Representation of total angular momentum states of beams through a four-parameter notation

It has been confirmed beams carrying total angular momentums (TAMs) that consist of spin angular momentums (SAMs) and orbital angular momentums (OAMs) are widely used in classical and quantum optics. Here we propose and demonstrate a new kind of representation consisting of four real numbers to describe the TAM states of arbitrary beams. It is shown that any homogeneous polarization, scalar vortices and complex vectorial vortex field, all of which result from the TAMs of photons, can be well represented conveniently using our proposed four-parameter representation. Furthermore, the proposed representation can also reveal the internal change of TAMs as the conversion between SAMs and OAMs. The salient properties of the proposed representation is to give a universal form of TAMs associated with complicated polarizations and more exotic vectorial vortex beams, which offer an important basis for the future applications.