# Measurement of the Resonant Magneto-Optical Kerr Effect Using a Free Electron Laser
## 1 Introduction  
Femtomagnetism, which refers to magnetization dynamics on a femtosecond timescale, has been attracting research attention for more than two decades because of its fundamental physics and its potential for use in the development of novel spintronic devices [1]. The ultrafast dynamics of femtomagnetism can be accessed using ultrashort laser pulses to perturb magnetic materials via thermal and nonthermal effects. This produces a system of strongly-nonequilibrium states [2]. Immediately after a sudden disturbance, magnetic systems show demagnetization due to a transition from ferro-/ferri-magnetism to paramagnetism that is caused by impulsive heating. Additionally, in certain magnetic compounds, such as ferromagnetic alloys and ferrimagnets composed of rare-earth and transition metals, the magnetization is reversed on a femtosecond timescale [3]. Laser-induced phase transitions from antiferromagnetic to ferromagnetic phases [4] and optical control of the spin precession have also been reported [5]. These ultrafast demagnetization and magnetization reversal phenomena, and particularly the mechanism of spin-flips, have been interpreted from various phenomenological viewpoints [6; 1; 7]. Interactions between the electron, spins, lattices and photon fields have been incorporated in some models. In addition, superdiffusive spin transport [8] and interactions with collective excitations such as phonons and magnons [9] have also been considered in other models. While the magnetization dynamics, especially on the femtosecond timescale, remain controversial from the microscopic perspective, a number of experimental techniques for probing of the transient characteristics have emerged to meet the demand for probing methods that are suitable for a wide variety of magnetic materials. One of these techniques is the magneto-optical Kerr effect (MOKE) method, which has been applied to investigations of static bulk/surface magnetism, spin transport and magnetization dynamics using linearly-polarized light [10]. These experiments have mainlybeen conducted in the visible range. In previous studies, MOKE in the soft X-ray range, which involves tuning of the photon energy to the absorption edge of a target magnetic element, has been investigated, and we call this method resonant MOKE (RMOKE) (see Equation (1)) in this review [11; 12]. However, when it comes to the static magnetization measurements, the usefulness of RMOKE has not been focused on until now by the availability of other convenient soft X-ray magneto-optical measurement methods, such as X-ray magnetic circular dichroism (XMCD) spectroscopy [13; 14]. Following the development of femtosecond light sources such as synchrotron radiation (SR) sources using a laser slicing technique, the free electron laser (FEL) and the high harmonic generation (HHG) laser, the RMOKE technique used in combination with these state-of-the-art light sources is becoming increasingly important, particularly for temporal domain measurements in the subpicosecond timescale. We first demonstrated time-resolved RMOKE (TR-RMOKE) measurements using the FEL in 2015 [15]. In this review, we clarify the importance of the RMOKE technique when using an extreme ultraviolet (EUV) FEL from the methodological view point and demonstrate time-resolved RMOKE measurements using a seeded FEL at FERMI@Elettra (FERMI-I is generally called X-ray ultraviolet (XUV) FEL). In the next section, the MOKE is introduced in terms of two of its aspects: rotation and ellipticity. The experimental geometry of the MOKE is explained with respect to the magnetization components that can be detected when using each configuration. In Section 3, we give an overview of the magnetic probing techniques, including the MOKE, that have been used to investigate static magnetism and laser-induced magnetization dynamics in previous research. In this section, the characteristics of the TR-RMOKE technique are clarified. In Section 4, methods used to perform MOKE measurements in the visible and soft X-ray ranges are explained.