## 3 Polarization Tensor in the Area of Evanescent Waves
In this region, the first contributions to (8) take the form [52]  
\[\Pi_{00}^{(0)}(\omega,k;\Delta) =-\frac{2\alpha k^{2}}{cp^{2}(\omega,k)}\,\Phi(\omega,k,\Delta),\] \[\Pi^{(0)}(\omega,k;\Delta) =\frac{2\alpha k^{2}}{c}\Phi(\omega,k,\Delta), \tag{11}\]  
where  
\[p^{2}(\omega,k)=\frac{\omega^{2}}{c^{2}}-\frac{v_{\rm F}^{2}}{c^{2}}k^{2} \geqslant 0 \tag{12}\]  
and \(\alpha=e^{2}/(\hbar c)\) is the fine structure constant. The function \(\Phi\) is defined as  
\[\Phi(\omega,k,\Delta)=\Delta-\hbar cp(\omega,k)\left[1+\frac{\Delta^{2}}{\hbar^ {2}c^{2}p^{2}(\omega,k)}\right]\left[\text{arctanh}\frac{\Delta}{\hbar cp( \omega,k)}+i\frac{\pi}{2}\right] \tag{13}\]  
for \(\hbar cp(\omega,k)\geqslant\Delta\) and as  
\[\Phi(\omega,k,\Delta)=\Delta-\hbar cp(\omega,k)\left[1+\frac{\Delta^{2}}{\hbar ^{2}c^{2}p^{2}(\omega,k)}\right]\text{arctanh}\frac{\hbar cp(\omega,k)}{\Delta} \tag{14}\]  
for \(\hbar cp(\omega,k)<\Delta\). The second contributions to (8) in the plasmonic region are more complicated. It is convenient to define their real and imaginary parts separately. We start from defining the real parts of \(\Pi_{00}^{(1)}\) and \(\Pi^{(1)}\) which, in turn, have different forms under the conditions \(\hbar cp(\omega,k)\geqslant\Delta\) and \(\hbar cp(\omega,k)<\Delta\). Thus, if the condition \(\hbar cp(\omega,k)\geqslant\Delta\) is satisfied, one obtains from [52] after identical transformations,  
\[\text{Re}\,\Pi_{00}^{(1)}(\omega,k;\Delta,T_{g})=\frac{8\alpha\hbar c^{2}}{v_{ \rm F}^{2}}(I_{1}+I_{2}+I_{3}), \tag{15}\]where the following notations are introduced:  
\[\Pi(\omega,k;\Delta,T_{g})=\Pi^{(0)}(\omega,k;\Delta)+\Pi^{(1)}(\omega,k;\Delta,T_{g}). \tag{8}\]0 \[\Pi(\omega,k;\Delta,T_{g})=\Pi^{(0)}(\omega,k;\Delta)+\Pi^{(1)}(\omega,k;\Delta,T_{g}). \tag{8}\]1 \[\Pi(\omega,k;\Delta,T_{g})=\Pi^{(0)}(\omega,k;\Delta)+\Pi^{(1)}(\omega,k;\Delta,T_{g}). \tag{8}\]2  
Here,  
\[\Pi(\omega,k;\Delta,T_{g})=\Pi^{(0)}(\omega,k;\Delta)+\Pi^{(1)}(\omega,k;\Delta,T_{g}). \tag{8}\]3 \[\Pi(\omega,k;\Delta,T_{g})=\Pi^{(0)}(\omega,k;\Delta)+\Pi^{(1)}(\omega,k;\Delta,T_{g}). \tag{8}\]4  
It can be seen that all the integrals \(I_{j}\) are the functions of \(\omega\), \(k\), \(\Delta\), and \(T_{g}\). Under the same condition \(\hbar cp(\omega,k)\geqslant\Delta\), we obtain from [52]  
\[\Pi(\omega,k;\Delta,T_{g})=\Pi^{(0)}(\omega,k;\Delta)+\Pi^{(1)}(\omega,k;\Delta,T_{g}). \tag{8}\]5  
where the quantities \(J_{j}\) are given by  
\[\Pi(\omega,k;\Delta,T_{g})=\Pi^{(0)}(\omega,k;\Delta)+\Pi^{(1)}(\omega,k;\Delta,T_{g}). \tag{8}\]6 \[\Pi(\omega,k;\Delta,T_{g})=\Pi^{(0)}(\omega,k;\Delta)+\Pi^{(1)}(\omega,k;\Delta,T_{g}). \tag{8}\]7 \[\Pi(\omega,k;\Delta,T_{g})=\Pi^{(0)}(\omega,k;\Delta)+\Pi^{(1)}(\omega,k;\Delta,T_{g}). \tag{8}\]8  
Here, the function \(B_{2}(x)\) is defined as  
\[\Pi(\omega,k;\Delta,T_{g})=\Pi^{(0)}(\omega,k;\Delta)+\Pi^{(1)}(\omega,k;\Delta,T_{g}). \tag{8}\]9  
If the opposite condition, \(\hbar cp(\omega,k)<\Delta\), is satisfied, the real parts of \(\Pi^{(1)}_{00}\) and \(\Pi^{(1)}\) take the following form [52]\[ \begin{split}\text{Re}\,\Pi^{(1)}_{00}(\omega,k;\Delta,T_{g})=\frac{8 \alpha\hbar c^{2}\bar{p}(\omega,k)}{v_{F}^{2}}\int\limits_{\frac{\Delta}{2 mc}}^{ \infty}\frac{du}{e^{\bar{p}(T_{g})u}+1}\bigg{[}2-\frac{B_{1}(2 cu+\omega)-B_{1} (2 cu-\omega)}{cp(\omega,k)}\bigg{]},\\ \text{Re}\,\Pi^{(1)}(\omega,k;\Delta,T_{g})=\frac{8\alpha\hbar \omega^{2}}{v_{F}^{2}}\int\limits_{\frac{\Delta}{2 mc}}^{\infty}\frac{du}{e^{ \bar{p}(T_{g})u}+1}\bigg{\{}2-\frac{cp(\omega,k)[B_{2}(2 cu+\omega)-B_{2}(2 cu- \omega)]}{\omega^{2}}\bigg{\}}.\end{split}  \tag{21}\]  
This concludes the consideration of the real parts of \(\Pi^{(1)}_{00}\) and \(\Pi^{(1)}\) in the plasmonic region (9). The imaginary parts of \(\Pi^{(1)}_{00}\) and \(\Pi^{(1)}\) are given by the unified expressions  
\[