# Kerr-Lens Mode-Locking: Numerical Simulation of the Spatio-Temporal Dynamics on All Time Scales
## 3 Materials and Methods  
The simulation assumes a set of common parameter values for the cavity configuration, gain medium and Kerr medium, that are typical for KLM experiments [18; 19]. The values used in the simulation appear in Table 1.  
Our simulation focuses on KLM oscillators in a single transverse spatial-mode, which is time-dependent due to the Kerr-lens. \[E(r,z,t)=A_{\rm s}(r,z)A_{t}(t,\tau) \tag{1}\]  
where \(A_{\rm s}(r,z)\) is a transverse Gaussian-Hermite mode and \(A_{t}(t,\tau)\) is a slowly-varying time dependent envelope that we divide into two time-scales, as shown in Figure 6. This product separation of space and time is justified on two accounts--(1) the oscillation spectrum is relatively small compared to the carrier frequency (for pulses down to 10 fs or so), indicating that spectral variations are small; (2) The Kerr-lens, which dominates the dynamics on the fast time scale, acts instantaneously without memory, which corresponds to a very weak (nearly non-existent) frequency dependence. Furthermore, the approximation that the spatial profile is a single transverse Gaussian mode is justified since the nonlinear losses strongly drive the laser toward a single, high-intensity spatial beam. Generally, this approximation is only correct for paraxial propagation, which is well justified for the small divergence angles of the optical beams in our simulation cavity. Aberrations of the Kerr-lens can, in principle, couple energy into higher spatial-modes; however, these aberrations do not build up in the cavity, but rather lead to diffraction outside the cavity, which is encapsulated in the total diffraction loss term. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline NL refractive index—\(n_{2}\) & \(3\times 10^{-16}\) cm\({}^{2}\)/W \\ Kerr (gain) medium thickness—\(L_{c}\) & 3 mm \\ Length of short arm—\(L_{1}\) & 50 cm \\ Length of long arm—\(L_{2}\) & 90 cm \\ Focus of cavity lenses/mirrors—\(f\) & 75 mm \\ pump waist in gain medium—\(w_{p}\) & 30 \(\upmu\)m \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameter values for the cavity configuration and the Kerr medium. Figure 5: The width and peak power of the pulse as a function of the overall dispersion. The pulse reaches maximum power and minimum width for a specific, nonzero value, due to the dispersion induced by the nonlinear Kerreffect. KLM oscillators can be thought of as passive cavities, with an intensity-dependent lens (or lenses) that couples the temporal and spatial dynamics. When the diffraction losses are also included, the Kerr-lensing effect generates an effective saturable absorber with an instantaneous response, able to generate extremely short pulses [4; 20; 21; 22]. As such, the KLM oscillator can be divided conceptually into two parts--linear and non-linear. The linear part accounts for the gain spectrum and saturation, dispersion, and linear loss, which can be encapsulated into a total spectral transfer function:  
\[E(\omega)\to g(\omega)T(\omega)E(\omega), \tag{2}\]  
where \(g(\omega)\) is the spectral gain of the active medium and \(T(\omega)\) is the transfer function of the passive elements in the cavity. The spectral phase of \(T(\omega)\) reflects the chromatic dispersion and its amplitude reflects the cavity loss \(\left\|T\right\|^{2}=1-\) loss. To take into account the gain depletion, which leads to non-linear saturation and mode competition, we reduce the gain \(g(\omega)\) as the intra-cavity pulse energy \(U\) increases according to the standard saturation formula,  
\[g(\omega)=\frac{g_{0}(\omega)}{1+U/U_{\text{sat}}}, \tag{3}\]  
where \(U_{\text{sat}}\) is the saturation energy of the gain medium, taken to be \(\tau\cdot 2.6\) W for our simulation (\(\tau\) is the round trip time and the saturation power of 2.6 W is based on the saturation intensity of Ti:Sapphire in the literature, calculated for a 3 mm long crystal with 0.25% doping and 20 \(\upmu\)m beam waist for the pump) [23].