Nonspecular effects in the vicinity of a photonic Dirac point

Since the construction of various topological photonic metamaterials, the Dirac degeneracy has been realized and provides a crucial opportunity to investigate the nonspecular effects [Goos-Hänchen (GH) and Imbert-Fedorov (IF) shifts] of light at the unique optical interface. In this paper, we furnish a general and precise model to explore how photonic Dirac point affects nonspecular effects in Dirac metamaterial. Based on this model, the giant nonspecular effects are discovered when a Gaussian beam reflects in the vicinity of a photonic Dirac point. We confirm that the giant nonspecular effects are the consequence of degeneracy between Brewster angles and critical angle caused by the Dirac point. Furthermore, the ability to generate a vortex beam at the Dirac point has also been simply verified. We believe that our work is of underlying significance, and may be served as a reference for the measurement of Dirac point, even other degenerate points. Moreover, our precise model can also be used to describe the nonspecular effects in other analogous photonic systems.

Figure 1

(a) Schematic illustrating the Cartesian coordinate system in Dirac metamaterial. (b) Band structure on the $k_x\text{−}{k}_z$ plane with ${\varepsilon }_y={\mu }_y=0.5$. (c) The dispersion relation along the $k_x$ and $k_z$ directions, respectively. In both (b) and (c), the Dirac points are marked with red spheres.

Figure 2

The GH and IF shifts are plotted as the functions of the distance $\delta {\theta }_i$ away from Dirac angle ${\theta }_D$ and the distance $\delta \omega$ away from the Dirac frequency ${\omega }_D$ on the surface of Dirac metamaterial. (a) GH spatial shift with HP incidence. (b) GH angular shift with HP incidence. (c) IF spatial shift with LCP incidence. (d) IF spatial shift with RCP incidence. We set the incident beam with horizontal polarization for GH shift and circular polarization for IF shift, waist radius $w_0=100\times \lambda$, and $\lambda =2\pi c/\omega$. Parameters for the Dirac metamaterial are chosen as ${\varepsilon }_y={\mu }_y=0.5$ and structural coefficient $f_1=0.5$.

Figure 3

The reflection coefficients, GH and IF shifts near the Dirac point. (a) Reflection amplitude. (b) Reflection phase. (c) GH spatial and angular shifts with HP incidence. (d) IF spatial shifts with LCP and RCP incidence. The frequency of the incident beam is $\delta \omega ={10}^{-3}{\omega }_D$. Other parameters are the same as those in Fig. 2.

Figure 4

The influence mechanism of Dirac point on nonspecular effects. (a) The connection between Dirac point, Brewster angles, and critical angle. The Dirac point $({\theta }_D,{\omega }_D)$ is marked with a green spot. (b) Brewster angle ${\theta }_B$ with the different frequencies. (c) GH spatial shift. (d) GH angular shift. (e) IF spatial shift. We set the frequency change as $\delta \omega /{\omega }_D=0.01,0.02,0.03,0.04$. Other parameters are the same as those in Fig. 2.

Figure 5

The reflected field intensity distributions and phase. (a) and (b) show the field intensity distribution with HP incidence for GH shift and LCP incidence for IF shift, respectively. (c) and (d) give the reflected helical phase and intensity distribution of reflected vortex field near the Dirac point($\delta {\theta }_i=-0.{0012}^{\circ },\delta \omega ={10}^{-5}{\omega }_D$) with LCP incidence.

Figure 6

The GH and IF spatial shifts are obtained from two different expressions for cross-polarization components considered (solid blue line) and not considered (dashed red line). (a) GH spatial shift. (b) IF spatial shift. Insets show the intensity distributions of the reflected field for two different expressions. The position pointed by the arrow corresponds to the incident condition and also the result of it. We set the change $\delta \omega /{\omega }_D={10}^{-4}$. Other parameters are the same as those in Fig. 2.

Figure 7

The GH spatial shift (a), GH angular shift (b), and IF spatial shift (c) change as a function of the incident beam waist for two different incident angles. We choose the frequency of beam as $\delta \omega /{\omega }_D={10}^{-3}$.