## 3 Dispersion-Managed Versatile Ultrafast Fibre Laser
### Numerical Simulations  
To gain deeper understanding of the laser dynamics, we performed numerical simulations of the laser based on the non-distributed model described in Section 2. The laser configuration used in the simulations was the same as the experimental setup. We used a rectangular spectral profile for the filter \(R(f)=\mathrm{rect}(f/B)\) to operate the laser at anomalous or slightly normal dispersion and a Gaussian spectral profile \(R(f)=\exp[(-f^{2})/((2B^{2}))]\) for the strong net-normal dispersion regime, in accordance with the experiment. To account for the additional length of anomalous-dispersion fibre in the cavity, we included two segments of SMF after the SA element and the filter, respectively. Linear losses of 10% were imposed after the SMF segment following the SA and after the filter, which summarised intrinsic losses and output coupling. The laser output was monitored behind the coupler at the exit of the SMF segment following the SA, after further propagation in a 3 m-long segment of SMF, as in the case of the experiment. The parameters used in the numerical simulations were similar to their nominal or estimated experimental values. Typical evolutions of pulse duration and spectral width in the cavity in the three pulse-generation regimes are shown in Figure 8. The respective net GVDs of the cavity were \(-0.397\ \mathrm{ps^{2}}\), \(0\ \mathrm{ps^{2}}\) and \(0.502\ \mathrm{ps^{2}}\), which are similar values to those of the experimental results in Figure 7. The corresponding simulated pulses and spectra were in qualitative fairly good agreement with their experimental counterparts, with TBP values close to the experimental ones. Distinctly different types of evolutions can be seen from Figure 8. The soliton-like pulse is nearly static in the cavity, with temporal and spectral breathing ratios (defined as the ratios of maximum and minimum rms widths within a round-trip in the cavity) of only 1.3 and 1.2, respectively. The DM soliton experiences the largest temporal and spectral breathing [50], with the respective breathing ratios of 7.5 and 1.8. This pulse temporally stretches and compresses three times per round-trip, reaches a minimum duration in the middle of each of the EDF and the two SMF segments and acquires both signs of chirp. The mode-locked pulse at large net-normal dispersion exhibits the features of a DS with an evolution defined by the dispersion map [15]. The evolution in each fibre segment is monotonic; the pulse duration increases in the EDF segment and decreases in the SMF segments; and the spectrum, cut away by the filter, grows back in the fibre sections. The pulse is positively chirped throughout the cavity, and it depends strongly on dissipative effects, such as the spectral filter and the SA. The different roles played by spectral filtering in the DM soliton and DS mode-locking regimes are evident: in the DM soliton regime, the large filter has nearly no effect on the pulse spectrum, while the pulse is temporally compressed by the effect of the normal dispersion applied on the filter; in the DS regime, by contrast, the narrow filter cuts away the spectral structure, and consequently, the pulse duration is increased.