## 1 Introduction  
Manipulation of magnetism using ultrafast laser pulses without any external magnetic field, also called All Optical Switching (AOS), is fundamentally interesting and promises for low-power and high-speed spintronic devices. After the pioneer observations of ultrafast demagnetization in a Nickel film submitted to a femtosecond laser pulse [1], other experiments were performed to manipulate the magnetization with laser pulses. The first studies of AOS appeared in ferrimagnetic alloys [2; 3], synthetic ferrimagnets [4; 5], and more recently, in ferromagnetic materials [6; 7; 8; 9], which have relevance for ultrafast magnetic recording applications. The optical control of the magnetic state in a ferromagnetic sample usually requires series of laser pulses with circular polarization. This multi-shot helicity-dependent control of the magnetization has been observed in three different types of experiments. (i) The local area under the laser beam of a ferromagnetic thin film with perpendicular magnetic anisotropy (PMA) can be reversed depending on the initial magnetic state and on the helicity of the laser pulses. This phenomenon is called Helicity-Dependent All-Optical Switching [3; 10]. (ii) Trains of laser pulses have been also applied on a thin film initially containing a domain wall (DW) separating up and down magnetic states. In some of these experiments, the center of the laser beam is located away from the DW center, and multi-shot laser pulses promote a displacement of the DW position if the laser helicity is properly chosen. These processes are commonly referred to as Helicity-Dependent All-Optical DW Motion (HD-DWM) [8; 9; 10; 11]. (iii) In other experimental setups, the laser beam is sweep along a given path over the ferromagnetic thin film. Similarly to the HD-AOS case for a fixed laser beam location on top of the ferromagnetic thin-film, the switching of the local magnetic state along the laser beam path can only be achieved for a given helicity of the laser pulses, which depends on the initial magnetic state [10]. The full comprehensive role of the ultrafast laser pulses in all these multi-shot helicity dependent processes is still under debate, and in general, there are several effects which need to be considered. The heating and eventual demagnetization of the sample at the femtosecond scale is a well established effect [12; 13]. During the application of a laser pulse, the temperature of the sample can overcome the Curie threshold resulting in a local demagnetized state where the ferromagnetic order is lost. Linear and circular polarizations of the laser pulse heats the sample, but if we consider the MCD, there are different absorption rates for each helicities, typically differing in some units of percent. The helicity-dependent optical switching of the initial magnetic state or domain wall motion may be determined by the Inverse Faraday Effect (IFE) [3; 14; 15; 16], the Magnetic Circular Dichroism (MCD) [17; 18], or the laser-induced spin currents [19; 20]. According to the IFE, an angular momentum density is associated to the circularly polarized laser pulse [21] due to the photon spins that leads to the induction of an out-of-plane effective field [22]. The direction of this magneto-optical effective field (\(\overset{\rightarrow}{B}_{MO}\)) is determined from the laser helicity (\(\sigma\)) [23], which favors a direction for the magnetic state [3]. On the contrary, if the laser beam is linearly polarized, no out-of-plane field is induced. In the framework of the MCD, it is assumed that the energy absorption of the magnetic sample from the laser beam depends on the laser helicity (right- or left-handed helicity, \(\sigma^{\pm}\ =\ \pm 1\)) and the local magnetic state (up \(\overset{\rightarrow}{m}\ \uparrow\), or down \(\overset{\rightarrow}{m}\ \downarrow\)).