## 1 Introduction
### Complex Dielectric Permittivity
\begin{split}\mathrm{Re}\left[\stackrel{{\rightarrow}}{{\mathrm{E} }}_{\mathrm{pol}}(\mathrm{t})\right]\cdot\mathrm{Im}\left[\stackrel{{ \rightarrow}}{{j}}^{(\Omega)}(\mathrm{t})\right]=\sigma\stackrel{{ \rightarrow}}{{\mathrm{\sum}}}_{\lambda^{\prime}=0}^{\infty}\stackrel{{ \rightarrow}}{{\mathrm{E}}}_{0}^{\rightarrow 2\lambda^{\prime}+2}\cos\Bigl{(}\Bigl{(}2\lambda^{ \prime}+1\Bigr{)}\omega t\Bigr{)}\sin(\omega\,t)+\\ +\omega\varepsilon_{0}\varepsilon_{\infty}\stackrel{{ \rightarrow}}{{\mathrm{\sum}}}_{\lambda^{\prime}=0}^{\infty}\stackrel{{ \rightarrow}}{{\mathrm{\sum}}}_{0}^{\rightarrow 2\lambda^{\prime}+2}\cos\Bigl{(}\Bigl{(}2\lambda^{\prime}+1\Bigr{)}\omega t \Bigr{)}\cos(\omega\,t)+\\ +\omega\stackrel{{\rightarrow}}{{\mathrm{\sum}}}_{ \lambda^{\prime}=0}^{\infty}\stackrel{{\rightarrow}}{{\mathrm{ \sum}}}_{\lambda^{\prime}=0}^{\infty}(2\lambda+1)\varepsilon_{0}\stackrel{{ \rightarrow}}{{\mathrm{E}}}_{0}^{\rightarrow 2(\lambda^{\prime}+1)+2}\Bigl{\{}\mathrm{Re}\Bigl{[} \stackrel{{\rightarrow}}{{\mathrm{\sum}}}_{2\lambda+1}\Bigr{)} \omega t\Bigr{)}\times\cos\Bigl{(}\Bigl{(}2\lambda^{\prime}+1\Bigr{)}\omega t \Bigr{)}+\\ -\mathrm{Im}\left\lfloor\stackrel{{\rightarrow}}{{ \mathrm{\sum}}}_{2\lambda}^{(\Omega_{2\lambda+1})}\right\rfloor\times\sin((2 \lambda+1)\omega t)\cdot\cos\Bigl{(}\Bigl{(}2\lambda^{\prime}+1\Bigr{)}\omega t \Bigr{)}\Bigr{\}}.\end{split}  \tag{112d}\]  
Averaging expressions (112a)-(112b) over the variable field period yields  
\[ \begin{split}\left\langle\mathrm{Re}\left[\stackrel{{ \rightarrow}}{{\mathrm{E}}}_{\mathrm{pol}}(\mathrm{t})\right]\cdot\mathrm{Re} \left[\stackrel{{\rightarrow}}{{j}}^{(\Omega)}(\mathrm{t}) \right]\right\rangle=\frac{1}{2}\sigma\stackrel{{\rightarrow}}{{ \mathrm{E}}}_{0}^{\rightarrow 2}+\frac{1}{2}\omega\stackrel{{\rightarrow}}{{\mathrm{\sum}}}_{ \lambda=0}^{\infty}(2\lambda+1)\varepsilon_{0}\stackrel{{ \rightarrow}}{{\mathrm{\sum}}}_{0}^{\rightarrow 4\lambda+2}\cdot\mathrm{Im}\Bigl{[}\stackrel{{ \rightarrow}}{{\mathrm{\sum}}}_{2\lambda}^{(\Omega_{2\lambda+1})}\Bigr{]}, \end{split}  \tag{113a}\] \[ \begin{split}\left\langle\mathrm{Im}\left[\stackrel{{ \rightarrow}}{{\mathrm{E}}}_{\mathrm{pol}}(\mathrm{t})\right]\cdot\mathrm{Im} \left[\stackrel{{\rightarrow}}{{j}}^{(\Omega)}(\mathrm{t}) \right]\right\rangle=-\frac{1}{2}\omega\varepsilon_{0}\varepsilon_{\infty} \stackrel{{\rightarrow}}{{\mathrm{E}}}_{0}^{\rightarrow 2}-\frac{1}{2}\omega\stackrel{{\rightarrow}}{{\mathrm{\sum}}}_{ \lambda=0}^{\infty}(2\lambda+1)\varepsilon_{0}\stackrel{{ \rightarrow}}{{\mathrm{\sum}}}_{0}^{\rightarrow 4\lambda+2}\cdot\mathrm{Re}\Bigl{[}\stackrel{{ \leftarrow}}{{\mathrm{\sum}}}_{2\lambda}^{(\Omega_{2\lambda+1})}\Bigr{]}, \end{split} \] (113c) \[ \begin{split}\left\langle\mathrm{Re}\left[\stackrel{{ \rightarrow}}{{\mathrm{E}}}_{\mathrm{pol}}(\mathrm{t})\right]\cdot\mathrm{Im} \left[\stackrel{{\rightarrow}}{{j}}^{(\Omega)}(\mathrm{t}) \right]\right\rangle=\frac{1}{2}\omega\varepsilon_{0}\varepsilon_{\infty} \stackrel{{\rightarrow}}{{\mathrm{E}}}_{0}^{\rightarrow 2}+\frac{1}{2}\omega\stackrel{{ \rightarrow}}{{\mathrm{\sum}}}_{\lambda=0}^{\infty}(2\lambda+1)\varepsilon_{0} \stackrel{{\rightarrow}}{{\mathrm{\sum}}}_{0}^{\rightarrow 4\lambda+2}\cdot\mathrm{Re}\Bigl{[}\stackrel{{ \leftarrow}}{{\mathrm{\sum}}}_{2\lambda}^{(\Omega_{2\lambda+1})}\Bigr{]}. \end{split}  \tag{113d}\]  
From (113a)-(113d), the identities are obvious  
\[ \begin{split}\left\langle\mathrm{Re}\left[\stackrel{{ \rightarrow}}{{\mathrm{E}}}_{\mathrm{pol}}^{\rightarrow}(\Omega)\right]\cdot \mathrm{Re}\left[\stackrel{{\rightarrow}}{{j}}^{(\Omega)}( \mathrm{t})\right]\right\rangle=\left\langle\mathrm{Im}\left[\stackrel{{ \rightarrow}}{{\mathrm{E}}}_{\mathrm{pol}}^{\rightarrow}(\Omega)\right] \cdot\mathrm{Im}\left[\stackrel{{\rightarrow}}{{j}}^{(\Omega)}( \mathrm{t})\right]\right\rangle,\end{split}  \tag{114}\]  
\[