## 3 Validation on a Case Study
### Thomson Scattering Simulations  
The numerical simulations to validate the Thomson scattering model were performed changing the geometry according to Figure 7. The light beam, generated by the laser, crosses the plasma. The laser wavelength is 1064 nm, while the peak power is \(5\times 10^{8}\) W (note that the TS diagnostic requires a pulsed laser with high peak power). Four different scattering volumes were placed along the direction of the laser beam and accumulate the variables needed for the scattering simulation. For each wavelength, the scattering volumes release rays with a specific solid angle aperture. The rays cross the plasma, exit the vacuum vessel, interact with the spherical mirror on the top of the tokamak and the rays are collected by the spectrometers, one for each scattering volume.  
The numerical simulations were conducted changing the \(N_{0}\), \(T_{0}\) and \(I_{p}\). The density goes from 0 to \(20\times 10^{20}\) m\({}^{-3}\), the electron temperature from 0 to 20 keV and the plasma current from 0 to 15 MA. Figure 8 compares the expected and the simulated spectral density (defined as spectral power divided by the total power, considering the wavelength step equal to 50 nm). For each channel, four different spectra are reported using 0.25, 0.5, 0.75 and 1 times the maximum values of electron density, electron temperature and plasma current. The simulated spectra are perfectly overlapped to the expected spectra at the temperature of each scattering volume, showing that the model works properly.  
Regarding the polarimetric Thomson scattering, the analyses were performed on the same scattering volumes with the same electron, temperature, plasma current and magnetic field. The reduced Stokes vector of the laser beam was placed equal to [1, 0, 0] (linear polarisation with an angle at 0\({}^{\circ}\)), which is the polarisation normally used in TS diagnostics. In Figure 7, the colour of the ray trajectories shows the first component of the scattered Stokes vector, which decreases as a function of the local temperature. The measurement of the temperature through the polarimetric TS can be performed by using Equation (22).  
Figure 6: Validation of the Cotton–Mouton phase shift by comparing the electron density line integral with the CM electron density.  
Figure 7: Thomson scattering geometry (**left**); Electron temperatures, ray trajectories of the primary beam (green) and scattered radiation with the first scattered reduced Stokes component \(s_{\text{sc,1}}\) (**right**).  
In the specific case of the numerical simulation, it can be demonstrated that the following equation is valid:  
\[4\tau^{2}(1-u)^{2}(3k-4)+2\tau\Big{(}u^{2}s_{sc,1}+u^{2}-4 us_{sc,1}+4u+k-3\Big{)} +2(s_{sc,1}-1)\ =\ 0 \tag{24}\]  
Thus, knowing the scattering angle and measuring the first component of the scattered Stokes vector, the temperature can be directly measured. Figure 9 (left) shows the first component of the reduced Stokes vector of the scattered light for the four channels as a function of the electron temperature. Note that the points do not lie on the same line because of the different scattering angle. The other two Stokes components are not shown since they are equal to zero (as expected when the input polarisation has a Stokes vector equal to [1, 0, 0]). Figure 9 (right) shows the expected electron temperature vs. the \"measured\" temperature by Equation (24) using the first component of the reduced Stokes vector. Even in this case, the measured temperature coincides with the expected one, validating the proper functioning of the simulation.  
Figure 8: Thomson scattering expected and simulated adimensional spectra of the four channels as a function of the temperature.