Multimode Nonlinear Fiber Optics: Massively Parallel Numerical Solver, Tutorial … ·...

JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 1

Multimode Nonlinear Fiber Optics: Massively Parallel NumericalSolver, Tutorial and Outlook

Logan G. Wright1, Zachary M. Ziegler1, Pavel M. Lushnikov2, Zimu Zhu1, M. Amin Eftekhar3,Demetrios N. Christodoulides3 , and Frank W. Wise1

1School of Applied and Engineering Physics, Cornell University, Ithaca, New York 14853, USA2Department of Mathematics and Statistics, University of New Mexico, Alberquerque, New Mexico 87131, USA

2CREOL, College of Optics and Photonics, University of Central Florida, Orlando, Florida 32816, USA

Building on the scientific understanding and technological infrastructure of single-mode fibers, multimode fibers are beingexplored as a means of adding new degrees of freedom to optical technologies such as telecommunications, fiber lasers, imaging, andmeasurement. Here, starting from a baseline of single-mode nonlinear fiber optics, we introduce the growing topic of multimodenonlinear fiber optics. We demonstrate a new numerical solution method for the system of equations that describes nonlinearmultimode propagation, the generalized multimode nonlinear Schrodinger equation. This numerical solver is freely available,implemented in MATLAB® and includes a number of multimode fiber analysis tools. It features a significant parallel computingspeed-up on modern graphical processing units, translating to orders-of-magnitude speed-up over the split-step Fourier method.We demonstrate its use with several examples in graded- and step-index multimode fibers. Finally, we discuss several key opendirections and questions, whose answers could have significant scientific and technological impact.

Index Terms—Nonlinear optics, optical fibers, multimode waveguides, ultrafast optics

I. INTRODUCTION

THE modern interconnected world has been constructedaround a network of single-mode optical fiber. Nowadays,

this foundation is beginning to crack [1]–[3]. A so-called ”ca-pacity crunch” is anticipated, whereby the single-mode fibersystems of today will be unable to meet increasing demand -except through the expensive undertaking of adding more andmore fibers. A promising solution to this potential bandwidthcrisis is spatial division multiplexing in multimode fibers [4]–[8]. While standard single-mode optical fibers support just thelowest-order LP01 mode, multimode optical fibers, having alarger core, can support a multitude of transverse eigenmodes,each with different spatial shapes and propagation constants(Fig 1). Inside multimode fibers, these modes can interactthrough the influence of optical nonlinearity, disorder and lasergain. Although these interactions make multimode fibers muchmore complex than their single-mode counterparts, they alsooffer a world of new physics and potential applications thatextends far beyond communications.

Optical fibers are the basis of several other importanttechnologies, for which multimode fibers provide a roadmapfor improvement. Today, high average power (>100 W) fiberlasers are a fast-growing industry. Since multimode fiberarchitectures spread guided light over a much larger area, theysupport the highest power levels and so are already widelyused in these products. Femtosecond-pulse fiber lasers, whichhave historically only used single-mode components, are nowemerging as commercial products. These devices have promisefor enabling mainstream applications of ultrafast pulses, suchas clinical applications of nonlinear optical microscopy andprecision materials processing. For widespread adoption, these

Manuscript received December 1, 2012; revised August 26, 2015. Corre-sponding author: L.G. Wright (email: [email protected]).

sources need to match or exceed the performance of the currentstandard solid-state femtosecond lasers such as the Ti:sapphirelaser and do so at significantly lower cost. Today, fiber sourcesstill mostly lag behind in peak power, or lack the necessarywavelength coverage. Consequently, despite the availability ofreliable, low-cost instruments, many users cannot adopt thefiber platform. In this respect, multimode waveguides offeraltogether new routes in generating new colors, and to scalingthe peak power of these sources.

Looking to the future, multimode waveguides can providea new dimension in optical wave propagation, which can leadto qualitatively-new physical behaviors, and qualitatively-newkinds of applications compared to those supported by single-mode devices. For example, multimode fibers may serve ascompact (even sub-mm) endoscopes, through which a laserbeam can be focused, and even scanned [9]–[12]. Such a capa-bility could one day enable minimally-invasive in vivo biopsy,and biological studies deep inside scattering tissue. In addition,nonlinear optical devices based on multimode waveguides canexploit multimode and spatiotemporal processes to generatenew colors or shape complex electromagnetic fields(e.g., [13]–[46]).

On the other hand, the complexity of multimode fiber opticspresents major challenges. With many modes, all of whichcan be coupled in principle, the physics can be conceptually-difficult. As will be described, numerical calculations can beextremely helpful in isolating the roles of different physicalprocesses, but are frequently limited by computational cost.Experiments generally require diagnostics of the spatial, tem-poral, and spectral content of a field, and all of the degreesof freedom may be coupled. On the other hand, multimodefibers provide a rich environment for the investigation of wavephenomena, where nonlinearity, dissipation, and disorder canbe controllably included. While recent efforts provide some

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Photonic Crystal

Hollow-core

Single-core Multimode Fibers Other Multimode Fibers

LP01 LP11a LP11b LP02 LP21a LP21b

0 5 10 15 20 25Mode (#)

-40

-30

-20

-10

0

0 (1/m

m)

-10

-5

0

0 (1/m

m)

x

y

|Fp(x

,y)|2

Graded-index

Step-index

claddingcore

r

n

r

n

Graded-index

Step-index

|E(x

,y)|2

(a) (b)

(c) (d)

(e)

Fig. 1. Modes in multimode fiber. Many studies consider fibers that support up to hundreds of modes. The linear-polarized (LP) modes are Fp(x, y) =ecFp(x, y), where c = x or y for each linear polarization. Although the LP modes are usually referred to by a pair of radial and azimuthal indices (LP01,LP11a, LP11b...), for compactness and generality throughout the rest of the text, we mainly use one index to refer to the modes, numbered starting from thefundamental mode (largest propagation constant, β = 2πneff/λ). As a result of the many supported modes, multimode beams can appear quite complex, since

they are superpositions of the spatial eigenmodes of the fiber. (a) shows an intensity profile |E(x, y)|2= |6∑papFp(x, y)|2 that is a random superposition of

the first 6 modes of a fiber whose intensity profiles |Fp(x, y)|2 are shown in (b). (c) shows the respective index profiles of graded-and step-index multimodefibers for comparison, and (d) the corresponding propagation constants of the first 25 modes for each fiber type (with identical radius and index contrast).Step-index fibers support many more modes, all of which are close to one another in propagation constant and effective index, while graded-index fiberssupport discrete mode groups: modes in the same group have near-identical propagation constants, while different groups are separated by large propagationconstant mismatch. (e) shows various other types of multimode fibers. More detail and information is provided in section III.B.

vindication that the complexity of multimode fibers can beunderstood and exploited, the science of multimode nonlinearfiber optics itself is still in its infancy.

II. OUTLINE AND PURPOSE

Our goal in this article is to provide readers the background,perspective and necessary tools for exploring multimode non-linear optical pulse propagation. Nonlinear optics in single-mode fiber is treated exhaustively in Ref. [47], and basicaspects of multimode waveguides can be readily found inthe literature (see for example Ref. [48]). To analyze thenonlinear behavior of multimode structures, we first intro-duce the most common theoretical model used for this task,the generalized multimode nonlinear Schrodinger equations(GMMNLSE, Section III). One reason we have chosen thisapproach is that it allows a modular dissection of all processesinvolved in multimode fibers [49], [50]. Many limitations ofthis model can be reduced by adding terms that correspond tonew processes, which in turn can be examined in relation toother individual effects. Hence in interpreting experiments orsimulations, we can build from the “bottom-up”.

We have employed the GMMNLSE to model several ex-periments in our laboratory over the past few years. Due toits computational cost, it has generally necessary to restrictthe calculations to a small subset of the actual modes of afiber, and/or smaller time windows or propagation lengths thandesired. This motivated the development of faster codes formultimode propagation. Here we introduce a new, massively-parallel numerical solver for the GMMNLSE. This solver is

implemented for efficiency and easy editing in the MATLAB®

computing platform, and is freely available [51]. By leveraginggraphical processing units (GPUs), the solver can be used forrapid modeling of multimode fiber propagation on personalcomputers. We demonstrate this numerical tool with severalsimple and concrete examples. Building on the basic conceptsdescribed by the GMMNLSE, we then describe several im-portant effects in multimode waveguides that go beyond theGMMNLSE model as presented, such as linear mode-coupling(including disorder), and gain and loss. Finally, we highlightsome important open questions for the physics and probableapplications of multimode nonlinear fiber optics.

III. THE GENERALIZED MULTIMODE NONLINEARSCHRODINGER EQUATIONS (GMMNLSE)

The GMMNLSE was first derived by Poletti and Horak[52]. The later, simplified version [49] will be consideredhere, mainly for the sake of reducing the complexity ofthis description. While this equation has turned out to beremarkably useful, it does have many limitations that will bediscussed later in the article. The GMMNLSE are a system ofcoupled NLSE-type equations for the electric field temporal


envelope for spatial mode p, Ap(z, t):

∂zAp(z, t) =

iδβ(p)0 Ap − δβ(p)

1 ∂tAp +

Nd∑m=2

im+1 β(p)m

m!∂mt Ap

+ in2ωoc

(1 +i

ωo∂t)

N∑l,m,n

[(1− fR)SKplmnAlAmA∗n

+ fRSRplmnAl

∫ t

−∞dτhR(τ)Am(z, t− τ)A∗n(z, t− τ)] (1)

We admit that this is not what would normally be referredto as a simple equation. But what we have here is not thesolo performance of one mode, but instead the orchestra of amultitude of modes. This analogy is apt in that it underlines thenecessity of understanding first the concepts of single-modepropagation, the single notes and rhythms, on which we haveto build the chords, the counterpoint, the interacting layersof complexity within a multimode symphony. In subsequentsections we will examine each term in Eqn. 1, attempting toisolate the component concepts of multimode wave propaga-tion, which are each expressed simply there.

Briefly, terms 1 through 3 on the right-hand side (RHS) ofEqn. 1 are the result of approximating the dispersion operatorin the mode p by a Taylor series expansion about ωo, thentransforming the terms into the time domain. The first twoterms are expressed relative to the lowest-order longitudinalphase evolution (δβ(p)

0 ) and group velocity (δβ(p)1 ) of first

mode (usually the fundamental). The third term representshigher-order dispersion effects (group velocity dispersion,third-order dispersion, etc.) up to order Nd. hR is the Ramanresponse of the fiber medium, fR ≈ 0.18 (in fused silica)is the Raman contribution to the Kerr effect, and n2 isthe nonlinear index of refraction. SRplmn and SKplmn are thenonlinear coupling coefficients for the Raman and Kerr effectrespectively [49]. In keeping with our goal of simplicity,we can significantly simplify the tensors by assuming thatthe modes excited are in a single linear polarization, andneglecting spontaneous processes which may cause couplinginto these modes. In this case, we have:

SRplmn = SKplmn = ∫dxdy[FpFlFmFn]

[∫

dxdyF 2p

∫dxdyF 2

l

∫dxdyF 2

m

∫dxdyF 2

n ]1/2(2)

where given our last assumption, it is sufficient to expresseach mode in terms of a real, scalar function Fp(x, y).

In many situations of interest, further simplifications canbe made. Self-steepening can be neglected by taking (1 +iωo∂t) → 1, and stimulated Raman scattering can be ignored

by letting fR → 0. We can also ignore the higher-orderdispersion, considering only the group velocity dispersion. Inthis case, we have equations which account for many recently-observed interesting phenomena, the MMNLSE:

∂zAp(z, t) = iδβ(p)0 Ap − δβ(p)

1 ∂tAp − iβ(p)2

2∂2tAp

+ in2ωoc

N∑l,m,n

SKplmnAlAmA∗n (3)

One perspective on Eqn. 3 is worth mentioning for intuitionand a complementary view of dynamics. If we undo the de-composition into modes and apply the paraxial approximation,Eqn. 3 can be written as:

∂zE(x, y, z, t) =i

2ke(ωo)∇2TE − i

β22∂2tE

+ ike(ωo)

2((n(x, y)/no)

2 − 1)E + in2ωoc|E|2E (4)

This is the 3D NLSE for the evolution of the electric fieldenvelope E(z, x, y, t) along z in the presence of a transversely-inhomogeneous refractive index n(x, y). This version of theequation makes it clear that what we have is just 3D wavepropagation (the LHS plus the first two terms on the RHScomprise a 3D wave equation), with perturbations from therefractive index inhomogeneity and a local nonlinear indexshift. Spatial modes originate as eigensolutions to the equationincluding only the first and third terms (diffraction and inho-mogeneous refractive index). This equation looks very similarto the Gross-Piteaevskii equation that is a well-used model forthe time evolution of a trapped Bose-Einstein condensate [53].It can naturally incorporate many types of effects that wouldbe difficult to express in Eqn. 1. In situations where manyguided modes must be considered, this equation will offercomputational advantages over the MMNLSE or GMMNLSE.When it is adequate to consider a small number of guidedmodes, or when the paraxial approximation is violated, itis faster to solve the coupled NLSEs. We usually find thatthe modal decomposition translates to higher accuracy, and toconceptually-clearer interpretation of observed dynamics. Thisis particularly true in practice, because the limited memorysize of consumer-grade GPUs restricts the maximum variablesize used to describe the propagated field. In most situationswith a small number of modes, the modal basis is much morememory efficient (the number of data points needed to fullydescribe the field is smaller).

The following sections introduce various processes andeffects that occur in a multimode fiber along with the parallelalgorithm for solving the appropriate equations. The treatmentis intentionally tutorial, and we emphasize that many ofthe presented results are well-known from prior work. Wehope that this will clarify the roles of different processes forreaders new to this area, and that introductory descriptions ofthe simplest processes will facilitate understanding of more-complex nonlinear phenomena considered later. We also hopethat consideration of the simplest cases first will set the stagefor description of the parallel algorithm.

A. Linear propagation in multimode fibersFigure 2 shows the main features of linear propagation in

multimode fiber. These behaviors can be described by the first


3 terms in the GMMNLSE. For simplicity we let Nd = 2, sothe equations are just

∂zAp(z, t) = iδβ(p)0 Ap − δβ(p)

1 ∂tAp − iβ(p)2

2∂2tAp

(5)

The eigenmodes of the multimode fiber are a set of or-thonormal electromagnetic field patterns (Figure 1). Eachmode has a different propagation constant, which determinesthe phase velocity of the electromagnetic field in that mode.There can be families of quasi-degenerate modes, such asLP11a and LP11b, which have very nearly the same prop-agation constant. In considering how modes interact, it can beuseful to refer to the “effective index”, neff, that an individualmode experiences. In most fibers, the difference between themaximum and minimum refractive indices is small, and as aresult the eigenmodes of the of the fiber are polarized alongaxes (x,y) orthogonal to the fiber axis (z). In this case, it istypical to write the modes of the fiber as “linearly polarized”LP modes, which can be treated as quasi-scalar fields with oneof two polarizations.

As is done for single-mode propagation, the effects ofchromatic dispersion are incorporated by approximating thefrequency dependence of the propagation constant in eacheigenmode by a Taylor series, which can then be expressed inthe time domain. The pulse propagation is then examined inthe reference frame of the fundamental mode (usually; anotherframe can be chosen if convenient), and we factor out thepropagation constant - the global longitudinal phase shift - ofthat mode as well. Since we can only choose one reference,most of the equations in the MMNLSE have two additionalterms to describe the difference in propagation constant andgroup velocity of each mode.

1) Propagation constant mismatchThe first term in Eq. 5 represents the propagation constant

mismatch, and is responsible for multimode interference ormode beating. As an example of the first term’s effects, weconsider multimode interference in parabolic-index fiber. In aparabolic fiber, such as is depicted in Figure 1, the propagationconstants of the modes are equally spaced, with modes occu-pying “mode groups” whose populations grow with decreasingpropagation constant. As a result, δβ(p)

0 = nP , where P is thepropagation constant mismatch between the mode groups, andn is an integer 0,1,2,3... equal to the difference in the modegroup between the two modes.

Over short distances, the second and third terms of Eqn. 5can be ignored. For the approximations made earlier, we canwrite the full electric field envelope as a composition over themodes:

E(x, y, z, t) =

N∑p

Fp(x, y)

[∫

dxdyF 2p ]1/2

Ap(z, t) (6)

Given that we are considering only the first term in 5, it iseasy to solve for the Ap(z, t) to find:

Fig. 2. Linear propagation effects in multimode fiber. The figure shows 3LP0N modes, LP01 (mode 1), LP02 (2) and LP03 (3) from a parabolicGRIN fiber. The pulse in each mode broadens according due to group velocitydispersion (Eqn. 9) and the pulses in each mode move away from the pulse inthe fundamental LP01 mode in time by δtp(L) = δβ

(p)1 L. When the pulses

are overlapped, we see periodic spatial evolution of the entire multimode field,whose periodic beating depends on the first term in Eqn. 5.

E(x, y, z, t) =

N∑p

Fp(x, y)

[∫

dxdyF 2p ]1/2

Apoei2πnpPz (7)

The field undergoes a periodic evolution along z. The exactdetails of the periodic evolution (including the period) dependon the modes involved. In step-index or other types of fibers,this lowest-order evolution is more complex since δβ

(p)0 is

not uniform. It is still generically referred to as multimodeinterference. An example of this rapid linear evolution, oversome millimeters of fiber after a multimode excitation of 3LP0N (radially-symmetric modes) is shown in Figure 2. Wewill see this lowest-order effect has a very important role whennonlinearity is considered.

2) Modal dispersionThe second term in Eqn. 5 describes modal dispersion, or

modal walk-off. Each mode has a slightly different groupvelocity, and so a pulse which is launched into multiple modeswill break up into sub-pulses whose separation grows linearlywith the fiber length. If we just consider the modal dispersionterm, and we assume we have some pulses in each mode witha temporal envelope S(t), then the field looks like:

E(x, y, z, t) =

N∑p

Fp(x, y)

[∫

dxdyF 2p ]1/2

S(t− δβ(p)1 z) (8)

We therefore see the electric field envelope break upinto pulses separated from the lowest-order mode pulseby δtp(z) = δβ

(p)1 z, whose spatiotemporal shapes are

Fp(x, y)S(t− δβ(p)1 z).

3) Chromatic dispersionThe third term in Eqn. 5 describes chromatic dispersion. In

any particular single mode of the fiber, the effects of chromaticdispersion are the same as in propagation in a single-modefiber: typically, short pulses broaden as they propagate. If a


transform-limited Gaussian pulse of duration To is launchedinto mode p, the pulse duration T broadens as [47], [54]:

T (z) = To

√1 + (z|β(p)

2 |/T 2o )2 (9)

In many situations that have been studied so far, thechromatic dispersion of all relevant modes is similar (i.e.,β(p)2 of all modes is about the same value). However, in

some cases the waveguide dispersion of particular modes (suchas the highest-order modes in solid, single-core multimodefibers) can be very strong. In single-mode fibers, waveguidedispersion requires a trade-off with the effective area. As aresult, dispersion engineering can make reaching high peakpowers a challenge. In multimode fibers, this trade-off iseliminated, and the unusual dispersion of higher-order modescan lead to interesting and useful effects [15]–[17], [55].

B. Types of multimode fibers

There are many types of multimode fiber. As we go onto consider nonlinear processes, it will be useful to examinehow each might play out in different kinds of multimodewaveguides. Scientifically, each kind of multimode waveguideprovides different opportunities for nonlinear effects. Forapplications, each offers unique properties and advantages.First, we examine the two basic kinds of solid single-corefibers, graded-index and step-index, then multicore fibers, andthen finally in more cursory detail the many other kindsof multimode waveguides to which the GMMNLSE can beapplied.

1) Graded-indexIn a graded-index (GRIN) fiber, the refractive index

smoothly changes through the core, reaching a maximum atthe center (Fig. 1(c)). Usually the profile is optimized to beclose to a parabola. As shown in Fig. 1(d), in a GRIN fiberthe modes cluster into nearly degenerate groups. Modes thatbelong to the same group have minimal modal dispersionbetween one another. As a result, the modal dispersion ingraded-index fibers is the minimum possible with a single-core design. The magnitude of the modal walk-off parameterswithin a given mode group in GRIN fiber can be about 10-100 times smaller than those in step-index fibers of similarspecifications (radius, index contrast, etc.), and the sign caneven be controlled by slightly modifying the design from aperfect parabola [56]. In practice, this means that one canobserve strong nonlinear intermodal interactions with veryshort (∼100 fs) pulses. As seen above, the regularity of themode spacing means there is a strong periodic component tothe longitudinal evolution of the field. This has been the sourceof several interesting observations, such as resonant radiationfrom oscillating multimode solitons [25], and the geometricparametric instability [38], [39], [57]–[59]. While intragroupmodal dispersions are very low, the effective index spacingbetween adjacent mode groups in GRIN fiber is actually quitelarge - larger, in fact, than the spacing between adjacentmodes in a similar step-index fiber. Linear mode coupling(such as from disorder in the fiber, see Section F1) decreasesrapidly with the difference in propagation constants, so the

fundamental mode of a graded-index fiber is the most stablemode of any multimode fiber (although the situation becomesmuch more interesting when nonlinearity is considered [41]).

2) Step-indexIn a step-index fiber (Fig. 1(c)), there is not quite as much

regularity in the mode structure as in GRIN fiber, and themodal dispersion is usually much stronger. The first feature hasproven useful, for example, in exciting high-order modes inthese fibers (primarily the LP0N family). The radial symmetryand the large difference in propagation constant from nearbymodes mean that these modes are weakly affected by disorderand can propagate over long distances [60]. Compared toGRIN fibers, there are fewer degeneracies. A step indexfiber supports more modes than a GRIN fiber with similarspecifications. For example, the GRIN and step-index fibersshown in Fig. 1(d) have equal core radius and index contrast,but support 170 and 330 spatial modes, respectively. Theeffective area of modes in step-index fiber tend to be largerthan in GRIN fiber. Nevertheless, unlike in GRIN fiber theeffective area of high-order LP0N modes actually decreaseswith mode order.

The strong modal dispersion in step-index fibers poses achallenge for observing interactions between short pulses indifferent modes, especially with anomalous dispersion in anyof the modes. To date, multimode dynamics in step-indexfibers have been primarily limited to broadband intermodalfour-wave mixing processes (e.g., [13], [22], [27], [42], [43],[61]). However, using longer pulses and particularly in thenormal dispersion regime, we expect that step-index fibersshould support a wider range of behaviors that involve otherprocesses. An example is considered in Section E3.

3) Multicore fibersMulticore fibers are multimode fibers formed by an array

of separate single-core fibers (the cores may be single-mode,multimode, graded-index, or step-index, etc.). An examplewith 7 cores is depicted in Fig. 1e. In the context of nonlinearmultimode dynamics, multicore fibers are another class of fiberthat is under-explored. When the cores are strongly coupled(i.e., a few diameters apart), the modes of a multicore fiberare the so-called “supermodes”. These can be interpreted ashybridizations of the individual modes of the separate cores.The supermodes have qualitatively different dispersive prop-erties than the modes of the individual cores. The propertiesof multicore fibers can be engineered much more easily thansingle-core multimode fibers. The modal dispersion can besimilar to GRIN fibers, and waveguide dispersion can bestrong even for the lowest-order modes. The effective areaof the supermodes can be large, and the overlap between thedifferent supermodes can be adjusted. When the cores arerelatively weakly or moderately coupled, an N-core multicorefiber is sensitive to disorder, since the supermodes of thefiber are very nearly degenerate [62], [63]. Finally, multicorewaveguides are attractive for their potential ease of integrationwith multiple single-mode fibers.

4) Other multimode waveguidesAdding spatial modes is a technique that can be applied

to virtually any optical guided-wave system to achieve newand/or better behaviors. For example, solid-core and hollow-


core photonic crystal fibers (Fig. 1e) can guide multiple modes.In hollow-core photonic crystal fiber, interactions betweenspatial modes have been used for phase-matching many uniqueprocesses [23], [64] however to date only fibers with a smallnumber of modes have been considered. Waveguides writteninto planar or bulk media (fused silica, or SiN, for example)[65], [66] may also support a few or many spatial modes. Manymicroresonators (and “macro” resonators like Herriot cells[67], [68]) support multiple spatial modes, and the opportuni-ties afford by multimode processes have just recently begun tobe explored in these systems [69]–[73]. Last, multimode “rodfibers” fabricated with very large claddings, or with very largedimensions offer an opportunity for studying highly multimodedynamics. By making the waveguide inflexible, these rodswould sacrifice some practical benefits of fiber. However, formultimode operation this inflexibility may be very importantfor realizing environmental stability.

C. Nonlinear propagation in multimode fibers

Nonlinearity underlies many surprises, opportunities, andproblems in multimode propagation. Fundamentally, nonlin-earity introduces coupling between the different modes, andis incorporated in the last term(s) of the GMMNLSE andMMNLSE.

1) Self phase modulationSelf-phase modulation (SPM) corresponds to SKpppp and

terms of the form i|Ap|2Ap. The fact that SKpppp is distinctfor each mode can lead to some interesting outcomes. Modeshave different effective areas (Aeff = 1/SKpppp). The effectivearea can be very large, so for applications such as high-powerlasers, this means that higher power can be guided beforenonlinear distortions become an issue. The differential rate ofnonlinear phase accumulation can also be important for inter-modal processes, since it leads to a decorrelation of the modalphases. An example is the observation of irreversibility withoutdissipation that occurs in Kerr beam cleanup, where unequalself-phase modulation arrests the periodic backconversion ofenergy through four-wave mixing [35].

2) Cross phase modulationTerms with SKpnpn or SKppnn, where n 6= p, are of the form

i|An|2Ap. These are cross-phase modulations (XPM). Therole of these terms in the formation of multimode solitonsis examined in the example in Section E2. XPM can lead toasymmetric spectral broadening when two pulses are travelingat different speeds. SPM and XPM are pure phase modula-tions, and cannot cause energy exchange between modes.

3) Four wave mixingAll other nonlinear coupling terms can be described as

“four-wave mixing” (FWM), which we will define as “termsthat can cause transfer of energy”. Examining the nonlinearterms in the GMMNLSE, we see that all terms are technically“four-wave mixing” in that they involve a mixture of four,possibly different, waves. But within our definition, four-wavemixing terms are ones that can take on complex values. We canunderstand FWM as the scattering between four waves due tothe nonlinear index modulations created by their interference.Significant energy exchange in a FWM process occurs only

when energy and momentum are conserved (the latter usuallybeing referred to as phase matching, and expressed in termsof the phase velocity of the interacting waves). In MMF, eachof the four interacting modes can have a different frequencyand be in a different spatial mode, so the conditions for FWMprocesses (except third-harmonic generation) are that:

ω1 + ω2 − ω3 − ω4 = 0;

∆β = β3(ω3) + β4(ω4)− β1(ω1)− β2(ω2) + ∆βNL = 0;(10)

Here, the final term in ∆β is a nonlinear contribution to thepropagation constant. In most cases, this has a small influenceon broadband multimode processes

Practically, the phase-matching requirements show thatthere are many degrees of freedom for controlling a FWMprocess in a highly-multimode fiber. We may be interestedin controlling the generated frequency, the generated mode,the bandwidth of the four-wave mixing process, or somefeatures of the generated photon’s correlations/entanglement.By adjusting the spatial mode, and frequency of two or threeof the waves involved, and/or by engineering the dispersionof the modes involved by design of the fiber, one has asignificant degree of control over the modes or frequencies ofthe remaining waves. One can also control the rate of changeof ∆β near 0, so as to affect the bandwidth, or range offrequencies, generated by the FWM process.

4) Self-steepeningThe self-steepening terms are proportional to −n2

c ∂t. Asin single-mode fiber, self-steepening in a multimode settingsis expected to lead to the formation of an increasingly steeptrailing edge (i.e., the side of the pulse where the t-variable inthe GMMNLSE is highest) and the corresponding broadeningof the blue side of the spectrum. The self-steepening termsmay also be responsible for intermode energy transfer, since∂tAlAmA

∗n may be complex. To date, self-steepening has

not been responsible for any major features of propagationin multimode fiber. However, it regularly has a noticeableeffect and it is often worth repeating simulations with andwithout this term to examine how it affects the results. Insingle-mode fiber, one would estimate self-steepening to beimportant when the fiber considered is comparable to the shockdistance, ≈ 0.39zNLω0T0, where zNL = Aeff/(n2ω0P0) is thenonlinear length, and where P0 is the peak power, and T0 isthe pulse duration. This will usually be a good guideline inmultimode fiber too. However, like many such characteristiclengths, it is only a loose guideline for how the solution to theGMMNLSE behaves.

5) Raman scatteringRaman scattering is a dissipative process where a phonon

from the fiber medium interacts with the electric field, causinga spectral redshift that is either continuous (in the case ofsolitons and multimode solitons [24], [74]) or discrete (moretypical in the normal dispersion regime [19], [41], but alsoobserved for group-velocity matched modes with anomalousdispersion [55], [71]). Raman scattering underlies multimodespectral incoherent solitons [75]. It can cause energy transfer


between modes, which occurs for example in Raman beamcleanup [76]. Within the GMMNLSE, stimulated Raman scat-tering is modeled using a phenomenological medium responsefunction, hR.

D. Parallel algorithm for solving the GMMNLSE

The GMMNLSE can be solved using the same numericalschemes used to solve the NLSE and GNLSE. Aspects andimplementation of the numerical solution to the GMMNLSEhave been considered by Poletti and Horak [52], and laterKhakimov et al. [77]. The most-common approach is basedon a split-step (SS) algorithm, where the nonlinear terms areintegrated in the time-domain and the linear dispersive termsare evaluated as multiplications in the spectral domain. For theGMMNLSE, this approach works well enough, but the non-linear step quickly becomes the limit as the number of modesincreases. For each mode p, propagating one step requirescalculating three nested sums, each over all modes. The overallcomputational complexity associated with this is thereforeO(P 4) for P modes, so the computational complexity growsextremely quickly for more than a few modes. For typicalresolutions, we find that this usually means the GMMNLSE isthe most efficient model when the number of modes is smallerthan about 10-30, beyond which the full-field techniques suchas the (3+1)-D GNLSE or the unidirectional pulse propagationequation [29], [78] become increasingly advantageous.

Here, we solve the GMMNLSE using a Massively ParallelAlgorithm (MPA) [79]–[81]. This is still subject to the poorscaling of the nonlinear term, but uses a parallel method tospeed up the calculations. In a typical split-step solution tothe GMMNLSE, the equation is separated into the linear andnonlinear terms, and integrated across each small step size inz over which these effects are assumed to be approximatelyindependent. The same is done over the next step in z, inessence “beam propagation” along z. In order to leveragethe increasingly parallel capabilities of modern computers,however, the MPA considers many subsequent small steps,which are computed in parallel and then iterates the until theerror is below a given tolerance. For each step, the linearterm is solved exactly, while the nonlinear term is numericallyintegrated, much like with the split-step method. As longas the number of iterations is smaller than the extent ofparallelization, which can always be the case, this algorithmcan be faster than a more traditional serial split-step algorithm.

Before continuing to explain the MPA, it is worth pointingout that the MPA provides a means of parallelizing manysimilar equations/numerical schemes, including the (3+1)-DGNLSE, and the methods recently developed by Lægsgaard[82], [83] and Conforti et al. [84]. These latter schemes arenot discussed in more detail here due to their recency. Briefly,the the former approach appears to provide a significant benefitfor highly-multimode simulations, both in terms of accuracyand speed, and the latter should provide very large speed-ups,albeit with good accuracy only for quite restricted conditions.Of course, the GMMNLSE reduces for one mode to theGNLSE. Since for one mode propagation is not limited bya short beat length, the MPA is less useful.

The numerical codes released in conjunction with this articleinclude an implementation of the MPA for the GMMNLSEas described in earlier sections [51], as well as the split-stepimplemented with and without GPU functionality, and multipletools for multimode fiber calculations, such as a mode solverfrom the MATLAB® file exchange [85]. In a future work,we hope to incorporate a broader set of codes, which will bediscussed later in this article.

Here we provide an overview of the MPA. A more sys-tematic description of the MPA GMMNLSE algorithm isincluded as part of the documentation for the numericalcode package [51], as well as within the commenting ofthe code itself. Starting with the GMMNLSE, we group thedispersion term (including mode-beating, modal and chromaticdispersions) into the linear operator D(t) = iδβ

(p)0 −δβ

(p)1

∂∂t+

i∑n≥2

β(p)n

n!

(i ∂∂t)n

and using the notation F [D(t)Ap(t, z)] =DωAp(z, ω), where F represents the Fourier Transform, MPAdefines the change-of-variables

Ap(z, ω) = ψp(z, ω)exp[Dω(z − z0)] (11)

Here ψp(ω, z) would be independent of z for purely lin-ear propagation. Adding nonlinearity makes ψp(ω, z) a slowfunction of z since the nonlinearity is weak. Applying thechange of variables, transforming to the Fourier domain, andintegrating converts the GMMNLSE, Eqn. 1, into

(12)

ψp(z, ω) = ψp(z0, ω) +

in2ω0

c

(1 +

ω

ω0

)∫ z

z0

∑l,m,n

F

{(1

− fR)SKplmnAl(z′, t)Am(z′, t)A∗n(z′, t)

+ fRSRplmnAl(z

′, t)

∫ t

−∞hR(τ)Am(z′, t

− τ)A∗n(z′, t− τ)dτ

}e−Dω(z′−z0)dz′

It should be noted that this is still an exact form of theGMMNLSE, no discretization has been applied yet. Fromthis form, however, one can see that the solution could becomputed efficiently in parallel if the space along z werebroken up into a number of discrete steps and at each pointz′ the integrand was computed in parallel. This is beneficialbecause by a large margin the most computationally expensivecalculation is the O(P 4) sum, which exists in the integrand.Because of the nonlinear nature of the integrand, in generalsuch a parallelization can not be done; however, by inspectingthe length scales of the problem we can formulate a solution.

We begin by defining two step sizes along z. A largestep size, L is broken up into M small steps ∆z in orderto compute the integrand in parallel. The two longitudinalstep sizes, the small step size ∆z and the longer step sizeL = M∆z, are determined by the two important length scalesin the problem, the nonlinear length, zNL, and the intermodebeat lengths, zIM = 1/δβ

(p)o . The nonlinear length depends

on the intensity, but is often no less than 50-100 µm andfrequently is orders of magnitude larger. While we typicallyestimate zNL = Aeff/(n2ω0P0) as in single-mode fiber, it


should be noted that, due to mode interference (e.g., the insetin Fig. 2), the length scale for nonlinear effects in MMFoften differs significantly from the single-mode estimate. Thebeat lengths between the modes depends on the fiber itself,however, and can be as small as a few µm. Given thesetwo lengths, therefore, we can choose the large step size Lsuch that L � zNL and the small step size ∆z such that∆z � zIM . Of course, these must be related by M∆z = L(i.e. M small steps fit into one large step). The z grid can thenbe denoted as the set of equally spaced points z0, z1, ..., zM ,where zM = z0 + L.

Each parallel step is computed as follows. Over a distanceof L, ψp(z, ω), which represents the result of nonlinear phaseaccumulation, is almost constant. To a first approximation,therefore, MPA sets ψ(n=1)

p (zj , ω) = ψp(z0, ω), where n isthe iteration number, for j = 1...M , which also establishesAn=1p (zj , ω) through Eqn. 11. Next each integrand is calcu-

lated in parallel, and then summed to get a more accurateapproximation of ψp(zj , ω). This process is then repeated inan iterative fashion. Each iteration, the algorithm recalculatesψ(n)p (zj , ω) by computing each integrand in parallel and then

summing over z′ using the by trapezoid rule. ψ(n)p (zj , ω) is

converted to A(n)p (zj , ω) through Eqn. 11, and then the process

repeats, each time giving A(n)p (zj , ω) with a higher and higher

accuracy. Once the accuracy reaches a given threshold, theprocess is considered converged and A

(ntot)p (zM , ω) is taken

as the final value at the end of the large step. This finishesone step from z0 to z0 +L, after which the next large step Lcan be taken using A(ntot)

p (zM , ω) as the new Ap(z0, ω). Therelative error from such iterations ∼ (L/znl)

n+1 as found inRefs. [80], [81].

In practice, even if L ≈ 110zNL this method converges to

a high degree of accuracy in ntot = 2− 3 iterations at most.Therefore, it is typically the case for simulations in MM fiberthat the MPA provides a significant parallel speedup. It shouldbe noted that such a speedup depends not only on ntot beingmuch smaller than M , but also heavily on the effectivenessof the implementation of parallelization. If M = 10 butcomputing the integrand for 10 points “in parallel” computeseach at 1/10th the speed that it would take to compute a singleintegrand in serial, this algorithm will not be effective andin fact it will be slower than a split-step method due to theadded overhead. As a result, a high degree of parallelizationis important, which motivates our use of GPUs.

Figure 3 shows the time comparison between the differentalgorithms, all implemented in MATLAB, for a simulation of10 modes of a graded-index fiber, under conditions for whichthe ratio znl/∆z ≈ 200, which is typical of our usage. Forthe MPA, we vary the M parameter. For our system (a high-end PC, with the top modern commercial GPU), simply usingthe GPU creates a significant speed-up. This is not surprisinggiven the numerous summations required. Above this, MPAprovides another speed-up, approaching an additional order-of-magnitude. As M increases, M∆z approaches znl andincreasingly higher values of ntot are required for convergenceover each large step. Hence, the value of M for whichmaximum speed-up occurs depends on the ratio of znl to the

5 10 15 20 25 30

M

0

20

40

60

80

100

120

time

on G

PU

(s)

0

1000

2000

3000

4000

5000

6000

time

on C

PU

(s)

MPA acceleration vs M(Note different axes)

MPA GPUSS GPUMPA CPUSS CPU

Fig. 3. Time comparison for simulations with the GMMNLSE solved usingsplit-step method (SS) and the MPA with the same numerical accuracy. Foreach case, we consider varying parallelism in the MPA (parameter M ). Asignificant speed-up is also observed from utilizing GPU functionality withinMATLAB.

required small step size ∆z, as well the overhead of the MPArelative to the split-step. The code used to generate Figure3 is included with the numerical code package. We verifiedthe accuracy of the MPA code by comparison to analyticexpressions, such as from Ref. [47], to older codes in ourgroup for MM and single-mode propagation, and to predictionsof the 3D NLSE, Eqn. 4.

E. Examples calculated with the GMMNLSE

In the numerical code package available online, we have in-cluded three examples that walk through representative studiesof pulse propagation with the GMMNLSE. We summarize thefindings of these studies here, but their main purposes are toillustrate 1) the use of the various codes (including calculationof the modes [85], dispersion and coupling tensors), 2) the useof the modularity of the GMMNLSE to understand complexnonlinear pulse propagation, 3) to emphasize the equation’slimits, and where to be cautious about its accuracy, and 4)to provide some some guidelines for avoiding and identifyingnumerical artifacts.

1) Example 1: linear propagation in a multimode fiberIn this example, a short pulse is launched into all the modes

of the fiber, with very small peak power. The results are similarto those shown in Fig. 2.

2) Example 2: Multimode soliton formationIn this example, we consider propagation in a GRIN fiber

modeled after standard telecommunication-grade GRIN mul-timode fiber, and look at the formation and propagation ofmultimode solitons [20], [24], [25], [74], [86]–[92]. Fig. 4illustrates the basic concepts underlying multimode solitons.In contrast to the linear propagation shown in Fig. 2, withnonlinearity we may observe a multimode pulse that does notbroaden in time, nor break apart into multiple pulses in eachconstitutent mode family. Anomalous chromatic dispersion isbalanced by self-phase and cross-phase modulations.

Since the modes have different group velocities, we expectthat cross-phase modulations cause asymmetric spectral broad-ening of pulses in different modes. This process can shift the


Fig. 4. Soliton propagation in multimode fiber. The figure shows 3 LP0Nmodes, LP01 (mode 1), LP02 (2) and LP03 (3) from a parabolic GRINfiber. Multiple spatial modes lock together in a single pulse, which evolvesover short distances due to the different modal propagation constants. Theformation of MM solitons is considered in Example 2.

average frequency of the pulse in each mode, and thereforecause pulses in the different modes to have a common groupvelocity, where the changes in chromatic dispersion compen-sate the mismatch in modal dispersion. Intuitively, this processcan be thought of as a mutual trapping of each pulse by theothers and itself.

Hence, as illustrations of use of the codes for numericalinvestigations, in this example we roughly aim to answer thequestions “What is the role of the initial pulse duration on theexcitation of highly multimode solitons?”, and “Is cross-phasemodulation the only important process for the formation of themultimode soliton?”

We consider the first 6 modes of the fiber, plus 2 ad-ditional LP0N modes. In our first simulation, we launch6 nJ, equally distributed among those modes, and monitorpropagation through 15 m. With a 50-fs initial pulse duration,a pristine multimode soliton emerges and experiences a solitonself-frequency shift. Similar to experimental work [24], [74],we filter out the shifted soliton in order to examine it inisolation (Fig. 5a-b). With Raman and four-wave-mixing termsincluded, we see that energy shifts from higher-order to low-order modes, also in agreement with experimental findings inthis kind of fiber. As expected due to XPM, the spectrum ineach mode shifts slightly to help equalize the group velocityacross the modes. When we neglect self-steepening and Ra-man, and include only the Kerr SPM and XPM terms, a MMsoliton also forms for the same initial condition (Fig. 5c-d),which illustrates that XPM alone is sufficient for MM solitonsto form. With all terms included, a 1-ps initial pulse breaks upinto multiple pulses, and at 15 m propagation no clear MMsoliton has formed (Fig. 5e-f). However, at around 3 ps in Fig.5e, we see evidence of a MM soliton comprising a subset ofthe launched modes (see inset Fig. 5f). If the reader cares toextend the propagation to well beyond 15 m, they will observeformation of several MM solitons, each with a unique (andperhaps surprising) composition. In summary, we can answerour earlier questions as “a shorter pulse can produce a more-

35.6 36

Time (ps)

5

10

15

Inte

nsity

(kW

)

1600 1700 1800

Wavelength (nm)

0

2

4

6

8

Inte

nsity

(a.

u.)

2.05 nJ1.41 nJ0.28 nJ0.38 nJ0.23 nJ0.2 nJ0.08 nJ0.07 nJ

3.4 3.6 3.8 4 4.2

Time (ps)

2

4

6

8

10

Inte

nsity

(kW

)

1500155016001650

Wavelength (nm)

2

4

6

8

Inte

nsity

(a.

u.)


0 10 20

Time (ps)

0.5

1

1.5

2In

tens

ity (

kW)

1550 1600

Wavelength (nm)

2

4

6

8

Inte

nsity

(a.

u.)


All terms, bandpass filtered

All terms, 1 ps launch pulse

(a) (b)

(c) (d)

(e) (g)

SPM and XPM only

3 4Time (ps)

0.5

1

1.5

2

Inte

nsity

(kW

) (f)

Fig. 5. Examination of multimode soliton formation and propagation. (a-b) show the propagated field after 15 m and bandpass spectral filtering toisolate the Raman-shifted soliton. The pulse is initially 50 fs long, with 6nJ equally distributed into the 8 modes considered. (c-d) show the result ofpropagation for the same initial condition, but with only the Kerr SPM andXPM terms included. (e-g) show the same simulation as (a-b), except with a1 ps duration initial pulse. The inset, f, shows a zoom-in of e to emphasizea forming multimode soliton. The energy values plotted in b, d, and g referto the total energy within the respective spatial mode plotted (starting withmode 1, the fundamental mode, at the bottom).

multimode soliton,” and “qualitatively yes, but quantitativelyno.”

3) Example 3: Generation of 1300-nm pulse through self-phase modulation

For biomedical nonlinear optical microscopy, excitation at1300-nm is highly desired [93]. Considering absorption andscattering together, this window corresponds to an attenuationminimum in typical tissue. Since well-developed fiber lasersemit around 1030 nm, 1550 nm, or 1900 nm, nonlinearfrequency conversion has been explored to reach 1300 nm.One simple approach has been to apply self-phase modulationto transform-limited pulses in short fibers [94], [95]. If thepropagation is dominated by SPM, we expect well-defined


spectral sidelobes. Since the outermost sidelobes correspondto inflections in the phase of the pulse, when filtered by abandpass filter, they are nearly transform-limited pulses [94],[95]. In order for this filtered sidelobe to have a high pulseenergy, we’d like to be able launch a very high energy pulse,and in a reasonable length of fiber (several centimeters at least)accumulate the spectral broadening. In this example, we areconcerned with the question “Can spectral broadening of apulse launched into the fundamental mode of a graded-indexor step-index fiber provide a Gaussian pulse at 1300 nm?”

Figure 6 shows the results of this investigation. Briefly, wefind that there is strong energy transfer due to the high peakpowers. In GRIN fiber, energy is relatively stable within thefundamental mode, while in the step-index fiber (which wedesign to have a similar fundamental mode area), significantenergy is transferred into the LP02 mode (i.e, mode 6 in Fig.6). Both these observations can be understood simply in termsof the effects of self-focusing (although the peak power hereis below the critical power for collapse, non-catastrophic self-focusing nonetheless occurs). We see that energy transfer inthe GRIN fiber occurs primarily on the trailing edge of thepulse, while in the step-index fiber, substantial energy is alsotransferred at the peak. In both cases, however, an energeticpulse can be obtained at 1300-nm. The pulses are slightlychirped. Given the very high peak power here, readers shouldquestion if such results could be obtained experimentallywithout damaging the fiber. Nonetheless (and in part becauseof this question), the problem is a good example of explorationusing the GMMNLSE. Within the context of these importantuncertainties, we can answer our earlier question with “yes”.

F. Other effects that can be treated within the GMMNLSE

1) Linear mode coupling including disorderAny small deviation from the ideal waveguide (for which the

modes used in the MMNLSE were calculated) can be modeledas causing coupling between the ideal guided modes of theunperturbed fiber, i.e. perturbative coupled mode theory. TheGMMNLSE can then be supplemented by an additional termon the right hand side [63], [96], C = i

∑Nn QnpAn, where

the coupling coefficients are given by:

Qnp(z)

=ko

2neff

∫dxdy[n2(x, y, z)− n2p(x, y, z)]Fn(x, y)F ∗p (x, y)

[∫

dxdyF 2n

∫dxdyF 2

p ]1/2

(13)

In this equation, np(x, y, z) is the perturbed index profileand n(x, y, z) is the ideal profile. If we consider just the longi-tudinal component of this, taking [n2(x, y, z)−n2p(x, y, z)] =

ε(x, y)f(z), then the Fourier transform f(k) of the longitudi-nal component tells what longitudinal momentum the mediumperturbations are providing to coupling processes. In otherwords, f(k) has a large component at some k = δβ, then twospatial modes whose propagation constant separation is δβwill be phase matched through the perturbation for efficientcoupling. For example, in the case of a long-period Bragggrating, a non-zero virtual momentum corresponding to the

GRIN fiber

Step-index fiber

After 1300-nm bandpass filtering

(a) (b)

(c) (d)

(e) (f)

Fig. 6. Generation of ∼ MW pulses at 1300-nm by self-phase modulation inthe fundamental mode of GRIN and step-index multimode fibers. (a-b) showthe result of propagation after 3.6 cm for a 600-nJ, 200 fs pulse launchedinto the fundamental mode of a GRIN fiber. (c-d) show the same for 2.8cm step-index fiber. For clarity, in (a-d), we are only showing modes whichhave more than 1% of the total energy. (e-f) show the result after bandpassfiltering the above fields at 1300-nm. Nearly-transform-limited pulses in thefundamental mode are obtained.

grating period can allow coupling between modes whose beatlengths are near that period.

In the case of disorder, f(z) is reasonably assumed to bea Gaussian stochastic function. This disorder would typicallynot have any particular periodicity, so f(k) is large only neark = 0 and therefore we expect that only modes with similarpropagation constants are coupled together by disorder. Insingle-core fibers these are the quasi-degenerate mode groups.The disorder is also usually described by its correlation length,which refers to the distance through which index perturbations,or equivalently linearly propagating fields in the coupledmodes, become uncorrelated. Most estimates for this lengthin typical glass fibers are in the 10-100 m range [47].

When the correlation length is shorter than the fiber length,or the length scale over which other physical processes oc-cur (e.g. modal dispersion, four-wave mixing), the effect ofdisorder qualitatively changes the pulse propagation physics.


In the case of modal dispersion, disordered coupling leads toa diffusive modal pulse broadening as the energy undergoesa random walk between different modes. In other words,instead of temporally broadening linearly with fiber length,as δβp1L as in Eqn. 8, pulses broaden instead in proportionto√L. The averaging, or wash-out, caused by disordered

mode coupling can also reduce the cross-phase modulationand reduce or eliminate the four-wave mixing terms in 3[63], [92], [96]–[99]. If the modes considered all occupy asingle quasi-degenerate mode group and relevant length scales(the dispersion, nonlinear lengths) all exceed the correlationlength, then the modes are in the so-called strong couplinglimit of disorder. In this case, Eqn. 3 reduces to a multimodeManakov equation [97], for which exact soliton solutionsexist. If multiple degenerate mode groups are present, a setof coupled Manakov equations [92] is obtained. On onehand, this elimination/reduction of four-wave mixing can bedetrimental, such as for parametric amplification [99]. Onthe other hand, for multimode fiber communications, strongdisordered mode coupling leads a remarkable outcome: Dueto nonlinear crosstalk averaging, for a given total power, thesignal-to-noise ratio for a set of N randomly coupled modes isactually lower than for N uncoupled single-mode fibers [98].

For more details on modeling disorder within the MMNLSEframework, we refer readers to Refs. [63], [99], to Ref. [100]for analytic expressions of disorder-induced mode-couplingcoefficients for typical fiber manufacturing errors, and to Ref.[98] for propagation with the multimode Manakov and coupledManakov equations.

2) Saturating gainIn multimode fibers with gain, the different modes not only

receive gain, but necessarily compete for available invertedpopulation since they overlap with the same gain medium.To lowest order, gain can be added into the GMMNLSE byan additional term gpAp, where gp is the small signal gainfor the electric field in mode p. In most fiber amplifiers,however, saturation will be important. For ultrashort pulses,a common model for this is to let the gain depend on thetime-integrated power, g(I) = go/[1 +

∫dtI(x, y, z, t)/Isat].

Incorporation of the full gain saturation will be describedin a future publication. A first approximation to the satu-ration (which is often adequate) is to expand the saturat-ing gain in a Taylor series, multiply on each side by aspatial eigenmode and integrate over space (i.e., perform achange of basis from the Cartesian coordinates to the spa-tial eigenmodes). This process yields the additional terms:

gpAp −N∑

l,m,n

SRplmnAl∫∞−∞ dτAm(z, τ)A∗n(z, τ)/Isat.

G. Important future directions

If you were overwhelmed by the GMMNLSE, we hope thatthe tutorial above, and walking through the included examples,has helped to convert this bewilderment into an appreciationof the beautiful orchestral arrangements of multimode non-linear optics. At this stage, however, we have to warn youof yet a second wave of whelming as we briefly describeimportant open directions. These directions include several

that are relevant to relatively near-term applications in fiberlasers, telecommunications, and imaging, but there are alsocompelling directions for basic research. This echoes what hashappened with the study of single-mode fiber, where appliedscience and engineering has supported, and also benefitedfrom, research that uses single-mode fiber as an experimentaltest-bed for basic physics.

1) Integrating disorder and nonlinearityFor applications, understanding the role of disorder in the

context of both linear and nonlinear multimode propagationwill be critical. Disorder is not only inevitable in long fibers,but as mentioned above, strong linear mode coupling can havean almost fantastic benefit for telecommunications. That said,how and if this will actually be achieved is still unknown. Theanswer will need to consider not only the interaction betweenlinear and nonlinear coupling effects, but also the practicalitiesand economics of a transition to multimode transmission [6].For now, experimental demonstrations of multimode fiberswith strong linear mode coupling (especially verifying theimpact on nonlinearity) are severely lacking. Several worksthat consider the complex intermediate coupling regime haveinteresting findings already [41], [96], [99], [101]. Meanwhile,several powerful techniques are being used to characterizecoupling within multimode fibers [102]–[109]. In comingyears, these theoretical and experimental developments willneed to come together.

2) Gain, amplifiers, and lasersVery recently, several works have shown the potential

for multimode propagation in fiber amplifiers and lasers togenerate unprecedented high power, and even spatially- orspatiotemporally-engineered light [31], [32], [110]. Beforethis, multimode fibers were considered as a means of imple-menting a fiber-format saturable absorber (e.g., [111]–[114]).The possibility for strong interactions between longitudinalmodes of many different transverse eigenmode families meansthat mode-locked lasers with multimode fiber can display astunning range of behaviors, including potentially orders-of-magnitude higher peak power than those based on single-mode, large-mode-area fibers. While these features are com-pelling, practical deployment of multimode fiber ultrafastoscillators is sure to be limited in the short term by com-plications of disorder, and from peak- and average-powerrelated damage to the fiber medium. Multimode fiber ampli-fiers present a more near-term possibility to obtain spatially-engineered, high-power laser sources. The potential impactof just being able to harness the ultra-large mode area ofmodes in multimode fiber is significant [115]–[117]. Furtherimprovements will probably require an increasing attentionto intermodal effects (as Example 3 suggests, for example).However, multimode fiber amplifiers like this already yieldpractical instruments with very compelling performance. Theyare likely to lead the way to a new generation of multimodefiber laser technologies.

A related direction concerns amplifiers for multimode fibertransmission systems. Nonlinear optical technologies may beparticularly interesting here, as the unique capabilities of engi-neering the parametric gain suggest a route to ultrabroadbandparametric amplifiers [45]. If this can be accomplished, it


could mean multimode fibers expand not only the spatialchannel density of telecom lines, but also the number ofspectral channels. Given that each spectral channel wouldrepresent N more total channels (for each spatial degree offreedom), the impact of this technology could be substantial.

3) Wave turbulence and other fundamental nonlinear dy-namics

The notion of turbulence in nonlinear optical wave prop-agation has produced many interesting results in the realmof quasi-1D systems and 2D systems [118]–[122]. Pulsepropagation in multimode waveguides can include a numberof compelling ingredients beyond dispersion and nonlinearity,namely vortex formation in multiple different dimensionalarrangements, different kinds of disorder [123], [124], anddissipation. For this reason, we expect the study of opticalwave turbulence in multimode fibers to find new phenomena,and to deepen the connection to fluid systems. Much likesingle-mode fibers have provided an unrivaled platform forstudying nonlinear waves in one dimension, multimode fibersmay provide a platform of equal utility for studying nonlineardynamics in higher dimensions, and with controllable com-plexity. Just as single-mode fiber studies were enabled bythe availability of cheap, high-quality fibers and measurementtools, scientific studies in multimode fiber will increasinglybenefit from coming improvements in multimode fiber mea-surement tools, theoretical techniques, and other multimodetechnologies developed for applications like fiber lasers andtelecommunications. It is difficult to estimate the long termimpact of such fundamental science technologically. However,if single mode waveguides provide a historical template,then there is good reason to be excited about uncoveringnew physics, and reaching deeper understanding of importantconcepts within nonlinear multimode waveguides.

4) Optical signal processing and other nonlinear opticalinformation technologies

As we move nearer to spatial division multiplexing, anotherarea where multimode fibers may make a difference is insignal processing [33], [34], [125]–[127]. Since intermodalinteractions can be controlled with many more degrees offreedom than single mode fiber, and because spatial divi-sion multiplexing will require greater signal processing thansingle-mode transmission, inline signal processing, signal rout-ing, etc. in multimode fibers could be important and seewidespread use. More speculative applications of multimodefiber for computing are possible [41], [128]. Quantum opticsand information processing may also benefit from multimodewaveguides [129]–[131], including possibly high-dimensionalentanglement with lower losses than most integrated platforms.

5) Multimode nonlinear optics beyond fused silica fibersWhile we have exclusively considered multimode optical

fibers here, there are many other platforms where we ex-pect multimode nonlinear dynamics to occur, and wherethe treatment using the GMMNLSE will be valid. Recently,multimode effects are being explored in microresonators inthe context of frequency combs [66], [69]–[73]. Multimodedegrees of freedom allow qualitatively new kinds of solitons,which may allow for combs at different wavelength regimes,integrated dual comb functionality, compatibility with spatial-

division multiplexing, with higher conversion efficiency, andso on. Bulk waveguides, or large stiff “rod” fibers may makemultimode fibers much more environmentally stable, and willallow highly multimode propagation with very large modeareas. While these devices will naturally sacrifice importantpractical features of fiber, such as flexibility or efficient heatdissipation, these downsides can be managed. If multimodefiber lasers and amplifiers relying on inflexible waveguidesprovide exceptional performance rivalling solid-state regenera-tive amplifiers, etc., the loss of these features will not diminishtheir attractiveness for applications compared to contempo-rary solutions. Finally, while multimode effects in hollowwaveguides have been considered already [23], [64], [132],expanding these studies to highly multimode propagation mayyield interesting results and practical benefits, such as ultrahighpower capacity, and increased flexibility of phase-matching thenumerous unique effects possible in guided-wave gas-basednonlinear optics.

Finally, there is much interest in generating supercontinuumin the infrared, using fibers comprised of chalcogenide glasses,for example. Generation of broad IR continua in few-modefibers has been analyzed [21], [36], [133], but experimentalwork has mostly focused on fundamental mode behavior [135](a notable exception is Ref. [134]). Recent developments inGRIN multimode chalcogenide fibers [136], [137] may allowfuture continuum sources with higher power and more conve-nient pumping wavelengths. The numerous four-wave mixingprocesses in multimode fiber may allow broad supercontinuumgeneration to be seeded in the normal dispersion regime(typically below 4.5 µm in chalcogenide fibers). Since lasersources above 4.5 µm are rare and underdeveloped, this wouldbe worthwhile.

6) Engineering excitation, spatiotemporal and mode-resolved measurements

To date, most studies of nonlinear optics in multimodefibers have utilized relatively simple techniques to excite thefibers, and to make measurements. When interpreted correctly,simple measurements can be useful, especially for initialexplorations. However, for studying specific phenomena andcertainly for developing practical instruments, more sophisti-cated techniques are needed. More powerful techniques, suchas spatial-light modulators, can not only allow better controlof the field, but can also allow fascinating exploratory studies[31], [138]. Given the huge parameter space of a multimodefiber, we expect these kind of studies to produce surprisingresults. As multimode optics develops for telecommunications,the need for reliable spatial mode-resolved and spatiotemporalpulse measurement tools is increasing. Work along theselines has made impressive demonstrations [43], [102]–[109].If these techniques can be made accessible and widespread,they will significantly enhance scientific and technologicaldevelopments in multimode waveguides and beyond.

IV. CONCLUSION

While in the beginning fiber optics emerged in the mul-timode domain, the development of low-loss single-modefibers eventually underpinned the scientific and technological


advances that have entirely transformed the internet (andthe world). In the past few years, technological motivationshave reinvigorated the study of multimode fibers. For non-linear optics, this motivation led to theoretical developments,including notably the GMMNLSE, which we employ hereas a tool to understand, both conceptually and numerically,multimode nonlinear wave propagation. The main featuresof multimode nonlinear wave propagation emerge from thedifferent kinds of dispersion and the many different possiblenonlinear interactions among modes. Linear mode coupling(often from disorder) and laser gain will likely increase inimportance in the future.

Just as the introduction of more and different instrumentscreates rich possibilities for composers, multiple modes addnew spatial and temporal degrees of freedom to nonlinearoptical wave propagation. As a result, qualitatively new opticalcapabilities and phenomena are being discovered. Applica-tions include high-bandwidth telecommunications, imaging,and high-power laser sources. For the most part, these appli-cations still require major developments before they becomepractical, but rapid progress is being made on a number offronts. Fundamental scientific studies in multimode fibers mayconnect deeply to other topics throughout science, and mayplay the first notes of entirely new ideas ultimately paving theway for entirely new technologies.

ACKNOWLEDGMENT

This work was supported by Office of Naval Research grantN00014-13-1-0649 and National Sciences Foundation grantECCS-1609129. The work of P.L. was partially supported bythe National Science Foundation DMS-1412140. LGW thanksWalter Fu for helpful discussions. The codes described in thisarticle are available for download online [51]. These codesinclude a freely-available mode solver from the MATLAB®

file exchange [85].

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Logan G. Wright received the B.S. degree in Engineering Physics fromQueen’s University, Kingston, Canada, in 2012. He is currently workingtoward the Ph.D. degree in applied physics at Cornell University, Ithaca, NY.His research interests include complex nonlinear optics and lasers.

Zachary M. Ziegler received the B.S. degree in Engineering Physics fromCornell University, Ithaca, NY in 2017. He is currently a research scientist atFeatureX in Boston, MA.

Pavel M. Lushnikov received the M.S. degree from Moscow Institute ofPhysics and Technology, Moscow, Russia in 1994, and the Ph.D. degree fromLandau Institute for Theoretical Physics, Moscow, Russia in 1997. He was thepostdoctoral research at Landau Institute in 1998-1999 and at the TheoreticalDivision of Los Alamos National Laboratory in 1999-2003. In 2004-2006he was Kenna Assistant Professor, Department of Mathematics, University ofNotre Dame. He has been at the Faculty in the department of Mathematics andStatistics at University of New Mexico since 2006 as the Associate Professor(2006-2012) and the Full Professor since 2012 until now. He is also currentlythe visiting member of Landau Institute for Theoretical Physics and VisitingScholar at the Theoretical Division of Los Alamos National Laboratory.

Zimu Zhu received his B.S. degree in Engineering Science specializing inPhysics from the University of Toronto, Canada, in 2013. He is currentlycompleting his Ph.D. degree at Cornell University, in Ithaca, NY. His researchinterests include the dynamics of nonlinear pulse propagation in multimodefibers.

M. Amin Eftekhar received the B.S. degree from Shiraz University ofTechnology, Shiraz, Iran, and the M.S. degree from the University of Tehran,Tehran, Iran. He is currently working toward his Ph.D. degree in Optics andPhotonicsat the College of Optics and Photonics, University of Central Florida.His research interests include nonlinear optics and fiber optics.

Demetrios N. Christodoulides received the Ph.D. degree from The JohnsHopkins University, Baltimore, MD, USA, in 1986. He is the Cobb FamilyEndowed Chair and Pegasus Professor of Optics at the College of Optics andPhotonics, University of Central Florida. He subsequently joined Bellcore asa Post-Doctoral Fellow at Murray Hill. Between 1988 and 2002, he was withthe Faculty of the Department of Electrical Engineering, Lehigh University.His research interests include linear and nonlinear optical beam interactions,synthetic optical materials, optical solitons, and quantum electronics. Hisresearch initiated new innovation within the field, including the discovery ofoptical discrete solitons, Bragg and vector solitons in fibers, nonlinear surfacewaves, and the discovery of self-accelerating optical (Airy) beams. He hasauthored and coauthored more than 300 papers. He is a Fellow of the OpticalSociety of America and the American Physical Society. In 2011, he receivedthe R.W. Wood Prize of OSA.

Frank W. Wise received the B.S. degree from Princeton University, Princeton,NJ, the M.S. degree from the University of California, Berkeley, and the Ph.D.degree from Cornell University, Ithaca, NY. Since receiving the Ph.D. degreein 1988, he has been on the Faculty in the School of Engineering and AppliedPhysics at Cornell University.


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