## 6 Cavity Analysis in Mode-Locked Operation
### A Gaussian Beam in a Kerr Medium  
The additional Kerr lens in the pulsed laser mode (Equation (2)) must be taken into account in order to calculate the Gaussian mode for ML operation. The presence of a lens-like effect inside the crystal, similar to thermal lensing [44], dramatically changes the stability behavior of the cavity. As for the Kerr effect in the laser crystal, several methods have been proposed to solve \(q_{ML}\) for ML operation [45; 46; 10]. These methods construct a nonlinear ABCD matrix to treat the propagation of Gaussian beams in the Kerr medium, where the nonlinear matrix depends on the mode intensity inside the crystal. Then, one can solve \(q_{ML}\) (for a given peak power of the pulse, \(P\)) in an iterative manner: starting with the solution of \(q_{cw}\) (at \(P=0\)) as an initial guess, one calculates the nonlinear ABCD matrix based on the mode size in the crystal given by \(q_{cw}\); this matrix defines, then, a new resonator with the additional Kerr lens, which yields a new solution of the laser mode, which allows recalculation of the nonlinear matrix, and so forth. The iterations continue, until a steady state is reached for the assumed intra-cavity peak power. A useful direct approach to solve \(q_{ML}\) along this line is to divide the Ti:sapphire crystal into many thin slices, considerably shorter than the Rayleigh range of the laser mode, \(z_{R}\). The crystal is then represented as a stack of many thin-lens matrices, where each matrix, \(M_{slice}\), consists of a nonlinear lens and propagation through a single slice with thickness \(l\):  
\[M_{slice}= \begin{bmatrix}1&0\\ -\frac{1}{f_{NL}(w)}&1\end{bmatrix}\begin{bmatrix}1&l\\ 0&1\end{bmatrix} = \begin{bmatrix}1&l\\ -\frac{1}{f_{NL}(w)}&1\end{bmatrix}  \tag{11}\]  
where \(f_{NL}(w)\) is the nonlinear focus given by Equation (2) for a Kerr medium with thickness \(l\) (calculated for the mode size at that slice. One can then calculate \(q_{ML}\) in an iterative manner, as described above. Note that the element, \(D\), in the matrix, \(M_{slice}\), equals \(D=1+O(l^{2})\). In the limit, where \(l\ll z_{R}\), one can use \(D=1\). Another useful numerical, but not iterative, method was given in [47], where it was shown that dividing the crystal into many infinitely thin lens-like slices with thickness \(dz\) can be represented as a differential equation for the inverse complex beam parameter, \(q^{-1}\):  
\[\frac{d}{dz}\left(\frac{1}{q}\right)+\left(\frac{1}{q}\right)^{2}+K\mathrm{Im}^ {2}\left(\frac{1}{q}\right)=0 \tag{12}\]  
Here, the nonlinear Kerr lensing process depends on a normalized parameter: \(K=P/P_{c}\), where \(P\) is the pulse peak power and \(P_{c}\) is the critical power for catastrophic self-focusing, given by:  
\[P_{c}=\frac{\lambda^{2}}{4n_{2}\pi} \tag{13}\]  
Note that the nonlinear process now depends on the normalized parameter \(K\). Consequently, other theoretical and experimental values for \(P_{c}\) can be defined [48]. For \(K=0\), Equation (12) reduces back to propagation of a Gaussian beam in a linear material with refractive index \(n\). By scaling the imaginary part of \(q^{-1}\) by \(\sqrt{1-K}\), the propagation of the new complex beam parameter \(\tilde{q}^{-1}=\mathrm{Re}[q^{-1}]+i\mathrm{Im}[q^{-1}]\sqrt{1-K}\) through a Kerr medium can be reduced back into a free-space propagation (_i.e._, K = 0). Thus, the effect of the nonlinearity is fully contained in the transformation of \(q^{-1}\), and a problem involving Kerr nonlinearity can be analyzed using linear ABCD matrices, like free space propagation. At the end of the Kerr medium, \(\tilde{q}^{-1}\) must be re-transformed by re-scaling the imaginary part of \(\tilde{q}^{-1}\) by \(1/\sqrt{1-K}\). Since there is no ABCD matrix that represents this transformation, one can no longer represent the entire cavity by a single ABCD matrix.