Topological photonics with anisotropic materials

Abstract

In this thesis we investigate several aspects of the topological characterisation of light propagating through matter, or topological photonics. Throughout we will focus on anisotropic dielectrics in which the optical response depends on the polarisation and direction of propagation of the incident light. We will explore both homogeneous dielectric media and periodically patterned optical materials known as photonic crystals. The investigations will be conducted in either the weak coupling regime, where the solutions of the wave equation describe photons, and in the strong coupling regime, where the solutions describe mixed light-matter polaritons.

Our first investigation considers the topological classification of the refractive index surfaces of homogeneous optical materials. This classification is done by computing the Chern number topological invariant. The particular materials we focus on are biaxial dielectrics which additionally exhibit either of two forms of optical activity. In the absence of any optical activity the refractive index surfaces of biaxial materials exhibit four Dirac point degeneracies in direction space. Once optical activity is introduced the degeneracies of the refractive index surface are lifted. We discover that one of these forms of optical activity can lift the degeneracies in such a way as to produce a non-zero Chern number. We also use the paraxial approximation to derive a Hamiltonian which evolves light through biaxial optically active materials in directions close to one of the lifted degeneracies.

Next we turn to assessing the topological classification of photonic crystals composed of anisotropic optically active materials. We do this by adapting the derived Hamiltonian to describe periodic optical media with a definite patterning geometry. The two patterning geometries we consider are square and triangular lattices. We find that, for any combination of lattice and optical activity variant, topologically non-trivial iso-frequency surfaces are possible. We compare and contrast these results with each other and with the conclusions on homogeneous materials. We explore the implications of the non-zero Chern number for the periodic systems. In particular, the prospect of edge states for the anisotropic photonic crystals is investigated.

Our final investigation considers the topologically protected degeneracies of the dispersion relation of polaritons. In particular, we are interested in the polaritons that form from the coupling of light to excitons in bulk magnetically biased semiconductors. In direct band-gap zincblende semiconductors the introduction of an external magnetic field results in an anisotropic and multiply-resonant optical response. We derive this optical response and use it to determine the polariton dispersion relation. We find that the resulting polariton dispersion features an abundance of topologically protected degeneracies, both in the absence and presence of dissipation.