## 2 Terahertz Time-Domain Spectroscopy of High-Permittivity Substrates
### Results  
The methodology described in the previous subsection was applied to the dielectric characterization of the three investigated Rogers substrates. The obtained results are summarized in Figure 4, where the first two columns report the relative permittivity and loss tangent frequency dispersion, as per the optimized fit to the Lorentz model, and the third column shows a comparison between the measured and theoretical fit of the power reflectance coefficients \(|R_{\text{meas}}|^{2}\) and \(|R_{\text{th}}|^{2}\). The Lorentz parameters for the three materials are summarized in Table 2. We highlight here that the dispersion model in this study is phenomenological, aiming to accurately describe the dielectric frequency dispersion of the  
Figure 3: (**a**) One-dimensional section of the backmetallized substrate under test and (**b**) its transmission-line equivalent model. Figure 2: (**a**) Time-domain signal \(s(t)\) measured for the reference (reflection from the metallic baseplate) and in the presence of the RO3010 substrate. The pulse around 1200 ps (light blue curve) preceding the reference one stems from reflection from the top surface of the substrate. A zoomed inset is reported at the top-right corner to better show the shape of the pulses. (**b**) Fourier transformed power spectra of the time-domain signals. The three major dips in the reference spectrum are due to water absorption. three investigated materials with a minimum number of terms so as to facilitate its use in the design flow of THz components. In the case of RO3010 in Figure 4a, we also included the tabulated data for the measurements reported in [11], showing good overall agreement. We note that the simulated reflectance spectra based on the reference data do not accurately reproduce our measurements, as shown in the right panel of Figure 4a. A comparison of the complex relative permittivity data for RO/duroid 6010.2 LM is also in line with available literature information [10]. In the reflectance spectra one can notice an oscillating part with a diminishing amplitude towards the higher frequencies. This effect stems from the Fabry-Perot (FP) multiple reflections of the THz wave that enters and propagates in the substrate. The periodicity of these oscillations, also known as free spectral range (FSR) [30], can be estimated at a first approximation by the following formula:  
\[\text{FSR}=\frac{c_{0}}{2d\cos\theta_{\text{i}}\sqrt{\epsilon_{\text{r}}^{t}}} \tag{4}\]  
The position of the FP maxima/minima depends directly on the thickness and relative permittivity, whereas the modulation depth depends on the loss tangent of the substrate. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Name** & \(\epsilon_{\infty}\) & \(\epsilon_{s}\) & \(f_{0}\) (THz) & \(\tau\) (ps) \\ \hline RO3010 & 3.223 & 11.464 & 1.663 & 0.779 \\ \hline RT/duroid 6010.2 LM & 1.001 & 13.502 & 2.083 & 0.654 \\ \hline TMM10i & 7.923 & 10.611 & 2.008 & 0.424 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Lorentz model parameters calculated for the investigated Rogers high-permittivity substrates. Figure 4: Relative dielectric permittivity, loss tangent, and comparison between measured and simulated spectra based on the Lorentz-fitted permittivities of the three investigated substrates: (**a**) RO3010, (**b**) RT/duroid 6010.2 LM, and (**c**) TMM10i. The insets in the second column show photos of the characterized samples with an area of approximately \(5\times 5\) cm\({}^{2}\). In (**a**), results are compared to the data provided in [11]. Thus, the oscillating spectrum contains the necessary information for the dispersion model fitting. At higher frequencies the reflectance converges towards a constant value, which corresponds to the part of the THz wave that is reflected from the top surface of the substrate and depends also on its relative permittivity. It can be noticed that the measured high-frequency reflectance is a bit lower than the theoretical prediction.