## 2 Modelling
### Thomson Scattering (TS)  
Thomson scattering consists in measuring the scattered radiation of a laser beam. The electromagnetic wave, interacting with the charged particles, forces them to oscillate accordingly to the electromagnetic field, causing the emission of radiation at the same frequency of the incident wave. The thermal motion of an electron in a thermonuclear plasma makes the scattered spectrum broader than the incident radiation spectrum due to the Doppler effect. Moreover, due to the relativistic aberration (or headlight effect), the spectrum is blue shifted. The spectral power, \(P_{s}\) (\(\mathbf{R}\), \(\omega_{s}\)), of the Thomson scattering radiation due to electrons can be written as follows [19]:  
\[P_{s}(\mathbf{R},\omega_{s})d\Omega d\omega_{s}\ =\ P_{i}r_{0}^{2}N_{e}L\ d\Omega\ \left(1+\frac{2\omega}{\omega_{i}}\right)|s\times(s\times E_{i0})|^{2}\frac{e^ {-\left(\frac{\omega}{\omega_{s}}\right)^{2}}}{\pi^{1/2}\ ka}d\omega_{s} \tag{18}\]  
where \(\omega_{s}\) is the frequency of the scattered radiation, \(P_{i}\) is the power of the incident radiation, \(r_{0}\) is the classical radius of the electron, \(L\) is a typical length associated with the scattering volume, \(N_{e}\) is the electron density, \(d\Omega\) is the scattering solid angle, \(\omega\) is the difference between the scattered and the incident radiation frequencies, \(s\) is the unit vector of the scattered radiation direction, \(E_{i0}\) is the electric field vector of the incident wavelength, and \(k\) is equal to \(2k_{i}\text{sin}(\theta/2)\), where \(k_{i}\) is the wavenumber of the incident wavelength and \(\theta\) is the scattering angle. The variable \(a\) is the mode velocity of the electrons in the scattering volume, defined as \(a=(2k_{b}T_{e}/m_{e})^{1/2}\), where \(k_{b}\) is the Boltzmann constant, \(T_{e}\) is the electron density and \(m_{e}\) is the electron rest mass. In practical application, the scattering spectrum is usually written in terms of radiation wavelength, using the scattered wavelength \(\lambda_{s}\) and incident wavelength \(\lambda_{i}\), with \(\Delta\lambda=\lambda_{s}-\lambda_{i}\). Equation (18), for probing a beam linearly polarised at \(\varphi=0\), becomes [19]:  
\[P_{s}(\mathbf{R},\lambda_{s})d\Omega d\lambda_{s}\ =\ \text{{corr}}\frac{P_{i}r_{0}^{2}N_{e}L}{ \left(2\right)\left(1-\cos\theta\right)^{1/2}a}\frac{c}{\pi^{1/2}}\ e^{-\frac{ \Delta\lambda^{2}_{s}}{\lambda^{2}_{s}}\frac{c^{2}}{a^{2}}\frac{1}{2\left(1- \cos\theta\right)}}d\Omega d\lambda_{s} \tag{19}\]  
\[ \begin{split}\mathbf{M}_{\text{TS}}\ =\ C_{0}\left[\begin{array}{cccc}m_{00}&m_{01}&0&0\\ m_{10}&m_{11}&0&0\\ 0&0&m_{22}&0\\ 0&0&0&m_{33}\end{array}\right]\\ m_{00}\ =\ 1+u^{2}-2G(\mu)\ \big{(}u^{2}+4u-3\big{)}+\frac{16}{\mu^{2}}(1-u )^{2}\\ m_{01}\ =\ m_{10}\ =\ 1-u^{2}\\ m_{11}\ =\ 1+u^{2}+2G(\mu)\big{(}u^{2}-4u+1\big{)}+\frac{12}{\mu^{2}}(1-u )^{2}\\ m_{22}\ =\ 2u-4G(\mu)\big{(}u^{2}-u+1\big{)}-\frac{12}{\mu^{2}}(1-u)^{2}\\ m_{33}\ =\ 2u-4G(\mu)u(2u-1)-\frac{8}{\mu^{2}}(1-u)^{2}\\ C_{0}\ =\ \frac{\tau_{0}^{2}E_{0}^{2}N}{2r^{2}}\end{split}  \tag{23}\]  
where \(u\ =\ \cos\theta\), \(\mu\ =\ \frac{1}{r}\ =\ m_{e}c^{2}/T_{e}\) and \(G(\mu)\ =\ K_{1}(\mu)/\left(\mu K_{2}\ (\mu)\right)\), where \(K_{1}\) and \(K_{2}\) are the modified Bessel functions of the second kind. The definition of the scattered Stokes vector needs the accumulation of three other variables from the first step: the Stokes vector components (\(S_{in,0}\) is the total energy, yet calculated from the ray density). Please note that this is the frequency integrated Mueller matrix, which does not take into account the different polarisations at different wavelengths. This problem could be solved just using the frequency-resolved Mueller matrix [33]. However, theoretical studies showed that, for wavelength relevant for nuclear fusion, the polarisation difference is very small and completely negligible for the measurement purposes.