General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.

Users may download and print one copy of any publication from the public portal for the purpose of private study or research.

You may not further distribute the material or use it for any profit-making activity or commercial gain

You may freely distribute the URL identifying the publication in the public portal If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim.

Downloaded from orbit.dtu.dk on: Jun 16, 2021

Efficient simulation of multimodal nonlinear propagation in step-index fibers

Lægsgaard, Jesper

Published in:Journal of the Optical Society of America B-optical Physics

Link to article, DOI:10.1364/JOSAB.34.002266

Publication date:2017

Document VersionPublisher's PDF, also known as Version of record

Link back to DTU Orbit

Citation (APA):Lægsgaard, J. (2017). Efficient simulation of multimodal nonlinear propagation in step-index fibers. Journal ofthe Optical Society of America B-optical Physics, 34(10), 2266-2273. https://doi.org/10.1364/JOSAB.34.002266

https://doi.org/10.1364/JOSAB.34.002266https://orbit.dtu.dk/en/publications/744abddf-75a1-457b-bb91-c9f1161e800bhttps://doi.org/10.1364/JOSAB.34.002266

Efficient simulation of multimodal nonlinearpropagation in step-index fibersJESPER LÆGSGAARDDTU Fotonik, Department of Photonics Engineering, Technical University of Denmark, Ørsteds Plads 343,DK-2800 Kongens Lyngby, Denmark (jlag@fotonik.dtu.dk)

Received 22 June 2017; revised 5 September 2017; accepted 6 September 2017; posted 7 September 2017 (Doc. ID 300785);published 28 September 2017

A numerical approach to nonlinear propagation in waveguides based on real-space Gaussian quadrature integra-tion of the nonlinear polarization during propagation is investigated and compared with the more conventionalapproach based on expressing the nonlinear polarization by a sum of mode overlap integrals. Using the step-indexfiber geometry as an example, it is shown that the Gaussian quadrature approach scales linearly or at mostquadratically with the number of guided modes and that it can account for mode profile dispersion withoutadditional computational overhead. These properties make it superior for multimode nonlinear simulationsextending over wide frequency ranges. © 2017 Optical Society of America

OCIS codes: (060.4370) Nonlinear optics, fibers; (060.5530) Pulse propagation and temporal solitons; (190.0190) Nonlinear optics.

https://doi.org/10.1364/JOSAB.34.002266

1. INTRODUCTION

Nonlinear propagation in multimoded fibers is currently ofhigh interest in connection with both space-division multiplex-ing in telecommunication and nonlinear frequency conversion.In telecommunication, the use of different fiber modes to carrydifferent data streams is envisioned to boost the signal trans-mitting capacity of single fiber cores [1,2]. Because of the hightotal field intensities obtained, and long propagation lengths, itis crucial to understand intermodal nonlinear interactions insuch applications [3]. In addition, multimode waveguides havebeen shown to open up new opportunities for frequencyconversion by nonlinear effects. Interactions between differentguided modes can lead to phase-matched four-wave mixingover wide frequency ranges and with pump wavelengths inthe normal-dispersion regime of the fibers [4–6]. Pumpinglarge-core multimode fibers with a parabolic index variation tominimize walkoff between guided modes has been shown toenable complex intermodal nonlinear interactions, in particularenabling short-wavelength dispersive-wave generation from sol-iton collapse and spatiotemporal soliton oscillations at elevatedpulse energies [7,8], and sideband formation from geometricparametric instabilities [9]. Also, in the emerging low-losshollow-core fibers guiding by antiresonant effects, intermodalnonlinearities can be important for the highly interestingfrequency-conversion processes enabled by these waveguides[10]. Because of these emerging applications and the continu-ing development of efficient methods for manipulating inputpulse profiles, it is of great importance to devise efficient

numerical methods for describing nonlinear propagation ofhigh-power ultrafast pulses distributed over many spatialmodes, and possibly spanning a wide frequency range duringpropagation. However, multimode formulations of the gener-alized nonlinear Schrödinger equation based on an expansion ofthe nonlinear polarization via nonlinear mode overlap integralsscale badly with the number of modes in terms of computa-tional efficiency [11]. In addition, an accurate and well-controlled treatment of mode profile variations with frequencyis not straightforward with such methods [12,13]. The purposeof this paper is to investigate an alternative approach using real-space integration of the spatial nonlinear polarization on aGaussian quadrature (GQ) grid during propagation [14], inthe context of multimodal propagation in optical fibers. A sim-ilar method was recently used by Tani et al. to treat ionizationeffects in gas-filled hollow-core fibers [10]. The approach istested by studying generation of dispersive waves, commonlydenoted “Cherenkov radiation” (CR) [15] in multimodestep-index fibers pumped by ultrafast high-power pulses.The GQ is shown to have superior scaling properties whenthe number of guided modes is increased, to handle mode pro-file dispersion efficiently, and to be comparable in numericalcomplexity to an approach based on mode overlap integralsalready for a test case with six fiber modes and neglect of modeprofile dispersion. Although the application testing is here lim-ited to CR generation in short fibers, the method described isexpected to be applicable to and relevant for a wide range ofcontemporary nonlinear propagation problems.

2266 Vol. 34, No. 10 / October 2017 / Journal of the Optical Society of America B Research Article

0740-3224/17/102266-08 Journal © 2017 Optical Society of America

https://orcid.org/0000-0002-5879-9092https://orcid.org/0000-0002-5879-9092https://orcid.org/0000-0002-5879-9092mailto:jlag@fotonik.dtu.dkmailto:jlag@fotonik.dtu.dkmailto:jlag@fotonik.dtu.dkhttps://doi.org/10.1364/JOSAB.34.002266https://crossmark.crossref.org/dialog/?doi=10.1364/JOSAB.34.002266&domain=pdf&date_stamp=2017-09-26

2. FORMAL THEORY

A. Propagation Equations

Expanding the propagating electric field in the fiber core as

E�r; t� � 12π

Xm

ZdωG̃m�z;ω�em�r⊥;ω�e�i�βm�ω�z−ωt��; (1)

with a corresponding expansion for the magnetic field, thenonlinear propagation equation may be written as [12,14]

∂G̃m�z;ω�∂z

� ε0χ�3�2πiω

2Nm�ω�e−iβm�ω�z

Zdteiωt

Zdr⊥

× e�m�r⊥;ω� ·E�r; t�Z

dt 0R�t − t 0�jE�r; t 0�j2: (2)

In these equations, em�r⊥;ω�, hm�r⊥;ω� are the transverse fielddistributions of the guided modes, indexed bym, and βm�ω� aretheir frequency-dependent propagation constants. Nm�ω� is anormalization parameter given byZ

dr⊥�em × h�n � e�n × hm� � 2Nm�ω�δmn; (3)

and R�t� is the nonlinear response function, which is acombination of a delta-function Kerr term and a delayedRaman term. In deriving Eq. (2), cross-polarization Ramanscattering has been neglected, and the nonlinear susceptibilityχ�3� has been assumed isotropic and position-independent. Acommon way to approach Eq. (2) or similar formulations isto insert the expansion in Eq. (1) for the electric field andperform the transverse integrations [10,11,16,17]. This overlapintegral (OI) approach leads to the expressions

∂G̃m�z;ω�∂z

� iωε04πNm�ω�

e−iβm�ω�zXnqp

Zdω1dω2

× Gn�z;ω1�G�q �z;ω1 � ω2 − ω�Gp�z;ω2�× R�ω − ω1�K mnqp�ω;ω1;ω2� (4)

Gm�z;ω� � G̃m�z;ω�eiβm�ω�z (5)

K mnqp�ω;ω1;ω2� ≡ χ�3�Z

dr⊥e�m�r⊥;ω�

· en�r⊥;ω1�e�q �r⊥;ω1 � ω2 − ω� · ep�r⊥;ω2�: (6)Numerically, the double frequency integral in Eq. (4) is verydemanding when a general frequency dependence of the modesis allowed. Assuming frequency-independent em, one obtainsthe more tractable expression

∂G̃m�z;ω�∂z

� iωε04πNm�ω�

e−iβm�ω�zZ

dteiωt ; (7)

Xnqp

K mnqpGm�z; t�Z

dt 0G�q �z; t 0�Gp�z; t 0�R�t − t 0�

Gm�z; t� �1

2π

Z∞

−∞dωGm�z;ω�e−iωt : (8)

The overlap integrals K mnqp are given by Eq. (6) with allfrequencies set equal to some chosen base frequency ω0.With M modes in the field expansion, and a Fourier gridwith N points, evaluation of the RHS of Eq. (7) requires 2M

complex fast Fourier transform (FFT) operations to transformthe Gm from frequency to time and back, as well as 2M 2 realFFTs to evaluate the convolution integrals in time. Thenumerical complexity of the FFT operations is O�N log N �.In addition, multiplication of the resulting functions adds anoverhead of O�M 4N �. This unfavorable scaling of computa-tional effort with M is a major impediment for simulationof multimoded nonlinear propagation.

An alternative approach to Eq. (2) is to avoid the modaloverlap integrals and evaluate the spatial integral “on thefly,” as

∂G̃m�z;ω�∂z

� iωπε0χ�3�

Nm�ω�e−iβm�ω�z

Zdteiωt ; (9)

XPk�1

wke�m�r⊥k� · E�r⊥k; z; t�Z

dt 0jE�r⊥k; z; t 0�j2R�t − t 0�

E�r⊥k; z; t� �Xm

Gm�z; t�em�r⊥k�: (10)

Here r⊥k are a set of P suitably chosen integration points, wkare the corresponding integration weights, and the frequencydependence of the fields is still neglected. The numerical com-plexity can now be estimated as 2M complex FFTs, 2P realFFTs, as well as O�PM � order-N operations multiplying thefields. The central question is therefore how large P must be toensure an adequate accuracy of the nonlinear simulation.

The frequency dependence of the modes is straightforwardlytaken into account by modifying Eq. (10) to

E�r⊥k; z; t� �1

2π

Zdωe−iωt

XmGm�z;ω�em�r⊥k;ω�: (11)

Indeed, the GQ approach has previously been used by Kolesikand Moloney to accurately describe the frequency dispersion ofa single mode [14]. Note that when using Eq. (11) instead ofEq. (10), P complex FFT operations are needed to establishE�r⊥k; z; t�. Similarly, the FFT back into the frequency domainmust be done at each point on the GQ grid rather than after thespatial integration. The total numerical complexity then be-comes 2P complex FFTs, as well as 2P real FFTs, and O�PM�order-N operations.

It may also be noted that if delayed (Raman) response can beneglected, the time convolution integral is replaced by a simpleproduct, and the real FFTs are not needed. If mode profiledispersion is also neglected, the complexity of the OI and GQmethods scale as O�MN log N � and O�2PN log N � if theFFTs dominate. However, the penalty for including modeprofile dispersion is now large, going from O�MN log N �to O�PN log N � because, as will be shown in Section 3, Pis typically an order of magnitude larger than M for a step-index fiber. For the operations linear in N , the complexity stillscales as O�M 4N � for the OI method and as O�MPN � for theGQ method.

In this paper, linearly polarized scalar modes will be consid-ered for simplicity. Inclusion of different polarization statesin the GQ approach is, however, straightforward. It will doublethe number of modes, whereas the number of FFT operationsto be performed per gridpoint will be doubled if onlytransverse polarization components are considered, or tripled

Research Article Vol. 34, No. 10 / October 2017 / Journal of the Optical Society of America B 2267

if longitudinal polarization components need to be included inthe total electric field, as may be relevant in some high-indexcontrast waveguides.

Equation (9) and its counterpart with frequency-dependenteigenmodes conserve the classical photon number in the ab-sence of propagation losses. Taking Nm�ω� � 1 for all modesfor simplicity, the photon number N ph may be written as

N ph �Xm

Zdω

jG̃m�z;ω�j2ℏω

(12)

and its z-derivative as

dN phdz

�Xm

Zdω

2Re�G̃�m�z;ω� ∂G̃m�z;ω�∂z

�ℏω

(13)

� Re�X

m

ZdωG̃�m�z;ω�

i2πε0χ�3�

ℏe−iβm�ω�z

×Z

dteiωtXPk�1

wke�m�r⊥k;ω� · E�r⊥k; z; t�

×Z

dt 0jE�r⊥k; z; t 0�j2R�t − t 0��: (14)

Here ℏ denotes Planck’s constant. Using Eqs. (5) and (11), thisexpression can be rewritten as

dN phdz

� Re�i�2π�2ε0χ�3�

ℏ

Zdt

XPk�1

wkjE�r⊥k; z; t�j2

×Z

dt 0jE�r⊥k; z; t 0�j2R�t − t 0��� 0; (15)

since χ�3� and R�t� are real functions in the absence of nonlin-ear losses. Eq. (15) shows that photon number conservation is ageneric feature of the propagation equation and does notdepend on the GQ integration accuracy.

B. Analysis of Parabolic-Index Fiber

The numerical complexity of Eqs. (9) and (10) may be analyzedprecisely for the special case of scalar modes in an ideal para-bolic-index fiber with a circularly symmetric index distribution

of the form n�r� � n0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 − � rr0�

2q

. Linearly polarized scalar

modes in such a fiber can be written as

ekl �r⊥;ω� � p̂e−ax2pk−1�x�e−ay2pl−1�y�; a �

n0ω2cr0

; (16)

where pk denotes a polynomial of order k and p̂ is a unit polari-zation vector. In the following, p̂ � x̂ will be assumed for allmodes. Evaluating the RHS of Eq. (9) now amounts to radiallyintegrating a sum of products of four mode profiles. Since theexponential prefactor is the same for all modes, any such prod-uct can be written as e−4a�x2�y2� multiplied by polynomials in xand y of degree ≤ 4�lmax − 1�, where lmax is the highest possiblevalue of the k, l indices of the eigenmodes. The technique ofGQ allows for determination of a set of 2�lmax − 1� integrationpoints rk and weights wk such that

Z∞

−∞dxe−4ax2pm�x� �

X2�lmax−1�k�1

wkpm�xk� (17)

for all pm with m ≤ 4�lmax − 1� [18]. In this way, the RHS ofEq. (9) can be evaluated as

iωε04πNm�ω�

e−iβm�ω�zZ

dteiωtX2lmax−1p�1

X2lmax−1q�1

wpqem�r⊥pq�

E�r⊥pq; z; t�Z

dt 0jE�r⊥pq; z; t 0�j2R�t − t 0� (18)

r⊥pq � �xp; xq�; wpq � wpwq: (19)Thus, P � 4�lmax − 1�2, and the total number of guided modesthat can be treated accurately is M � l 2max. This means that Pscales linearly with M , so that the numerical complexity of thenonlinear propagation becomes O�MN log N � if FFTs domi-nate, and O�M 2N � if order-N operations dominate. In a prob-lem with many modes, this is a huge benefit compared with theO�M 2N log N � or O�M 4N � complexity of the OI approach.C. Step-Index Fiber

An important question is whether the linear scaling of P withM is a special feature of the parabolic index distribution, due tothe particular mathematical properties of its eigenmodes, orwhether it holds more generally. The most important examplestructure in standard fiber optics is the step-index fiber.The eigenmodes of this fiber cannot be separated in x- andy-dependent functions as the eigenmodes of the parabolic-index fibers. However, like the eigenmodes of any indexdistribution with cylindrical symmetry, they can be split intoproducts of radial and angular functions in a cylindrical coor-dinate system:

Ψml �r⊥� � Rml �r�f m�ϕ�; (20)where r is the distance from the fiber center, and ϕ is, say, theangle between r⊥ and the cartesian x-axis. f m may be chosen aseither complex exponentials, f m�ϕ� � eimϕ, or trigonometricfunctions, f m�ϕ� � cos�mϕ� or f m�ϕ� � sin�mϕ�. Clearly, mmust be an integer to make f m�ϕ�2π-periodic. For each m,there may be several guided eigenmodes, which are countedby the index l .

In line with these modal properties, a radial-angular integra-tion grid is chosen, with points given by

r⊥pq � rp�cos

�2π

q − 14M � 1

�; sin

�2π

q − 14M � 1

��: (21)

Here, q runs from 1 to 4mmax � 1, and mmax is the maximalvalue of m occurring among the guided modes of interest. Thisangular grid is sufficient to integrate any product of four f mfunctions of the kind described previously. The radial grid isagain chosen as GQ points, in this case determined to exactlyintegrate low-order polynomials (Gauss–Legendre integration[18]) over an interval from r � 0 to r � rs, where the outerradius rs is a free parameter to be optimized. If rs is too small,integration of poorly confined modes will be inaccurate. If it istoo large, an excessive number of radial GQ points will berequired for well-confined modes.

2268 Vol. 34, No. 10 / October 2017 / Journal of the Optical Society of America B Research Article

Since Rml �r� for a step-index fiber cannot be written as afinite polynomial, the number of rp points required cannotbe determined from an exact mathematical analysis. This prob-lem must instead be analyzed by numerical calculations. Oneoption is to do direct convergence checks for a specific nonlin-ear problem by gradually increasing the number of rp points,but a convenient proxy also exists. Consider the overlap integralZ

dr⊥Ψ4ml �r⊥�; (22)

which equals the inverse of the classical effective area of the mlmode if Ψml is properly normalized. If Ψml is the mode withthe most complicated radial structure, this product must beexpected to require the largest number of radial GQ points.Therefore, the accuracy of the GQ integration can be estimatedby comparing effective areas calculated on a specific GQ gridwith the exact values calculated on a fine grid, and this analysismay be performed ahead of any nonlinear simulations.

3. NUMERICAL RESULTS

A. Linear Scaling of P versus M

The most important question is to determine whether the totalnumber of radial-angular grid points P needed for an accuratenonlinear simulation scales linearly with the number of modesM , as was the case for a parabolic-index fiber. This question ishere addressed in the following way: for a step-index fiber withcore radius rc � 7 μm and index difference Δn � 0.01 be-tween core and cladding, the guided-mode fields were deter-mined over a range of wavelengths. Although analyticalsolutions do exist for the step-index fiber geometry, the fieldswere calculated using a radial finite-difference approach, whichcan be readily generalized to other cylindrically symmetric fiberstructures and which treats guided and non-guided modes onthe same footing. A zero-field boundary condition was appliedat an outer radius of 62.5 μm, meaning that even non-guidedmodes would be confined within this radius. A more realisticdescription of a coated fiber might be obtained by replacing thissimple boundary condition with an absorbing cladding layer,but this was not investigated here. Guided modes were deter-mined simply as those which that effective index neff � cβ∕ωlarger than the cladding index. As the wavelength gets shorter,the number of guided modes increases. For each wavelength,GQ grids were determined as described earlier, withmmax beingthe highest m-value occurring among the guided modes. Thenumber of radial meshpoints, n, was increased until all effectiveareas calculated using GQ integration were accurate withinsome specified criterion and for a particular value of the outerintegration radius rs. The results are shown in Fig. 1. Two rsvalues of 1.3rc and 2rc have been considered. For each rs value,the maximal relative deviation of effective areas, ΔAeff , is re-quired to be less than 10%, 5%, or 1% for all guided modes,and the data points show the values of P � n�4mmax � 1�needed to match these requirements. P is plotted as a functionof M , which means there may be several results for each M ,because some of the investigated wavelengths had the samenumber of guided modes. For a few wavelengths, it was notpossible to reduce ΔAeff to all required thresholds. To makethese cases stand out in the plot, P was set to zero, so that

the data points fall on the x-axis. This problem occurs forpoorly confined modes close to cutoff, when rs is not largeenough for a proper description of the mode profile. As rs isincreased from 1.3rc to 2rc , the incidence of these points isstrongly reduced. On the other hand, an increased value of rsleads to a higher slope of the P versusM curves. Nevertheless, alinear scaling of P with M is apparent in all cases. It cantherefore be expected that the numerical complexity ofnonlinear simulations using the GQ approach will scale asO�MN log N � or O�M 2N � depending on which part ofthe calculation dominates. Compared with the mode overlapapproach, this saves 1 to 2 orders ofM , which will make a hugedifference as M becomes large.

B. Comparison between Mode Overlapand GQ Approach

Having established the superior scaling properties of the GQ ap-proach, the next question is to estimate at what value of M thetwo approaches are equal. Sincemore than oneGQpoint will beneeded to correctly integrate the fundamental mode of a single-mode fiber, clearly the mode overlap approach is superior in thiscase. The next relevant case for a step-index fiber is three guidedmodes above the LP11 cutoff (not counting polarization states),and the following is six guided modes above the LP21∕LP02cutoff. The latter case will be considered in this subsection.

The fiber geometry discussed in the previous subsection isconsidered. As a test case, soliton compression and breakup offemtosecond (fs) pulses at 1.55 μm wavelength is investigated.This is highly relevant for recent frequency-conversion experi-ments and critically tests the model’s ability to describe strongnonlinear processes over a broad frequency range. On the otherhand, because of the short propagation lengths, it is still a nu-merically tractable problem that allows for careful convergencechecks with a reasonable computational effort.

To further simplify the problem, mode profile dispersion isinitially neglected, with guided-mode profiles being calculated

0 50 100 150Number of modes

0

1000

2000

3000

4000

5000

Num

ber

of g

ridp

oint

s

ΔAeff

at the input wavelength of 1.55 μm. In that case, the modeoverlap method is exact, within the overall approximationsmade in deriving Eq. (2), and may serve as a benchmark fortesting GQ grids of different sizes. In Fig. 2, the output spec-trum of a test calculation using the mode overlap approach isshown. A Gaussian input pulse having a peak power of 1 MWand a full width at half-maximum (FWHM) of 50 fs is distrib-uted equally, and with equal phase, over the six guided modesand propagated over a distance of 2.1 cm. The calculations weredone on a Fourier grid with 2048 points spanning frequenciesfrom 100 to 750 THz (3.0–0.4 μm wavelengths). As can beseen from the output spectrum, this is sufficient to ensure non-linear compression and breakup with Raman redshifted solitonsand CR emission around 800 nm. Because of the technical fo-cus of this paper, the detailed nonlinear dynamics leading to thefinal spectrum will not be discussed here. An important pointis, however, that both the short-wavelength CR and the spectralweight appearing beyond 2 μm is mainly in the LP21 modes.

To validate the GQ approach, simulations are performedwith a varying number of radial gridpoints, n, whereas thenumber of angular gridpoints is kept fixed at nine, which allowsfor exact integration in this coordinate. In Fig. 3, the total spec-tral density, summed over all modes, is shown for GQ calcu-lations with n � 5 and n � 6, compared with the result of anOI calculation. It can be seen that the n � 5 result deviatessignificantly from the OI result, whereas the differences areslight for the n � 6 case. In Fig. 4, the integral of the absolutevalue of the deviation in total spectral density between the OIand GQ methods is plotted, normalized to the integral of totalspectral density for the OI calculation. Curves are given for GQcalculations with rs � 1.3rc and rs � 2rc , respectively. For thelow value of rs, a deviation slightly below 1% can be obtained.For n > 9, there is no significant improvement. For rs � 2rc,the deviation could be driven below the per mill level forn � 40 (not shown), which indicates that the remaining error

for rs � 1.3rc is due to the finite integration radius. On theother hand, the larger value of rs requires higher n values toreach a specific level of accuracy.

To connect back to the ΔAeff metric discussed inSubsection 3.A, it can be mentioned that the maximal ΔAefffor n � 5 is ∼10%, whereas it is ∼1% for n � 7. Thus, theaccuracy levels achieved for the integrated spectral deviationsappear to resemble those found for the effective areas, althougha more comprehensive simulation program would be needed tosubstantiate this as a general claim. As may be appreciated bycomparing with Fig. 3, an accuracy level of a few percent inthese metrics only implies slight differences in the calculatedspectra. Therefore, rs � 1.3rc appears to be an acceptablechoice unless in exceptional situations, e.g., if the major partof the input power is coupled into a poorly confined mode.Regarding numerical complexity, the runtime for the GQmethod with n � 5 was similar to that for the overlap integralmethod. The GQ runtime scales roughly linearly with n as

1 1.5 2 2.5 3Wavelength (μm)

0

20

40

60

Spec

tral

ene

rgy

dens

ity (

pJ/n

m) LP01

LP02

LP11c

LP11s

LP21c

LP21s

Fig. 2. Spectral densities for the six modes of a fiber with core radiusrc � 7 μm, core-cladding index difference Δn � 0.01, after propaga-tion over 2.1 cm of a 50 fs input pulse with 1 MW peak power, evenlydistributed over the six modes, with fixed linear polarization. Indexlabels c and s on the LPm1 modes with m > 0 indicate modes withangular dependence of cos�mϕ�, sin�mϕ�, respectively. The resultswere found using the OI method with mode profiles calculated ata wavelength of 1.55 μm.

1 1.5 2 2.5 3λ (μm)

0

50

100

150

Spec

tral

den

sity

(pJ

/nm

)

Overlap integralsGQ n=5GQ n=6

Fig. 3. Total spectral densities, summed over all modes, for the OIcalculation shown in Fig. 2, as well as corresponding GQ calculationswith rs � 1.3rc , n � 5 − 6.

5 10 15 20 25n

0.001

0.01

0.1R

elat

ive

spec

tral

dev

iatio

n

rs=1.3r

c

rs=2r

c

Fig. 4. Integrated absolute differences in total spectral densitybetween OI and GQ calculations as a function of the number of radialgridpoints n used in the GQ calculations. Results are divided by theintegral of the total OI spectral density.

2270 Vol. 34, No. 10 / October 2017 / Journal of the Optical Society of America B Research Article

expected. Therefore, the OI method is still slightly better thanthe GQ method for this case of six modes but can be expectedto become unfavorable at significantly higher mode numbers.

Of course, the accuracy required of a nonlinear simulation isproblem-specific and may in some cases be very high, e.g., ifone is interested in modes or frequencies carrying a small part ofthe total spectral weight. In the present calculations, the smallfrequency components in the visible spectrum were found toconverge with a relative accuracy similar to, or even better than,the spectra around the pump wavelength. In any event, it is anadvantage of the GQ approach that numerical convergence canbe checked in a straightforward and systematic manner simplyby increasing the number of integration points.

C. Mode Profile Dispersion

In Section 2, it was described how mode profile dispersion canbe accounted for in a straightforward manner using the GQapproach. For the OI approach, on the other hand, a correcttreatment in the multimode case is not easy. A systematicapproach is to expand the frequency-dependent guided modesin a larger set of frequency-independent basis states, withfrequency-dependent expansion coefficients. Even for a single-mode fiber, this is a cumbersome strategy [12] because thenumerical complexity increases strongly with the number ofbasis states used. For the multimode case, using a polynomialexpansion as in Refs. [12,13], it was found practically impos-sible to obtain well-converged results. In particular, the descrip-tion of modes with cutoffs within the frequency window ischallenging with such an expansion. An alternative approachcould be to generalize the modified effective area approachdescribed in Refs. [12,19], but the most straightforward gen-eralization to the multimode case will increase the numericalcomplexity from O�M 2N log�N �� to O�M 4N log�N ��,which is still a major drawback, and it is difficult to control theapproximation in a systematic way. On the other hand, theaccuracy of the GQ approach can still be checked simply byincreasing n and/or rs.

Mode profile dispersion is a particularly important phe-nomenon for multimode fibers because some modes may havecutoffs appearing within the frequency window of interest. Astriking example of this may be appreciated by comparingFig. 2 with Fig. 5, which shows a similar calculation usingthe GQ approach with frequency-dependent mode profiles.It can be seen that the inclusion of frequency dispersion sup-presses the spectral weight beyond 2 μm and also changes theshape of the CR spectrum between 700 and 800 nm. The rea-son is straightforward: in the calculation without mode profiledispersion, the spectrum beyond 2 μm was mainly in the LP21modes, but these modes (together with the LP02 mode) actuallycut off just below 2 μm in the fiber under study, meaning thatthe simulation with fixed mode profiles does not give an accu-rate description of the nonlinear dynamics, especially since thepresent calculations do not account for modal losses. In the GQcalculation, the cutoff means that the modes expand strongly,leaving only a little weight in the integration region close to thecore; therefore, nonlinear coupling to longer wavelengths inthese modes is suppressed, which is physically reasonable. Thesmall spectral weight observed on the LP21 modes just beyond

2 μm appears because the unguided modes still have somefraction of power inside the core due to the boundary conditionon the modal field calculations discussed in Subsection 3.A;however, this fraction rapidly goes down with increasingwavelength.

In Fig. 6, the total spectral density, summed over all modes,is plotted for the OI calculation without mode profiledispersion, as well as for GQ calculations with rs � 1.3rc ,and n � 5–8 including mode profile dispersion. Apart fromshowing again the impact of mode profile dispersion, the figureshows that a fair convergence is reached for n � 6, whereas thecurves for n � 7 and n � 8 are almost indistinguishable.

To further validate the GQ approach, calculations withrs � 1.3rc and low n-values are compared with a reference cal-culation with rs � 2rc , n � 40 in Fig. 7, using the same metricas in Fig. 4. An accuracy at the percent level is readily reachedalready for n � 7. This indicates that mode profile dispersiondoes not have a strong impact on GQ convergence, andalso that the use of a low rs value is justified, even in the case

1 1.5 2Wavelength (μm)

0

20

40

60

80

Spec

tral

ene

rgy

dens

ity (

pJ/n

m) LP01

LP02

LP11c

LP11s

LP21c

LP21s

Fig. 5. Final spectral density for the same pulse and fiber parametersas in Fig. 2, when using a GQ approach including mode profiledispersion. rs � 1.3rc and n � 8.

1 1.5 2 2.5 3λ (μm)

0

50

100

150

200

250

Spec

tral

den

sity

(pJ

/nm

)

Overlap integralsGQ n=5GQ n=6GQ n=7GQ n=8

Fig. 6. Total spectral densities, summed over all modes, for the OIcalculation without mode profile dispersion shown in Fig. 2, as well ascorresponding GQ calculations with rs � 1.3rc , n � 5–8.

Research Article Vol. 34, No. 10 / October 2017 / Journal of the Optical Society of America B 2271

where some modes become unguided in the relevant frequencywindow.

As discussed in Subsection 3.A, the numerical overhead forincluding mode profile dispersion is a factor of ∼2 for the FFTcontribution, whereas there is no change for the terms linear inN . In the present example, the computation time was found toincrease by 50%–60% when including mode profile dispersion,indicating that both O�N � and O�N log N � terms contributesignificantly to the computational effort.

4. DISCUSSION

The GQ method as implemented in this work relies on real-space grids optimized for integrating low-order polynomials,i.e., smooth mode profile distributions. The chosen test case ofmodes in fibers with weak index contrast is therefore favorablefor the method, and further work will be needed to establish itsmerits in other situations. For waveguides with high index con-trasts, such as photonic crystal fibers, or high-index integrated-optics technology, one typically has a situation where both thefields and the value of the nonlinear coefficient change discon-tinuously over the waveguide cross-section. In such a situation,direct application of a GQ grid is obviously not a good idea. Forsimple geometries, however, it may be possible to apply the GQapproach by sectioning the nonlinear polarization integral witha GQ grid for each material region. A particularly simple ex-ample would be a silica nanofiber, where the method presentedin this work could be implemented directly using rs � rc , sincethe nonlinear coefficient of the (air) cladding in this case is neg-ligible [14]. In more complicated geometries, such as solid-corephotonic crystal fibers, the choice of a suitable integration gridmay be more difficult. A particularly complex example may behollow-core photonic bandgap fibers, where both silica and airnonlinearity play a significant role, and the silica parts of thefiber have a complex geometry. On the other hand, the morerecent class of hollow-core fibers guiding by antiresonant effectsin a core with negative-curvature surround features modes witha negligible silica overlap, and a GQ grid confined to the air

core may be useful for obtaining nonlinear simulation results[10]. In the absence of ionization effects, this will, of course,only be relevant in fibers where a substantial number of higherorder modes are well-confined to the air core with negligiblesilica overlap, which is perhaps not quite characteristic of cur-rent fibers optimized for effective single-mode operation.

As the number of guided modes increases, one must even-tually expect the numerical complexity of the method to scaleas O�MNP�. One can improve this to O�NP� by abandoningthe modal expansion entirely [9], evaluating the lineardispersion operator in real space by finite-differences, or pos-sibly 2D FFTs (in which case one has O�PN log N � scaling).In this case, however, the dispersion operator is approximated,and P must be chosen large enough to obtain a satisfactoryaccuracy. The comparison between such methods and themodal GQ integration approach investigated here is thereforenot straightforward and is likely to be highly problem-specific.

5. CONCLUSION

In conclusion, a numerical approach to nonlinear propagationproblems in multimode fibers using on-the-fly real-space inte-gration of the nonlinear polarization on a GQ grid has beeninvestigated. It has been shown that the numerical complexityof this approach scales linearly or at most quadratically with thenumber of guided modes and that mode profile dispersionmay be taken into account with modest computational over-head (except in cases where Raman scattering is neglected).The approach was compared with the widely used approachbased on evaluating the nonlinear polarization by summingmode overlap integrals and was found to have a comparablenumerical complexity for the particular case of a step-indexfiber with six guided modes. Since the overlap integral approachhas less favorable scaling properties and is more problematicwith respect to inclusion of mode profile dispersion effects,the GQ method will be superior for cases with a large numberof guided modes as well as cases where mode profile dispersionis important.

REFERENCES

1. H. R. Stuart, “Dispersive multiplexing in multimode optical fiber,”Science 289, 281–283 (2000).

2. R. Ryf, H. Chen, N. K. Fontaine, A. M. Velazquez-Benitez, J. Antonio-Lopez, C. Jin, B. Huang, M. Bigot-Astruc, D. Molin, F. Achten, P.Sillard, and R. Amezcua-Correa, “10-mode mode-multiplexed trans-mission over 125-km single-span multimode fiber,” in EuropeanConference on Optical Communication (ECOC) (2015), pp. 1–3.

3. C. Antonelli, M. Shtaif, and A. Mecozzi, “Modeling of nonlinearpropagation in space-division multiplexed fiber-optic transmission,”J. Lightwave Technol. 34, 36–54 (2016).

4. Y. Ding, J. Xu, H. Ou, and C. Peucheret, “Mode-selective wavelengthconversion based on four-wave mixing in a multimode silicon wave-guide,” Opt. Express 22, 127–135 (2014).

5. J. Demas, P. Steinvurzel, B. Tai, L. Rishoj, Y. Chen, and S.Ramachandran, “Intermodal nonlinear mixing with Bessel beams inoptical fiber,” Optica 2, 14–17 (2015).

6. S. R. Petersen, T. T. Alkeskjold, C. B. Olausson, and J. Lægsgaard,“Intermodal and cross-polarization four-wave mixing in large-corehybrid photonic crystal fibers,” Opt. Express 23, 5954–5971 (2015).

7. L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Controllable spa-tiotemporal nonlinear effects in multimode fibres,” Nat. Photonics 9,306–310 (2015).

5 6 7 8 9 10 11 12n

0.01

0.1

Rel

ativ

e sp

ectr

al d

evia

tion

Fig. 7. Integrated absolute differences in total spectral density be-tween a GQ calculation with rs � 2rc and n � 40 and GQ calcula-tions with rs � 1.3rc as a function of the number of radial gridpoints nused in the latter case. Results are divided by the integral of the totalOI spectral density.

2272 Vol. 34, No. 10 / October 2017 / Journal of the Optical Society of America B Research Article

8. L. G. Wright, S. Wabnitz, D. N. Christodoulides, and F. W. Wise,“Ultrabroadband dispersive radiation by spatiotemporal oscillationof multimode waves,” Phys. Rev. Lett. 115, 223902 (2015).

9. K. Krupa, A. Tonello, A. Barthélémy, V. Couderc, B. M. Shalaby, A.Bendahmane, G. Millot, and S. Wabnitz, “Observation of geometricparametric instability induced by the periodic spatial self-imaging ofmultimode waves,” Phys. Rev. Lett. 116, 183901 (2016).

10. F. Tani, J. C. Travers, and P. S. Russell, “Multimode ultrafastnonlinear optics in optical waveguides: numerical modeling andexperiments in kagome photonic-crystal fiber,” J. Opt. Soc. Am. B31, 311–320 (2014).

11. F. Poletti and P. Horak, “Description of ultrashort pulse propagationin multimode optical fibers,” J. Opt. Soc. Am. B 25, 1645–1654(2008).

12. J. Lægsgaard, “Mode profile dispersion in the generalisednonlinear Schrödinger equation,” Opt. Express 15, 16110–16123(2007).

13. T. X. Tran and F. Biancalana, “An accurate envelope equation for lightpropagation in photonic nanowires: new nonlinear effects,” Opt.Express 17, 17934–17949 (2009).

14. M. Kolesik, E. M. Wright, and J. V. Moloney, “Simulation of femtosec-ond pulse propagation in sub-micron diameter tapered fibers,” Appl.Phys. B 79, 293–300 (2004).

15. N. Akhmediev and M. Karlsson, “Cherenkov radiation emitted bysolitons in optical fibers,” Phys. Rev. A 51, 2602–2607 (1995).

16. A. Mafi, “Pulse propagation in a short nonlinear graded-index multi-mode optical fiber,” J. Lightwave Technol. 30, 2803–2811 (2012).

17. L. G. Wright, W. H. Renninger, D. N. Christodoulides, and F. W. Wise,“Spatiotemporal dynamics of multimode optical solitons,” Opt.Express 23, 3492–3506 (2015).

18. W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling,Numerical Recipes in Fortran 77 (Cambridge University, 2001).

19. P. V. Mamyshev and S. V. Chernikov, “Ultrashort-pulse propagation inoptical fibers,” Opt. Lett. 15, 1076–1078 (1990).

Research Article Vol. 34, No. 10 / October 2017 / Journal of the Optical Society of America B 2273