## 4 Analysis of KH Stability Near Its Threshold Around R \(\approx\) 1
### Experimental Observation of These Unstable Behaviors
#### 4.3.1 Welding Experiments
[33] that take into account ray-tracing of the laser beam propagation inside the KH, show that due to this multi-reflections process, the rear KH wall can be directly irradiated by the reflected beam from the KH front wall when its inclination is about 45\({}^{\circ}\). The resulting recoil pressure on this rear wall of the KH prevents the KH from closing due to surface tension and controls the position of the rear wall. However, this rear wall is subject to many fluctuations due to this unstable balance of the inner geometry of the KH, which is controlled by the distribution of the surface tension pressure and the recoil pressure resulting from these multi-reflections. These simulations were realized for operating conditions of additive manufacturing (V \(\approx\) 0.8 m/s, spot diameter d \(\approx\) 50 \(\upmu\)m), where the KH length is rather elongated (about 250 \(\upmu\)m) and strongly fluctuates inside the melt pool of about 600 \(\upmu\)m length (see an example in Figure 11a). These conditions are of course very different from the ones we used. But this mechanism could also be applied to our conditions, and it is possible to propose another scenario based on multi-reflections for explaining the previous experimental results. One starts with a given incident laser power that is high enough to generate a corresponding KH of low aspect ratio, (typically R \(\approx\) 1-2); therefore, with an inclination angle lower than \(\uppi\)/4 (Figure 11b). At some point, probably due to fluctuations within the melting front we have seen that for the same operating conditions, the inclination of the KH front can be switched to the other solution described in Section 4.1 for which R \(<\) 1, and thus an inclination \(\alpha\) \(>\)\(\uppi\)/4 exists. This incident laser power which is capable of generating the KH inside the solid sample is therefore suddenly mainly redirected toward the melt (Figure 11c). A strong evaporation must therefore occur locally at the impact, as the video shows that the melt pool surface rises in a few milliseconds, like a bubble, which eventually bursts, but with practically no loss of melt through this ejection. Then, the melt returns to the inside of the cavity thus produced, and during the reorganization of this melt--the laser beam being ever-present and continuing to move forward--a blow-hole is then left on the rear side of the melt pool (Figure 11d). It is obvious that the use of a time-resolved X-ray radiography would be much more appropriate to visualize the internal geometry of this liquid bubble during its expansion. These experiments have been reproduced at welding speeds of 5 and 9 m/min with higher incident laser powers varying from 0.5 to 10 kW, (the corresponding power thresholds P\({}_{\text{t}}\) being estimated respectively to 1250 and 2000 W). In fact, no significant appearance of similar blow-holes was observed in these cases. Because of these higher welding speeds and the large focal spot diameter used here, the velocity of the melt ejected backwards is higher than at V\({}_{\text{w}}\) = 1 m/min, and consequently makes the KH aperture appear rather elongated, particularly for V\({}_{\text{w}}\) = 9 m/min. Moreover, for these high welding speeds, the width of the melt pool is narrower; this geometry should be therefore less prone to being lifted by the reflected beam. Therefore, this result seems to confirm that this unstable regime should mainly occur when the KH rear wall is rather close to the KH front, and more generally, when the velocity of ejected melt backwards is low (see corresponding conditions on Figure 7).