## 2 Projections of the Electric and Magnetic Field Strength Vectors at the Focus  
Let us consider that an initial light field with a Jones vector satisfies:  
\[\mathbf{E}=\frac{\exp(in\varphi)A(\theta)}{\sqrt{2(1+\alpha^{2})}}\left( \begin{array}{c}1+\alpha\\ i(1-\alpha)\end{array} \right)=\frac{\exp(in\varphi)A(\theta)}{\sqrt{(1+\alpha^ {2})}}\bigg{[}\frac{1}{\sqrt{2}}\left( \begin{array}{c}1\\ i\end{array} \right)+\frac{\alpha}{\sqrt{2}}\left( \begin{array}{c}1\\ -i\end{array} \right)\bigg{]}. \tag{1}\]  
Here (\(r\), \(\varphi\)) are the polar coordinates in the beam section (\(r=f\sin\!\theta\),\(f\) is the focal length of a spherical lens), \(n\) is an integer topological charge of the optical vortex, \(A(\theta)\) is the real function, the amplitude of the radially symmetric initial light field (1). The radial coordinate \(r\) on the spherical front of the converging wave is related to the angle \(\theta\) between the optical axis and the segment connecting the center of the focus with a point on the spherical front, and \(0<\alpha<1\) is a real number that regulates the contribution to the superposition of light with left- and right-handed circular polarization. It can be seen from (1) that this light field is an axial superposition of two optical vortices with identical topological charges and left- and right-handed circular polarizations. At \(\alpha=0\) the beam (1) has right-handed circular polarization, and at \(\alpha=1\) the polarization is linear. The beam (1) can be considered as a generalized Poincare beam [21]:  
\[ \begin{array}{l}\mathbf{E}_{p}=a\frac{\exp(ip\varphi)}{\sqrt{2}}\bigg{(} \begin{array}{c}1\\ i\end{array}\bigg{)}+b\frac{\exp(iq\varphi)}{\sqrt{2}}\bigg{(}\begin{array}{c }1\\ -i\end{array}\bigg{)},\\ a^{2}+b^{2}=1,\end{array}  \tag{2}\]at \(p=q=n\) and \(a=\left(1+\alpha^{2}\right)^{-1/2}\), \(b=\alpha\left(1+\alpha^{2}\right)^{-1/2}\). In practice, two light modulators are required to form a field (2) with different topological charges \(p\) and \(q\). Therefore, we restrict our consideration to field (1); when \(p=q\), this beam (1) can be formed by only one light modulator. Using the theory [34], we can write components of the electric and magnetic fields vectors in the sharp focus of the beam (1) as follows:  
\[ \begin{array}{l}E_{x}=\frac{\mu^{n-1}\exp(inq)}{\sqrt{2(1+\alpha^{2})}} \big{(}(1+\alpha)I_{0,n}+\alpha e^{-2i\varphi}I_{2,n-2}+e^{2i\varphi}I_{2,n+2} \big{)},\\ E_{y}=\frac{\mu^{n}\exp(inq)}{\sqrt{2(1+\alpha^{2})}}\big{(}(1-\alpha)I_{0,n} +\alpha e^{-2i\varphi}I_{2,n-2}-e^{2i\varphi}I_{2,n+2}\big{)},\\ E_{z}=\frac{\sqrt{2}\mu^{n}\exp(inq)}{\sqrt{(1+\alpha^{2})}}\big{(}\alpha e^{- i\varphi}I_{1,n-1}-e^{i\varphi}I_{1,n+1}\big{)},\\ H_{x}=-\frac{\mu^{n}\exp(inq)}{\sqrt{2(1+\alpha^{2})}}\big{(}(1-\alpha)I_{0,n} -\alpha e^{-2i\varphi}I_{2,n-2}+e^{2i\varphi}I_{2,n+2}\big{)},\\ H_{y}=\frac{\mu^{n-1}\exp(inq)}{\sqrt{2(1+\alpha^{2})}}\big{(}(1+\alpha)I_{0,n} -\alpha e^{-2i\varphi}I_{2,n-2}-e^{2i\varphi}I_{2,n+2}\big{)},\\ H_{z}=\frac{\sqrt{2}\mu^{n+1}\exp(inq)}{\sqrt{(1+\alpha^{2})}}\big{(}\alpha e^{ -i\varphi}I_{1,n-1}+e^{i\varphi}I_{1,n+1}\big{)}.\end{array}  \tag{3}\]  
Functions \(I_{\
u,\mu}\) in (3) are defined by the following:  
\[I_{\
u,\mu}=2 kf\int\limits_{0}^{\theta_{0}}\sin^{\
u+1}\bigg{(}\frac{\theta}{ 2}\bigg{)}\cos^{3-\
u}\bigg{(}\frac{\theta}{2}\bigg{)}\cos^{1/2}(\theta)A( \theta)e^{ikz\cos\theta}J_{\mu}(kr\sin\theta)d\theta, \tag{4}\]  
where \(k=2\pi/\lambda\) is the wavenumber of the monochromatic light with the wavelength \(\lambda,f\) is a focal distance of the lens, \(\theta_{0}\) is the maximum tilt angle of the rays to the optical axis, which determines the numerical aperture of the aplanatic lens NA = sin \(\theta_{0}\), \(I_{\mu}(x)\) is the Bessel function of the first kind and \(\mu\)-th order, \(z\) is a longitudinal coordinate, and the focal plane is situated at \(z=0\). Function (4) depends on the radial and longitudinal coordinates \(I_{\
u,\mu}(r,z)\).