# Kerr-Lens Mode-Locking: Numerical Simulation of the Spatio-Temporal Dynamics on All Time Scales
## 4 Conclusions  
We introduced a complete algorithm, accompanied by an open-source MATLAB package for simulating the real-time dynamics of a Kerr-lens mode-locked laser. We simulate from basic principles the complete coupled spatio-temporal dynamics of the laser cavity, under only the assumption of a single Gaussian spatial mode, allowing to observe the complete dynamical evolution of the pulse field in the cavity from the initial noise  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Simulation Parameters** & **Description** \\ \hline \(E(t)\) & Field in time, \(M\)-vector. \\ \(\hat{E}(\omega)\) & Field in frequency, \(M\)-vector. \\ \(N\) & Total number of cavity round-trips in the simulation. \\ \(M\) & Total number of time-points within each round-trip. \\ \(w(t)\) & Instantaneous beam waist, \(M\)-vector for each round. \\ \(R(t)\) & Beam radius of curvature, \(M\)-vector. \\ \(P(t)\) & Beam power, \(M\)-vector. \\ \(I(t)\) & Beam intensity at the waist, \(M\)-vector. \\ \(q(t)\) & Complex beam radius, \(M\)-vector. \\ \(g_{0}\) & Small-signal gain (at center frequency). \\ \(g(P,w(t))\) & Saturated gain (depends on waist and average power). \\ \(s_{0}\) & Cavity loss rate (at center frequency). \\ \(\sigma_{G}\) & Gain/loss bandwidth (assuming Lorentzian spectrum). \\ \(\sigma_{D}\) & Linear dispersion (GVD). \\ \(L_{1}\) & Right cavity arm-length. \\ \(L_{2}\) & Left cavity arm-length. \\ \(L_{c}\) & Crystal’s length. \\ \hline \hline \end{tabular}
\end{table}
Table 2: Variable names and descriptions.  
Figure 8: Artificial aperture for numerical convergence. In order to maintain a reasonable beam size during all stages of the simulation, we implemented an artificial aperture to prevent the beam size from diverging during the initial stage of cavity amplification, when the intra-cavity power is too low to support a significant Kerr lens. This aperture was implemented as a lens with an imaginary focus, which is equivalent to a Gaussian aperture that limits the beam width, but keeps its total energy. To verify that the aperture does not affect the ML results, we ran the simulation for various artificial aperture sizes, recording the beam waist at steady state (**left axis**), as well as the number of round trips required to reach steady state (**right axis**). Note that within the convergence range (green background), the actual beam waist in steady-state is not affected by the artificial aperture. The run-time, however, diverges for wider aperture sizes until the algorithm stops converging at all.  
field to the final steady state. The simulation was thoroughly validated by reproducing all the basic properties of Kerr-lens mode-locked lasers, such as the dependence of the pulse duration in the gain bandwidth, the dispersion, and the self-phase modulation with reasonable agreement to the documented behavior of KLM in Ti:Sapphire. We demonstrate the evolution of pulse energy and duration, as well as the gain dynamics that determine them. Furthermore, the simulation also predicted novel phenomena of space-time coupling in linear KLM cavities [15; 17], which demonstrates the power of the simulation as a tool for the design and analysis of ultrafast lasers.  
Conceptualization, I.P., L.B. and M.-E.M.; project administration, I.P.; supervision, A.P.; writing--original draft, I.P. and L.B.; writing--review and editing, M.-E.M. All authors have read and agreed to the published version of the manuscript.  
This research was funded in part by the Israeli innovation authority, under the MAGNET project ALTIA.  
Not applicable.  
The authors declare no conflict of interest.