Angular momentum-carrying radially-polarised twisted light

It is well-known that the fields of pure radially-polarised optical vortex modes do not possess the usual phase function eiℓϕ of typical twisted light and so they do not carry angular momentum. However, recent work has confirmed that radially-polarised modes with helical wave fronts have been created in the laboratory by several groups and they can now be readily generated using commercially available devices. They are endowed with the phase function eiℓϕ for arbitrary values of the winding number ℓ. Crucially, the original theoretical treatments disregard the longitudinal electric and magnetic field components. We show here that the longitudinal components must be included, not just for the consistency of the treatment, but because only this can lead to angular momentum. We show that for such created modes the cycle-averaged angular momentum is αħℓ where α=(Pw02)/(ħc2) is a dimensionless constant, with P the power in the mode and w0 the mode width at focus.