Goos-Hänchen and Imbert-Fedorov effects in Weyl semimetals

The emergence of Weyl semimetals provides an important opportunity to reexamine the Goos-Hänchen (GH) and Imbert-Fedorov (IF) effects at the unique optical interface. In this paper, we establish a general model to describe the GH and IF shifts at the interface of Weyl semimetals. Based on this model, both the GH and IF shifts are obtained when a light beam impinges on the interface of Weyl semimetals. It is demonstrated that the GH and IF shifts are sensitive to the separation of two Weyl points in momentum space. We believe this result to be of fundamental significance and may provide a possible scheme for the direct optical measurement of the parameters of Weyl semimetals. Our model can also be applied to describe the GH and IF shifts in other topological materials due to their similar topological origins.

Figure 1

Schematic illustrating a geometry beam reflection on the surface of a Weyl semimetal. Both GH and IF shifts occur for a linearly polarized incident beam.

Figure 2

Optical conductivities are plotted as a function of (a) photon energies ($b=0.2$) and (b) separation of two Weyl points ($\lambda =633$ nm). The real and imaginary parts of the optical conductivity are in units of ${\sigma }_0=e^2/\hslash$. Parameters for the Weyl semimetal were chosen as $d=20$ nm, $v_F=1\times {10}^6$ $\mathrm{m}/\mathrm{s}$, and $a=10$ $\AA$. The refractive index of the substrate is $n=1.5$.

Figure 3

The Goos-Hänchen effect on the surface of a Weyl semimetal. The spatial shift (a) and angular shift (b) are plotted as the function of incident angle ${\theta }_i$ and parameter $b$. We assume an incident beam with horizontal polarization, $w_0=10$ $\mu \mathrm{m}$, and $\lambda =633$ $\mathrm{nm}$. Other parameters are the same as those in Fig. 2.

Figure 4

The GH shifts are obtained from three different expressions. The angular shift (a) and spatial shift (b) are plotted as a function of the separation of two Weyl points $b$. The angle of incidence of the light beam is set as ${\theta }_i={55}^{\circ }$. Other parameters are the same as those in Fig. 2.

Figure 5

The IF shifts on the surface of the Weyl semimetal for incident circular polarization. (a), (b) The spatial shifts of the incident beams with left- and right-handed circular polarization, respectively. The spatial shifts are plotted as the function of incident angle ${\theta }_i$ and parameter $b$. Other parameters are the same as those in Fig. 2.

Figure 6

The IF shifts at the interface of the Weyl semimetal for incident linear polarization. The spatial shifts (a) and the angular shifts (b) are plotted as a function of the separation of the two Weyl points $b$ for different incident angles. Other parameters are the same as those in Fig. 2.