Polarization evolution on the higher-order Poincaré sphere via photonic Dirac points

In this work, we report a general reflection model to describe the reflected behaviors of a Gaussian beam near photonic Dirac point at the optical interface. With spin polarized electromagnetic waves strikes near the single Dirac point on Dirac material interface, vortical phase distribution and spin inversion are demonstrated for reflected state, and we further develop a higher-order Poincaré sphere to describe the evolution of polarization states. It is possible to generate any desired Poincaré vortex beam on the sphere by modulating the incident polarization state of light. Moreover, we discuss the case of normal incidence around two adjacent Dirac points, reflection spectrum involves a couple of spin flip vortices corresponding to two points with the corresponding topological invariants, which are consistent with the Berry curvature and topological charge. These results can be extended to similar hyperbolic metamaterials with degenerate point, and inspire a different perspective for manipulating polarized vortex beam with photonic Dirac point.

Figure 1

(a) The effective bulk band structure of Dirac metamaterials on a wave vector plane. Two bands are nearly overlapping with each other, and there are fourfold band degeneracies at two points (marked by the red spheres), which are symmetrically placed. (b) Schematic illustrating the wave reflected vortex beams near the Dirac point at the interface of Dirac hyperbolic metamaterials. We set $\mathit{xy}$ plane as an interface between air ($z>0$) and Dirac metamaterials ($z<0$) characterized by $\mathit{\varepsilon },\mathit{\mu }$.

Figure 2

Schematic illustrating the homogeneous polarization state distribution on the fundamental Poincaré sphere. The north and south poles represent left- and right-handed circular polarization, respectively. Any polarization state on the sphere can be regarded as their linear combination. The signs of $\text{a}\sim \mathrm{f}$ are the reference points with coordinates (0, 0, 1), (1, 0, 0), (0, 0, $-1$), ($\sqrt{2}/2,\sqrt{2}/2$, 0), (0, 1, 0), and ($-1$, 0, 0). We set an incident beam with waist radius $w_0=100\lambda$, and $\lambda =2\pi c/\omega$.

Figure 3

The reflection amplitude (a) and imaginary part of reflection coefficients (b) near the Dirac point ($\theta \to {30}^{\circ }$). The parameters $r_{pp}^{\prime }$ and $r_{ss}^{\prime }$ also as the function of incident angle. (c) and (d) show phase distribution of left-handed and right-handed circular polarizations light incidence in momentum space, respectively. We set an incident beam with frequency $\omega ={\omega }_D(1+0.002)$, parameters for the Dirac metamaterials are chosen as ${\varepsilon }_y={\mu }_y=0.5$, the resonance frequency ${\omega }_0={10}^{14}\text{Hz}$ and structural coefficients $f_1=1$.

Figure 4

The higher-order Poincaré sphere representation of the reflected field near the Dirac point illustrating the local polarization vectors states at various positions on the sphere. The location of six points ${\mathrm{a}}^{\prime }\sim {\mathrm{f}}^{\prime }$ on the sphere with coordinates (0, 0, $-1$), (1, 0, 0), (0, 0, 1), ($\sqrt{2}/2$,-$\sqrt{2}/2$, 0), (0, $-1$, 0), and ($-1$, 0, 0), which correspond to points $\mathrm{a}\sim \mathrm{f}$ on the fundamental sphere. The incident frequency is chosen as $\omega ={\omega }_D(1+{10}^{-12})$ and incident angle is chosen as $\theta ={30}^{\circ }$. Parameters for the Dirac metacrystal are chosen as ${\varepsilon }_y={\mu }_y=0.5$, the resonance frequency ${\omega }_0={10}^{14}\text{Hz}$ and structural coefficients $f_1=1$.

Figure 5

Illustration of the Gaussian beam vertically incident at the interface of Dirac metamaterials with a paired Dirac points, which are placed center in momentum space. $\vartheta$ is the propagation angle of the plane wave components in the spherical coordinate system, $k_{\perp }=\sqrt{k_{ix}^2+k_{iy}^2}$ is the transverse wave vector. Combining the unit incident plane vector $\mathcal{I}$, thus the angle $\alpha$ can be indicated by arbitrary wave vector: $cos\alpha =k_{ix}/k_{\perp }$, $sin\alpha =k_{iy}/k_{\perp }$.

Figure 6

Normalized reflectance amplitude and two-dimensional (2D) contour of left-handed circular polarization beam vertically incident at the interface with paired Dirac points.

Figure 7

Equifrequency contours of (a) isotropic system (air) and (b) Dirac metamaterials. In the transformation from air to degenerate Dirac metamaterials, the spin Chern numbers are split into $\pm 1$. (c) and (d) represent the transition Chern numbers for spin up ($\sigma =1$) and spin down ($\sigma =-1$) state in $k_x-k_y$ plane. (e) and (f) shows the interference patterns in left- and right-handed incident polarization respectively. For different spin input state, the reflected field occur the corresponding topological charge (Chern number) at Dirac points.