with the transversality condition.

Energy flow approach
====================

Remarkably, the same results, Eq. (\[eqn:10\])–(\[eqn:14\]), can be derived from an approach based on the separation of the spin and orbital parts in the Poynting energy flow [@Berry2009; @Li]. Let us consider a monochromatic beam-like field propagating in the positive $z$-direction. Pecularities of the $(2+1)D$ formalism for such a problem are discussed in the Appendix, Eqs. (\[eqn:A6\])–(\[eqn:A9\]).

The transverse center of gravity (\[eqn:A9\]) obtained in the momentum representation from operator equation (\[eqn:12\]) can be equally derived from the traditional coordinate-representation definition $$\label{eqn:31}
\mathbf{R}_ \bot  \left( z \right) = \frac{1}{g^2} \int {\mathbf{r}_ \bot  \left| {\mathbf{E}\left( {\mathbf{r}_ \bot  ,z} \right)} \right|^2 \,} d^2 \mathbf{r}_ \bot~,$$ where ${\bf r}_{\bot}=(x,y)$, $g=\sqrt{2\omega/\varepsilon_0}$, and we used normalization $\int {  \left| {\mathbf{E}\left( {\mathbf{r}_ \bot  ,z} \right)} \right|^2 \,} d^2 \mathbf{r}_ \bot = g^2$ corresponding to $N=1$ (see Appendix). Substituting here Fourier representation (\[eqn:A6\]) with the helicity-basis expansion (\[eqn:A2\]), and using $\int {\mathbf{r}_ \bot  e^{i\mathbf{k}_ \bot   \cdot \mathbf{r}_ \bot  } d^2 \mathbf{r}_ \bot  }  = \left( {2\pi } \right)^2 \delta ^2 \left( {\mathbf{k}_ \bot  } \right)\partial _{\mathbf{k}_ \bot  }$ (${\bf k}_\bot =(k_x , k_y )$) together with expression (\[eqn:9\]) for the Berry connection, we arrive at Eq. (\[eqn:A9\]) which is 