Photons in restricted topologies

Abstract

In this work we discuss three different aspects of the topological classification of propagating beams of light. This topological classification relies on global properties of the light beam, which are insensitive to local disorder or perturbations. However, topological invariants calculated for scalar fields may not fully describe a beam of light, which consists of a vector field which points in a particular direction at a given time. We consider the extension of the topological classification of light beams to cases where the polarisation is allowed to vary. When the phase of a scalar electric field varies around a point, that field carries angular momentum, proportional to the magnitude of the change of phase. When the polarisation also varies, we show that there is a new angular momentum which is carried by such a beam. This generalised angular momentum accounts for the possible winding in both the direction of polarisation and the phase around a point. We show that the spectrum of this angular momentum can be a half-integer or an integer multiple of Planck's constant. To confirm these predictions, we measure the angular momentum current in a beam with varying polarisation. As well as the classical current, we measure quantum fluctuations due to the discrete nature of the photons which carry this current. This experiment shows that the generalised angular momentum is indeed quantised in half-integer multiples of h, and provides a general method for sorting and detecting beams according to their generalised angular momentum. Next, we study a new class of material, the hyperbolic metamaterial, in the generic case where all three principal dielectric constants are unequal. These materials have negative dielectric constant in one direction, and positive but unequal in the other two. We show that the iso-frequency surface, that is the surface of wave vectors at which light of a constant frequency can propagate, consists of two sheets which meet at four linear intersection points. We derive a geometrical optics, and then a full diffraction theory of light propagating close to one of these directions. We also include the effects of absorption and discuss how such a material could be realised in practice. Finally, we examine the topological classification of light beams in periodic structures, in terms of their crystal wave vector k. Again we show that although it is possible to describe a single polarisation using the classification of scalar fields, when the polarisation is allowed to vary then this classification is no longer suitable. Instead we show that the propagation of the field can be described by a local non-Abelian gauge field. This non-Abelian field has an integer associated with it which classifies pairs of bands describing two orthogonal polarisations. We give a simple formula to compute the non-Abelian field and the invariant which classifies the winding of this field around loops in reciprocal space.