Rigorous higher-order Poincaré optical vortex modes

The state of polarization of a general form of an optical vortex mode is represented by the vector ε̂m, which is associated with a vector light mode of order m > 0. It is formed as a linear combination of two product terms involving the phase functions e±imφ times the optical spin unit vectors σ∓. Any such state of polarization corresponds to a unique point (ΘP, ΦP) on the surface of the order m unit Poincaré sphere. However, albeit a key property, the general form of the vector potential in the Lorenz gauge A = ε̂mψm, from which the fields are derived, including the longitudinal fields, has neither been considered nor has had its consequences been explored. Here, we show that the spatial dependence of ψm can be found by rigorously demanding that the product ε̂mψm satisfies the vector paraxial equation. For a given order m this leads to a unique ψm, which has no azimuthal phase of the kind eiℓφ, and it is a solution of a scalar partial differential equation with ρ and zas the only variables. The theory is employed to evaluate the angular momentum for a general Poincaré mode of order m yielding the angular momentum for right- and left- circularly polarized, elliptically polarized, linearly polarized and radially and azimuthally polarized higher-order modes. We find that in applications involving Laguerre–Gaussian modes, only the modes of order m ≥ 2 have non-zero angular momentum. All modes have zero angular momentum for points on the equatorial circle for which cos ΘP = 0.