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Instabilities, breathers and rogue waves in optics
John M. Dudley1, Frédéric Dias2, Miro Erkintalo3, Goëry Genty4
1. Institut FEMTO-ST, UMR 6174 CNRS-Université de Franche-Comté, Besançon,
France
2. School of Mathematical Sciences, University College Dublin, Belfield, Dublin 4,
Ireland
3. Department of Physics, University of Auckland, Auckland, New Zealand
4. Department of Physics, Tampere University of Technology, Tampere, Finland
Optical rogue waves are rare yet extreme fluctuations in the value of an optical
field. The terminology was first used in the context of an analogy between pulse
propagation in optical fibre and wave group propagation on deep water, but has since
been generalized to describe many other processes in optics. This paper provides an
overview of this field, concentrating primarily on propagation in optical fibre systems
that exhibit nonlinear breather and soliton dynamics, but also discussing other optical
systems where extreme events have been reported. Although statistical features such as
long-tailed probability distributions are often considered the defining feature of rogue
waves, we emphasise the underlying physical processes that drive the appearance of
extreme optical structures.
Many physical systems exhibit behaviour associated with the emergence of high
amplitude events that occur with low probability but that have dramatic impact. Perhaps the
most celebrated examples of such processes are the giant oceanic “rogue waves” that emerge
unexpectedly from the sea with great destructive power [1]. There is general agreement that
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the emergence of giant waves involves physics different from that generating the usual
population of ocean waves, but equally there is a consensus that one unique causative
mechanism is unlikely. Indeed, oceanic rogue waves have been shown to arise in many
different ways: from linear effects such as directional focusing or random superposition of
independent wave trains, to nonlinear effects associated with the growth of surface noise to
form localized wave structures [1,2].
The analogous physics of nonlinear wave propagation in optics and in hydrodynamics
has been known for decades, and the focusing nonlinear Schrödinger equation (NLSE)
applies to both systems in certain limits (Box 1). The description of instabilities in optics as
“rogue waves” is recent, however, first used in 2007 when shot-to-shot measurements of fibre
supercontinuum (SC) spectra by Solli et al. yielded long-tailed histograms for intensity
fluctuations at long wavelengths [3]. An analogy between this optical instability and oceanic
rogue waves was suggested for two reasons. Firstly, highly skewed distributions are often
considered to define extreme processes, since they predict that high amplitude events far
from the median are still observed with non-negligible probability [4]. And secondly, the
particular regime of SC generation being studied developed from modulation instability (MI),
a nonlinear process associated with exponential amplification of noise that had previously
been proposed as an ocean rogue wave generating mechanism [2].
These pioneering results enabled for the first time a quantitative analysis of the
fluctuations at the spectral edge of a broadband supercontinuum, and motivated many
subsequent studies into how large amplitude structures could emerge in optical systems.
These studies attracted broad interest and have essentially opened up a new field of “optical
rogue wave physics”. Although most research since has focused on propagation in optical
fibres and in particular in regimes analogous to hydrodynamics, the terminology “optical
rogue wave” has now been generalized to describe other noisy processes in optics with long
3
tailed probability distributions, irrespective of whether they are observed in systems with a
possible oceanic analogy. Moreover, particular analytic solutions of the NLSE describing
solitons on a finite background or “breathers” are now also widely referred to as “rogue
wave” solutions, even when studied outside a statistical context for mathematical interest, or
when generated experimentally from controlled initial conditions. These wider definitions
have become well-established, but can unfortunately lead to difficulty for the non-specialist.
Our aim here is to remove any possible confusion by presenting a synthetic review of
the field, but we do so not in terms of its chronological development which has been
discussed elsewhere [5,6]. Rather, we organise our presentation by classifying rogue waves
in terms of their generating physical mechanisms. We begin by discussing rogue waves in
the regime of NLSE fibre propagation where MI and breather evolution dominate the
dynamics, and we then discuss how the effect of perturbations to the NLSE can lead to the
emergence of background-free solitons. This provides a natural lead-in to discuss the physics
and measurement techniques of rogue solitons in fibre supercontinuum generation. We then
describe techniques used to control rogue waves dynamics in fibre systems, and this is
followed by a survey of results in other systems: lasers and amplifiers where dissipative
effects are central to the dynamics, and spatial systems where both nonlinear and linear
dynamics can play a role.
ROGUE WAVES AND STATISTICS
Before considering specific examples of optical rogue waves, we first briefly review
how rogue wave events are manifested in the statistics of the particular system under study.
In optics, statistical properties are defining features of light sources. For example, the
random intensity fluctuations of polarized thermal light follow an exponential probability
4
distribution, and the intensity fluctuations of a laser above threshold follow a Gaussian
probability distribution [7]. It was the experimental observation of “L-shaped” long tailed
distributions in Ref. [3] that linked for the first time nonlinear optics with the wider theory of
extreme events. In a sense, seeing long-tailed distributions in optics is not surprising, since it
is well-known that a nonlinear transfer function will modify the probability distribution of an
input signal. Indeed, an exponential probability distribution in intensity is transformed under
exponential gain to a power law Pareto-distribution. There are other cases, however, where a
functional nonlinear transformation of an input field cannot be identified, and the emergence
of high amplitude events arises from more complex dynamics. Optical rogue waves and
long-tailed statistics have been observed in systems exhibiting both types of behaviour.
Rogue waves in optics have been identified in different ways. One approach uses the
idea from probability theory that associates rogue events with particular extreme-value
probability distributions, and such functions have provided good fits to the tails of histograms
of optical intensity fluctuations in several studies [8-10]. Another approach has adapted the
oceanographic definition, where rogue waves are those with trough-to-crest height HRW
satisfying HRW 2 H1/3. Here H1/3 is the significant wave height, the mean height of the
highest third of waves [11]. In optics, however, the accessible data is not the field amplitude
but rather the intensity, and indeed such data can take a variety of forms: an intensity time
series; the levels of a two-dimensional camera image; the space-time intensity evolution of an
optical field. From this data, the intensity peaks are analysed statistically to compute a
histogram, and the oceanographic definition is modified to: IRW ≥ 2 I1/3 where the “significant
intensity” I1/3 is the mean intensity of the highest third of events. Although somewhat arbitrary,
this definition has been applied in several studies [12-15].
From a general perspective, what is of most interest is to consider whether events in
the distribution tail for a particular system arise from different physics compared to the
5
events closer to the distribution median. This is also relevant in a practical context, since the
ability to identify the conditions causing extreme events is key to their prediction and control.
MODULATION INSTABILITY AND BREATHER DYNAMICS
Modulation instability is a fundamental property of many nonlinear dispersive
systems, associated with the growth of periodic perturbations on a continuous wave (CW)
background [16]. In the initial evolution, the spectral sidebands associated with the
instability experience exponential amplification at the expense of the pump, but the
subsequent dynamics are more complex and display cyclic energy exchange between
multiple spectral modes. In optics, MI seeded from noise results in a series of high-contrast
peaks of random intensity [17,18], and it is these localised peaks that have been compared
with similar structures seen in studies of ocean rogue waves [2,19,20]. Significantly, the MI
growth and decay dynamics in the NLSE have exact solutions in the form of various types of
“breather” or “soliton on finite background” (SFB) [21], and this fact has motivated much
research to gain analytic insight into conditions favouring rogue wave emergence.
SFBs constitute a class of NLSE solutions whose real and imaginary parts are linearly
related, and they include Akhmediev Breathers (ABs), Kuznetsov-Ma (KM) solitons, the
Peregrine soliton (PS), and even more generally the bi-periodic solutions of the NLSE
described by Jacobi elliptic functions [21-23]. Many of these solutions have been labelled as
“rogue waves” [24] although this interpretation must be made with great care. As we will see
below, the statistical criterion for rogue waves in an MI field seeded by noise is generally
only satisfied by particular higher-order SFB solutions (sometimes referred to as “multi-
rogue waves” or “higher-order rogue waves”) [25-28].
6
Before discussing noise-seeded MI in detail, we first describe these SFBs and breather
solutions by referring to the dimensionless NLSE:
. (1)
The envelope is a function of ξ (propagation distance) and
τ
(co-moving time).
Equation (1) can be related to the dimensional fibre-optic NLSE in Fig. B1 by defining
timescale T0 = (|
β
2| LNL)1/2 and nonlinear length LNL = (
γ
P0)-1, where P0 is optical background
power in W. The dimensional field A(z,T) [W1/2] is ; dimensional time T [s] is T =
τ
T0 and dimensional distance z [m] is . Analytic solutions of MI dynamics have
been obtained by several authors [20-23,29-31], with the relevance to optics first pointed out
in Ref. [30]. The particular solution that describes MI growth and decay is [21,30]:
(2)
The solution’s properties are determined by one positive parameter a (a ≠ 1/2) through
arguments b = [8a (1-2a)]1/2 and
ω
= 2(1-2a)1/2. Over the range 0 < a < ½ the solution is the
Akhmediev breather (AB) where we observe evolution from the trivial plane wave (a = 0) to
a train of localised pulses with temporal period Δ
τ
= π/(1-2a) ½ [30]. The AB solution
provides MI’s analytic framework, with the real parameter
ω
corresponding to the
modulation frequency, and the real parameter b giving the parametric gain coefficient [32].
Figure 1(a) plots solutions for different a as indicated, and it is the spatial and
temporal localization properties of these solutions which have led to their association with
rogue waves. The MI instability growth rate is maximal at a = ¼, but increasing a actually
leads to stronger localization in both dimensions until the limit a → ½ which gives the
Peregrine Soliton [23]. The PS, given by:
ψ
(
ξ
,
τ
) = [1 − 4(1+2i
ξ
)/(1 + 4
τ
2 + 4
ξ
2)] ei
ξ
,
corresponds to a single pulse with localization in time (
τ
) as well as along the propagation
direction (
ξ
) as shown, and it has maximum intensity amongst the AB solutions of |
ψ
PS|2 = 9.
2
2
2
10
2
i
ψψ
ψψ
ξτ
∂∂
++=
∂∂
(,)
ψξτ
0
AP
ψ
=
NL
zL
ξ
=
2(1 2 )cosh( ) sinh( )
(,) 1 2cos( ) cosh( )
ia b ib b
e
ab
ξξξ
ψξτ ωτ ξ
%&
−+
=+
()
−
*+
7
When a > ½, the parameters
ω
and b become imaginary, and the solution exhibits localisation
in the temporal dimension
τ
but periodicity along the propagation direction
ξ
. This is the
Kuznetsov-Ma soliton [22,29] which is shown in Fig. 1(a) for a = 0.7. In this regard, we note
that it was the KM result in Ref. [22] which was the first mathematical SFB solution of the
NLSE reported.
Each of the solutions described by Eq. (2) is a special case of a more general family
of solutions that exhibit periodicity in both transverse time
τ
and longitudinal propagation
direction
ξ
[21]. More complex higher-order solutions also exist with even stronger
localisation and higher intensities than the PS [21], such as higher-order rational solitons [25-
28,33] and AB collisions [34]. Figure 1(a) shows examples of both types of solutions
[25,35,36].
The analytical results above have been used to design experiments with controlled
initial conditions to excite particular SFB dynamics in fibre optics. The use of optical fibres
provides an especially convenient experimental platform as the dispersion and nonlinearity
parameters can be conveniently matched to available optical sources to yield a propagation
regime where the NLSE is a valid model for the dynamics. Experiments typically inject a
multi-frequency field into a nonlinear fibre, similar to the method developed for coherent
pulse train generation in telecommunications [37,38]. Figure 1(b-d) shows a selection of
results obtained. The first experiments in 2010 used frequency-resolved optical gating to
show localisation in the PS regime [39], with measured intensity and phase agreeing well
with numerical and analytical predictions [Fig. 1(b)]. Later experiments examined the
growth and decay of spectral amplitudes during AB evolution in more detail [40]. As pointed
out by Van Simaeys et al., these reversible dynamics of MI is a clear manifestation of Fermi-
Pasta-Ulam recurrence in optics [41].
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Experiments exciting KM-like evolution along the propagation direction have also
been realised, and Fig. 1(c) shows results illustrating the growth and decay of the peak
temporal intensity [42]. In another experiment, spectral shaping of an optical frequency comb
synthesised initial conditions to excite the collision of two ABs [35]. Figure 1(d) shows an
example of the results obtained, comparing the measured collision profile at the fibre output
with numerical simulation. These results are significant in highlighting how collisions can
yield significantly larger intensities than the elementary ABs alone, even exceeding the PS
limit. Note that excited AB collision dynamics can be considered as a particular case of
higher-order MI, where the simultaneous excitation of multiple instability modes within the
MI gain bandwidth gives rise to the nonlinear superposition of ABs [21,43-45].
Note that the use of a modulated input field in these experiments means that the initial
conditions correspond to a truncated Taylor-series expansion of the analytic AB or PS far
from the point of maximum localization, or an approximation to the ideal KM profile at its
point of minimal intensity in its evolution cycle. The use of non-ideal initial conditions
induces differences compared to the ideal dynamics (e.g. the occurrence of multiple Fermi-
Pasta-Ulam recurrence cycles for an AB), but simulations have shown that the spatio-
temporal localisation and the field properties at point of highest intensity remain very well
described by the corresponding analytic SFB solution [46].
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FIGURE 1 | SFB solutions of the NLSE. a, Analytical SFB solutions of Eq. (2) for varying parameter a.
From left to right: an Akhmediev breather (AB); Peregrine soliton (PS); Kuznetsov-Ma (KM) soliton [42]. An
example of an AB collision and the second-order rational soliton (or second-order PS) are also shown. b-d,
Experimental results. b shows temporal PS properties asymptotically approached for a = 0.42 [39], c shows
KM dynamics along the propagation direction for a = 1 [42] with experiments, simulations and theory
compared in each case. Here zp = 5.3 km corresponds to one period of the KM cycle. d compares experiments
and simulations of a second-order solution consisting of the collision of two ABs (a = 0.14 and a’ = 0.34) [35].
The experiments above give insight into how appropriate initial conditions can excite
a range of analytic breather or SFB structures in an NLSE system. The reason why these
results are important in the study of rogue waves is that structures very similar to those
described by Eq. (2) (and their higher-order extensions) also appear in chaotic fields when
MI develops from noise [33,34,47,48]. We illustrate this in Fig. 2 using numerical
simulations of the dimensionless NLSE where clear signatures of the SFB solutions can be
10
identified (i.e. periodicity in
τ
or along
ξ
,
τ
-
ξ
localization, value of peak intensity) when MI
is triggered from a broadband noise background [49]. Figure 2(a) plots a density map of the
evolution as a function of distance
ξ
, showing emergence of an irregular series of temporal
2
π
, corresponding to the reciprocal of the frequency of maximum MI gain. After the initial
emergence of these localized peaks, we see more complex periodic growth and decay
behavior along
ξ
.
Examining particular features of the evolution map reveals signatures of the analytic
KM, AB, and PS solutions described above, as plotted in Fig. 2(b). For the region marked
KM, a line profile of the evolution along
ξ
agrees well with the analytic result expected for a
KM soliton, whilst in the regions AB and PS, the temporal localization characteristics also
agree well with corresponding analytic predictions. Of course, observing ideal analytic SFB
structures is not expected given the random initial conditions, but it is remarkable how the
analytic solutions can be mapped closely to the noise-generated structures. In this context,
we remark that recent results have also considered different initial forms of small
perturbations to the CW background and found similar signatures of AB structures [50].
These results confirm that SFB solutions can provide analytical insight into structures
emerging from noise-seeded MI, but understanding their significance to the physics of rogue
waves requires analysis of the associated statistics [33,34]. To this end, Fig. 2(c) plots the
histogram of the intensities of the localized peaks in a larger computational window (around
106 peaks in the
τ
−
ξ
plane). We note firstly that the histogram maximum corresponds to
intensity |
ψ
|2 = 5.6, close to the AB intensity at maximal gain when a = ¼. The fact that
these structures appear more frequently than others in the chaotic regime is consistent with
experiments studying the spectral characteristics of spontaneous MI [18].
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Secondly, we note an exponential tail (linear on a semi-logarithmic scale) for higher-
intensities, and the dashed line indicates the point in the tail IRW ~ 13 corresponding to the
rogue wave “significant intensity”. It is interesting to remark here that the elementary AB
and PS structures (with intensity less than IPS = |
ψ
PS|2 = 9) actually have intensities below the
significant intensity IRW, suggesting caution in the description of these solutions as rogue
waves [24,39]. Approximately 2.5% of the total peaks have intensity exceeding the Peregrine
soliton limit IPS = 9, and these events correspond to AB collisions arising physically from the
continuous range of frequencies amplified by MI [33,34]. In our simulations, the largest
among them (which make up only 0.1% of the total) do have intensities exceeding the rogue
wave threshold IRW and would very clearly be described as rogue wave events according to
any criteria used [2,51]. Note that the figures below the histogram in Fig. 2(c) show
simulation results comparing a typical AB solution with a third-order breather collision in the
tail. Finally, we note that these results were obtained with a one-photon per mode noise
background, and the exact fraction of highest-intensity events in the distribution tails is
expected to vary with the noise level used to seed MI [33,34].
12
FIGURE 2 | Numerical simulation showing signatures of analytic NLSE solutions in chaotic MI. a,
Density map showing the long term temporal evolution of a chaotic field triggered by one photon per mode
noise superimposed on a CW background. Signatures of SFB solutions can be observed in the dynamics as
indicated. b, Line profiles extracted from the regions of the chaotic field indicated by white dashed rectangles in
a, compared with analytical NLSE solutions (red solid line). c, Peak-intensity statistics using an 8-connected
neighbourhood regional maximum search to identify 2-D peaks from a wider simulation window. The
maximum of the probability density corresponds to AB-like solutions at the peak of the MI gain, whilst the most
extreme outliers arise from higher-order ABs superposition; the bottom subfigures show the evolution of two
events from different regions of the histogram (i) and (ii) as indicated.
13
SUPERCONTINUUM GENERATION AND SOLITONS
The studies above show how SFB dynamics during fibre propagation can lead to
strongly-localised structures with rogue wave statistics. Interestingly, however, the first
observations of rogue waves in optics were not made in the regime of SFB dynamics at all,
but in “long-pulse” supercontinuum generation. In this regime, higher-order effects beyond
the basic NLSE played an important role [3], and although approximate SFB solutions can
exist even under these conditions [52], the first observed rogue wave characteristics arose
instead from the dynamics of background-free hyperbolic secant solitons.
Noise-seeded MI dynamics dominate the initial stages of long-pulse and continuous
wave SC generation, but the dynamics are significantly modified with propagation by higher-
order dispersion and stimulated Raman scattering [32,49]. It is this perturbed dynamical
evolution that drives the emergence of the rogue wave soliton characteristics of SC
generation. Specifically, after initial evolution that is statistically dominated by the AB with
maximum gain (see Fig. 2), perturbations cause the associated temporal peaks to reshape with
further propagation into fundamental background-free hyperbolic secant solitons [18,53].
These sech-solitons then experience additional dynamics of dispersive wave generation [54]
and a continuous shift to longer wavelengths through the Raman effect [55]. Because the
solitons emerge from a stochastic field of breather-like structures, their durations, amplitudes
and wavelengths show considerable statistical variation. The strong dependence of the
Raman self-frequency shift on duration [55], coupled with effects of group velocity
dispersion (GVD), therefore induce complex subsequent evolution involving multiple
(stochastic) collisions and energy exchange between the solitons [56-59].
The fact that these chaotic soliton dynamics could cause significant shot-to-shot noise
in the SC spectrum has been known for many years [56], but it was the real-time
14
measurement of these fluctuations in Ref. [3] that first highlighted links to the field of
extreme events. The experiments in Ref. [3] used a long-pass wavelength filter to select out
the portion of the SC spectrum where strong soliton intensity variations were expected to
occur, and then used dispersive Fourier transformation [60,61] to perform shot-to-shot
measurements of the associated spectral fluctuations. It was the peaks in the resulting
intensity time-series that showed striking long-tailed statistics. Figure 3(a) shows the
experimental setup used, and a selection of recorded histograms at three different pump
levels. The main conclusion drawn from these experiments was that the largest events in the
tails of the histograms corresponded to a small number of “rogue solitons” (RS) which had
experienced extremely large Raman frequency shift such that their central wavelength was
shifted completely within the filter transmission band (i.e. they appeared as high intensity
events in the time series). More recently it has become possible to measure the full-
bandwidth shot-to-shot SC spectra [62], and these experiments [see Fig. 3(b)] have
highlighted more directly the small number of extremely red-shifted RSs.
Subsequent modelling and experiments examined in detail the dynamics leading to
such extreme frequency shifts, clarifying the central role of soliton collisions [8,12,63-68].
Figure 3(c) shows numerical simulations highlighting the frequency and time-domain
properties of a particular rogue soliton event [69]. In the frequency domain, we see the
transition from initial MI dynamics with symmetric growth of noise-driven sidebands to a
regime where distinct sech-soliton structures appear in the spectrum. A rogue soliton
emerges at a distance of z ~ 9 m, and the snapshot of the time-domain evolution plot clearly
illustrates a two-soliton collision at this point. The collision is associated with significant
energy exchange (mediated by stimulated Raman scattering) that yields one higher energy
soliton (which experiences a much greater Raman self-frequency shift) as well as a lower
amplitude residual pulse [12,63,69,70].
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Note that although stimulated Raman scattering plays a central role in a fibre context,
any perturbation that breaks NLSE integrability can cause a homogeneous initial state to self-
organize into a large-scale, coherent rogue soliton as a result of multiple interactions with
other solitons and dispersive waves [71]. Indeed, numerical studies have shown that higher-
order dispersive perturbations alone can give rise to rogue solitons [59,68] provided that the
incoherence in the system is not too large [48,72]. Although it may seem surprising, energy
exchange between colliding solitons can occur even in this case, owing to resonant coupling
between the solitons and radiation emitted during the collision [73].
Other numerical studies have investigated the statistical properties of SC rogue
solitons in more detail. Several authors have considered the variation in local intensity along
the propagation dimension, showing that the intensity of colliding solitons at the point of
collision can in fact be much higher than that of the rogue soliton at the fibre output
[12,63,67]. Figure 3(d) shows this for the case of continuous wave SC generation [67],
where it is clear that the maximum intensity at certain points in the fibre is much higher than
at the output. This suggests that significant differences may be observed between the
statistical properties measured over the full field at all points of propagation and those
measured at the fibre output. A detailed study of these differences (including a discussion of
the effect of spectral filtering on the statistics) has been reported [12]. We also note recent
experimental work using dispersive Fourier transformation to examine the intensity
correlation properties of both MI and SC to yield further insights into the underlying
dynamics [61,62,74]. Finally, we remark that although most studies in optics have focused
on perturbation-induced collisions as a primary generating mechanism for extreme-frequency
shifting rogue solitons, the random emergence of coherent structures has been seen in
numerical simulations of a basic NLSE model during the evolution of multimode CW fields
with initial random phases [75].
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FIGURE 3 | Selection of experimental and numerical results on supercontinuum rogue solitons.
a, Experimental setup for the first optical rogue wave soliton measurements, and the long-wavelength
intensity statistics at three different pump power levels [3]. b, One thousand single-shot SC spectra
measured using the dispersive Fourier transformation (gray); the computed mean (black); and the
average spectrum measured with an optical spectrum analyser (OSA, red) [62]. c, Numerical
simulations showing the spectral and temporal evolution of a rogue soliton in picosecond SC
generation. Parameters from [69]. A rogue soliton emerges from the main spectrum at a propagation
distance of 8.8 m. In the time-domain two solitons collide at the same distance, and it is the energy
transfer to one of the soliton that yields the enhanced redshift. d, Evolution of peak intensity in
numerical simulations of continuous wave SC generation. The most intense events correspond to
collisions of solitons, with the time-frequency diagram shown in the inset illustrating a particular
collision example [67].
17
CONTROLLING ROGUE WAVES IN FIBRE SYSTEMS
The fact that SC rogue solitons have their origin in MI suggests the possibility to
control their propagation using a dual-frequency input field such that the instability develops
from a coherent modulation on the input envelope rather than from noise [32]. This approach
is similar to that used to excite SFBs under controlled conditions.
The potential of seeding to stabilize rogue wave dynamics was first demonstrated
experimentally by controlling picosecond SC generation with a frequency-shifted replica
derived from the pump pulses [76]. Although the seed pulses had only 0.01% of the pump
intensity, there was dramatic improvement in the SC stability. Related numerical studies
showed how an appropriate choice of seed frequency could significantly decrease the rogue
soliton wavelength jitter at the same time as increasing the overall SC bandwidth [77]. These
studies showed the potential of rogue soliton control in improving the performance of
practical SC sources [8,77-79]. They also stimulated ideas to enhance spectral broadening in
silicon waveguides [80]. Other experiments considered how seeding can improve the spectral
properties of spontaneous MI, and remarkable spectral control has been demonstrated using
CW seeding at only the 10-6 level [81]. A benefit of this technique is that it does not rely on
the time-delay tuning required for picosecond seed pulses [82,83].
As well as controlling input conditions, other studies have shown that longitudinal
variation in fibre dispersive and nonlinear properties can modify intensity fluctuations in both
supercontinuum generation [84], and the dynamics of an evolving AB [85]. These studies are
significant in that they show how a modified fibre propagation environment can mimic the
way in which ocean topography influences water wave propagation [1].
18
AMPLIFIERS AND LASERS
In addition to the examples in conservative (or weakly dissipative) systems discussed
above, rogue waves in systems with strong dissipation (gain or loss) have also been observed.
Of course, dissipative dynamics in optics have been studied for decades, and resonators,
amplifiers, and multimode lasers are well-known to exhibit a wide range of chaotic and self-
organization effects [86]. However, the studies of optical rogue waves have now motivated
the interpretation of these noise characteristics in terms of extreme value processes.
Although dissipative systems do not generally have NLSE-governed hydrodynamic
counterparts, exploring regimes of long-tailed statistics in such systems has provided new
insights into the underlying physical processes. The first observation of extreme events in a
highly dissipative system reported long-tailed intensity statistics from Raman amplification
of a coherent signal using an incoherent Raman pump [87]. The long-tailed statistics were
attributed to the transfer of pump intensity fluctuations onto the coherent signal due to the
exponential dependence of Raman gain on pump intensity. Similar nonlinear noise transfer
underlies the emergence of extreme-value statistics in fibre parametric amplifiers [88], and
silicon waveguide Raman amplifiers [89,90]. Reference [89] is notable for explicitly
calculating the power-law probability distribution function of the amplified signal intensity.
The complex noise characteristics of lasers in systems with external injection, mode-
locking, and delayed feedback is well known, and it is not surprising that laser noise spiking
behaviour can exhibit long-tailed statistics. Experiments have reported such properties in an
Erbium fibre laser with harmonic pump modulation [91], a CW Raman fibre laser [92],
mode-locked Ti:Sapphire and fibre lasers [14,93,94] as well as passively-Q switched lasers
[95]. In Fig. 4 we illustrate an experimental result, where a sequence of highly localized
temporal noise bursts was recorded from an Erbium-doped mode-locked fibre laser; the
19
associated intensity statistics showed significant deviation from exponentially-bounded
distributions [14]. These experiments confirmed previous numerical studies predicting
intensity spikes in passively mode-locked fibre lasers through chaotic pulse bunching [13,96].
Rogue wave behavior has also been seen from an optically-injected semiconductor laser,
where it was shown that the rogue wave dynamics could be described deterministically, with
noise influencing only the probability of their observation [97] (similar to noise-seeded MI.)
These results are inspiring significant theoretical efforts to identify the mechanisms
that induce the extreme temporal localization [13,96] and to experimentally characterize the
instabilities in real-time [14,94,98]. However, it should be noted that many of the reported
features of extreme laser fluctuations had likely been seen in earlier experiments without
being recognized as a separate class of rogue wave instability [99,100].
20
FIGURE 4 | Dissipative rogue waves. a, Schematic of an Er-doped mode-locked fibre laser
generating a train of temporally localised bursts of noise. b, Time-domain experimental measurement
showing an example of a chaotic pulse cluster emitted by the laser over a single roundtrip. c, Peak
intensity statistics over nearly 10 million roundtrips. The red line shows the significant peak intensity,
while the dashed yellow line corresponds to an exponentially-bounded distribution. Adapted from
[14].
21
SPATIAL INSTABILITIES
There has also been extensive interest interpreting spatial instabilities in terms of
extreme events. The first such study considered intensity noise in the output spatial mode of
a cavity with nonlinear gain from an optically-pumped liquid-crystal light valve [101].
Depending on the system parameters, the cavity exhibited complex transverse mode
dynamics which, in a high-finesse limit, showed highly unstable oscillation behavior with
long-tailed statistics. The optical rogue wave-like events were attributed to the nonlinear
feedback and symmetry-breaking in the cavity design.
Another example of a spatially-extended system exhibiting rogue wave statistics is
optical filamentation. Filamentation is a complex process involving self-focussing, plasma
formation and temporal shaping dynamics and, depending on the particular parameter regime
investigated, different statistical behaviour has been reported. In the single filament regime,
shot-to-shot spectral fluctuations arising from pump noise transfer through self-phase
modulation have shown long-tailed statistics [9,102]. In the multi-filament regime, where the
transverse profile of the input beam is broken into multiple strands through spatial MI,
localized structures obeying non-Gaussian statistics have been predicted [103]. Experiments
and simulations studying filamentation in gas have revealed that local refractive index
variations driven by nonlinearity can induce merging of individual filament strings, giving
rise to short-lived spatial rogue waves at the gas cell output [104]. Figure 5(a) shows the
experimental setup in this work, with numerical and experimental results shown in Figs. 5(b)
and (c), respectively.
Interestingly, long-tailed statistics have also been seen in linear spatial systems. In
one experiment studying emission from a GHz microwave emitter array, it was possible to
probe the emitted electromagnetic field as a function of time and spatial position across the
array [105]. Although the system contained no (obvious) nonlinear element, long-tailed
22
distributions in the microwave intensity were observed. Another example of a linear system
exhibiting rogue wave statistics in the transverse spatial plane (but here at visible
frequencies) was found to be the speckle pattern observed at the output of a strongly
multimode fibre [106]. In this system the asymmetry and inhomogeneity in the injected beam
profile [see Fig. 5(d-f)] yielded a sub-exponential intensity distribution, driving the
emergence of rare high-intensity spots in the speckle pattern.
FIGURE 5 | Rogue waves in the transverse spatial plane of optical beams. a, Experimental setup
used to study multiple filamentation in a gas cell and an example of observed fluence statistics. b,
Results of numerical simulations, showing how highly localised rogue structures can arise from the
merging of individual filaments. c, Two-dimensional multifilament fluence profiles experimentally
measured with a high-speed camera reveals a spatio-temporal rogue wave appearing 4.89 s after the
start of the recording. d, Experimental setup used to investigate rogue waves in the speckle pattern at
the output of a multimode fibre. A spatial light modulator (SLM) is used to control the input beam
profile. e,f Experimentally measured speckle patterns (centre) and corresponding intensity
distributions taken at a selected y-coordinate (right) when the SLM transmission mask (left) is
uniform (e) and inhomogeneous (f). An optical rogue wave is observed in f. a-c adapted from [104]
and d-f adapted from [106].
23
DISCUSSION AND OUTLOOK
After the initial results of Solli et al. in 2007, the science of extreme events is now
firmly embedded within the domain of optics, and we have seen how very different optical
systems can exhibit strong localization and long-tailed statistics. However, a major
conclusion of this review is that the mechanisms driving the emergence of rogue wave
behaviour in optics can be very different depending on the particular system studied, and we
hope that the categorisation we have provided here will assist in structuring future work in
this field. We also remark that great care must be taken when comparing results obtained in
different contexts. The particular example of optical fibre propagation is a case in point. The
physics driving the excitation of analytical “rogue wave” breather solutions in a regime of
propagation governed by the basic NLSE is significantly different from the perturbed-NLSE
dynamics of extreme red-shifting rogue solitons in supercontinuum generation, and it is
essential to stress this distinction.
Although there is clearly much intrinsic interest in studying extreme instabilities in
optics, much of the motivation to study rogue waves in optical systems has been to gain
insight into the origin of their oceanic counterparts. In this regard, however, it is also
essential to recognize that not all experiments in optics directly yield insight into ocean wave
propagation. There are certain regimes of wave propagation on the ocean and in optical fibre
that are both well-described by a basic NLSE model and, provided experiments are
performed in these regimes before the onset of any perturbations, insights obtained in the
different environments can be shared. In other cases, whilst the observation of long-tailed
statistics in optics may be linked to the wider theory of extreme events in physics, it is simply
incorrect to compare such instabilities with oceanic wave shaping processes. Analogies can
be powerful tools in physics, but they must be used with care [107].
24
That said, in regimes where the optical-ocean analogy is valid, there is of course
intense interest to use experiments in optics to improve understanding of rogue waves in
general, and experiments in optics have indeed motivated similar studies in water wave tanks
(Box 1). Although rogue waves on the ocean may arise in a number of different ways, these
experiments provide convincing evidence that nonlinearity in the ocean can play a role in
extreme wave emergence and must be included a priori in any consideration of potential high
amplitude ocean wave shaping mechanisms. A particular advantage of optical systems is the
fact that the high repetition rate of optical sources allows the generation of large data sets
such that even events occurring with extremely low probabilities can be studied [108].
Even in a strictly optical context, there remain many open directions of research.
Experiments and modelling of optical rogue wave dynamics are providing new insights into
how noise drives (and/or stabilises) the dynamics of nonlinear optical systems, how energy
exchange occurs during soliton interactions, and how novel measurement techniques can
reveal noise correlations even in the very complex supercontinuum. It is interesting to remark
in closing that there is also significant effort in optics to understand how effects like
dispersion and nonlinearity engineering can allow optical systems to model different aspects
of the ocean environment [109]. Indeed, the recent development of “topographic” fibres have
already been shown to be able to introduce control of both optical modulation instability and
soliton dynamics [110,111]. As research continues to progress in this field, it is likely that the
noise properties in a wide range of optical systems may find analogies with areas of physics
other than oceanography, allowing the use of a convenient optical testbed with which to study
a wide range of different physical processes.
25
Acknowledgements
JMD and FD acknowledge the European Research Council Advanced Grant ERC-2011-
AdG-290562 MULTIWAVE. ME acknowledges the Marsden Fund of the Royal Society of
New Zealand. GG acknowledges the Academy of Finland Grants 130 099 and 132 279.
Competing Interests Statement
The authors declare no competing financial interests.
Correspondence
Correspondence and requests for materials should be addressed to G.G. (email: go-
ery.genty@tut.fi).
26
BOX 1 | THE OPTICAL FIBRE-OCEAN WAVE ANALOGY
The analogy between the dynamics of ocean waves and pulse propagation in optics
arises from the central role of the NLSE in both systems. Figure B1 gives the governing
equations and illustrates characteristic soliton solutions for both cases. In optics, the NLSE
describes the evolution of a light pulse envelope modulating an electric field whilst for deep
water, it describes the evolution of a group envelope modulating surface waves. It is
important not to over-simplify or exaggerate this analogy. For example, the deep water
NLSE in oceanography does not describe the shape of individual wave cycles but only their
modulating envelope. Thus, specific envelope solutions of the deep water NLSE cannot be
considered physically as individual “rogue waves”; within the narrowband approximation of
the NLSE, there will always be multiple surface waves underneath this envelope.
The recent work studying rogue waves in optics in the MI and breather regime has
motivated similar water wave experiments [112,113], even to the extent of testing the
resistance of scale models of maritime vessels to particular NLSE breather solutions [114].
Interestingly, higher-order effects described by an extended NLSE can also be present in
deep water [115-117], and wave tank experiments have even shown a form of hydrodynamic
supercontinuum and soliton fission [118]. Note, however, that while a clear analogy between
ocean wave and optical propagation exists in the unperturbed NLSE regime, there is no such
rigorous analogy for the extended NLSE since the physical forms of the higher-order
perturbations are very different.
27
Figure B1 | The NLSE describes evolution in a frame of reference moving at the group velocity of: (a) wave
group envelopes u on deep water; (b) light pulse envelopes A in optical fibre with anomalous GVD. The figures
illustrate solitons on finite background (top) and solitons on zero background (bottom). For the ocean wave
case, there is always deep water underneath u(z,t). For the water wave NLSE, k0 is the wavenumber [m-1],
ω
0 is
the carrier frequency [rad s-1]; for the fibre NLSE,
β
2 < 0 is the GVD [ps2 m-1],
γ
is the nonlinear coefficient in
[W-1 m-1]. A water wave NLSE with time and space interchanged is also encountered but in this case the
coefficients need to be adapted [2].
28
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