Geometry of 3D monochromatic light: local wavevectors, phases, curl forces, and superoscillations

This is an exploration of some geometric properties, and their interconnections, for nonparaxial electromagnetic waves. The properties involve the electric field alone, and also in combination with the magnetic field ('electric-magnetic democracy'). The orbital part of the Poynting vector, represented in terms of the electric field, is proportional to the nonconservative 'curl force' exerted on small absorbing polarisable particles; its circuit integral represents the work done transporting a particle. Normalised versions of the electric, magnetic, and electric-magnetic orbital Poynting vectors are natural candidates for wavevectors representing the direction of the wave at each point, generalising the phase gradient for scalar waves; there are associated total, dynamical and geometric phases. For isotropic ensembles of random waves, the probability distributions of the different wavevectors are estimated by codimension arguments and calculated exactly for statistically isotropic superpositions of plane waves. The superoscillation probability, that the magnitude of the local wavevector exceeds that of the waves in the superposition, is 0.031 40 for the electric wavevector and 0.000 109 for the electric-magnetic wavevector. Both values are much smaller than the known superoscillation probability 0.350 48 for scalar waves in three-dimensions, illustrating a general phenomenon: interference detail is weaker for electric properties than for scalar, and weaker still for electric-magnetic properties.