## 3 Results
It can be seen that the results of the calculation at \(d\leq 1700\) nm almost coincide with the curves obtained by the approximate Formula (7), which indicates the possibility of its effective application for evaluation calculations of the complex refractive index of THz SPPs in a three-layer structure in the presence of a dielectric coating layer of subwavelength thickness on the metal. It was already noted above that the presence of a dielectric layer on the metal surface leads to an increase in the fraction of the SPP field energy transferred into the metal. In this case, the depth \(\delta_{\rm m}\) of the penetration of the SPP field into the metal is practically independent of the coating thickness, as evidenced by the graphs in Figure 6b, as obtained from the solution of Equation (5) for the \"gold-ZnS layer of thickness \(d\)-air\" structure. It can be seen that with the gold permeability found via the approximation method from the experimental results, the \(\delta_{\rm m}\) value is approximately six times greater than that for the permeability calculated by the Drude model. Therefore, for the deposited gold layer to be non-transparent for the THz SPP field (which excludes the effect of the substrate on the SPP characteristics), its thickness must be 200-300 nm at least. Figure 6: (**a**) Ratio of the radiative and Joule losses of the SPPs, as calculated using Formula (8) at \(\lambda_{0}\) = 141 \(\upmu\)m vs. thickness \(d\) of the ZnS layer in the “deposited gold–ZnS layer–air” structure; and (**b**) penetration depth \(\delta_{\rm m}\) (in terms of intensity) of the SPP field into metal for the “gold–ZnS layer with thickness \(d\)–air” structure at \(\lambda_{0}\) = 141 \(\upmu\)m and gold permeability \(\varepsilon_{\rm m}\) calculated according to Drude model (black line) and \(\varepsilon_{\rm m}\) = \(-7000\) + \(i\):3000, as determined via the approximation method from characteristics of the SPPs (blue line). Dotted lines: corresponding values of the skin layer thickness. From the plot of \(\kappa_{\rm s}(d)\) in Figure 6a, it can be seen that at small thicknesses \(d<300\) nm, the loss of the SPPs in the experiment is much greater than the calculations predict. This is explained by the presence of the radiative losses of the SPPs, which are not taken into account either by the approximate formulas or by the dispersion equation. Starting from \(d\approx 300\) nm (where the minimum of the integral losses of the SPPs is reached and their radiative losses become relatively small compared with the Joule losses), the measured values of the absorption coefficient of the SPPs approach the calculated dependence. Based on this interpretation of the existence of the minimum of the losses \(\kappa_{\rm s}(d)\) and the assumption that other mechanisms of losses contribute negligibly to the total SPP decay, the SPP radiative losses can be estimated by the following formula:  
\[\kappa_{\rm rad}=\kappa_{\rm s}-\kappa_{\rm Joule} \tag{8}\]  
where \(\kappa_{\rm s}\) is the total SPP losses measured in the experiment; and \(\kappa_{\rm Joule}\) is the Joule SPP losses determined from the solution of Equation (5). Figure 6a shows the ratio of the radiative losses estimated by Formula (8) to the Joule losses at different thicknesses \(d\) of the ZnS layer, as calculated from the solution of Equation (5) with the substitution of \(\varepsilon_{\rm m}=-7000+i\):3000 (found via the approximation method) into it. In this case, in the absence of a dielectric layer on the metal surface, the radiative losses exceed the Joule losses by about four-fold. However, as the \(d\) grows, this ratio decreases monotonically to zero at \(d\geq 960\) nm, when the SPP decay is determined mainly by the losses in the metal.