Soliton-pair dynamical transition in mode-locked lasers

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.

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 [21,[29][30][31][32][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 modula- † The first two authors contributed equally to this work. tion 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 wellseparated, 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.
2 Dynamical transition in a mode-locked fiber laser

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- 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.
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 intensitydependent. 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. Our experimental protocol starts by generating stationary states of soliton pairs by increasing the pumping current to 124 mA, beyond the threshold for stable singlepulse operation. We then decrease the pumping current 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. 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 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.
function of the pumping current ( Figure 3(c)). The central wavelength is 1561 nm, and the full width at half maximum (FWHM) bandwidth is ∼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.

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: 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, β is the group velocity dispersion, g, κ are the gain coefficients to be described below, γ is the Kerr coefficient, Ω g is the gain bandwidth, and δ, σ are the fast saturable absorption constants, modeling bleaching and saturation of the absorber, respectively. η(z, t) is a complex Gaus- The noise term generates a non-homogeneous quasi-CW noise floor in the waveform that is the medium of the noisemediated interaction mechanism.
Since the amplifier gain is depleted by the passage of a pulse, and gain recovery is slow, the depleted gain profile g(u)+κ(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 κ 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: where g 0 is the unsaturated gain, P sat is the saturation power of the amplifier, and P av (u) is the mean intracavity power in which t R is the round-trip time. The inhomogenous gain is governed by the equation where g d is the gain depletion coefficient. To ensure that κ(z, t) has a temporal mean of zero, we subtract its mean 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 phaselocking each spectrum was averaged over 86 sequential traces spanning 8.6 · 10 5 cavity lengths.
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 · 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 · 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 ∼0.4. Narrow sidebands appear at the Gordon-Kelly resonances, of which the primary ones are detuned by ±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.

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 nonoverlapping 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 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 = | t 2 − t 1 | reaches a steady state value that minimizes the effective potential [34] (see supplementary information for more details):t wheret s = t s /t R ,w = w/t R , andẼ p = E p /(E s + E p ). In contrast, ifẼ p < 2w, the noise-mediated interaction is always attractive, so that the minimum of the effective potential appears at zero separation timet 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 timet s as a function of the normalized pedestal widthw and the normalized pedestal energyẼ 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.

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. Figure 5: Soliton steady states analysis. Theoretically predicted normalized separation timet 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 oft s are associated with loosely bound states, whilet 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 thet s > 0 and t s = 0 marks a dynamical transition line between loosely and tightly bound states.
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.

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 noisemediated 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. [9] Xueming Liu and Meng Pang.
Revealing the transition dynamics from q switching to mode locking in a soliton laser. Physical review letters, 123(9):093901, 2019.
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 8GHz oscilloscope (Tektronix MSO70804C).

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 · 10 −2 is the linear loss, β = −2 is the group velocity dispersion, γ = 4 is the Kerr coefficient, Ω g = √ 12 is the gain bandwidth, δ = 0.02 and σ = 0.01 are the fast saturable absorber constants. g d = 5 · 10 −5 is the gain depletion coefficient. ζ = 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 · 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.     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 ≤ k ≤ 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 equa- 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 δ(t) + 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.
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 kth 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