of 31
applied
sciences
Review
Soliton Molecules and Multisoliton States in Ultrafast
Fibre Lasers: Intrinsic Complexes in
Dissipative Systems
Lili Gui
1,
*, Pan Wang
2
, Yihang Ding
2
, Kangjun Zhao
2
, Chengying Bao
3
, Xiaosheng Xiao
2
and Changxi Yang
2,
*
1
4th Physics Institute and Research Center SCoPE, University of Stuttgart, Pfaffenwaldring 57,
70569 Stuttgart, Germany
2
State Key Laboratory of Precision Measurement Technology and Instruments, Department of Precision
Instruments, Tsinghua University, Beijing 100084, China; p-wang14@mails.tsinghua.edu.cn (P.W.);
dingyh17@mails.tsinghua.edu.cn (Y.D.); zhaokj16@mails.tsinghua.edu.cn (K.Z.);
xsxiao@tsinghua.edu.cn (X.X.)
3
Watson Laboratory of Applied Physics, California Institute of Technology, Pasadena, CA 91125, USA;
cbao@caltech.edu
*
Correspondence: l.gui@pi4.uni-stuttgart.de (L.G.); cxyang@tsinghua.edu.cn (C.Y.);
Tel.: +86-10-6277-2824 (C.Y.)
Received: 14 December 2017; Accepted: 23 January 2018; Published: 29 January 2018
Featured Application: High-capacity telecommunication, advanced time-resolved spectroscopy,
self-organization and chaos in dissipative systems.
Abstract:
Benefiting from ultrafast temporal resolution, broadband spectral bandwidth, as well as
high peak power, passively mode-locked fibre lasers have attracted growing interest and exhibited
great potential from fundamental sciences to industrial and military applications. As a nonlinear
system containing complex interactions from gain, loss, nonlinearity, dispersion, etc., ultrafast fibre
lasers deliver not only conventional single soliton but also soliton bunching with different types.
In analogy to molecules consisting of several atoms in chemistry, soliton molecules (in other words,
bound solitons) in fibre lasers are of vital importance for in-depth understanding of the nonlinear
interaction mechanism and further exploration for high-capacity fibre-optic communications. In this
Review, we summarize the state-of-the-art advances on soliton molecules in ultrafast fibre lasers.
A variety of soliton molecules with different numbers of soliton, phase-differences and pulse
separations were experimentally observed owing to the flexibility of parameters such as mode-locking
techniques and dispersion control. Numerical simulations clearly unravel how different nonlinear
interactions contribute to formation of soliton molecules. Analysis of the stability and the underlying
physical mechanisms of bound solitons bring important insights to this field. For a complete view of
nonlinear optical phenomena in fibre lasers, other dissipative states such as vibrating soliton pairs,
soliton rains, rogue waves and coexisting dissipative solitons are also discussed. With development of
advanced real-time detection techniques, the internal motion of different pulsing states is anticipated
to be characterized, rendering fibre lasers a versatile platform for nonlinear complex dynamics and
various practical applications.
Keywords:
soliton molecules; bound states; fibre lasers; nonlinear interaction; ultrafast nonlinear
optics; fibre optics
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2018
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www.mdpi.com/journal/applsci
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1. Introduction
Passively mode-locked fibre lasers generate ultrashort pulses with high peak power and broad
optical spectrum, which have attracted growing attention in the past decades due to the advantages of
compactness, stability, easy handling and portability [
1
,
2
]. The great potential has enabled a wealth of
applications in military, industry, fundamental science, biomedicine, etc. In most cases, one single pulse
travels through the entire round trip, with a repetition rate inversely proportional to the optical length
of the laser cavity. However, higher pumping can lead to pulse splitting and coexistence of several
pulses inside the cavity because a fibre laser can only withstand a certain amount of nonlinear phase
shift. Depending on a comprehensive interaction via nonlinearity, dispersion, gain, loss, etc. inside
the cavity, a fibre laser serves as a complex dissipative system and a versatile platform for plentiful
pulse states such as harmonic mode-locking, pulse bunching and soliton molecules [
3
]. In particular,
recent advances on saturable absorbers and dispersion schemes are of important benefit to control
the experimental parameters and therefore enable observation of various states of soliton molecules.
In analogy to molecules in chemistry which achieve balance of atoms via strong chemical bond,
soliton molecules as a boundary soliton state are comprised of several individual pulses and result
from balance of repulsive and attraction forces between solitons caused by many effects including
nonlinear [for instance, cross phase modulation (XPM), gain saturation, saturable absorption, etc.] and
dispersive effects [
4
]. Such a balance causes that several identical pulses (exactly the same intensity,
as well as the same spectral and temporal profiles) travel through the cavity and keep a constant
temporal separation and a fixed phase difference between neighbouring pulses. Although solitons
pairs exhibiting variant characteristic parameters such as evolving phase [
5
] are sometimes referred
to soliton molecules too, we focus on “soliton molecules” in this Review as stationary soliton pairs
preserving all details for at least thousands of round trips in a cavity. Such soliton molecules are of
vital significance to understand and explore. The reason is twofold: on the one hand, they are the
most frequently observed in fibre lasers as an outcome of the complex nonlinear dynamics, which help
us to retrieve the underlying dissipation nature; on the other hand, they are particularly useful for
developing larger telecommunication capacity in optical fibre transmission lines [
6
8
] and advancing
ultrafast characterization approaches such as real-time spectroscopy [9,10].
Soliton molecules were first predicted theoretically by Malomed within the framework
of nonlinear Schrödinger-Ginzburg-Landau equation [
11
] and coupled nonlinear Schrödinger
equations [
12
]. Afanasjev et al. and Akhmediev et al. also investigated analytically and numerically
the existence and stability of bound states with
π
, 0 and
±
π
/2 phase differences [
13
15
]. So far bound
states have been studied by simulated and/or experimental methods in types of fibre laser cavities,
i.e., in soliton [
16
,
17
], stretched-pulse [
18
], gain-guided [
19
] and self-similar [
20
] regimes, as well as
dissipative soliton regime associating with large net-normal dispersion and spectral filtering [
21
].
A lot of experimental observations of bound solitons have been reported in fibre lasers mode-locked
by nonlinear polarization evolution (NPE) [
22
24
], nonlinear amplifying loop mirror (NALM) [
8
,
25
]
and some real saturable absorbers (e.g., semiconductor saturable absorber mirror (SESAM) [
26
],
carbon nanotubes (CNT) [
27
29
], graphene [
30
,
31
], topological insulator [
32
], molybdenum disulphide
(MoS
2
) [
33
] and black phosphorus [
34
]). Although a lot of experimental findings have been made in
many different cavities, in-phase two-soliton bound states and a systematic observation of soliton
molecules with different phase differences were only reported several years ago [
28
], due to large
degree of freedom of manipulation on the experimental configurations in a fibre laser using a real
saturable absorber. Later, the rapid development of sorts of mode-lockers promotes discovery and
research on bound states and enables complete collection of the stable solutions with
π
, 0 and
±
π
/2
phase differences predicted in theory.
In this review, we will summarize the state-of-the-art developments on bound solitons in
ultrashort fibre lasers. We mainly concentrate on soliton molecules in both experiment and simulation.
We will first derive the analytical expression for characterization of soliton molecules in both frequency
and time domain. Second, we summarize typical experimental results of soliton molecules with
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distinct temporal separations, phase differences and numbers of solitons that appear in kinds of laser
cavities. Third, simulation procedures and corresponding findings are intensively addressed. Fourth,
we discuss guides and tricks to generation of various bound states and analyse their stability, with
nonlinear interaction mechanisms interpreted. Fifth, we extend the content of multisoliton states to
other more sophisticated situations and briefly take several cases (soliton rains, rogue waves and
coexisting dissipative solitons) which our group has studied as an example, so that a general picture
of possible multisoliton complexes is captured. In the end, we give a short conclusion, involve some
unexplored parts and mention possible challenges and promising improvement.
2. Analytical Expression of Soliton Molecules for Optical Characterization in Frequency and
Time Domain
2.1. Frequency Domain
2.1.1. Two-Soliton Bound States
We assume that the complex amplitude of slowly-varying envelope of a single soliton in time
domain can be described by
f
(
t
) and the complex amplitude of its optical spectrum can be described
by
F
(
ν
), with
ν
denoting the frequency difference compared to the central optical frequency. It means
that Fourier transform of
f
(
t
) corresponds to
F
(
ν
).
Consider a two-soliton bound state with pulse separation of
T
and phase difference of
θ
1
, which
reads
f
(
t
) +
f
(
t
T
)exp(
j
θ
1
) in time domain. According to Fourier transform, the spectral amplitude
of the bound state is expressed by
F
(
ν
) +
F
(
ν
)
exp
(
j
2
πν
T
)
exp
(
j
θ
1
) =
F
(
ν
)
exp
[
j
(
πν
T
θ
1
2
)]
·
2 cos
(
πν
T
θ
1
2
)
(1)
The spectral intensity is therefore proportional to
|
F
(
ν
) +
F
(
ν
)
exp
(
j
2
πν
T
)
exp
(
j
θ
1
)
|
2
=
2
|
F
(
ν
)
|
2
·
[
1
+
cos
(
2
πν
T
θ
1
)]
(2)
Equation (2) indicates that the bound state manifests itself by a modulated optical spectrum under the
envelope of 4 times single-soliton counterpart. The modulation period is
ν
= 1/
T
and the modulation
depth is 100%. Importantly, the intensity at the centre of the spectrum depends on
θ
1
, which provides us
a rather unique method to characterize the phase difference. As an example, the spectra for
π
, 0 and
±
π
/2
phase differences are shown in Figure 1a–d respectively (the envelope is normalized).
From Figure 1, we can see that out-of-phase and in-phase bound states have symmetrical spectra
with respect to the centre wavelength, while
±
π
/2 phase-difference bound states have asymmetrical
spectra. In detail, out-of-phase bound state has a minimum at the centre, while in-phase bound state
has a maximum at the centre. For comparison,
π
/2 phase-difference bound state has a second
maximum peak on the left side (i.e., blue shift) of the first maximum peak, while
π
/2 phase-difference
bound state has a second maximum peak on the right side (i.e., red shift) of the first maximum peak.
Due to close-to-linear leading and trailing edges of the single-soliton spectrum, the spectral peaks of
the
±
π
/2 phase-difference bound states satisfy the approximate relationship when modulation period
is relatively small
I
k
(
I
k
1
+
I
k
+1
)/2 (
k
= 2, 3,...),
(3)
where
I
k
denotes the intensity of the
k
th maximum peak.
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Figure 1.
Spectra of two-soliton bound states with
θ
1
= (
a
)
π
; (
b
) 0; (
c
)
π
/2; and (
d
)
π
/2 according
to Equation (2). Blue dotted curve is for single soliton (unbound) and black solid curve is for bound
solitons. For all figures, 0.8-ps chirp-free sech-shaped pulses with centre wavelength of 1560 nm and
pulse separation of 6 ps are used. (Reproduced with permission from [
28
].
©
The Optical Society of
America (Washington, DC, USA), 2013).
2.1.2. Triple-Soliton Bound States
We express a triple-soliton bound state with complex amplitude of solitons in time domain as
f
(
t
),
f
(
t
T
)exp(
j
θ
1
) and
f
(
t
2
T
)exp(
j
θ
2
). Then, its spectral amplitude reads
F
(
ν
) +
F
(
ν
)
exp
(
j
2
πν
T
)
exp
(
j
θ
1
) +
F
(
ν
)
exp
[
j
2
πν
·
(
2
T
)]
exp
(
j
θ
2
)
=
F
(
ν
)
exp
[
j
(
2
πν
T
θ
2
2
)]
·{
2 cos
(
2
πν
T
θ
2
2
) +
exp
[
j
(
θ
1
θ
2
2
)]
}
.
(4)
If
θ
1
=
π
and
θ
2
= 0, its spectral intensity is then proportional to
F
(
ν
)
2
·
[
2 cos
(
2
πν
T
)
1
]
2
=
F
(
ν
)
2
·
[
2 cos
(
4
πν
T
)
4 cos
(
2
πν
T
) +
3
]
.
(5)
For a guide to eyes, we plot an exemplary optical spectrum of a triple-soliton bound state shown
in Figure 2. It exhibits a symmetrical spectrum with respect to the centre wavelength and meanwhile
two high peaks are located on both sides of the low central peak.
Figure 2.
Spectrum of a triple-soliton bound state with
θ
1
=
π
and
θ
2
= 0 according to Equation (5).
Blue dotted curve is for single soliton (unbound) and black solid curve is for bound solitons. Here
0.8-ps chirp-free sech-shaped pulses with centre wavelength of 1560 nm and pulse separation of 4 ps
are employed. (Reproduced with permission from [
28
].
©
The Optical Society of America (Washington,
DC, USA), 2013).
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2.2. Time Domain
Second-harmonic-generation autocorrelation trace records the intensity autocorrelation function
of repetitive ultrashort laser pulses, usually obtained by measuring background-free (noncollinear)
second-harmonic generation of nonlinear crystals.
It is described by the formula
I
AC
(
τ
)
=
I
(
t
)
I
(
t
τ
)
dt
, where
I
(
t
)
is the pulse intensity and
τ
denotes a time delay. In the experiment, the
impinging optical pulses are split into two identical beams and then mechanical translation or rotation
introduces continuous scan of the time delay of the two beams. The second-harmonic-generation
intensity is closely related to the temporal overlap of the two beams. Hence its change with time delay
enables us to retrieve the information of the optical pulses in time domain. Autocorrelation trace
is the most commonly used measure to characterize the temporal length of an ultrashort optical
pulse, by assuming an appropriate temporal profile. Similarly, we can use it for obtaining the
temporal details of soliton molecules as complementary information. Figure 3b,c depicts the typical
autocorrelation traces of a two- and a triple-soliton molecule respectively, obtained by numerical
calculation. For comparison, the autocorrelation trace of the individual soliton atom is illustrated in
Figure 3a. For a soliton molecule which is comprised of
n
identical solitons (only phase might differ),
its autocorrelation trace shows (2
n
1) peaks and the peak intensity jumps from 1 to
n
and then back to
1 again step by step. Similar to the situation of conventional single solitons, the width of one individual
peak in autocorrelation trace is proportional to the width of one single optical pulse, see Figure 3a,b.
Note that autocorrelation trace enables detection of not only temporal separation between adjacent
pulses but also the amplitude ratio of them, which are reflected by modulation period and modulation
depth in frequency domain. Autocorrelation measurement is not able to reveal the phase relationship
between neighbouring solitons, therefore unfortunately less powerful than optical spectrum recording
in terms of complete characterization of a soliton molecule.
Figure 3.
Autocorrelation traces of (
a
) a single soliton; (
b
) a twin-soliton bound state; and (
c
) a
triple-soliton bound state. Chirp-free sech-shaped pulses with pulse width of 0.8 ps and temporal
distance of 4 ps are utilized.
3. Experimental Observation of Various Soliton Molecules in Different Fibre Lasers
3.1. Anomalous Dispersion Regime
Fibre laser cavities operating in anomalous dispersion regime allow solutions of conventional
sech-profiled solitons and have been widely investigated due to the great potential in abundant
applications such as fibre optical communications and military security. The typical spectral working
bands lie in the range of 1.5 and 2 microns. In this section, main experimental findings on soliton
molecules at this spectral window are given, which simultaneously indicate what we have studied in
the past several years.
3.1.1. 1.5-Micron Wavelength Regime
Figure 4 shows the laser cavity schematically. A segment of 48-cm long erbium-doped fibre
(EDF) is forward pumped by a 980-nm laser diode. The pump laser is coupled into the cavity through
a wavelength division multiplexer (WDM). A polarization-independent isolator (PI-ISO) ensures
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unidirectional operation of the fibre laser. A polarization controller (PC) is employed to adjust
birefringence and optimize operation of the cavity. A polyimide (PI) film incorporating 1 wt. %
single-walled carbon nanotubes (SWNT) [
35
] acts as the saturable absorber (SA), which is inserted
between PC and an output coupler. Ninety percent of the laser power is fed back into the cavity
to maintain sufficient gain, while 10% output is detected to investigate the generated soliton states.
An optical spectrum analyser with 0.06-nm resolution bandwidth, a 2-GHz oscilloscope and a 13.6-GHz
radio-frequency (RF) signal analyser with a 2-GHz photo-detector and a second-harmonic-generation
autocorrelator are used to characterize the output pulses in detail. The ring cavity is approximately
13 m in length and the net anomalous group-velocity dispersion (GVD) is ~
0.28 ps
2
.
Figure 4.
Schematic figure of the laser cavity. WDM: wavelength division multiplexer; EDF:
erbium-doped fibre;PI-ISO: polarization-independent isolator; PC: polarization controller; SWNT:
single-walled carbon nanotube; PI: polyimide; SA: saturable absorber. (Reproduced with permission
from [28]. © The Optical Society of America (Washington, DC, USA), 2013).
1.
Two-soliton bound states
Sufficient pump power ensures gain for stable operation of multi-soliton states within the cavity.
When polarization state is adjusted by rotating PC orientation carefully, kinds of bound states with
different phase differences could be observed. Figure 5a,b shows optical spectrum and autocorrelation
trace for an example of soliton molecules with
π
phase difference. The experimental result (black
solid curve of Figure 5a) agrees well with fitting result (red dashed curve of Figure 5a) of bound
solitons comprising two sech-profiled pulses with opposite phases. Some imperfection arises from
coexistence of continuous-wave (cw) component, generation of sidebands and a little asymmetry of the
spectrum. The first two factors are common in a soliton-type fibre laser. The last one is caused mainly
by asymmetry of gain spectrum of the erbium fibre and by the third-order dispersion. Modulation
period of 2.7 nm and ~100% modulation depth in the experimental spectrum imply that two solitons
with 2.8 ps pulse separation have an excellent coherence. Autocorrelation trace (Figure 5b) confirms
the temporal distance further, as well as exactly the same pulse width and amplitude of the two
bound solitons.
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Figure 5.
(
a
) The optical spectrum and (
b
) autocorrelation trace of an out-of-phase two-soliton bound state;
(
c
) The optical spectrum and (
d
) autocorrelation trace of an in-phase two-soliton bound state. (Reproduced
with permission from [28]. © The Optical Society of America (Washington, DC, USA), 2013).
Figure 5c,d shows another example for in-phase bound solitons. The black solid curve of Figure 5c
is the experimental optical spectrum with modulation period of 0.6 nm. It can be seen that there is
a large cw component existing at the centre of the spectrum and Kelly sidebands observable at trailing
edges, which result in slightly imperfect agreement between measurement and fitting by assuming
Gaussian-shaped pulses (red dashed). We emphasize that it does not influence the conclusion that the
two bound solitons are in phase. Modulation depth of the experimental spectrum seems to be lower
than 100%, which is just because a relatively coarse resolution bandwidth (0.2 nm) was used (an optical
spectrum analyser is sufficient to resolve the spectrum of a bound state only when modulation period
is at least 4 to 5 times the resolution bandwidth [
31
]). Figure 5d plots the autocorrelation of the bound
solitons, indicating pulse full-width at half-maximum (FWHM) of 0.9 ps and pulse separation of 13.5 ps.
The pulse separation is the same as that calculated through modulation period of the optical spectrum.
In addition,
±
π
/2 phase-difference bound states were obtained in this laser cavity, as shown by
the spectra in Figure 6. Both Figure 6a,b have a prominent asymmetry, which indicates that their phase
differences are neither
π
nor 0. The excellent fitting with sech-shaped pulses concludes the phase
differences in both figures, despite of a large cw component coinciding with the maximum peak in
Figure 6b.
Figure 6.
(
a
) Experimental and fitting spectra of a -
π
/2 phase-difference two-soliton bound state;
(
b
) Experimental and fitting spectra of a
π
/2 phase-difference two-soliton bound state. Fitting spectra
of the envelope (unbound) are also presented. (Reproduced with permission from [
28
].
©
The Optical
Society of America (Washington, DC, USA), 2013).