No general relation between phase vortices and orbital angular momentum

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 not equal 0, and ⟨l⟩ not equal 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.

Automated summary

Simple superpositions of Laguerre-Gauss beams highlight the difference between the component of orbital angular momentum in the propagation direction and the total topological charge. The examples demonstrate the contrasting situations where either the orbital angular momentum is zero but the total charge is not, or vice versa.