# Topology and Polarization of Optical Vortex Fields from Atomic Phased Arrays
## 2 Radiation of an Atomic Phased Array with Many Elements
\begin{split}&\sum_{j=0}^{N-1}e^{i(l+m_{z})\phi_{j}}e^{i\tilde{ \jmath}|\vec{r}-\vec{r}_{j}|}\\ &=e^{i\tilde{\jmath}(z+\frac{R^{2}}{z})}\sum_{j=0}^{N-1}e^{i(l+m _{z})\phi_{j}}e^{-i\kappa\rho\cos(\phi_{\rho}-\phi_{j})}\\ &=Ne^{i\tilde{\jmath}(z+\frac{R^{2}}{z})}\sum_{j=0}^{N-1}\sum_{n =-\infty}^{n=\infty}(-i)^{n}J_{n}(\kappa\rho)e^{in(\phi_{\rho}-\phi_{j})}e^{i( l+m_{z})\phi_{j}}.\end{split}  \tag{23}\]  
The above result corresponds to the \(\eta_{0}^{\mu}\) component of the vector field in Equation (22). For \(+1\) and \(-1\) spherical-vector components (i.e., the factors in front of \(\eta_{+1}^{\mu}\) and \(\eta_{-1}^{\mu}\) in Equation (22)), we have  
\[Ne^{i\tilde{\jmath}(z+\frac{R^{2}}{z})}\sum_{j=0}^{N-1}\sum_{n=-\infty}^{n= \infty}(-i)^{n}J_{n}(\kappa\rho)e^{in(\phi_{\rho}-\phi_{j})}e^{i(l+m_{z}-1) \phi_{j}} \tag{24}\]  
and  
\[Ne^{i\tilde{\jmath}(z+\frac{R^{2}}{z})}\sum_{j=0}^{N-1}\sum_{n=-\infty}^{n= \infty}(-i)^{n}J_{n}(\kappa\rho)e^{in(\phi_{\rho}-\phi_{j})}e^{i(l+m_{z}+1) \phi_{j}}. \tag{25}\]Summation over \(j\) selects certain orders of Bessel vortices \(n\), while other order terms are canceled, as was demonstrated for scalar fields in [16]. After the summation of radiation from individual atoms, the final result for the field is:  
\[ \begin{split}& A_{m_{z}}^{u}(\vec{r},t)=\frac{A}{16\pi}\frac{ \omega_{0}^{2}e^{2}}{e_{0}c^{2}}\frac{1}{z}e^{-i\omega t}e^{ik(z+\frac{R^{2}}{ \pm})}\\ &[(1+m_{z}\cos^{2}\theta_{k})C_{+1}\eta_{+1}^{u}+(1-m_{z}\cos^{ 2}\theta_{k})C_{-1}\eta_{-1}^{u}\\ &+\frac{1}{\sqrt{2}}\sin 2\theta_{k}C_{0}\eta_{0}^{u}],\end{split}  \tag{26}\]  
where the coefficient for the \(\eta_{0}^{u}\) field component is  
\[C_{0}=\sum_{m=-\infty}^{m=\infty}[(-i)^{u}J_{n}(\kappa\rho)e^{in\Phi_{P}}]_{n= l+m_{z}+mN}, \tag{27}\]  
and for the \(+1\) and \(-1\) field components, we obtain  
\[C_{\pm 1}=\sum_{m=-\infty}^{m=\infty}[(-i)^{u}J_{n}(\kappa\rho)e^{in\Phi_{P}}]_{ n=l+m_{z}\mp 1+mN}, \tag{28}\]  
where \(m\) is an integer that also serves as a summation index: \(m=0,\pm 1,\pm 2,\ldots\). The above Equations (26)-(28) represent a result for the total vector potential of a phased atomic array with \(N\) atoms and a phase parameter \(l\) expressed in terms of an infinite series of Bessel vortices. The quantity \(m=0\) corresponds to the leading order terms \(n=l+m_{z}\pm 1\) and \(n=l+m_{z}\) of each of the three spherical components of the vector vortex field. In general, we have an infinite series of Bessel vortices contributing. However, a closer look reveals a crucial role of the number of emitters \(N\): the next non-vanishing order corresponds to the addition/subtraction of an integer number of \(N\), so the larger \(N\) is, the wider the \"gap\" is between the orders of contributing Bessel vortices. The result also sets a requirement for the minimal number of atoms necessary to form a vortex. To generate a vortex of order \(l\) for each field component, we find that the necessary conditions \(N\geq 2(l+m_{z})-1\) for the positive-helicity component, \(N\geq 2(l+m_{z})+3\) for the negative-helicity component, and \(N\geq 2(l+m_{z})+1\) for the zeroth (or longitudinal) component are satisfied. The strongest of the above conditions is \(N\geq 2(l+m_{z})+3\). If it is satisfied, then all three components of the field have vortex structures (i.e., show a phase singularity). When \(m_{z}=-1\), the condition becomes \(N\geq 2l+1\), which is the condition for a scalar vortex field [16]. However, for \(m_{z}=+1\), the conditions are different. The difference is due to the different azimuthal variations in the phase \(e^{i(l+m_{z}-1)\Phi_{P}}\), \(e^{i(l+m_{z}+1)\Phi_{P}}\) and \(e^{i(l+m_{z})\Phi_{P}}\) in the three spherical components of the vector vortex field. If we only keep the leading-order terms in Equations (27) and (28), the vortex field exactly matches the continuous (\(N\rightarrow\infty\)) limit given by Equation (15). Let us discuss the role of spin in our approach. For a ray of twisted light, the helicity \(\Lambda=\pm 1\) plays the role of spin angular momentum projection.