Real-time characterization of spectral instabilities in a mode-locked fibre laser exhibiting soliton-similariton dynamics

The study of dissipative solitons in mode-locked lasers reveals a rich landscape of interaction dynamics resulting from the interplay of nonlinearity, dispersion and dissipation. Here, we characterize a range of instabilities in a dissipative soliton fibre laser in a regime where both conventional soliton and similariton propagation play significant roles in the intracavity pulse shaping. Specifically, we use the Dispersive Fourier Transform technique to perform real-time spectral measurements of buildup dynamics from noise to the generation of stable single pulses, phase evolution dynamics of bound state “similariton molecules”, and several examples of intermittent instability and explosion dynamics. These results show that the instabilities previously seen in other classes of passively mode-locked fibre lasers are also observed in the presence of strong nonlinear attraction of similariton evolution in an optical fibre amplifier.

two closely-spaced pulses in the cavity 18 . This work was significant in revealing the internal dynamics of soliton "molecules, " bound states of interacting pulses that exist in dissipative systems [19][20][21] . These results subsequently motivated many additional experiments using real-time techniques to characterize instabilities in mode-locked lasers, and particularly fibre lasers where a wide range of different dynamics can be observed depending on the intracavity group velocity dispersion (GVD) map 22 .
For fibre lasers with net anomalous GVD (which display average or dispersion-managed soliton dynamics), DFT experiments have reported chaotic evolution and soliton explosions 23,24 , soliton bunching and interactions during pulse buildup and Q-switching [25][26][27][28][29][30] , coherence degradation 31 , and internal motion of soliton molecules 28,32 . Real-time experiments using time-lens or combined time-lens and DFT characterization have also been reported for this class of laser 33,34 .
There has also been significant effort to use DFT to characterize mode-locking in lasers with net (or all) normal GVD. In such lasers, a circulating pulse generally undergoes significant changes during one round trip as it encounters fibre segments of normal or anomalous dispersion, as well as gain and spectral filtering elements 22,35 . Confusingly, it has become common in the fibre optics community to refer to such normal GVD designs as "dissipative soliton lasers" even though the terminology of dissipative soliton applies much more widely, essentially to all fibre lasers with any degree of nonlinearity. As with the experiments on lasers with net anomalous cavity dispersion, a range of dissipative soliton instabilities with normal dispersion have also been reported, including complex build-up dynamics 30 , rogue wave-like behaviour 36,37 , soliton molecules 38 , soliton explosions 39 and soliton oscillations 40 .
A design of mode-locked fibre laser that has attracted attention from a fundamental viewpoint exploits very different types of nonlinear propagation in distinct fibre segments. In particular, nonlinear self-similar propagation in a normal-GVD amplifier 41 combined with soliton shaping in anomalous-GVD fibre has been shown to lead to robust operation with excellent noise properties 42 . Significantly, although such an ideal "soliton-similariton" laser requires very careful design and optimisation, soliton shaping and similariton-like amplification can be exploited to obtain improved laser performance over a much wider parameter range 22 .
To our knowledge, however, the instability dynamics of a laser showing soliton and similariton characteristics have not been studied using any real-time technique, and this is a gap that we fill with this paper. In particular, we have performed an extensive series of experiments using DFT that have revealed a range of instabilities in a quasi soliton-similariton regime, including pulse build-up dynamics, chaotic evolution and "explosions", and both stable and unstable bound state "similariton molecules. " Our results show that whilst the laser design can indeed be configured to display highly stable operation (in both single pulse and molecule regimes), with detuning from steady-state it exhibits the same class of instabilities as observed in other mode-locked fibre laser cavities. These results point to the universality of such instabilities in dissipative systems.

Results
Our experiments studied the dynamics of an Er-doped fibre (EDF) amplifier based ring laser producing pulses centred on ~1555 nm with a repetition rate of 9.50 MHz (i.e. cavity round trip time of 105.0 ns). The cavity (shown in Fig. 1a) includes anomalous and normal dispersion fibre segments for soliton and self-similar propagation respectively 42 , and was mode-locked using nonlinear polarization evolution where a series of waveplates and a polarizing beam splitter act as a quasi-instantaneous saturable absorber 43 . A spectral filter in the saturable absorber module reduces the bandwidth after self-similar amplification in the EDF before injection in the soliton propagation segment. This spectral filtering step is a key feature of the soliton-similariton design that optimizes the properties of the input to the EDF so as to favour self-similar shaping during the amplification process.
We measured the output pulse characteristics at the 1% coupler (point D in Fig. 1a). In the stable mode-locking regime, we used an integrating optical spectrum analyser (OSA) for spectral measurements, and frequency-resolved optical gating (FROG) to access the complex field 44,45 . For measurements of unstable spectra, we used the DFT technique after propagation in a segment of dispersion compensating fibre. Further details are given in Methods.
To provide general insight into the laser operation, Fig. 1b-d show results from numerical simulations based on a nonlinear Schrödinger equation (NLSE) model 42 . Here, propagation in each distinct segment of the cavity is modelled using an NLSE suitably parameterized with appropriate dispersion and nonlinearity and, in the case of the EDF, an additional term describing gain. Simulation parameters were based on the experimental design (see Methods.) Note that this simplified scalar model is not expected to quantitatively reproduce the complex instability behaviour in our experiments, but it is nonetheless highly useful to illustrate the intracavity pulse evolution in stable operation. For example, Fig. 1b shows simulated intracavity evolution for the temporal and spectral widths, clearly illustrating the strong differences between propagation in EDF (A-B) and SMF-28 (G-H), as well as the abrupt spectral filtering in the saturable-absorber segment (F-G). Figure 1c,d show the simulated EDF output in single-pulse and molecule regimes plotting (i) temporal and (ii) spectral profiles. Although the simplified gain model does not show the spectral asymmetry seen in experiments, the temporal and spectral widths are close to experimental values (see Methods).

Stable single pulses and molecules.
Mode-locked operation with stable single pulses of duration ~7-10 ps was typically initiated at 80 mW of 976 nm pump power by adjusting the positions of the waveplates in the saturable absorber segment, and monitoring the laser output using a 5 GHz photodiode and an optical spectrum analyser. Once initiated, stable mode-locking could be sustained at reduced pump powers down to a ~30 mW pump power threshold. Stable bound state molecules could be excited with pump powers in the range 80-160 mW, but operation in the molecule regime was significantly more sensitive to waveplate orientation. Such molecule states consisted of two pulses with temporal separations in the range 20-50 ps with the exact separation depending greatly on the power level and waveplate positions. The appearance of double-pulse molecule states www.nature.com/scientificreports www.nature.com/scientificreports/ could be seen by monitoring the spectral fringes on the OSA, with the high contrast observed in stable operation indicative of pulses of high mutual coherence 18,21 . Figure 2 shows results of measured stable pulse characteristics after amplification (point D) showing: (a-c) stable single pulse operation and (d-g) a stable bound state "similariton molecule". The pump power was 80 mW in both cases with the difference in the observed behaviour resulting from the waveplate orientations. In both cases, the figure shows the FROG traces, retrieved intensity and chirp, and corresponding measured spectra from the OSA and DFT measurements. The results from the FROG measurements in Fig. 2b,e show the expected strong linear chirp and rapid fall-off in the temporal intensity over the pulse centre, characteristics of self-similar evolution [46][47][48] .
The OSA and DFT results show overall very good agreement, but the peaks at the spectral edges (1540 nm and 1565-1570 nm) measured with the DFT have slightly lower intensity than those measured with the OSA. This is attributed to the impulse response of the photodiode used in the DFT measurements (see Methods). We also note that the observed fringe contrast in the molecule case is not 100% as would be expected for an ideal soliton bound state, but is resolution-limited by the OSA (0.07 nm) and DFT (0.10 nm) to a reduced visibility of However, the coherence of the soliton pair in the stable regime was confirmed by checking the stability of the fringe pattern during a series of 4000 sequential DFT measurements.
These results are to our knowledge the first complete characterisation of molecule states in a soliton-similariton laser, but their main significance here is to confirm that the DFT measurements reproduce the broad bandwidth spectra from the laser measured in the stable regime. This confirmation provides confidence in the shot-to-shot DFT measurements in the unstable regime which is the main focus of the results below. www.nature.com/scientificreports www.nature.com/scientificreports/ From noise to stable single solitons. Our first experiments studying unstable laser operation characterized the emergence of stable single dissipative solitons (as shown in Fig. 2a-c) from noise-like operation just below the mode-locking theshold. To study this transition, the pump power was first set at 80 mW and the waveplates adjusted until stable mode-locking was observed. The pump power was then reduced in small increments until mode-locking ceased, and a DFT scan was initiated as the pump power was again increased above the threshold for stable mode-locking. The recorded DFT time series was then post-processed to extract each sequential spectrum on a roundtrip-to-roundtrip basis. Figure 3 shows the results obtained. The evolution from noise to stable operation over 2000 roundtrips is shown in Fig. 3a where we clearly see the evolution from a narrowband spectral peak to stable operation via a chaotic transition region. An expanded view of the transition region (the dashed white box) is also shown to highlight the complexity of this phase of evolution. The subfigure to the right shows the corresponding integrated energy and Fig. 3b shows corresponding single-shot spectra to illustrate particular dynamical features at the roundtrip number indicated.
We clearly see from Fig. 3a that the initial evolution up to ~900 roundtrips is associated with the appearance of multiple discrete lines in the DFT trace. The appearance of such lines in DFT measurements during pulse buildup has been previously reported in a mode-locked Ti:Sapphire laser 17 as well as in mode-locked fibre lasers 26,27,29 , and ref. 26 in particular has discussed their origin in terms of modulation-instability. In this context, we note that even in a dissipative soliton laser cavity with net normal dispersion, modulation instability effects could develop as a result of the periodic cavity boundary conditions 49,50 .
The transition region is associated with chaotic spectra (as shown in Fig. 3b at roundtrip 1000) and dramatic spectral broadening typical of "explosion" like dynamics 15,39 . Although similar transients have been reported in previous studies of build-up dynamics of several different designs of mode-locked fibre laser 30,51,52 , our results have been obtained in a cavity specifically favouring distinct soliton and similariton like propagation. The fact www.nature.com/scientificreports www.nature.com/scientificreports/ that such explosion dynamics still appear suggest they are a universal feature of all dissipative soliton systems transitioning from noise to stability.
Beyond the transition regime, the laser progressively stabilizes both in the spectral structure and the energy shown in the subfigure to the right. In fact, we actually recorded DFT spectra in these experiments over 4000 roundtrips and we plot as the dashed line the asymptotic stable energy at the end of the full measurement window. We can see how we begin to enter into the stable regime around 1500 roundtrips, and at Roundtrip 2000, the spectral characteristics (i.e. spectral width and structure) are similar to those previously shown in Fig. 2(c) during stable operation. Interestingly, in the intermediate phase of evolution before stabilisation (Roundtrip 1250), some small oscillatory structure can be seen on the spectrum. A detailed analysis using the autocorrelation function (see Methods) reveals that this is associated with a low amplitude temporal feature, associated with to the transitory appearance and decay of secondary pulses. We discuss this in more detail in the context of the results shown in Fig. 6 which show similar features.
Dynamically-evolving molecules. From a configuration associated with stable two-pulse molecule states as in Fig. 2d-g, rotation of the quarter-wave plate after the filter was found to trigger a range of evolution scenarios associated with the dynamic variation of the relative phase between the two bound pulses. Such phase variation leads to a roundtrip to roundtrip shift in the position of the modulation fringes in the spectrum which cannot be seen on an averaging detector such as an OSA, but which can be directly captured using the DFT 4 . Moreover, by analysing the fringe positions from the DFT measurements, we can extract the corresponding relative phase and plot its evolution directly 18 . Figure 4 shows typical DFT measurements at 80 mW pump power, and for three different waveplate positions (differing by only a few degrees.) Spectra are again measured after point D in the cavity, and we observe molecule states with ~50 nm bandwidth (as in Fig. 2f,g), but with spectral fringes whose position varied from roundtrip to roundtrip. Figure 4a shows direct DFT measurements over 200 roundtrips for the case of periodic fringe oscillation, where we plot over a limited 10 nm wavelength span to highlight the oscillatory behaviour. Applying the www.nature.com/scientificreports www.nature.com/scientificreports/ Wiener-Khinchin theorem allows us to determine the field autocorrelation from the Fourier transform of these DFT spectra (see Methods), and this is plotted in Fig. 4b, allowing us to see that the temporal separation between the constituent components of the molecule remains constant even while the phase is varying and inducing spectral oscillations. By analysing the fringe positions near the spectral centre using the method described in ref. 27 (see Methods), it is possible to determine the magnitude of the phase variation and this is shown in Fig. 4c. The results in Fig. 4d-f show similar evolution but for a larger phase excursion. These results which show constant temporal spacing yet periodic phase oscillation illustrate how dissipative soliton states in a soliton-similariton laser can exhibit internal motion analogous to diatomic molecular vibration. This behaviour is similar to that seen in other classes of fibre laser 32 , and our results show that it can also be seen in a soliton-similariton like cavity.
Although such bound state phase oscillations have been discussed in terms of molecular vibration, there is also a particular class of sliding phase evolution in fibre lasers that does not have any apparent molecular analogy 4 . We have also seen this behaviour in our laser, with these results are shown in Fig. 4g-i. This is an especially interesting phenomenon previously seen in an all-normal dispersion fibre laser 53 , as well as with lasers with negative net cavity dispersion 18,32 . Our results (which display a remarkable linear phase evolution over several 10's of 2π cycles) show that such dynamics persist even with soliton and similariton like dynamics in the cavity. This supports previous theoretical studies that associated such phase evolution with saturation dynamics in the gain medium, suggesting that all mode-locked laser systems capable of supporting bound molecule states could exhibit such behaviour 54 . www.nature.com/scientificreports www.nature.com/scientificreports/ Intermittency and Explosions. A central feature of nonlinear dissipative systems is intermittency, the irregular and aperiodic transition of a system between different stable and/or unstable dynamical states. Such intermittent behaviour has been reported in a number of previous DFT studies of mode-locked lasers, and includes notably the process of chaotic noise bursts and soliton explosion dynamics occurring aperiodically during phases of otherwise stable laser operation 15,39 .
This class of behaviour was observed in our laser system as the pump power was increased to 175 mW, above the upper limit of 160 mW corresponding to stable molecule formation. At this higher pump power, qualitatively different classes of instability were observed depending on the waveplate orientation, and the spectra captured via DFT for two typical regimes are shown in Figs 5 and 6.
The results in Fig. 5 show results over an extended measurement window of 4300 roundtrips, clearly illustrating a range of different dynamical features. Here Fig. 5a plots the spectra, the associated energy and the corresponding computed autocorrelation function, and Fig. 5b plots selected spectra at the roundtrips indicated. Note that a portion of this data sequence (in the soliton explosion regime) exhibited saturation in the measured DFT spectra (see Methods), and in this regime, it is not possible to compute the energy or autocorrelation.
Examining Fig. 5 in detail reveals an initial phase of unstable molecule evolution with fluctuating energy and unstable yet generally decreasing temporal separation (Roundtrips 1-1245). This is followed by a short-lived regime of stable molecule formation with constant energy (Roundtrips 1246-1360) followed by a series of periodic explosions (Roundtrips 1361-2550). After the explosions we see quasi-stable molecule propagation with slowly increasing energy and near-constant temporal separation in the autocorrelation (Roundtrips 2551-3320). This is then followed by another regime of unstable molecule evolution with increasing temporal separation (Roundtrips 3321-3670), quasi-stable molecule propagation (Roundtrips 3671-3940) and unstable molecule evolution with decreasing temporal separation (Roundtrips 3941-4300). For completeness, Fig. 5b also shows some selected spectra at particular roundtrips as indicated, and we note that all these spectra do not show saturation (the dashed line shows the saturation level.)   Fig. 6a which plots the DFT spectra, energy and autocorrelation over 4000 roundtrips, showing stable molecule propagation (40 ps temporal separation) up to ~1400 roundtrips, before the onset of very large fluctuations in energy ~50% and aperiodic spectral "explosions". The autocorrelation in particular shows how the explosion regime is associated by the two distinct pulses leaving their stable regime to approach each other and then collide, behaviour which has also been seen in ref. 39 . Related temporal evolution dynamics have been also seen in other systems 27,29,38 , and our results provide further confirmation of the ubiquitous nature of this behaviour in dissipative soliton lasers. Figure 6b plots expanded views of the chaotic dynamics showing how a spectral explosion is followed by the short lived emergence (50-100 roundtrips) of a distinct single pulse state. This pulsed state is clearly unstable, however, and suddenly collapses into a broadband noisy spectrum. Extracted spectral profiles Fig. 6c show this for several cases: evolution from roundtrip 1600 to roundtrip 1710; and a second example from 2120 evolving to roundtrip 2250. Of further interest in the expanded views in Fig. 6b is the even shorter-lived appearance (20 roundtrips) of an oscillatory structure on the spectrum around roundtrips 1710 and 2250. This is associated with a low amplitude temporal feature in the autocorrelation function centred around ~±10 ps in both cases, and we can therefore interpret the modulation as arising from the growth and decay of secondary pulses. A similar spectral feature was also seen in Fig. 3.
To our knowledge, such short-lived quasi-stable pulse dynamics appearing within a chaotic regime of spectral explosions has never been reported in any mode-locked laser, and our results highlight the great utility of the DFT technique to capture even highly complex transient processes in dissipative soliton systems. More generally, the results in Figs 5 and 6 show a rich landscape of multiscale nonlinear dynamics, but capturing this in full detail to explore effects such as period recurrence will require measurements over a significantly larger number of roundtrips. Particular behaviour that could form the focus of future work in this area could be correlating the explosion dynamics with the duration of the intervening quasi-stable regime, and possible quantization effects in the temporal pulse separation even in the unstable regime. www.nature.com/scientificreports www.nature.com/scientificreports/

Discussion and conclusions
The results in this paper provide further evidence of the tremendous utility of real-time spectral measurements using DFT to characterize instabilities in mode-locked lasers. Our results are the first such characterization of a soliton-similariton fibre laser system, and reveal a range of dynamical instabilities including pulse build-up dynamics, chaotic evolution and oscillation in the relative phase of bound-state molecules. We also reported what we believe to be a previously-unobserved regime of operation associated with the intermittent appearance of short-lived stable single pulses within a phase of otherwise chaotic explosion-like dynamics.
Although the soliton-similariton configuration is considered to be a particularly robust class of optical fibre laser 55 , our results show that it still yields regimes of unstable operation similar to those reported across a wide range of other laser designs. This suggests that instabilities such as soliton explosions and intermittence are a universal feature of dissipative soliton systems transitioning from noise to stability. We expect that the study of these universal instabilities using real-time techniques such as the DFT will provide even more insights into the underlying nonlinear dynamics, helping to more precisely identify the cavity regimes associated with stable operation suitable for applications.
A further important area of research will be to develop numerical models capable of reproducing the diverse range of dynamical scenarios seen in our experiments (and in other related work 38,39 ). In particular, although previous numerical simulations have reproduced certain phenomena such as periodic soliton explosions and soliton molecule evolution 15,32,56 , we anticipate that the modelling of the particularly rich intermittency dynamics seen here will constitute a very fruitful area of future work.

Methods
Experimental Setup. Our mode-locked laser ring cavity is shown in Fig. 1a The laser was mode-locked using the nonlinear polarisation evolution (NPE) technique where the transfer function of the polarization-selective bulk elements acts as a saturable absorber, adjusted by tuning the wave plate orientations 43 . The bandwidth of the pulses incident upon the bulk elements was ~25 nm and this was reduced by an intracavity filter (Andover 155FSX-1025, 80% peak transmission, bandwidth of 10 nm FWHM) before the soliton-reshaping segment G-A. In this context, we note that although a dissipative soliton laser can be readily constructed to lase with segments of both normal dispersion (usually the amplifier) and anomalous dispersion (usually passive fibre), this in itself does not guarantee that the laser will operate in a soliton-similariton regime. Achieving evolution that exhibits self-similar characteristics in the amplifier segment requires that the input pulse to the amplifier possess a particular peak power and pulse width, and it is here that the precise choice of the spectral filter bandwidth before the length of the passive anomalous dispersion fibre is especially important.
When adjusting the waveplate positions to observe mode-locked operation, we used a 5 GHz detector to monitor the developing pulse envelope on an ultrafast oscilloscope and an RF spectrum analyser to monitor the corresponding frequency comb. For the results in Fig. 2a, the waveplate which was adjusted to see the transition between stable and unstable operation was the quarter-wave plate positioned after the spectral filter (see Fig. 1a). However we observed that a similar transition could be observed by adjusting any of the waveplates in the saturable absorber section.
Pulse characterization in stable operation used a standard optical spectrum analyser (Anristu MS9710B) with 0.07 nm resolution and a second harmonic generation FROG setup. For the soliton molecule states with strong spectral modulation, the frequency-domain marginal computed from the fundamental spectrum was used to compensate for the bandwidth limitation of the FROG setup at the second harmonic wavelength 44 . The usual FROG retrieval errors for single pulses were typically < .
G 0 005 whereas for molecule states were typically < . G 0 015, the higher value in the latter case being consistent with the greater fraction of the computational grid occupied by non-zero data 44 .
In stable operation, the pulse durations (FWHM) after the EDF (measured at the 1% coupler, point D) were typically ~8.1 ps, and before injection to the EDF (measured at the 5% coupler) were typically ~1.4 ps At 80 mW pump power, average power measured at the 1% coupler (point D) was typically 0.32 mW, in point H was 2.98 mW and at the PBS output typically 5.32 mW. For the particular stable single pulse results shown in Fig. 2b, the temporal and spectral widths (FWHM) were respectively τ ∆ = . 7 2 ps and ν ∆ = .
3 1 THz (25 nm), with the highly-chirped nature of the pulse reflected in the large time-bandwidth product of τ ν ∆ ∆ = .
22 3. For the stable molecule state results in Fig. 2e, the retrieved pulse duration is 8.2 ps with a comparable spectral width to the single-pulse case, but with strong fringe contrast in the associated spectrum as seen in the measurements with both the OSA (Fig. 2f) and DFT (Fig. 2g). The temporal separation between the pulses is 39.75 ps, consistent with the 0. www.nature.com/scientificreports www.nature.com/scientificreports/ nm) can be readily calculated 6 , and is given by: δλ = | | = .
It can be seen from Fig. 2 that the peak heights at the edge of the spectra measured using the DFT technique are reduced relative to those measured using the OSA, and this is attributed to the effect of the impulse response of the photodiode used to detect the stretched pulse at the DCF output. In this context, we first note that with the sign of dispersion used in the DCF, the long wavelengths in the spectrum actually correspond to the leading edge of the stretched pulse at the DCF output. We have confirmed numerically that when such an asymmetric stretched pulse is convolved with a typical asymmetric photodiode impulse response (i.e. steep leading edge followed by a slower decay on the trailing edge), the detected pulse displays reduced peak heights at the edges in a similar way as seen in the DFT measurements.
When capturing the startup dynamics with pump power shown in Fig. 3, the triggering procedure was as follows. After determining the peak of the DFT signal corresponding to stable mode-locked operation, the oscilloscope trigger was set at 90% of this value. The pump power was then reduced below mode-locking threshold until mode-locking ceased, before being increased again to the value corresponding to stable operation. As mode-locking develops from noise, the DFT scan is initiated when a noise spike reaches the 90% trigger level, and we found that this typically occurred around 500-1000 roundtrips before the transition from noise to stable operation. We note, however, that this procedure does not allow us to quantitatively relate the timescale of the build up dynamics to any characteristic timescale of the pump or amplifier system. To study this in more detail would require the ability to synchronize the modification of the pump parameters with the triggering of the DFT measurement with the use of an external reference.
When analyzing unstable spectra of molecule structures to determine the relative phase between the bound temporal pulses (Fig. 4), we used the method described in ref. 27 to determine the phase from the position of fringes near the spectral centre. In particular, considering two bound pulses centred on frequency ω 0 with temporal separation T and relative phase ϕ, the corresponding power spectrum is: ) 0 is the spectral amplitude of each pulse and π Ω = T 2 / is the frequency separation between adjacent fringes in the modulated spectrum. Considering two fringe maxima at ω 1 and ω 2 bracketing ω 0 , the relative phase can be determined from: . Note that this approach was extensively tested using numerical data and cross-checked using direct least-squares fitting of the modulated spectrum.
The field autocorrelation of the soliton molecule bound state was computed from the Fourier transform of the DFT spectra via the Weiner-Khinchin theorem. Specifically, for temporal and spectral amplitudes that are Fourier transform pairs i.e. ω ↔  E t E ( ) ( ), the Weiner-Khinchin theorem gives: ). Note that we plot the magnitude of C(t) in Figs 5 and 6. Note that when recording measurements of unstable operating regimes, a tradeoff is needed between capturing low intensity events with sufficient dynamic range and avoiding saturation (manifested by truncation of the DFT spectra) during regimes of large energy fluctuations. In some instances, displaying the full range of dynamics requires that some regimes of data saturation are included, but it is important in these cases to clearly identify where saturation is occurring, such as indicated in the results shown in Fig. 5.
Finally for completeness, we give the experimental power values corresponding to the unstable modes of operation. For the regimes of dynamical phase evolution in the molecule results shown in Fig. 4 at 80 mW pump power, the output powers at the 1% coupler (point D) and at the PBS output respectively were: (a) 0.35 mW and 6.0 mW; (d) 0.32 mW and 5.5 mW; (g) 0.41 mW and 7.0 mW. The results in Fig. 5 were obtained with 170 mW pump power and with output powers at the 1% coupler (point D) and the PBS output respectively 0.52 mW and 8.4 mW. The results in Fig. 6 were obtained with 170 mW pump power and with output powers at the 1% coupler (point D) and the PBS output respectively 0.43 mW and 7.0 mW. Of course, for the unstable operation in Figs 5 and 6, these average power measurements cannot be reliably used to determine peak powers of any associated temporal structures because of the large variation in pulse characteristics from roundtrip to roundtrip. numerical Simulations. Our numerical model is based on a scalar modified nonlinear Schrödinger equation: ( , ) is the complex pulse envelope, z is the propagation coordinate, and τ denotes the usual co-moving time in the pulse frame. The equation is solved with different parameters corresponding to each distinct fibre segment in the cavity, using an initial condition corresponding to a 2 ps Gaussian pulse with 100% random noise. Numerical propagation through all the cavity segments was performed iteratively until steady-state was reached. This model is similar to that used in previous studies of mode-locked fibre lasers 22,42 . The dispersion and nonlinearity parameters for each fibre segment were as given in the preceding section. The Raman term in Eq. 1 used timescale τ = .
5 0 fs R but Raman effects were found to play a negligible role in the laser dynamics. We note in this context that although this Raman model is an approximation 58 , the negligible effect of the Raman term was confirmed by comparison with a full Raman model 59 .
The gain parameter for the EDF is given by 60 : www.nature.com/scientificreports www.nature.com/scientificreports/ where g 0 is the unsaturated small-signal gain, is the intracavity pulse energy in the EDF, and E sat is a gain saturation energy parameter. ω 0 is the central transition frequency and Ω g is the gain bandwidth. The saturable absorber transmittance is modelled by the transfer function: where q 0 denotes the unsaturated loss, τ τ = | | P Az ( ) ( , ) 2 is the instantaneous pulse power, and P 0 is a saturation power. The spectral filter has a near Gaussian transmission profile with 10 nm bandwidth. To model the stable single pulse regime in Fig. 2c, the EDF parameters used were = .