Optical signature of topological insulators

The axion coupling in topological insulators couples electric polarization with magnetic field, and magnetization with electric field. As a result, the usual laws of electromagnetic wave propagation are modified. We report on the Fresnel formula for the reflection of electromagnetic wave at the interface of materials with different axion couplings. The Brewster angle and the Goos-Hänchen effect are also studied. We find that, because of the axion coupling, in order to realize the Brewster-angle condition, the incident polarization should be rotated away from the plane of incidence. The maximum angle of rotation is $\pi /4$ when both materials have nearly the same refraction indices. This offers a convenient way to determine the axion angle by optical measurement.

Figure 1

(Color online) Optical reflection and refraction at the interface between two materials with refraction indices $n$ and $n^{\prime }$ and axion angles $\Theta$ and ${\Theta }^{\prime }$.

Figure 2

Changes in reflectance $\delta r_1^2$ and $\delta r_2^2$ (defined in the text) due to the axion coupling, plotted as a function of the incident angle $\theta$. Solid lines are for ${\Theta }^{\prime }-\Theta =\pi$; dashed lines are for ${\Theta }^{\prime }-\Theta =\pi /2$ (bold solid and dashed lines are for $\delta r_1^2$). The critical angle ${\theta }_c={\text{sin}}^{-1}\left(n^{\prime }/n\right)=71.805°$ for $\left(n,n^{\prime }\right)=\left(10,9.5\right)$.

Figure 3

Rotation angle $\gamma$ of the eigenmodes plotted as a function of $\left(n^{\prime }-n\right)/n$, where $n$ and $n^{\prime }$ are the refraction indices on the two sides of the interface. Solid lines are for ${\Theta }^{\prime }-\Theta =\pi$; dashed lines are for ${\Theta }^{\prime }-\Theta =\pi /2$.

Figure 4

Changes in the GH shift $\delta D_1$ and $\delta D_2$ (defined in the text) due to the axion coupling, plotted as a function of the incident angle $\theta$. Solid lines are for ${\Theta }^{\prime }-\Theta =\pi$; dashed lines are for ${\Theta }^{\prime }-\Theta =\pi /2$ (bold solid and dashed lines are for $\delta D_1$). The critical angle ${\theta }_c=71.805°$ for $\left(n,n^{\prime }\right)=\left(10,9.5\right)$.