# Kerr-Lens Mode-Locking: Numerical Simulation of the Spatio-Temporal Dynamics on All Time Scales
## 2 Results
Figure 2: Evolution of the laser field during the entire oscillation buildup: (**a**) The cavity (pulse) energy and (**b**) the pulse-duration (FWHM) as a function of round-trip number (slow time scale). To observe the evolution of the pulse field within the cavity round-trip, we sampled four specific round-trips, as shown in (**c**) for the instantaneous power and (**d**) for the instantaneous gain as a function of time within a single round-trip (fast time scale). The colors of the graphs in (**c,d**) correspond to the circles on the energy graph of (**a**). The temporal width of the pulse is dictated by the gain bandwidth and the chromatic dispersion (GVD). The gain-bandwidth sets a lower bound on the pulse duration (by Fourier uncertainty), and the dispersion distorts the temporal shape of the pulse (chirping), which affects shorter pulses more severely than longer pulses [16]. Thus, the net GVD of the cavity sets another lower bound on the pulse-duration. Our simulation correctly reproduces this behavior, as shown in Figure 4, where the pulse duration follows the gain bandwidth limit until it reaches the dispersion limit. So far, we considered only linear effects on the pulse duration due to GVD and gain bandwidth. The pulse, however, is highly affected by the nonlinear chirping due to SPM, which is critical in the soliton formation mechanism. Specifically, SPM chirping acts as another source of dispersion that shifts the optimal GVD value toward a small negative value. Interestingly, the simulation can explore this interplay between SPM and GVD in greater detail than possible in the experiment. Specifically, in the simulation, we can simply turn off the temporal SPM effects, while keeping the spatial Kerr-lensing effect, and compare the pulse performance in this (unphysical) situation to the one with SPM, a feat that is inherently impossible in the experiment. Figure 5 highlights exactly this comparison: we plot the pulse duration as a function of the linear GVD in the cavity with SPM (blue) and without (orange). Indeed, the simulation shows that SPM shifts the optimal pulse formation toward the negative GVD range as expected, but also that SPM enhances the peak power at the optimal GVD value. Thus, optimal operation is achieved at some dispersion, rather than the zero dispersion point. Beyond the validations presented above, this simulation already served two previous publications, where it was used to predict and analyze new details of KLM, with quantitative agreement with experimental results. Specifically, in [15], we used the simulation to observe and explore the effective saturable-absorption mechanism of KLM, where, although no actual absorption takes place, there are diffraction losses over an effective aperture form the saturation loss mechanism. Using the simulation, we identified how and where exactly these losses occur, which aided us to experimentally measure this time-dependent loss for the first time. In another paper [17], we used the simulation to show that KLM lasers can break the spatial symmetry between the forward and backward halves of the round-trip in a linear cavity when the pump power is increased above the mode-locking threshold. This symmetry breaking allows the laser to increase the energy of the pulse. Both of these phenomena would have been impossible to predict without a full dynamical simulation, which illustrates the utility of this numerical tool. Figure 4: The temporal width of the pulse in time as function of gain bandwidth. As expected, the pulse becomes shorter and shorter as bandwidth increases. Furthermore, we see that dispersion broadens the pulse, with the pulse duration in the high dispersion configuration being almost an order of magnitude longer than the low dispersion one.