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 $\epsilon$, permeability $\mu$, and the topological parameter $\theta$. As a particular case, crystals with periodic $\epsilon$ and $\mu$ 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 $R={\epsilon }_B/{\epsilon }_A$, layer thicknesses $X=d_B/d_A$, and topological parameters $\delta ={\pi }^2{({\theta }_A-{\theta }_B)}^2/{\alpha }^2$. Our numerical results reveal that the band-gap features (width, center position, upper and lower frequencies) depend on the three parameters $\epsilon$, $\mu$, and $\theta$, but with the dependence on $\theta$ 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 $\theta$. We show that the presence of $\theta$ produces a photonic band gap (PBG) which depends only on the $\delta$ term. The widening of this PBG is slightly asymmetrical and monotonic as a function of $\delta$. It was also found that $\delta$ 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.

中文翻译：

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轴离子光子绝缘体中圆偏振光的传递矩阵法

类轴离子系统（或电子拓扑绝缘体）的光子类似物称为轴离子光子晶体。这些材料按三种物理特性分类：介电常数$\epsilon$, 渗透性 $\mu$, 和拓扑参数 $\theta$. 作为一种特殊情况，具有周期性的晶体$\epsilon$ 和 $\mu$是所谓的光子晶体。在本文中，我们采用传递矩阵处理来研究光波在由交替构建层组成的轴子光子晶体中的传播$\mathrm{一种}$ 和 $乙$. 我们将光子能带结构的数值结果作为介电常数之比的函数$\mathrm{电阻}={\epsilon }_{乙}/{\epsilon }_{\mathrm{一种}}$, 层厚 $X=d_{乙}/d_{\mathrm{一种}}$, 和拓扑参数 $\delta ={\pi }^2{({\theta }_{\mathrm{一种}}-{\theta }_{乙})}^2/{\alpha }^2$. 我们的数值结果表明带隙特征（宽度、中心位置、上下频率）取决于三个参数$\epsilon$, $\mu$， 和 $\theta$，但依赖于 $\theta$变得更强。特别是，就带隙宽度而言，我们发现$X$ 和 $\mathrm{电阻}$互相对抗；也就是说，作为$X$ 增加， $\mathrm{电阻}$对于宽带隙出现，必须减小，反之亦然。然而，更有趣的是拓扑参数的结果$\theta$. 我们证明存在$\theta$ 产生光子带隙 (PBG)，它仅取决于 $\delta$学期。这个 PBG 的加宽是稍微不对称和单调的$\delta$. 还发现$\delta$对 PBG 的中心位置没有影响。我们的结果为轴离子光子晶体在光传播的控制和限制方面的技术应用开辟了可能性。