Optical vortex

Remarks

The integer $\ell$ does not depend on the choice of the small circle, by continuity of the winding number. A continuous field with a nonzero winding number around $p$ must vanish at $p$, which is why vortices are dark points. In space, the zeros of a field $U(x, y, z)$ generically form lines, called vortex lines.

Example: $U(x, y) = (x + iy)^\ell e^{-(x^2 + y^2)}$ with $\ell \ge 1$ equals $r^\ell e^{-r^2} e^{i\ell\varphi}$ in polar coordinates. Its only zero is the origin, and $\arg U = \ell\varphi$ increases by $2\pi\ell$ around it, so the origin is a vortex of charge $\ell$. The Laguerre–Gaussian mode $\mathrm{LG}_{p\ell}$ with $\ell \ne 0$ has a vortex of charge $\ell$ on its axis.

Non-example: $U(x, y) = x^2 + y^2$ vanishes at the origin, but it is real and positive elsewhere. Its phase is constant, so the winding number is $0$ and the origin is not a vortex.