,*

_{b}*σ*and

_{a}*σ*. In the case of Schemes 1 and 2 one can use Eqs. (3) and (4) to show that

_{b}*σ*

_{2}−

*σ*

_{1}∝ 〈

*ℓ*〉/2

*N*, which is inversely proportional to the number of modes, consistent with Fig. 3.

## 3. Capacity loss per mode vs. capacity ratio

In this section we discuss the role of quantifying the impact of MDL in terms of the capacity loss per mode *ℓ* = (*C*_{ref} − *C*)/2*N*, as opposed to the more frequently used capacity ratio *C/C*_{ref} [1–4, 6]. While both quantities are reliable figures of merit, the capacity loss per mode, in the relevant limit of high SNR, becomes independent of *S*_{0} and reflects only the MDL parameters of the system, which makes this metric attractive. This point is illustrated in Fig. 4, where we plot the average capacity loss per mode in the top left panel and the average relative capacity in top right panel, versus the average link MDL. It is evident from the figure that the curves in the left panel become practically indistinguishable when *S*_{0} exceeds 20 dB, whereas in the right panel the curves remain distinctively different. The curves are identical for all the considered amplification schemes. As shown in bottom panels of the same figure, when plotting the outage capacity loss per mode versus the average link MDL, a similar behavior is observed, namely the outage capacity loss per mode becomes independent of the SNR for large SNR values. However the outage capacity loss curves are different for the three amplification schemes.

## 4. Impact of specifying the SNR in MDL-impaired systems

The dependence of the MDL-induced penalty on the amplification scheme shown in Fig. 3 exemplifies the sensitivity of the MDL-induced capacity reduction to certain details of system operation. We show in what follows that this sensitivity also extends to two system modeling aspects. These are the definition of the MDL-free reference system, which is discussed in this section, and the modeling of the amplification noise, which is the subject of the next section.

A subtle aspect of MDL is that, when looking at the system information capacity, the randomness of mode-averaged signal and noise powers is as important as the fact that the individual eigenmodes are characterized by random and different attenuations. In our formalism, this is equivalent to claiming that the randomness of *γ*_{0} and *γ′*_{0} is as important as the randomness of Γ⃗ and Γ⃗′. In order to illustrate this, we recall the expression for the system information capacity which has been derived in the regime of large SNR in [5]

*C*

_{MA}, where the subscript MA stands for “mode-averaged.” The third term is the capacity penalty caused directly by the existence of differential losses and gains, which are described by the MDL vector Γ⃗ and by the noise coherency vector Γ⃗′, and we denote this term by Δ

*C*

_{MD}, where the subscript MD stands for “mode-dependent.” In the left panel of Fig. 5 we show the ratio Δ

*C*

_{MA}/Δ

*C*

_{MD}in the form of a scatter plot in a range of average link MDL values. It can be seen from the figure that the magnitude of Δ

*C*

_{MA}is non-negligible as it reaches values as large as one quarter of the mode-dependent term Δ

*C*

_{MD}.

In order to see why this is the case, we recall that the end-to-end MDL of a link is the result of many local perturbations such as imperfect splices, micro-bends, etc. These make the attenuation coefficients of the *local* eigenmodes random quantities, which can be characterized by a standard deviation *σ _{α}*. In the regime of small MDL, Γ⃗ and Γ⃗′ have been shown in [5] to scale with

*σ*, whereas we show in the appendix that

_{α}*γ*

_{0}= 1 + Δ

*γ*

_{0}and

*γ′*

_{0}= 1 + Δ

*γ′*

_{0}, where Δ

*γ*

_{0}and Δ

*γ′*

_{0}scale with ${\sigma}_{\alpha}^{2}$. In this regime Eq. (5) reduces to

*γ*

_{0}and Δ

*γ′*

_{0}, and quadratically on Γ and Γ′, so that all of the contributions scale with ${\sigma}_{\alpha}^{2}$, and they are therefore of similar importance. This situation is in contrast with MDL studies where the preferred metric is the deterioration in the worst channel’s signal-to noise-ratio, or in the average bit-error-rate (BER) [9,11]. There the relevant quantities depend linearly on all four parameters Δ

*γ*

_{0}, Δ

*γ′*

_{0}, Γ⃗, and Γ⃗′, and therefore the contribution of Δ

*γ*

_{0}and Δ

*γ′*

_{0}is negligible. Indeed it has been shown in [9] that the dependence of the BER on the fluctuations of the ratio between the mode-averaged signal and noise powers becomes noticeable only in the regime of very large MDL values.

The above discussion is closely related to the question of which reference system one should use to quantify the effect of MDL on capacity. In one case, one may opt to choose a reference system operating at an SNR value of *S*_{0} = *P _{s}/P_{n}*, which is the SNR that one would measure if it were possible to remove all of the local MDL sources [3]. Obviously this approach is not practical since the only measurable quantities at the system output are

*γ*

_{0}

*P*and

_{s}*γ′*

_{0}

*P*. Since these are random quantities, an acceptable definition would be that the reference SNR is set to ${S}_{\text{ref}}={S}_{0}\frac{\u3008{\gamma}_{0}\u3009}{\u3008{\gamma}_{0}^{\prime}\u3009}$, ad we assumed in this work. The ensemble averaging involved in the definition of

_{n}*S*

_{ref}can be approximated by averaging the aggregate signal and noise powers either in time or in frequency. On the other hand, if proper averaging cannot be performed (when the timescale of MDL dynamics is too slow [20], or if one does not have access to a sufficiently broad spectrum in an operating system), one can only resort to defining the reference SNR as ${S}_{\text{ref}}^{\prime}={S}_{0}\frac{{\gamma}_{0}}{{\gamma}_{0}^{\prime}}$, which is what was assumed in [5]. It is also customary to set the SNR to

*S′*

_{ref}when evaluating the relation between SNR and BER in simulated systems [9].

Interestingly, in a relevant range of MDL values, *S*_{ref} turns out to be practically identical to *S*_{0}, as can be seen in the right panel of Fig. 5, where we plot the ratio
$\frac{\u3008{\gamma}_{0}\u3009}{\u3008{\gamma}_{0}^{\prime}\u3009}$ as a function of the average link MDL. The situation changes drastically when setting the reference SNR to *S′*_{ref}, as we illustrate in Fig. 6. In the left and center panels of the figure we compare the PDF and the CCDF obtained for amplification scheme 1 by setting the reference SNR to *S′*_{ref} with the corresponding distributions of Fig. 2, which are shown by thin curves. While the average capacity loss is the same for the two choices of reference system, as we demonstrate in the appendix, the variance of the capacity loss per mode in this case is
${\sigma}_{1}^{\prime 2}=55{\sigma}_{1}^{2}$, consistent with the broadening of the distributions seen in the figure, and with the corresponding increase in outage capacity loss per mode (the computed ratio between
${\sigma}_{1}^{\prime 2}$ and
${\sigma}_{1}^{2}$ is approximately 62). A direct comparison of the outage capacity loss obtained for the two settings of the reference SNR (assuming an outage probability of 10^{−4}) is presented in the right panel. Finally, we note that when setting the reference SNR to *S′*_{ref} the MDL impact on system performance is practically unaffected by the assumed amplification scheme.

## 5. Noise loading

Another important and often overlooked aspect of modeling MDL has to do with whether or not the effect of MDL on the amplification noise is accounted for – an issue that has been addressed to some extent in [1, 5, 21]. In all of the above results, noise was assumed to be added to the propagating signal distributed at each amplification site. If instead spatially isotropic noise is assumed to be added at the receiver, as in [2,3], the average capacity loss per mode *ℓ* increases by a factor of up to 3/2 relative to the actual value.

In Fig. 7 we illustrate the importance of correctly accounting for the effect of MDL on the noise spatial coherency, where the amplifiers are operated according to scheme 1. Here, the average capacity loss when the noise is added at the end is significantly larger than the actual average capacity loss. This can clearly be seen in the left panel, which shows the PDF of capacity loss *ℓ*, with noise properly distributed (thin curve, identical to the corresponding curve in Fig. 2) and when spatially isotropic noise is added at the receiver (thick curve). Note that when the noise is added at the receiver, the PDF of the capacity loss approaches a delta function (we show in the appendix that the variance of *ℓ* reduces to zero to first order in
${\sigma}_{\alpha}^{2}$). The right panel of Fig. 7 shows the ratio between the average capacity loss calculated when assuming noise loading at the receiver and the actual capacity versus the average link MDL. The plot shows that the ratio of 1.5 derived in the regime of small MDL remains approximately valid for large MDL values.

## 6. Numerical simulations

In this section we provide the details concerning the simulations that have been performed in the context of this paper and whose results are displayed in Figs. 2 – 7. In all our simulations we assumed the regime of strong mode coupling.

The simulated system contained *N _{s}* = 20 amplified spans, with every span constructed of

*M*= 100 MDL elements, each characterized by a local MDL vector

*α⃗*. This choice ensures that each MDL element contributes incrementally to the overall MDL, consistently with the theory. Cases in which few MDL elements are responsible for dominant contributions to the overall MDL are not treated in this paper. We verified that varying the number of spans within a relevant range of values does not affect the results. Each vector

_{k,j}*α⃗*is independently drawn at random from an isotropic Gaussian distribution with the value of 〈

_{k,j}*α*〉

^{2}≡ 〈|

*α*|

_{k,j}^{2}〉 being a parameter. The transfer matrix

**A**

*of the*

_{k,j}*j*-th MDL element in the

*k*-th span is first set to the matrix $\text{exp}\left(\frac{1}{2}{\overrightarrow{\alpha}}_{k,j}\cdot \overrightarrow{\mathrm{\Lambda}}\right)$ and then normalized such that the average trace of ${\mathbf{A}}_{k,j}{\mathbf{A}}_{k,j}^{\u2020}$ is equal to 2

*N*. In the limit where the magnitude of the local MDL vector

*α⃗*is small, it can be shown that this normalization is equivalent to setting ${\mathbf{A}}_{k,j}=\text{exp}\left(-\frac{1}{4}\u3008{\alpha}^{2}\u3009\right)\text{exp}\left(\frac{1}{2}{\overrightarrow{\alpha}}_{k,j}\cdot \overrightarrow{\mathrm{\Lambda}}\right)$. The transfer matrix of the

_{k,j}*k*-th span is ${\mathbf{M}}_{k}=\sqrt{{G}_{k}}\text{exp}\left(-\frac{1}{2}\frac{{\alpha}_{0}L}{{N}_{s}}\right){\mathbf{A}}_{k,M}{\mathbf{A}}_{k,M-1}\dots {\mathbf{A}}_{k,1}$, where

*G*is the gain of the

_{k}*k*-th amplifier. The transfer matrix describing propagation from the link input to the end of the

*h*-th span is

**T**

*=*

_{h}**M**

_{h}**M**

_{h−1}...

**M**

_{1}.

When simulating amplification scheme 2, *G _{k}* is set so that the trace of
${\mathbf{T}}_{h}{\mathbf{T}}_{h}^{\u2020}=2N$ for all values of

*h*. Similarly, in the case of scheme 3,

*G*is set so that the trace of ${\mathbf{M}}_{k}{\mathbf{M}}_{k}^{\u2020}=2N$ for all values of

_{k}*k*. The case of scheme 1 is somewhat less straightforward to simulate, as it requires the computation of ensemble averages prior to setting the individual gain values. We show in the appendix that in the limit where every span contains a large number of small MDL contributions, this is equivalent to setting ${G}_{k}=\text{exp}\left(\frac{{\alpha}_{0}L}{{N}_{s}}\right)$.

The noise coherency matrix is computed as
$\mathbf{Q}=({G}_{{N}_{s}}-1)\mathbf{I}+{\sum}_{k=2}^{{N}_{s}}({G}_{k-1}-1){\mathbf{F}}_{k}{\mathbf{F}}_{k}^{\u2020}$, where **F*** _{k}* =

**M**

_{Ns}...

**M**

*is the transfer function of the last (*

_{k}*N*−

_{s}*k*+ 1) spans, and where we used the fact that the power of the amplification noise generated by an amplifier with gain

*G*is proportional to

*G*− 1. In all numerical examples we assumed that the mode-averaged loss coefficient is

*α*

_{0}= 0.0461 Np/km, corresponding to 0.2 dB/km.

We verified that simulations of the same system based on the numerical method presented in [1] yield results which are in full agreement with those presented here.

## 7. Conclusions

Mode-dependent loss is expected to be a major issue in MIMO-SDM systems. In this paper we have shown that the accurate assessment of its impact on system performance requires carefully addressing a number of aspects that have received little attention in previous studies.

To quantify the impact of MDL we assumed as a metric the capacity loss per mode, which is defined as *ℓ* = (*C*_{ref} − *C*_{0})/2*N*, where *C* is the capacity of the MDL-impaired system, *C*_{ref} is the capacity of an MDL-free system which is assumed as a reference, and 2*N* is the number of modes, including polarizations. We showed that the average value of *ℓ* is independent of how the inline optical amplifiers are operated. In contrast, its fluctuations - and hence the corresponding outage values - are proportional to 〈*ℓ*〉/2*N*, with the proportionality coefficient ranging between
$1/\sqrt{20}$ and
$\sqrt{55}/\sqrt{20}$, depending on the amplification scheme and on the assumptions that are made on the MDL-free reference system. Since the size of these fluctuations reduces with the number of modes, the impact of the amplification scheme is more significant in systems with low mode counts. In the regime of large mode counts the capacity loss fluctuations become very small and, as a result, no significant margin is to be allocated in this case for MDL-induced outages.

We also discussed the choice of metrics to quantify the impact of MDL on system performance. We showed that the capacity loss per mode *ℓ*, compared to the frequently used capacity ratio *C/C*_{ref}, has the advantage of depending very weakly on the SNR value in the regime of large SNR.

The sensitivity of the capacity loss on the amplification scheme arises from the MDL-induced fluctuations of the mode-averaged signal and noise powers. We showed that the same fluctu-ations are responsible for the dependence of *ℓ* on the choice of the MDL-free system that is assumed as a reference to evaluate the MDL-induced capacity reduction.

Finally, we quantified the error that is incurred when neglecting the effect of MDL on amplification noise. The capacity loss per mode is approximately 50% larger if one assumes, as is sometimes done in MDL studies, that isotropic noise is added to the signal at the receiver, as compared to the case where ASE noise is properly assumed to be loaded at each inline amplifier.

## Appendix: Analysis

The information capacity of an SDM system where 2*N* modes (including polarizations) are transmitted, in the absence of channel state information at the transmitter, is given by

*S*

_{0}=

*P*is the SNR that would be measured in the absence of MDL,

_{s}/P_{n}**T**is the 2

*N*× 2

*N*channel transfer matrix, and

**I**denotes the identity. The matrix

**Q**is the noise space coherency matrix normalized to the MDL-free noise power

*P*, such that its trace in the absence of MDL is equal to 2

_{n}*N*. We recall that the matrices

**TT**

^{†}and

**Q**are Hermitian, and hence they can be expressed as

*γ*

_{0}is the mode-averaged channel attenuation, Γ⃗ is the MDL vector,

*γ′*

_{0}is the mode-averaged noise power, and Γ⃗′ is the noise coherency vector. For an extensive description of the formalism used in this Appendix, the reader is referred to [5]. In the limit of large signal-to-noise ratio and small MDL, Eq. (7) can be approximated by

*γ*and

*γ′*denote the lengths of the vectors

*γ⃗*=

*γ*

_{0}Γ⃗ and

*γ⃗′*=

*γ′*

_{0}Γ⃗′, respectively, and where the second equality on the right-hand side of Eq. (9) is correct to first order in Γ

^{2}and Γ

*′*

^{2}. After expanding the logarithm, the above yields

*ℓ*, which in the limit of large SNR reduces to

*γ*

_{0}〉/〈

*γ′*

_{0}〉. The equation for ${\gamma}_{0}^{2}-{\gamma}^{2}$ follows from the equations for

*γ*

_{0}and

*γ⃗*in [5], in which we neglect terms that are of second order in MDL in the equation for

*γ⃗*and of third order in MDL in the equation for

*γ*

_{0}(as required for consistency since Eq. (11) involves the first power of

*γ*

_{0}, but only the second power of

*γ*),

*α′*. As will become evident in what follows, this term was introduced to ensure that 〈

*γ*

_{0}(

*z*)〉 which represents the ensemble average of the mode averaged gain, is equal to exp[(

*g*−

*α*

_{0})

*z*], as would be the case in an MDL-free system. We now approximate the local MDL vector

*α⃗*as a white Gaussian noise process, in which case

*α⃗*d

*z*is an increment d

*W⃗*of a three-dimensional Wiener process and its variance $\text{d}{W}^{2}={\sigma}_{\alpha}^{2}\text{d}z$ represents the magnitude of the local MDL along the system. Rewriting Eqs. (12) in the standard Ito form [22] yields

*α′*needs to be set to ${\alpha}^{\prime}={\sigma}_{\alpha}^{2}/2$. Substituting this value in Eqs. (13) and (14), conveniently yields

*γ*(0) = 0 and

*γ*

_{0}(0) = 1. Using the same procedure, the equations for noise parameters

*γ′*

_{0}and

*γ⃗′*that were introduced in [5] assume the form

*γ′*(0) =

*γ′*

_{0}(0) = 0 is given by

*γ′*

_{0}is the solution of the first of Eqs. (18),

*γ⃗′*(

*z′*) is obtained by solving the second of Eqs. (18),

*γ′*

_{0}with its zeroth order expression

*γ′*

_{0}=

*z/L*, obtained from the first of Eqs. (18) by setting

*g*=

*α*

_{0}in the absence of MDL.

In what follows we assume that the quantity *α*_{0} − *g*, and hence *I*, is first order in
${\sigma}_{\alpha}^{2}$. We will verify that for amplification schemes 1 and 2 this is indeed the case. Note that this is also consistent with the fact that the link net gain is to be close to one (in the absence of MDL we assume that this is exactly one). Under this assumption Eqs. (22) and (21) reduce to

*α*

_{0}−

*g*= −d

*I*(

*z′*,

*z*)/d

*z′*and

*I*(0,

*z′*) +

*I*(

*z′*,

*z*) =

*I*(0,

*z*), the expression for

*ℓ*becomes

*z*caused by gain nonuniformity along the link.

We are left with the calculation of 〈*γ′*_{0}〉 and 〈*γ*_{0}〉. For a given amplification scheme, the value of 〈*γ′*_{0}〉 can be evaluated by averaging the second of Eqs. (23), which involves evaluating the ensemble average of *I*(*z′*, *z*). This task can only be accomplished by assuming some amplification strategy. Finally, we can evaluate 〈*γ*_{0}〉 using for *γ*_{0} the following expression, which, along with that of *γ⃗*, is obtained through a procedure similar to the one followed in the derivation of Eqs. (23),

## Amplification scheme 1

In our formalism the operation of Scheme 1 corresponds to enforcing the condition 〈d*γ*_{0}(*z*)〉 = 0 in the first of Eqs. (16), which is obtained by setting *g* = *α*_{0}. In this case *I* = 0 and 〈*γ′*_{0}〉 = *z/L*, 〈*γ*_{0}〉 = 1, and the capacity loss per mode reduces to

*W⃗*and d

_{z′}*W⃗*never overlap [22], and hence the average capacity loss per mode is The fluctuations of the capacity loss per mode are given by

_{z″}*f*(

*z′*) = 0 can be used to obtain the desired result, In the derivation of (29) we used the properties

*g*= 0) the first of Eqs. (15) yields 〈

*γ*

_{0}(

*z*+

*L*)〉 =

_{s}*γ*

_{0}(

*z*) exp(−

*α*

_{0}

*L*). For this reason, in the simulations of Scheme 1 the gain of each amplifier was set to

_{s}*G*= exp(

_{k}*α*

_{0}

*L*), so that 〈

_{s}*γ*

_{0}〉 = 1 at the end of each amplified span of length

*L*.

_{s}#### 7.0.1. Equivalence between Scheme 1 and Scheme 4

We express the matrix
${\mathbf{T}}_{k}^{\u2020}{\mathbf{T}}_{k}$ of Table 1 as
${\mathbf{T}}_{k}^{\u2020}{\mathbf{T}}_{k}={t}_{k}\mathbf{I}+{\overrightarrow{\mathrm{\Gamma}}}_{k}\cdot \overrightarrow{\mathrm{\Lambda}}$, where
${t}_{k}=\text{Tr}\left[{\mathbf{T}}_{k-1}^{\u2020}{\mathbf{T}}_{k-1}\right]/2N$ and Γ⃗* _{k}* are the mode-averaged loss and the MDL vector of the

*k*-th span, respectively. This gives ${\mathbf{M}}_{k}^{\u2020}{\mathbf{M}}_{k}={\mathbf{M}}_{k-1}^{\u2020}{\mathbf{T}}_{k}^{\u2020}{\mathbf{T}}_{k}{\mathbf{M}}_{k-1}={t}_{k}{\mathbf{M}}_{k-1}^{\u2020}{\mathbf{M}}_{k-1}+{\mathbf{M}}_{k-1}^{\u2020}\left({\overrightarrow{\mathrm{\Gamma}}}_{k}\cdot \overrightarrow{\mathrm{\Lambda}}\right){\mathbf{M}}_{k-1}$. By averaging the two sides of this equation and taking their traces, we obtain $\u3008\text{Tr}\left[{\mathbf{M}}_{k}^{\u2020}{\mathbf{M}}_{k}\right]\u3009=\frac{\u3008\text{Tr}\left[{\mathbf{T}}_{k-1}^{\u2020}{\mathbf{T}}_{k-1}\right]\u3009}{2N}\u3008\text{Tr}\left[{\mathbf{M}}_{k-1}^{\u2020}{\mathbf{M}}_{k-1}\right]\u3009$, where we used the fact that different spans are statistically independent in terms of MDL, and in the strong coupling regime within each span 〈Γ⃗

*〉 = 0 [5]. The derived equality implies that $\u3008\text{Tr}\left[{\mathbf{M}}_{k}^{\u2020}{\mathbf{M}}_{k}\right]\u3009=2N$ if and only if $\u3008\text{Tr}\left[{\mathbf{T}}_{k}^{\u2020}{\mathbf{T}}_{k}\right]\u3009=2N$, which proves the equivalence between the two schemes.*

_{n}## Amplification scheme 2

In our formalism the operation of Scheme 2 corresponds to enforcing the condition d*γ*_{0} = 0 in the first of Eqs. (16). This gives

*I*〉 = 0, and hence 〈

*γ′*

_{0}〉 =

*z/L*, 〈

*γ*

_{0}〉 = 1, and the capacity loss per mode becomes

*ℓ*we use Eq. (29) with $f({z}^{\prime})=L-\frac{L}{{\alpha}_{0}(L-{z}^{\prime})}$, and obtain

## MDL-free reference system operating at S′_{ref}

In this case, the capacity of the MDL-free reference system for large SNR values is

*γ*

^{2}≃ Γ

^{2}and

*γ′*

^{2}≃ Γ′

^{2}. Using the firsts of Eqs. (23) and (25), the above can be expressed as

*I*and

*g*, shows that the capacity loss is in this case independent of the amplification scheme that is assumed. Using the relation $\u3008\text{d}{\overrightarrow{W}}_{{z}^{\prime}}\cdot \text{d}{\overrightarrow{W}}_{{z}^{\u2033}}\u3009={\sigma}_{\alpha}^{2}\delta ({z}^{\prime}-{z}^{\u2033})\text{d}{z}^{\prime}\text{d}{z}^{\u2033}$, one can see that the average of the first integral is ${\sigma}_{\alpha}^{2}L$, while that of the second integral is ${\sigma}_{\alpha}^{2}{L}^{3}/3$, which confirms that the average capacity loss is also in this case $\u3008\ell \u3009={\sigma}_{\alpha}^{2}L/3\text{ln}(2)$, as already found in [5]. The evaluation of the mean square value of

*ℓ*involves solving the following averages,

*ℓ*, which in this case we denote by ${\sigma}_{1}^{\prime 2}$, is found to be

## Noise at the receiver

Under the assumption of isotropic noise loading at the receiver the noise coherency matrix reduces to the identity, **Q** = **I**, with the result that *γ′*_{0} = 1 and *γ⃗′* = 0. The expression for the capacity loss per mode reduces in this case to

*γ*

_{0}〉 = 1, which holds for both amplification schemes 1 and 2. Equation (41) shows that for amplification schemes 1 and 2, owing to the relation 〈

*I*(0,

*L*)〉 = 0, the average capacity loss per mode is $\u3008{\ell}_{\text{RX}}\u3009=\frac{{\sigma}_{\alpha}^{2}L}{2\text{ln}(2)}=\frac{3}{2}\u3008\ell \u3009$. Moreover, for Scheme 1

*I*(0,

*L*) = 0, with the result that ${\ell}_{\text{RX}}=\frac{1}{2\text{ln}(2)}{\sigma}_{\alpha}^{2}L$, which shows that to first order in ${\sigma}_{\alpha}^{2}$ the capacity loss per is deterministic, consistent with the delta-like PDF in plotted in Fig. 7.

## Acknowledgments

C. Antonelli and A. Mecozzi acknowledge financial support from the Italian Ministry of University and Research through ROAD-NGN project ( PRIN2010–2011). M. Shtaif acknowledges financial support from Israel Science Foundation (grant 737/12) and the Terasanta consortium.

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