Formation of the reverse flow of energy in a sharp focus

Abstract

It was theoretically shown that in the interference pattern of four plane waves with specially selected directions of linear polarization it is formed a reverse flow of energy. The areas of direct and reverse flow alternate in a staggered order in the cross section of the interference pattern. The absolute value of the reverse flow directly depends on the angle of convergence of the plane waves (on the angle between the wave vector and the optical axis) and reach the maximum at an angle of convergence close to 90 degrees. The right-handed triples of the vectors of four plane waves (the wave vector with positive values of projection to optical axis and the vector of electric and magnetic fields) when added in certain areas of the interference pattern form an electromagnetic field described by the left-handed triple of vectors; however, the projection of wave vector to optical axis has negative values. In these areas, the light propagates in the opposite direction. A similar explanation of the mechanism of the formation of a reverse flow can be applied to the case of a sharp focusing of a laser beam with a second-order polarization singularity. It is also shown that if a spherical dielectric Rayleigh nanoparticle is placed in the backflow region, then a force directed in the opposite direction will act on it (the scattering force will be more than the gradient force).