Canonical representation of spherical functions: Sylvester's theorem, Maxwell's multipoles and Majorana's sphere

Geometric phases in 2D and 3D polarized fields: geometrical, dynamical, and topological aspects

Geometric phases are a universal concept that underpins numerous phenomena involving multi-component wave fields. These polarization-dependent phases are inherent in interference effects, spin-orbit interaction phenomena, and topological properties of vector wave fields. Geometric phases have been thoroughly studied in two-component fields, such as two-level quantum systems or paraxial optical waves. However, their description for fields with three or more components, such as generic nonparaxial optical fields routinely used in modern nano-optics, constitutes a nontrivial problem. Here we describe geometric, dynamical, and total phases calculated along a closed spatial contour in a multi-component complex field, with particular emphasis on 2D (paraxial) and 3D (nonparaxial) optical fields. We present several equivalent approaches: (i) an algebraic formalism, universal for any multi-component field; (ii) a dynamical approach using the Coriolis coupling between the spin angular momentum and reference-frame rotations; and (iii) a geometric representation, which unifies the Pancharatnam-Berry phase for the 2D polarization on the Poincaré sphere and the Majorana-sphere representation for the 3D polarized fields. Most importantly, we reveal close connections between geometric phases, angular-momentum properties of the field, and topological properties of polarization singularities in 2D and 3D fields, such as C-points and polarization Möbius strips.

Superoscillation phenomena in SO(3)

We prove a new asymptotic formula for the Wigner d -functions, as a consequence of the discovery of superoscillatory behaviour in SO(3).

A three-dimensional degree of polarization based on Rayleigh scattering

A measure of the degree of polarization for the three-dimensional polarization matrix (coherence matrix) of an electromagnetic field is proposed, based on Rayleigh scattering. The degree of polarization at a point is defined as an average, over all scattering directions, of an imagined dipole scattering of the three-dimensional state of polarization. This gives a well-defined purity measure, which, unlike other proposed measures of the three-dimensional degree of polarization, is not a unitary invariant of the matrix. This is demonstrated and discussed for several examples, including a partially polarized transverse beam.

Examination of evidence for a preferred axis in the cosmic radiation anisotropy

We examine previous claims for a preferred axis at (b,l) approximately (60,-100) in the cosmic radiation anisotropy, by generalizing the concept of multipole planarity to any shape preference. Contrary to earlier claims, we find that the amount of power concentrated in planar modes for l = 2,3 is not inconsistent with isotropy and Gaussianity. The multipoles' alignment, however, is indeed anomalous, and extends up to l = 5 rejecting statistical isotropy with a probability in excess of 99.9%. There is also an uncanny correlation of azimuthal phases between l = 3 and l = 5. We are unable to blame these effects on foreground contamination or large-scale systematic errors. This reappraisal may be crucial in identifying the theoretical model behind the anomaly.