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Optical Properties Chern-Simons (3+1)D $θ$-systems

Chern-Simons (CS) $θ$-systems are described by a $θ\int F\wedge F$ term in the action ($θ$ is an adimensional parameter), which does not change field equations in the bulk, but affects the system behaviour when it is bounded. When two of those $θ$-systems, each one characterized by a different value of $θ$ (even zero), share a common boundary, surface effects are then induced by a CS $θ$ term. Here, we study the consequences of the above in the propagation of electromagnetic radiation in $θ$-systems. In a previous paper, electromagnetic radiation properties traversing a $θ$-vacuum were analyzed where a number of interesting features arise related to polarization and energy distribution. Now, we investigate how electric and magnetic properties of the $θ$-system affect the optical response. Apart from the well-known Kerr polarization rotation found for the particular case of topological insulators, we examine in detail the issue and the results could be applied in other contexts where $θ$-term accounts for the system dynamics. In particular, we find two different Brewster angles, for $s$ and $p$ polarization of reflected radiation, respectively, with peculiar features derived from the $θ$ term influence. Possible applications of these results are discussed.