## Appendix A  
On the metal-air interface, the tangential component \(\mathbf{E}_{\tau}\) of the electric field vector is in the following relation to the magnetic field vector \(\mathbf{B}\):  
\[\mathbf{E}_{\tau}=\zeta\cdot[\mathbf{n}\cdot\mathbf{B}], \tag{10}\]  
where the unit vector \(\mathbf{n}\) is directed along the outer normal to the metal surface, and the parameter  
\[\xi=\frac{\zeta}{i\varepsilon_{\mathrm{a}}k_{0}} \tag{11}\]  
is referred to as the surface impedance (\(\zeta=k_{0}\sqrt{\widetilde{n}_{\mathrm{s}}{}^{2}-\varepsilon_{\mathrm{a}}}\) is the rate of decay of the SPP field in air, while \(\varepsilon_{\mathrm{a}}\) is the dielectric constant of air). In the presence on the metal surface of a dielectric layer with \(\varepsilon_{\mathrm{d}}\) and thickness \(d<<1/k_{0}\sqrt{\varepsilon_{\mathrm{m}}}\), the impedance of the \"metal-dielectric layer-air\" structure can be calculated by the following approximate formula [64]:  
\[\xi=\frac{1}{\sqrt{\varepsilon_{\mathrm{m}}}}-i\cdot\left(\frac{\varepsilon_{ \mathrm{d}}-\varepsilon_{\mathrm{a}}}{\varepsilon_{\mathrm{d}}}\right)\cdot k _{0}\cdot d \tag{12}\]  
As can be seen from Formula (12), for sufficiently thick layers (\(k_{0}d>1/\sqrt{\varepsilon_{\mathrm{m}}}\)), the correction to the impedance due to the dielectric layer is significant, and Formula (12) becomes inapplicable. When \(d=0\), dispersion Equation (5) for a three-layer structure has the root \(\widetilde{n}_{\mathrm{s}}^{0}=\sqrt{\varepsilon_{\mathrm{a}}\varepsilon_{ \mathrm{m}}/(\varepsilon_{\mathrm{a}}+\varepsilon_{\mathrm{m}})}\). Substituting \(\widetilde{n}_{\mathrm{s}}{}^{2}=\left(\widetilde{n}_{\mathrm{s}}^{0}\right) ^{2}+d\cdot x\) into Equation (5) and expanding both sides of this equation into the Taylor series to terms linear in \(d\), we obtain a linear (with respect to \(x\)) equation, and if \(\varepsilon_{\mathrm{m}}>>\varepsilon_{\mathrm{d}}\), \(\varepsilon_{\mathrm{a}}\), we come up to Formula (12). This expansion is valid if the correction \(d\cdot x\) is small compared with \(\left(\widetilde{n}_{\mathrm{s}}^{0}\right)^{2}-\varepsilon_{\mathrm{a}}\), \(\left(\widetilde{n}_{\mathrm{s}}^{0}\right)^{2}-\varepsilon_{\mathrm{d}}\), \(\left(\widetilde{n}_{\mathrm{s}}^{0}\right)^{2}-\varepsilon_{\mathrm{m}}\), the first condition  
\[d\cdot x<<\left(\widetilde{n}_{\mathrm{s}}^{0}\right)^{2}-\varepsilon_{\mathrm{ a}}=-\frac{\varepsilon_{\mathrm{a}}{}^{2}}{\varepsilon_{\mathrm{a}}+ \varepsilon_{\mathrm{m}}} \tag{13}\]  
being the most difficult to satisfy. As  
\[\xi=\frac{\sqrt{\widetilde{n}_{\mathrm{s}}{}^{2}-\varepsilon_{\mathrm{a}}}}{ i\cdot\varepsilon_{\mathrm{a}}}=\sqrt{-\frac{\varepsilon_{\mathrm{a}}{}^{2} }{\varepsilon_{\mathrm{a}}+\varepsilon_{\mathrm{m}}}+d\cdot x}/\left(i\cdot \varepsilon_{\mathrm{a}}\right) \tag{14}\]  
Condition (13) means that the second term in Formula (12) is small compared with the first one, i.e., \(k_{0}d<<1/\sqrt{\varepsilon_{\mathrm{m}}}\). If, on the contrary, \(k_{0}d>1/\sqrt{\varepsilon_{\mathrm{m}}}\), then we must take into account the fact that the correction to \(\widetilde{n}_{\mathrm{s}}{}^{2}\), which depends on \(k_{0}\cdot d\), significantly changes the terms that contain \(\sqrt{\widetilde{n}_{\mathrm{s}}{}^{2}-\varepsilon_{\mathrm{a}}}\). On the other hand, the terms in Equation (5) that contain \(\sqrt{\widetilde{n}_{\mathrm{s}}{}^{2}-\varepsilon_{\mathrm{d}}}\) and \(\sqrt{\widetilde{n}_{\mathrm{s}}{}^{2}-\varepsilon_{\mathrm{m}}}\) are much less sensitive to the correction \(k_{0}\cdot d\). Therefore, the zero approximation \(\widetilde{n}_{\mathrm{s}}{}^{2}=\widetilde{n}_{0}{}^{2}\) can be substituted into these terms. As a result, we obtain a simple equation for \(\xi\). Replacing the hyperbolic tangent on the left side of Equation (5) with its argument (since the argument is very small), we obtain an approximate expression for the surface impedance of the three-layer medium (see Formula (6)).