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

摘要

It is shown that the momentum density of free electromagnetic field splitsinto two parts. One has no contribution to the net momentum due to thetransversality condition. The other yields all the momentum. The angularmomentum that is associated with the former part is spin, and the angularmomentum that is associated with the latter part is orbital angular momentum.Expressions for the spin and orbital angular momentum are given in terms of theelectric vector in reciprocal space. The spin and orbital angular momentumdefined this way are used to investigate the angular momentum of nonparaxialbeams 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 ageneralized Jones vector and a new characteristic represented by a unit vector$\mathbf{I}$, though the spin depends only on the polarization. Thepolarization dependence of orbital angular momentum through the impact of$\mathbf{I}$ is obtained and discussed. Some applications of the resultobtained here are also made. The fact that the spin originates from themomentum density that has no contribution to the net momentum is used to showthat there does not exist the paradox on the spin of circularly polarized planewave. The polarization dependence of both spin and orbital angular momentum isshown to be the origin of conversion from the spin of a paraxialLaguerre-Gaussian beam into the orbital angular momentum of the focused beamthrough a high numerical aperture.