Vol. 26, No.

5 | 5 Mar 2018 | OPTICS EXPRESS 5564

Dynamics of soliton explosions in ultrafast

fiber lasers at normal-dispersion
YUEQING DU AND XUEWEN SHU*
Wuhan National Laboratory for Optoelectronics & School of Optical and Electronic Information,
Huazhong University of Science and Technology, Wuhan, 430074, China
*xshu@hust.edu.cn

Abstract: We found two kinds of soliton explosions based on the complex Ginzburg-Landau
equation without nonlinearity saturation and high-order effects, demonstrating the soliton
explosions as an intrinsic property of the dissipative systems. The two kinds of soliton
explosions are caused by the dual-pulsing instability and soliton erupting, respectively. The
transformation and relationship between the two kinds of soliton explosions are discussed.
The parameter space for the soliton explosion in a mode-locked laser cavity is found
numerically. Our results can help one to obtain or avoid the soliton explosions in mode-
locked fiber lasers and understand the nonlinear dynamics of the dissipative systems.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
OCIS codes: (060.5530) Pulse propagation and temporal solitons; (140.4050) Mode-locked lasers; (140.3510)
Lasers, fiber.

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#315459 https://doi.org/10.1364/OE.26.005564
Journal © 2018 Received 11 Dec 2017; revised 6 Feb 2018; accepted 14 Feb 2018; published 23 Feb 2018
 Vol. 26, No. 5 | 5 Mar 2018 | OPTICS EXPRESS 5565

20. A. F. J. Runge, N. G. R. Broderick, and M. Erkintalo, “Observation of soliton explosions in a passively mode-
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34(3), 553–558 (2017).

1. Introduction
Ultrafast fiber lasers have become important tools in scientific researches and industry [1–3].
They are also important test beds to research the nonlinear dynamics of dissipative systems
[4–6]. A better understanding of the nonlinear dynamics can help one to make a better design
of a ultrafast fiber laser. The complex cubic-quintic Ginzburg-Landau equation (CQGLE) and
its coupled version are widely used to describe the ultrafast dynamics of the mode-locked
fiber lasers [7–10]. The soliton explosion was numerically found for the first time in [11],
where the soliton explosion manifests itself as a chaotic and quasi-periodic process when the
dissipative system is in a meta-stable state. The soliton in the meta-stable dissipative system
erupts into pieces in temporal domain abruptly and gradually recover its original state after
the eruption, which is similar to exploding behavior and thus regarded as the so-called soliton
explosion [11]. The soliton explosion was experimentally demonstrated in [12] in a solid-state
Kerr-lens mode-locked laser. After the first theoretical and experimental finding of soliton
explosions, several theoretical works on this phenomenon have been done based on the
CQGLE [13–19]. Linear stability analysis was proposed to explain the origin of the soliton
explosion [13, 14]. High-order effects (H.O.Es) such as the third-order dispersion (TOD),
self-steepening (SST) and soliton self-frequency shifting (SSFS) were demonstrated to
control to dynamics of the soliton explosion [15–17]. Noise-induced soliton explosion was
demonstrated in [18]. For experimental findings, the dissipative soliton explosions
accompanied with Raman emission were found in a nonlinear amplifying loop mirror
(NALM) mode-locked fiber laser [20, 21]. The explosions in [20, 21] were linked to the
multi-pulsing instability, which is different from the typical soliton explosion caused by the
soliton eruption. The typical soliton explosion and the following rogue waves in a carbon-
nanotube (CNT) mode-locked fiber laser were found in [22, 23], where the explosions
manifested themselves as abrupt collapse of the soliton spectrum without Raman emission.
Very recently, the polarization dynamics of the soliton explosion was experimentally
researched in [6].
All the theoretical works on the soliton explosions were based on the distributed CQGLE
model [13–19] and sometimes H.O.E effects were added in the model [15–17]. Though the
distributed model helps one to understand the dynamics of the soliton explosions very well, it
is not straightforward for one to obtain or avoid the soliton explosions in a mode-locked laser
in the high dimension parameter space of the CQGLE. For an actual fiber laser system, it is
easy for one to adjust the net-dispersion, pump strength or the modulation depth of a saturable
absorber (SA) to obtain different mode-locking states. The birefringence of the laser cavity
can also be easily adjusted by the polarization controllers (PCs) used in the cavity, however,
the polarization properties of the solitons are not in the scope of this paper. To obtain the
qualitative and quantitative relationship between the soliton explosion and actual laser
parameters is not only important to understand the dynamics of the dissipative systems but
also important to the design of mode-locked lasers. The simulation of soliton explosions
based on a lumped model in [20, 21] reproduced the experimental results well, however, one
wants to know whether the soliton explosion can happen and how they evolve in the lumped
model without H. O. E effects and nonlinearity saturation.
 Vol. 26, No. 5 | 5 Mar 2018 | OPTICS EXPRESS 5566

In this paper, we report our numerical simulations of the soliton explosions based on the
complex Ginzburg-Landau equation (CGLE) in a lumped cavity model at normal-dispersion
without H. O. Es and nonlinearity saturation. The cavity length in our simulations ranges
from 2m-6.35m in order to demonstrate the universality of the soliton explosion with different
laser cavity length at normal-dispersion. We find that two kinds of explosions exist in the
laser system, one is the dual-pulsing instability (DPI) and the other is the typical erupting
soliton explosion. Pump strength, lumped loss and spectral filtering have strong effects on the
dynamics of the soliton explosions. The DPI explosion can be transformed into the erupting
soliton explosion by reducing the lumped loss. Both DPI and erupting soliton explosion can
be transformed into the stable multi-pulse state through spectral filtering. No soliton
explosion happens under small net-dispersion (below 0.064ps2 in our case) with 70%
modulation depth of the SA. The detailed behaviors of the two kinds of soliton explosions are
presented and discussed. Our results enrich the dynamics of the dissipative soliton explosions
and are useful for the design of ultrafast laser.
2. Model and parameters in simulations

Fig. 1. Schematic of the fiber laser in our simulation. LD:laser diode, WDM:wavelength
division multiplexer, OC: output coupler, SA: saturable absorber, EDF: Er-doped fiber, SMF:
single mode fiber.

The lumped cavity model that we use in our simulation is schematically shown in Fig. 1. The
main parts of the cavity include 2m Er-doped fiber (EDF), a segment of single-mode fiber
(SMF), a lumped SA and an output coupler (OC). Our simulation runs from the EDF to the
SMF in each round as shown in Fig. 1. We use the CGLE to describe the pulse propagating in
the laser cavity:

∂u 1 ∂ 2u 1  1 ∂2 
− iβ2
= + iγ u u + g 1 + 2
2
u (1)
∂z 2 ∂T 2
2  Ω ∂T 2 

where u represents the slow varying envelop of the optical field and T is the retarded time. β2
is the second-order dispersion, which is 50ps2/km in EDF and −23ps2/km in SMF. γ is the
nonlinearity of the fiber, which is 4.7 W−1km−1 and 1.3 W−1km−1 in EDF and SMF,
respectively. Ω represents the gain bandwidth of the EDF and the full width of half maximum
(FWHM) of the gain bandwidth is 20nm in our simulation. g is the gain of the EDF, which is
represented by:
2
=g g 0 exp(− ∫ u dt / Es ) (2)
 Vol. 26, No. 5 | 5 Mar 2018 | OPTICS EXPRESS 5567

where Es is the saturable energy of the EDF, which also represents the pump strength. g0 is
the small signal gain of the EDF, which is 2.5m−1 in our simulation. The transmission
function of the amplitude of the lumped SA is represented by:

T =1 − α / (1 + P / P0 ) (3)

where α is the modulation depth of the SA, which is varied in our simulations. P0 is the
saturation power of the SA, which is 100W in our simulations. The parameters are chosen to
make the simulation converge quickly. The parameters of the SA can be flexibly controlled
by the artificial SA of the nonlinear polarization rotation (NPR) [23]. For simplicity, we use
the monotonic model and the parameters of the SA have no obvious effects on the
conclusions of this paper. We have checked that the H. O. E effects do not have obvious
effects on the pulse properties under the pulse parameters in this paper. In fact, our paper here
is to demonstrate that the soliton explosion is an intrinsic property of the dissipative systems,
which is independent of the H. O. E effects. We vary the length of SMF to adjust the net-
dispersion of the laser cavity in the simulations. We use the symmetric split-step Fourier
method to implement our simulations. We run the programs for 50000 rounds for each case in
our simulations.
3. Results
3.1 Dual-pulsing instability induced soliton explosions
In our simulations, DPI happens when the net-dispersion is larger than 0.064ps2. For a certain
value of the net-dispersion, the DPI exists in a certain range of pump strength. Weaker or
stronger pump strength results in the single or double pulse mode-locking. The parameter
space will be summarized in section 3.3, and we present the typical results in sections 3.1 and
3.2. We set the net-dispersion, modulation depth of the SA and the output ratio of the OC to
be 0.091ps2, 0.7 and 10:90, respectively. By increasing the pump strength to 1020pJ, DPI
explosion happens and its dynamics is shown in Fig. 2. Figures 2(a) and 2(b) show the round-
to-round evolution of the dissipative soltion in temporal and spectral domains. We can get
from Fig. 2(a) that the soliton jumps chaotically in the temporal domain. The new pulse
originates from the background noise in temporal domain. One can get from Fig. 2(b) that
there are obvious spectral perturbations before the DPI soliton exploding. When the explosion
comes into being, the soliton spectrum becomes narrow as well as modulated because of the
dual-pulsing state in temporal domain. The pulse energy and its spectral width evolutions are
shown in Fig. 2(c) and 2(d), respectively. It is obvious in Fig. 2(c) that the soliton experiences
a transformation from the stationary state to the relaxation oscillating state before the abrupt
energy increasing. The pulse energy at the stationary and exploding states are 1.2nJ and 2.1nJ,
respectively. After the abrupt energy increasing, the pulse energy gradually recovers its
stationary state. The period of the DPI is ~995-1050 round trips. The spectrum of the
ultrashort pulse experiences an abrupt narrowing during the DPI explosion, which is shown in
Figs. 2(b) and 2(d). Figures 2(e) and 2(f) show four pulse states in one period of DPI
explosion. One can see from Fig. 2(e) that the new born pulse emerges at the background
during the explosion and the old pulse still exists at its original location. Due to the gain
competition between the old and new pulses, the old pulse gradually annihilates and new
soliton comes into being at the new location. We can get from Fig. 2(f) that the spectrum is
modulated due to the double-pulsing state and much narrower than the single soliton state.
We should note that the temporal jumping in this case is irregular, for example, the temporal
jumping in the first explosion is ~26ps while the value for the third explosion is ~50ps, which
means there is no specific position of the new pulse relative to the old pulse. Such DPI
explosion is similar to the multi-pulsing instability explosion in [21], however, the explosion
in [21] is accompanied with Raman emission and an additional-noise-like pulse, which is
different from the two smooth pulses in our DPI explosion. Considering the abrupt changing
 Vol. 26, No. 5 | 5 Mar 2018 | OPTICS EXPRESS 5568

(energy increasing, spectrum narrowing) as well as the quasi-periodic evolution of the soliton
state [11–19], we think the DPI is a new state of the soliton explosion.

Fig. 2. Pulse characteristics with 1020pJ pump strength, 70% modulation depth and 0.091ps2
net-dispersion: (a) pulse evolution in temporal domain, (b) pulse evolution in spectral domain,
(c) pulse energy evolution, (d) evolution of frequency width of the soliton, (e) pulse intensity
profiles at different stages, (f) pulse spectra at different stages.

With the pump strength increasing while keeping the other parameters fixed, the irregular
jumping of the soliton in temporal domain can be regular, limited at two fixed temporal
locations. Figure 3 shows the pulse characteristics under the pump strength of 1180pJ. We
can obtain from Fig. 3(a) that the soliton jumps back and forth at two fixed temporal locations
with ~22.7ps separation. Spectral perturbations exists on the spectrum of the soliton before its
exploding as one can see from Fig. 3(b). The pulse energy evolution in Fig. 3(c) has a shorter
period of ~75-77 round trips than that under the pump strength of 1020pJ (~995-1050 round
trips). We should note that the DPI explosion is not strictly periodic but chaotic, however, we
can still observe that the periods for the DPI explosion decrease as the pump strength
increasing. Spectral narrowing appears when the soliton explodes, which is similar to the case
in Fig. 2. There is no relaxation oscillation in Figs. 3(c) and 3(d) and the recovering stage
dominates during the explosion, which is accompanied by the spectrum broadening, soliton
narrowing and energy decreasing. Figures 3(e) and 3(f) show the pulse characteristics before,
during and after the DPI exploding with the insets showing their local enlargement portions.
We can see from Fig. 3(e) that the new pulse emerges at one side of the old pulse and the old
pulse annihilates after the exploding. From the inset in Fig. 3(e), we can note that a very weak
pulse emerges with a quite weak intensity of ~0.008W before the exploding. Such weak pulse
originates from the background noise and quickly develops into a new pulse during the
explosion. We can obtain from Fig. 3(f) as well as its inset that there is a small spectral
perturbation before the soliton exploding, manifesting itself as a localized and modulated part
in the center of the spectrum. This spectral perturbation corresponds to the weak new pulse
 Vol. 26, No. 5 | 5 Mar 2018 | OPTICS EXPRESS 5569

before exploding in Fig. 3(e). The coherent supposition of the weak new and old strong pulses
in temporal domain results in the localized and modulated spectral perturbation on the
spectrum. As the weak pulse gradually developing into a strong pulse, the small localized
spectral perturbation gradually spreads to the whole spectrum as shown in Fig. 3(f). The
spectrum is narrowed and modulated during the DPI exploding, which is shown in Figs. 3(d)
and 3(f). From the above simulations in Figs. 2 and Figs. 3, we can get that the DPI explosion
is accompanied with pulse energy increasing and spectrum narrowing, which is in agreement
with the erupting explosion [11] even there are some differences between the DPI and the
erupting explosion. The DPI is an intermediate state between the single pulse and dual-pulses,
while the results in [20, 21] locate in transition zone between the single pulse and noise-like
pulse (NLP). Visualization 1 in the supplementary shows the explosion process in Fig. 3. In
our simulations, we found that the pulse spacing between the new and old pulses in the DPI
decreases with the pump strength increasing. We estimate that the temporal location where
the total perturbation function of the old pulse has its maximum value depends on the pump
strength of the dissipative system, however, this should be strictly checked by the linearity
analysis [13].

Fig. 3. Pulse characteristics with 1180pJ pump strength, 70% modulation depth and 0.091ps2
net-dispersion: (a) pulse evolution in temporal domain (see Visualization 1), (b) pulse
evolution in spectral domain (see Visualization 1), (c) pulse energy evolution, (d) evolution of
frequency width of the soliton, (e) pulse intensity profiles at different stages, (f) pulse spectra
at different stages.

3.2 Pulse erupting induced soliton explosions

In our simulations, DPI happens when the lumped loss is large enough (in our case, an OC
with output ratio from 5:95-10:90) while the typical erupting soliton explosion happens when
the lumped loss is reduced (typically, 0:100-4:96 output ratio) in the cavity. We removed the
OC in the cavity model and set the net-dispersion and the modulation depth of the SA to be
0.1ps2 and 0.7, respectively, and gradually increase the pump strength, the soliton explosion
emerged when the pump strength was 1610pJ and its exploding period is ~657-680 round
 Vol. 26, No. 5 | 5 Mar 2018 | OPTICS EXPRESS 5570

trips. In order to show more periods of explosions in several hundreds of round trips, we set
the pump strength to be 1800pJ. The pulse characteristics are shown in Fig. 4. We can see
from Fig. 4(a) that the soliton erupts abruptly within ~25 round trips as shown in Fig. 4(a) and
its inset. The perturbations in temporal domain develops quickly into a broad pulse with many
dips in it, which is also regarded as the process of the soliton erupting into multi-pieces in
[13–19]. We can get from Fig. 4(b) that the spectrum of the soliton transforms from the flat-
top shape into the collapsing state during the soliton exploding. The pulse energy and spectral
width evolutions of the soltion in Figs. 4(c) and 4(d) are quasi-periodic as well as chaotic,
which also shows the characteristics of energy increasing and spectrum narrowing during the
soliton exploding. One can get from Fig. 4(e) that the soliton recovers from the erupting state
to its original state after the soliton explosion. The exploding soliton in Fig. 4(e) is cut into
pieces with many dips on it. The spectra of the soliton before and after explosion in Fig. 4(f)
have flat tops and near-rectangular shapes, which are characteristics of the dissipative soliton
at normal-dispersion. The spectrum of the exploding soliton has many peaks and stronger
intensity compared to those before and after explosion, the simulated spectrum in Fig. 4(f) is
similar to the experimentally measured shape in [22]. The results shown in Fig. 4 are typical
characteristics of the soliton explosion in [11], whose dynamic evolutions can be regarded as
the homoclinic while chaotic trajectories in the infinite phase space [13]. We can note that the
soliton and its exploding state are strictly symmetric in both temporal and spectra domains,
which is due to the symmetric perturbations. Visualization 2 in supplementary shows the
pulse explosion in Fig. 4.

Fig. 4. Pulse characteristics with 1800pJ pump strength, 70% modulation depth and 0.1ps2 net-
dispersion: (a) pulse evolution in temporal domain (see Visualization 2), (b) pulse evolution in
spectral domain (see Visualization 2), (c) pulse energy evolution, (d) evolution of frequency
width of the soliton, (e) pulse intensity profiles at different stages, (f) pulse spectra at different
stages.
 Vol. 26, No. 5 | 5 Mar 2018 | OPTICS EXPRESS 5571

With the pump strength increased to 2400pJ, a kind of asymmetric soliton explosion
happened, manifesting itself as a chaotic process accompanied with irregular temporal
jumping and soliton erupting. The results are shown in Fig. 5. One can see from Fig. 5(a) that
the explosion is asymmetric in temporal domain, which is different from the symmetric state
in Fig. 4. Such asymmetric explosion can be qualitatively explained by the linear stability
analysis in [13, 14], which is caused by the composition of different symmetric or asymmetric
eigen perturbation functions of the systems. Different from the cases of DPI in Figs. 2 and
Figs. 3, the background noise mixes with the old pulse, developing into a collapsed soliton
with many pieces during the explosion rather that the dual-pulse state. The spectrum
evolution in Fig. 5(b) shows that the exploding state dominates the soliton evolution in one
period of explosion and the spectra in Fig. 5(b) is no longer symmetric as the case in Fig.
4(b). One can obtain from Fig. 5(c) that the exploding soliton is cut into an asymmetric state
with multi-pieces. We can get from the inset of Fig. 5(c) that the perturbation before
explosion is asymmetric in temporal domain, which results in the asymmetric soliton
explosion and the temporal jumping of the solitons. The perturbations in spectral domain are
also asymmetric as one can get from Fig. 5(d). We can get from Fig. 5(e) that the explosion
process dominates the soliton evolution. The pulse energy increases from ~0.97nJ to ~7nJ
during the soliton explosion. Visualization 3 in the supplementary shows the pulse explosion
in Fig. 5.

Fig. 5. Pulse characteristics with 2400pJ pump strength, 70% modulation depth and 0.1ps2 net-
dispersion: (a) pulse evolution in temporal domain (see Visualization 3), (b) pulse evolution in
spectral domain (see Visualization 3), (c) pulse intensity profiles at different stages, (d) pulse
spectra at different stages, (e) pulse energy evolution.

The soliton explosion became more chaotic and successive when the pump strength was
increased to 10000pJ. We can get from Fig. 6(a) that the soliton evolution is quite chaotic
while localized in the temporal domain. The chaotic wave packet occasionally shrinks into a
short soliton during evolution, which can be regarded as a successive soliton explosion. The
soliton evolution in the spectral domain shown in Fig. 6(b) has similar behavior to Fig. 5(b),
which is dominated by the exploding states and recovers to the single pulse state
 Vol. 26, No. 5 | 5 Mar 2018 | OPTICS EXPRESS 5572

intermittently. One can get from Figs. 6(c) and 6(d) that the soliton is a smooth pulse with
strongly asymmetric perturbations before and after the explosion while it is a wave packet
with multi-pieces during the explosion. The pulse evolution in Fig. 6(e) shows that the soliton
explosion is successive as there are no pauses for the local minimum energy states. Further
increasing the pump strength results in NLP generation.

Fig. 6. Pulse characteristics with 10000pJ pump strength, 70% modulation depth and 0.1ps2
net-dispersion: (a) pulse evolution in temporal domain, (b) pulse evolution in spectral domain,
(c) pulse intensity profiles at different stages, (d) pulse spectra at different stages, (e) pulse
energy evolution.

3.3 Parameters space for the soliton explosion

The soliton explosion in [20–22] are regarded as an intermediate state between the stable
single soliton and NLP. In our case of the DPI, the soliton explosion is an intermediate state
between the single soliton and soliton molecule. Figures 7(a)-7(c) show the soliton evolutions
with 0.091ps2 dispersion under pump strength of 1180pJ, 1190pJ and 1200pJ, respectively.
We can get from Fig. 7(a) the soliton works at the DPI state with the pump strength of
1180pJ. With slightly increasing the pump strength to 1190pJ, the soliton transforms into an
asymmetric soliton pair, pulsating during its evolution. With the pump strength of 1200pJ, the
soliton transforms into the stable soliton molecule, which is shown in Fig. 7(c). For a specific
net-dispersion, the DPI explosion happens in a limited range of pump strength between the
single soliton and soliton molecule.The parameters space for the DPI explosion with 0.7
modulation depth of SA is shown in Fig. 8. We can get from Fig. 8 that for a specific net-
dispersion, the DPI happens in a limited region of pump strength and the region of pump
strength increases with the net-dispersion increasing. No DPI happens if the net-dispersion is
less than 0.064ps2 as we can obtain from Fig. 8. We also research the effect modulation depth
of the SA on the soliton explosion with a fixed net-dispersion of 0.091ps2. We find that no
explosion happens when the modulation depth is below 20%.
 Vol. 26, No. 5 | 5 Mar 2018 | OPTICS EXPRESS 5573

Fig. 7. Pulse evolutions (normalized) with 70% modulation depth and 0.091ps2 net-dispersion
under different pump strength: (a) 1180pJ, (b) 1190pJ, (c) 1200pJ.

Fig. 8. Upper and lower limits of the pump strength for the soliton explosion happening with
70% modulation depth under different net-dispersion.

We find that the spectral filtering can transform the erupting explosion as well as the DPI
into multi-pulse states. If we add a band-pass filter (BPF) at the end of the cavity with a net-
dispersion of 0.1ps2 and pump strength of 1700pJ, the soliton will transform its explosive
state to the stable multi-pulse state, which is shown in Fig. 9. We can get from Figs. 9(b) and
9(c) that stronger filtering results in more pulses, which means the spectral filtering has strong
effects on the soliton explosion dynamics. The pulse energy evolutions in Figs. 9(d)-9(f)
show that the multi-pulse states are quite stable compared to the soliton explosion. We think it
is easy for us explain this phenomenon: the explosion originates from the temporal
perturbation on the background of the soliton while a BPF can filter the perturbation to avoid
the explosion. As for the multi-pulse state, it has been explained in [24] that extra loss by the
spectral filtering results in the splitting of the DS. We think the spectral filtering is an
alternative method to manipulate the soliton explosion in an ultrafast laser. As has been
mentioned in section 3.2, the DPI explosion can be transformed into the typical erupting
soliton explosion by decreasing the lumped loss. The reason for such transformation remains
unclear, however, it can advise us that soliton erupting tends to be in the higher energy state
compared to the DPI.
 Vol. 26, No. 5 | 5 Mar 2018 | OPTICS EXPRESS 5574

Fig. 9. Pulse evolutions with 70% modulation depth, 0.1ps2 net-dispersion, 1700pJ pump
strength under different spectral filtering: (a) and (d) without BPF, (b) and (e) 10nm(FWHM)
BPF, (c) and (f) 5nm BPF.

4. Discussion
The lumped model in our case is different from the distributed model in [13–19]. The model
of CQGLE in [13–19] includes two quintic items, the saturation of the nonlinear gain (SNG)
and the saturation of the nonlinear index (SNI). It is easy for us to understand that Eqs. (2)
and 3 correspond to the SNG in the distributed CQGLE model, which are important for the
pulse generation and stabilization. However, there are no items in Eq. (1) corresponding to
the SNI in the distributed CQGLE model. As has been proposed in [19], a complex system
should have a minimum nonlinear complexity to obtain the robust pulsating solutions,
together with the results in our case, we think the SNG is complex enough for a dissipative
system to obtain the pulsating solutions such as the soliton explosions while SNI is not
necessary. Even the SNI as well as H. O. E are not necessary for the soliton explosion
generation, they can affect the behavior of the soliton explosions strongly [11, 16, 20].
The DPI and erupting soliton explosions in our paper seems to be different physical
phenomena, however, we can obtain clearly from Visualization 1, Visualization 2 and
Visualization 3 that their recovering processes have the very similar behaviors, which are the
pulse narrowing, spectrum broadening and energy decreasing. We think they both belong to
the soliton explosion proposed in [11], but with different states. It has been demonstrated in
experiments that the soliton explosions can have different states, such as Raman accompanied
explosion [20], successive explosion [22], blue shifted explosion [12] and vector soliton
explosion [6], and we believe more states of soliton explosions and their dynamics are
remained to be observed both experimentally and theoretically.
5. Conclusion
In conclusion, we have given a thorough investigation of the soliton explosion by simulations.
We find a new kind of soliton explosion, the dual-pulsing instability. The DPI is an
intermediate state between the single pulse and soliton molecule while the erupting soliton
explosion is an intermediate state between the single pulse and NLPs. The DPI can be
transformed into the typical erupting soliton explosion by decreasing the lumped loss under
certain parameters. Both the erupting explosion and DPI can be transformed into the multi-
pulse state by spectral filtering. For a normal-dispersion mode-locked laser, the DPI happens
in a certain range of pump strength when the net-dispersion is large than a threshold (0.064ps2
in our case). Both kinds of soliton explosions are sensitive to the dissipative processes such as
the nonlinear gain, lump loss and spectral filtering, so we think they can be regarded as the
recently proposed item, the incoherent dissipative soliton [6], considering their localized but
 Vol. 26, No. 5 | 5 Mar 2018 | OPTICS EXPRESS 5575

incoherent structures. For the applications of the laser systems, the soliton explosion might be
undesirable because it can result in time jittering, intensity floating and spectrum eruption,
our simulation may give some qualitative instructions for the ultrafast laser designs. Also, our
results can give an insight into the nonlinear dynamics of the ultrafast dissipative systems.
Funding
National Natural Science Foundation of China (NSFC) (61775074); National 1000 Young
Talents Program, China; 111 Project (No. B07038).