Mapping of focused Laguerre-Gauss beams: The interplay between spin and orbital angular momentum and its dependence on detector characteristics

Abstract

We show that propagating optical fields bearing an axial symmetry are not truly hollow in spite of a null electric field on axis. The result, obtained by general arguments based upon the vectorial nature of electromagnetic fields, is of particular significance in the situation of an extreme focusing, when the paraxial approximation no longer holds. The rapid spatial variations of fields with a ``complicated'' spatial structure are extensively analyzed in the general case and for a Laguerre-Gauss beam as well, notably for beams bearing a $|l|=2$ orbital angular momentum for which a magnetic field and a gradient of the electric field are present on axis. We thus analyze the behavior of an atomic size light detector, sensitive as well to quadrupole electric transitions and to magnetic dipole transitions, and apply it to the case of a Laguerre-Gauss beam. We detail how the mapping of such a beam depends on the nature and on the specific orientation of the detector. We show also that the interplay of mixing of polarization and topological charge, respectively associated to spin and orbital momentum when the paraxial approximation holds, modifies the apparent size of the beam in the focal plane. This even leads to a breaking of the cylindrical symmetry in the case of a linearly polarized transverse electric field.