## 7 Mode-Locking Techniques
### Virtual Hard Aperture Mode-Locking
As the beam refracts into the crystal, the mode size in the tangential plane, \(w_{t}\), expands, while the sagittal mode size, \(w_{s}\), remains the same, hence reducing the intensity and the nonlinear response in the tangential plane. The difference in the nonlinear response between the two planes will produce an intensity-dependent astigmatism, even if the linear astigmatism is fully corrected. In terms of the power-dependent stability limit, \(\delta_{1}\), of mode-locked operation illustrated in Figure 7a, both the sagittal and tangential stability limits will be pushed towards higher values of \(\delta\),but the sagittal limit will be pushed farther away, compared to the tangential limit. Note that only hard aperture mode-locking pushes the stability limits towards higher values of \(\delta\), in contrast to soft aperture mode-locking, where the stability limits are pushed towards lower values of \(\delta\). However, in both hard and soft aperture techniques, the relative variation of the power-dependent stability limit will be higher in the sagittal plane compared to the tangential plane. In terms of the Kerr lens strength, \(\gamma\), plotted in Figure 7b, both the tangential and sagittal planes will have a similar qualitative behavior, but the absolute values of \(\gamma\) will be reduced in the tangential plane compared to the sagittal. Therefore, a fully-corrected CW mode will not remain corrected after mode-locking. The standard solution to the problem is to pre-compensate for the nonlinear astigmatism [49] by introducing extra linear astigmatism for the CW mode in the opposite direction, as seen in Figure 8, such that the plane with the stronger \(|\gamma|\) will \"catch up\" with the weaker one. By changing the values of the angles, \(\theta_{1}\) and \(\theta_{2}\), away from perfect linear astigmatic correction, one can pre-compensate for nonlinear astigmatism at \(\delta_{1}\). Consequently, a deliberately non-circular CW mode will become circular after mode-locking. Note, however, that this compensation holds only for a specific value of \(K=P/P_{c}\), _i.e._, specific intra-cavity peak power. Increasing (lowering) \(K\) with the same folding angles (_i.e._, the same linear astigmatism) will result in over (under) compensating for the nonlinear astigmatism. Any change in parameters that keeps the CW astigmatism compensated for, but that affects the intra-cavity intensity, be it the peak power or mode size in the crystal, will require a change in the folding angles to match the precise CW astigmatism needed to converge into a non-astigmatic ML beam. This includes a change in: pump power, pump focusing, output coupler, short arm length and, also, \(Z\) or \(\delta\). This requires specific compensation for every time one changes cavity parameters, making nonlinear astigmatism a nuisance in standard cavity designs. Recently, a novel cavity design has been demonstrated that nulls nonlinear astigmatism completely [53], as will be elaborated in Section 11.2. Figure 7: **(a)** ML and CW mode size for both sagittal and tangential planes on the end mirror, \(EM1\) (short arm), near the CW stability limit, \(\delta_{1}\), for hard aperture mode-locking; **(b)** Kerr strength \(\gamma\) at \(EM1\) as a function of the crystal position, \(Z\), near the stability limit, \(\delta_{1}\).