## 7 Mode-Locking Techniques
### Virtual Hard Aperture Mode-Locking  
Maybe the most convenient technique for hard aperture mode-locking near the first stability limit, \(\delta_{1}\), without using a physical aperture, is to choose the mode-locking point at \(\delta\gtrsim\delta_{1}\), a little bit beyond the stability limit, as was experimentally demonstrated in [52]. As can be seen in Figure 4a, crossing the stability limit, \(\delta_{1}\), affects only CW operation, while the cavity can still be stable for ML operation. At \(\delta\gtrsim\delta_{1}\), though the cavity is unstable for CW, it will operate very well, only with an elevated threshold, due to increased diffraction losses. Diffraction losses are increased as \(\delta\) is increased beyond \(\delta_{1}\). The concept of re-stabilization is illustrated in Figure 6a. The Kerr lens images the focus of the ML mode forward, effectively shortening the distance between \(M1\) and \(M2\) compared to CW, thereby pushing the stability limit for mode-locking towards higher values of \(\delta\). The unstable CW passively suffers from diffraction losses, which is effectively equivalent to employing an aperture. Mode-locking can still be initiated by knocking on one of the end mirrors, just as with standard hard/soft aperture. On a practical note, thermal lensing in the Ti:sapphire crystal must be considered, as it may also shift the stability limits of the cavity. In the experiments shown in [52], the absence of thermal lensing was experimentally verified. Virtual hard aperture mode-locking can be achieved also near \(\delta_{2}\) (though far less effectively than soft aperture) by choosing the mode-locking point at \(\delta\lesssim\delta_{2}\). As illustrated in Figure 6b, CW operation is destabilized, while for ML operation, the cavity can still be stable. The Kerr lens then virtually images the focus of the ML mode backwards, thereby effectively lengthening the distance between \(M1\) and \(M2\) for the ML mode compared to CW. Thus, one can choose a mode-locking point at \(\delta\lesssim\delta_{2}\), which is stable for ML operation, but not stable for CW operation. This, however, is much less efficient than mode-locking at \(\delta\gtrsim\delta_{1}\), since it requires a weaker Kerr lensing effect in order to create a virtual image of the focus point rather than a real image. **Figure 6.** Illustration of the Kerr lensing effect as a mode-locking stabilizing mechanism in virtual hard aperture technique: (**a**) increasing \(\delta\) beyond the stability limit, \(\delta_{1}\), destabilizes CW operation, while ML operation remains stable as the effective distance between \(f_{1}\) and \(f_{2}\) becomes shorter, by the nonlinear forward imaging of the focus point; (**b**) decreasing \(\delta\) below the stability limit, \(\delta_{2}\), destabilizes CW operation, while ML operation remains stable as the effective distance between \(f_{1}\) and \(f_{2}\) becomes larger by nonlinear virtual imaging of the focus point. Last, we note that hard/soft aperture mode-locking can be achieved in the same manner as explained above near the stability limits, \(\delta_{3}\) and \(\delta_{0}\). At \(\delta_{3}\), hard aperture has the stronger nonlinear response, similar to \(\delta_{1}\). However, due to experimental considerations, one usually prefers to have a collimated beam at one of the cavity arms, which is not the case at \(\delta_{3}\) (see Figure 2). At \(\delta_{0}\), soft aperture has the stronger nonlinear response, similar to \(\delta_{2}\), yet cavity operation near the plane-plane limit, \(\delta_{0}\), is usually less convenient compared to the point-plane limit, \(\delta_{2}\), since it is more sensitive to the misalignment of the end mirrors. **8. Nonlinear Kerr Lens Astigmatism**  
Kerr lensing in the Brewster-cut Ti:sapphire crystal introduces an additional source of nonlinear astigmatism into the cavity.