arX

iv:1

207.

6645

v1 [

phys

ics.

optic

s] 2

7 Ju

l 201

2NONLINEAR PROPAGATION IN MULTIMODE FIBERS 1

Nonlinear Propagation in Multimode and MulticoreFibers: Generalization of the Manakov Equations

Sami Mumtaz, Rene-Jean Essiambre and Govind P. Agrawal, Fellow IEEE

Abstract—This paper starts by an investigation of nonlineartransmission in space-division multiplexed (SDM) systemsusingmultimode fibers exhibiting a rapidly varying birefringenc e. Aprimary objective is to generalize the Manakov equations, wellknown in the case of single-mode fibers. We first investigate areference case where linear coupling among the spatial modes ofthe fiber is weak and after averaging over birefringence fluctu-ations, we obtain new Manakov equations for multimode fibers.Such an averaging reduces the number of intermodal nonlinearterms drastically since all four-wave-mixing terms average out.Cross-phase modulation terms still affect multimode transmissionbut their effectiveness is reduced. We then verify the accuracy ofour new Manakov equations by transmitting multiple 114-Gb/sbit streams in the PDM-QPSK format over different modes of amultimode fiber and comparing the numerical results with thoseobtained by solving the full stochastic equation. The agreementis excellent in all cases studied. A great benefit of the newequations is to reduce the computation time by a factor of 10 ormore. Another important feature observed is that birefringencefluctuations improve system performance by reducing the impactof fiber nonlinearities. The extent of improvement depends onthe fiber design and how many spatial modes are used for SDMtransmission. We investigated both step-index and graded-indexmultimode fibers and considered up to three spatial modes in eachcase. In this paper, we also consider multicore fibers and showthat the Manakov equations remain valid as long as the lengthscale of birefringence fluctuations is about 100 times shorterthan that of linear coupling. Finally multimode fibers with strongrandom coupling among all spatial modes are considered. Linearcoupling is modeled using the random matrix theory approach.We derive new Manakov equations for multimode fibers in thatregime and show that such fibers can perform better than single-modes fiber for large number of propagating spatial modes.

Index Terms—Birefringence, Manakov equation, MultimodeFiber, Multicore Fiber, Space-division multiplexing, Fiber Non-linearity.

I. I NTRODUCTION

Recent work on the optical fiber capacity limit [1]–[3] hasindicated that the continued use of single-mode fibers formodern telecommunication systems may not be able to supportthe growing data-traffic demand in the near future [4], [5].Although more single-mode fibers can be deployed as a short-term solution, it is important to look for alternative solutionsfor the next generation of optical transmission systems. Theimpressive bit-rate increase of the last two decades wasmade possible by exploiting diverse properties of the elec-tromagnetic field through wavelength-division multiplexing

Sami Mumtaz and Govind P. Agrawal are with The Institute of Optics,University of Rochester, Rochester, NY,14627,USA

Rene-Jean Essiambre is with Bell Laboratories, Alcatel-Lucent, 791Holmdel, NJ, 07733, USA

Manuscript received January xx, 2012; revised March xx, 2012. Firstpublished May xx, 2012.

This work was funded by Alcatel-Lucent in the framework of GreenTouch.

(WDM) in combination with phase modulation formats andpolarization-division multiplexing (PDM). Recently, multi-mode fibers have become the focus of attention [6], [7] as theypermit simultaneous propagation of multiple spatial modes.Multiplexing signals using different spatial modes of an opticalfiber is referred as space-division multiplexing (SDM). Recentwork on multimode fiber has focused on demonstrating thefeasibility of the SDM technique and how digital signalprocessing (DSP) can help to compensate linear distortionsofthe signal [8], [9]. However, nonlinear propagation effects arethe principal limitation of optical fiber communication systemsand their understanding in the case of multimode fibers is stillvery limited compared to the case of single-mode fibers.

A set of nonlinear propagation equations has been derivedfor multimode fibers in [10] by expanding the optical fieldin terms of vectorial fiber modes. It is difficult to use theseequations in their most general form to study birefringenceeffects. In the case of silica fibers used for telecommunicationsystems, we can simplify these equations by making use ofthe so-called weakly-guiding approximation. In this paperwe derive in sectionII a new set of multimode nonlinearequations, which represents the two polarization componentsof each spatial mode through a Jones vector and includeslinear coupling between spatial modes. We then use them tostudy the effects of fiber’s random modal birefringence andlinear mode coupling using random matrices in two importantcases of practical interest that we refer to as the weak andstrong coupling regimes. In the weak-coupling regime, linearcoupling among distinct spatial modes is weak compared to thebirefringence-induced coupling between the two polarizationcomponents of the same spatial mode. In the strong-couplingregime, both types of coupling occur in a random fashion withthe same order of magnitude. In practice, certain spatial modesof multimode fibers may be weakly coupled whereas others,may be more strongly coupled. For instance, it was observed in[8] that the LP11a spatial mode is strongly coupled with LP11bbut weakly coupled with LP01. The strong and weak spatialmode coupling regimes studied in this work represent twoextreme cases for practical systems. Multimode fibers alwayshave, in general, some degree of coupling between any pair ofspatial modes. Important factors impacting how much linearcoupling there is between spatial modes are: fiber design andfabrication, cabling, bending and twisting.

It is well known in the case of single-mode fibers thatrapidly varying birefringence results in an averaging effectthat reduces the strength of nonlinearities: mathematically,this averaging is described by the Manakov equations [11]–[13]. Its use also reduces the computation time considerably

NONLINEAR PROPAGATION IN MULTIMODE FIBERS 2

when the nonlinear propagation is performed numerically. Ingeneral, the standard Manakov equations cannot be used inthe case of multimode fibers since they only describe howbirefringence impactsintramodal nonlinear effects and do notinclude intermodal nonlinear effects. In sectionIII , we derivea new set of Manakov equations for nonlinear propagationin multimode fibers in the regime of weak coupling betweenspatial modes, following the ideas in [11] and averaging overall possible polarization states. In sectionIV, the impact ofrapidly varying birefringence on a specific SDM transmissionsystem is studied through numerical simulations. We show thatthe new set of Manakov equations successfully predict theeffects of random modal birefringence, while also reducingthe computational complexity of the problem. In sectionV,we apply the Manakov equations to the case of a multicorefiber whose cores are linearly coupled with a constant couplingcoefficient. We show that Manakov equations remain validin the weak coupling regime as long as the length scale ofthe birefringence fluctuations is 100 times shorter than that oflinear coupling.

Finally, we consider in sectionVI a case with large randomcoupling between spatial modes, referred to as the strong-coupling regime. Although current transmission experimentsover multimode fibers are not fully in the strong-couplingregime [6], this case has attracted theoretical attention toobtain an understanding of stochastic effects such as ran-dom differential group delays among various modes [14]or random mode-dependent loss within the multimode fiber[15]. Recently Manakov equation have been generalized formultimode fibers in the high coupling regime [16]. Followinga different approach based on random matrix theory, we obtainsimilar results within the weakly guiding approximation. Acomparison of the weak- and strong-coupling regimes is alsopresented in this paper.

II. N ONLINEAR PROPAGATION IN MULTIMODE FIBERS

We simplify the analysis in [10] by making the weaklyguiding approximation, known to be quite accurate for silicafibers with a relatively low difference in the refractive indicesbetween core and cladding. In this approximation, the modalfields are linearly polarized in the transverse plane, as thefieldcomponents along the direction of propagation (z axis) are as-sumed to be negligible. Moreover, the spatial modes associatedwith the two polarization components are assumed to have thesame spatial distribution. With these simplifications, we writethe total electric field as a sum over theM distinct spatialmodes (we use the same definition of the fiber modes as in[8]) of the fiber in the spectral domain:

E(x, y, z, ω) =

M∑

m

eıβm(ω)zAm(z, ω)Fm(x, y)/√Nm, (1)

where Am(z, ω) = [Amx(z, ω) Amy(z, ω)]T is the Fourier

transform of the time-domain slowly varying field envelopeof the mth mode represented in the Jones vector notation.The superscriptT denotes the transpose operation and thetilde denotes a frequency-domain variable. It includes theslowly varying amplitudes of both polarization components

for the spatial modem with the spatial distributionFm(x, y)and the propagation constantβm(ω), expressed in the formof a diagonal matrix to account for fiber birefringence, i.e.,βm = diag[βmx βmy]; eıβmz corresponds to the usual defi-nition of an exponential matrix. Note thatFm(x, y) is real inthe weakly guiding approximation, and the phase of the opticalenvelope is contained in the complex fieldAm. Finally, thenormalization constantNm in (1) represents the power carriedby themth mode. It can be expressed asNm = 1

2 ǫ0 neff c Im,where Im = nm/neff

∫∫F2m(x, y)dx dy, ǫ0 is the vacuum

permittivity, neff is the fiber effective index of the fundamentalmode andnm the fiber effective index of the modem.

Each frequency component of the optical field satisfies thefrequency-domain Maxwell equations,

∇2E(ω) + n2(x, y, z)ω2

c2E(ω) = −ω2µ0P

NL(ω), (2)

wheren(x, y, z) is the refractive index distribution within thefiber whosez-dependence accounts for small perturbationsalong the propagation distance, i.e.,n2(x, y, z) = n2

0(x, y) +∆n2(x, y, z) [17]. PNL(ω) is the Fourier transform of thethird-order nonlinear response [18]

PNL(t) =ǫ04χ(3)

([ETE]E∗ + 2[EHE]E

). (3)

Here ∗ denotes the complex conjugate and the superscriptH denotes the Hermitian conjugate.χ(3) is the instantaneousnonlinear response of silica.

We assume that the spatial distribution of a mode is notperturbed by nonlinearities or by small variations of therefractive index along the propagation distancez. In that case,Fm(x, y) satisfies the eigenvalue equation

∇2Fm(x, y) +

(ω2

c2n20(x, y)− β2

mi

)Fm(x, y) = 0, (4)

where the subscripti = x, y refers to the two orthogonalpolarizations of themth spatial mode. In fibers with per-fect cylindrically symmetric refractive index profilen0(x, y),βmx = βmy. In practice, because of imperfections, fibersexhibit some polarization birefringence andβmx 6= βmy.In the remainder of this section we consider a sufficientlyshort fiber section such that the birefringence remains constantalong the fiber length. The case of accumulation of randomlyoriented birefringent fibers segments is treated in subsequentsections. Note that the various solutions of (4) are chosen tobe orthogonal to each other and are normalized such that

∫∫Fm(x, y)Fp(x, y)dx dy = Imδmp, (5)

δmp being the Kronecker function.To isolate the evolution of a specific spatial mode, say the

pth mode, we use their orthogonal nature. We substitute (1)in (2), multiply both sides withFp, and integrate over thetransversex-y plane. The resulting equation is then convertedto the time domain by following a standard procedure and ex-pandingβp(ω) in a Taylor series around the carrier frequency

NONLINEAR PROPAGATION IN MULTIMODE FIBERS 3

ω0 [19]. The final result can be written in the form

∂Ap

∂z= ı(β0p−βr)Ap − (β1p−

1

vgr)∂Ap

∂t− ı

β2p

2

∂2Ap

∂t2

+ ı∑

lmn

flmnpγ

3

[(AT

l Am)A∗n + 2(AH

l Am)An

]

+ ı∑

m

qmpAm, (6)

whereAp(z, t) is the time-domain slowly varying envelopeof the pth mode expressed in a reference moving frame atthe group velocityvgr . βr is an arbitrary reference propa-gation constant for all the spatial modes andβ0p = βp(ω0),β1p = ∂βp/∂ω|ω0

, β2p = ∂2βp/∂ω2|ω0

are respectivelythe propagation constant, inverse group velocity and group-velocity dispersion (GVD) of thepth spatial mode. We assumehere that the two polarization components of a spatial modemay have different group velocities but have the same GVD.

The nonlinear parameter in (6) is defined in a similar fashionas in single-mode fibers, byγ = 3k0χ

(3)/(4ǫ0cn2effAeff),

where Aeff is the effective area of the fundamental mode.The linear and nonlinear coupling among spatial modes isgoverned, respectively, by

qmp(z)=k0

2neff(ImIp)1/2

∫∫∆n2(x, y, z)FmFp dx dy, (7)

flmnp=Aeff

(IlImInIp)1/2

∫∫FlFmFnFp dx dy. (8)

Note that linear coupling terms appear as a result of theperturbation of the refractive index∆n2(x, y, z). In the caseof an ideal fiber,∆n2(x, y, z) = 0, no linear coupling occursbetween the spatial modes.

Equation (6) governs propagation of arbitrarily polarizedlight in the spatial modep within the Jones-matrix formalism.It includes all third-order nonlinear effects (both intramodaland intermodal kinds) as well as dispersive and birefringenceeffects. Fiber losses can also be included by adding the term−(αp/2)Ap on its right side, where the loss parameterαp canbe different for different modes.

III. M ANAKOV EQUATION IN WEAK-COUPLING REGIME

Equation (6) assumes constant birefringence along the fiberlength. In practice, owing to fiber imperfections such as anonuniform core whose shape varies along the fiber, bire-fringence varies randomly and rapidly on a length scale thatis short compared to the effective lengths associated withthe GVD and various nonlinearities. This feature can beimplemented numerically by rotating the principal axes of thefiber periodically after a distance shorter than the length scaleof birefringence fluctuations. However, a numerical solutionof Eq. 6 then requires a relatively small step size and is quitetime-consuming.

In the case of single-mode fibers, the problem has beensolved by adopting the well-known Manakov equation [11][12]. The idea is that, as a rapidly varying birefringencechanges continually the state of polarization (SOP) of propa-gating light in a random fashion, one can average the prop-agation equation itself over all polarization states. We can

follow the same procedure for multimode fibers by assumingthat the SOP of each spatial mode evolves, randomly andindependently, i.e., we assume that the SOP of each modeevolves in an independent fashion. This can be justified bynoting that, as the spatial distributions of the modes aredifferent, the influence of local stress and fiber imperfectionsmay also be different from one spatial mode to another.

A rapidly varying fiber birefringence can be tracked usingthe transformation

Ap(z) = Rp(z)Ap(z) (9)

whereRp(z), a unitary matrix belonging to the group U(2) isa Jones matrix of the form

Rp(z) =

[r11p r12pr21p r22p

](10)

RHp Rp = I2 (11)

where I2 is the 2 × 2 identity matrix andrijp are randomvariables. An important issue is how to constructRp to ensurethat the SOP probability for thepth mode is uniformly dis-tributed over all possible SOPs. We make use of the followingprocedure applicable to the general case of n×n matrices[26]: pick a matrix M whose all elements are normallydistributed (with zero mean) and apply a QR decomposition,M = RT, such thatT is an upper triangular matrix withpositive diagonal entries. Then,R is a Haar matrix whoseelements are uniformly distributed over U(n).

The use of (9) in (6), leads to the following equation forAp:

∂Ap

∂z=ı δβ0p Ap − δβ1p

∂Ap

∂t− ı

β2p

2

∂2Ap

∂t2

+ı∑

lmn

flmnpγ

3

([AT

l RTl RmAm

]RH

p R∗nA

∗n

+ 2[AH

l RHl RmAm

]RH

p RnAn

)

+ı∑

m

qmpAm (12)

where

δβ0p=RHp (β0p−βr)Rp − ıRH

p

∂Rp

∂z, (13)

δβ1p=RHp (β1p−

1

vgr)Rp, (14)

qmp=qmpRHp Rm. (15)

Equation (12) is stochastic becauseRm(z) is a randommatrix that changes along the fiber length on a length scaleassociated with birefringence fluctuations. As a result, thebirefringence parameters appearing there vary randomly. Inaddition, the intramodal and intermodal nonlinear couplings,as well as the linear coupling, also become random.

In the following analysis, we average the propagation equa-tion (12) over all possible realizations of the matrixRm.Averaging over birefringence amounts to assuming that otherphenomena that producez-dependent variations, in particularthe linear coupling between spatial modes (qmp(z) in (15)),occur over a much longer length scale in comparison to the

NONLINEAR PROPAGATION IN MULTIMODE FIBERS 4

length scale of polarization fluctuations within each spatialmode in the weak-coupling regime considered here.

Let us focus first on the nonlinear terms in (12) by writingthem as

Np= ı∑

lmn

fplmnγ

3

([AT

l RTl RmAm

]RH

p R∗nA

∗n

+ 2[AH

l RHl RmAm

]RH

p RnAn

). (16)

The matrix multiplications appearing there can be expressedas

RTl Rm =

[alm blmbml clm

]RH

l Rm =

[alm blmb∗ml clm,

](17)

where we have defined the following quantities:

alm=r∗11l r11m + r∗21l r21m (18)

blm=r∗11l r12m + r∗21l r22m (19)

clm=r∗21l r21m + r∗22l r22m (20)

alm=r11l r11m + r21l r21m (21)

blm=r11l r12m + r21l r22m (22)

clm=r21l r21m + r22l r22m. (23)

Using (17) in (16), the nonlinear terms for thex-polarizedpth mode become:

Npx=ı∑

lmn

fplmnγ

3

(

2[almapnA

∗lxAmxAnx + b∗mlapnA

∗lyAmxAnx

+blmapnA∗lxAmyAnx + clmapnA

∗lyAmyAnx

+almb∗pnA∗lxAmxAny + b∗mlb

∗pnA

∗lyAmxAny

+ blmbpnA∗lxAmyAny + clma∗pnA

∗lyAmyAny

]

+[alma∗pnAlxAmxA

∗nx + bmla

∗pnAlyAmxA

∗nx

+ blma∗pnAlxAmyA∗nx + clma∗pnAlyAmyA

∗nx

+ almb∗pnAlxAmxA∗ny + b∗mlb

∗pnAlyAmxA

∗ny

+blmb∗pnAlxAmyA∗ny + clmb∗pnAlyAmyA

∗ny

] )

(24)

We are interested in finding the ensemble average, or theexpectation value, that we denote by〈 〉 of the variouscoefficients appearing in (24) under random unitary transfor-mations. When the subscriptsl,m, p, n correspond to differentspatial modes (l 6= m 6= p 6= n), the coefficientalmapnis a sum of products of functionsr11l, r11m, r11n, r11p andr21l, r21m, r21n, r21p, all of which are independent randomvariables with zero mean. As a result,〈almapn〉 = 0. Withthe same kind of reasoning, we find that all terms in (24)average to zero, except those containing the combinations

〈bppb∗pp〉 = 1− 〈appa

∗pp〉=

1

3(25)

cmmapp = cmmapp = appapp = cppapp=1 (26)

〈apma∗pm〉 = 〈bpmb∗pm〉=1

2(27)

〈ampapm〉 = 〈bpmb∗pm〉=1

2. (28)

These equations are obtained using the properties of Haarmatrices, which are given by Eq. (37) to (39) in the case of2M × 2M matrices and assuming thatRi are independentrandom unitary matrices. Finally, by gathering the nonzeroterms together inNpx and Npy , we obtain the followingaveraged propagation equation:

∂Ap

∂z+〈δβ0p〉Ap + 〈δβ1p〉

∂Ap

∂t+ ı

β2p

2

∂2Ap

∂t2

=ıγ(fpppp

8

9|Ap|

2 +∑

m 6=p

fmmpp4

3|Am|2

)Ap, (29)

with

〈δβ0p〉=1

2(βpx + βpy)− βg (30)

〈δβ1p〉=1

2(βpx

∂ω

∣∣∣ω0

+βpy

∂ω

∣∣∣ω0

)−1

vg(31)

These generalized Manakov equations are deterministic, asthey contain no rapidly fluctuating random terms. Equations29represents an extension of the standard Manakov equations,for multimode fibers with birefringence. The first nonlinearterm represents theintramodal nonlinear effects, resultingfrom self-phase modulation (SPM), and it occurs for single-mode fibers as well with the same coefficient 8/9. The secondnonlinear term is new. Its origin lies in theintermodal nonlin-earities resulting from cross-phase modulation (XPM) amongvarious fiber spatial modes. The averaging over birefringencefluctuations reduces the factor of 2 that is associated with XPMto a lower value of 4/3. Note that the random quantityqmp

averages to zero.The new Manakov equations (29) obtained for multimode

fibers can be solved numerically much faster than (12), andit should help considerably in designing future SDM systems.However, before it can be used with confidence, we need toensure that its predictions agree with those of (12). In the nexttwo sections we verify this by considering a specific SDMsystem.

IV. N UMERICAL SIMULATIONS

In this section we compare the performance of a specificSDM system for multimode fibers with and without randombirefringence. Propagation equations are solved numericallywith the split-step Fourier method. The case with no bire-fringence is obtained as a reference by solving (6). The casewith birefringence is obtained either by solving explicitly (12)applying new random matrices at each step [20] or by solvingthe new set of generalized Manakov Equations (29). Thecomparison of the numerical results represents a test of thenew Manakov equations.

For simplicity, we assume that each spatial mode carriesa single-wavelength channel (no WDM) at a bit rate of114 Gb/s. We make use of polarization multiplexing (PDM)together with the QPSK format so that the symbol rate isRs = 28.5 Gbaud. Each PDM-QPSK signal is constructedfrom an independent random bit stream that is Gray-mappedonto the QPSK symbols. The pulse shape corresponds to araised-cosine spectrum with a roll-off factor of 0.2. EachPDM-QPSK signal is made of215 modulated symbols with

NONLINEAR PROPAGATION IN MULTIMODE FIBERS 5

TABLE I: General shape of spatial mode distributions sup-ported by the step-index and graded-index fibers used innumerical simulations.

Spatial distribution shapes Step-index fiber Graded-index fiber

LP01 HG00

LP11a, LP11b HG01, HG10

LP02 HG02+HG20

LP21a, LP21b HG11a, HG11b

217 samples per polarization. The injected power per spatialmode isPin = 7 dBm. This choice is deliberately high in orderto observe large nonlinear impairments so as to represent avery stringent test of the new Manakov equations. .

The 1000-km transmission line consists of 10 sections of100-km multimode fiber, each section being followed by anideal erbium-doped fiber amplifier (EDFA) that can com-pensate for the span losses of 0.2 dB/km (assumed to bethe same for each spatial mode). At the end of the fiberlink, we artificially degrade the signal by adding noise toaccount for the amplified spontaneous emission (ASE) addedby all EDFAs. This is justified since the nonlinear interactionbetween ASE and the signal has been verified to be muchsmaller than the nonlinear interaction of the signal on itself.

We focus on a step-index multimode fiber with a core diam-eter of12.3 µm and a numerical aperture of 0.2 (∆ = 0.01).Such a fiber has the V parameter of 5 at 1550 nm, andit supports LP01, LP02, LP11, LP21 modes, resulting in atotal of 6 spatial modes when we take into account 2-folddegeneracy of the LP11 and LP21 modes [17]. TableI presentsthe spatial distributionsFm(x, y) of the different spatial modessupported by the fiber.

Each spatial mode has its own propagation constant, modalgroup velocity and GVD parameterD (see Tab.II ). Thefundamental-mode nonlinear coefficient isγ = 1.4W−1km−1.We consider the case where there is no coupling between thespatial modes, i.e.qmp = 0.

We solve numerically (12) and (29) with a step size of100 m in all cases (larger step sizes could have been usedin the Manakov case). For simplicity, we chooseδβ1p = 0 inthe case of (12). Smaller step sizes were tested and shown togive the same results.

At the receiving end, the symbol stream is coherentlydetected by polarization diverse 90-degree hybrid followedby analog to digital converters. We assume an ideal DSPscheme that compensates perfectly all linear impairmentssuch as chromatic dispersion, group-velocity mismatch, mode

TABLE II: Differential modal group delay (compared tothe fundamental mode) and group velocity dispersionD ofeach spatial mode of the step-index multimode fiber used insectionIV

DMGD [ns/km] D [ps/(km-nm)]LP01 0 25LP11 6.5 27.3LP02 9.9 -2.3LP21 12 20.8

Fig. 1:BER averaged over the co-propagating spatial modes of a stepindex multimode fiber for 1000 km transmission versus OSNR. 114-Gb/s PDM-QPSK signals are transmitted over different combinationsof spatial modes indicated in the upper right inset. Solid curvesobtained solving Manakov Eq. (29) are superposed with dashedcurves representing the case with random birefringence obtainedsolving (12); pointed curves represent the zero-birefringence case (6).Back-to-back case (no fiber) is displayed as a reference.

mixing, and polarization mixing. This is justified in practicesince modern equalizers can remove most, if not all, linearimpairments [21]. It also lets us present our results in a waythat is independent of the type of equalizer employed.

Figure 1 shows the bit-error rate (BER) averaged over allthe propagating spatial modes as a function of the opticalsignal-to-noise ratio (OSNR) [1] after 1000 km multimodefiber when 114-Gb/s bit streams are transmitted on differentcombinations of spatial modes. Clearly, the predictions ofournew Manakov equation (29) agree very well with those ofthe full stochastic equation (12). As discussed later, the useof Manakov equations reduces considerably the computationtime. To study the impact of random birefringence on systemperformance, we also consider the case of an ideal multimodefiber with no birefringence by solving Eq. (6). It is evidentthat birefringence actually improves system performance inall cases by lowering the BER. It should be stressed thatthis performance improvement occurs in practice only ifthe random polarization mixing resulting from birefringencecan be efficiently compensated at the receiver. The systemperformance is easily understood if we recall that the averagingover rapidly varying birefringence leads to (29) in which thenonlinear coefficient associated with the SPM is reduced from

NONLINEAR PROPAGATION IN MULTIMODE FIBERS 6

Fig. 2: BER of each spatial mode versus OSNR after 1000 kmtransmission on step-index fiber with rapidly varying birefringence.Solid curves represent the case where 6 spatial modes co-propagate;dashed curves represent the cases where spatial modes propagateindividually.

1 to 8/9. Note that for the chosen configurations of transmittedspatial modes, inter-modal XPM effects are averaged outbecause of the large differential group delay between spatialmodes [22]. The reason of the large performance degradationobserved in Fig.1 in the case of 3 co-propagating spatialmodes (LP01+LP11+LP02) is related to a small magnitudeof the dispersion parameterD of the LP02 spatial mode forthe chosen fiber design parameters. Figure2 presents thereceived BER of each spatial mode. In one case, the 6 spatialmodes are transmitted together. In the other case, each spatialmode is transmitted individually in the multimode fiber. Theperformance degradation between these two cases is due tothe inter-modal XPM. We observe that LP02 and LP01 spatialmode suffer very little from inter-modal XPM because of theirlarge differential group delay compared to the other spatialmodes. On the other hand, performance of both LP11 andLP21 modes are largely degraded by the inter-modal XPMbecause the degenerated spatial modes (for instance LP11aand LP11b) propagate at the same group velocity. Note thattwo degenerated spatial mode perform the same. The nonlinearinter-modal XPM coefficient isf11a,11a,11b,11b = 0.3 betweenLP11a and LP11b andf21a,21a,21b,21b = 1 between LP21aand LP21b.

In a step-index multimode fiber, there is a spread ofgroup velocities between spatial modes. This group-velocitymismatch introduces linear degradations (similar to PMD)that must be compensated at the receiver. In practice, theequalization complexity grows in proportion to the magnitudeof the differential group delays among modes. This problemcan be solved using a graded-index fiber for which group-velocity mismatch nearly vanishes. It is thus important to studyhow the system performance in Fig.1 will change if the step-index multimode fiber used there were replaced with a graded-index multimode fiber. In particular, we focus on a multimodefiber whose core has a quadratic graded refractive index (inthe shape of a parabola). Such fibers are designed to have all

Fig. 3:BER averaged over the co-propagating spatial modes of a stepindex multimode fiber for 1000 km transmission versus OSNR. 114-Gb/s PDM-QPSK signals are transmitted over different combinationsof spatial modes indicated in the upper right inset. Solid curvesobtained solving Manakov Eq. (29) are superposed with dashedcurves representing the case with random birefringence obtainedsolving (12); pointed curves represent the zero-birefringence case (6).Back-to-back case (no fiber) is displayed as a reference.

spatial modes propagate at the same group velocity [17].For numerical simulations, we consider a graded-index fiber

with a core diameter of 17.4µm, D = 21.5 ps/(km-nm)andγ = 1.4 W−1/km and designed such that it supports the(Hermite-Gauss) modes presented in TableI. Data transmittedon different combinations of these spatial modes and weassume no linear coupling between them, i.e.qmp = 0.For similarity with LP02 spatial mode of step-index fibers,we consider a spatial distribution corresponding to the sumof the HG02 and HG20 spatial mode distributions. Figure3 shows the BER as a function of OSNR under the sameconditions used in Fig.1 except that the graded-index fiberreplaces the step-index fiber. A comparison of two figuresshows several interesting features. First, a rapidly varyingbirefringence results in a larger improvement in the case ofgraded-index fibers than in the case of step-index fibers. Here,spatial modes have the same group velocity so, there arenonlinear interactions between them due to inter-modal XPM.The effect of rapidly varying birefringence results to reductionof the XPM coefficient from 2 to 4/3, which improves theperformance. Second, nonlinear penalties, for graded-indexfibers, are larger in the two-mode case but become consid-erably smaller in the three-mode case. These results suggestthat graded-index multimode fibers may perform better whena large number of spatial modes are used to simultaneouslytransmit data.

Finally, we come to a main advantage of using the newManakov equations (29). We have already seen that its nu-merical predictions are virtually identical to those obtainedby solving Eq. (12). From a practical standpoint, the use ofManakov equations is preferable because it requires muchless computational time. One reason is that the number ofnonlinear terms is drastically reduced. For instance in thecase of 3 propagating modes, there are 81 nonlinear terms in

NONLINEAR PROPAGATION IN MULTIMODE FIBERS 7

TABLE III: Computing time in seconds for 1000-km transmissionof 114-GB/s channels over multiple modes with a 100-m step size.

Number of modes,M 1 2 3 4Manakov Eq. (29) 270 800 1600 2800

unaveraged Eq. (12) 720 7600 16000 31000

Eq. (12) but only 3 terms are needed in Eq. (29). In general,there areM3 − 1 intermodal nonlinear terms for aM -modefiber while there areM − 1 intermodal nonlinear terms foreach generalized Manakov equation. A second reason is thata solution of Eq. (12) requires computation of an exponentialmatrix of size2M × 2M , M being the number of spatialmodes, which is a time-consuming numerical operation. Asa result, the reduction in computing time depends on thevalue of M . This is evident in TableIII , which shows thecomputation time obtained on a desktop computer using theMATLAB software (version 2007b) for 1000-km propagationof 114-Gb/s channels with a step size of 100 m and a temporalgrid size of 217. Computation time is reduced by nearly afactor of 10 even forM = 2, with larger reductions occurringfor larger values ofM . Identical step sizes were used inboth cases. Solving Eq. (12) numerically requires a step sizethat is determined by birefringence fluctuations. In the caseof Manakov equations (29), the step size is mainly set bynonlinear effects which usually occur on a length scale muchlarger than that of birefringence fluctuations. Consequently itis usually possible to choose a much larger step size. Takingthis into account, the use of the new Manakov equations shouldreduce the computing time easily by more than a factor of 100in most cases of practical interest.

V. M ULTICORE FIBERS

Manakov equations (29) in SectionIII assume weak cou-pling among distinct spatial modes of a multimode fiber. Inpractice, linear mode coupling can stem from such effects ascore irregularities [23] or fiber bending [25]. Linear couplingbetween distinct spatial modes can be relatively weak incomparison to the coupling between the two polarizationcomponents of each mode. In our theory, linear mode couplingresulting from fiber imperfections is governed by the parameterqmp(z) appearing in Eq. (6). This coupling coefficient variesrandomly along the propagation distance and its strengthdepends on the degree of fiber imperfections and the levelof stress applied to the fiber.

Future SDM systems may also make use of multicore fibersin which each core supports a single mode, but modes indifferent cores can couple strongly if these cores are relativelyclose to each other. In this case, the extent of coupling dependson the physical distance among its cores and its magnitudevaries exponentially with this distance. The coupling coeffi-cient qmp(z) may vary withz in practice due to waveguideimperfections but in order to simplify the analysis, we assumeideal fiber so thatqmp is a constant in the multicore study.

For numerical simulations, we consider an ideal 3-core fiberhaving identical, and equidistant single-mode cores, i.e., in

1 10 100 1000

−2.5

−2

Lc [km]

log 10

BE

R

No birefringence: 1−Core " : 3−CoreRandom birefringence: 1−Core " : 3−Core

Fig. 4: BER averaged over the cores as a function of couplinglength Lc for 1000 km transmission. 114-Gb/s PDM-QPSK withindependent random bit streams are transmitted over a single-core(solid horizontal line) and a 3-core (solid curve) fiber. Dashed curvesshow these two cases for fibers without birefringence.

an equilateral triangular configuration. There are three spatialmodesFm each associated with the field propagating in onecore. The nonlinear propagation equation is still given by (6)but the coupling coefficients are now given by

qmp=k20

2β0p(ImIp)1/2

∫∫(n2 − n2

m)FmFp dx dy, (32)

where nm(x, y) is the spatial distribution of the refractiveindex of the isolated cores in which the spatial modem prop-agates. This propagation equation is obtained using a coupled-core approach, which assumes that the spatial distributionof a field propagating in one core, is not perturbed by thepresence of other cores [23]. Because of the symmetry of thethree-core fiber considered here, all coupling coefficientsareidentical and we can replaceqmp with a single parameterq.Each core of the fiber has a core diameter of 8.2µm, with∆ = 0.003 and γ = 1.4 W−1km−1. With these designparameters, intermodal nonlinear coefficients are negligible[24]. We modify the strength of linear coupling by adjustingthe distance between fiber cores. It is useful to present theresults in terms of the coupling lengthLc = π/(3q), whichrepresents the propagation distance at which maximum linearpower transfer occurs among the cores. The rapidly varyingbirefringence is included by solving the propagation Eq. (12)with a step size of 100 m and changing the rotation matrixrandomly after each step.

Figure 4 shows the BER as a function of the couplinglength (solid blue curve) when 114-Gb/s independent randombit streams are simultaneously transmitted through all threecores. The case of a single-core fiber is shown for comparisonby a solid black line. Also shown for comparison are the twodashed curves for which the fiber link has no birefringence.The injected power per core isPin = 8 dBm and theOSNR at the receiver is set to20 dB in all cases. Whenthe coupling length is comparable to the length scaleLf

of birefringence fluctuations (here 0.1 km), there is a largeperformance improvement in the case of the 3-core fiber.

NONLINEAR PROPAGATION IN MULTIMODE FIBERS 8

This improvement can be understood by noting that a rapidpower transfer among various modes induced by a stronglinear coupling and birefringence-induced polarization mixinghelps to mitigate nonlinear effects. As predicted by the newManakov equations (29) (where the termqmp in (15) averagesto zero), when the coupling length becomes much largerthan Lf , the effects of linear coupling are averaged out byrapidly varying birefringence, and the 3-core fiber behavessimilarly to a single-core fiber. This occurs in Figure4 forLc > 10 km. Indeed, when Eq. (29) is used, the numericalresults correspond to the case of a single-core fiber. The mainmessage from Figure4 is that the new Manakov equations(29) can be used with success for multicore fibers as longas the cores are far enough apart that the coupling length isabout 100 times larger than the length scale of birefringencefluctuations.

These results also suggest that in the case of multimodefibers, Manakov equations (29) are valid when the coupling be-tween spatial modes, resulting from fiber imperfections, occurson a length scale 100 times larger than that of birefringencefluctuations.

VI. M ANAKOV EQUATIONS IN STRONG-COUPLING

REGIME

The Manakov equations (29) obtained in the weak-couplingregime fail if linear coupling among various spatial modesbecomes comparable to birefringence-induced coupling. Inthis section, we derive new Manakov equations in the caseof strong random coupling amongM spatial modes of amultimode or multicore fiber. In this strong-coupling regime,we need to use2M×2M random unitary matrices to accountfor random coupling amongM spatial modes, with twopolarization states per mode.

Before applying such matrices to the nonlinear propagationequation, we need to rewrite the set (6) of M equations (p =1 . . .M ) in a vector form. By introducingA = [AT

1 . . .ATM ]T

as the column vector containing2M field envelopes, we obtain

∂A

∂z=ıδB0A− B1

∂A

∂t− ı

B2

2

∂2A

∂t2

+

∫∫ıγ

3[(AFA)F∗A∗ +2(AHFA)FA

]dx dy

(33)

whereF is a 2M × 2M matrix whose diagonal elements areM diagonal2× 2 matrices such that

Fij=Fi Fj I2, i, j = 1 . . .M. (34)

B0, B1 andB2 are2M×2M diagonal matrices containing re-spectively the propagation constant, the inverse group velocity,and the dispersion parameter of each mode. Further,

δB0 = B0 −1

2MTr (B0) (35)

represents deviations of the propagation constants from theaverage value of all spatial modes. Note that the couplingterm qmp does not appear in (33) as it is included throughthe random unitary matrixR.

Fig. 5: BER averaged over the co-propagating modes versus OSNRafter 1000 km over graded-index fibers. 114-Gb/s bit PDM-QPSKsignals are transmitted on one, three and six spatial modes.Squareand diamond markers represent the weakly and strong couplingregimes, respectively. The dotted curve shows the back-to-back case(no fiber).

Following the procedure in sectionIII , we apply the substi-tution

A = RA , (36)

whereR is a random rotation matrix of dimension2M × 2Mwhose elementsrij satisfy the following relations [27]:

〈|rik|2|rik′ |2〉=

1 + δkk′

2M(2M + 1)(37)

〈|rkm|2|rk′m′ |2〉=1

(2M + 1)(2M − 1)k 6= k′

m 6= m′

(38)

〈r∗kirkjrk′ir∗k′j〉=

−1

(2M + 1)2M(2M − 1)k 6= k′

i 6= j(39)

By applying the substitution (36) and averaging the propa-gation equation (33), we obtain the following generalizationof the Manakov equations in the strong-coupling regime:

∂A

∂z+1

v

∂A

∂t+ ı

β2

2

∂2A

∂t2= ıγκ|A|2A (40)

where

κ =

M∑

k≤l

32

2δkl

fkkll6M(2M + 1)

, (41)

1/v = trace (B1) /2M is the average inverse group velocity,and β2 = trace (B2) /2M is the average group velocitydispersion. Equations (40) and (41) are in agreement with theresults obtained in [16] using a different procedure.

A comparison of the magnitudes of various nonlinearcoefficients appearing in (29) and (40) reveals that systemperformance are improved in the high-coupling regime as thenumber of spatial modes increases. The nonlinear coefficientκ in Eq. (41) tends to zero withM increasing, which predictsthat multimode fibers in the high coupling regime can performbetter than single-mode fibers for large number of propagating

NONLINEAR PROPAGATION IN MULTIMODE FIBERS 9

spatial modes. Using the same parameter values used insectionIV, Figure 5 presents numerical results for 1000-kmtransmission over a graded-index fibers supporting one, threeand seven Hermite-Gaussian HGmn spatial modes. Each fiberis designed in order to haveγ = 1.4 W−1/km. With thethree mode fiber, data are transmitted on the HG00, HG01and HG10 spatial modes. With the seven mode fiber, data aretransmitted on the six spatial modes presented in Table.I.Comparison of the weak and strong coupling regime showsthat, as expected, the BER curves in the two cases coincidewhen only one spatial mode of the fiber is used. However,the high-coupling regime results in better BER performancewhen data is transmitted using multiple spatial modes. Anotherdifference is that the performance is degraded in the weak-coupling regime as the number of spatial modes increaseswhereas, somewhat surprisingly, performance is nearly thesame for three or six spatial modes in the high-couplingregime.

VII. C ONCLUSIONS

In this paper we have focused on SDM systems designed byusing multimode or multicore fibers and derived a nonlinearpropagation equation satisfied by the bit stream transmittedthrough each optical mode in the presence of fiber disper-sion, birefringence, and nonlinearity. We first expressed thisequation in Jones vector form in the slowly varying envelopeapproximation so that both polarization components of eachmode can be treated simultaneously and we then used it toinvestigate the performance of SDM systems designed withmultimode fibers exhibiting modal birefringence that variesrandomly on a length scale smaller than the length over whichnonlinear effects become important. The effects of fluctuatingbirefringence were included through random Jones matrices.As expected, the XPM and FWM effects resulting fromintermodal nonlinearities cannot be ignored for multimodefibers. There are as many asM3 − 1 intermodal nonlinearterms for anM -mode fiber.

Since numerical simulations based on such stochastic non-linear propagation equations are generally time-consuming, wegeneralized the Manakov equations, well known in the case ofsingle-mode fibers, to the case of multimode fibers. We con-sidered the weak linear coupling case first and averaged overbirefringence fluctuations following a procedure similar to thatused in the original derivation of the Manakov equations forsingle-mode fibers. We found that averaging over birefringencefluctuations reduces theM3−1 intermodal nonlinear terms toonly M − 1 terms because all FWM-type terms disappear.The XPM-type terms remain but their effectiveness is reducedbecause the well-known factor of 2 is reduced to 4/3.

We verified the validity of our new Manakov equationsby simultaneously transmitting multiple 114-Gb/s bit streamsin the PDM-QPSK format over different modes of a multi-mode fiber and comparing the numerical results with thoseobtained by solving the full stochastic equation explicitly.The agreement between the two methods was excellent in allcases studied. We found that the use of the new Manakovequations can reduce computation time by at least a factor

of 10 and by a factor in excess of 100 as the number ofmodes grows. Our numerical results show that birefringencefluctuations improve system performance by reducing theimpact of fiber nonlinearities. The extent of improvementdepends on the fiber design and how many modes are usedfor SDM transmission. We investigated both the step-indexand graded-index multimode fibers and considered up to threespatial modes in each case.

We extended our theory to the case of multicore fibersand discussed the validity of the new Manakov equations bystudying the impact of linear coupling among various cores.We found that the new Manakov equations remain valid whenthe length scale of random birefringence of the cores is about100 times smaller than the linear coupling length among thecores of a multicore fiber.

Finally we studied the case of strong random couplingamong various spatial modes of a multimode fiber and derivenew Manakov equations for this regime. We showed thoughnumerical simulations that the high-coupling regime leadstobetter system performance than the weak-coupling regimewhen the number of spatial modes is at least two. Theoryalso predicts that performance improves for large number ofpropagating spatial modes so that such multimode fibers canperform even better than single-mode fiber.

REFERENCES

[1] R. -J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini,and B. Goebel,”Capacity limits of optical fiber networks,”J. Lightw. Technol., vol. 28,no. 4, pp. 662-701, Feb. 2010.

[2] R.-J. Essiambre, ”Impact of Fiber Parameters on Nonlinear Fiber Ca-pacity”, in Proc. Opt. Fiber Commun. Conf. Expo./ Nat. Fiber Opt. Eng.Conf., OTuJ1, March 2011.

[3] A. Mecozzi and R.-J. Essiambre, ”Nonlinear shannon limit in pseudo-linear coherent systems”, Lightwave Technology, Journal of , vol.30, no.12, pp. 2010-2024, 2012.

[4] R. W. Tkach, ”Scaling Optical Communications for the Next Decadeand Beyond,”Bell Labs Tech. J., vol. 14, no. 4, pp. 3-9, 2010.

[5] R.-J. Essiambre and R.W.Tkach, ”Capacity Trends and Limits of OpticalCommunication Networks,” Proceedings of the IEEE , vol. 100, no. 5,pp. 1035-1055, May 2012

[6] R. Ryf, S. Randel, A. H. Gnauck, C. Bolle, R.-J. Essiambre, P. J.Winzer, D. W. Peckham, A. McCurdy and R. Lingle, ”Space-divisionmultiplexing over 10 km of three-mode fiber using coherent 6×6 MIMOprocessing,” inProc. Opt. Fiber Commun. Conf. Expo./ Nat. Fiber Opt.Eng. Conf., pp. 1-3, March 2011.

[7] E. Ip, E.; N Bai, Y.-K. Huang, E. Mateo, F. Yaman, M.-J. Li;S. Bickham,S. Ten, J. Linares, C. Montero, V. Moreno, X. Prieto, V. Tse, K. Chung,A. Lau, H.-Y. Tam, C. Lu, Y. Luo, G.-D. Peng and G. Li; , ”88×3×112-Gb/s WDM transmission over 50 km of three-mode fiber with inline few-mode fiber amplifier,” inProc. 37th Eur. Conf. Opt. Commun., pp.1-3,Sept. 2011.

[8] R. Ryf, S. Randel, A. H. Gnauck, C. Bolle, A. Sierra, S. Mumtaz, M.Esmaeelpour, E.C. Burrows, R. Essiambre, P.J. Winzer, D.W.Peckham,A.H. McCurdy and R. Lingle, ”Mode-Division Multiplexing Over 96 kmof Few-Mode Fiber Using Coherent 6×6 MIMO Processing,” LightwaveTechnology, Journal of , vol.30, no.4, pp. 521-531, Feb. 2012.

[9] C. Koebele, M. Salsi, L. Milord, R. Ryf, C. Bolle, P. Sillard, S. Bigoand G. Charlet , ”40km transmission of five mode division multiplexeddata streams at 100Gb/s with low MIMO-DSP complexity,” inProc.37th Eur. Conf. Opt. Commun., pp.1-3, Sept. 2011.

[10] F. Poletti and P. Horak, ”Description of ultrashort pulse propagation inmultimode optical fibers”,J. Opt. Soc. Am. B, vol. 25, pp. 1645-1654,Sep. 2008.

[11] P. K. A. Wai and C. R. Menyuk, ”Polarization mode dispersion,decorrelation, and diffusion in optical fibers with randomly varyingbirefringence”,J. Lightw. Technol., vol.14, no. 2, pp. 148-157, 1997.

NONLINEAR PROPAGATION IN MULTIMODE FIBERS 10

[12] D. Marcuse, C. R. Manyuk and P. K. A. Wai, ”Application oftheManakov-PMD equation to studies of signal propagation in optical fiberswith randomly varying birefringence”,J. Lightw. Technol., vol. 15, no.9, pp. 1735-1746, 1997.

[13] S. V. Manakov, ”On the theory of two-dimensional stationary self-focusing of electromagnetic waves”,Sov. Phys. JETP, vol. 38, pp. 248-253, 1974.

[14] K. Ho, J. Kahn, ”Statistics of Group Delays in MultimodeFiber WithStrong Mode Coupling”, Lightwave Technology, Journal of , vol.29,no.21, pp.3119-3128, Nov. 2011.

[15] K. Ho, J. Kahn, ”Mode-dependent loss and gain: statistics and effect onmode-division multiplexing,” Opt. Express 19, pp. 16612-16635, 2011.

[16] A. Mecozzi, C. Antonelli, M. Shtaif ”Nonlinear propagation in multi-mode fiber in the strong coupling regime”, arXiv:1203.6275v2, March2012.

[17] A. Ghatak and K. Thyagarajan,Introduction to Fiber Optics, Cambridgeuniversity press, 1998.

[18] R. W. Boyd ”Nonlinear Optics”, 2nd ed., Academic Press,2003.[19] G. P. Agrawal, ”Nonlinear Fiber Optics”, 4th ed. San Diego: Academic,

2007.[20] S. G. Evangelides, L. F. Mollenauer, J. P. Gordon, N. S. Bergano,

”Polarization multiplexing with solitons,”J. Lightw. Technol., vol. 10,no. 1, pp. 28-35, 1992.

[21] R. Ryf, R. -J. Essiambre, S. Randel, A. H. Gnauck, P. J. Winzer, T.Hayashi, T. Taru and T. Sasaki, ”MIMO-Based Crosstalk Suppressionin Spatially Multiplexed 3 56-Gb/s PDM-QPSK Signals for StronglyCoupled Three-Core Fiber”, IEEE Photon. Technol. Lett., vol. 23, no.20, pp. 1469-1471, Oct., 2011.

[22] C. Koebele, M. Salsi, G. Charlet, S. Bigo, Member, IEEE,”NonlinearEffects in Mode-Division-Multiplexed Transmission Over Few-ModeOptical Fiber”, IEEE Photon. Technol. Lett., vol. 23, no. 18, pp. 1316-1318, Sep., 2011

[23] D. Marcuse,Theory of Dielectric Optical Waveguides, 2nd ed., Aca-demic, New York, 1991.

[24] S. Mumtaz R. -J. Essiambre, G. P. Agrawal ”Reduction of NonlinearPenalties due to Linear Coupling in Multicore Optical Fibers”, acceptedto IEEE Photon. Technol. Lett.

[25] M. B. Shemirani, W. Mao, R.A. Panicker and J.M. Kahn, ”PrincipalModes in Graded-Index Multimode Fiber in Presence of Spatial- andPolarization-Mode Coupling”,J. Lightw. Technol., vol. 27, no. 10, pp.1248-1261, May 2009.

[26] F. Mezzadri, ”How to Generate Random Matrices from the ClassicalCompact Groups”,Notices of AMS, vol. 54, no. 5, pp. 592-604, May2007.

[27] A. M. Tulino, S. Verdu ”Random matrix theory and wireless communi-cations”, Communications and Information Theory, vol. 1 np. 1, p.1-182,June 2004.