## 4 Analysis of KH Stability Near Its Threshold Around R \(\approx\) 1
### Dependance of the KH Front Inclination for a Given Incident Laser Power  
We have previously seen that both regimes can be described by an equation similar to Equation (13) that relates the aspect ratio R as a function of incident power P, the corresponding absorptivity A(R) and the power threshold A(1)P\({}_{\rm t}\approx\) d(a + b(V\({}_{\rm w}\)d)). But because rather similar parameters a and b have been obtained for both regimes, for the sake of simplicity, we will assume the same power threshold A(1)P\({}_{\rm t}\) in what follows. Therefore, for a given incident power P, the resulting aspect ratio R will then be a solution to Equation (18):  
\[\frac{\rm R}{\rm A(R)}=\frac{\rm P}{\rm A(1)P_{\rm t}}=\rm P_{\rm n}, \tag{18}\]  
In Equation (18), a normalized incident power P\({}_{\rm n}\) = P/(A(1)P\({}_{\rm t}\)) has been defined. For a given incident normalized power P\({}_{\rm n}\), we have examined the solution of Equation (18) by using three different variations of A(R) around R = 1. These three cases are reported on Figure 9a. They all have the same constant absorptivity A\({}_{0}\) from for 0 \(<\) R \(<\) 1\({}^{-}\). The first one considers a discontinuous (non-realistic) variation at R = 1 from A\({}_{0}\) to A\({}_{\rm G}\)(1) (A\({}_{\rm G}\)(1) is given by the [PERSON]'s model for R = 1). The two others show more realistic evolutions, with a continuous transition being rather sharp (red curve) or smooth (green curve) for 1\({}^{-}<\) R \(<\) 1\({}^{+}\). For R \(>\) 1\({}^{+}\), these three variations follow the [PERSON]'s model of absorptivity A\({}_{\rm G}\)(R). The corresponding evolutions of R/A(R) have been reported on Figure 9b, and different values of P\({}_{\rm n}\) have been considered. In the case of the discontinuous variation of A(R) (blue curves in Figure 9), it is easy to see that R/A(R) varies linearly from 0 to 1/A\({}_{0}\) for 0 \(<\) R \(<\)1. For R \(>\) 1, it increases quite linearly from 1/A\({}_{\rm G}\)(1) with a slope locally defined by the [PERSON]'s model. One can see that the discontinuity of A(R) at R = 1 induces for R/A(R) two separate branches on each side of R = 1. Therefore, if P\({}_{\rm n}\) \(<\) 1/A\({}_{\rm G}\)(1) or P\({}_{\rm n}\) \(>\) 1/A\({}_{0}\), there is a unique solution of Equation (18) for R. However if 1/A\({}_{\rm G}\)(1) \(<\) P\({}_{\rm n}\) \(<\) 1/A\({}_{0}\), two solutions for R are obtained: one being such that R\({}_{1}\) \(<\) 1, and for the other one, R\({}_{2}\) \(>\)1 (see Figure 9a). One must note  
Figure 9: **(a)** Three examples of variation of the absorptivity A(R) around R = 1. Blue curve: discontinuous variation from A\({}_{0}\) to A\({}_{\rm G}\)(1) at R = 1. Red curve: smooth but rather sharp transition around R = 1. Green curve: much smoother transition. Dotted curve: [PERSON]’s absorptivity for a cone (for steel, \(\lambda\) = 1.03 \(\upmu\)m). **(b)** Corresponding evolutions of R/A(R) for these three cases of variation of the absorptivity. (A\({}_{0}\)\(\approx\) 0.31 for steel at \(\lambda\) = 1.06 \(\upmu\)m is used, and A\({}_{\rm G}\)(1) is the absorptivity given by the Gouffe’s law at R = 1). that these two solutions are physically possible because they both correspond to positive derivatives of R with incident power P\({}_{n}\). In order to estimate the corresponding normalized incident power P/P\({}_{t}\) instead of P\({}_{n}\), we assume here that A(1) \(\approx\) 1 - (1 - A\({}_{0}\))\({}^{2}\) = 0.52, which would be the KH absorptivity when only two reflections occur inside (see Figure 2c). Therefore, the previous power ranges can be also written as (for steel):  
* For P/P\({}_{t}\) < (P/P\({}_{t}\))\({}_{min}\) = A(1)/A\({}_{G}\)(1) \(\approx\) 0.8, only one solution with R \(<\) 1 is obtained, which does not correspond to a KH. It can be considered as some \"forced\" conductive regime, where the melt pool surface is depressed with an inclination angle \(\alpha\) > \(\pi\)/4.