forward Laplace transformations, denoted respectively, by $\Gamma^b(s)$ and $ \Gamma^f(\rho)$ are defined as $$\begin{aligned}
\label{c39}
\Gamma^b(s)&=&\int_0^\infty dt\ e^{-s t}\
\Gamma(-t)\nonumber\\
\Gamma^f(s)&=&\int_0^\infty dt\ e^{-s t}\ \Gamma(t)\end{aligned}$$ Carrying out the backward and forward Laplace transformations of Eqs.(\[c27\]) , (\[c27.1\]), (\[c31\]) , (\[c36\]) and $
\imath\vec{k}\times \underline{\vec{E}}(\vec{k},t)= -
\underline{\dot{\vec{B}}}(\vec{k},t)$ and then their combination, for the electric field operator we find $$\begin{aligned}
\label{c40}
&&\int d^3q \left[ \vec{k}\times \underline{\tilde{\mu}}(\vec{k},
\vec{q},s)\left(\vec{q}\times\underline{\vec{E}}^{ b}(\vec{q}
,s)\right)\right] -\frac{s^2}{c^2}\int d^3q \
\underline{\tilde{\varepsilon}}( \vec{k},
\vec{q},s)\underline{\vec{E}}^{b}(\vec{q}
,s)=\underline{\vec{J}}^{ b}_N(\vec{k},s)\nonumber\\
&&\int d^3q \left[ \vec{k}\times \underline{\tilde{\mu}}(\vec{k},
\vec{q},s)\left(\vec{q}\times\underline{\vec{E}}^{ f}(\vec{q}
,s)\right)\right] -\frac{s^2}{c^2}\int d^3q \
\underline{\tilde{\varepsilon}}( \vec{k},
\vec{q},s)\underline{\vec{E}}^{f}(\vec{q}
,s)=\underline{\vec{J}}^{ f}_N(\vec{k},s)\nonumber\\\end{aligned}$$ where $$\begin{aligned}
\label{c41}
\underline{\tilde{\varepsilon}}( \vec{k},
\vec{q},s)&=&\delta(\vec{k}-\vec{q})I+\underline{\tilde{\chi}}^e(\vec{k},
\vec{q},s)\nonumber\\
\underline{\tilde{\mu}}( \vec{k},
\vec{q},s)&=&\delta(\vec{k}-\vec{q})I-\underline{\tilde{\chi}}^m(\vec{k},
\vec{q},s)\end{aligned}$$ are respectively the permitivity and permeability tensors of the medium in Laplace language. The source terms in the right hand of the inhomogeneous Eqs.(\[c40\]) are the backward and forward Laplace transformations of the noise current density and are expressed in terms of initial conditions of the dynamical variables of the total system at $t=0$ as $$\begin{aligned}
\label{c42}
\underline{\vec{J}}^{ b}_N(\vec{k},s)&=&\mu_0
s^2\underline{\vec{P}}^b_N(\vec{k},s)-\mu_0 s\
\imath\vec{k}\times\underline{M}^b_N(\vec{k},s)+\nonumber\\
&&\imath\vec{k}\times \int d^3q \ \underline{\tilde{\mu}}(\vec{k},
\vec{q},s)\
\underline{\vec{B}}(\vec{q},0)-\mu_0 s \underline{\vec{D}}(\vec{k},0)\nonumber\\
\underline{\vec{J}}^{ f}_N(\vec{k},s)&=&\mu_0 s^2\
\underline{\vec{P}}^f_N(\vec{k},s)+\mu_0 s\
\imath\vec{k}\times\underline{M}^f_N(\vec{k},s)\nonumber\\
&&-\imath\vec{k}\times \int d^3q \ \underline{\tilde{\mu}}(\vec{k},
\vec{q},s)\ \underline{\vec{B}}(\vec{q},0)-\mu_0 s
\underline{\vec{D}}(\vec{k},0)\end{aligned}$$ Eqs. (\[c40\]) can be rewritten in a compact form as $$\begin{aligned}
\label{c43}
