Vector wave solutions in electrodynamics: the Heaviside-Larmor symmetry and tiered potential invariance

The tiered symmetric structure of the electromagnetic equations, in particular the Heaviside Larmor (HL) symmetry, is used to obtain vector wave solutions of the wave equation obtained from Maxwell's equations. In this HL vector solutions procedure (HL-VS), the seed vector need not be constant but must satisfy an inhomogeneous Helmholtz equation. This seed can be used either to obtain the polarization basis in a given coordinate system or a particular polarization state. The spherical vector waves (SVW) basis is economically derived retaining the physical meaning throughout the procedure. Examples of low order modes are expounded. The cylindrical vector wave basis is also derived and particular polarization states are compared with previous results. Plane waves are used to elucidate the HL symmetry terms and their controlled manipulation in order to tailor specific structured light states. The angular momentum of plane waves with elliptical polarization is obtained. The controversy as to whether circularly polarized plane waves carry angular momentum is answered in the affirmative.