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LETTERS
Optical rogue waves
D. R. Solli
1
, C. Ropers
1,2
, P. Koonath
1
& B. Jalali
1
Recent observations show that the probability of encountering an
extremely large rogue wave in the open ocean is much larger than
expected from ordinary wave-amplitude statistics
1–3
. Although
considerable effort has been directed towards understanding the
physics behind these mysterious and potentially destructive events,
the complete picture remains uncertain. Furthermore, rogue waves
have not yet been observed in other physical systems. Here, we
introduce the concept of optical rogue waves, a counterpart of
the infamous rare water waves. Using a new real-time detection
technique, we study a system that exposes extremely steep, large
waves as rare outcomes from an almost identically prepared initial
population of waves. Specifically, we report the observation of
rogue waves in an optical system, based on a microstructured
optical fibre, near the threshold of soliton-fission supercontinuum
generation
4,5
—a noise-sensitive
5–7
nonlinear process in which
extremely broadband radiation is generated from a narrowband
input
8
. We model the generation of these rogue waves using the
generalized nonlinear Schro
¨
dinger equation
9
and demonstrate that
they arise infrequently from initially smooth pulses owing to power
transfer seeded by a small noise perturbation.
For centuries, seafarers have told tales of giant waves that can
appear without warning on the high seas. These mountainous waves
were said to be capable of destroying a vessel or swallowing it beneath
the surface, and then disappearing without the slightest trace. Until
recently, these tales were thought to be mythical. In the mid-1990s,
however, freak waves proved very real when recorded for the first
time by scientific measurements during an encounter at the
Draupner oil platform in the North Sea
3
. Although they are elusive
and intrinsically difficult to monitor because of their fleeting exis-
tences, satellite surveillance has confirmed that rogue waves roam the
open oceans, occasionally encountering a ship or sea platform, some-
times with devastating results
1
. It is now believed that a number of
infamous maritime disasters were caused by such encounters
10
.
The unusual statistics of rogue waves represent one of their defin-
ing characteristics. Conventional models of ocean waves indicate that
the probability of observing large waves should diminish extremely
rapidly with wave height, suggesting that the likelihood of observing
even a single freak wave in hundreds of years should be essentially
non-existent. In reality, however, ocean waves appear to follow ‘L-
shaped’ statistics: most waves have small amplitudes, but extreme
outliers also occur much more frequently than expected in ordinary
(for example, gaussian or Rayleigh) wave statistics
11–13
.
It is likely that more than one process can produce occasional
extreme waves with small but non-negligible probability
14,15
.
Possible mechanisms that have been suggested to explain oceanic
rogue waves include effects such as nonlinear focusing via modu-
lation instability in one dimension
16,17
and in two-dimensional
crossings
18,19
, nonlinear spectral instability
20
, focusing with caustic
currents
21
and anomalous wind excitation
12
. Nonlinear mechanisms
have attracted particular attention because they possess the requisite
extreme sensitivity to initial conditions.
Although the physics behind rogue waves is still under investiga-
tion, observations indicate that they have unusually steep, solitary or
tightly grouped profiles, which appear like ‘‘walls of water’’
10
. These
features imply that rogue waves have relatively broadband frequency
content compared with normal waves, and also suggest a possible
connection with solitons—solitary waves, first observed by J. S.
Russell in the nineteenth century, that propagate without spreading
in water because of a balance between dispersion and nonlinearity. As
rogue waves are exceedingly difficult to study directly, the relation-
ship between rogue waves and solitons has not yet been definitively
established, but it is believed that they are connected.
So far, the study of rogue waves in the scientific literature has
focused on hydrodynamic studies and experiments. Intriguingly,
there are other physical systems that possess similar nonlinear char-
acteristics and may also support rogue waves. Here we report the
observation and numerical modelling of optical rogue waves in a
system based on probabilistic supercontinuum generation in a highly
nonlinear microstructured optical fibre. We coin the term ‘optical
rogue waves’ based on striking phenomenological and physical simi-
larities between the extreme events of this optical system and oceanic
rogue waves.
Supercontinuum generation has received a great deal of attention in
recent years for its complex physics and wealth of potential applica-
tions
5,8
. An extremely broadband supercontinuum source can be cre-
ated by launching intense seed pulses into a nonlinear fibre at or near
its zero-dispersion wavelength
22
. In this situation, supercontinuum
production involves generation of high-order solitons—the optical
counterparts of Russell’s solitary water waves—which fission into red-
shifted solitonic and blueshifted non-solitonic components at differ-
ent frequencies
4,5
. The solitonic pulses shift further towards the red as
they propagate through the nonlinear medium because of the Raman-
induced self-frequency shift
9
. Interestingly, frequency downshifting
effects are also known to occur in water wave propagation. It has been
noted that the aforementioned Raman self-frequency shift represents
an analogous effect in optics
23
. The nonlinear processes responsible for
supercontinuum generation amplify the noise present in the initial
laser pulse
6,7
. Especially for long pulses and continuous-wave input
radiation, modulation instability—an incoherent nonlinear wave-
mixing process—broadens the spectrum from seed noise in the initial
stages of propagation and, as a result, the output spectrum is highly
sensitive to the initial conditions
24,25
.
A critical challenge in observing optical rogue waves is the lack of
real-time instruments that can capture a large number of very short
random events in a single shot. To solve this problem, we use a
wavelength-to-time transformation technique inspired by the con-
cept of photonic time-stretch analog-to-digital conversion
26
. In the
present technique, group-velocity dispersion (GVD) is used to
stretch the waves temporally so that many thousands of random
ultra-short events can be captured in real time. A different single-
shot technique has been used to study isolated supercontinuum
pulses
27
; however, the real-time capture of a large number of random
1
Department of Electrical Engineering, University of California, Los Angeles 90095, USA.
2
Max Born Institute for Nonlinear Optics and Short Pulse Spectroscopy, D-12489 Berlin,
Germany.
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events has not been reported. Using the present method, a small but
statistically significant fraction of extreme waves can be discerned
from a large number of ordinary events, permitting the first obser-
vation of optical rogue waves.
The supercontinuum radiation used in our experiments is gener-
ated by sending picosecond seed pulses at 1,064 nm through a length
of highly nonlinear microstructured optical fibre with matched zero-
dispersion wavelength. The output is red-pass filtered at 1,450 nm
and stretched as described above so that many thousands of events
can be captured with high resolution in a single-shot measurement. A
schematic of the experimental apparatus is displayed in Fig. 1 and
additional details are provided in the Methods Summary.
Using this setup, we acquire large sets of pulses in real time for very
low seed pulse power levels—power levels below the threshold
required to produce appreciable supercontinuum. We find that the
pulse-height distributions are sharply peaked with a well-defined
mean, but contrary to expectation, rare events with far greater inten-
sities also appear. In Fig. 1, we show representative single-shot time
traces and histograms for three different low power levels. In these
traces, the vast majority of events are concentrated in a small number
of bins and are so weak that they are buried beneath the noise floor of
the measurement process; however, the most extreme ones reach
intensities at least 30–40 times the average. The histograms display
a clear L-shaped profile, with extreme events occurring rarely, yet
much more frequently than expected based on the relatively narrow
distribution of typical events.
Because the red-pass filter transmits only a spectral region that is
nearly dark in the vast majority of events, the rare events clearly have
extremely broadband, frequency-downshifted spectral content. The
data also show that the frequency of occurrence of the rogue events
increases with the average power, but the maximum height of a freak
pulse remains relatively constant. These features indicate that the
extreme events are sporadic, single solitons.
The nonlinear Schro
¨
dinger equation (NLSE) models soliton
dynamics and has also been used to study hydrodynamic rogue waves
generated by nonlinear energy transfer in the open ocean
16–19
. As the
NLSE also describes optical pulse propagation in nonlinear media, it
is certainly plausible that this equation could predict optical rogue
waves. We investigate this numerically using the generalized NLSE
(neglecting absorption), which is widely used for broadband optical
pulse propagation in nonlinear fibres
9
. The generalized NLSE incor-
porates dispersion and the Kerr nonlinearity, as well as approxima-
tions for self-steepening and the vibrational Raman response of the
medium. This equation has been successfully used to model super-
continuum generation in the presence of noise
28,29
and, as we demon-
strate here, is capable of qualitatively explaining our experimental
results. In anticipation of broadband application, we include several
higher orders of dispersion in the nonlinear fibre, which we calcu-
lated from the manufacturer’s test data (see Methods). Similarly,
higher-order dispersion has also been used to extend the validity of
the NLSE for broadband calculations in hydrodynamics
30
.
As expected, the present model shows that a high-power, smooth
input pulse ejects multiple redshifted solitons and blueshifted non-
solitonic components, and a tiny amount of input noise varies their
spectral content
5,25
. On the other hand, for low power levels, the
spectral content of the pulse broadens, but no sharp soliton is shed.
In this case, the situation changes markedly when a tiny amount of
noise is added. This perturbation is amplified by nonlinear inter-
actions including modulation instability, which dramatically
lowers the soliton-fission threshold and permits unpredictable freak
events to develop. Interestingly, the hydrodynamic equivalent—the
Benjamin–Feir modulation instability—is also thought to initiate
hydrodynamic rogue waves
16–19
. This instability spreads spectral con-
tent from a narrow bandwidth to a broader range in the initial stages
of water wave propagation, just as it does in this optical system.
We include a stochastic perturbation in our simulations by adding
to the initial pulse envelope a small amount of bandwidth-restricted
random noise with amplitude proportional to the instantaneous field
strength. We then solve the NLSE repeatedly for a large number of
independent events. For a small fraction of events, the spectrum
becomes exceptionally broad with a clear redshifted solitonic shoulder.
Figure 2a shows the time trace and histogram of peak heights for a
trial of 1,000 events after red-pass filtering each output pulse at
the start of the solitonic shoulder illustrated in Fig. 2b. Clearly, the
histogram of heights is sharply peaked but has extended tails, as
observed in the experiment, and the distribution contains rogue
events more than 50 times as large as the mean. The same rogue
events are identified regardless of where the filter is located within
the smooth solitonic shoulder and can also be identified from the
complementary non-solitonic blue side of the spectrum.
The rogue pulses have exceptionally steep leading and trailing
edges compared with the initial pulses and the typical events, as
shown in Fig. 2c. The wide bandwidth and abrupt temporal profile
of an optical rogue wave is also highlighted in Fig. 3 where the power
is displayed as a function of both wavelength and time using a short-
time Fourier transform. Because there are no apparent features in the
perturbations that lead to the development of the rogue events, their
appearance seems unpredictable.
To pinpoint the underlying feature of the noise that produces
rogue waves, we have closely analysed the temporal and spectral
properties of the initial conditions. Examining the correlations
Mode-locked
laser
Yb-doped
fibre amplifier
Nonlinear fibre
Filter
–1,300 ps nm
–1
GVD
Photodetector
Real-time oscilloscope
Supercontinuum
~ 15,000 events
b
c
d
a
Time
Intensity
Number of events
20
10
0
0 100 200 300 0 100 200 300 0 100 200 300
Intensity bins (arbitrary units)
Figure 1
|
Experimental observation of optical rogue waves. a, Schematic of
experimental apparatus.
b–d, Single-shot time traces containing roughly
15,000 pulses each and associated histograms (bottom of figure: left,
b; middle, c; right, d) for average power levels 0.8 mW (red), 3.2 mW (blue)
and 12.8 mW (green), respectively. The grey shaded area in each histogram
demarcates the noise floor of the measurement process. In each
measurement, the vast majority of events (.99.5% for the lowest power) are
buried in this low intensity range, and the rogue events reach intensities of at
least 30–40 times the average value. These distributions are very different
from those encountered in most stochastic processes.
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between the initial conditions and their respective output waveforms,
we find that if the random noise happens to contain energy with a
frequency shift of about 8 THz within a 0.5-ps window centred about
1.4 ps before the pulse peak (Fig. 2c), a rogue wave is born. Noise at
this particular frequency shift and on a leading portion of the pulse
envelope efficiently seeds modulation instability, reshaping the pulse
to hasten its breakup. The output wave height correlates in a highly
nonlinear way with this specific aspect of the initial conditions. Thus,
the normal statistics of the input noise are transformed into an
extremely skewed, L-shaped distribution of output wave heights.
Further study is needed to explain precisely why the pulse is so highly
sensitive to these particular noise parameters. Nevertheless, the spe-
cific feature we have identified in the initial conditions offers some
predictive power for optical rogue waves, and may offer clues to the
oceanic phenomenon.
The rogue waves have a number of other intriguing properties
warranting further study. For example, they propagate without
noticeable broadening for some time, but have a finite, seemingly
unpredictable lifetime before they suddenly collapse owing to cumu-
lative effects of Raman scattering. This scattering seeded by noise
dissipates energy or otherwise perturbs the soliton pulse beyond
the critical threshold for its survival
9
. The decay parallels the unpre-
dictable lifetimes of oceanic rogue waves. The rogue optical solitons
are also able to absorb energy from other wavepackets they pass
through, which causes them to grow in amplitude, but appears to
reduce their lifetime. A similar effect may help to explain the develop-
ment of especially large rogues in the ocean.
In conclusion, we have observed extreme soliton-like pulses that
are the optical equivalent of oceanic rogue waves. These rare optical
events possess the hallmark phenomenological features of oceanic
rogue waves—they are extremely large and seemingly unpredictable,
follow unusual L-shaped statistics, occur in a nonlinear medium, and
are broadband and temporally steep compared with typical events.
On a physical level, the similarities also abound, with modulation
instability, solitons, frequency downshifting and higher-order dis-
persion as striking points of connection. Intriguingly, the rogue
waves of both systems can be modelled with the nonlinear
Schro
¨
dinger equation. Although the parameters that characterize this
optical system are of course very different from those describing
waves on the open ocean, the rogue waves generated in the two cases
bear some remarkable similarities.
METHODS SUMMARY
Our supercontinuum source consists of a master oscillator, a fibre amplifier, and
a 15-m length of highly nonlinear microstructured fibre whose zero-dispersion
point matches the seed wavelength. The master oscillator is a mode-locked
ytterbium-doped fibre laser producing picosecond pulses at about 1,064 nm with
a repetition rate of 20 MHz. The output pulses are amplified to a desired level in a
large-mode-area ytterbium-doped-fibre amplifier. This amplification process
yields chirped pulses of ,5-nm bandwidth and a few picoseconds temporal
width.
The wavelength-to-time transformation for real-time detection is accomp-
lished using a highly dispersive optical fibre producing about 21,300 ps nm
21
of
GVD over the wavelength range of interest. Because the supercontinuum output
is red-pass filtered with a cut-on wavelength of 1,450 nm, adjacent pulses do not
overlap in time after being stretched. The GVD-stretched signal is then fed to a
fast photodetector and captured by a real-time 20-gigasample-per-second oscil-
loscope, which records sequences of ,15,000 pulses with high temporal resolu-
tion in a single-shot measurement.
The detection of rogue events is insensitive to the filter window, so the specific
choice of the red-pass cut-on wavelength is not critical. The soliton shoulder
shown in Fig. 2b is smooth and extends to very long wavelengths, so a freak
soliton can be detected by examining any section of this extended region.
Because of experimental constraints, we limit the measurements to the long-
wavelength tail of the soliton shoulder, whereas, in the simulations, it is instruc-
tive to capture the entire soliton spectrum. The simulations show that it is
acceptable to detect the rogue events experimentally by their red tails because
the same rogue events are identified no matter where the filter is located through-
out this spectral region.
Full Methods and any associated references are available in the online version of
the paper at www.nature.com/nature.
Received 22 February; accepted 11 October 2007.
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W
a
v
e
le
n
gth (nm
)
Time (ps)
900
1,000
1,100
1,200
1,300
0
2
4
6
Figure 3
|
Time
–
wavelength profile of an optical rogue wave obtained from
a short-time Fourier transform.
The optical wave has broad bandwidth and
has extremely steep slopes in the time domain compared with the typical
events. It appears as a ‘wall of light’ analogous to the ‘wall of water’
description of oceanic rogue waves. The rogue wave travels a curved path in
time–wavelength space because of the Raman self-frequency shift and group
velocity dispersion, separating from non-solitonic fragments and remnants
of the seed pulse at shorter wavelengths. The grey traces show the full time
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20
10
0
0
500
1,000
1,500
a
Power
1,000 events
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0
–20
–40
–60
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1,000 1,100 1,200 1,300
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b
c
Time (ps)
0
2
4
6
Number of events
Relative spectral
density (dB nm
–1
)
Power bins (W)
8
Figure 2
|
Simulation of optical rogue waves using the generalized
nonlinear Schro
¨
dinger equation. a
, The time trace and histogram of 1,000
events with red-pass filtering from 1,155 nm. The initial (seed) pulses have
width 3 ps, peak power 150 W, fractional noise 0.1%, and noise bandwidth
50 THz. The vertical axis of the histogram contains a scale break to make it
easier to see the disparity between the most common events at low peak
power and the rogue events at high peak power.
b, The complete relative
spectral densities of the initial pulse (black line), a typical event (grey line)
and the rare event shown in
c (red line). c, The markedly different temporal
profiles of the seed pulse and the rare event indicated in the histogram. The
typical events from the histogram are so tiny that they are not visible on this
linear power scale. The shaded blue region on the seed pulse delineates the
time window that is highly sensitive to perturbation.
LETTERS NATURE
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Author Information Reprints and permissions information is available at
www.nature.com/reprints. Correspondence and requests for materials should be
addressed to D.R.S. (solli@ucla.edu).
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METHODS
Our simulations are based on the nonlinear Schro
¨
dinger equation (NLSE),
which governs the propagation of optical pulses and has been widely used to
model supercontinuum generation in optical fibres. The equation describes the
evolution of the slowly varying electric field envelope, A(z,t), in the presence of
temporal dispersion and nonlinearity. In its generalized form, the NLSE
accounts for dispersion as well as both the electronic (instantaneous) and vibra-
tional (delayed) nonlinearities in silica glass. For many applications, it is suf-
ficient to use approximations for these nonlinearities that are physically intuitive
and efficient for numerical computations using the well-known split-step
method. Relative to a reference frame co-moving with the optical pulse, this
form of the equation can be expressed as
LA
Lz
{i
X
m~2
i
m
b
m
m!
L
m
A
Lt
m
~ic Ajj
2
Az
i
v
0
L
Lt
Ajj
2
A
{T
R
A
L A
jj
2
Lt
where b
m
are values that characterize the fibre dispersion, c is the nonlinear
coefficient of the fibre, v
0
is the central carrier frequency of the field, and T
R
is a parameter that characterizes the delayed nonlinear response of silica fibre
9
.
The bracketed terms on the right-hand side of the equation describe the Kerr
nonlinearity, self-steepening and the vibrational Raman response of the med-
ium, respectively. For completeness, we include the self-steepening term in our
simulations, but we have found that it is not required for rogue wave generation.
The Kerr term produces self-phase modulation, and the Raman term causes
frequency downshifting of the carrier wave.
This form of the NLSE has been successfully used in the literature to model
supercontinuum generation in the presence of noise and is capable of qualita-
tively explaining our experimental results. In our calculations, we include dis-
persion up to sixth order, which we calculated from the manufacturer’s test data
(see Crystal Fibre NL-5.0-1065 for fibre specifications). Operating at the zero
dispersion wavelength of the fibre, we use the dispersion parameters b
2
< 0,
b
3
< 7.67 3 10
–5
ps
3
m
21
, b
4
< 21.37 3 10
–7
ps
4
m
21
, b
5
< 3.61 3 10
–10
ps
5
m
21
,
and b
6
< 5.06 3 10
–13
ps
3
m
21
. The nonlinear coefficient and the Raman res-
ponse parameter are given by c 5 11 W
21
km
21
and T
R
5 5 fs. These numbers
model the experimental situation, but we find that the NLSE can also produce
rogue wave solutions with other values of the parameters.
To generate rogue waves, we perturb the input pulse by adding a very small
amount of amplitude noise directly to its temporal envelope. Specifically, at each
point in time, a small random number is added to the input field envelope. The
noise amplitude at each point is proportional to the instantaneous amplitude of
the pulse. The peak power of the unperturbed pulse is chosen to be small enough
that the pulse will not break up without the noise perturbation. We then apply a
frequency bandpass filter to limit the input noise to a relatively narrow band-
width around the seed wavelength, sufficient to mimic the optical noise band-
width of the input field in the experiment. The specific noise amplitude and peak
power of the pulse are not critical, but influence the rogue wave generation rate.
Noise amplitudes of the order of 0.1% of the pulse amplitude or even signifi-
cantly less are adequate to observe a rare, but reasonable generation rate.
Although we include noise in this particular way, this specific form is not
required to create rogue waves—other perturbations produce similar results.
This particular form of noise serves as a conceptually simple perturbation that
qualitatively accounts for our experimental results. When we solve the NLSE
repeatedly given these conditions, rogue waves are produced as statistically rare
events from members of an initial population that are nearly indistinguishable.
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