## 3 Numerical and Experimental Results  
We first look at the manner in which characteristic parameters of the output pulses under mode-locking operations would evolve with the increase of pump level. The results are shown \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameter** & **Units** & **Er-Fiber** & **SMF** & **DCF** \\ \hline Length & m & 1.2 & 0.6 & 10 \\ Mode field diameter & \(\mu\)m & 4.9 & 10 & 6 \\ GVD (\(\beta_{2}\)@1550 nm) & ps\({}^{2}\)/m & +0.06 & \(-\)0.023 & +0.116 \\ Effective area (\(A_{eff}\)) & \(\mu\)m\({}^{2}\) & 18.8 & 78.5 & 28.26 \\ Nonlinear coefficient (\(\gamma\)) & W\({}^{-1}\cdot\)m\({}^{-1}\) & 6.4 \(\times\) 10\({}^{-3}\) & 1.5 \(\times\) 10\({}^{-3}\) & 4.2 \(\times\) 10\({}^{-3}\) \\ Peak small signal gain (g0) & m\({}^{-1}\) & 4.6 & 0 & 0 \\ Gain bandwidth (\(\Delta\lambda_{g}\)) & nm & 10 & - & - \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Fiber parameters used in the simulation. Figure 2: Schematic representation of the unfolded cavity, as used in the numerical simulations. SMF: single-mode fiber. in Figure 3, where the pump power is represented by the gain saturation energy of the doped fiber. Pump powers beyond the limit shown in the figures are also emulated, and the results are not included because only undesired multi-pulse operations are retrievable. We can see that the pulse energy, duration, and spectral bandwidth monotonically increased while the duration of the de-chirped pulse monotonically decreased. The simulation predicted pulse energies up to 70 nJ for this cavity design, and with a spectral bandwidth of \(\approx\)16 nm, the de-chirped pulse duration could reach less than 400 fs. It should be noticed that at higher energy levels, single and multiple pulse operations coexist, meaning that mode transition is possible in our configuration, which uses an NPE-SAM cooperated mode-locking mechanism. We then characterized the pulse properties under different GDD configuration (i.e., for different lengths of DCF) so that the impact of the dispersion management to the energy scaling could be compared and studied. As we focused on high-energy pulse delivery, the results and discussion concerning the laser with similar configuration but with very low normal GDD are not discussed and can be found in Ref. [31]. Motivated by the results of the simulations, we first built the oscillator with 10 m of DCF inside the cavity, corresponding to a cavity length of \(\approx\)12.5 m and a GDD around +1.22 ps\({}^{2}\). Stable mode-locking was achievable by adjusting the focusing conditions of the coupling lenses and fine-tuning of the polarization components. Figure 4 shows the output pulse characterizations of a typical regime under this configuration: the pulse energy was about 22 nJ derived from the measured average power of 176 mW at a repetition rate of 7.9 MHz. The pulse duration was 12.4 ps (NB: measured pulse durations were inferred from the AC trace full width at half maximum (FWHM), and the same criterion applies to all following measurements unless otherwise mentioned. The values are \(\sqrt{2}\) times the actual pulses' durations.) and could be de-chirped down to 940 fs outside the cavity using a pair of transmission gratings with 1000 lines/mm and 92% diffraction efficiency (Figure 4b), which is very close to the Fourier limit of its 8.1 nm bandwidth spectrum. The required de-chirping dispersion was measured to be about \(-\)0.416 ps\({}^{2}\). We also observe the transitions between single-pulse and multi-pulse operations when the pump power was increased, a phenomenon predicted by the simulation, which was mitigated by careful tuning of the polarization components; this is an indication of NPE being a part of the mode-locking mechanism. We then increased the DCF length to 15 m, consequently increasing the cavity GDD to +1.83 ps\({}^{2}\), its length to \(\approx\)20 m, and reducing the repetition rate to 5.33 MHz for the single pulse mode-locked operations.