## 4 The Optical Kerr Effect  
The optical Kerr effect is the nonlinear mechanism at the core of ultrashort pulse formation in many solid-state lasers. We provide a quantitative description of the nonlinear response and discuss how it affects the laser light in both the spatial and time/frequency domains. Since both the spatial and temporal effects are nonlinear (_i.e._, intensity-dependent), they are coupled together through the master equation of pulse propagation. A more detailed and mathematical review of the master equation and pulse dynamics, laser noise, the theory of active/passive mode-locking, along with slow/fast saturable absorbers and Kerr media can be found in [27; 28; 29; 30; 31]. The optical Kerr effect is a third-order nonlinear process, where the refractive index of the material is intensity-dependent [32], given by:  
\[n(I)=n_{0}+n_{2}I \tag{1}\]  
where \(n_{0}\) and \(n_{2}\)[33] are the linear and nonlinear refractive indices, respectively, and \(I\) is the light intensity. As explained below, the Kerr effect can effectively introduce an artificial saturable absorber to the cavity, placing it under the category of passive mode-locking [34]. Since the Kerr effect in transparentmedia is practically instantaneous, this effective saturable absorber is exceedingly fast, allowing the generation of pulses as short as a few femtoseconds, far beyond what can be achieved with \"real\" saturable absorbers (in the picosecond scale). In practice, the pulse duration of Kerr lens mode-locked lasers will be limited by the dispersion management in the cavity, the gain bandwidth of the active medium and the optical period of the carrier wave. Since \(n_{2}\) for the sapphire crystal is relatively low (\(\approx\)\(3\times 10^{-16}\) cm\({}^{2}\)/W) [35], only high peak power pulses will be sensitive to this effect. The preference for pulses via the optical Kerr effect is rooted in its influence on the pulse profile in both space and time. In the transverse spatial domain, due to the Gaussian-shaped laser beam, the center of the beam experiences a larger refractive index compared to the wings, producing an effective focusing lens for the laser. The effective nonlinear lens can be calculated by considering the phase accumulated after propagating through a Kerr medium with thickness \(z\) and a nonlinear coefficient, \(n_{2}\), given by: \(\phi=k_{0}n(I)z\) (to be considered as thin, the material thickness, \(z\), should be considerably smaller than the Rayleigh range of the mode). By comparing the result to the well-known phase of a simple lens, the dioptric power of the nonlinear Kerr lens can be easily identified as:  
\[\frac{1}{f_{NL}}=\frac{4n_{2}z}{\pi}\frac{P}{w^{4}} \tag{2}\]  
where \(P\) is the pulse peak power and \(w\) is the mode radius. Note that Equation (2) holds only for a circular beam where \(w_{x}=w_{y}=w\). For a non-circular beam with \(w_{x}\
eq w_{y}\), the dioptric power will be different for each plane, and both planes will be coupled by the nonlinear interaction. A generalized expression for the dioptric power of each plane can be found in [36]. In combination with an intra-cavity aperture, this intensity-dependent nonlinear lens can be used as an intensity-dependent loss element that generates self amplitude modulation (SAM), which favors pulsed operation over CW operation. In general, strong SAM is desired in order to achieve robust mode-locking over a wide pump power range. Two major concepts are used when introducing an aperture into a cavity: hard and soft aperture; where hard aperture exploits a real, physical aperture in the cavity, while soft aperture exploits the pump beam size inside the laser crystal as the effective aperture. Both methods are explained in detail in Section 7. In the temporal domain, the intensity-dependent refractive index modulates the instantaneous phase of the pulse according to the instantaneous intensity by: \(\phi(t)=\omega_{0}t+\phi_{NL}=\omega_{0}t+k_{0}n(I)L\); where \(L\) is the thickness of the Kerr medium.