Orbital angular momentum of light

Definition

Let a monochromatic beam of angular frequency $\omega$ move along the $z$ axis and obey the paraxial wave equation. Let $u(x, y)$ be its transverse complex amplitude in a plane $z = \text{const}$, smooth and not identically zero, with $u$ and $\partial_\varphi u$ square integrable. Here $\partial_\varphi = x \, \partial_y - y \, \partial_x$ is the derivative in the polar angle $\varphi$. The orbital angular momentum per photon of the beam, in units of the reduced Planck constant $\hbar$, is $$\bar\ell = \frac{\displaystyle\int_{\R^2} \overline{u} \, (-i \, \partial_\varphi u) \, dx \, dy}{\displaystyle\int_{\R^2} \abs{u}^2 \, dx \, dy}.$$ The orbital angular momentum of the beam about the $z$ axis is $\bar\ell / \omega$ times its energy, in the same section of beam.