# Soliton-pair dynamical transition in mode-locked lasers

Kfir Sulimany  Racah Institute of Physics, The Hebrew University of Jerusalem,
Jerusalem 91904, Israel  Offek Tziperman  Racah Institute of Physics, The
Hebrew University of Jerusalem, Jerusalem 91904, Israel  Yaron Bromberg  Racah
Institute of Physics, The Hebrew University of Jerusalem, Jerusalem 91904,
Israel  Omri Gat  Racah Institute of Physics, The Hebrew University of
Jerusalem, Jerusalem 91904, Israel  Corresponding author:
<EMAIL_ADDRESS>
###### Abstract

Multi-soliton mode-locked laser waveforms are much sought as a complex light
source for research and applications, but are difficult to manipulate
effectively because of the elaborate and diverse interactions present. Here we
present an experimental, numerical, and theoretical study of the interaction
and control of the internal dynamics of a two-soliton waveform in a mode-
locked fiber laser. Using the pumping current as a control agent, we
demonstrate experimentally a two-orders-of-magnitude reduction in the
separation of a bound soliton pair, inducing a dynamical transition between a
loosely bound, phase-incoherent pair, and a tightly bound phase-locked pair.
We show on the basis of a Haus-model numerical simulation of the recently-
proposed noise-mediated interaction theory, that the pulse separation and
dynamical transition are governed by the shape of the dispersive-wave
pedestals. We explain the dynamical transition by showing analytically, within
a simplified theory, that the noise-mediated interaction becomes purely
attractive when the pedestals energy drops below a threshold. This work
demonstrates the ability to control the waveform through the interaction
forces, without external intervention in the light propagation in the laser.

##  1 Introduction

Dissipative solitons are localized waves in open systems far from equilibrium,
whose existence results from a balance of dissipative, nonlinear, and
dispersive effects. They appear in numerous physical areas including reaction-
diffusion systems, neurological and ecological sciences, fluid dynamics, and
photonics [1, 2, 3, 4, 5, 6, 7]. These nonlinear systems, which exhibit a
plethora of unique nonlinear phenomena, are unified by the presence of
solitons. Photonics provides an outstanding platform to study complex
nonlinear interactions of dissipative solitons, which naturally emerge in
strongly-pumped passively mode-locked lasers [8, 9, 10, 11, 12, 13, 14]. In
particular, collective nonlinear phenomena such as the aggregation of solitons
into bunches have created great interest due to their potential applications
in optical communication systems, and their resemblance to molecules [15, 16,
17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28]. ††$\dagger$ The first two
authors contributed equally to this work.

Soliton molecules and other multi-soliton patterns exhibit an extensive pallet
of short- and long-range pulse interactions. Short-range interactions take
place when pulse tails overlap [29, 30, 31, 32, 21, 33]. However, interactions
over distances that are orders of magnitude longer than the width of
individual solitons must be mediated by, for example, the gain medium [19, 34]
or by an acoustic response [17, 35, 36]. Furthermore, non-soliton components
of optical pulses, such as dispersive-wave pedestals can give rise to
interactions with an intermediate range, significantly longer than the soliton
width, but much shorter than the cavity length [18, 37]. Recently, optical
modulation components have been utilized to induce transitions between complex
multi-soliton patterns [36, 38, 39]. However, a dynamical quasi-static
transition between incoherently coupled solitons, separated by hundreds of
times their width, and coherently coupled overlapping solitons, has not been
reported to date. It is desirable to realize a laser with a straightforward
and reproducible protocol for driving such transitions, supported by a
theoretical underpinning.

Recently, we introduced a long-range interaction mechanism arising from the
effect of gain depletion in the presence of noisy quasi-CW light in mode-
locked lasers [34, 40]. The interaction results from the reduction of optical
fluctuations due to gain depletion following the passage of a pulse through
the gain medium. Suppression of fluctuations decreases the temporal jitter of
subsequent pulses and in this way biases the jitter and sets a pulse drift
motion. This noise-mediated interaction (NMI) mechanism shares some intriguing
properties with the Casimir effect in quantum electrodynamics, where
macroscopic objects experience an effective interaction due to the suppression
of electromagnetic field fluctuations [41]. In both the NMI and the Casimir
effect, distant objects inhibit microscopic fluctuations in extended
electromagnetic modes. The consequent breaking of spatiotemporal homogeneity
gives rise to the weak interactions among these objects.

Optical solitons in anomalous-dispersion mode-locked lasers are often
accompanied by weak and broad dispersive-wave pedestals [42]. When solitons
are well-separated, the NMI is generally attractive [34], but when two
solitons are close enough that their pedestals overlap, the interaction can
become repulsive; consequently, NMI can lead to the formation of bound states
of solitons whose steady state separation is comparable with the pedestal
width. The phases of solitons in such bound states are not coherently locked,
so they can be viewed as _loosely bound_. In contrast, when both the temporal
separation and the relative phase of soliton pairs are locked, they are
_tightly bound_ by coherent overlap interactions. In practice, the complex
(1+1)-dimensional space-time nonlinear dynamics is driven by multiple
interaction mechanisms where both time scales and space scales span many
orders of magnitudes. Therefore, a precise control over the separation of
bunched solitons is still lacking.

Here, we experimentally observe a dynamical transition between loosely- and
tightly-bound soliton states in a passively mode-locked fiber laser. We use
the pumping current as a control knob that drives the transition as
illustrated in Figure 1. When we pump the laser above the gain threshold for
single-pulse operation (inner arc), a pair of solitons is formed [43] (central
and outer arcs). At high gain values the solitons are loosely bound (outer
arc) by the NMI, at a separation time dictated by their pedestals as shown in
Figure 1. We then lower the pumping current and observe a reduction of the
solitons’ separation time as a result of the reduction of the pedestals
energy. We achieve a control of the separation time over a range of hundreds
of times the solitons duration with an accuracy higher than the soliton
duration. When the soliton separation drops under a threshold, the coherent
interaction becomes dominant and drives the dynamical transition to a tightly
bound state (central arc).

We show that the transition is induced by NMI using a stochastic master
equation that we developed for capturing the NMI mechanism, which reproduces
the experimentally observed dynamical transition. Furthermore, we use a
simplified analytical model of NMI to demonstrate that the transition point is
set by the energy carried by the pedestals and by their temporal width. Since
both parameters depend on the gain coefficient, which itself depends on the
pumping current, it is possible to actively switch between loosely- and
tightly-bound solitons by simply tuning the pumping current of the laser,
providing a significant step towards applications of multipulse laser
waveforms.

Figure 1: Soliton pair dynamical transition in a mode-locked laser. The laser
gain is used as a knob to control a dynamical transition between loosely and
tightly bound soliton pairs (central and outer arcs). Above the gain threshold
for stable single-pulse operation (inner arc), a steady state of two solitons
is obtained. At high gain values, a loosely bound state (outer arc) is
obtained as a result of the noise mediated interaction mechanism, which is
driven by the pedestals around the solitons. By decreasing the gain, the
soliton separation time decreases over a range spanning hundreds of times
their duration and at an accuracy higher than the soliton duration. At lower
gain values, the soliton separation becomes comparable to the duration of each
soliton, and a dynamical transition to a tightly bound state (central arc) is
observed.

##  2 Dynamical transition in a mode-locked fiber laser

###  2.1 Experimental observation

To experimentally study the dynamics of soliton pairs in laser cavities we
constructed a mode-locked laser that is based on an all-fiber integrated ring
cavity operated in the anomalous dispersion regime [44]. Passive mode-locking
is achieved by adding a segment of a multimode fiber (MMF) to the single-mode
fiber cavity, as depicted in Figure 2 [45, 46, 47, 48, 49, 50, 51, 52]. Since
light entering the MMF can excite multiple fiber modes, the fraction of the
light that is coupled back to the single mode fiber depends on the
interference of the excited fiber modes. At high intensities the interference
is modified by Kerr nonlinearity, making the transmission through the MMF
intensity-dependent. By tweaking the properties and conformation of the MMF,
the intensity-dependent transmission can be adjusted to realize a saturable
absorber, with a controllable amount of linear loss, saturated loss, and
saturation power. These additional degrees of freedom in the saturable
absorber enable us to realize a wide range of multi-soliton waveforms [53].
For more details regarding the experimental methods and components used in
this work please see supplementary information.

Figure 2: The mode-locked fiber laser used for studying the dynamical
transition between loosely and tightly bound solitons. The cavity length is 14
m, including a 2.7 m Erbium-doped fiber gain medium. The Erbium-doped fiber is
core pumped by a 976 nm diode laser through a wavelength-division multiplexer.
Passive mode-locking is achieved by adding a 1.5 m long segment of a graded-
index multimode fiber. A polarization-independent optical isolator ensures
unidirectional lasing, two fiber polarization controllers tune the overall low
cavity birefringence and a 90/10 coupler provides the laser output. The laser
output power is measured by a fast photodiode and analyzed with an optical
spectrum analyzer and a real-time oscilloscope. SA: saturable absorber, SMF:
single mode fiber, MMF: multimode fiber, EDF: Erbium-doped fiber, WDM:
wavelength-division multiplexer, ISO: optical isolator, PC: polarization
controller, OSA: optical spectrum analyzer, OSC: oscilloscope. Figure 3:
Soliton dynamical transition in a mode-locked fiber laser. (a) The laser
output temporal intensity profile measured as a function of the pumping
current using a fast photodetector. Cross-sections at pumping currents of 123
mA (blue arrows) and 120 mA (red arrows), plotted in panel (b) with
corresponding colors, demonstrate that the total energy in the soliton
waveform remains nearly constant even when the inter-soliton separation is too
small to be resolved by the oscilloscope. (c) The laser output spectrum
measured as a function of the pumping current. Cross-sections at pumping
currents of 123 mA (blue arrows) and 120 mA (red arrows), plotted in panel (d)
with corresponding colors, demonstrate the two soliton binding regimes:
loosely bound pairs with fluctuating relative phase and tightly bound pairs
with coherently locked phases.

Our experimental protocol starts by generating stationary states of soliton
pairs by increasing the pumping current to 124 mA, beyond the threshold for
stable single-pulse operation. We then decrease the pumping current and study
the dynamics of the soliton pairs. We start by studying the time-domain traces
of the output intensity (Figure 3(a)), exhibiting two peaks that approach
each-other as the pumping current is decreased. The soliton duration is sub-
picosecond, estimated by the spectral profile presented below. However, the
width of the pulses in Figure 3 (a) is determined by the temporal resolution
of our detection apparatus, which is limited by the 8 GHz bandwidth of the
oscilloscope. For pumping currents below 121 mA the soliton separation is too
small to be resolved in Figure 3 (a), yet the approximate conservation of
total energy in the soliton waveform indicates that the number of solitons has
not changed, as expected from the clamping of the soliton energy in anomalous-
dispersion passively mode-locked lasers [54, 55, 56]. Figure 3 (b) shows
examples of temporally resolved (blue) and unresolved (red) two-pulse
waveform.

We continue by studying the output spectrum as a function of the pumping
current (Figure 3(c)). The central wavelength is 1561 nm, and the full width
at half maximum (FWHM) bandwidth is $\sim$3.8 nm. At pumping currents above
120 mA we do not observe spectral interference fringes, indicating that the
solitons are not phase locked and they form a loosely bound state. At pumping
currents below 120 mA, we observe an abrupt appearance of spectral fringes,
indicating that the solitons interact coherently so that their phases lock,
i.e. the solitons are tightly bound. In both regimes, narrow sidebands appear
at the primary Gordon-Kelly resonances [57, 58], 1553 nm and 1567 nm, and
their power grows with increasing pumping current. Figure 3(d) presents two
spectral cross-sections of Figure 3(c), at pumping currents of 123 mA (blue)
and 120 mA (red), exhibiting the Kelly sidebands and the interference fringes
in the latter. Below 120 mA in particular, the spacing of the spectral fringes
implies that waveform consists of two solitons, with a separation time of a
few picoseconds.

###  2.2 Numerical simulations

The temporal dynamics of dissipative solitons in fiber lasers is conveniently
modelled by the Haus master equation for mode-locked lasers [59]. To show that
the dynamical transition is caused by the NMI mechanism, we derived a
generalized master equation that includes all of the physical processes needed
to observe both the NMI and the coherent nonlinear interaction:

$\begin{gathered}\frac{\partial u}{\partial
z}=-\frac{l}{2}u-\frac{i\beta^{\prime\prime}}{2}\frac{\partial^{2}u}{\partial
t^{2}}+\frac{(g(u)+\kappa(z,t))}{2}(u+\frac{1}{\Omega_{g}^{2}}\frac{\partial^{2}u}{\partial
t^{2}})\\\ +(i\gamma+\delta-\sigma|u|^{2})|u|^{2}u+\eta(z,t)\ ,\end{gathered}$
(1)

where $u=u(z,t)$ is the complex scalar envelope of the electric field, $t$ is
the retarded time, $z$ is the propagation distance, $l$ is the linear loss
coefficient, $\beta^{\prime\prime}$ is the group velocity dispersion,
$g,\kappa$ are the gain coefficients to be described below, $\gamma$ is the
Kerr coefficient, $\Omega_{g}$ is the gain bandwidth, and $\delta,\,\sigma$
are the fast saturable absorption constants, modeling bleaching and saturation
of the absorber, respectively. $\eta(z,t)$ is a complex Gaussian white
uncorrelated noise term,
$\langle\eta(z,t)^{*}\eta(z^{\prime},t^{\prime})\rangle=\zeta\delta(z-z^{\prime})\delta(t-t^{\prime})$,
where $\zeta$ is the noise strength. The noise term generates a non-
homogeneous quasi-CW noise floor in the waveform that is the medium of the
noise-mediated interaction mechanism.

Figure 4: Soliton dynamical transition simulated by a stochastic Haus master
equation. (a) The simulated laser temporal intensity waveform as a function of
propagation distance. After $8.7\cdot 10^{5}$ roundtrips we reduced abruptly
the unsaturated gain from $g_{0}=0.65$ to $g_{0}=0.58$; the high and low gain
regimes are separated by the white dashed lines. Temporal cross-sections from
before the unsaturated gain was changed (blue arrow) and after the tightly
bound pair was formed (red arrow), are plotted in panel (b) with corresponding
colors. The time traces exhibit around each soliton a symmetric, exponentially
decaying pedestal, $\sim 100\,$ times wider than the solitons. (c) The
simulated laser output spectrum as a function of propagation distance. Cross-
sections of the spectrum before the gain was changed (blue arrow) and after
the solitons reached a tightly bound state (red arrow) are plotted in panel
(d) with corresponding colors, demonstrating the two binding regimes: loosely
bound pairs with fluctuating relative phase and tightly bound pairs with
coherently locked phases. To depict the phase-locking each spectrum was
averaged over 86 sequential traces spanning $8.6\cdot 10^{5}$ cavity lengths.

Since the amplifier gain is depleted by the passage of a pulse, and gain
recovery is slow, the depleted gain profile $g(u)+\kappa(z,t)$ is nonuniform,
which means that the quasi-CW intensity is not uniform either. The depleted
gain consists of two terms; the temporally-homogenous mean saturated gain
$g(u)$ and the inhomogenous gain $\kappa$ that has zero mean and is a key
element of the NMI interaction. If the relaxation of the gain medium is much
slower than the pulse repetition rate, the mean saturated gain $g(u)$ can be
well approximated by:

$g(u)=\frac{g_{0}}{1+P_{\text{av}}(u)/P_{\text{sat}}}$ (2)

where $g_{0}$ is the unsaturated gain, $P_{\text{sat}}$ is the saturation
power of the amplifier, and $P_{\text{av}}(u)$ is the mean intracavity power

$P_{\text{av}}(u)=\frac{1}{t_{R}}\int_{-t_{R}/2}^{t_{R}/2}|u(z,t)|^{2}dt$ (3)

in which $t_{R}$ is the round-trip time. The inhomogenous gain is governed by
the equation

$\kappa(z,t)=-g_{d}\int_{-t_{R}/2}^{t}|u(z,t)|^{2}dt+(t+\frac{t_{R}}{2})g_{d}P_{\text{av}}(u)$
(4)

where $g_{d}$ is the gain depletion coefficient. To ensure that $\kappa(z,t)$
has a temporal mean of zero, we subtract its mean after using (4). In Eq. (4)
we approximate the recovery rate by a constant because the recovery time of
the Erbium-doped fiber gain medium is much longer than the round-trip time
[60].

In the Haus model, the effects of the laser elements (nonlinear propagation,
dispersion, gain, and saturable absorption) are averaged over one round trip
in the laser cavity to give a continuous-variable equation that allows for
faster simulation. However, it cannot reproduce the Gordon-Kelly sidebands
[57, 58] that are essential for the dynamical transition. Thus, although we
are not attempting to make a one-to-one numerical model of the experiment, we
introduce an additional discrete loss component in each round trip, that
models the cavity inhomogeneities that produce the dispersive-wave pedestals
[42].

We use the split-step method to numerically integrate Eq. (1); the numerical
method is based on [61, 62] with the additional elements that drive the noise-
mediated interaction as described below. The simulation parameters and a
sketch of the code are given in the supplementary information, in addition to
the complete open source Matlab package used in this work, which is available
online [63].

To observe the soliton-pair dynamical transition we apply the following
protocol. First, we initiate the cavity with two solitons at an estimated
steady state separation time. This estimation is based on simulations with
different initial separations and unsaturated gain values. The unsaturated
gain values are chosen well above the threshold for a stable soliton-pair
state and slightly below the value for a formation of a third soliton. Next,
we propagate the input field many cavity lengths, and then abruptly decrease
the unsaturated gain to a value slightly above the value for which one of the
solitons is annihilated. Such relatively low gain values support coherently
locked states of soliton-pairs.

Figure 4(a) presents time-domain traces of the cavity waveform intensity as a
function of the propagation distance. At the beginning of the propagation
$g_{0}=0.65$, and the distance between the solitons jitters around their
initial separation, approximately 200 times larger than their width, but there
is no significant drift. The phases of the two solitons are not locked (see
below), implying that the calculated waveform is in a loosely bound steady
state. After $8.7\cdot 10^{5}$ cavity lengths, we decrease abruptly the
unsaturated gain to $g_{0}=0.58$, and the solitons start to drift towards each
other, until a new, tightly bound, steady state is reached at $1.4\cdot
10^{6}$ cavity lengths. Note that the simulated waveforms are snapshots of a
trajectory between two steady states at two values of the unsaturated gain, in
contrast to the experimental waveforms that are all in steady states

Decreasing the unsaturated gain also changes the amount of energy carried by
the pedestals that surround each soliton. The pedestal energy drops by 75%
after the abrupt reduction in $g_{0}$ (Figure 4(b)). This drop is consistent
with the experimentally observed changes in the Gordon-Kelly sidebands as the
pumping current was reduced (see below), although the pedestal intensity was
too weak to measure in our experiments.

Figure 4(c) presents the simulated output spectrum as a function of the
propagation distance. The FWHM bandwidth of the soliton spectrum is $\sim$0.4.
Narrow sidebands appear at the Gordon-Kelly resonances, of which the primary
ones are detuned by $\pm 0.2$ from the pulse peak. Their frequency detuning
increases and their peak power decreases the moment we lower the unsaturated
gain, as was also observed experimentally. In contrast with the experimental
data, where the spectrum is averaged over a few milliseconds, the numerical
spectral traces are instantaneous, and therefore all of them exhibit fringes
of interference between the two soliton spectra. Therefore, to distinguish
between phase-locked and unlocked states we compare averages of successive
spectral traces in Figure 4(d). For the low unsaturated gain value we observe
stable spectral fringes in the steady state, indicating that the solitons are
tightly bound (red curve). In contrast, for the high unsaturated gain value we
do not observe stable interference fringes (blue curve), so the pulse phases
are not coherently locked; since the mean separation in this state is
constant, the solitons are loosely bound.

Our simulation paves the way for investigating many complex phenomena where
the noise mediated interaction takes a major role. Here, we focus on the
dynamical transition between the loosely and tightly bound states. We found
this task to be much harder than simulating scenarios where only one of the
interactions plays a significant role, such as when observing either a tightly
or a loosely bound state, because of the large range of both the fast time
scale and the slow time (space) scale, associated with the two types of bound
states.

###  2.3 Analytic model

In order to obtain a simple expression for the loosely bound steady state
separation time, we approximate the laser waveform by an incoherent sum of two
non-overlapping soliton waveforms, each of which is accompanied by a weak and
broad dispersive-wave pedestal generated by the Gordon-Kelly resonances [42].
The soliton intensity profile is approximated by a delta function because its
duration is much shorter than all other time scales, and the pedestal waveform
is modeled by an exponential envelope decaying over a time scale $w$ that is
much longer than the pulse width but much shorter than the cavity round-trip
time $t_{R}$. The waveform intensity profile can therefore be approximated by

$\displaystyle I(t)$ $\displaystyle=I_{p}(t-t_{1}(z))+I_{p}(t-t_{2}(z))\ ,$
(5) $\displaystyle I_{p}(t)$
$\displaystyle=E_{s}\delta(t)+\frac{E_{p}}{2w}e^{-|t|/w}\ ,$ (6)

$E_{s}$, $E_{p}$ being the pulse and pedestal energy, respectively. The NMI
makes $t_{1}$ and $t_{2}$ drift under an effective potential until the mean
separation $t_{s}=|\langle t_{2}-t_{1}\rangle|$ reaches a steady state value
that minimizes the effective potential [34] (see supplementary information for
more details):

$\tilde{t}_{s}=\tilde{w}\ln\Bigl{(}\frac{\tilde{E}_{p}}{2\tilde{w}}\Bigr{)}\
,\qquad\textnormal{if }\tilde{E}_{p}>2\tilde{w}$ (7)

where $\tilde{t}_{s}=t_{s}/t_{R}$, $\tilde{w}=w/t_{R}$, and
$\tilde{E}_{p}=E_{p}/(E_{s}+E_{p})$. In contrast, if
$\tilde{E}_{p}<2\tilde{w}$, the noise-mediated interaction is always
attractive, so that the minimum of the effective potential appears at zero
separation time $\tilde{t}_{s}=0$. Notice that Eq. (7) corrects an algebraic
mistake in the separation time formula that originally appeared in [34].
Figure 5 is a color map showing the normalized steady state separation time
$\tilde{t}_{s}$ as a function of the normalized pedestal width $\tilde{w}$ and
the normalized pedestal energy $\tilde{E}_{p}$, for the typically small values
of these variables. The black area represents zero steady state separation
time, where the NMI is always attractive.

We next use these results to explain the dynamical transition between loosely
and tightly bound soliton steady states observed in the experiments when the
pumping current is lowered: when we reduce the pumping current, the normalized
pedestal energy decreases, which in turn shortens the separation time of the
soliton pair, according to Eq. (7). As the pumping current is lowered,
therefore, two scenarios may occur. The first scenario is that the separation
time decreases until the coherent interaction takes over at short separations.
In this scenario, a dynamical transition occurs, after which the separation
between the pulses, together with their relative phase, are locked by the
coherent overlap interaction, as explained in the previous section. In the
second scenario, when the pumping current is lowered, the soliton-pair state
becomes unstable to pulse annihilation before the threshold for the transition
between the loosely and tightly bound states is reached.

Figure 5:  Soliton steady states analysis. Theoretically predicted normalized
separation time $\tilde{t}_{s}$ of the steady state of two solitons,
normalized by the round trip time, plotted as a function of the pedestal width
and energy, normalized by the total pulse energy, and the cavity round trip
time respectively. Non-zero values of $\tilde{t}_{s}$ are associated with
loosely bound states, while $\tilde{t}_{s}=0$ corresponds to the regime where
the NMI is always attractive and hence tightly bound steady states are
observed (black region). The boundary between the $\tilde{t}_{s}>0$ and
$\tilde{t}_{s}=0$ marks a dynamical transition line between loosely and
tightly bound states.

##  3 Discussion

The principles of NMI are well in agreement with the experimental
observations. Specifically, decreasing the pump power results in weaker
pedestal energies and hence shorter separations of loosely bound solitons,
where below some pumping threshold tightly bound pairs are formed. This
hypothesis is strongly supported by the numerical simulation results, that are
able to reproduce both the NMI drift of solitons toward loosely bound
configurations for large gain values and the sharp diminishing of pedestal
intensity profile, as well as the ensuing NMI attraction leading to the onset
of overlap soliton interaction and the formation of tightly bound states when
the gain is lowered.

The NMI mechanism is probably the main source of long-range inter-pulse
interactions in most multipulse mode-locked fiber lasers. Thus, we seek to
explain the dynamical transition between loosely and tightly bound soliton
states in the NMI framework. In most complex dissipative systems, a
theoretical modeling cannot provide accurate prediction for the transition
parameters, since it often oversimplifies the complex dynamics in the system.
In the present work, the NMI theory makes quantitative predictions for the
soliton separation times in terms of the pedestal properties, but direct
quantitative comparison of theory and experiment is not possible. One issue is
that the existing NMI calculations are based on averaged pedestal width and
energy, which in practice can oscillate significantly during a laser round
trip, making the measurement of the average pedestal energy at the output
coupler unreliable. More importantly, in our setup we could only estimate the
pedestal duration indirectly from the width of Gordon-Kelly sidebands,
obtaining a poor accuracy. Alternatively, the pedestal waveforms can be
estimated for different pumping currents using the model presented in [42].
Unfortunately, the prediction of this mode disagrees with our experimental and
numerical simulations: While the theoretical model predicts that with weaker
pumping the pedestal becomes temporally narrower, so that its energy decreases
while its peak power remains fixed, in experiments and simulations we find
that when the pumping current is decreased, the pedestal duration hardly
changes, while the peak power decreases by a large factor. We attribute this
discrepancy to the assumption of fixed soliton amplitude in [42]. While it is
well established that the amplitudes of solitons in multipulse mode-locked
lasers are clamped to an optimal operation point [64], residual variations
around it may remain, yielding large variations in the pedestal peak power.

##  4 Conclusions and outlook

This paper presents a dynamical transition of two-soliton waveforms in a mode-
locked laser, in which the pulse separation time is governed by the long-range
noise-mediated interaction (NMI) mechanism and a coherent overlap interaction.
In the loosely bound pair regime, where the separation between the pulses
ranges from a few tens to a few hundreds of picoseconds, the overlap
interaction is negligible so that the soliton phases are not coherently
locked, and the steady state separation is determined by the balance between
long-range attraction and repulsion due to the dispersive-wave pedestals of
the solitons.

When the pumping current is lowered, the pedestals become weaker, and the
steady state separation between the solitons decreases; when the separation
drops below a few dozen picoseconds, the coherent overlap interaction becomes
effective and dominates over the NMI, binding the solitons tightly, so that
their relative phases are locked and their separation is fixed. By tweaking
the properties of the saturable absorber, one can control whether a dynamical
transition between loosely and tightly bound states can be observed or not: if
the pumping current required for forming tightly bound states is too low for
supporting two solitons in the cavity, the transition will not be observed. In
this way, the NMI provides a systematic way to control the separation between
the pulses in a two-pulse mode-locked laser configuration, and in particular,
the transition between loosely and tightly bound configurations.

The theoretical explanation of the dynamical transition is based on numerical
results and on analytic results derived from the NMI model of [34]. The full
nonlinear dynamics are complex, and neither the numerical nor the analytic
model capture all its details. Nevertheless, the key effects that we describe,
namely the long-range NMI attraction, the pedestal-induced repulsion, the
reduction of pedestal energy for smaller gain and consequent destabilization
of the loosely bound configurations are independent of the fine details of the
experiment, and have all been confirmed numerically. We used the simplified
NMI model to show that the steady state soliton pair separation decreases to
zero when the pedestal energy decreases to a finite value, confirming that
unless a pulse is annihilated, reduction of pumping current eventually leads
to the dynamical transition from a loosely to a tightly bound configuration.

Our experimental and numerical results show that the current theoretical
understanding of dispersive wave dynamics in laser cavities, and the NMI
effective interactions, are incomplete. We believe that the theoretical models
can be improved by taking into account effects that have been neglected so
far, such as the variations of the pulse amplitude as a function of the gain,
oscillations of pedestal waveforms, and the finite response time of the quasi-
CW to the gain depletion, and in this way achieve quantitative agreement
between theory, numerical simulations and experiments. This will enable
precise control of the soliton separation and the dynamical transition with
the pump current. Furthermore, our numerical model provides a generic tool for
simulating the dynamics of multipulse configurations in mode-locked lasers
under different mechanisms of interaction.

Experimentally, better control of multi-soliton waveforms can be achieved by
directly modifying the pedestal waveform with a reconfigurable spectral filter
introduced into the laser cavity. Recently we have developed such a filter by
engineering the transmission matrix of a multimode fiber, in an all-fiber
configuration [65]. Such technology may allow fine control of the pulse
separations without modifying the laser gain. The dynamical transition, along
with the new understanding presented here of the interplay between the long
and short range interactions will allow new novel ways to control multipulse
bunches, providing a significant step towards the goal of multipulse waveform
engineering.

##  5 Acknowledgments

The authors kindly thank Ori Katz for fruitful discussions and suggestions.
This research was supported by the United States-Israel Binational Science
Foundation (BSF) (Grant No. 2017694) and the Israel Science Foundation (ISF)
(Grant No. 2403/20). KS and YB acknowledge the support of the Israeli Council
for Higher Education, and the Zuckerman STEM Leadership Program.

## References

  * [1] H-G Purwins, HU Bödeker, and Sh Amiranashvili.  Dissipative solitons.  Advances in Physics, 59(5):485–701, 2010.
  * [2] Neil Akhmediev and Adrian Ankiewicz.  Dissipative solitons in the complex ginzburg-landau and swift-hohenberg equations.  In Dissipative solitons, pages 1–17. Springer, 2005.
  * [3] Andreas Liehr.  Dissipative solitons in reaction diffusion systems.  Springer, 2013.
  * [4] Philippe Grelu and Nail Akhmediev.  Dissipative solitons for mode-locked lasers.  Nature photonics, 6(2):84–92, 2012.
  * [5] Philippe Grelu.  Nonlinear optical cavity dynamics: from microresonators to fiber lasers.  John Wiley & Sons, 2015.
  * [6] Yufeng Song, Xujie Shi, Chengfa Wu, Dingyuan Tang, and Han Zhang.  Recent progress of study on optical solitons in fiber lasers.  Applied Physics Reviews, 6(2):021313, 2019.
  * [7] J Javaloyes, P Camelin, M Marconi, and M Giudici.  Dynamics of localized structures in systems with broken parity symmetry.  Physical review letters, 116(13):133901, 2016.
  * [8] Piotr Ryczkowski, M Närhi, Cyril Billet, J-M Merolla, Goëry Genty, and John Michaël Dudley.  Real-time full-field characterization of transient dissipative soliton dynamics in a mode-locked laser.  Nature Photonics, 12(4):221–227, 2018.
  * [9] Xueming Liu and Meng Pang.  Revealing the buildup dynamics of harmonic mode-locking states in ultrafast lasers.  Laser & Photonics Reviews, 13(9):1800333, 2019.
  * [10] Xueming Liu, Daniel Popa, and Nail Akhmediev.  Revealing the transition dynamics from q switching to mode locking in a soliton laser.  Physical review letters, 123(9):093901, 2019.
  * [11] Junsong Peng and Heping Zeng.  Soliton collision induced explosions in a mode-locked fibre laser.  Communications Physics, 2(1):1–8, 2019.
  * [12] Qiu-Yi Ning, Hao Liu, Xu-Wu Zheng, Wei Yu, Ai-Ping Luo, Xu-Guang Huang, Zhi-Chao Luo, Wen-Cheng Xu, Shan-Hui Xu, and Zhong-Min Yang.  Vector nature of multi-soliton patterns in a passively mode-locked figure-eight fiber laser.  Optics express, 22(10):11900–11911, 2014.
  * [13] Zhenhong Wang, Zhi Wang, Yange Liu, Ruijing He, Jian Zhao, Guangdou Wang, and Guang Yang.  Self-organized compound pattern and pulsation of dissipative solitons in a passively mode-locked fiber laser.  Optics letters, 43(3):478–481, 2018.
  * [14] Mathias Marconi, Julien Javaloyes, Salvador Balle, and Massimo Giudici.  How lasing localized structures evolve out of passive mode locking.  Physical review letters, 112(22):223901, 2014.
  * [15] D. J. Richardson, R. I. Laming, D. N. Payne, M. W. Phillips, and V. J. Matsas.  320 fs soliton generation with passively mode-locked erbium fibre laser.  Electronics Letters, 27(9):730–732, April 1991.
  * [16] A. B. Grudinin, D. J. Richardson, and D. N. Payne.  Passive harmonic modelocking of a fibre soliton ring laser.  Electronics Letters, 29(21):1860–1861, October 1993.
  * [17] AN Pilipetskii, EA Golovchenko, and CR Menyuk.  Acoustic effect in passively mode-locked fiber ring lasers.  Optics letters, 20(8):907–909, 1995.
  * [18] AB Grudinin and S Gray.  Passive harmonic mode locking in soliton fiber lasers.  JOSA B, 14(1):144–154, 1997.
  * [19] J Nathan Kutz, BC Collings, K Bergman, and WH Knox.  Stabilized pulse spacing in soliton lasers due to gain depletion and recovery.  IEEE journal of quantum electronics, 34(9):1749–1757, 1998.
  * [20] D. Y. Tang, B. Zhao, D. Y. Shen, C. Lu, W. S. Man, and H. Y. Tam.  Compound pulse solitons in a fiber ring laser.  Physical Review A, 68(1):013816, July 2003.
  * [21] DY Tang, B Zhao, LM Zhao, and Hwa Yaw Tam.  Soliton interaction in a fiber ring laser.  Physical Review E, 72(1):016616, 2005.
  * [22] Veronika Tsatourian, Sergey V Sergeyev, Chengbo Mou, Alex Rozhin, Vitaly Mikhailov, Bryan Rabin, Paul S Westbrook, and Sergei K Turitsyn.  Polarisation dynamics of vector soliton molecules in mode locked fibre laser.  Scientific reports, 3(1):1–8, 2013.
  * [23] ZQ Wang, K Nithyanandan, A Coillet, P Tchofo-Dinda, and Ph Grelu.  Optical soliton molecular complexes in a passively mode-locked fibre laser.  Nature communications, 10(1):1–11, 2019.
  * [24] Andrey Komarov, Konstantin Komarov, and François Sanchez.  Quantization of binding energy of structural solitons in passive mode-locked fiber lasers.  Physical Review A, 79(3):033807, 2009.
  * [25] LM Zhao, DY Tang, Han Zhang, and X Wu.  Bunch of restless vector solitons in a fiber laser with sesam.  Optics express, 17(10):8103–8108, 2009.
  * [26] YF Song, H Zhang, LM Zhao, DY Shen, and DY Tang.  Coexistence and interaction of vector and bound vector solitons in a dispersion-managed fiber laser mode locked by graphene.  Optics express, 24(2):1814–1822, 2016.
  * [27] Chong Wang, Lu Wang, Xiaohui Li, Wenfeng Luo, Tianci Feng, Ying Zhang, Penglai Guo, and Yanqi Ge.  Few-layer bismuthene for femtosecond soliton molecules generation in er-doped fiber laser.  Nanotechnology, 30(2):025204, 2018.
  * [28] Chengbo Mou, Sergey V Sergeyev, Aleksey G Rozhin, and Sergei K Turitsyn.  Bound state vector solitons with locked and precessing states of polarization.  Optics express, 21(22):26868–26875, 2013.
  * [29] BA Malomed.  Bound states in a gas of solitons supported by a randomly fluctuating force.  EPL (Europhysics Letters), 30(9):507, 1995.
  * [30] NN Akhmediev, A Ankiewicz, and JM Soto-Crespo.  Stable soliton pairs in optical transmission lines and fiber lasers.  JOSA B, 15(2):515–523, 1998.
  * [31] Ph Grelu, F Belhache, F Gutty, and J-M Soto-Crespo.  Phase-locked soliton pairs in a stretched-pulse fiber laser.  Optics letters, 27(11):966–968, 2002.
  * [32] MJ Lederer, Barry Luther-Davies, HH Tan, C Jagadish, NN Akhmediev, and JM Soto-Crespo.  Multipulse operation of a ti: sapphire laser mode locked by an ion-implanted semiconductor saturable-absorber mirror.  JOSA B, 16(6):895–904, 1999.
  * [33] Michel Olivier and Michel Piché.  Origin of the bound states of pulses in the stretched-pulse fiber laser.  Optics express, 17(2):405–418, 2009.
  * [34] Rafi Weill, Alexander Bekker, Vladimir Smulakovsky, Baruch Fischer, and Omri Gat.  Noise-mediated casimir-like pulse interaction mechanism in lasers.  Optica, 3(2):189–192, 2016.
  * [35] Jae K Jang, Miro Erkintalo, Stuart G Murdoch, and Stéphane Coen.  Ultraweak long-range interactions of solitons observed over astronomical distances.  Nature Photonics, 7(8):657–663, 2013.
  * [36] Wenbin He, Meng Pang, Dung-Han Yeh, Jiapeng Huang, CR Menyuk, and P St J Russell.  Formation of optical supramolecular structures in a fibre laser by tailoring long-range soliton interactions.  Nature communications, 10(1):1–9, 2019.
  * [37] Jose M Soto-Crespo, Nail Akhmediev, Ph Grelu, and F Belhache.  Quantized separations of phase-locked soliton pairs in fiber lasers.  Optics letters, 28(19):1757–1759, 2003.
  * [38] Yi Zhou, Yu-Xuan Ren, Jiawei Shi, Huade Mao, and Kenneth KY Wong.  Buildup and dissociation dynamics of dissipative optical soliton molecules.  Optica, 7(8):965–972, 2020.
  * [39] Luca Nimmesgern, Cornelius Beckh, Hannes Kempf, Alfred Leitenstorfer, and Georg Herink.  Soliton molecules in femtosecond fiber lasers: universal binding mechanism and direct electronic control.  Optica, 8(10):1334–1339, 2021.
  * [40] Kfir Sulimany, Ohad Lib, Gilad Masri, Avi Klein, Moti Fridman, Philippe Grelu, Omri Gat, and Hadar Steinberg.  Bidirectional soliton rain dynamics induced by casimir-like interactions in a graphene mode-locked fiber laser.  Physical review letters, 121(13):133902, 2018.
  * [41] Hendrik BG Casimir and Dirk Polder.  The influence of retardation on the london-van der waals forces.  Physical Review, 73(4):360, 1948.
  * [42] Rafi Weill, Alexander Bekker, Vladimir Smulakovsky, Baruch Fischer, and Omri Gat.  Spectral sidebands and multipulse formation in passively mode-locked lasers.  Physical Review A, 83(4):043831, 2011.
  * [43] Brandon G Bale, Khanh Kieu, J Nathan Kutz, and Frank Wise.  Transition dynamics for multi-pulsing in mode-locked lasers.  Optics express, 17(25):23137–23146, 2009.
  * [44] Fengyan Zhao, Hushan Wang, Yishan Wang, Xiaohong Hu, Ting Zhang, and Ran Pan.  Observation of various bound states based on a graded index multimode fiber saturable absorber.  Laser Physics Letters, 17(2):025105, 2020.
  * [45] Shijie Fu, Quan Sheng, Xiushan Zhu, Wei Shi, Jianquan Yao, Guannan Shi, Robert A Norwood, and N Peyghambarian.  Passive q-switching of an all-fiber laser induced by the kerr effect of multimode interference.  Optics express, 23(13):17255–17262, 2015.
  * [46] Huanhuan Li, Zhaokun Wang, Can Li, Junjie Zhang, and Shiqing Xu.  Mode-locked tm fiber laser using smf-simf-gimf-smf fiber structure as a saturable absorber.  Optics express, 25(22):26546–26553, 2017.
  * [47] Zhaokun Wang, DN Wang, Fan Yang, Liujiang Li, Chunliu Zhao, Ben Xu, Shangzhong Jin, Shiying Cao, and Zhanjun Fang.  Er-doped mode-locked fiber laser with a hybrid structure of a step-index–graded-index multimode fiber as the saturable absorber.  Journal of Lightwave Technology, 35(24):5280–5285, 2017.
  * [48] Zhaokun Wang, DN Wang, Fan Yang, Liujiang Li, Chun-Liu Zhao, Ben Xu, Shangzhong Jin, Shi-Ying Cao, and Zhan-Jun Fang.  Stretched graded-index multimode optical fiber as a saturable absorber for erbium-doped fiber laser mode locking.  Optics letters, 43(9):2078–2081, 2018.
  * [49] Uğur Teğin and Bülend Ortaç.  All-fiber all-normal-dispersion femtosecond laser with a nonlinear multimodal interference-based saturable absorber.  Optics letters, 43(7):1611–1614, 2018.
  * [50] Fan Yang, DN Wang, Zhaokun Wang, Liujiang Li, Chun-Liu Zhao, Ben Xu, Shangzhong Jin, Shi-Ying Cao, and Zhan-Jun Fang.  Saturable absorber based on a single mode fiber–graded index fiber–single mode fiber structure with inner micro-cavity.  Optics express, 26(2):927–934, 2018.
  * [51] Tao Chen, Qiaoli Zhang, Yaping Zhang, Xin Li, Haikun Zhang, and Wei Xia.  All-fiber passively mode-locked laser using nonlinear multimode interference of step-index multimode fiber.  Photonics Research, 6(11):1033–1039, 2018.
  * [52] Fengyan Zhao, Ning Li, and Hushan Wang.  Generation of a noise-like pulse from an erbium-doped fiber laser based on nonlinear multimode interference.  Laser Physics, 30(12):125102, 2020.
  * [53] Fengyan Zhao, Hushan Wang, Xiaohong Hu, Yishan Wang, Wei Zhang, Ting Zhang, Chuandong Sun, and Zhijun Yan.  Experimental observation of bound solitons with a nonlinear multimode interference-based saturable absorber.  Laser Physics Letters, 15(11):115106, 2018.
  * [54] A.B. Grudinin, D.J. Richardson, and D.N. Payne.  Energy quantisation in figure eight fibre laser.  Electronics Letters, 28(1):67–68, January 1992.
  * [55] D. Y. Tang, L. M. Zhao, B. Zhao, and A. Q. Liu.  Mechanism of multisoliton formation and soliton energy quantization in passively mode-locked fiber lasers.  Phys. Rev. A, 72(4):043816, 2005.
  * [56] Rafi Weill, Boris Vodonos, Ariel Gordon, Omri Gat, and Baruch Fischer.  Statistical light-mode dynamics of multipulse passive mode locking.  Physical Review E, 76(3):031112, September 2007.
  * [57] SMJ Kelly.  Characteristic sideband instability of periodically amplified average soliton.  Electronics Letters, 28(8):806–807, 1992.
  * [58] J. P. Gordon.  Dispersive perturbations of solitons of the nonlinear Schrödinger equation.  Journal of the Optical Society of America B, 9(1):91, January 1992\.
  * [59] Herman A Haus.  Mode-locking of lasers.  IEEE Journal of Selected Topics in Quantum Electronics, 6(6):1173–1185, 2000.
  * [60] Heman A Haus.  Mode-locking of lasers.  IEEE Journal of Selected Topics in Quantum Electronics, November/December 2000.
  * [61] Shaokang Wang, Andrew Docherty, Brian S Marks, and Curtis R Menyuk.  Comparison of numerical methods for modeling laser mode locking with saturable gain.  JOSA B, 30(11):3064–3074, 2013.
  * [62] Feng Li, Ping Kong Alexander Wai, and J Nathan Kutz.  Geometrical description of the onset of multi-pulsing in mode-locked laser cavities.  JOSA B, 27(10):2068–2077, 2010.
  * [63] Kfir Sulimany, Offek Tziperman, Yaron Bromberg, and Omri Gat.  Soliton-pair dynamical transition.  https://github.com/kfirsuli/Soliton-pair-dynamical-transition, 2022\.
  * [64] Rafi Weill, Boris Vodonos, Ariel Gordon, Omri Gat, and Baruch Fischer.  Statistical light-mode dynamics of multipulse passive mode locking.  Physical Review E, 76(3):031112, 2007.
  * [65] Zohar Finkelstein, Kfir Sulimany, Shachar Resisi, and Yaron Bromberg.  Spectral shaping in a multimode fiber by transmission matrix engineering.  In Frontiers in Optics, pages JTu1A–116. Optical Society of America, 2021.
  * [66] H.A. Haus and A. Mecozzi.  Noise of mode-locked lasers.  IEEE Journal of Quantum Electronics, 29(3):983–996, March 1993\.
  * [67] Michael Katz, Omri Gat, and Baruch Fischer.  Noise-induced oscillations in fluctuations of passively mode-locked pulses.  Optics Letters, 35(3):297–299, February 2010.

##  6 Supplementary Information

###  6.1 Experimental setup

The laser setup is an all-fiber 14m integrated ring cavity, utilizing a 2.7-m
Erbium-doped fiber (Thorlabs ER16-8/125) gain medium, based on the design
reported in [44]. The EDF is core pumped by a 976-nm diode (Thorlabs
BL976-SAG300) laser through a wavelength-division multiplexer (Thorlabs
WD9850FA). A polarization-independent optical isolator (Thorlabs IO-H-1550APC)
ensures unidirectional lasing, and two fiber polarization controllers
(Thorlabs FPC560) tune the overall low cavity birefringence. The nonlinear
interference saturable absorber consists of a 1.5m long multimode fiber (OM1,
62.5 $\mu$m core diameter, $\textrm{NA}=0.275$)) spliced to two single mode
fibers (SMF-28) at both ends. A 90/10 fiber coupler (TW1550R2A2) provides the
laser output.

The laser output power is measured by a fast photodiode (Thorlabs DET08CFC)
and analyzed with an optical spectrum analyzer (Yokogawa AQ6374) and a real-
time $8$GHz oscilloscope (Tektronix MSO70804C).

###  6.2 Matlab code for the stochastic numerical simulation

Here we provide our Matlab code of the physical core utilized to calculate the
propagation of the waveform presented in section 2. The complete Matlab
package used to generate Figure 4 is available at [63].

The simulation parameters are as follows: $l=1.636\cdot 10^{-2}$ is the linear
loss, $\beta^{\prime\prime}=-2$ is the group velocity dispersion, $\gamma=4$
is the Kerr coefficient, $\Omega_{g}=\sqrt{12}$ is the gain bandwidth,
$\delta=0.02$ and $\sigma=0.01$ are the fast saturable absorber constants.
$g_{d}=5\cdot 10^{-5}$ is the gain depletion coefficient. $\zeta=10^{-4}$ is
the noise amplitude. In Eq. 4, $t_{R}=100000$ is the round trip time. The
linear gain is initially $g_{0}=0.65$ and then $g_{0}=0.58$ after $8.7\cdot
10^{5}$ cavity lengths and the saturation power is $P_{sat}=1.8$. We
periodically perturb the waveform to account for inhomogeneities and attenuate
it by 5% every cavity length.

⬇

1% Experimental ParametersatTR =1.8; Omega = sqrt(12); g0=0.58; %Gain
parameters

2Cooupling_ratio = 0.95; CavityLength = 1.45; Loss =
0.2+log(Cooupling_ratio)/CavityLength; % Local and distributed loss

3del = 0.02; sigma = 0.01; %Saturaber absorber parameters

4gam = 4; beta2 = -2; % Fiber nonlinearity and loss

5NOISE_AMP = 0.01; gd = 0.00005; Tr = 100000;

6

7% Numerical Parameters

8Nt = 2^12; T = 2^10; dt = T/Nt;

9t = (-Nt/2:1:Nt/2-1)'*dt; dw = 2*pi/T; w = [0:Nt/2-1 -Nt/2:-1]'*dw;

10Z = 4000000; h = 0.05; NumSteps = round(Z/h);

11SaveDistance = CavityLength*100 ; SaveInterval = round(SaveDistance/h);

12raundInterval = round(CavityLength/h);

13

14%Operators

15separation = [];

16L = (1i*beta2*w.^2-loss)/2;

17K = (1 - (w/Omega).^2)/2;

18

19%Initial condition

20init_seperations = [40];

21u0 = 1*exp(-((t-init_seperations(1)/2)/1).^2)...

22 +1*exp(-((t+init_seperations(1)/2)/1).^2);

23pulse_index_0 = find(t>-init_seperations(1)/2,1); shift = [];

24averaged_spectrum = u0; Etot = norm(u0).^2*dt; uf = fft(u0);

25uplot = abs(u0).'; uplot_w = abs(fftshift(uf)).';

26gtplot= [-gd*cumsum(abs(u0).^2)*dt+...

27 Etot*gd/Tr*dt.*(1:length(u0))']';

28Psatf = PsatTR/dt*Nt;

29zplot = 0; PULSE_POWER_THREASHOLD = 0.1; WIDTH_THREASHOLD = 50;

30

31% Propagator

32for istep = 1:NumSteps

33 uf = uf + sqrt((h/0.05)*(dt/(1/4)))*(1/sqrt(2)*...

34 (randn(1,length(uf))'*NOISE_AMP...

35 +1i*randn(1,length(uf))'*NOISE_AMP));

36 if mod(istep,raundInterval)==0

37 uf = uf*Cooupling_ratio;

38 end

39 g1 = g0/(1+norm(uf)^2/Psatf); %Saturated gain

40 g2 = -2*(g1^2/g0/Psatf)*real(dot(uf,(L + g1*K).*uf)); %frequency dependence

41 u=ifft(exp(L*h/2+(g1*h/2+g2/8*h^2)*K).*uf);

42 Etot =norm(u).^2*dt;

43 Etot_w_noise = Etot;

44 [pks,pulse_indexes,width,promince] = ...

45 findpeaks(abs(u).^2,'MinPeakProminence'...

46 ,PULSE_POWER_THREASHOLD);

47 if length(pulse_indexes)$\geq$2

48 current_separation = ...

49 (abs(pulse_indexes(2)-...

50 pulse_indexes(1))*dt);

51 end

52 if length(pulse_indexes)$\geq$2

53 indeces = (1:length(u));

54 dist_from_pulses = min(abs(indeces - pulse_indexes));

55 max_width = max(width);

56 noise_avg_pow = ...

57 mean(abs(u(dist_from_pulses>...

58 (WIDTH_THREASHOLD*max_width))).^2);

59 Etot_w_noise = Etot + noise_avg_pow*dt*length(u)*(Tr-T)/T;

60 cavity_indexes = ...

61 (Tr/2/dt-length(u)/2:...

62 Tr/2/dt+length(u)/2-1)';

63 gt = -gd*cumsum(abs(u).^2)*dt - ...

64 gd*noise_avg_pow*dt*(Tr/2/dt-...

65 length(u)/2)+Etot_w_noise*...

66 gd/Tr*dt.*cavity_indexes;

67 else

68 current_separation = 0;

69 cavity_indexes = (Tr/2/dt-length(u)/2:...

70 Tr/2/dt+length(u)/2-1)';

71 gt = -gd*cumsum(abs(u).^2)*dt - ...

72 Etot*gd/Tr*dt.*cavity_indexes;

73 end

74 gt = gt - mean(gt)*T/Tr ;

75 uf = fft(exp(+(del+1i*gam)*h*abs(u).^2-...

76 sigma*h*abs(u).^4+gt.*h).*u);

77 g1 = g0/(1+norm(uf)^2/Psatf);

78 uf = exp(L*h/2 + (g1*h/2+g2/8*h^2)*K).*uf;

79end

###  6.3 Analytical model for noise mediated interaction mechanism

The NMI mechanism was first proposed in [34], together with the basic
equations of motion governing the pulse dynamics. The interaction is based on
fluctuations of pulse timing generated by its overlap with the quasi-CW
permeating the cavity. Since NMI is mediated by the gain dynamics, it is
expected to act in any multi-pulse mode-locked laser, although its importance
in comparison with other interaction mechanisms can vary widely between
different systems. We briefly explain the principles of NMI with an emphasis
on the aspects relevant to the present work and refer the reader to [34] for
more details. Notice that here we correct an algebraic mistake in the
separation time formula that originally appeared in [34].

We approximate the laser waveform by an incoherent sum of $n$ non-overlapping
soliton waveforms centered at time points $t_{k},\;1\leq k\leq n$ (in the
frame moving with the group velocity of a single soliton). Each soliton is
accompanied by a weak and broad dispersive-wave pedestal generated by the
Gordon-Kelly resonances [42], and an extended quasi-CW waveform. The quasi-CW
is generated by noise buildup, and therefore has a random phase, as indicated
in Figure S1. The growth of the quasi-CW is limited by the small-signal net
loss, and therefore the quasi-CW intensity is inversely proportional to the
square root of the net loss in a uniform system. However, since the amplifier
gain is depleted by passage of a pulse through the gain medium, and gain
recovery is slow, the gain profile $g(t)$ is nonuniform, and therefore the
quasi-CW intensity is not uniform either.

The gain profile is governed by the differential equation

$\frac{dg}{dt}=g_{r}-g_{d}\sum_{k}I_{p}(t-t_{k})\ ,$ (S1)

where $g_{r}$ and $g_{d}$ are the gain recovery and gain depletion
coefficients, respectively, and $I_{p}(t)$ is the intensity profile of a
soliton with pedestal centered at zero, neglecting the quasi-CW gain
depletion, and approximating the gain recovery by a constant as explained
following Eq. 4.

Of the two pulse components, the soliton waveform is approximated by a delta
function because its duration is much shorter than all other time scales, and
the pedestal waveform is modeled by an exponential decaying on a time scale
$w$ that is much longer than the pulse width but much shorter than the cavity
round-trip time $t_{R}$, so that

$I_{p}(t)=E_{s}\delta(t)+\frac{E_{p}}{2w}e^{-|t|/w}\ ,$ (S2)

$E_{s}$, $E_{p}$ being the pulse and pedestal energy, respectively.
Consequently, the gain profile $g(t)$ is a smoothed-corner sawtooth shaped
function, as shown in Figure S1, overlaid with the pulse-intensity profile.
Note that the gain recovery rate $g_{r}$, which is fixed in Equation S1 by
continuity and the total pulse energy, sets the saturation depth of the gain.

The overlap of a pulse and the noisy quasi-CW is a source of timing jitter
[66, 67], which can be modeled as a diffusion process. The diffusion
coefficient $D_{k}$ of pulse number $k$ is proportional to the instantaneous
quasi-CW power at time $t_{k}$, which is in turn determined by the net gain
profile as [34],

Figure S1:  Illustration of the NMI mechanism. Passage of solitons through the
gain medium depletes the instantaneous gain. A pair of pulses yields a
smoothed sawtooth gain profile. The quasi-CW light that runs in the cavity is
determined by the gain profile, and thus also exhibits a smoothed saw-tooth
profile. The NMI is a byproduct of the pulse timing jitter caused by nonlinear
overlap interaction between the pulses and the quasi-CW light. The inset shows
the effective potential for the noise-mediated interaction of the loosely
bound (blue) and tightly bound (red) states.

$D_{k}\propto(\frac{1}{\sqrt{l-g_{k}^{-}}}+\frac{1}{\sqrt{l-g_{k}^{+}}})$ (S3)

where $l$ is the total small signal loss and $g_{k}^{-}$, $g_{k}^{+}$ are the
values of the gain coefficient just before and just after the $k$th pulse,
respectively. Note that the $D_{k}$ depend on the pulse separations, since the
separations determine the gain profile. Consequently, even though the noise
itself is symmetrically distributed, the pulse jitter is biased, with an
effective noise-induced drift of

$\frac{d\langle t_{k}\rangle}{dz}=\frac{1}{2}\frac{\partial D_{k}}{\partial
t_{k}}\ ,$ (S4)

where the brackets indicate noise-averaging.

From this point we consider two-soliton waveforms; the drift equations S4 can
be expressed as overdamped motion in an effective potential [34] that is
attractive at large separations, but may have a minimum at a separation value
$t_{s}$ of the order of the pedestal width, corresponding to a stable loosely
bound state at this separation (see inset of Figure S1).

In a stable steady state configuration, $D_{k}$ will be maximized with respect
to the pulse times $t_{k}$, as can be seen directly from S4. To find the
steady state pulse separation we begin by finding the gain profile. We then
use S3 to find the diffusion coefficients and maximize them with respect to
the pulse times. Integrating S1, and finding the recovery rate using the
periodic boundary conditions yields

$g(t)=g_{0}+g_{d}\int_{0}^{t}(P-I_{p}(t^{\prime}-t_{1})-I_{p}(t^{\prime}-t_{2}))dt^{\prime},$
(S5)

where $g_{0}$ is the mean gain and $P$ is the average power of the waveform:

$P=\frac{1}{t_{R}}\int_{0}^{t_{R}}(I_{p}(t^{\prime}-t_{1})+I_{p}(t^{\prime}-t_{2}))dt^{\prime}.$
(S6)

Using now equation 5 we find the gain profile:

$\begin{gathered}\frac{l-g(t)}{\bar{g}}=1-\frac{g_{d}}{\bar{g}}(E_{s}+E_{p})(\frac{2t}{t_{R}}-1)\\\
-\frac{g_{d}}{\bar{g}}\begin{cases}E_{s}+E_{p}-\frac{E_{p}}{2}(e^{\frac{t-t_{1}}{w}}+e^{\frac{t-t_{2}}{w}})&t<t_{1}\\\
\frac{E_{p}}{2}(e^{\frac{t_{1}-t}{w}}-e^{\frac{t-t_{2}}{w}})&t_{1}<t<t_{2}\\\
-(E_{s}+E_{p})+\frac{E_{p}}{2}(e^{\frac{t_{1}-t}{w}}+e^{\frac{t_{2}-t}{w}})&t_{2}<t\end{cases}\end{gathered}$
(S7)

where $\bar{g}$ is absolute value of the mean net gain. We can assume without
loss of generality that the pulse times obey $t_{R}=t_{1}+t_{2}$, and define
$t_{s}=t_{2}-t_{1}$. The relation between the gain and the diffusion
coefficients S3 gives the following expression

$\begin{gathered}D_{1}\propto\frac{1}{\sqrt{1+\frac{g_{d}}{\bar{g}}(\frac{t_{s}}{t_{R}}(E_{s}+E_{p})+\frac{E_{p}}{2}(-1+e^{-\frac{t_{s}}{w}}))}}\\\
+\frac{1}{\sqrt{1+\frac{g_{d}}{\bar{g}}((\frac{t_{s}}{t_{R}}-1)(E_{s}+E_{p})+\frac{E_{p}}{2}(1+e^{-\frac{t_{s}}{w}}))}}\\\
D_{2}\propto\frac{1}{\sqrt{1+\frac{g_{d}}{\bar{g}}(-\frac{t_{s}}{t_{R}}(E_{s}+E_{p})+\frac{E_{p}}{2}(1-e^{-\frac{t_{s}}{w}}))}}\\\
+\frac{1}{\sqrt{1+\frac{g_{d}}{\bar{g}}((1-\frac{t_{s}}{t_{R}})(E_{s}+E_{p})-\frac{E_{p}}{2}(1+e^{-\frac{t_{s}}{w}}))}}.\\\
\end{gathered}$ (S8)

The diffusion coefficients in the model do not depend on the absolute pulse
timings but only on the separation $t_{s}$, thus S4 in a steady state turns
into:

$\frac{\partial{D_{s}}}{\partial{t_{s}}}=0,\quad\textrm{where
$D_{s}=D_{1}+D_{2}$},\ $ (S9)

from here, some algebra leads to

$\tilde{t}_{s}=\tilde{w}\ln(\frac{E_{p}t_{R}}{2(E_{p}+E_{s})w})=\tilde{w}\ln(\frac{\tilde{E}_{p}}{2\tilde{w}}),\
$ (S10)

where we introduced the notations $\tilde{E}_{p}=\frac{E_{p}}{E_{p}+E_{s}}$,
$\tilde{t}_{s}=\frac{t_{s}}{t_{R}}$ and $\tilde{w}=\frac{w}{t_{R}}$ which
completes the derivation of equation 7.