Angular momentum dynamics of a paraxial beam in a uniaxial crystal

The conservation law governing the dynamics of the radiation angular momentum component along the optical axis (z axis) of a uniaxial crystal is derived from Maxwell's equations; the existence of this law is physically related to the rotational invariance of the crystal around the optical axis. Specializing the obtained general expression for the z component of the angular momentum flux to the case of a paraxial beam propagating along the optical axis, we find that the expression is the same as the corresponding one for a paraxial beam propagating in an isotropic medium of refractive index n(o) (ordinary refractive index of the crystal); besides, we show that the flux is conserved during propagation and that it decomposes into the sum of an intrinsic and an orbital contribution. Investigating their dynamics we demonstrate that they are coupled and, during propagation, an exchange between them exists. This exchange asymptotically exhibits a saturation process leading, for z--> infinity, the intrinsic part to vanish and the orbital one equates the total amount of angular momentum flux. As an example, the evolution of the intrinsic and the orbital contributions to the flux is investigated in the case of circularly polarized beams. Besides, the radiation angular momentum stored in the crystal is also investigated, in the paraxial regime, showing that it is simply given by the product of the total angular momentum flux by the time the radiation takes in passing through the crystal.