## 6 Cavity Analysis in Mode-Locked Operation
### A Gaussian Beam in a Kerr Medium
\(q_{ML}\) of the stable mode must, thus, be obtained by a numerical solution of the stability condition \(|A+D|<2\) for the general cavity configuration illustrated in Figure 1. Analytical solutions, however, can still be achieved for a ring cavity [49] or for a symmetrical linear cavity [50] (equal cavity arms). The result of the calculation is a different mode size for \(w_{0}^{(ML)}\) as a function of \(\delta\) compared to \(w_{0}^{(CW)}\) (see Figure 4). **Figure 4.** Mode-locked (ML) and CW mode size on the end mirror, \(EM1\) (short arm), near the CW stability limit, \(\delta_{1}\), for the cavity illustrated in Figure 1. (**a**) The crystal is at the CW focus with pulse parameter \(K\approx 0.1\); the ML stability limit is pushed towards higher values of \(\delta\) by the Kerr lens. In (**b**), the crystal is away from focus with pulse parameter \(K\approx 0.2\); the ML stability limit is pulled down by the Kerr lens to lower values of \(\delta\). The dotted-dashed lines (black) represent positions where hard and soft aperture mode-locking techniques can be experimentally employed (these techniques are discussed in detail in Section 7). The long-dashed lines (blue) represent the second stability limit in CW operation. It is important to note that all calculations are performed within the aberration-free approximation for the Kerr lens, in which the transverse variation of the refractive index is approximated to be parabolic, so that the beam maintains its Gaussian shape during propagation, and the ABCD method to analyze the cavity can be applied. A detailed discussion regarding the limits of the aberration-free approximation can be found in [51]. Note, in addition, that in Figure 4, astigmatism was neglected for simplicity. The effects of nonlinear astigmatism on the ML mode are discussed in Section 8. The solution for the ML mode size presented in Figure 4 highly depends on the Kerr medium (crystal) position. Changing the position of the crystal along the beam affects the mode intensity in the crystal and, hence, the nonlinear response, leading to a different (sometimes dramatically) solution of \(w_{0}^{(ML)}\). In Figure 4a, the crystal was positioned at the CW focus point between the curved mirrors, whereas in Figure 4b, the crystal is located away from the focus, showing the completely opposite behavior. While in Figure 4a, the Kerr lens pushes the stability limit for ML operation towards higher values of \(\delta\), in Figure 4b, the stability limit is pulled towards lower values. The position of the crystal accordingly determines which method will be employed in order to mode-lock the cavity (hard or soft aperture), as will be discussed in Section 7. It is important to note that the above method ([47] and Equation (12)) assume a circular mode inside the crystal and do not include the effects of the more realistic elliptical (astigmatic) mode in the Brewster -cut crystal. One may try to calculate \(q_{ML}\) for each plane separately by assuming a circular mode with the corresponding mode size (\(w_{s}/w_{t}\) for the sagittal/tangential plane, respectively), but this is oversimplified in most cases. While such a simple separation of the planes may produce a qualitative understanding of the mode, it cannot provide a quantitative solution, since the planes in reality are coupled by the Kerr lensing effect (the mode size in one plane affects the peak intensity, which then affects the mode of the other plane). A detailed description of the coupled propagation equations of a Gaussian beam through a Kerr medium is provided in [36]. We further discuss the effects of nonlinear astigmatism in Sections 8 and 11.2.