Chern numbers for the index surfaces of photonic crystals: Conical refraction as a basis for topological materials

The classification of band structures by topological invariants provides a powerful tool for understanding phenomena such as the quantum Hall effect. This classification was originally developed in the context of electrons but can also be applied to photonic crystals. In this paper we study the topological classification of the refractive index surfaces of two-dimensional photonic crystals. We consider crystals formed from birefringent materials in which the constitutive relation provides an optical spin-orbit coupling. We show that this coupling, in conjunction with optical activity, can lead to a gapped set of index surfaces with nonzero Chern numbers. This method for designing photonic Chern insulators exploits birefringence rather than lattice structure and does not require band crossings originating from specific lattice geometries.

Figure 1

Illustration of the formation of topologically nontrivial index surfaces in two-dimensional photonic crystals. Each panel is a schematic of a section of a refractive index (isofrequency) surface for (a) a biaxial dielectric, (b) a biaxial dielectric with optical activity, (c) a periodic biaxial dielectric, and (d) a periodic biaxial dielectric with optical activity. Each panel is centered on the wave vector corresponding to the conical singularity in (a). The periodicity is taken to be in the plane perpendicular to this wave vector, forming a two-dimensional Brillouin zone (BZ).

Figure 2

Polarization structure of the index surface over the first Brillouin zone for a biaxial material and a square lattice, predicted by Eqs. (2) and (3). The arrows show the field $(h_x,h_y)$. The zeros of $h_x$ and $h_y$ lie along the lines indicated in turquoise (dark gray) and green (light gray), respectively. In the absence of optical activity $h_z=0$, and there are conical singularities at the intersections of the contours shown. The first Brillouin zone is the square bounded by the two black lines and the upper and lower green contours. The parameters are ${\epsilon }_1=2.25, {\epsilon }_2=2.5, {\epsilon }_3=2.75$, and $ka=1.6\pi$, where $k=\omega /\left(c\sqrt{{\epsilon }_2}\right)$.

Figure 3

Chern number $C$ (shading) of a biaxial material on a square lattice, from the paraxial Hamiltonian model, with optical activity due to the Faraday effect. The spherical coordinates $(\theta ,\varphi )$ give the direction of the axis of optical activity, i.e., the magnetic field. The polar angle $\theta$ is measured from the optic axis and the azimuthal angle $\varphi$ from the $x$ axis of the rotated coordinate system described in the Appendix. The shadings represent $C=0$ (white) and $C=\pm 1$ (dark or light gray). The parameters are those given in Fig. 2.

Figure 4

Comparison of the locations of the Dirac points given by the paraxial theory and a frequency-domain plane-wave simulation as $k$ is varied. In both cases Dirac points occur along the lines $k_y=0$ (top) and $k_y=\pi /a$ (bottom). Solid lines: $k_x$ for the Dirac points from the paraxial theory. Shading: bounds on $k_x$ for the Dirac points from the plane-wave simulations, where the magnitude of the splitting between the index surfaces is smaller than $a\mathrm{\Delta }{k}_z=2\pi \times {10}^{-6}$. The dielectric parameters are those in Fig. 2. The plane-wave simulation has cylindrical air holes in the dielectric background, with a filling factor of 0.15.

Figure 5

Index surfaces along high-symmetry directions of the first Brillouin zone from the paraxial model (top panel) and a frequency-domain plane-wave calculation (lower panel). The relevant high-symmetry points are indicated in Fig. 2. $ka\simeq 4\pi$, and all other parameters are as in Figs. 2 and 4.