# Topology and Polarization of Optical Vortex Fields from Atomic Phased Arrays
## 5 Numerical Results and Discussion  
Let us supplement the above results with numerical examples. Assume that the wavelength is the unit of length and the atoms are prepared in \(m_{z}=-1\) states. The atomic ring array has a radius of \(R=1000\lambda\). We evaluate the energy flux density,  
\[f(\vec{\rho},z)=\cos(\theta_{k})(|E|^{2}+|B|^{2})/4,\]  
and polarization parameters \(p_{z}\) and \(p_{zz}\) for several distances \(z=z_{R},1.5z_{R},2z_{R}\) from the array, where \(z_{R}=\frac{mv_{0}^{2}}{\lambda}=3.14\) is the Rayleigh range and \(w_{0}=R=1000\lambda\) is the beam's waist equal to the array radius. For each value of phase parameter \(l=1,2,3\), the results are analyzed for different numbers of radiating atoms, \(N=3,6,12\). As shown in Figures 2 and 3, the peaks of light energy flux are located in a transverse plane in the range of a few waist radii for \(z<2z_{R}\), that is in 1000 s of wavelengths. We can also see in Figure 3 that increasing the number of emitters \(N\) results in a transition from a lattice of peaks to a ring-like structure (described by three Bessel functions of different orders in the limit \(N\to\infty\)). For the scalar fields, this transition was previously demonstrated in Ref. [16]. The present paper fully considers a vector nature of the electromagnetic fields. In contrast to the spatial inhomogeneity of the flux, the spatial scales for the change in polarizations \(p_{z}\) and \(p_{zz}\) are about one wavelength around the phase singularity (Figure 4), and, quite remarkably, are independent of the propagation distance \(z\). As shown in Figure 5, the position dependence of the polarization parameters near the vortex center is affected by the phase parameter \(l\), which, for large \(N\), may be associated with a topological charge of the electromagnetic vortex. Note from the same figure that the vector polarization \(p_{z}\) turns to zero in the vortex center if \(l\) and \(m_{z}\) are equal in magnitude but opposite in sign. The effect on polarization of varying the number of emitters \(N\) can be seen in Figure 6. It should be noted that for \(l=2\) and \(N=3\), the leading-order vortex contribution has a topological charge \(n=-1\) (equal in sign to \(m_{z}\)), according to Equations (27) and (28), which results in a different behavior of polarizations (shown in Figure 6) as opposed to large values of \(N=6,12\). In other words, if the condition for the number of sources needed to generate a vortex vector field is not met, the topological charge changes its sign with respect to the applied phase parameter \(l\), the spin and OAM become aligned, and the longitudinal component of the field is suppressed. As evidence, \(p_{zz}=1\), which is dominated by the transverse field. At the same time, \(p_{z}\) is fully determined throughout the entire wave front by the sign of \(m_{z}\), which shows left- and right-circularly polarized light. Figure 3: Flux density of field Equation (26) at a distance \(z=2z_{R}\) from the array plotted against transverse position \(\rho\) for different numbers of sources (**a**) \(N=\)3, (**b**) \(N=\)6, (**c**) \(N=\)12, (**d**) \(N=\infty\). Here, \(l=1\) and \(m_{z}=-1\). Figure 2: (**a**) Flux density (arbitrary units) of field with \(N=12\), \(l=1\), and \(m_{z}=-1\) for three different locations of projection plane along \(z\)-axis \(z=z_{R}\) (red dotted), \(z=1.5z_{R}\) (blue dotted), and \(z=2z_{R}\) (black dashed). (**b**) Same plot as (**a**), but for different number of sources \(N=3,6,12,\infty\) (\(z=2z_{R}\)). Equation (26) against \(\rho_{x}\) on the \(x\)-axis. Figure 4: Polarization parameter (**a**) \(p_{z}\) and (**b**) \(p_{zz}\) against \(\rho_{x}\) on the \(x\)-axis in the vicinity of the vortex center (\(N=12,l=1,m_{z}=-1\)). The plots are obtained by substituting the expressions (26) into Equation (30).