# Transverse and Longitudinal Energy Flows in a Sharp Focus of Vortex and Cylindrical Vector Beams
## 2 Spin and Energy Flows in the Focal Plane for a Circularly Polarized Optical Vortex  
Following the works [36; 37] for the convenience, we shortly present expressions for the characteristics of the light beam in the focal plane. In the incident plane, the Jones vector of a vortex beam with right-handed circular polarization can be written as follows:  
\[\mathbf{F}(\theta,\varphi)=\frac{B(\theta)}{\sqrt{2}}\exp(im\varphi)\left( \begin{array}{c}1\\ i\end{array} \right) \tag{1}\]  
Here, \(\varphi\) is an azimuthal angle in the plane of an exit pupil of an aplanatic optical system. \(\theta\) is a polar angle, which equals the inclination of light rays to the optical axis, \(B(\theta)\) is a real function and is an initial beam amplitude, radially symmetrical about the optical axis, and \(m\) is an integer topological charge of an optical vortex. Using the Richards-Wolf formalism [1], expressions for all projections of the energy flux vector in the focal plane can be found. In general, the Poynting vector \(\mathbf{P}\) is determined by the well-known expression \(\mathbf{P}=c\mathrm{Re}(\mathbf{E}^{*}\times\mathbf{H})/(2\pi)\), where \(\mathbf{H}\) and \(\mathbf{E}\) are magnetic and electric field vectors, signs \"\(\star\)\" and \"\(\times\)\" mean complex conjugate and vector multiplication, \(\mathrm{Re}\) is a real part of a complex number and \(c\) is the vacuum light speed (we will omit later the constant \(c/(2\pi)\)). Projections of the energy flow vector at the focus can be written as follows [36; 37]:  
\[ \begin{array}{l}P_{x}=-\Omega(r)\sin\varphi,\\ P_{y}=\Omega(r)\cos\varphi,\\ P_{z}=J_{0,m}^{2}-J_{2,m+2}^{2},\\ \Omega(r)=J_{1,m+1}(J_{0,m}+J_{2,m+2}).\end{array}  \tag{2}\]  
Functions \(J_{\alpha,\beta}\) in (2) denote the following integrals, where the first index is the type of the integral (\(\alpha=0,1,2\)), and the second index (\(\beta=0,\,\pm 1,\,\pm 2,\,\ldots\)) equals the Bessel function order \(J_{\beta}(\xi)\) under the integral:  
\[J_{\alpha,\beta}=\left(\frac{4\pi f}{\lambda}\right)\int\limits_{0}^{\theta_{0} }\sin^{\alpha+1}\left(\frac{\theta}{2}\right)\cos^{3-\alpha}\left(\frac{\theta }{2}\right)\cos^{1/2}(\theta)B(\theta)e^{ik\mathrm{z}\cos\theta}J_{\beta}(\xi )d\theta, \tag{3}\]  
where \(k=2\pi/\lambda\) is the wave number, \(\lambda\) is the light wavelength, (\(r,\,\varphi,\,z\)) are cylindrical coordinates in focus, \(f\) is a focal distance of the aplanatic system and \(\theta_{0}\) is the maximal inclination angle of rays to the optical axis, which defines a numerical aperture of the aplanatic lens. The numerical aperture is \(NA=\sin(\theta_{0})\), \(\zeta=kr\) sin(\(\theta\)). Similarly to (2), expressions for the projections of the spin angular momentum (SAM) vector in the focal plane for the initial field (1) can be obtained. The SAM vector is determined by the well-known formula \(\mathbf{S}=\text{Im}(\mathbf{E}^{*}\times\mathbf{E})/(8\pi\omega)\), where Im is an imaginary part of a complex number, and \(\omega\) is an angular frequency of light. Next, we will omit the constant \(1/(8\pi\omega)\). In this case, expressions for SAM vector components at the focus of the initial field (1) are completely similar to the Poynting vector components (2):  
\[ \begin{array}{l}S_{x}=-\Omega(r)\sin\varphi,\\ S_{y}=\Omega(r)\cos\varphi,\\ S_{z}=J_{0,m}^{2}-J_{2,m+2}^{2},\\ \Omega(r)=J_{1,m+1}(J_{0,m}+J_{2,m+2}).\end{array}  \tag{4}\]  
For the orbital angular momentum (OAM) vector in the focal plane of the field (1), we find a formula only for the longitudinal projection [36]:  
\[ \begin{array}{l}L_{z} =\text{Im}\left(F_{x}^{z}\frac{\partial}{\partial\varphi}F_{x}+F_ {y}^{*}\frac{\partial}{\partial\varphi}F_{y}+F_{z}^{*}\frac{\partial}{\partial \varphi}F_{z}\right)\\ =mJ_{0,n}^{2}+(m+2)J_{2,n+2}^{2}+2(m+1)J_{1,n+1}^{2}.\end{array}  \tag{5}\]  
In (5), \(F_{x}\), \(F_{y}\) and \(F_{z}\) are projection of the electric field vector \(\mathbf{F}\) in the focus of the field (1).