Rigorous higher-order Poincaré optical vortex modes

Abstract

The state of polarization of a general form of an optical vortex mode is represented by the vector ϵ^ m , 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 (Θ P ,Φ P ) 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=ϵ^ m Ψ m , 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 Ψ m can be found by rigorously demanding that the product ϵ^ m Ψ m satisfies the vector paraxial equation. For a given order m this leads to a unique Ψ m , which has no azimuthal phase of the kind e i ℓ ϕ , and it is a solution of a scalar partial differential equation with ρ and z as 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⁡Θ P =0.