Orbital energy flow in the paraxial focus of a plane wave, bounded by a circular aperture.

Abstract

We investigate two cases of paraxial focusing of a plane wave-by a spherical lens with a circular aperture and by a cylindrical lens with a rectangular aperture. For these two cases, we obtain explicit analytical expressions for the longitudinal component of the canonical energy flow vector. These expressions indicate that the intensity in the focus has isolated nulls in longitudinal planes. Around these nulls, screw phase dislocations exist (longitudinal optical vortices) and the reverse energy flow arises. However, in contrast to nonparaxial light beams, the reverse energy flow is generated in a significantly subwavelength region. If the focusing lens has a numerical aperture of 0.1, this region has a transverse size of nearly 0.005 wavelengths, and the magnitude of the energy flow is nearly 10^-7 of the forward energy flow in the beam focus. Such a small reverse energy flow cannot be observed in an experiment and thus it can be neglected in practice.