Angular momentum of non-paraxial light beam: Dependence of orbital angular momentum on polarization

TL;DR

This paper investigates how the orbital angular momentum of non-paraxial light beams depends on polarization and a new characteristic vector, revealing polarization effects and resolving the spin paradox in electromagnetic fields.

Contribution

It introduces a new expression for orbital angular momentum that depends on polarization and a novel vector, clarifying the polarization influence on non-paraxial beams.

Findings

Orbital angular momentum depends on polarization and a new vector I.

Spin angular momentum depends only on polarization.

Polarization influences spin-orbital conversion in focused beams.

Abstract

It is shown that the momentum density of free electromagnetic field splits into two parts. One has no contribution to the net momentum due to the transversality condition. The other yields all the momentum. The angular momentum that is associated with the former part is spin, and the angular momentum that is associated with the latter part is orbital angular momentum. Expressions for the spin and orbital angular momentum are given in terms of the electric vector in reciprocal space. The spin and orbital angular momentum defined this way are used to investigate the angular momentum of nonparaxial beams that are described in a recently published paper [Phys. Rev. A 78, 063831 (2008)]. It is found that the orbital angular momentum depends, apart from an $l$-dependent term, on two global quantities, the polarization represented by a generalized Jones vector and a new characteristic represented by a unit vector $\mathbf{I}$, though the spin depends only on the polarization. The polarization dependence of orbital angular momentum through the impact of $\mathbf{I}$ is obtained and discussed. Some applications of the result obtained here are also made. The fact that the spin originates from the momentum density that has no contribution to the net momentum is used to show that there does not exist the paradox on the spin of circularly polarized plane wave. The polarization dependence of both spin and orbital angular momentum is shown to be the origin of conversion from the spin of a paraxial Laguerre-Gaussian beam into the orbital angular momentum of the focused beam through a high numerical aperture.