## 2 Terahertz Time-Domain Spectroscopy of High-Permittivity Substrates
### Methods  
First, the samples were prepared for the measurement by etching the 35 um electrodeposited copper film from one side of the substrates. This was done by depositing a 15 um thick resin FP415 by ElgaEurope using a low temperature (120 \({}^{\circ}\)C) lamination technique (Bungard RLM 419P) as a protection layer. Then, the exposed copper side was etched by immersing the samples in a copper wet etching solution (100 mL H\({}_{2}\)O, 60 ml HCL, 20 mL H\({}_{2}\)O\({}_{2}\)) at 25 \({}^{\circ}\)C for 4 min. A dry resin stripper was then applied for 3 min to remove the resist and finally the sample was rinsed in isopropanol and deionized water. The processed Rogers substrates with a bare top side and a backmetallized side were investigated in reflection mode using TeraFlash Pro [28], a THz-TDS instrument which allows the broadband characterization of the samples in the sub-TTE range. The measurement procedure has been recently presented in detail [17] and its main aspects are briefly reported next for the readers' convenience. The schematic of the employed standard reflection-mode configuration of the THz-TDS setup is shown in Figure 1a. The laser source of the instrument is the FemtoFErb THz FD6.5 laser, which generates 50 fs pulses at 1560 um with a repetition rate of 100 MHz [28]; optical fibers bring these laser pulses to the emitter, which radiates an electromagnetic THz signal through a photoconductive antenna. The terahertz wave impinges on the sample at a fixed oblique angle of \(\theta_{\mathrm{i}}=8^{\circ}\). A photo of \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Name** & **Composition** & **Thickness (mm)** & \(\varepsilon_{\mathrm{r}}^{\prime}\) & **tan\(\delta\)** \\ \hline RO3010 & Ceramic-filled PTFE composites & \(1.270\pm 0.051\) & \(10.2\pm 0.25\) & 0.0022 \\ \hline RT/duroid 6010.2 LM & Ceramic-filled PTFE composites & \(1.905\pm 0.102\) & \(10.2\pm 0.25\) & 0.0023 \\ \hline TMM10i &
\begin{tabular}{c} Ceramic, hydrocarbon, thermoset \\ polymer composites \\ \end{tabular} & \(1.270\pm 0.038\) & \(9.8\pm 0.245\) & 0.002 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Properties of the investigated high-permittivity substrates. Relative dielectric constant and loss tangent values were provided by Rogers Corporation at 10 GHz. the THz wave propagation module of the setup with a sample placed for measurement is shown in Figure 1b. In all cases, we averaged 100 pulses for each time-domain pulse trace \(s(t)\) over a 200 ps time window (see Figure 2a) to obtain a spectral resolution of 10 GHz. We then analyzed the frequency-domain reflection spectra over the range 0.3-1 THz (see Figure 2b), which ensures more than 80 dB of dynamic range with respect to the noise floor. We started from a preliminary analysis of the signal in the time domain. Here, we estimated the thickness of the substrates from the arrival times of the zeroth-order echo (reflection from the top side of the substrate) with the formula provided in ([17], [Equation (1)]). Results were consistent with the nominal thickness values and their tolerance, as provided by the manufacturer and summarized in Table 1.  
However, we were interested in an accurate material characterization of the substrates, which took into account the losses and frequency dispersion. To this aim, in analogy with the procedure carried out in [17], we developed an equivalent circuit model (see Figure 3) to derive a theoretical amplitude reflection coefficient \(R_{\text{th}}\) (a time dependence \(\exp(j\omega t)\) of the fields was assumed) at the air-substrate interface, viz., \(z=0\). Thus, by definition (see, e.g., [29]) we had \(R_{\text{th}}=|Y_{\text{c}}-Y_{\text{in}}|/|Y_{\text{c}}-Y_{\text{in}}|\), where \(Y_{\text{c}}\) and \(Y_{\text{in}}\) are the characteristic admittance at \(z=0^{+}\) and the input admittance looking downwards at \(z=0^{-}\), respectively.