# What is the relation between polarization and wave front?

I'm reading about the angular momentum of light, in which wave fronts are like helix. When I checked for a circular polarization on the web, shapes look very similar to these helical structures, So I'm curious to know is it possible for a wave to be linearly or circularly polarized but also have angular momentum. In another way, is there any relation between polarization and wave front or these are two different concepts?

• The linearly polarized wave cannot be simultaneously circuarly polarized. Circularly polarized carries a nonzero angular momentum, linearly carries 0 angular momentum (at least in quantum average). The helix is the simplest left-right-asymmetric picture so it appears in graphs of circular polarization as well as other places. – Luboš Motl Jun 2 '16 at 13:17
• @Luboš The OP seems to be referring to the orbital angular momentum of light, which does stem from helical wavefronts that stem from the spatial dependence of the beam, and which is independent of the polarization degree of freedom. – Emilio Pisanty Jun 2 '16 at 13:46

The distinction between the two types of angular momentum is that the amount of orbital angular momentum in a system depends on the coordinate origin (i.e. if you shift the origin by $\mathbf r_0$, the angular momentum of a massive particle $\mathbf L=\mathbf r\times\mathbf p$ will shift to $$\mathbf L=\mathbf r\times\mathbf p+\mathbf r_0\times\mathbf p,$$ to account for the angular momentum of the centre of mass), whereas spin angular momentum does not (i.e. the amount of angular momentum in the Earth's rotation is independent of where you observe it from). The angular momentum in the spatial and polarization degrees of freedom obey precisely that distinction.
It's also important to emphasize that both types of angular momentum are pretty much completely independent. Thus you can have a beam with orbital angular momentum and linear or circular polarization equally well, and you can even have superposition beams with more complicated properties such as a right-circular beam of $-1$ OAM plus a left-circular beam of $+1$ OAM, which will make a pretty complex spatial polarization dependence across the beam but which will nevertheless have quite definite angular momentum properties.