## 10 Typical Characteristics of Pulsed Operation  
The spectroscopic and laser characteristics in CW operation were well studied in the past and can be found in [73]. Pulsed operation of the Ti:sapphire laser was also extensively studied, leading toremarkable results in terms of pulse duration [14], repetition rate [74], average power [75], mode-locking threshold [76] and pulse energy [77]. Here, we provide some of the basic characteristics and typical qualitative behavior of a Ti:sapphire oscillator in ML operation, especially focused on the onset of ML and the experimental procedure to obtain ML. Using virtual hard aperture mode-locking (Section 7.2), pulsed operation can be achieved in a band of \(\delta\) values slightly beyond the stability limit, \(\delta_{1}\), showing a qualitative behavior, as plotted in Figure 10a. Pulsed operation can be achieved only after the pump power is raised beyond the CW threshold up to a certain ML threshold value, where the appearance of pulses is abrupt in terms of pump power. The threshold-like behavior of mode-locking was elegantly explained, both theoretically and experimentally, as a first order phase transition by the theory of statistical light mode dynamics [78,79]. At the ML threshold, the output power of the pulse will be higher than the CW output power, since the pulse enjoys lower diffraction losses than the CW. Figure 10b plots the ratio between ML output power  
Figure 10: **(a)** Qualitative behavior of pulsed operation for a given mode-locking point at \(\delta\gtrsim\delta_{1}\). **(b)** ML threshold and ML-to-CW ratio of output powers as a function of \(Z=\delta-\delta_{1}\), while mode-locking with the virtual hard aperture technique at \(\delta>\delta_{1}\). Measurements were taken from a typical configuration of a Ti:sapphire cavity, including a \(3\) mm-long Brewster-cut crystal, curved mirrors of \(f_{1}=f_{2}\) = 5 cm, \(L_{1}\) = 20 cm and \(L_{2}\) = 75 cm, an output coupler of \(95\%\) reflectivity and the pump focused to a diameter of \(\approx\)\(22\)\(\upmu\)m in the Ti:sapphire crystal. **(c)** Typical measured spectrum (not optimized) of an ultrashort pulse with a bandwidth of \(>\)200 nm at full width at half maximum (FWHM). **(d)** Measured spectrum in the double pulse regime where the spectral interference pattern can be observed. and the CW output power as \(\delta\) is increased beyond the stability limit (\(\delta\gtrsim\delta_{1}\)). The ratio \(\gamma_{e}=P_{ML}/P_{CW}\) is an experimental measure of the Kerr lensing strength, similar to \(\gamma\) of Equation 14. In addition, the mode-locking threshold as a function of \(\delta\) is also plotted in Figure 10b. One finds that both the ML threshold and the ML power increase monotonically with increasing \(\delta\), due to the need to overcome increasing diffraction losses, but the Kerr lensing strength has a maximum efficiency point, where ML is optimal. Figure 10c plots a typical measured broadband spectrum of an ultrashort pulse at the point of maximum Kerr lensing efficiency. The typical bandwidth of \(>200 nm\) at FWHM can be achieved even without the use of double-chirped mirrors or additional bandwidth maximization techniques (these techniques will be discussed in Section 11.1). Several techniques for temporal characterization of the pulse were developed in the past, such as interferometric autocorrelation (IAC), frequency-resolved optical gating (FROG) and spectral phase interferometry for direct electric-field reconstruction (SPIDER) [80, 81, 82, 83]. For a given mode-locking point at \(\delta\gtrsim\delta_{1}\), further increasing of the pump above the ML threshold, as illustrated in Figure 10a, will increase the CW output power, while the ML output power will remain approximately the same. A qualitative and intuitive explanation for this phenomenon is that the soliton-like pulse peak power is \"quantized\", _i.e._, a unique peak power stabilizes the cavity for ML operation, to which the laser clutches.