## 4 Discussion and Application  
### Intracavity Dynamics  
Simulations were carried out to better understand the intracavity evolution of the pulse. The filtering effect was measured to be equivalent to a 3.6 nm FWHM Gaussian profile bandpass filter by splicing a 2 \(\times\) 2-10/90 coupler to the DCF and comparing the measured spectra from both 10% output ports. The coupler was minimized in length and has negligible influence on GDD, which was confirmed by the unchanged pulse parameters measured after its incorporation. With a 10-m long DCF, the numerical simulations were in good agreement with the experimental results shown in Figure 4, as the main pulse features were reproduced. Figure 6 indeed shows the simulated pulse spectrum and temporal profile at a gain saturation energy of 15 nJ, corresponding to a pulse energy of 20 nJ.  
Figure 4: Pulse characterization of configuration with a 10-m DCF in the cavity. (**a**) Spectrum (inset: AC trace). (**b**) AC trace of the de-chirped pulse and calculated Fourier limit of the pulse.  
Figure 5: Characterization of a mode-locking regime with 15-m-DCF-in-cavity configuration. (**a**) spectrum and AC trace (inset), where the brown dashed line indicates the filtering effect of the grating-lens combination. (**b**) AC trace of the de-chirped pulse and its comparison to the Fourier limit.  
The simulated pulse had a spectral bandwidth of 7.7 nm and a quasi-parabolic temporal shape with a FWHM of 10.8 ps, and could be de-chirped to 862 fs. The fact that our simulation was in good agreement with the experimental results proved the numerical modeling of this cavity under this energy level was viable and can be used to have a better comprehension of the pulse evolution inside the cavity.  
Figure 7 depicts the pulse's temporal and spectral evolution along the cavity under stable mode-locking operation. We can notice that the DCF contributes the most to the pulse stretching as expected, but also contributed the most to the SPM-induced spectral broadening, outweighing the effect from the EDF. This meant that the DCF was the main source of nonlinearity in the cavity, and explained why increasing its length to 15 m resulted in a regime where pulses suffered excessive nonlinear phase and could not be linearly de-chirped. It is also worth noticing that the spectral filter (SF) played an important role in the mode-locking by ensuring the pulse self-consistency. While the effect of the R-SAM was negligible at steady state, it was crucial for the mode-locking initiation.  
The pulse shown in Figure 6b was fitted assuming a parabolic shape as the common Gaussian shape was not matching. This made it worth investigating the pulse shape evolution inside the cavity. We could represent such an evolution in terms of pulse kurtosis (denoted by \"\(\kappa\)\"). The kurtosis is defined as \(\kappa=\mu_{4}/\sigma^{4}-3\), where \(\mu_{4}\) is the fourth centered moment and \(\sigma^{4}\) is the square of the variance. For instance, \(\kappa=-0.86\) corresponds to a parabolic pulse shape and \(\kappa=0\) to a Gaussian shape. This metric has thereby been used to analyze laser pulse temporal shapes [30; 32]. Figure 8 shows the evolution of \(\kappa\) and highlights the fact that the pulse shape evolves from a Gaussian to a parabola when propagating along the cavity. The parabolic shape was reached at the end of the DCF in the forward direction, just before the spectral filter. It was also clear that the pulse shape did not evolve significantly in the EDF and SMF, which exhibited anomalous dispersion.  
Figure 6: Simulated pulse characteristics from a cavity with a 10-m DCF and spectral filtering of 4.5 nm.  
Figure 7: Pulse duration and bandwidth evolutions along the cavity elements.  
The performances in terms of output energy were, however, satisfying for the objective of pumping a fiber OPO based on a small-core dispersion-shifted fiber, with the extra advantage of this laser directly generating highly-chirped pulses, then enabling broad OPO tunability, as detailed in the following section.