Higher-Order Modulation Instability in Nonlinear Fiber Optics
Miro Erkintalo,
1
Kamal Hammani,
2
Bertrand Kibler,
2
Christophe Finot,
2
Nail Akhmediev,
3
John M. Dudley,
4
and Goe
¨
ry Genty
1
1
Tampere University of Technology, Optics Laboratory, FI-33101 Tampere, Finland
2
Laboratoire Interdisciplinaire Carnot de Bourgogne, UMR 5209 CNRS/Universite
´
de Bourgogne, 21078 Dijon, France
3
Research School of Physics and Engineering, The Australian National University, Canberra ACT 0200, Australia
4
Institut FEMTO-ST UMR 6174 CNRS/Universite
´
de Franche-Comte
´
, Besanc¸on, France
(Received 29 July 2011; published 13 December 2011)
We report theoretical, numerical, and experimental studies of higher-order modulation instability in the
focusing nonlinear Schro
¨
dinger equation. This higher-order instability arises from the nonlinear super-
position of elementary instabilities, associated with initial single breather evolution followed by a regime
of complex, yet deterministic, pulse splitting. We analytically describe the process using the Darboux
transformation and compare with experiments in optical fiber. We show how a suitably low frequency
modulation on a continuous wave field induces higher-order modulation instability splitting with the pulse
characteristics at different phases of evolution related by a simple scaling relationship. We anticipate that
similar processes are likely to be observed in many other systems including plasmas, Bose-Einstein
condensates, and deep water waves.
DOI: 10.1103/PhysRevLett.107.253901 PACS numbers: 42.65.Tg, 42.65.Re, 42.65.Sf, 42.81.Dp
The Benjamin-Feir or modulation instability (MI) is a
central process of physical systems described by the focus-
ing nonlinear Schro
¨
dinger equation (NLSE). MI dynamics
are associated with energy exchange between a periodic
perturbation and a continuous background, and have been
extensively studied since the 1960s [1]. Despite this long
history, MI is once again the subject of significant interest
as a mechanism to describe the emergence of strongly
localized ‘‘rogue wave’’ structures in hydrodynamics and
optics [2–4]. Of particular importance has been the real-
ization that many properties of MI previously described
only approximately (via numerical or truncated mode ap-
proaches) can in fact be described almost exactly using
known analytic NLSE Akhmediev-breather solutions [5].
This has led to successful studies of the spectral character-
istics of MI excited from noise [6] and experiments excit-
ing the previously unobserved Peregrine soliton limit of the
NLSE [7–9].
For ideal initial conditions, NLSE breathers exhibit only
one growth-return cycle. Such evolution, however, requires
a precisely constructed initial field consisting of an infinite
number of frequency modes with appropriate phases [10].
More realistic studies (both numerical and in physical
systems such as optics) have shown that such ideal single
growth-return evolution is not generally observed, but
rather the initial phase of breather localization is followed
by a regime of significantly more complex evolution
[11]. Specifically, when the second harmonic of the initial
modulation frequency falls under the elementary MI gain
curve it has been shown both numerically [11,12] and
experimentally [8] that the breather undergoes decompo-
sition and splits into two subpulses. In this Letter, we show
that these dynamics are in fact the manifestation of a
particular regime of NLSE propagation associated with
the excitation of higher-order modulation instability, and
we present what is to our knowledge the first combined
experimental, theoretical, and numerical study of this pro-
cess in any NLSE-described system.
The defining physics of higher-order MI is the simulta-
neous excitation of multiple instability modes, each mode
associated with a corresponding nonlinear breather. The
rich dynamics that is observed then arises from the non-
linear superposition of the individual constituent breathers.
Here, we demonstrate how the excitation of higher-order MI
can be observed readily in experiments using only a single
initial frequency modulation on a plane wave, provided
that the modulation frequency is below a critical low fre-
quency limit such that multiple instability harmonics fall
under the elementary gain curve. We apply the Darboux
transformation [11,13] to analytically describe the complex
splitting dynamics, and we show excellent agreement with
numerical simulation and experiment. Another significant
originality of our work is that it represents to our knowledge
the first example in any NLSE system where the Darboux
transformation is used for the analysis and interpretation of
physical experiments, and thus our results validate this
powerful analytical technique. In addition, insights from
the Darboux transformation approach lead us to develop a
simple scaling criterion relating pulse characteristics at
various phases of the instability, making an important link
with previous studies of self-similar dynamics in NLSE
systems [14,15]. Finally, we note that because the
Darboux transformation allows for an infinite number of
new solutions to be constructed, its experimental validation
opens up possibilities to observe a large class of novel high-
order instabilities at very high nonlinearity, a regime that is
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generally intractable using numerical techniques. Our re-
sults could therefore represent a major step towards under-
standing extreme wave localization dynamics in nonlinear
systems including plasmas, Bose-Einstein condensates,
and deep water waves [11], as well as providing a theoreti-
cal framework for the interpretation of various regimes of
nonlinear turbulence [16].
We first discuss the NLSE breather solutions and the
Darboux transformation. The dimensionless focusing
NLSE is
i
@
c
@
þ
1
2
@
2
c
@
2
þj
c
j
2
c
¼ 0; (1)
where and are normalized distance and time, respec-
tively. The modulation instability dynamics of a harmonic
perturbation on an initial plane wave are well described
by the breather solution of Akhmediev [5]:
c
ð;Þ¼
ð14aÞcosh½bð
0
Þþibsinh½bð
0
Þ
cosh½bð
0
Þ
ffiffiffiffiffiffi
2a
p
cos½ð
0
Þ
þ
ffiffiffiffiffiffi
2a
p
cos½ð
0
Þ
cosh½bð
0
Þ
ffiffiffiffiffiffi
2a
p
cos½ð
0
Þ
e
i
: (2)
Here is a dimensionless frequency related to the pa-
rameter a through a ¼ 1=2ð1
2
=4Þ, and the instability
growth is b ¼½8að1 2aÞ
1=2
¼ ð1
2
=4Þ
1=2
. There
is MI gain for 0 <a<1=2, corresponding to frequencies
max
> > 0 with
max
¼ 2. Real parameters
0
and
0
describe the spatial and temporal centers of the solution.
Transformations to dimensional quantities in the context
of fiber optics are given later when we describe our
experiments.
The solution in Eq. (2) was originally obtained using a
physically motivated ansatz [5], but similar solutions have
been obtained using the inverse scattering technique
[17,18]. Another powerful approach to NLSE analysis is
the Darboux transformation, a linear algebraic dressing
method allowing successive construction of higher-order
solutions from a known solution [13]. The approach is
based on representing the NLSE as the compatibility con-
dition between the following linear set [11]:
R
¼JR þ UR;
R
¼JR
2
þ UR
1
2
ðJU
2
JU
ÞR;
(3)
where R, J, U, and are 2 2 matrices defined by
¼
0
0
"#
; J ¼
i 0
0 i
"#
;
U ¼
0
c
c
0
"#
; RðÞ¼
rðÞ s
ðÞ
sðÞr
ðÞ
"#
:
(4)
Here, the functions rðÞ¼rð; ; Þ and sðÞ¼sð;;Þ
depend on a set of three arbitrary constants: the
spectral parameter and two integration constants. The
compatibility condition R
¼ R
is equivalent to
Eq. (1), which means that the set of Eq. (3) is simulta-
neously satisfied for all if and only if
c
ð; Þ is a
solution of the NLSE. The linear set (3) is invariant under
the Darboux transformation:
R ðÞ!R
1
ð
1
;Þ¼RðÞ ð
1
ÞRðÞ; (5)
U ! U
1
¼ U Jð
1
Þþð
1
ÞJ; (6)
where ð
1
Þ¼Rð
1
Þ
1
R
1
ð
1
Þ. This invariance means
that if a particular solution
c
0
of the NLSE and its corre-
sponding complex functions r
0
ðÞ and s
0
ðÞ are known,
then a new solution can be obtained from Eq. (6). (Recall
that the matrix U is a function of the desired solution
c
.)
One can then apply the technique on the new solution, as
the Darboux transformation also provides the correspond-
ing functions satisfying the linear set (3) through Eq. (5). In
this way one can construct iteratively and via only alge-
braic transformations an entire hierarchy of solutions up to
arbitrary order, obtaining the complete range of higher-
order MI solutions from the trivial NLSE plane wave
solution
c
0
¼ e
i
[19].
In physical terms, the first-order solution of this type
gives the elementary breather of Eq. (2). The N th-order
solution constructed from further iteration corresponds to
the nonlinear superposition of N such elementary breath-
ers, each associated with unique parameters
k
¼
fa
k
;
0k
;
0k
g; a distinct modulation frequency (determined
by a
k
) and temporal and spatial origins
0k
and
0k
(which
depend on initial conditions and can be considered as free
parameters).
Although general solutions can be constructed for arbi-
trary unequally spaced modulation frequencies within the
gain band, higher-order MI solutions can also be excited by
a single initial modulation frequency, provided that it is
below a critical low frequency limit. In particular, for a
primary modulation frequency , Nth-order MI (N 2)
is excited when N <
max
¼ 2 such that the initial
frequency generates harmonics at
k
¼ k, where k ¼
1; ...;N and all modes experience gain under the elemen-
tary MI gain curve. In terms of the input modulation
parameter a, this constraint is N
2
< ð1 2aÞ
1
so that
we can readily determine the low frequency limit for
higher-order MI as < 1, equivalent to a>0:375. The
condition between N and a allows the integral order of the
solution expected for given input conditions to be readily
determined (see Fig. 2 below).
We now apply this analysis to interpret numerical simu-
lations of typical higher-order MI dynamics. Figure 1(a)
shows simulation results from integrating numerically the
NLSE for initial conditions that do not correspond to the
exact mathematical form required for perfect breather
growth and decay [10]. We consider an initial field
c
ð ¼
0;Þ¼½1 þ a
mod
cosðÞ, where a ¼ 0:43 is near the
limit of recent experiments studying Peregrine soliton
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evolution, and in the regime a>0:375 discussed above
where we expect higher-order MI dynamics. In particular,
for this a, two frequencies ¼ 0:75 (¼
1
) and
2 ¼ 1:5 (¼
2
) fall within the MI gain band ( < 2)
so we expect to see N ¼ 2 higher-order MI evolution.
The figure shows results for two values of modulation
amplitude a
mod
where we see an initial phase of typical
breather evolution (growth along and temporal compres-
sion along ), but in both cases evolution beyond the point
of maximum amplitude is associated with temporal split-
ting into subsidiary pulses. Note that the value of a
mod
influences both the exact point of temporal compression as
well as the nature of subsequent splitting. Although similar
dynamics have been previously seen in numerical simula-
tions [6,12], we now use the Darboux transformation to
analytically describe this behavior. In particular, we
construct through the scheme described above the analyti-
cal superposition of the two elementary breathers associ-
ated with the frequencies
1
¼ 0:75 and
2
¼ 1:5.
The constructed solution in this way is shown in
Fig. 1(b) for parameters
1
¼f0:430; 0; 13:41g,
2
¼
f0:219; 0; 18:425g for the low modulation amplitude
a
mod
¼ 2 10
4
, and
1
¼f0:430; 0; 3:4g,
2
¼
f0:219; 0; 4:47g for the higher modulation amplitude
a
mod
¼ 2 10
1
. Here subscripts 1 and 2 refer to the
elementary breather solutions associated with frequencies
1
and
2
, respectively.
We first note that the solutions constructed from the
Darboux transformation are in excellent agreement with
those obtained from the numerical integration of the
NLSE. Our analysis also allows us to obtain significantly
further physical insight into the dynamics because the
construction of the higher-order solutions strongly depends
on the relative separation in between the centers of the
constituent elementary solutions. When the separation is
large, the superposition solution is simply the linear sum of
elementary solutions of the form (2). In this case, the first
stage of propagation corresponds to that of the primary
breather defined by
1
followed by a secondary stage that
is dominated by the dynamics of the secondary breather
defined by
2
. Such independent dynamics are seen for
small modulation amplitudes, and the left-hand panel of
Fig. 1(c) shows this clearly by plotting the intensity profiles
at the points of maximum amplitude during the initial and
secondary evolution phases. Note that because of the non-
linear nature of the superposition of the constituent solu-
tions, the distances of maximum amplitude are close but
not identical to the corresponding center parameters of the
Darboux transformation given by
1
and
2
.
Significantly, however, provided that the nonlinear cou-
pling is weak such that the splitting can be distinctly
observed, the manner in which the higher-order solutions
are constructed from elementary solutions suggests that the
subsidiary peaks upon splitting can be rescaled to the
primary pulse that generates them. Indeed, it is straightfor-
ward to show that, for a>1=8, the breather peaks at the
point of maximum amplitude obey an area theorem such
that peaks at step k and step k þ 1 are related through
q
c
kþ1
ðt=qÞ¼
c
k
ðtÞ with the scale parameter q ¼
maxj
c
k
j= maxj
c
kþ1
j > 1. This result is an important
link with previous numerical studies of scaling effects in
pulse splitting dynamics [14].
On the other hand, for increased input modulation am-
plitude the nonlinear coupling between the constituent
solutions becomes more significant, and it is not possible
to consider the evolution as an independent superposition
of constituent breathers. This coupling affects the intensity
profile of both individual breathers as well as the temporal
evolution of the secondary breather phase, and is seen more
clearly in the right-hand panel of Fig. 1(c).
For even lower modulation frequencies and MI of in-
creasingly higher orders, simulations show a series of
progressive temporal bifurcation dynamics. Yet, even for
this case the Darboux transformation can be used to accu-
rately reconstruct and interpret the dynamics. This is illus-
trated in Fig. 2. We first show in Fig. 2(a) the bifurcation
points corresponding to the range of values of a where
order N splitting is observed based on the relation between
N and a discussed above. This graph is especially useful in
allowing the dynamics complexity in terms of N to be
FIG. 1 (color online). Spatiotemporal evolution of a weakly
modulated cw field with a ¼ 0:43 and for values of a
mod
as
shown. The left column (a) shows results from numerical simu-
lations; the right column (b) shows results from the Darboux
transformation (DT) solutions. Results in (c) are line profiles at
selected distances for the two values of modulation amplitude.
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quantified for a specific input a. Figures 2(b) and 2(c) show
simulation results for N ¼ 3 (a ¼ 0:468) and
N ¼ 5 (a ¼ 0:486), respectively. For the latter case we
also show in Fig. 2(d) the spectral evolution which clearly
exhibits a complexity not seen in simple MI. Simulation
results in Fig. 2(e) are for N ¼ 6 (a ¼ 0:4895) to show
even higher-order splitting. For this case, we also explicitly
show the power of the Darboux transformation to recon-
struct these dynamics, with the analytic results in Fig. 2(f).
All simulations used a
mod
¼ 0:02.
The above analysis of the complex dynamics in terms of
higher-order MI processes has allowed us to design experi-
ments to excite the nonlinear superposition breather solu-
tions using a modulated plane wave optical field injected
into optical fiber. Our setup is similar to that of Ref. [8] and
consists of a 16 GHz intensity modulated 1550 nm laser
diode of 0.8 W average power after an erbium-doped fiber
amplifier. A phase modulator is also used to inhibit
Brillouin scattering. The optical fiber is an 8.34 km long
segment of SMF-28 with group velocity dispersion and
third-order dispersion at 1550 nm:
2
¼21:4ps
2
=km
and
3
¼ 0:12 ps
3
=km, respectively. The nonlinear pa-
rameter ¼ 1W
1
km
1
and the fiber losses are
0:19 dB=km. The third-order dispersion is included in
simulations for completeness but has negligible effect on
the dynamics which are well described by the NLSE of
Eq. (1). The output signal is characterized in the time
domain by means of an ultrafast optical sampling oscillo-
scope (Picosolve) which, when combined with cutback
measurements, allows direct characterization of the longi-
tudinal field evolution. Numerical simulations are based
on the dimensional variant of Eq. (1) using experi-
mental parameters. The initial condition is Aðz ¼ 0;TÞ¼
ffiffiffiffiffiffi
P
0
p
½1 þ
mod
cosð!
mod
TÞ
0:5
, where P
0
is the power
of the cw field, A ¼
c
ffiffiffiffiffiffi
P
0
p
, z ¼ =ðP
0
Þ, T ¼ T
0
,
!
mod
¼ =T
0
. and T
0
¼½j
2
j=ðP
0
Þ
1=2
. The param-
eter
mod
¼ 0:58 here is the intensity modulation
contrast. Note that with these parameters, the normalized
input modulation frequency is ¼ 0:539, parameter a ¼
0:464, and N ¼ 2= ¼ 3:7 consistent with the observed
excitation of 3 subpulses at the fiber output.
Our experimental results are shown in Fig. 3(a), and
clearly exhibit the expected higher-order MI dynamics
with splitting of the initial modulated field into two,
then three subpulses with propagation. Numerical simula-
tions including all experimental parameters are shown
in Fig. 3(b) and are in excellent agreement with
experiment. The analytic construction of the corresponding
field using the Darboux transformation is shown in
Fig. 3(c). Since the Darboux transformation applies
meaningfully only to the ideal NLSE, we do not include
loss in the analytic construction, and thus the calculated
field displays a higher amplitude than in experiments,
but the calculated field nonetheless displays identical
temporal splitting to experiment at the fiber output.
The analytically constructed third-order solution
used
1
¼f0:464; 0 ; 4:5g,
2
¼f0:355; 0; 4:9g,
3
¼
f0:173; 0; 7g.
These results represent the first quantitative study of
higher-order MI dynamics in any physical NLSE
system. The analysis of breather dynamics in terms of
the Darboux transformation allows us to naturally interpret
the complex evolution at low instability modulation
FIG. 2 (color online). Pulse splitting with higher-order MI.
(a) Bifurcation points showing the range of a where order-N
splitting is observed. (b),(c) NLSE simulations for N ¼ 3
(a ¼ 0:468) and N ¼ 5 (a ¼ 0:486), respectively. For the latter
case (d) shows the associated complex spectral evolution.
(e) Simulations for N ¼ 6 (a ¼ 0:4895) to show even higher-
order splitting. (f) Corresponding analytical construction using
the Darboux transformation for case (e).
FIG. 3 (color online). Top: Spatiotemporal evolution of cw
modulated field for a ¼ 0:464. Bottom: Temporal profile at a
distance of 8.34 km. (a) Experiments, (b) NLSE simulation, and
(c) analytical solution from the Darboux transformation.
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frequencies in terms of the superposition of multiple
elementary instabilities governed by distinct dynamical
parameter sets. We have shown that higher-order
MI dynamics can be readily excited by injecting a
modulated plane wave with modulation frequency
selected so that its harmonics are also located within the
modulation instability gain curve. Our ability to quantita-
tively describe the measured third-order pulse splitting
using the Darboux construction is a remarkable illustra-
tion of the power of this analytical method, and we antici-
pate that these results will motivate its use for similar
analysis of wave localization dynamics in other NLSE
governed systems such as deep water hydrodynamics,
Bose-Einstein condensates, etc. We further expect that
signatures of higher-order modulation instability will be
found in cases where the entire range of unstable modes
spontaneously grows from noise. It should also be clear
that the Darboux transformation allows for exactly
constructing solutions of very high order which are
otherwise extremely difficult to generate through numeri-
cal simulations because of prohibitive computational
effort. In this context, it is especially important to note
that the analytical approach allows us to identify the
appropriate initial conditions necessary to generate a
particular high-order excitation state; determining the
initial conditions would not be possible numerically.
This makes the Darboux transformation particularly
adapted for the (numerical and experimental) search
and decomposition of new forms of instabilities and local-
ization dynamics in various nonlinear regimes including
turbulence [16], Anderson localization [20], and self-
similarity [15]. Finally, we anticipate that the analysis
developed may also have important technical applications
in providing insights into optimizing the generation
of high repetition rate pulse trains and broadband fiber
parametric amplifiers.
We thank the Academy of Finland (Research Grants
No. 121953, No. 130099, and No. 132279), the graduate
school of Tampere University of Technology, the Institut
Universitaire de France, the French Agence Nationale de la
Recherche projects IMFINI (ANR-09-BLAN-0065) and
MANUREVA (ANR-08-SYSC-019), the Conseil
Re
´
gional de Bourgogne, and the Australian Research
Council (Discovery Project No. DP110102068).
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