No general relation between phase vortices and orbital angular momentum

Source abstract

Abstract Simple superpositions of Laguerre–Gauss beams illustrate, counterintuitively, the difference between two quantities that are commonly conflated: the component of orbital angular momentum ⟨ l ⟩ in the propagation direction z , and the total topological charge S , which is the algebraic sum of the charges of vortices piercing any plane perpendicular to z . The examples illustrate two contrasting situations: ⟨ l ⟩ = 0, S ≠ 0, and ⟨ l ⟩ ≠ 0, S = 0. In the second situation, not only is the total charge zero but also there are no vortices in the infinite half-space beyond the beam waist plane z = 0.