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 cosΘ
P
=0.
Subject
Atomic and Molecular Physics, and Optics,Statistical and Nonlinear Physics