# Transverse and Longitudinal Energy Flows in a Sharp Focus of Vortex and Cylindrical Vector Beams
## 4 Simulation
### Energy Flow near the Focus of a Cylindrical Vector Beam  
The simulation results are obtained using the finite difference time domain method of solving the Maxwell's equations (FDTD method). Simulation parameters are as follows: \(\lambda=633\) nm, simulation grid along all three axes is \(\lambda/30\), the incident field size is \(8\times 8\)\(\upmu\)m, the focal length is \(f=1.31\)\(\upmu\)m (NA = 0.95), the aperture of the incident field is \(8\)\(\upmu\)m (the radius is \(4\)\(\upmu\)m). Arrows in the figures below indicate the direction of the Poynting vector, and the arrow's length is proportional to the magnitude of it. The arrows are constructed utilizing projections of the Poynting vector \(P_{x}\), \(P_{y}\) in the transverse XY plane, and according to the projections \(P_{x}\), \(P_{z}\) in the longitudinal plane XZ. The background (halftones) is an intensity of the electric vector \(\mid\)F\(\mid\)\({}^{2}\) (all three projections of the light field are taken into account). Figure 6 shows the transverse Poynting vector distribution near the focus (at a distance of \(1\) wavelength from the focal plane) for the initial field:  
\[\mathbf{E}=\left( \begin{array}{c}\cos n\varphi\\ \sin n\varphi\end{array} \right)\exp(i(m\varphi-kl)),\;l=\sqrt{x^{2}+y^{2}+f^{2 }},\;k=2\pi/\lambda. \tag{30}\]  
Before the focus, the transverse energy flow is seen from Figure 6 to converge to the optical axis (\(m=0\)) (a). After the focus, the transverse energy flow diverges (b) and, almost at the focus itself, partly converges and partly diverges (c). Figure 6d shows the distribution of transverse energy flow in the focal plane: black color--the flow is directed towards the center, white color--from the center. The energy flow in the longitudinal XZ plane before the focus, after the focus and at the focus (e-g) has toroidal form when part of the energy propagates in the opposite direction. Shown in Figure 6g is an enlarged part of the picture in Figure 6e. Figure 6 confirms the prediction of the theory that part of the total initial energy flow \(W\) before and after the focus transforms into a transverse energy flow 2(\(W_{2}\) + \(W_{1}\)) (light converges at the focus and then diverges). And, at the focus itself, the light converges in some places of the focal plane, and, in some, diverges (c, d). On average, the transverse flow in the focal plane is zero. And part of the energy is directed in the backward direction to the optical axis (reverse or toroidal energy flows). The maximum value of the transverse flow is \(4\) a.u. before (Figure 6a) and after the focus (Figure 6b), and, at the focus, the maximum value of the transverse flow is almost \(10\) times less--\(0.5\) a.u. (Figure 6c), though the maximum value of the longitudinal energy flow is \(25\) a.u. (Figure 6f). Figure 5: The same as in Figure 3, but for \(m=-2\). Transverse planes are \(z=-\lambda\) (before the focus) (**a**), \(z=0\) (at the focus) (**b**) and \(z=+\lambda\) (behind the focus) (**c**). Figure 7 shows the longitudinal intensity and directions of energy flow in the XZ plane, as in Figure 6e, but for the initial field (30) with parameters \(m=0\), \(n=2\). The reverse energy flow is seen from Figure 7 to take place near the optical axis. And, in addition, horizontal arrows directed towards the optical axis and away from the optical axis are visible in the focal plane. Both the backward flow and the horizontal arrows shown in Figure 7 prove that not all energy of the initial beam (21), (30) crosses the focal plane. Part of the energy propagates along the surface itself (transverse flow: diverges and converges in different areas), and part in the backward direction to the optical axis. Figure 6: _Cont._  
Figure 6: Intensity distributions near the focus for the incident field (30) at \(m=0\), \(n=1\) in color. Arrows show the energy flow direction in the XY plane (**a–c**) at a distance of \(f-\lambda\) (**a**), \(f+\lambda\) (**b**), \(z=1.41\) μm (**c**) and in the XZ plane (**e**,**g**).