## 1 Introduction  
Laser welding has become an extremely widely used process in the industrial world because it allows the easy assembly of metal parts having a great range of thicknesses and with a very high productivity. All these processes operate in the so-called keyhole (KH) mode; i.e., the laser radiation penetrates the material to a depth at least greater than the focal spot. Thanks to this effect, in a regime that can be described as conventional \"macro-welding,\" it is possible to assemble thicknesses of about ten millimeters, at welding speeds of a few m/min, using lasers delivering powers of a few tens of kW and focal spots of several hundred microns. At the other end of the range of operating parameters, there is a more dynamic regime, the so-called \"micro-welding\" regime, which uses lasers of a few hundred watts and focal spots typically smaller than a hundred microns. Sub-millimeter thicknesses can then be welded, but at very high welding speeds, typically of several m/s. One must also note that these conditions are met for example, in additive powder bed manufacturing processes [1, 2]. A crucial point in the implementation of this process is the knowledge of the depths of the KH obtained, as a function of these operating and thermo-physical parameters of the material used. In order to estimate these depths, many semi-analytical [3, 4, 5, 6, 7, 8] or numerical [9, 10, 11, 12, 13, 14, 15] approaches have been considered. We had also recently proposed a relatively efficient and simple model [16] for estimating these KH depths based on a purely thermal analysis of the mechanisms involved, which makes it possible to reproduce quite satisfactorily the experimental results of this \"macro-welding\" regime, obtained with various operating parameters and on materials with very different thermo-physical properties. However, it turned out that this model could also reproduce the experimental results obtained in the \"micro-welding\" regime, in particular, those obtained in the context of additive manufacturing experiments [17; 18; 19], which seemed rather surprising, given that in this regime the geometries of the KHs present are very different from those observed in a \"macro-welding\" regime. Indeed, [PERSON] and [PERSON] [20] have shown that at high welding speeds, which could also characterize this \"micro-welding\" regime, the interaction of the incident laser must mainly take place on the inclined front wall of the KH and not inside the quasi-vertical cylindrical KH characteristic of the macro-welding regime. Experimental visualizations of the KH shapes by time-resolved X-ray radiography have directly confirmed this geometry [21; 22]. For this reason, in order to explain these results, an original model adapted to this inclined KH front is proposed here. It is based on a generalization of the well-known \"piston model\" developed by [PERSON] [23] and allows for reproducing and interpreting the experimental results of [PERSON] et al. [21]. It also makes it possible to estimate all the relevant parameters of this interaction regime--the incident power dependence on the resulting aspect ratio, the inclination of the KH front and the velocity of rearward liquid ejection--and to quantify the threshold power, allowing for the appearance of KH; i.e., when the aspect ratio becomes greater or equal to one. As a consequence, the analysis of these KH depths scaling laws also shows that for these threshold conditions, an unstable behavior of the melt pool which has never been shown before should appear. The influences of the main parameters controlling this unstable behavior, particularly the laser absorptivity of the material, are also analyzed. This paper is organized as follows: the model describing the thermal approach used for the analysis of KH depths in the macro-welding regime is briefly recalled in Section 2. It allows comparing its resulting scaling laws with those obtained in the dynamic regime, as described in Section 3, where the incident laser beam mainly impinges the inclined KH front.