Conditions for walk-off soliton generation in a
Mario Zitelli1✉, Fabio Mangini 2, Mario Ferraro 1, Oleg Sidelnikov3& Stefan Wabnitz1,3
It has been recently demonstrated that multimode solitons are unstable objects which evolve,
in the range of hundreds of nonlinearity lengths, into stable single-mode solitons carried by
the fundamental mode. We show experimentally and by numerical simulations that femto-
second multimode solitons composed by non-degenerate modes have unique properties:
when propagating in graded-index ﬁbers, their pulsewidth and energy do not depend on the
input pulsewidth, but only on input coupling conditions and linear dispersive properties of the
ﬁber, hence on their wavelength. Because of these properties, spatiotemporal solitons
composed by non-degenerate modes with pulsewidths longer than a few hundreds of fem-
toseconds cannot be generated in graded-index ﬁbers.
1Department of Information Engineering, Electronics and Telecommunications (DIET), Sapienza University of Rome, Rome, Italy. 2Department of Information
Engineering (DII), University of Brescia, Brescia, Italy. 3Novosibirsk State University, Novosibirsk, Russia. ✉email: firstname.lastname@example.org
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Pioneering works on multimode (MM) ﬁber transmission1,2,
dated 40 years ago, predicted the existence of MM solitons,
providing conditions for the temporal trapping of the input
optical modes, in order to form a MM or spatiotemporal
soliton3,4. Research on multimode ﬁbers (MMFs) is a topic of
renewed interest, motivated by their potential for increasing the
transmission capacity of long-distance optical ﬁber transmissions
via the mode-division-multiplexing technique, based on multiple
spatial modes as information carriers. Moreover, the capacity of
MMFs to carry high-energy beams permits the upscaling of high-
power ﬁber lasers. However, MM ﬁber solitons have only recently
been systematically experimentally investigated in graded-index
(GRIN) MMFs, unveiling the complexity of a new, previously
uncharted ﬁeld5–11. For example, ref. 9has shown that the spa-
tiotemporal oscillations of solitons, due to self-imaging in a GRIN
ﬁber, generate MM dispersive waves over an ultrabroadband
spectral range, leading to a new source of coherent light with
unprecedented spectral range. Moreover, in ref. 10, a study of the
ﬁssion of high-order MM solitons has revealed that the generated
fundamental solitons have a nearly constant Raman wavelength
shift and equal pulsewidth over a wide range of soliton energies.
In a recent study11, we observed that the beam content of fem-
tosecond MM solitons propagating over relatively long spans of
GRIN ﬁber is irreversibly attracted toward the fundamental
mode of the MMF. This is due to the combined action of inter-
modal four-wave mixing (IM-FWM) and stimulated Raman
In this work, we extensively study the dynamics of the gen-
eration of these femtosecond MM soliton beams. We reveal the
existence of previously unexpected dynamics, which make these
MM solitons very different from their well-known single-mode
counterparts. In single-mode ﬁbers, a soliton forms when chro-
matic dispersion pulse broadening is compensated for by self-
phase modulation-induced pulse compression. Speciﬁcally, the
nonlinear and the chromatic dispersion distances must be equal.
This means that a single-mode soliton can have an arbitrary
temporal duration, provided that its peak power or energy is
properly adjusted. In contrast, for obtaining a MM soliton in a
MMF, it is additionally required to compensate for modal dis-
persion or temporal walk-off. As a result, it is necessary to impose
the additional condition that the walk-off distance is of the same
order of the nonlinear and chromatic dispersion lengths. The
presence of this extra condition leads to a new class, to the best of
our knowledge, of “walk-off”MM solitons composed of non-
degenerate modes: they have a pulsewidth and energy, which are
independent of the input pulse duration, and only depend on the
ﬁber dispersive parameters, hence the soliton wavelength.
Experimental evidence. The transmission of ultrashort pulses
(input pulsewidth and wavelength were 60–240 fs, and
1300–1700 nm, respectively) was tested over long spans of graded-
index (GRIN) optical ﬁber (see “Methods”—“Experiments”).
When coupling exactly on the ﬁber axis, with 15 µm input beam
waist, we could excite three nondegenerate, axial-symmetric
Laguerre–Gauss modes, that will be addressed from now on as
(0,0), (1,0), (2,0) or LG
. The fraction of power
carried by these modes was calculated by a speciﬁc software (see12
and Supplementary note 1) to be 52%, 30%, and 18%, respectively.
The input laser pulse energy ranged between 0.1 and 20 nJ.
What we observed did not appear to obey the predictions of the
variational theory for spatiotemporal solitons13–18:Figs.1–3
provide the experimental evidence when using 120 m of GRIN
ﬁber. By testing different input wavelengths (1420–1550 nm) and
input pulsewidths (67–245 fs), spatiotemporal solitons with a
common minimum pulsewidth of 260 fs were observed at the ﬁber
output, for values of the input pulse energy ranging from 2 up to 4
nJ (Fig. 1). The case of a 1300 nm input wavelength represented an
exception, which resulted into a minimum pulsewidth of 200 fs at
The output beam waist (Fig. 2) was severely reduced from its
input value (down to a value of 8.5 µm, which is close to the
theoretical value of 7.7 µm for the fundamental ﬁber mode), in
correspondence of the input energy leading to minimum output
pulsewidth. In this regime, the output beam shape was
substantially monomodal, with a measured M2=1.45, against
the value of 1.3 of the input beam. The curve of the beam waist vs.
energy was much narrower for input short (67fs) pulses than for
long (235 fs) pulses; also for the beam waist, the case of 1300 nm
represented an exception, with no signiﬁcant beam reduction. The
output soliton wavelength (Fig. 3) was severely affected by Raman
soliton self-frequency shift (SSFS)3,7. Still, the case of 1300 nm
represented an exception, that provided reduced values of SSFS.
The explanation to the relatively sharp dependence of the
output beam diameter on the input pulse energy in Fig. 2, for 67
fs pulses, is found in linear ﬁber losses and wavelength red-shift.
As a matter of fact, shorter pulses suffer faster soliton SSFS for
increasing energy, and experience increased linear ﬁber losses
when the wavelength overcomes 1700 nm11(Figs. 1and 3). As a
result, while the propagating soliton is attenuated, dispersive
waves provide a more signiﬁcant relative contribution to the
Fig. 1 Measured soliton pulsewidth vs. input pulse energy, after 120 m of
graded-index (GRIN) ﬁber, for different input wavelengths and pulse
durations. The insets are the measured output autocorrelation traces, for
an input wavelength 1550 nm and pulsewidth 67 fs. Measurement error
comes directly from the instrument accuracy, that is below 1%, i.e., too
small to be visible at this scale.
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output beam image, which integrates energy over all wavelengths
within the camera response window. On the other hand, longer
pulses suffer a slower SSFS, as seen when comparing, in Fig. 3,
cases at 1550 nm and 235 or 67 fs input pulsewidth. Therefore,
the output beam waist reduction or beam cleaning process
appears smoother when the input energy changes. The same
considerations also apply to the fundamental beam waist itself,
1=2, being λthe wavelength, r
radius, Δthe relative index difference, and n
the modal index;
the waist scales with the square root of the increasing wavelength,
resulting in a broadened fundamental beam at the ﬁber output. In
addition, it should be considered that the response of the InGaAs
camera used in our experiments sharply drops for wavelengths
above 1850 nm, which impairs its ability to record the red-
shifting soliton above this wavelength.
In all cases, numerical simulations performed with a coupled-
mode equations model (see “Methods”—“Simulations”) fully
conﬁrmed the experimental results (empty dots).
Experimental evidence with either shorter or longer spans of
GRIN ﬁber, ranging from 2 to 850 m (Fig. 4), shows that
requirements for the optimal soliton energy as the ﬁber length is
reduced are less stringent. At 850 m distance, 1550 nm and 67 fs
input pulsewidth, a sharp input energy of 1.5 nJ is required to obtain
a minimum pulsewidth of 470 fs at output; at 120 m, a minimum
pulsewidth of 260 fs is measured for energy range between 2 and 4
nJ; at 10 and 2 m distance, pulsewidth remains minimum (110 and
60 fs, respectively), for input energies larger than 2.5 nJ. Soliton
pulsewidth increases with distance, as a consequence of the
wavelength red-shift due to Raman SSFS, and the need to conserve
the soliton energy condition E1¼λβ
T0¼TFWHM=1:763, n2(m2W1) the nonlinear index coefﬁcient,
β2ðλÞthe chromatic dispersion, and w
the effective beam waist.
Note that the “sharp”resonance for 67 fs input temporal
duration at 1550 nm is not seen in Figs. 1and 3, because with the
Fig. 2 Measured soliton beam waist vs. input pulse energy, after 120 m of
graded-index (GRIN), in the same conditions of Fig. 1.The insets show
the measured output near-ﬁelds, for an input wavelength 1550 nm and
pulsewidth 67 fs. Infrared (IR) camera resolution is 0.25 μm, which is too
small to be visible at this scale.
Fig. 3 Measured soliton wavelength vs. input pulse energy, after 120 m of
graded-index (GRIN) ﬁber, in the cases of Fig. 1.The insets show the
measured output spectra, for an input wavelength 1550 nm and pulsewidth
67 fs. Spectrometer resolution is 0.2nm, which is too small to be visible at
Fig. 4 Measured soliton pulsewidth vs. input energy, for input pulses
with 1550 nm wavelength, 67 fs pulsewidth, and 15 µm input waist,
respectively, with graded-index (GRIN) ﬁber spans of length 2, 10, 120,
and 850 m. Measurement error comes directly from the instrument
accuracy, that is below 1%, too small to be visible at this scale.
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autocorrelator and the optical spectrum analyzer, one directly
monitors the pulse duration and the wavelength of the Raman
soliton, even if its relative amplitude drops with respect to the
residual dispersive waves remaining around 1550 nm. Numerical
simulations (empty dots) substantially conﬁrmed the experi-
mental observations. In our experiments, the minimum output
pulsewidth, at 120 m distance, appeared to be independent of the
input pulsewidth and wavelength, and it was obtained at
comparable input energies in all cases. The case at 1300 nm
showed a different behavior. We started investigating our
observations from numerical simulations.
Numerical simulations. Figure 5provides an explanation to the
observed spatiotemporal effects; it is a numerical representation of
the evolution of an input 235 fs pulse, composed of three axial
modes, propagating over 120 m of GRIN ﬁber, at the optimal input
energy of 2.5 nJ. During their propagation, the three nondegenerate
modes remain temporally trapped; the mode LG
acts as an
attractor for other modes, owing to non-phase matched, asymme-
trical IM-FWM, and to intermodal SRS11; at the output, a mono-
modal bullet remains. The pulse carried by the fundamental or LG
mode experiences SSFS, while it traps and captures a large portion
of energy carried by higher-order modes. The observed process of
spatial beam cleaning becomes more efﬁcient when the soliton
wavelength shift reaches roughly 100 nm with respect to the resi-
dual dispersive wave; it is therefore principally a Raman beam
cleanup process, but not of a pump-probe type as discussed in
ref. 19 because the pump and the Stokes probe are indeed co-
propagating within the spectrum of the same soliton pulse, and the
residual dispersive wave may become negligible in some conditions.
In order to explain the observed temporal effects, we present in
Fig. 6numerical simulation results that shed light on the process of
soliton formation. Figure 6a compares the evolution of the temporal
duration of pulses launched with either 67 or 235 fs input
pulsewidth, for different input wavelengths (1350, 1550, or 1680
nm). In each case, we picked the optimal input energy that
produces a minimum pulsewidth at 120 m. Input energies range
between 2 and 3 nJ when going from shorter to longer wavelengths,
and remain the same for both initial pulsewidths of 67 or 235 fs,
respectively. Input pulses with the same wavelength and different
durations always form a soliton with identical pulsewidth T
necessary propagation distance for soliton formation is 1 m at
1350 nm, and 6 m for both 1550 and 1680 nm. As the soliton
propagates, its pulsewidth grows larger because of SSFS, which
increases its wavelength, and local ﬁber dispersion, so that the
soliton condition is maintained10,20,21. For longer distances, the
pulsewidth differences, which are observed in Fig. 6a for different
input wavelengths, approach to a common value, as experimentally
conﬁrmed at 120 m (see Fig. 2). Correspondingly, one observes a
redistribution of the input modal powers (Fig. 6b) toward the
fundamental mode of the GRIN ﬁber, as in the example of Fig. 6c,
which is obtained after 10m of propagation.
All input pulses with the same wavelength, comparable energies,
and different pulsewidths, generated after a few meters of pro-
pagation a spatiotemporal soliton with common pulsewidth T
increased for growing values of the input wavelength. Once
that the spatiotemporal soliton was formed, a slow energy transfer
into the LG
mode was experimentally observed, while the
soliton simultaneously experiences SSFS (Fig. 3). For distances
larger than 100 m, the generated soliton appeared to be intrinsi-
cally monomodal, with a near-ﬁeld waist approaching that of the
fundamental mode of the MMF (Fig. 2). For all tested wave-
lengths and pulsewidths, a long-distance soliton was always
Fig. 5 Spatiotemporal soliton formed from an input 235 fs pulse. The soliton is composed by three axial modes, propagating over 120 m of graded-index
(GRIN) ﬁber, for the optimal input energy of 2.5 nJ. (a) modes power, (b) modes spectra.
Fig. 6 Evolution with ﬁber length of simulated soliton pulsewidths for
different input pulse durations and wavelengths. Simulations for input
pulses of 67 and 235 fs duration, and wavelengths of 1350, 1550, 1680 nm,
respectively (a). The input modes are LG
. Modal powers at
the ﬁber input (b), and after 10 m of propagation (c), for input 1550 nm
input wavelength and 235 fs input pulsewidth.
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observed for comparable input energies (i.e., between 2 and 3 nJ
in the case of initial excitation of three axial modes). Numerical
simulations (See Fig. 1in Supplementary note 2) and
experiments11 conﬁrmed that similar monomodal solitons are
observed when launching a larger number of degenerate and
nondegenerate modes at the ﬁber input, by varying the input
beam size or angle.
A possible analytical explanation for the invariance of
the observed pulsewidth T
of the generated solitons is provided
considering the modal walk-off length. From numerical simula-
tions, we know the wavelength dependence of the ﬁber dispersion
parameters of different ﬁber modes. Speciﬁcally, we know the
group velocities, say, β11 λðÞ;β12 λðÞ;β13 λðÞ(ps km1) for the three
axial modes LG01;LG02 ;LG03, as well as their group velocity
dispersion β2ðλÞ(ps2km1), which can be assumed to be equal
for all of the three modes. The resulting weighted mean of the
group velocity difference can be written as: Δβ1¼
0:30 β12 λðÞβ11 λðÞ
þ0:18 β13 λðÞβ11 λðÞ
Let us recall now the following characteristic lengths: the mean
modal walk-off length of the forming soliton LW¼T0=Δβ1,with
T0¼Ts0=1:763, is deﬁned as the distance where, in the linear
regime, the modes separate temporally. The pulse nonlinearity
and its dispersion length LD¼T2
as the characteristic length scales for Kerr nonlinearity and chro-
matic dispersion, respectively. The random mode coupling and
birefringence correlation lengths, L
, are the characteristic
length scales associated with linear coupling between degenerate
modes or between polarizations, respectively22–24.
In order to explain our observations, we may assume that,
when nonlinearity acts over distances shorter than those asso-
ciated with random mode coupling and birefringence, i.e., for
cm;Lcp , it is possible to observe a spatiotemporal soliton,
which is attracted into an effectively single-mode soliton11.A
second requirement to be considered is that both of the disper-
sion and nonlinearity lengths are comparable with the ﬁber walk-
off length: LD¼LNL ¼const LW, being const an adjustment
constant close to one. A similar condition for temporal trapping
of the optical modes was initially predicted, for the jth mode, by
refs. 1,2as LWj ≥LD¼LNL.
Based on the above considerations, we may ﬁnd the condition
to be respected by the soliton pulsewidth at the distance of initial
TS0ðλÞ¼const 1:763 β2ðλÞ
By performing a cut-back experiment, we measured the forming
soliton pulsewidth at 1 m of distance (at 1300, 1350, and 1420 nm)
and at 6 m of distance (at 1550, 1680 nm), for the optimal input
energies of 2–3 nJ, and input pulsewidths ranging between 61 and
96 fs (depending on the input wavelength). We compared experi-
mental results with numerical simulations at different input wave-
lengths and input pulsewidths of 67 and 235 fs, and with the
theoretical curve of T
upon wavelength as it is obtained from Eq.
(1), by using the dispersion curves of a GRIN ﬁber. The corre-
sponding results are shown in Fig. 7,conﬁrming the good agree-
ment between theory and experiments/simulations, provided the
adjustment constant is set to const =0.87, and for wavelengths
above 1350 nm (the const value may change with the coupling
conditions). Whereas for wavelengths below 1350 nm, the (anom-
alous) dispersion becomes very small, and the theory fails in pre-
dicting solitons with unphysical short pulse durations, also because
we neglect the presence of higher-order dispersion terms.
Therefore, Eq. (1)conﬁrms that a spatiotemporal soliton, includ-
ing nondegenerate modes, may be formed from the initial pulse,
eventually leaving behind a certain amount of energy in dispersive
waves. The soliton initial pulsewidth, and therefore energy, depends
on the ﬁber dispersion parameters. Our solitonic object appears to be
clamped to the ﬁber walk-off length L
, in the sense that the walk-
off, dispersion, and nonlinearity lengths must be related by an exact
equation, with const dependent on the MM soliton modal distribu-
tion. The soliton pulsewidth T
, at its formation distance, varies with
the wavelength of the input pump pulse, but its value turns out to be
independent of input pulse duration. From Fig. 7,weﬁnd that a
cannot arise from nondegenerate modes.
For relatively long input pulses (e.g., a 10 ps input pulse carried
by 15 modes, see Fig. 2of Supplementary note 2), groups of
modes separate temporally. However, in this case, it is still pos-
sible to inject the proper energy in each group of degenerate
modes, in order to obtain the generation of several independent
MM solitons composed by nondegenerate modes have unique
properties of evolving, in the range of hundreds of nonlinearity
lengths, into stable single-mode pulses. Their pulsewidth and
energy do not depend on the input pulsewidth, but only on the
coupling conditions and the input wavelength. Therefore, spa-
tiotemporal solitons composed by nondegenerate modes with
pulsewidth larger than a few hundreds of femtoseconds cannot be
generated. The unique properties of walk-off solitons can be
advantageously used to develop high-power spatiotemporal
mode-locked MM ﬁber lasers, with pulses of ﬁxed duration at a
given wavelength. The additional ability to form a single-mode
beam can be used for high-energy beam delivery applications.
Simulations. Numerical simulations are based on a coupled-mode equations
approach25,26, which requires the preliminary knowledge of the input power dis-
tribution among ﬁber modes. The modelcouples the propagating mode ﬁelds via Kerr
nonlinearities, by four-wave mixing terms of the type QplmnAlAmA*
coupling coefﬁcients, proportional to the overlap integrals of the transverse modal
ﬁeld distributions, and by SRS with same coupling coefﬁcients. Fiber dispersion and
nonlinearity parameters are estimated to be β
=−28.8 ps2km1at 1550 nm, β
0.142 ps3km1; nonlinear index n
=2.7 × 10−27 m2W1, Raman response h
typical times of 12.2 and 32 fs27,28. Wavelength-dependent linear losses of silica were
Fig. 7 Theoretical curve of soliton pulse duration vs. wavelength,
compared with measured and simulated soliton pulsewidth. Experiments
carried out with 1 m of graded-index (GRIN) ﬁber, with input wavelength
and pulsewidth: 1300 nm and 61 fs, 1350 nm and 61 fs, or 1420 nm and
70 fs, respectively. Experiment carried out with 6 m of GRIN ﬁber, with
input wavelength and pulsewidth: 1550 nm and 67 fs, 1680 nm and 96 fs,
respectively. Measurement error comes directly from the instrument
accuracy, which is below 1%. Simulations were carried out with same
distances and input wavelengths as in the experiments, with pulsewidths of
67 and 235 fs, respectively.
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Experiments. The experimental setup included an ultrashort pulse laser system,
composed by a hybrid optical parametric ampliﬁer (Lightconversion ORPHEUS-F),
pumped by a femtosecond Yb-based laser (Lightconversion PHAROS-SP-HP),
generating pulses at 100 kHz repetition rate with Gaussian beam shape (M2¼1:3);
the central wavelength was tunable between 1300 and 1700nm, and the pulsewidth
ranged between 60 and 240 fs, depending on the wavelength and the insertion of
pass-band ﬁlters. The laser beam was focused by a 50 mm lens into the ﬁber, with a
1=e2input diameter of ~30 µm (15 µm beam waist). The laser pulse input energy
was controlled by means of an external attenuator, and varied between 0.1 and
20 nJ. Care was taken during the input alignment in order to observe, in the linear
regime, an output near-ﬁeld that was composed by axial modes only; this could be
particularly appreciated for long lengths of GRIN ﬁber (120 m and more). The used
ﬁber was a span (from 1 to 850 m) of parabolic GRIN ﬁber, with core radius
=25 µm, cladding radius 62.5 µm, cladding index nclad ¼1:444 at 1550 nm, and
relative index difference Δ=0.0103. At the ﬁber output, a micro-lens focused the
near ﬁeld on an InGaAs camera (Hamamatsu C12741-03); a second lens focused the
beam into an optical spectrum analyzer (Yokogawa AQ6370D) with wavelength
range of 600–1700 nm, and to a real-time multiple octave spectrum analyzer
(Fastlite Mozza) with a spectral detection range of 1100–5000 nm. The output pulse
temporal shape was inspected by using an infrared fast photodiode, and an oscil-
loscope (Teledyne Lecroy WavePro 804HD) with 30 ps overall time response, and
an intensity autocorrelator (APE pulseCheck 50) with femtosecond resolution.
The data that underlie the plots within this paper and other ﬁndings of this study are
available from the corresponding authors on reasonable request.
The code used to generate simulated data and plots is available from the corresponding
authors on reasonable request.
Received: 22 April 2021; Accepted: 27 July 2021;
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We wish to thank Dr Cristiana Angelista for her valuable help in narrative restyling of
this paper. We acknowledge the ﬁnancial support from the European Research Council
Advanced Grants Nos. 874596 and 740355 (STEMS), the Italian Ministry of University
and Research (R18SPB8227), and the Russian Ministry of Science and Education Grant
M.Z., F.M. and M.F. carried out the experiments. S.W., M.Z. and O.S. developed the
theory and performed the numerical simulations. All authors analyzed the obtained
results, and participated in the discussions and in the writing of the manuscript.
The authors declare no competing interests.
Supplementary information The online version contains supplementary material
available at https://doi.org/10.1038/s42005-021-00687-0.
Correspondence and requests for materials should be addressed to M.Z.
Peer review information Communications Physics thanks the anonymous reviewers for
their contribution to the peer review of this work. Peer reviewer reports are available.
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ARTICLE COMMUNICATIONS PHYSICS | https://doi.org/10.1038/s42005-021-00687-0
6COMMUNICATIONS PHYSICS | (2021) 4:182 | https://doi.org/10.1038/s42005-021-00687-0 | www.nature.com/commsphys
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