\mathbf{r}_ \bot  }~,~~ 
\mathbf{S} = \int {\mathbf{r} \times \bm{\pi }^{\rm s}  d^2 \mathbf{r}_ \bot  }~.$$ Substituting Eqs. (\[eqn:34\]) and (\[eqn:35\]) into Eqs. (\[eqn:37\]) and employing properties of Fourier integrals, we arrive at $$\label{eqn:38}
\mathbf{S} = i\int { ({\bf e}^{\sigma}\times {\bf e}^{\sigma *}) \left| {\tilde E^\sigma  } \right|^2 d^2 \mathbf{k}_ \bot  }
= \int {\sigma \bm{\kappa }\left| {\tilde E^\sigma  } \right|^2 d^2 \mathbf{k}_ \bot  }~,$$

$$\label{eqn:39}
\mathbf{L} = \int {\tilde E^{\sigma *} \mathbf{e}^{\sigma *}  \cdot \left( { - i\mathbf{k} \times \partial _\mathbf{k} } \right)\tilde E^\sigma  \mathbf{e}^\sigma  d^2 \mathbf{k}_ \bot  }  
   =  \int {\tilde E^{\sigma *} \left( -i{\mathbf{k} \times \partial _\mathbf{k} } \right)\tilde E^\sigma  d^2 \mathbf{k}_{\bot}  }  - \int { \sigma (\mathbf{A}_B  \times \mathbf{k}) \left| {\tilde E^\sigma  } \right|^2  d^2 \mathbf{k}_{\bot} },$$

where identity ${\bf e}^{\sigma *}\times {\bf e}^{\sigma}=i\sigma{\bm \kappa}$ and Eq. (\[eqn:9\]) were used. Clearly, the values of SAM and OAM, Eqs. (\[eqn:38\]) and (\[eqn:39\]), derived from the Poynting energy flows are in perfect agreement with our operator formalism, Eqs. (\[eqn:10\]) and (\[eqn:11\]) (see also Appendix).

![(Color online) (a) Bessel-beam distribution (\[eqn:11\]) on the sphere in ${\bf k}$-space with the azimuthal phase $2\pi\ell$. (b) Cylindrical caustic in the real space, an example of the closed orbit $\Gamma$ on it, and the 