# Spin-Orbital Transformation in a Tight Focus of an Optical Vortex with Circular Polarization
## 2 The Denseness of Lengthwise Projections of the SAM and OAM  
The Jones vector for the electric and magnetic fields of the source beam have the form:  
\[\overrightarrow{E}(\psi,\vartheta)=\frac{A(\psi)}{\sqrt{2}}\exp(im\theta) \left( \begin{array}{c}1\\ i\end{array} \right),\ \ \overrightarrow{H}(\psi,\vartheta)=\frac{A(\psi)}{\sqrt{2}}\exp(im \phi)\left( \begin{array}{c}-i\\ 1\end{array} \right), \tag{1}\]  
where \(A(\psi)\) is an amplitude of the radially symmetric launch field, (\(\rho\),\(\vartheta\)) are polar coordinates in the cross-section of the light beam, \(\rho=\)\(f\)cos\(\psi,f\) is a focal length of a spherical lens. Figure 1 shows the optical scheme, which is studied in this work. The light beam from the laser acquires a linear polarization after the polarizer P\({}_{1}\) and enters the light modulator SLM. The modulator has a transmission in the form of a diffraction grating with a fork. The number of teeth at the fork is equal to the topological charge of the optical vortex, which is formed after SLM. After a quarter wave plate, the optical vortex acquires circular polarization. Aperture D cuts out the working +1 order of diffraction that is formed by the grating. A microobjective focuses light onto a CCD-camera. Thus, the diagram in Figure 1 makes it possible to register the distribution of light intensity in the focus of the light field (1). In Figure 1, for definiteness, a wavelength of 633 nm is used. Simulation results will be also given for this wavelength, but the effect of spin-orbital transformation takes place for any wavelength. The amplitudes in the proposed launch field (1) are given for right-hand circular polarization. At the source, the lengthwise projection of the SAM \(S_{z}=2\text{Im}\big{(}E_{x}^{*}E_{y}\big{)}\) of the field (1) is equal to \(S_{z,0}=A^{2}(\psi)\), and the total spin in the source plane is equal to the total energy (power) of the beam \(W\):  
\[\hat{S}_{z,0}=2\pi\int\limits_{0}^{\infty}\int\limits_{0}^{2\pi}A^{2}(\psi) \rho d\rho=W \tag{2}\]  
In reference [34], using the Richards-Wolf formalism [35], formulas for the intensity of the lengthwise component of the SAM in the focal plane were derived. The distributions of the intensity and the SAM lengthwise component of the focused field (1) will have the form:  
\[I(\rho,\vartheta)=\left|E_{x}\right|^{2}+\left|E_{y}\right|^{2}+\left|E_{z} \right|^{2}=\gamma_{0,m}^{2}+\gamma_{2,m+2}^{2}+2\gamma_{1,m+1}^{2}, \tag{3}\]  
\[S_{z}=2\text{Im}\big{(}E_{x}^{*}E_{y}\big{)}=\gamma_{0,m}^{2}-\gamma_{2,m+2}^ {2}. \tag{4}\]  
Formulas (3) and (4) include functions \(\Upsilon_{\xi,\chi}\) which rely only on the radial variable \(\rho\):  
\[\gamma_{\xi,\chi}=2 kf\int\limits_{0}^{\beta}\sin^{\xi+1}\left(\frac{\psi}{2} \right)\cos^{3-\xi}\left(\frac{\psi}{2}\right)\cos^{1/2}(\psi)A(\psi)e^{ikz \cos\psi}J_{\chi}(k\rho\sin\psi)d\psi, \tag{5}\]  
where \(k=2\pi/\lambda\) is the wavenumber of light with the wavelength of \(\lambda\);\(f\) is the focal length; \(\beta\) is the maximum inclination angle of the rays to the optical axis, which determines the numerical aperture of the aplanatic lens \(NA=\sin\beta\); \(J_{\chi}(k\rho\sin\psi)\) is the Bessel function of the first kind of \(\chi\) -th order. In Expression (5) and below, the indices \(\chi\) and \(\xi\) can get the next values: \(\xi=0\), \(1\), \(2\); \(\chi=m-2\), \(m-1\), \(m+1\), \(m+2\). By virtue of the circular polarization of the launch field in the focus, both the intensity (3) and the spin denseness (4) distributions have circular symmetry.