## 3 Results
Expanding both parts of Equation (5) into a Taylor series with respect to a small correction to the refractive index \(\widetilde{n}_{\mathrm{s}}\) because of the dielectric coating on the metal, we obtain an approximate formula for calculating the surface impedance of the three-layer structure (see Appendix A):  
\[\xi\approx\frac{i\times\left(\varepsilon_{\mathrm{d}}\cdot\sqrt{-\frac{ \varepsilon_{\mathrm{m}}{}^{2}}{\varepsilon_{\mathrm{a}}+\varepsilon_{\mathrm{ m}}}}\times(\varepsilon_{\mathrm{a}}+\varepsilon_{\mathrm{m}})+k_{0}d \cdot\varepsilon_{\mathrm{m}}\times[\varepsilon_{\mathrm{a}}(\varepsilon_{ \mathrm{m}}-\varepsilon_{\mathrm{d}})-\varepsilon_{\mathrm{m}}\varepsilon_{ \mathrm{d}}]\right)}{\varepsilon_{\mathrm{d}}\cdot(\varepsilon_{\mathrm{a}}+ \varepsilon_{\mathrm{m}})\times\left(k_{0}d\cdot\varepsilon_{\mathrm{d}}\sqrt{- \frac{\varepsilon_{\mathrm{m}}{}^{2}}{\varepsilon_{\mathrm{a}}+\varepsilon_{ \mathrm{m}}}}+\varepsilon_{\mathrm{m}}\right)} \tag{6}\]  
where  
\[\widetilde{n}_{\mathrm{s}}=\sqrt{\varepsilon_{\mathrm{a}}-(\varepsilon_{ \mathrm{d}}\xi)^{2}} \tag{7}\]  
The dependences of \(n_{\mathrm{s}}-1\) and \(\kappa_{\mathrm{s}}\) on the ZnS layer thickness \(d\) were found using Formulas (6) and (7) for the gold permittivity \(\varepsilon_{\mathrm{m}}\) calculated according to the Drude model (blue lines in Figure 5). In the figure, the red lines show the results of the numerical solution of Equation (5). It can be seen that at \(d\leq 1500\) nm, the approximate and exact solutions practically coincide, while at \(1500\) nm \(<d\leq 3000\) nm, they differ by no more than \(8\%\). Figure 5: Real part (without unity) \(n_{\mathrm{s}}\) (a) and imaginary part \(\kappa_{\mathrm{s}}\) (b) of the SPP refractive index in the structure “gold–ZnS layer of thickness \(d\)–air” vs. \(d\) calculated via the approximate Formulas (6) and (7) (blue lines) and by the numerical solution of dispersion Equation (5) (black lines). The calculations used gold permittivity \(\varepsilon_{\mathrm{m}}=-104,700+i\cdot 317,180\), as calculated by the Drude model at \(\lambda_{0}=141\) μm. Then, the experimental dependences (\(n_{\rm s}-1\))(\(d\)) and \(\kappa_{\rm s}(d)\) were approximated by means of a MATLAB program, using the least squares method to find the minimum deviations of the experimental values from the corresponding functions \(\mathrm{Re}(\vec{n}_{\rm s}-1)\) and \(\mathrm{Im}(\vec{n}_{\rm s})\) (\(\vec{n}_{\rm s}\) was found by Formula (7)). The real and imaginary parts of the gold permittivity \(\varepsilon_{\rm m}\) were varied as parameters. The search for the solution for both functions was performed in the range of ZnS layer thicknesses 300 nm \(\leq d\leq\) 1670 nm. The domain of the functions did not include small thicknesses (\(d<\) 300 nm) when the SPP decay is governed by the radiative losses, which are not taken into account by Equation (5). At \(d=\) 3000 nm, the approximate formula gives noticeable deviations from the exact solution. As a result of the approximation, the permeability of the deposited gold was found: the real part was \(\mathrm{Re}(\varepsilon_{\rm m})\approx-7000\) \(\pm\) 3000, while the imaginary was \(\mathrm{Im}(\varepsilon_{\rm m})\approx 3000\)\(\pm\) 1300. The approximation dependences (\(n_{\rm s}-1\))(\(d\)) and \(\kappa_{\rm s}(d)\) are shown by the gray lines in Figure 4. The large errors (of up to 40%) in the determination of the \(\varepsilon_{\rm m}\) by this method are due to the closeness of the refractive index \(n_{\rm s}\) of the THz SPPs to the refractive index of air, which necessitates a higher accuracy in finding the \(n_{\rm s}\) value. In Figure 4, the green squares show the results for the numerical solution of the dispersion Equation (5) via the downhill method [55] at \(\varepsilon_{\rm m}=-7000+i\):3000.