## 4 Analysis of KH Stability Near Its Threshold Around R \(\approx\) 1  
### KH Absortitivity Variations  
In the previous equations, one sees that the absorptivity A(R) of a KH is a determining parameter that controls the resulting aspect ratio R, and therefore, it must be known for any R. For a rather wide range of aspect ratios, the KHs absorptivity has been measured [30; 31]. For example, [PERSON] et al. [30] realized calorimetric time integrated measurements on SS 316L steel. In Figure 8a, an example of their measured absorptivity for a welding speed of 0.5 m/s has been reproduced with the corresponding aspect ratio. A clear transition occurs at KH threshold when R \(\approx\) 1. For comparison, for each aspect ratio R, the corresponding absorptivity using the [PERSON]'s model [32] of beam trapping inside a cone-shape geometry (with A\({}_{0}\approx\) 0.31) has been also reported. If the general trend is relatively well reproduced, we can nevertheless see that [PERSON]'s model overestimates the absorptivity, and the transition around R = 1, which is quite different from the one observed experimentally. These deviations must mainly result from the difference in geometry between the real KH and the cone shape used in the [PERSON]'s model. In fact, if one considers the geometry of KH when R \(<\) 1, A(R) is rather well known, because of the well-defined KH front geometry that allows for the use of the Fresnel equations. But when R \(>\) 1, because the reflected beam is directed downwards and then the multiple reflections process starts, A(R) should increase and the total absorptivity of the KH should be higher than that given by the Fresnel equation, and this discrepancy increases with the aspect ratio. Now for 1 \(<\) R \(<\) 2-3, the reflected beam from the KH front is not very strongly deflected downwards [29]; it is quite horizontal and it irradiates the KH rear wall, the height of which is equivalent to that of the KH front. (See for example, numerical simulations of [PERSON] et al. [33]). But the effect of this interaction on the KH rear wall depends of the location of this wall, which is defined by the length of the KH aperture itself controlled by the welding speed. At high welding speed, as the melt ejection velocity V\({}_{\rm m}\) is high (see Figure 7), the KH is elongated and its geometry is controlled by this high melt velocity, and the KH aperture is then much larger than the focal spot (confirmed by X-ray radiographies of the KH at these high welding speeds [21; 22]). This is a regime where humping is likely to start in the melt flow behind the KH. As a consequence, for this elongated KH, the interaction of the reflected beam with the KH rear wall does not modify its overall depth or the KH geometry, which is then stable. But at low welding speed, the KH aperture is quite circular with a dimension of the order of the spot diameter. Therefore, when R begins to be greater than one, the interaction of the reflected beam on this rather close KH rear wall increases the overall absorptivity of the incoming laser beam. Typically, for these conditions, after this second interaction of the incoming beam, if one considers that the reflected beam exits from this KH aperture (because of some inclination of the KH rear wall towards the rear melt pool due to the melt flow [33]), the total absorptivity after these two reflections would be \(1-(1-A_{0})^{2}\), where A\({}_{0}\) is a characteristic absorptivity of the melted KH surface. Then from there, if R continues to increase, the KH absorptivity increases further due to the increasing number of multi-reflections. Therefore, it is only inside a transition region located around R = 1 that A(R) cannot be clearly defined; however, one can expect that A(R) increases from A\({}_{\rm F}\)(1-), (when R is slightly  
Figure 8: (a) From [PERSON] et al.