Transfer-matrix method of circular polarization light in an axionic photonic insulator

Abstract

The photonic analog of an axioniclike system (or electronic topological insulator) is called an axionic photonic crystal. These materials are classified by three physical properties: permittivity $\ensuremath{\epsilon}$, permeability $\ensuremath{\mu}$, and the topological parameter $\ensuremath{\theta}$. As a particular case, crystals with periodic $\ensuremath{\epsilon}$ and $\ensuremath{\mu}$ are the so-called photonic crystals. In this paper, we employ a transfer-matrix treatment to study the propagation of light waves in an axionic photonic crystal composed of alternating building layers $A$ and $B$. We present numerical results for the photonic band structure as a function of the ratio between permittivities $R={\ensuremath{\epsilon}}_{B}/{\ensuremath{\epsilon}}_{A}$, layer thicknesses $X={d}_{B}/{d}_{A}$, and topological parameters $\ensuremath{\delta}={\ensuremath{\pi}}^{2}{({\ensuremath{\theta}}_{A}\ensuremath{-}{\ensuremath{\theta}}_{B})}^{2}/{\ensuremath{\alpha}}^{2}$. Our numerical results reveal that the band-gap features (width, center position, upper and lower frequencies) depend on the three parameters $\ensuremath{\epsilon}$, $\ensuremath{\mu}$, and $\ensuremath{\theta}$, but with the dependence on $\ensuremath{\theta}$ being stronger. In particular, as far as the band-gap width is concerned, we find that $X$ and $R$ work against each other; that is, as $X$ increases, $R$ must decreases for a wide band-gap emergence and vice versa. More interesting, however, are the results for the topological parameter $\ensuremath{\theta}$. We show that the presence of $\ensuremath{\theta}$ produces a photonic band gap (PBG) which depends only on the $\ensuremath{\delta}$ term. The widening of this PBG is slightly asymmetrical and monotonic as a function of $\ensuremath{\delta}$. It was also found that $\ensuremath{\delta}$ has no influence on the center position of the PBG. Our results open possibilities for technological applications of axionic photonic crystals with regard to the controlling and confinement of the propagation of light.