in a non-absorbing medium [@20; @22].\
In order to prevent some difficulties with a non-local Lagrangian such as in (\[c4\]), it is the easiest way to work in the reciprocal space and write all the fields and the coupling tensors $f\ ,\ g$ in terms of their spatial Fourier transforms. For example the dynamical variable $ \vec{X}_\omega $ can be written as $$\label{c5}
\vec{X}_\omega(\vec{r},t)=\frac{1}{\sqrt{(2\pi)^3)}}\int d^3k\
\vec{\underline{X}}_\omega(\vec{k},t)\ e^{\imath \vec{k}\cdot
\vec{r}}$$ Since we are concerned with real valued fields in the total Lagrangian (\[c1\])-(\[c4\]), we have $\vec{\underline{X}}_\omega^*(\vec{k},t)=\vec{\underline{X}}_\omega(-\vec{k},t)$ for the field $ \vec{X}_\omega(\vec{r},t)$ and the other dynamical fields in this Lagrangian. Similarly the real valued coupling tensors $f$ and $g$ can be expressed in reciprocal space as $$\begin{aligned}
\label{c6}
f_{ij}(\omega,\vec{r},\vec{r'})&=&\frac{1}{(2\pi)^3} \int d^3k
\int d^3k'  \underline{f}_{ij}(\omega , \vec{k} , \vec{k'})
e^{\imath \vec{k}\cdot \vec{r}- \imath \vec{k'}\cdot
\vec{r'}}\nonumber\\
g_{ij}(\omega,\vec{r},\vec{r'})&=&\frac{1}{(2\pi)^3} \int d^3k
\int d^3k'  \underline{g}_{ij}(\omega , \vec{k} , \vec{k'})
e^{\imath \vec{k}\cdot \vec{r}- \imath \vec{k'}\cdot \vec{r'}}\end{aligned}$$ which obey the following conditions $$\begin{aligned}
\label{c6.1}
\underline{f}_{ij}(\omega , \vec{k} ,
\vec{k'})&=&\underline{f}_{ij}^*(\omega , -\vec{k} ,
-\vec{k'})\nonumber\\
\underline{g}_{ij}(\omega , \vec{k} ,
\vec{k'})&=&\underline{g}_{ij}^*(\omega , -\vec{k} , -\vec{k'})\end{aligned}$$ The number of independent variables can be recovered by restricting the integrations to the half space $k_z\geq 0$. The total Lagrangian (\[c1\])-(\[c4\]) is then obtained as $$\label{c7}
 