# Pulsating dissipative solitons in a Mamyshev oscillator

Bo Cao  Kangjun Zhao  Chenxin Gao  Chengying Bao<EMAIL_ADDRESS>Changxi Yang<EMAIL_ADDRESS>State Key Laboratory of Precision
Measurement Technology and Instruments, Department of Precision Instruments,
Tsinghua University, Beijing 100084, China  Xiaosheng Xiao  State Key
Laboratory of Information Photonics and Optical Communications, School of
Electronic Engineering, Beijing University of Posts and Telecommunications,
Beijing 100876, China

###### Abstract

Mode-locked fiber lasers provide a versatile playground to study dissipative
soliton (DS) dynamics. The corresponding studies not only give insights into
soliton dynamics in dissipative systems, but also contribute to femtosecond
fiber laser design. Recently, Mamyshev oscillators (MOs), which rely upon a
pair of narrow filters with offset passing frequencies, have emerged as a
promising candidate for high power, ultrabroad bandwidth pulse generation. To
date, only stable mode-locking states in MOs have been reported. Here, we
present a comprehensive experimental and numerical investigation of pulsating
DSs in an ytterbium MO. By reducing the filter separation down to 4 nm, we
observe pulsation in a single pulse state as well as a soliton molecule state.
In the single pulse state, the output pulse energy can vary as large as 40
times in our MO. Single-shot spectra measured by the dispersive Fourier
transform (DFT) method reveal the spectral bandwidth breathing during
pulsation and enables the observation of soliton explosion in a pulsation
state. In addition, pulsation with a period lasting 9 round trips and even a
chaotic pulsation state are also observed. Numerical simulations based on a
lumped model qualitatively agree with our observation. Our results enrich DS
dynamics in MOs and show the impact of filter separation on the stability of
MOs.

††preprint: APS/123-QED

##  I Introduction

Passively mode-locked fiber lasers can support dissipative solitons (DSs) via
the delicate balance between nonlinearity, dispersion, gain and loss [1]. The
stability endowed by DS formation makes mode-locked fiber lasers invaluable
tools for applications spanning from nonlinear microscopy [2], precise timing
dissemination [3] to optical frequency combs [4]. DS dynamics in mode-locked
lasers can be governed by the cubic-quintic Ginzburg-Landau equation (CGLE)
[1]. In addition to stable DSs, pulsating DSs, where DSs vary periodically in
propagation, is also an eigenstate [5, 6, 7, 8]. Extreme pulse energy
variation exceeding two orders of magnitude has been theoretically predicted
for pulsating DSs [9]. The CGLE averages the DS propagation within a single
round trip. When considering the round trip nature of lasers, pulsation of DSs
manifests as period multiplication (period-$N$ pulsation means DSs only repeat
after $N$ round trips) in mode-locked fiber lasers [10, 11, 12, 13, 14]. Due
to the bandwidth limitations of electronic digitizers, typically only pulse
energy variation was measured in early reports onDS pulsation and it was quite
challenging to measure fast pulsation dynamics regarding other parameters in
real-time. With the emergence of the dispersive Fourier transform (DFT) method
[15], single-shot spectra during pulsation have been measured recently [16,
17, 18, 19]. These measurements reveal that pulsating DSs can be associated
with strong spectral bandwidth breathing [18] or even invisible if only time
domain pulse train is measured [16].

To date, pulsating DSs have been observed in fiber lasers mode-locked by
material-based saturable absorbers [20, 17, 16] and by nonlinear polarization
rotation (NPR) [11, 12, 13, 14, 18]. Recently, a new saturable absorption
mechanism based on the Mamyshev regenerators [21] have been leveraged for DS
mode-locking in fiber lasers [22, 23, 24, 25, 26, 27]. This effective
saturable absorber relies upon spectral broadening induced by self-phase
modulation (SPM) and a pair of filters with offset transmission frequencies
[21]. Thus, high power optical field will pass the filter pair with higher
transmission than low power field, resulting in an effective saturable
absorber [28, 29, 30]. Mode-locked fiber lasers based on this mechanism,
referred to as Mamyshev oscillators (MOs), show great prospects in reaching
high output pulse energy [22, 23, 25]. For example, a MO delivering 50 nJ
pulses with a peak power exceeding 10 MW has been reported [25]. Moreover, MOs
can enable the generation of ultrabroad spectra and few-cycle pulses. Ma et
al. have demonstrated a MO with an output spectrum spanning $\sim$400 nm that
can be externally dechirped to $\sim$20 fs [31].

Besides high energy ultrashort pulse generation, the unique architecture of
MOs makes them a new playground to study DS dynamics. Indeed, multi-soliton
states including harmonic mode-locking [32, 33, 34] and soliton molecule [35]
have been observed in MOs in addition to stable single-pulse states. However,
these DSs remain stable and pulsating DSs in MOs have not been reported to our
knowledge.

Here, we report experimental and numerical observation of pulsating DSs in a
MO operating around 1030 nm. Pulsating DSs are observed by controlling the
pump power as well as the polarization state under a relatively small filter
separation (4 nm). Single-shot spectra of these pulsating DSs are also
measured by the DFT method, which indicate that our MO admits various
pulsation dynamics. Spectral breathing associated with pulse energy pulsation
occurs for both a single pulse state or a soliton molecule state in period-2
pulsation states. In the single pulse state, pulsating DSs whose output energy
varies as large as 40 times during pulsation are observed. Period-3 pulsation
is observed for two wide spaced DSs pulsating in different ways. Soliton
explosion [36, 37, 19, 17, 38] is also observed in this state, which
constitutes the first observation of soliton explosion in MOs, to our
knowledge. In addition to these short period pulsation, pulsation lasting a
relatively long period (e.g., 9 round trips) and even chaotic pulsation (no
regular period) also exist in the MO. Numerical simulations based on a lumped
model including the polarization dynamics are in reasonably qualitative
agreement with the experimental observation and verify the existence of
various pulsation dynamics in MOs. Our observation not only adds to the
possible DS states in MOs but also aids the design of MOs.

##  II Laser design and stable operation

Figure 1:  Experimental setup and laser output in a stable state. (a)
Schematic of the Mamyshev oscillator (MO) and the measurement system. The top
inset illustrates a train of dissipative solitons in pulsation. OC: optical
coupler, WDM: wavelength division multiplexer, COL: collimator, ISO: isolator
HWP: half waveplate, QWP: quater waveplate, PBS: polatization beam splitter,
SMF: single mode fiber, SPC: signal-pump combiner, PD: photodetector. (b)
Optical spectrum of the MO outoput with a filter separation of 7 nm. Dashed
blue and red curves correspond to the transmission of the filters. (c) Pulse
train of the MO output; (d) RF spectrum; (e) autocorrelation trace, showing a
pulse width of 4.6 ps. Figure 2: Single soliton and soliton molecule
pulsation. (a) and (d) Pulsation pulse trains of the two typical operations.
(b) and (e) DFT measured single-shot spectra corresponding to (a) and (d),
respectively. The white lines are the integrated pulse energy for the
pulsating DS. (c) and (f) Single-shot spectra in two successive round trips.
(g) Pulsating pulse train of a soliton molecule. (h) Single-shot spectra of
the pulsation state. (i) Spectra of two successive round trips. The spectra
show modulation due to the interference between two closely spaced pulses. (j)
Fourier transform of panel (i). Peaks at 4.65 ps illustrates the pulse
separation of the soliton molecule. Figure 3: Period-3 pulsation and explosion
for two wide spaced solitons. (a) The soliton trains showing pulse energy
during pulsation. (b) RF spectrum of the pulse train exhibiting additional RF
tones besides the fundamental repetition rate 11.8 MHz. (c) Single-shot
spectra in 6000 round trips. Note that the y-axis of this panel is labeled as
time rather as wavelength derived from DFT (i.e., time to wavelength mapping).
This is because the two pulses are plotted with increasing time but they
operate around the same wavelength. (d) Zoom in of the data marked by the
rectangular shape in panel (c) showing a regular period-3 pulsation. The light
blue and white curves are the pulse energy change for dissipative soliton I
and II, respectively. (e1)-(e3) Spectra of the pulsating dissipative solitons
within a single pulsation period. Dissipative soliton I and II both undergo
spectra A, B, C in pulsation, but experience them in different orders.
(f1)-(f4) Zoom in of the data marked by the ellipse shape in panel (c). The
single-shot spectra show abrupt spectral collapse in soliton explosion, as
indicated by the arrows. The while line shows the integrated pulse energy for
soliton in explosion. Figure 4: Long-period pulsation and bifurcation to a
chaotic state. (a) A soliton train showing period-9 pulsation. (b) Single-shot
spectra of the period-9 pulsation state. The white line shows the integrated
during pulsation. (c) Details of the spectral evolution within one pulsation
period. (d) Pulse energy bifurcation diagram when rotating the quarter
waveplate. (e) Pulse train in a chaotic state under an waveplate orientation
of 148∘. Figure 5: Simulation of the pulsating DSs in a vector model. (a)
Transition from stable mode-locking to pulsating DS by reducing the filter
separation. (b) Transition into pulsation by rotating the waveplates. The
white curves in panels (a, b) show the integrated change. (c) Spectral
dynamics of a period-2 pulsation state with a relatively large pulse energy
change. (d) Single-shot spectra of the period-2 pulsating DS in panel (c). (e)
Temporal and (f) spectral dynamics within two successive cavity round trips
for the pulsation state in panel (c). The white solid line in panel (e) is the
peak power of the pulse in the cavity, while the insets of panel (f) depict
the zoom in spectra right after the filter 1 of two successive round trips.
(g) Period-2 pulsating DS simulated by a scalar model (the other panels are
all based on the vector model). The white curve is the integrated showing a
very small change in pulsation. (h) Evolution of spectrum of a soliton
molecule in period-2 pulsation. (i) Single-shot spectra of the period-2
pulsating bound soliton state in panel (g). (j) Zoom in of the data in panel
(i). (k) Fourier transform of panel (i) shows the pulse separation remains
unchanged in pulsation. Figure 6: Simulations of the pulsating soliton
explosion and the long-period pulsation. (a) Evolution of the spectra over
2000 round trips in a period-3 pulsation with soliton explosion, marked by the
arrows. (b) Zoom in of the data marked by the rectangular shape in panel (a),
showing regular period-3 pulsation. The white curve shows the integrated . (c)
Single-shot spectra of the period-3 pulsating DS. (d) and (e) Zoom in of the
data marked by the arrows in panel (a). The white curves show the pulse energy
change during explosion. (f1) and (f2) Temporal shift of the pulsating soliton
explosion depicted in panels (d) and (e), respectively. (g) Evolution of the
spectra of a long-period (period-14) pulsation state. The white curve stands
for the pulse energy change. (h) Three representative single-shot spectra
within a pulsation period shown in panel (g). (i) Spectral evolution of a
chaotic pulsation state; the corresponding pulse energy (white curve)
evolution loses periodicity.

Scheme of the experimental setup is shown in Fig. 1(a). The laser cavity
consists of two Mamyshev regenerators. The upper arm comprises a 2 m core
pumped Yb-fiber with a small core diameter of 6 $\mu$m so as to enhance the
nonlinearity. Thus, a 976 nm single mode pump is sufficient to amplify the
seeding pulses to pass filter 2 via SPM induced spectral broadening (see Fig.
1(a)). For the second arm, a 1.5 m double-cladding Yb-fiber (LMA-10/125-YDF,
Nufern) is pumped by a 976 nm multimode pump so that a higher output power is
possible. All the remaining passive fibers in the cavity are HI1060 fibers. An
isolator is used to ensure unidirectional propagation. To realize offset
filtering, a pair of 600 lines/mm reflective diffraction gratings (GR13-0610,
Thorlabs) are used to form two filters whose passing band can be adjusted by
the grating orientation and the collimator position (dotted squares in Fig.
1(a)) [23]. Two half waveplates are placed before the gratings to maximize the
diffraction efficiency. The filter bandwidth is measured to be $\sim$2 nm. The
other waveplates and a polarizing beam splitter (PBS) are used to create a
NPR-like transmission in the cavity that can assist initialization of the
mode-locking. Note that the waveplates can be adjusted over a wide range (in
some cases 360∘ for the QWP) without losing mode-locking after pulses
generated, which elucidates the effect of Mamyshev regenerators on stabilizing
the mode-locked pulses.

The pulses are sampled by the PBS as output. The pulse train is detected by a
5 GHz photodetector (PD 1) and a 12 GHz photodetector (PD 2), and digitized by
a real-time oscilloscope (Keysight DSOS804A) with an effective bandwidth of 4
GHz, see Fig. 1(a). The averaged optical spectra are recorded by an optical
spectrum analyzer (OSA, Agilent 86142B) with a resolution of 0.06 nm. The DFT
path includes a 12 km SMF, corresponding to a group delay dispersion of
$\sim$360 ps/nm around 1030 nm. The resolution of single-shot DFT spectra is
estimated to be 0.58 nm [15].

To initiate mode-locking in the MO, we inject seed pulses from a home-built
all normal dispersion fiber laser [39] into the MO. The seeding pulses has a
$\sim$10 nm bandwidth that covers the filter separation. Filter 1 is set to
1030 nm while filter 2 is adjusted between 1034 nm and 1038 nm. When the
single mode pump and multi-mode pump are adjusted to 400 mW and 1.1 W,
respectively, stable single pulse mode-locking can be achieved by adjusting
the waveplates. Once initiated, the mode-locked pulse circulates stably in the
MO after blocking the seed laser. By increasing the filter separation to 7 nm
and increasing the pump (multimode pump, the same below) power to 1.3 W after
starting mode-locking [40], we get a stable mode-locking state whose
characteristics are depicted in Figs. 1(b-e). The pulse spectrum spans about
50 nm at $-$20 dB level and the mode-locked pulse train has a repetition rate
of 11.81 MHz. The corresponding radio-frequency (RF) spectrum of the pulse
train exhibits a signal-to-noise of $\sim$70 dB with a resolution bandwidth of
1 Hz (see Fig. 1(d)). The chirped pulse duration in this state is $\sim$4.6 ps
as determined by an intensity autocorrelator (see Fig. 1(e)).

##  III Observation of pulsating dissipative solitons

###  III.1 Single soliton and soliton molecule period-2 pulsation

The laser tends to operate in the stable single pulse regime when the filter
separation is relatively large. When decreasing the filter separation to 4 nm
by tuning filter 2 to $\sim$1034 nm under a pump power of 1.1 W, pulsating DSs
can be observed frequently in the MO. Operation regime of the MO can be
adjusted by the polarization dynamics. Various pulsation states can be
accessed by rotating the waveplates under this relatively narrow filter
separation. This also enables a comprehensive study of pulsation dynamics in
our MO. We present two typical examples of single pulse period-2 pulsation
states in Figs. 2 (a-f). A pulse train is plotted in Fig. 2(a), showing that
pulses in successive round trips have significant intensity change. Figure
2(b) shows the periodic change of the DFT measured single-shot spectra in 50
round trips; pulse energy derived by integrating the single-shot spectra power
also exhibits period-doubling. To show more details of the pulsation, spectra
of two successive round trips (i.e., 1st and 2nd round trip) are shown in Fig.
2(c). These two spectra show variations in 3 dB bandwidth (changing between 5
and 11 nm), spectral fringe features and average power. The large spectral
bandwidth breathing can be attributed to the different spectral broadening
levels in the Mamyshev regenerators when the pulse energy changes in the
pulsation. Note that the narrower spectrum still covers the two filters (4 nm
separated). Thus, it can be regenerated in the cavity to retain mode-locking.
By rotating the waveplates, the output pulse energy variation can be very
large in some cases. Figures 2(d-f) show an extreme example, where one of the
output DSs can be very weak. The output pulse energy change (defined as the
ratio between the strong and the weak one) is as large as 40 times within one
pulsation cycle by integrating the single-shot spectra (note that the time
domain measurement does not show such a high ratio as the detector is
saturated for the strong pulse in order to measure the weak one). This
corresponds to the largest pulse energy variation during pulsation in mode-
locked lasers, to our knowledge. As a caveat, the PBS output port is a
nonlinear output port and the intracavity pulse energy may not change as large
as the output (see also simulations below). In other words, the large output
energy variation between two successive round trips may arise from significant
nonlinear polarization rotation in two round trips. This may help the
intracavity pulse to retain an energy level required to keep consistent
spectral broadening in the MO. Therefore, the large output energy variation
may not provide direct experimental evidence for the extreme pulsation yet
[9].

By increasing the pump power to 1.2 W, a pulsation state comprising two
bounded DSs (i.e., a soliton molecule) can be obtained by rotating the
waveplates. Figure 2(g) shows a pulse train measured without DFT, which
indicates period-2 pulsation. The associated single-shot spectra are presented
in Fig. 2(h). Stretching and compression of the spectra is evident and the
weak pulse has an energy about half of the strong one.

To compare the pulsating spectra, two consecutive round trips spectra are
shown in Fig. 2(i). The measured spectra exhibit evident spectral modulation,
which arises from the interference between two closely spaced DSs [41]. The
contrast of the spectral fringe is lower than unity due to the finite
resolution of the DFT method. Similar to the single soliton pulsation in Fig.
2, the 3 dB bandwidth of the soliton molecule spectra change from 25 nm to 16
nm in the pulsation. Fourier transform of the single-shot spectra in Fig. 2(i)
is plotted in Fig. 2(j). Peaks at non-zero time delay can be observed and they
indicate that the 2 pulsating DSs are separated by 4.65 ps. This separation
remains the same during pulsation, which differs from the vibrating solitons
whose pulse separation changes periodically [42, 43, 44].

###  III.2 Period-3 pulsation and soliton explosion

For higher pump power (e.g., exceeding 1.2 W), multi-soliton state occurs
frequently. We also observe pulsation in a two-DS state with the pulses widely
spaced. Figure 3(a) plots the nonuniform pulse train in the time domain. This
state shows period-3 pulsation instead of period-2 pulsation. The RF spectrum
of the pulse train shows RF tones at the harmonics of 11.8 MHz/3 besides the
fundamental 11.8 MHz (Fig. 3(b)), which further validates the period-3
pulsation of this state.

Single-shot spectra over 6000 round trips are presented in Fig. 3(c). We first
zoom in the data of 50 round trips around the 3000th round trip in Fig. 3(d)
(rectangular shaped region in Fig. 3(c)). Although the two DSs both have
period-3 pulsation, they pulsate in two ways. The single-shot spectra within 3
successive round trips are depicted in Fig. 3(e1-e3). DSs show three different
spectral shapes marked as A, B and C. DS I pulsates in a route as A-¿B-¿C,
while DS II pulsates as C-¿A-¿B. In other words, both DSs exhibit three
spectral shapes in pulsation, but experience them in different orders. This
also corroborates the pulse energy change delay in Fig. 3(d). Such a delayed
spectral evolution can reduce the total pulse energy change than a
synchronized spectral evolution.

Besides the periodic pulsation, there are some round trips where the DSs
experience soliton explosion (see ellipse regions in Fig. 3(c)). These soliton
explosions associated abrupt collapses occur randomly and details of the
explosion are indicated by arrows in Figs. 3(f1-f4). Typically, DSs recover to
a steady state in soliton explosion [36, 37], and soliton explosion in
pulsating DSs has only been recently reported for a NPR mode-locked fiber
laser [19] and carbon-nanotube mode-locked fiber lasers [17, 38]. Here,
soliton explosion for pulsating DSs is observed in a MO. Instead of collapsing
into chaotic and wide spectra as shown in Ref. [19], the explosions in our
case are mild in both time and frequency domain. We attribute this relatively
mild pulsation to the strong shaping effect on pulses of Mamyshev
regenerators; extreme explosion cannot survive in concatenated offset
frequency filtering. The explosion lasts for about 5 round trips ($\sim$400
ns) which is much smaller than the $\sim$20 $\mu$s explosion time scale
reported in ref. [19]. The pump power for this state (1.3 W) is higher than
the mode-locking threshold (1.1 W) and we believe that strong nonlinearity is
essential to the observation of explosion in pulsation in our MO. As an aside,
when one of the DSs undergoes the explosion, the other one pulsates regularly,
which suggests that the interaction is relatively weak for these two widely
spaced DSs.

###  III.3 Long period and chaotic pulsation

In addition to the short period pulsation shown above, pulsation with a
relatively long period can also be possible in fiber lasers. For example,
long-period pulsation (up to 170 round trips) associated with spectral
bandwidth breathing has been reported in NPR mode-locked lasers [18]. This
type of long-period pulsation can also be observed by careful control of the
polarization state in our MO under a pump power of 1.15 W. Figure 4(a) plots
the pulse train of an observed period-9 pulsation. Figure 4(b) records the
spectral evolution of 200 round trips, where we also notice abrupt spectral
breathing during long-period pulsation. The spectrum widens about 1.5 times
compared to the narrowest one (from 10 nm to 15 nm for 3 dB bandwidth), which
creates significant spectral content in the wings of the spectrum. We plot the
spectral evolution within one pulsation period in Fig. 4(c). The figure shows
that the gradually decreases and transfers to the spectral wings during
pulsation and this evolution repeats in 9 round trips. Note that the
periodicity is not very robust in this long period pulsation, which can be
seen from the power variation in Fig. 4(b).

Beyond this long period pulsation, chaotic state can also be realized in our
MO. The route towards the chaotic state can also be controlled by fine control
of the waveplates (for example, QWP in Fig. 1(a)). When the angle of the
quarter waveplate is about 165∘, the MO operated in the stable single pulse
state (see Fig. 4(d)). The pulse energy decreases as the waveplate rotates
counterclockwise. Period-2 pulsation appears at a angle of 153∘. Further
tuning of the quarter waveplates leads to a chaotic state (Fig. 4(e)). The
pulse train can no longer repeat itself in this state. Further experiments
show that all soliton forms (single soliton, multi-soliton and soliton
molecule) can bifurcate from stable operation to multi-period pulsating and
even to chaos, which suggests that chaotic pulsation is a ubiquitous
phenomenon in our MO.

##  IV Simulation of pulsation in Mamyshev oscillators

###  IV.1 Period-2 pulsation in simulations

To have a deeper understanding of our experimental observations, we use a
lumped model to simulate the pulsating DS dynamics in the MO. Details of the
simulations are provided in Appendix, which also includes parameters used in
generating the pulsation dynamics. Since polarization dynamics is important in
our experiment, we use a vector generalized nonlinear
Schr${\rm\ddot{o}}$dinger equation (NLSE) to simulate pulse propagation in the
fibers and Jones matrix to model the waveplates and PBS. Consistent with
experiments, small filter separation can lead to pulsation in simulations.
Figure 5(a) shows an example of generating pulsating DSs by reducing the
filter separation from 6 nm to 3 nm from a stable mode-locking state. In
agreement with the experiment, the pulsating DSs are also associated with
spectral breathing, with the 3 dB bandwidth varying from 10 nm to 12 nm in the
pulsation (see inset in Fig. 5(a)). In another case, pulsation can also
transition from stable mode-locking by adjusting the waveplates with the
filter separation and bandwidth being 3 nm and 0.8 nm, respectively (see Fig.
5(b)).

The pulse energy change during pulsation can be adjusted by increasing gain of
the active fibers and rotating the waveplates in simulations (see Appendix).
Figures 5(c, d) show a pulsation state with a relatively large pulse energy
change and the corresponding normalized spectra. The corresponding temporal
and spectral dynamics within two successive round trips are shown in Figs.
5(e, f). Pulsating DSs have slightly different peak powers at the same
position in these two round trips (see white line in Fig. 5(e)), resulting in
different spectral broadening in Fig. 5(f). The insets of Fig. 5(f) show the
spectra right after filter 1 whose relative spectral power can be referred to
the colorbar. The simulated pulse energy change is 3.3 times in Figs. 5(c-f).
Although this is still much smaller than the 40 times observed in Fig. 2(e),
we believe a larger energy change can be possible by further exploration of
the parameters. Simulation shows that polarization effects and using PBS as
the output port are important to have a relatively large pulse energy change
in pulsation. As a comparison, a typical pulsation state simulated using a
scalar model (see Appendix) only have a pulse energy change of $\sim$1.006
times (Fig. 5(g)). As noted in Section 3A, the intracavity pulse energy change
may be smaller than the PBS output. This is verified in the simulation; the
intracavity pulse energy only changes 1.3 times in the state of Fig. 5(c). As
an aside, QWP can be rotated over 360∘ in some filter and pump power settings
without losing mode-locking, which is also consistent with the observation
noted in Section 2.

Pulsating soliton molecule can also be simulated by increasing the gain
saturation energy $E_{\rm sat}$ (corresponding to increasing pump power in
experiments, see Appendix), as shown in Fig. 5(h). Single-shot spectra in two
consecutive round trips (Fig. 5(i)) and their zoom-in (Fig. 5(j)) show an
imperceptible change of spectral structure along with a mild energy change.
Figure 5(k) is the Fourier transform of the simulated spectra, indicating that
the pulse separation remains unchanged in this state.

###  IV.2 Explosion and chaotic pulsation in simulations

Period-3 pulsation and DS explosion can also be attained by increasing the
small signal gain $g_{0}$ as well as saturation energy $E_{\rm sat}$ and
finely tuning the waveplates (see Appendix). Figures 6(a-c), (b), and (c) show
the spectral evolution over 2000 round trips, over 50 round trips and single-
shot spectra within a pulsation period, respectively. Note that our simulation
time window is limited to 100 ps (only one DS included in the window) so as to
reduce the computation complexity. This simplification would not impact the
pulsation dynamics significantly, since the interaction between 2 DSs in Fig.
3 is negligible. Similar to experimental observation, the simulated period-3
pulsation is associated with soliton explosion (indicated by arrows in Fig.
6(a)). Zoom-in of the explosion is shown in Figs. 6(d, e). The spectrum
collapses into a distorted one (around the 922 nd round trip) and the pulse
energy changes irregularly during the explosion (white line in Fig. 6(d)). The
DS recovers to the regular pulsation state after about 5 round trips. This
explosion time scale is close to the measurements in Fig. 3. Figure 6(e) shows
another example of soliton explosion during pulsation where the explosion
lasts slightly longer ($\sim$30 round trips). Figures 6(f1, f2) show the
simulated temporal shift during explosion shown in Figs. 6(d, e),
respectively. Such a temporal shift during explosion was reported in refs.
[37, 19]. And it may arise from the spectral center frequency and group
velocity change during explosion (power variation during explosion can lead to
different spectral broadening in the MO). Note that the period-3 pulsation can
be obtained under the similar conditions as used in Fig. 5, while DS explosion
can only be observed by increasing $g_{0}$ and $E_{\rm sat}$ to relatively
large values (see Table 2). Therefore, we believe that high pump power is
essential to the emergence of soliton explosion in MOs, which is consistent
with the experimental observations. And it is in contrast to the explosion
observed in ref. [19], which occurs at a relatively low pump power.

Long period and chaotic pulsation can also be simulated by rotating the
waveplates (see Appendix for used parameters). Figure 6(g) shows the spectral
dynamics for a period-14 pulsation, while Fig. 6(h) shows three normalized
representative spectra in a pulsation period (the 1st spectrum corresponds to
the narrowest spectra in Fig. 6(g)). Further rotation of the waveplates can
generate chaotic pulsation as shown in Fig. 6(i). The corresponding pulse
energy change loses periodicity. These simulations are in qualitative
agreement with the observation in Fig. 4.

##  V Discussion and conclusion

The spectral regeneration and filtering in MOs make them distinct from other
mode-locked lasers. Our measurements unveil that short-period, long-period and
chaotic pulsations all exist in MOs. Under relatively high pump powers, DSs
can even explode in pulsation. Numerical simulation verifies the existence of
these dynamics in MOs. Our observation confirms that pulsating DSs are an
eigen-solution of the master equation governing the DS dynamics in MOs. Since
nonlinear gain/loss effects (e.g., quintic dissipative terms in CGLE) are
essential for the generation of pulsation in mode-locked lasers [7], the
saturable absorption based on the Mamyshev regenerator may provide these
effects (slow gain dynamics are not included in our simulation). However, NPR-
like effects may also contribute to needed nonlinear gain/loss terms and
impact the observed dynamics. It can be interesting to build a polarization
maintaining MO to eliminate the polarization dynamics and study the scalar
pulsation dynamics in MOs. Combining DFT with the time lens method [45] may
reveal more pulsation dynamics in MOs. Such future work can be helpful to
further reveal the role of Mamyshev regenerator in pulsation, experimentally
and theoretically. Finally, it can contribute to a better understanding of
complex pulsation dynamics including the DS explosion in pulsation. Moreover,
our work is based on an all-normal-dispersion MO and it can be interesting to
explore the pulsation dynamics in MOs with other dispersion configurations.

In conclusion, we have experimentally and numerically observed pulsating DSs
in a MO. These pulsating DSs arise under a relatively high pump powers and
small filter separation and bandwidths. We observe various pulsation dynamics
for mode-locking states comprising a single DS or 2 DSs. By controlling the
polarization state, dramatic pulse energy change with output pulse energy
change up to 40 times is possible. Soliton explosion during pulsation has also
been observed in the MO under a relatively high pump power. Chaotic pulsation
state also exists in the MO. Simulations are in qualitative agreement with the
observations. Our results not only show that narrow filter separation should
be avoided for stable operation of high power MOs but also add to the
pulsation dynamics in dissipative optical systems.

## Appendix

Table 1: Parameters of the optical fibers Fiber | Length (m) |  $\beta_{2}$ (ps2 km-1) |  $\beta_{3}$ (ps3 km-1 ) |  $\gamma$ (W-1 km-1) |  $L_{B}$ (m) | Gain bandwidth (nm)
---|---|---|---|---|---|---
YDF1 | 3.1 | 24.9 | 59 | 4.8 | 1 | 30
YDF2 | 3.1 | 24.9 | 59 | 4.8 | 1 | 30
SMF1 | 0.7 | 22.2 | 63.8 | 4.4 | 1 | NA
SMF2 | 2.1 | 22.2 | 63.8 | 4.4 | 1 | NA
Table 2: Other parameters used in the simulations Figure | Filter separation (nm) | Filter bandwidth (nm) | $g_{0}\rm(m^{-1})$ | $E_{\rm sat}\rm(nJ)$ | $\phi_{\rm QWP}(^{\circ})$ | $\phi_{\rm HWP1}(^{\circ})$
---|---|---|---|---|---|---
Fig. 5(a) | 6 to 3 | 0.8 | 10 | 1 | 90 | 20
Fig. 5(b) | 3 | 0.8 | 9 | 1 | 90 to 50 | 50
Fig. 5(c) | 3 | 0.7 | 13 | 1.1 | 83 | 23
Fig. 5(g) | 2.5 | 1 | 10 | 1 | NA | NA
Fig. 5(k) | 4 | 0.9 | 5.8 | 12.2 | 19.1 | 20
Fig. 6(a) | 3 | 0.7 | 31.3 | 3.8 | 89.86 | 20
Fig. 6(g) | 2.5 | 0.8 | 10 | 1 | 90 | 26
Fig. 6(i) | 2.5 | 0.8 | 10 | 1 | 90 | 24

Simulations is based on a lumped model. In this model, pulse propagation in
the fibers is simulated by the coupled generalized nonlinear
Schr${\rm\ddot{o}}$dinger equation (NLSE) [46]:

$\displaystyle\frac{\partial u}{\partial z}=i\beta u-\delta\frac{\partial
u}{\partial t}-\frac{i\beta_{2}}{2}\frac{\partial^{2}u}{\partial
t^{2}}+i\gamma\left(|u|^{2}+\frac{2}{3}|v|^{2}\right)u+$
$\displaystyle\frac{i\gamma}{3}v^{2}u^{*}+\frac{g}{2}u+\frac{g}{2\Omega_{g}^{2}}\frac{\partial^{2}u}{\partial
t^{2}}$ (1) $\displaystyle\frac{\partial v}{\partial z}=-i\beta
v+\delta\frac{\partial v}{\partial
t}-\frac{i\beta_{2}}{2}\frac{\partial^{2}v}{\partial
t^{2}}+i\gamma\left(|v|^{2}+\frac{2}{3}|u|^{2}\right)v+$
$\displaystyle\frac{i\gamma}{3}u^{2}v^{*}+\frac{g}{2}v+\frac{g}{2\Omega_{g}^{2}}\frac{\partial^{2}v}{\partial
t^{2}}$ (2)

where $u$ and $v$ are the slowly varying pulse envelopes of fast and slow axes
of the cavity, respectively. $2\beta=2\pi/(\lambda\Delta n)=2\pi/L_{B}$ and
$2\delta=\beta\lambda/(\pi c)$ are the wave number difference and inverse
group velocity difference, respectively ($L_{B}$ is the beat length).
$\beta_{2}$ is the group velocity dispersion, $\gamma$ is the Kerr
nonlinearity coefficient,
$g=g_{0}\exp\left[-\int\left(|u|^{2}+|v|^{2}\right)dt/E_{\rm sat}\right]$ is
the gain, with $g_{0}$ and $E_{\rm sat}$ being the small signal gain and gain
saturation energy, respectively; $\Omega_{g}$ is the gain bandwidth. For
passive fibers, small signal gain is chosen as $g_{0}$=0 m-1. The parameters
of the fibers are given in Table 1. The fiber parameters are fixed for all
simulations.

Simulations start from two weak pulses along the fast and slow axes of the
cavity (a weak pulse for the scalar model) to speed up the convergence of our
simulations. Starting from the Yb-doped gain fiber (YDF1), the weak pulses are
amplified and experience spectral broadening due to SPM. The spectrum further
broadens in the passive fiber (SMF1) before filtered at both axes by filter 1
with a gaussian profile. The filtered pulse experiences amplification and
spectral broadening for the second time in YDF2 and SMF2. Then the pulse
passes a QWP, a PBS and two HWPs, which are modeled by the Jones matrixes
(angles are defined with respect to the x-axis, which is assumed as one of
fiber eigen-polarization axes). The PBS is inserted before the second HWP
(HWP2) to output the x-polarized component. Finally, the pulse is filtered by
filter 2, whose transmission window is shifted to longer wavelengths from
filter 1. Then the loop is closed. HWP 2 is fixed to have an orientation of
90∘. The parameters used to generate the various pulsation dynamics are listed
in Table 2. As for the scalar model used in Fig. 5(g), we used a model similar
to ref. [32]. We neglect the polarization effect of all components and get the
result from a linear output port with a coupling ratio of 20 $\%$ after the
SMF2.

The parameters used in simulations are in reasonable agreement with
experimental measurements. Some conditions may differ from the experiments;
for example, the active fibers in two arms share the same gain parameters
($g_{0},E_{\rm sat}$) and the filter properties are adjusted in simulations
(fixed in experiments). These slight discrepancies arise from the difficulty
in modeling the complex MO accurately and the need to reduce the parameter
space to explore and computation complexity. However, the discrepancies will
not cause the loss of generality. For example, the simulated pulsating DS and
other results we discussed above can be attained under different lengths of
YDF and SMF by adjusting $g_{0}$ and $E_{\rm sat}$ as well as filter
parameters.

###### Acknowledgements.

This work is supported by the National Natural Science Foundation of China
(NSFC) (51527901, 61575106), and fundamental Research Funds for the Central
Universities (BUPT 2021RC08). The authors declare no conflicts of interest.
Data underlying the results presented in this paper are not publicly available
at this time but may be obtained from the authors upon reasonable request. ….

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