## 4 Analysis of KH Stability Near Its Threshold Around R \(\approx\) 1
### Experimental Observation of These Unstable Behaviors
#### 4.3.2 Spot-Welding Experiments  
The previous absorptivity measurements, based on time-integrated measurements with calorimetry techniques, are not representative of the time-dependent variations of the absorptivity that should occur inside the KH for R \(\approx\) 1. We have seen that these two solutions R\({}_{1}\) or R\({}_{2}\) are obtained when an absorptivity A(R\({}_{1}\)) or A(R\({}_{2}\)) of about A\({}_{0}\) and A\({}_{G}\)(1) respectively occurs. Only time-dependent measurement of the absorptivity, obtained for example by measuring the reflected laser power from the sample by using an integrating sphere, can be suitable to follow its evolution. This technique has been employed by [PERSON] et al. [36], but during a \"spot welding\" (SP) process, where the welding speed V\({}_{w}\) = 0, so the sample was irradiated locally in a pulsed mode in order to generate a weld nugget. In fact, one can also apply the previous scheme of KH formation by considering that there is also a threshold for generating the keyhole for these static conditions. Indeed, when the melt pool surface with a cylindrical symmetry is depressed from the beginning of the interaction, its inclination also decreases, and the reflected beam that was initially reflected upwards is gradually deflected downwards and can be refocused downwards on its axis, when the mean inclination of the melt pool surface becomes greater than \(\pi\)/4. It then follows that the depression increases sharply and oscillates (similar conditions have also been visualized using coherent X-ray radiography by [PERSON] et al. [21]), and leads to generating a deep KH that induces an increase of the absorptivity [37]. Experimentally, by using an integrating sphere, the time dependence of the reflected beam was measured for different incident laser powers, and a power threshold P\({}_{t-SP}\) (defined such that the melt pool depth equals the spot diameter) has been estimated [36]. As expected, for P \(<\) P\({}_{t-SP}\), the absorptivity of the melt pool surface is constant at about 0.30, and for P \(>\) P\({}_{t-SP}\), because of the multiple reflections inside the increasing depth of the cylindrical KH, the absorptivity increases with the incident laser power while remaining stable. But for P \(\approx\) P\({}_{t-SP}\), the time-dependent absorptivity during the pulse duration shows several humps where the absorptivity varies from about 0.30 (the typical absorptivity of the melt pool surface) to about 0.50 (some absorptivity of a non-completely developed KH). The time interval between the observed oscillations peaks of absorptivity probably corresponds to the delay for achieving the maximum penetration of the KH that follows its closure due to the surface tension effect. One can understand that for these threshold conditions, the intensity distribution at the bottom of the depression that controls the recoil pressure, which roughly equilibrates the local surface tension pressure, can be easily perturbed by the unavoidable fluctuations of the KH surface. Figure 11: (**a**) Example of multi-physics simulations of the multi-reflections mechanism inside the KH [33]. Sequence of events leading to melt pool perturbation: (**b**) A low aspect ratio KH is first generated. (**c**) Due to fluctuations that modify the KH front inclination, the incident laser beam is redirected toward the melt pool whose surface is lifted and then tears; a transient cavity forms and then fills (**d**) A new melt pool is reconstructed but with a blow-hole left behind. Experimentally, the threshold seems to appear for an incident pulse energy range from \(\approx\)2 J to \(\approx\)2.9 J, delivered with a pulse duration of 10 ms, which corresponds to a power range \(\approx\)200 W \(<\) P \(<\)\(\approx\)290 W. But these experiments that use a short pulse duration are not characteristic of a stationary regime.