## 3 Results and Discussion  
### Data Analysis  
Using Fresnel equations, the THz-TDS approach allows us to find the complex index of refraction \(\hat{n}(\omega)=n(\omega)+i\kappa(\omega)\), or the dielectric function \(\hat{\varepsilon}(\omega)=\varepsilon_{1}(\omega)+i\varepsilon_{2}(\omega)\), of a tested material. The \(\hat{\varepsilon}(\omega)\)  
Figure 5: (a) Time-resolved transient signals for the tested thin and thick polymers and 1-wt% graphene nanocomposite samples and the empty spectrometer (nitrogen). (b) The corresponding power spectra showing absorption of the samples. Note: The thin samples consist of the PET-DLA 4060 copolymer, while the thick ones consist of PET-DLA 6040.  
can be represented as the square of \(\hat{n}(\omega)\), i.e., \(\hat{\varepsilon}(\omega)=\left[\hat{n}(\omega)\right]^{2}\). When the sample is placed between the THz emitter and detector, the recorded electric field signal \(E_{\text{sam}}(t)\) is the THz transient that is transmitted through sample. The reference signal, \(E_{\text{ref}}(t)\), is registered when the same measurement is taken, but without the sample. By taking FFT of time-domain transients, we can obtain the frequency-domain spectra of both the sample and reference. When these two spectra are compared, the \(\hat{\varepsilon}(\omega)\) and \(\hat{n}(\omega)\) dependencies, as well as the absorption coefficient \(\alpha(\omega)\), can be calculated.  
In agreement with our measurements, we can treat our samples as optically thick materials. In this case, internal reflections do not overlap with the main THz pulse, so, for the sake of simplicity, the data before and after the main pulse can be truncated and only the main THz pulse, analyzed [31]. Consequently, the transmission amplitude ratio \(A(\omega)\) and phase difference \(\varphi(\omega)\) of sample and reference fields can be calculated as [32],  
\[\frac{E_{\text{sam}}(\omega)}{E_{\text{ref}}(\omega)}=\frac{4n(\omega)}{ \left[n(\omega)+1\right]^{2}}e^{-\frac{ad}{2}+i\omega\frac{\left[n(\omega)-1 \right]d}{\varepsilon}}=A(\omega)e^{i\varphi(\omega)}, \tag{1}\]  
where \(c\) is the speed of light, \(d\) represents the sample thickness, and \(\omega\) is the angular frequency. From Equation (1) we can derive the absorption coefficient \(\alpha(\omega)\) and the real component \(n(\omega)\) of \(\hat{n}(\omega)\) :  
\[\alpha(\omega)=-\frac{2}{d}ln\Bigg{\{}A(\omega)\frac{\left[n(\omega)+1\right]^{ 2}}{4n(\omega)}\Bigg{\}}, \tag{2}\]  
\[n(\omega)=1+\frac{c}{\omega d}\varphi(\omega). \tag{3}\]  
In addition, the extinction coefficient \(\kappa(\xi)\) is related to the absorption coefficient as,  
\[\kappa(\omega)=\frac{c\alpha(\omega)}{2\omega}. \tag{4}\]  
After finding the components of \(\hat{n}(\omega)\) and \(\hat{\varepsilon}(\omega)\), the complex conductivity \(\hat{\sigma}(\omega)=\sigma_{1}(\omega)+i\sigma_{2}(\omega)\) of the sample can be subsequently calculated using Equations (5)-(8) [33]:  
\[\varepsilon_{1}(\omega)=n(\omega)^{2}-\kappa(\omega)^{2}, \tag{5}\]  
\[\varepsilon_{2}(\omega)=2n(\omega)\kappa(\omega), \tag{6}\]  
\[\sigma_{1}(\omega)=2\varepsilon_{0}\omega n(\omega)\kappa(\omega), \tag{7}\]  
\[\sigma_{2}(\omega)=\Big{[}\varepsilon_{\text{pl}}(\omega)-n(\omega)^{2}+ \kappa(\omega)^{2}\Big{]}\varepsilon_{0}\omega, \tag{8}\]  
where \(\varepsilon_{0}\) is the permittivity of the free space, and \(\varepsilon_{\text{pl}}(\omega)\) is the dielectric constant of a neat polymer matrix. We assume that imaginary part \(\varepsilon_{\text{p2}}\) is negligibly small since the neat polymer is an insulating material. \(\varepsilon_{\text{pl}}(\omega)\) can be extracted from THz-TDS measurements of a neat polymer in dry nitrogen using the same routine as presented above for the graphene-polymer nanocomposite (using Equations (1)-(5)).