The angular spectrum decomposition of vector laser beams: polarization control of the electromagnetic field vectors

The angular spectrum representation as method to solve the electric vector Helmholtz equation in 3D is well known, in particular for beam propagation studies, where the field is predominantly directed along the optical axis. The inversion formulas then transform the electric field into its angular spectrum ( and vice versa). By using an intrinsic coordinate system instead of the usual 3D Cartesian system, we arrive at compact 2D inversion formulas as solutions. Furthermore, next to the 2D intrinsic version, a 2D Cartesian version of the 3D field vectors is derived, by considering the fields in the 2D sections transverse to the optical axis. It is shown that this polarization representation is rigorously correct and simultaneously describes 3D fields without paraxial approximation . The ability to switch between the field and its spectrum shows that studying polarization in the Fourier domain can be advantageous to studying it in the more familiar spatial domain.