## 2 Experimental Setup
The system is composed of a fiber-coupled emitter (FCE), two off-axis parabolic mirrors (OAP), four apertures (AP), two wire grid polarizers (WGP) mounted on rotary stages, a sample (S) illuminated at \(\varphi=70^{\circ}\) angle of incidence and a fiber-coupled detector (FCD). The beam path is housed (H) for dry air purging; (**b**) Photograph of the designed THz time-domain ellipsometer. \(\Psi\) and \(\Delta\), which are the amplitude ratio and the phase difference of the reflected p- and s-polarized THz-radiation. The acquired temporal signal of the detector is transferred into the frequency domain using a Fourier transform. The resulting complex frequency spectrum allows for calculating the frequency dependent amplitude and phase of the signal. The complex parameter \(\rho\) is the ratio of the detector signals for azimuthal analyzer settings \(\theta_{A}=0^{\circ}\) (p-polarization) and \(\theta_{A}=90^{\circ}\) (s-polarization):  
\[\rho(\Psi,\Delta,\theta_{P},NON)=\frac{I_{exp}(\theta_{A}=0^{\circ})}{I_{exp} (\theta_{A}=90^{\circ})} \tag{1}\]  
where \(I_{exp}\) refers to the measured complex spectra and \(\rho\) to the theoretical spectrum as a function of the ellipsometric parameters \(\Psi\) and \(\Delta\), the polarizer azimuthal angle \(\theta_{P}\) (usually \(45^{\circ}\)) and all non-idealities NON of the components used. The latter are the extinction ratio and phase retardance of polarizer and analyzer, the polarization dependent sensitivity of the detector and the source polarization of the emitter. For ideal components and \(\theta_{P}=45^{\circ}\), the ellipsometric parameters can be calculated as  
\[\Psi=\tan^{-1}\lvert\rho\rvert,\ \Delta=\arg(\rho)\ \, \tag{2}\]  
where \(\arg(\rho)\) denotes the argument or phase angle of \(\rho\). In the THz spectral range, the aforementioned non-idealities cannot be neglected. Hence, the frequency dependent values of these non-idealities must be calibrated in advance in order to measure the ellipsometric parameters accurately. In addition, the errors of the azimuthal angles \(\theta_{P}\) and \(\theta_{A}\) of the motorized polarizer and analyzer must be determined accurately. The calibration procedure was carried out according to [37; 38] using a gold coated metal substrate and a fused silica substrate for maximum accuracy. The fundamental ellipsometric equation [39] is:  
\[\rho=\tan\Psi e^{i\Delta}=\frac{r_{p}}{r_{s}}\ \, \tag{3}\]  
where \(r_{p}\) and \(r_{s}\) denote the Fresnel coefficients in reflection for \(p\)- and \(s\)-polarization, respectively. It connects the measured ellipsometric parameters \(\Psi\) and \(\Delta\) with the sample properties. The Fresnel coefficients depend on the angle of incidence, the complex refractive indices and the thicknesses of each layer and the substrate. By solving Equation (3), the dielectric function of a bare substrate can be calculated directly from the measured ellipsometric parameters \(\Psi_{\rm exp}\) and \(\Delta_{\rm exp}\)[39]:  
\[\varepsilon_{1}+i\varepsilon_{2}=\sin^{2}\varphi\Bigg{\{}1+\left[\frac{1- \rho}{1+\rho}\right]^{2}\tan^{2}\varphi\Bigg{\}}, \tag{4}\]  
where \(\varphi\) is the angle of incidence. Therefore, only the exact knowledge of the angle of incidence is needed to calculate the optical parameters of a substrate from the ellipsometric parameters. For all the other cases, like thin films on a substrate (see Figure 2), a surface roughness on a bare substrate or a transparent substrate with thickness T, a more complicated model of the sample and data analysis is required. For a three-layer model on a transparent substrate, as shown in Figure 2, the thickness of each layer and of the substrate, and the angle of incidence as well as the frequency-dependent complex refractive indices of each layer and the substrate must be known to be able to obtain the Fresnel coefficients in reflection for p- and s-polarization. Using Equation (3), the model ellipsometric parameters \(\Psi_{\rm mod}\) and \(\Delta_{\rm mod}\) can then be derived.