How to calculate topological charge?

For a complex vector field in two dimensions with one or more phase singularity - a point where the field amplitude is zero and the phase is undefined - how do you explicitly calculate the total topological charge (specifically, winding number) of the field?

For a complex scalar field $F$, the winding number can be calculated using $$q = \frac{1}{2\pi} \oint_\Gamma (\nabla \arg(F)) \cdot d\textbf{l}$$ where $\Gamma$ is a closed curve around some singularity. You then sum the charges of all singularities to get the total. I'm wondering if there is an analogous expression for a complex vector field?

• What is your definition of topological charge? It's a vague term that is often given to any topologically invariant quantity. – ACuriousMind Oct 23 '14 at 18:41
• I haven't been able to find a good definition, but I see it in literature pretty often. Here's an example: journals.aps.org/pra/pdf/10.1103/PhysRevA.56.4064 – David Ding Oct 24 '14 at 3:53
• As I said, it is a name given to any topologically invariant quantity. Only given the equations your field obeys - or, equivalently, its Lagrangian - we could have a look at the possible solutions and see if there are topological invariants characterizing them. There's no general procedure to do topology, it is kind of an art form (as is all math). I believe what you are searching for is done in this paper. (Note: I cannot access the article you linked, so sorry if this is redundant) – ACuriousMind Oct 24 '14 at 13:00
• Thanks for the paper. I think I should be more specific. I'm trying to explicitly calculate the topological charge of an electromagnetic field carrying orbital angular momentum in a waveguide. I can't find any good sources for this. – David Ding Oct 25 '14 at 21:59