# Kerr-Lens Mode-Locking: Numerical Simulation of the Spatio-Temporal Dynamics on All Time Scales
## 2 Results  
Before we delve into the intricacies of the algorithm of the numerical simulation, let us demonstrate the performance of the simulation by reconstructing the complete dynamical evolution of the pulse, from the initial spontaneous noise seed, through the entire cavity buildup, to the final steady state. We will show how the spatial mode of the cavity is formed in steady state by self-focusing and how the soliton pulse is dictated by the interplay between gain-bandwidth, dispersion, and SPM. The simulation assumes the standard linear cavity layout of Figure 1, and the evolution of the intra-cavity field during the laser buildup is highlighted in Figure 2. The cavity (pulse) energy (Figure 2a) and the pulse-duration (FWHM, Figure 2b) are shown as a function of round-trip number (the slow time scale), where three stages of evolution are evident **(I)** the initial exponential growth, **(II)** the nonlinear mode competition, and **(III)** the final steady-state. To understand the dynamical operation during each stage, we sampled four specific round-trips from the different stages, which are shown in Figure 2c (instantaneous power) and Figure 2d (instantaneous power gain) as a function of time within a single round-trip (fast time scale). The colors of the graphs in Figure 2c,d correspond to the circles on the energy graph of Figure 2a. The field at the end of the initial exponential growth (black) is highly noisy and multi-pulsed, with noisy, yet positive gain at all times. This picture changes during the mode competition (red and green). We observe a number of pulses competing for the gain, before all pulses, except one, diminish. During that phase, we also observe variations in the shape of the main pulse. Finally, the purple graph shows the stable pulse near steady state. It is worth noting that our simulated cavity shows a _self-starting_ mode-lock dynamic under suitable conditions, with no special starting mechanism. This is due to the assumption of ideally slow gain saturation, which is calculated only over the slow time scale of cavity round-trips. Specifically, we assume that the gain within each round trip depends only on the total energy of that round-trip, not to be influenced by the temporal profile of the field within the round trip. Thus, a noise seed at the first round trip will basically grow undisturbed and maintain its temporal profile until the Kerr effect kicks in to favor one specific temporal fluctuation (usually the highest one) to form a single pulse. Although this assumption of slow gain saturation is well validated for the Ti:Sapphire gain medium with  
Figure 1: The cavity layout of a standard KLM laser in a linear cavity: The cavity focus is formed by two spherical mirrors/lenses, where the nonlinear Kerr medium (and gain) is placed. Two cavity arms of lengths \(L_{1}\), \(L_{2}\) exist around the focus. One of the arms includes components for tuned dispersion control (e.g., a double-prism). ultrashort pulses, it is not correct for a continuous field, where the finite memory of the gain saturation leads to relaxation oscillations and slow damping of temporal fluctuations. Consequently, an external noise mechanism may be needed in reality to induce sufficiently large fluctuations to enjoy the benefit of the Kerr-lens before they are damped out by the gain dynamics. Our simulation is fully spatio-temporal, and calculates the beam propagation through the cavity in space as well as in time. Figure 3 shows the propagation of the beam through a single run of the cavity. The core function of the Kerr-lens is shown in the inset, where the simulation captures the lensing effect that creates an effective \"wave-guide\" that stabilizes the cavity and counteracts the diffraction losses. Figure 3: The beam propagation through the cavity in a representative round-trip. The non-linear focus bends the beam back into stable operation, creating an effective saturable absorber—low-power pulses are unstable, while high-power pulses are focused back into stability.