## 2 Terahertz Time-Domain Spectroscopy of High-Permittivity Substrates
### Methods
According to Figure 2b, \(Y_{\text{c}}=Y_{0}\), \(Y_{0}\) is the wave admittance in the air, and \(Y_{\text{in}}=-jY_{\text{c}}\cot(k_{zx}d)\), \(Y_{\text{c}}\) and \(k_{zx}\) are the wave admittance and the propagating wavenumber in the dielectric substrate of thickness \(d\). The amplitude reflection coefficient finally reads:  
\[R_{\text{th}}=\left|\frac{\cos\theta_{\text{i}}-j\sqrt{\varepsilon_{\text{r}}- \sin^{2}\theta_{\text{i}}}\cot\left(k_{0}d\sqrt{\varepsilon_{\text{r}}-\sin^{ 2}\theta_{\text{i}}}\right)}{\cos\theta_{\text{i}}+j\sqrt{\varepsilon_{\text{ r}}-\sin^{2}\theta_{\text{i}}}\cot\left(k_{0}d\sqrt{\varepsilon_{\text{r}}-\sin^{ 2}\theta_{\text{i}}}\right)}\right| \tag{1}\]  
where \(k_{0}\) is the vacuum wavenumber. \(R_{\text{th}}\) was then compared with that of the measured amplitude reflection coefficient \(R_{\text{meas}}\), which was obtained from the frequency-domain measurements as the ratio between the amplitude of the reflection spectrum \(|S(f)|\) from the substrate over the reference measurement in its absence. Figure 2b shows a couple of indicative power spectra \(|S(f)|^{2}\), reference and sample, measured for the substrate RO3010. For the calculation of \(R_{\text{th}}\), we considered that the substrate relative permittivity was described by a standard single-term Lorentz model [25]:  
\[\varepsilon_{\text{r}}(f)=\varepsilon_{\infty}+\frac{(\varepsilon_{\text{s} }-\varepsilon_{\infty})f_{0}^{2}}{f_{0}^{2}-f^{2}+j\frac{f}{2\pi\tau}} \tag{2}\]  
Figure 1: **(a)** Schematic of the employed THz-TDS setup, configured for sample characterization in reflection mode. The setup consists of an fs-pulse, near-infrared laser, photoconductive transmit (Tx) and receive (Rx) antennas, an optical delay line, a set of mirrors, and the sample under test placed on a metallic baseplate. The optical beam path is marked with red arrows and the THz wave path by red-shaded elements. **(b)** Photo showing the THz-wave propagation module of the TDS setup with a sample placed on the metallic baseplate. where \(\varepsilon_{s},\varepsilon_{\infty}\) are the relative permittivity values at DC and infinite frequency, respectively, and \(f_{0}\) and \(\tau\) are the Lorentz resonant frequency and decay time, respectively. These four parameters were introduced as free variables in a standard optimization process based on the Nelder-Mead simplex method implemented in MATLAB(r), which minimized the \(\ell^{2}\)-norm of the difference \(R_{\text{th}}-R_{\text{meas}}\) over the 0.3-1 THz spectral window of investigation. In other words, the four-dimensional (4D) optimization problem consisted in seeking the optimum values of \(\varepsilon_{s},\varepsilon_{\infty},f_{0},\tau\) such that:  
\[\min_{\{\varepsilon_{s},\varepsilon_{\infty},f_{0},\tau\}}\left(\sqrt{\int_{ f_{\min}}^{f_{\max}}|R_{\text{th}}-R_{\text{meas}}|^{2}\text{d}f}\right) \tag{3}\]  
where \(f_{\min}=0.3\) THz and \(f_{\max}=1\) THz. This minimization process provided a 4-tuple of optimum values to be applied into (2) to get the best fit of \(R_{\text{meas}}\). In Section 2.3, these values are provided for all Rogers substrates analyzed here, namely the RO3010, the 6010.2 LM and the TMM10i.