## 2 Modelling
### Polarimetry  
While the presence of the magnetic field can be neglected to evaluate the propagation of the ray, it is important for the evolution of the polarisation of the electromagnetic wave. In fact, the presence of the two refractive indexes, even if similar, changes the polarisation of the beam crossing the plasma. Plasma polarimetry is based on measuring the change of the polarisation to obtain some integral parameters directly correlated with the electron density and the magnetic field. The Stokes notation has been used to model the polarisation variation in plasma [15; 29]. The reduced Stokes vector has three components that are directly correlated to four polarisation angles: the phase shift \(\phi\), the electric field component ratio \(\tan(\alpha)\), the ellipticity \(\tan(\chi)\) and the ellipse orientation \(\psi\), as follows [15]:  
\[\mathbf{s}\ =\ \left[ \begin{array}{c}s_{1}\\ s_{2}\\ s_{3}\end{array} \right]\ =\ \left[ \begin{array}{c}\cos 2\chi\cos 2\psi\\ \cos 2\chi\sin 2\psi\\ \sin 2\chi\end{array} \right]\ =\ \left[ \begin{array}{c}\cos(2\alpha)\\ \sin(2\alpha)\cos(\phi)\\ \sin(2\alpha)\sin(\phi)\end{array} \right] \tag{11}\]  
The evolution of the Stokes vector in a magnetised plasma can be evaluated by the following differential vector equation [15]:  
\[\frac{ds(z)}{dz}\ =\ \mathbf{\Omega}(z)\times\mathbf{s}(z) \tag{12}\]  
The vector \(\Omega\) takes into account the properties of the plasma and magnetic field [15]:  
\[\mathbf{\Omega}(z)\ =\ \left[ \begin{array}{c}\Omega_{1}\\ \Omega_{2}\\ \Omega_{3}\end{array} \right]\ =\ \left[ \begin{array}{c}C_{1}\lambda^{3}N_{e} \left(B_{x}^{2}-B_{y}^{2}\right)\\ 2C_{1}\ \lambda^{3}N_{e}B_{x}B_{y}\\ C_{3}\lambda^{2}N_{e}B_{p}\end{array} \right] \tag{13}\]  
where \(B_{p}\) is the component of the magnetic field along the beam propagation direction, while \(B_{x}\) and \(B_{y}\) are the orthogonal components, the constant \(C_{1}\) is \(2.45\times 10^{-11}\) and \(C_{3}\) is \(5.26\times 10^{-13}\)[15]. The Faraday rotation (\(\Delta\psi\)) and the Cotton-Mouton phase shift (\(\Delta\phi\)) can be directly correlated to the line integral of the electron density and magnetic field. The most common approximation, known as type 1, links the polarisation effects with the plasma parameters as follows [17]:  
\[\Delta\psi\ =\ \int\frac{1}{2}\Omega_{3}dz\ =\ 2.63\times 10^{-13}\lambda^{2} \int N_{e}B_{p}dz \tag{14}\]  
\[\Delta\phi\ =\ \int\Omega_{1}dz\ =\ 2.45\times 10^{-11}\lambda^{3}\int N_{e} \left(B_{x}^{2}-B_{y}^{2}\right)dz \tag{15}\]  
The propagation of the Stokes vector in the model has been considered by means of three new auxiliary variables, one for each reduced Stokes component (s\({}_{1}\), \(s_{2}\) and \(s_{3}\)), using Equation (12). Attention must be paid to the reference frame used to describe the Stokes vector and the magnetic field. In nuclear fusion plasmas, the magnetic field is usually given in terms of a toroidal field (\(B_{t}\)), radial field (\(B_{r}\)) and vertical field (\(B_{z}\)), where r is the horizontal direction and z is the vertical direction on the poloidal section, while the Stokespropagation, described by Equation (12), is written in the reference frame of the probing beam. The direction of propagation of each beam is described by its unit vector as:  
\[\mathbf{n_{p}}\ =\ \frac{\mathbf{k}}{k}\ =\ \left[ \begin{array}{c}n_{p,r}\\ n_{p,z}\\ n_{p,t}\end{array} \right] \tag{16}\]  
One of the two orthogonal unit vectors, defined as the \"\(x\)\" direction, is calculated as \(\mathbf{n_{x}}=\ \left[ \begin{array}{c}n_{p,t}\\ -n_{p,r}\\ n_{p,z}\end{array} \right]\), which is always orthogonal to the \(\mathbf{n_{p}}\) unit vector. The last vector, which describes the \"y\" direction, is calculated by the cross-product between \(\mathbf{n_{p}}\) and \(\mathbf{n_{x}}\). Thus, the magnetic field is rotated from the reference frame of the reactor to the reference frame of the electromagnetic wave.