## 3 Results
Determination of the Effective Dielectric Constant of the Metal Surface from SPP Characteristics on One Sample  
In Section 3.1, we described an estimation of the dielectric permittivity of metal \(\varepsilon_{\rm m}\) via the application of the approximation method to the measurements of the characteristics of the THz SPPs on a set of samples with different thicknesses of dielectric layers on the metal surface. Despite its rather high reliability, this method is very laborious for the rapid assessment of the dielectric constant of metal (metallized) optical surfaces. Therefore, it is desirable (in view of practical applications) to have a method for the determination of \(\varepsilon_{\rm m}\) from the complex refractive index \(\tilde{n}_{\rm s}=n_{\rm s}+i\kappa_{\rm s}\) of the THz SPPs, as found from the measurements of its real and imaginary parts on only one sample. From Equation (5), we obtained an approximate formula for calculating the \(\varepsilon_{\rm m}\) from the measured characteristics of the SPPs (see Appendix A):  
\[\varepsilon_{\rm m}(\omega)=1-\frac{{\omega_{p}}^{2}}{\omega^{2}+{\omega_{\rm \tau}}^{2}}+i\,\frac{{\omega_{\rm\tau}}\cdot{\omega_{p}}^{2}}{\omega\cdot({ \omega^{2}+{\omega_{\rm\tau}}^{2}})} \tag{4}\]0  
where the surface impedance \(\xi\) is found from Formula (6). As was established above, the radiative losses make a predominant contribution to the SPP decay on a metal without a dielectric coating layer, which leads to large errors in the determination of the \(\varepsilon_{\rm m}\). Therefore, it is necessary to find a layer thickness \(d\) range in which the error in the determination of the \(\varepsilon_{\rm m}\) is not so large. For this purpose, the permittivity \(\varepsilon_{\rm m}\) of the deposited gold was calculated using Formula (9) with the substitution of the experimentally determined values of \(\widetilde{n}_{\rm s}\) for samples with different thicknesses \(d\) of the ZnS layer. The found values of the real Re(\(\varepsilon_{\rm m}\)) and imaginary Im(\(\varepsilon_{\rm m}\)) parts of the gold permittivity are shown by the blue circles in \[\varepsilon_{\rm m}(\omega)=1-\frac{{\omega_{p}}^{2}}{\omega^{2}+{\omega_{\rm \tau}}^{2}}+i\,\frac{{\omega_{\rm\tau}}\cdot{\omega_{p}}^{2}}{\omega\cdot({ \omega^{2}+{\omega_{\rm\tau}}^{2}})} \tag{4}\]1 The errors on the graphs correspond to the maximum permittivity deviations calculated in the ranges of the experimental values \(n_{\rm s}\pm\Delta n_{\rm s}\) and \(\kappa_{\rm s}\pm\Delta\kappa_{\rm s}\), where \(\Delta n_{\rm s}\) and \(\Delta\kappa_{\rm s}\) are the corresponding measurement errors. It can be seen that the values of the Re(\(\varepsilon_{\rm m}\)) and Im(\(\varepsilon_{\rm m}\)) at small thicknesses of the ZnS layer (\(d<300\) nm) differ greatly from the approximation values, while at \(d\geq 300\) nm, they approach the approximation values asymptotically (except for the Im(\(\varepsilon_{\rm m}\)) at \(d=3000\) nm, where already Formula (9) gives insufficiently reliable results compared with the exact solution of the dispersion Equation (5) and the increased inaccuracy of the measurements of the imaginary part of the refractive index of the SPPs due to the inhomogeneity of the ZnS film over the thickness). This character of the dependence \(\varepsilon_{\text{m}}(d)\) can be explained by the fact that Formula (9) does not take into account the radiative losses of the SPPs. At larger thicknesses of the dielectric layer, when the radiative losses become much less than the thermal losses, Formula (9) describes the experiment much better. Note that according to the performed calculations, the real part \(\text{Re}(\varepsilon_{\text{m}})\) of the permittivity of gold without a ZnS coating layer (\(d=0\)) turned out to be equal to \(-800\) (see Figure 7).