Exceptional points in anisotropic planar microcavities

Planar microcavities allow the control and manipulation of spin polarization, manifested in phenomena like the optical spin Hall effect due to the intrinsic polarization mode splitting. Here, we study a transparent microcavity with broken rotational symmetry, realized by aligning the optic axis of a uniaxial cavity material in the cavity plane, giving rise to exceptional points in the dispersion relation. These occur pairwise, are circularly polarized, and are cores of polarization vortices. The exceptional points are result of the non-Hermitian character of the system and are in close relationship to singular optic axes in absorptive biaxial systems.

Figure 1

Cavity photon modes for $d_c=130$ nm, i.e., $n_c{d}_c>{\lambda }_{\text{DBR}}/2$, the cavity layer's optic axis along $y$, and 15 DBR layer pairs on each side: (a) photon mode energy $E$ and (b) broadening $\gamma$ for three propagation directions which are depicted in (c): along $x$ (solid black curves), $y$ (dashed blue curves), and crossing an EP (solid red curves). Differences in the (c) mode energies and (d) mode broadenings in momentum space. Black contours illustrate the degeneracy of $E$ and $\gamma$, respectively. Both are simultaneously degenerate at the EPs (black points). Mode broadening $\gamma$ of (e) mode 1 and (f) mode 2 showing the discontinuities in momentum space. For all color bars, the units are meV. The shape of the edge of the light cone differs for the two modes due to the optical anisotropy.

Figure 2

Mode polarization in terms of the pseudospin according to Eq. (2) for the microcavity in Fig. 1. EPs appear as circularly polarized spots and polarization discontinuities occur for ${\tilde{P}}_1$.

Figure 3

Detailed view of the polarization of the two modes in the momentum-space region around a pair of EPs (cf. Fig. 2). Arrows represent the linear polarization vector, i.e., ${\tilde{P}}_1$ pointing along the $k_x$ axis and ${\tilde{P}}_2$ pointing along the $k_y$ axis. ${\tilde{P}}_3$ is represented by the arrows' colors and also by the contour lines. The color scale is the same as in Fig. 2. The gray-scaled background represents the mode splitting $E_2-E_1$, which is 0 (black) along the connection line between the two EPs.

Figure 4

Positions of EPs (dots) and lines of degenerate real ($E$; thick lines) and imaginary ($\gamma$; narrow lines) parts of the complex mode energy in momentum space. (a) Varying cavity layer thicknesses $d_c$ as given by the color bar (in nm), with DBRs of 15 layer pairs. (b) Varying number of layer pairs according to the color bar for $d_c=130$ nm.

Figure 5

Definition of the sample coordinate system.

Figure 6

Mode polarization in terms of the pseudospin components for a microcavity with an optically uniaxial cavity layer, the optic axis of which is aligned along $y$ (cf. Fig. 2 in the text). DBRs of 15 layer pairs are assumed at the top and bottom. Two cavity thicknesses $d_c$ are shown. For comparison, Fig. 2 in the text depicts $d_c=130$ nm.