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Relying on the time averaged Poynting vector expressed in terms of beam shape coefficients, dark and superdark theorems are discussed generalizing a previously published dark theorem. The superdark theorem allows one to characterize helical beams (that is to say beams possessing a topological charge) in terms of beam shape coefficients, and to demonstrate that beams with a topological charge equal to ± 2 are not necessarily vortex beams, although it is erroneously traditional to state that helical beams are all vortex beams. • Dark and superdark theorems are established in terms of beam shape coefficients of GLMT. • They allow one to decide on the properties of light intensities (null or not) on the axis of a beam. • And then to decide whether a beam with topological charge is a vortex beam or not. • We may then exhibit examples of beams with topological charges which are not vortex beams. [ABSTRACT FROM AUTHOR]

Previous works have been devoted to the symmetry properties of the Poynting vector for longitudinal axisymmetric beams expressed in terms of beam shape coefficients (BSCs). In the present paper, we examine the case of circularly axisymmetric beams. The former is defined as beams whose longitudinal component of the Poynting vector does not depend on the azimuthal angle. The latter is defined as beams whose all components of the Poynting vector do not depend on the azimuthal angle. We establish that two previously studied families of longitudinal axisymmetric beams are actually circularly axisymmetric beams as well. Examples are provided. • Symmetries of Poynting vector may be expressed in terms of beam shape coefficients of the GLMT. • Such symmetries are studied in the case of circularly symmetric beams. • Two families of such circularly symmetric beams are examined. • Examples and counter-examples are provided. [ABSTRACT FROM AUTHOR]