\int d^3q \ \underline{\Lambda}(\vec{k},\vec{q},s)\ \cdot\
\underline{\vec{E}}^{b}(\vec{q},s)&=&\underline{\vec{J}}^{b}_N(\vec{k},s)\nonumber\\
\int d^3q \ \underline{\Lambda}(\vec{k},\vec{q},s)\ \cdot\
\underline{\vec{E}}^{f}(\vec{q},s)&=&\underline{\vec{J}}^{f}_N(\vec{k},s)\end{aligned}$$ with $$\label{c44}
\underline {\Lambda}_{ij}(\vec{k},\vec{q},s)=
\varepsilon_{i\delta\gamma}\varepsilon_{\alpha\beta j}k^\delta
q^\beta
\underline{\tilde{\mu}}^{\gamma\alpha}(\vec{k},\vec{q},s)-\frac{s^2}{c^2}\underline{\tilde{\varepsilon}}_{ij}(\vec{k},\vec{q},s)$$ From (\[c43\]) we see that the backward and forward Laplace transformations of the electric field operator satisfy an inhomogeneous equation with a source term. To solve such equations it is common to use the Green function method. The suitable Green function associated to the Eq. (\[c43\]) is defined as the solution of an algebraic equation as follows $$\label{c45}
\int d^3q \ \underline{\Lambda}(\vec{k},\vec{q},s)\ \cdot\
\underline{G}(\vec{q}, \vec{p},s)= I \ \delta(\vec{k}-\vec{p})$$ Accordingly the solution of Eqs. (\[c43\]) in terms of the Green function defined in (\[c45\]) can be written as $$\begin{aligned}
\label{c46}
\underline{\vec{E}}^{b}(\vec{k},s)&=& \int d^3p\
\underline{G}(\vec{k},\vec{p},s) \cdot\
\underline{\vec{J}}^b_N(\vec{p},s)\nonumber\\
\underline{\vec{E}}^{f}(\vec{k},s)&=& \int d^3p\
\underline{G}(\vec{k},\vec{p},s) \cdot\
\underline{\vec{J}}^f_N(\vec{p},s)\end{aligned}$$ In the special case of a homogeneous medium for which the electric and magnetic susceptibility tensors are translationally invariant, that is when the tensors $ \chi^e(\vec{r},\vec{r'},t)$ and $
\chi^m(\vec{r},\vec{r'},t)$ are a function of the difference $
\vec{r}-\vec{r'}$ and so is then, the permitivity and permeability tensors $ \varepsilon (\vec{r},\vec{r'},t)$ and $
\mu(\vec{r},\vec{r'},t) $, we deduce $$\begin{aligned}
\label{c47}
 \underline{\tilde{\varepsilon}}(\vec{k},\vec{q} ,s)&=&
\left(I+\underline{\tilde{\chi}}^e(\vec{k},s)\right)\delta(\vec{k}-\vec{q})\nonumber\\
\underline{\tilde{\mu}}(\vec{k},\vec{q}
,s)&=&\left(I-\underline{\tilde{\chi}}^m(\vec{k},s)\right)\delta(\vec{k}-\vec{q})\end{aligned}$$ and therefore from the Eqs. (\[c44\]) and (\[c45\]) we find $$\begin{aligned}
\label{c48}
&&\underline{G}(\vec{k},\vec{p},s)=
\underline{T}^{-1}(\vec{k},s)\delta(\vec{k}-\vec{p})\nonumber\\
&&\underline{T}_{ij}=\left[\varepsilon_{i\delta\gamma}\varepsilon_{\alpha\beta
j}k_\delta k_\beta 