Beam shifts for pairs of plane waves

Following Hans Wolter's treatment of the spatial Goos–Hänchen shift of a totally internally reflected light beam by the superposition of two plane waves, polarized perpendicular to the plane of incidence, we consider the reflection and refraction of several similar pairs of plane waves, with varying geometry and incident polarization. We consider explicitly the partial reflection analogue and the in-plane polarized analogue to Wolter's example, as well as a pair of plane waves propagating slightly out of their mutual plane of incidence, revealing the transverse, Imbert–Fedorov shift. We find these simple cases have a complicated polarization structure, with a range of polarization singularities and complex orbital and spin current flows, generalizing Wolter's discovery of an optical vortex and circulating energy flow at the heart of the net scalar interference pattern