Capacity Enhancement of Few-Mode Fiber Transmission Systems Impaired by Mode-Dependent Loss

Space-division multiplexing over few-mode fibers is a promising solution to increase the capacity of the future generation of optical transmission systems. Mode-dependent loss (MDL) is known to have a detrimental impact on the capacity of few-mode fiber systems. In the presence of MDL, spatial modes experience different attenuations which results in capacity reduction. In this work, we propose a digital signal processing solution and an optical solution to mitigate the impact of MDL and improve the channel capacity. First, we show that statistical channel state information can be used for a better power allocation for spatial modes instead of equal launch power to increase the system capacity. Afterwards, we propose a deterministic mode scrambling strategy to efficiently reduces the impact of MDL and improves few-mode fiber systems capacity. This scrambling strategy can be efficiently combined with the optimal power allocation to further enhance the capacity. Through numerical simulations of the average and outage capacities, we show that the proposed techniques bring significant capacity gains.


Introduction
The last two decades have known an exponential growth in the demand for optical network bandwidth.Since single-mode fiber (SMF) systems are approaching the nonlinear Shannon limit, space-division multiplexing (SDM) holds the promise to increase the capacity of future optical transmission systems [1,2].SDM can be realized either by few-mode fibers (FMFs) that allow the propagation of more than one spatial mode or multi-core fibers (MCFs) where each core can be single mode or few-mode [3].In all schemes, cross-talk is inevitable especially if cores are close in MCFs or if the differential modal group delay (DMGD) is close to zero in FMFs.In both cases, it is compulsory to use multiple-input multiple-output (MIMO) approaches (already used in wireless communications) to recover the signals.In our work, we are interested in FMF sysytems.The impact of DMGD and unitary modal coupling in FMFs is very important to determine the MIMO equalizer complexity at the receiver [4,5].Moreover, in FMF systems, propagating modes can also be affected by a non-unitary cross-talk known as mode-dependent loss (MDL) that can be either distributed through the fiber or local at the insertion of optical components.MDL arising from the fiber line is due to manufacturing imperfections such as splices or micro-bends.Accumulated splice losses as low as 0.03 dB may lead to system outage after only a few hundred of kilometers [6].Another source of MDL comes from optical components such as multiplexers/de-multiplexers (MUX/DEMUX) [7,8] and few-mode optical amplifiers [9,10].
To mitigate the MDL effect, different optical and digital signal processing (DSP) solutions were proposed [11,12].The availability of information about the FMF channel at the transmitter known as channel state information (CSI) is an important factor that governs the capacity of FMF systems impaired by MDL.In the absence of CSI, all modes are excited with equal launch power which results in capacity reduction when modes have different attenuations.However, with the availability of CSI at the transmitter, capacity can be improved by allocating optimal powers to spatial modes [12].Additionally, strong mode coupling was reported to reduce the accumulated MDL and hence results in a better system capacity [11].Moreover, modal coupling can be intentionally added at local points thanks to mode scramblers [13].
In this work, we investigate the FMF channel capacity impaired by MDL.First, we review our proposed statistical CSI that improves the channel capacity independently of channel variations [14] .Afterwards, we present a deterministic scrambling strategy to better average losses between spatial modes and reduce the accumulated MDL.Finally, we show that mode scrambling and the statistical CSI can be used together to reach higher capacities.

Few-Mode Fiber Channel Model
Optical devices such as few-mode amplifiers and multiplexers represent the main sources of MDL [7][8][9][10].Nonetheless misaligned fiber splices also lead to an important end-to-end accumulated MDL [6].In our work, we focus on distributed MDL arising from fiber splices and micro-bends.We consider graded-index FMFs (GI-FMFs) having parabolic profile with core radius r c and a numerical aperture 205, where n c (resp.n cl ) is the refractive index of the core of the fiber (resp. of the cladding).GI-FMFs have a low DMGD and more modal coupling is obtained than step index fibers [15].The wavelength is set to λ = 1.55 µm, the number of propagating modes depends on the core radius r c , the wavelength λ and the numerical aperture N A. In our work we fix λ and N A and we increase r c to allow the propagation of higher order modes.In Table 1, we report the parameters of the GI-FMFs [6].We can express the transverse field distribution of the linearly-polarized (LP) modes with a good approximation by Laguerre-Gauss polynomials as in [16]: The field distribution of the last equation is written in polar coordinates given by r and φ. l denotes the circumferential order and q is the radial order of the LP lq mode.L (l) q (x) is the generalized Laguerre polynomial of order l and degree q and w = r c /(k 0 N A) is the spot size of the fundamental mode LP 01 with k 0 = 2π/λ is the free space wavenumber.C l,q is a normalization factor chosen to fulfill the orthogonal relation between spatial modes A E i E j dA = δ i,j .
In our capacity analysis, we consider a physical description of the propagation through GI-FMFs with realistic non-unitary modal coupling generated by fiber misalignments.This modelization is based on the electrical field distributions of spatial modes as described before rather than a mathematical modeling based on random matrices as in [12,17].Neglecting fiber non-linearities, the resulting linear MIMO transmission system is given by: where X M×1 (resp.Y M×1 ) is the vector of transmitted (resp.received) symbols, N M×1 is an additive white Gaussian noise vector with zero mean and variance 2σ 2 per mode.
The matrix H M×M represents the spliced FMF composed of K fiber sections.The factor L represents the mode averaged propagation loss.T k is a diagonal matrix with random phase entries exp(iφ m ) and φ m ∈ [0, 2π].The dispersive effect of DMGD is not considered in our work since it does not impact the capacity of the system.The impact of DMGD only affects the complexity of the channel equalization.In the case of using OFDM modulation format, the size of the cyclic prefix should be longer enough to absorb the DMGD.Non-unitary modal coupling due to fiber misalignments at a splice point is given by an M × M coupling matrix C k , with entries computed using an overlap integral of the electrical fields of propagating modes at fiber cross sections as in [6]: where is the normalized complex amplitude of the of the ith mode before the splice and (x + ∆x, y + ∆y) is the normalized complex amplitude of the jth mode after the splice.∆x and ∆y are fiber misalignments assumed to be independent Gaussian distributed in x and y directions with zero mean and standard deviation (std) σ x,y .In our channel model, a non-unitary coupling occurs between modes at each fiber misalignment, hence the effects of modal coupling and MDL cannot be separated.In [18], a more appropriate description of linear modal coupling of spliced FMFs was given, however MDL was not taken into account.In our simulations, we consider FMFs made of K = 300 concatenations of misaligned fiber sections.To emulate different MDL levels, we vary σ x,y as a percentage of the fiber core radius r c .The accumulated effect from all misalignments results in a end-to-end MDL defined as the ratio in decibels (dB) of the maximum to the minimum eigenvalues of HH † .MDL(dB) = 10 log 10 λ max λ min (4)

Capacity Enhancement by Channel State Information
Assuming that the channel matrix H is known to the receiver (e.g., through training sequences) but is unknown at the transmitter due to the long round-trip delays in optical transport networks.In this case, the transmitter sends uncorrelated signals of equal power on all modes.The average capacity of the FMF channel with equal power allocation C ep is then given by [12,17]: P t is the total power at the transmitter.In the last equation, the average is taken over the eigenvalues of the channel H.The analytical computation of the average capacity requires the knowledge of the distributions of λ i 's which is not always available, consequently we compute the capacity by extensive numerical simulations.Note that this is a linear capacity that increases with the amount of power, if nonlinearities are taken into account the capacity drops after a non-linear power threshold [19].

Channel State Information
By computing the singular value decomposition (SVD) of the channel matrix as H = UΛV † , where Λ = diag( √ λ 1 , ..., √ λ M ) and U, V are unitary matrices.CSI refers to the matrices Λ and V that can be sent back from the receiver to the transmitter.In this case, V is used by the transmitter to send correlated symbols vector VX M .The matrix Λ is used to compute the optimal powers allocated to each mode.This can be done by the iterative water-filling algorithm known from MIMO wireless communications [20,21].The average channel capacity with channel state information is given by [12,20,21]: p * i is the optimal power allocated to the mode i and is given by: µ is a constant that satisfies the total power constraint ∑ M i=1 (µ − 2σ 2 λ i ) = P t .This power allocation strategy faces two major limitations.First, the CSI must be provided to the transmitter to send the data before the channel changes.Second, the optimal powers must be updated at the minimum as fast as the channel variations.These constrains cannot be satisfied for the moment for FMF optical systems where the channel dynamics change faster than round-trip delays.For example, in [22], the authors found that for a 26-km two-mode fiber, the channel variations at a wavelength λ = 1550 nm were of the order of 40 µs.In this case, the round-trip delay, is equal to 250 µs which is more than six-times longer than channel variations.

Statistical Channel State Information
To overcome the round-trip delay issue, we propose that the receiver computes an average of the channel that we note H = U Λ V † , with U, V are unitary matrices, Λ = diag( λ 1 , ..., λ M ) contains the long term eigenvalues of the channel.V and Λ represent the statistical channel state information.
The matrix Λ is used to compute the optimal powers p * 1 , ..., p * M per excited modes only once, and the matrix V will be used by the transmitter to send correlated symbols vector independently of the channel variations.We refer to this capacity by C CSI : To have an insight on the FMF channel capacity behavior, we simulate the previous capacities for the 6-mode and 10-mode fibers.Plotted results are averaged over 10 5 channel realizations.Figure 1a shows the complementary cumulative distribution function (CCDF) of the capacities given by Equations ( 5), ( 6) and ( 8) for the 6-mode fiber for MDL = 15 dB.We notice that both C CSI and C CSI are higher than C ep and as the SNR increases the gain brought by CSI decreases.In Figure 1b we plotted the CSI gain defined as the ratio of C CSI (resp.C CSI ) to C ep as a function of the SNR for different MDL values.We notice that the CSI gain increases with MDL and for higher SNRs the CSI gain vanishes.
To focus on the proposed statistical CSI, we compare the average capacities of equal power allocation to the long term power allocation.The average channel H is computed using 10 4 realizations of the instantaneous channel H. Figure 2a shows the average capacities C ep and C CSI for the 6-mode and 10-mode fibers as a function of SNR for MDL = 20 dB.We notice that the proposed statistical CSI increases the capacity for both fibers.Outage considerations are very important for the design of FMF optical systems.Since the channel changes randomly, transmitters are configured to encode for a fixed capacity that is higher than the worst case capacity called outage capacity to avoid system outage.The outage capacity C out is related to an outage probability p out as: where C inst (H) is the instantaneous capacity defined withing the coherence time of the channel during which channel coefficients h ij remain constants.In Figure 2b, we plotted the outage capacities for an outage probability of 10 −3 for the 6-mode and 10-mode fibers as a function of MDL for SNR = 10 dB.We compare the capacity with equal launch power to the capacity when using the long term optimal powers based on the statistical CSI.We notice that both capacities are decreasing with MDL, however C CSI is better than C ep .At MDL = 20 dB we have 0.3 bit/s gain for the 6-mode fiber and 0.5 bit/s gain for the 10-mode fiber.Consequently, since the availability of instantaneous CSI at the transmitter achieves the maximum capacity but is unfortunately unfeasible in practical systems, the proposed statistical CSI can be a solution to increase the capacity rather than using equal launch power at the transmitter.This method has a low-cost complexity since it works independently of the channel variations, thus it can be feasible for real implementation.Nonetheless, it is important to determine how many channel acquisitions are enough to properly use the average channel.Also, the time duration of the use of a given statistical CSI before having to compute a new one.

Mode Scrambling for Few-Mode Fibers
The principle of mode scrambling (MS) consists in introducing a random mode permutation after K scr splices.The application of MS to reduce MDL and increase FMF systems capacity was first proposed in [13].The authors showed that randomly scrambling all modes after each splice (K scr = 1) leads to a completely uncorrelated modal coupling.This strategy can reduce the MDL of the link but a large number of scramblers is required.The MIMO transmission system including MS is given by: In the last equation, we added the scrambling matrix P k corresponding to a random permutation matrix if k is a multiple of the scrambling period K scr , or an identity matrix otherwise.

Deterministic Mode Scrambling
In our scrambling approach, we propose to use a deterministic mode scrambler that permutes modes having more received power with modes having less received power.To have an insight into this strategy, we consider the previous 6-mode and 10-mode fibers with a misalignment std σ x,y = 4%r c .At the transmitter we launch all modes with unit energy E s = 1 and we compute the received energy per mode at the receiver side averaged over 10 5 channel realizations.
From Figure 3a,b, we clearly notice that some modes arrive at the receiver with more power than the others.For the 6-mode fiber the less attenuated modes are the LP 01 and LP 11a,b and the most attenuated modes are the LP 02 and LP 21a,b .Hence, we propose to permute the LP 01 with the LP 02 and the LP 11a,b with the LP 21a,b .For the 10-mode fiber, modes arriving with more power are the LP 01 , LP 11a,b and LP 21a,b , these modes will be permuted respectively with modes arriving with less power which are the LP 02 , LP 31a,b and LP 12a,b .Our scrambling strategy, avoids permutations between modes arriving with close levels of power especially between degenerate modes that arrive with the same amount of power.However, this situations can occur if random mode scrambling is considered, which do not average power effectively between all modes.
To compare the different scrambling strategies, we add 6 scramblers (K scr = 50) to the previous fibers (made of K = 300 sections).In Figure 4, we plot the PDF of MDL for 10 5 channel realizations and for a misalignment std σ x,y = 4%r c .We can see that MS reduces the impact of MDL, and that the deterministic MS outperforms the random MS.For the 6-mode fiber, the average MDL without scramblers in the line is 25 dB, with random scrambling the MDL is reduced to 15 dB and for the proposed scrambling strategy is reduced to 9 dB.For the 10-mode fiber, the average MDL without scramblers in the line is 36 dB, with random scrambling the MDL is 15 dB and for the proposed scrambling strategy is reduced to 10 dB.To study the impact of MS, we simulate the system average and outage capacities based on 10 5 channel realizations of the same previous fibers made of K = 300 sections and using 6 scramblers.In Figure 5, we plot the average capacities for the 6-mode and 10-mode fibers for different scrambling strategies.We notice that MS increases the system capacity and that the deterministic MS outperforms the random MS.For an SNR = 15 dB and a std σ x,y = 4%r c , the capacity gain brought by using deterministic MS to a strategy where no scrambler is used is 4.5 bits/s for the 6-mode fiber and 3.5 bits/s for the 10-mode fiber.Consequently, deterministic MS averages power more efficiently between all modes, which reduces significantly MDL and results in a better average capacity.Furthermore, we also notice that for low SNRs the capacity gain provided by MS is very low.This is due to the impact of noise that dominates the effect of MDL (given by the different eigenvalues λ i 's) in the capacity expression of Equation (5).However, for high SNRs the gain provided by MS becomes more important.This behavior is different from the CSI gain that is more important for low SNR levels.
In Figure 6a,b, we simulate the outage capacity as a function of the misalignment std σ x,y for the different scrambling strategies and for an outage probability of 10 −3 .We notice that as std σ x,y increases which increases also MDL, the outage capacities are decreasing.However, the use of MS enhances these capacities significantly and the deterministic MS outperforms the random MS.For an std σ x,y = 4%r c .
The capacity gain brought by using deterministic MS to a strategy where no scrambler is used is 7.8 bit/s for the 6-mode fiber and 5.2 bit/s for the 10-mode fiber.

Combined Statistical CSI and Deterministic MS
To further enhance the FMFs capacity, we combine the deterministic MS with the statistical CSI.In Figure 7a, we plotted the average capacity with different techniques for the 10-mode fiber and a misalignment std σ x,y = 4%r c .From the figure, we notice that the statistical CSI provides an important capacity gain for low SNRs, however this gain decreases as the SNR increases.The deterministic MS increases the capacity as the SNR increases.The combination of both techniques enhances the average capacity for all levels of SNR.In Figure 6b, we simulate the outage capacities for the same previous techniques and an outage probability of 10 −3 and SNR = 20 dB.From the figure, we notice that the capacity with deterministic MS is better than the capacity with statistical CSI, this is due the high SNR level of 20 dB.Furthermore, the outage capacity with the combination of both techniques is the highest one.

Conclusions
Capacity of FMF systems impaired by MDL can be improved by appropriate techniques.A long term knowledge of the optical channel can help improve the capacity by allocating optimal powers to each mode.Depending on channel variations that are commonly longer than the round-trip delays, the statistics of the FMF channel can be used.Moreover, deterministic scramblers can be placed in the optical link to reduce the MDL penalty by averaging power between modes at local points.Permuting modes with more power with modes having low power results in a better power averaging at the link end.The two techniques can be complementary for a further capacity improvement.