\nabla \times \mathbf{B}-\frac{\partial \mathbf{E}}{\partial t} &=&\tilde{%
\theta}\delta \left( \Sigma \right) \mathbf{E}\times \mathbf{n}+4\pi \mathbf{%
J},  \label{Ampere}\end{aligned}$$while the homogeneous equations are included in the Bianchi identity $\partial _{\mu }\tilde{F}^{\mu \nu }=0$. Here $\mathbf{n}$ is the unit normal to $\Sigma $ shown in the Fig. [regions]{}. In this work we consider a simple geometry corresponding to a surface $\Sigma$ taken as the plane $z=a$.

As we see from Eqs. (\[GaussE\]-\[Ampere\]) the behavior of $\theta $-ED in the bulk regions $\mathcal{U}_{1}$ and $\mathcal{U}_{2}$ is the same as in standard electrodynamics. The $\theta $-term modifies Maxwell equations only at the surface $\Sigma $. Here $F^{i0}=E^{i}$, $F^{ij}=-\varepsilon
^{ijk}B^{k}$ and $\tilde{F}^{i0}=B^{i}$, $\tilde{F}^{ij}=\varepsilon
^{ijk}E^{k}$. Equations (\[GaussE\]-\[Ampere\]) also suggest that the electromagnetic response of a system in the presence of a $\theta $-term can be described in terms of Maxwell equations in matter $$\nabla \cdot \textbf{D} = 4 \pi \rho \;\;\;\ , \;\;\;\ \nabla \times \textbf{H} = 4 \pi \textbf{J} , \label{MaxEqMatter}$$ with constitutive relations $$\mathbf{D}=\mathbf{E}+\frac{\alpha }{\pi }\theta \left(z\right) \mathbf{B}%
,\;\;\;\;\;\;\;\;\;\mathbf{H}=\mathbf{B}-\frac{\alpha }{\pi }\theta \left(
z\right) \mathbf{E},  \label{CONST_REL}$$where $\theta \left(z\right) $ is given in Eq. (\[theta\]). If $\theta (z)$ is globally constant in $\mathcal{M}$, there is no contribution to Maxwell equations from the $\theta $-term in the action, even though 