# Topology and Polarization of Optical Vortex Fields from Atomic Phased Arrays
## 2 Radiation of an Atomic Phased Array with Many Elements
\begin{split}& A_{m_{z}}^{\mu}=e^{-i(\omega t-kz)}\sqrt{\frac{ \kappa}{2\pi}}A\\ &(-\frac{i}{2}e^{i(l+m_{z}-1)\phi_{p}}J_{l+m_{z}-1}(\kappa\rho)(1 +m_{z}\cos^{2}\theta_{k})\eta_{1}^{\mu}\\ &+\frac{i}{2}e^{i(l+m_{z}+1)\phi_{p}}J_{l+m_{z}+1}(\kappa\rho)(1 -m_{z}\cos^{2}\theta_{k})\eta_{-1}^{\mu}\\ &+\frac{m_{z}}{2\sqrt{2}}e^{i(l+m_{z})\phi_{p}}J_{l+m_{z}}(\kappa \rho)\sin 2\theta_{k}\eta_{0}^{\mu}).\end{split}  \tag{15}\]  
**3. Radiation Field from a Finite Number of Atomic Sources**  
In the discrete case, recall the expression Equation (2). The light emitted by the \(j\)-th atom has the same angular distribution as the light emitted by an oscillating dipole [19]. \[ \begin{split}\langle 0|\hat{A}^{\mu}|m_{z}\rangle_{j}& =\frac{A_{N}}{16\pi}\frac{\omega_{0}^{2}e^{2}}{\epsilon_{0}c^{2}} \frac{1}{|\vec{r}-\vec{r}_{j}|}e^{ik|\vec{r}-\vec{r}_{j}|}e^{-i\omega t}\\ &\qquad[(1+m_{z}\cos^{2}\theta_{j})e^{i(m_{z}-1)\phi_{j}}\eta_{+1 }^{\mu}\\ &\qquad+(1-m_{z}\cos^{2}\theta_{j})e^{i(m_{z}+1)\phi_{j}}\eta_{-1 }^{\mu}\\ &\qquad\qquad+\frac{1}{\sqrt{2}}\sin 2\theta_{j}e^{im_{z}\phi_{j}} \eta_{0}^{\mu}],\end{split}  \tag{16}\]  
where the normalization constant is \(A_{N}=A/N\). The angles are  
\[ \begin{split}&\cos\phi_{j}^{\prime}=\frac{\rho\cos\phi_{p}-R \cos\phi_{j}}{|\vec{r}-\vec{r}_{j}|}\\ &\sin\phi_{j}^{\prime}=\frac{\rho\sin\phi_{p}-R\sin\phi_{j}}{| \vec{r}-\vec{r}_{j}|}\\ &\cos\theta_{j}^{\prime}=\frac{z}{|\vec{r}-\vec{r}_{j}|}\\ &\sin\theta_{j}^{\prime}=\frac{|\vec{r}-\vec{r}_{j}|}{|\vec{r}- \vec{r}_{j}|}.\end{split}  \tag{17}\]  
We are interested in large distances \(z\); therefore, we assume that \(\frac{1}{|\vec{r}-\vec{r}_{j}|}\approx\frac{1}{z}\). The field is  
\[ \begin{split}&\langle 0|\hat{A}^{\mu}|m_{z}\rangle_{j}=A\frac{ \omega_{0}^{2}e^{2}}{3\epsilon_{0}c^{2}}\frac{1}{z}e^{ik|\vec{r}-\vec{r}|}e^{- i\omega t}\\ &\qquad[\frac{3}{16\pi}(1+m_{z}\cos^{2}\theta_{j}^{\prime})e^{i(m_ {z}-1)\phi_{j}^{\prime}}\eta_{+1}^{\mu}\\ &+\frac{3}{16\pi}(1-m_{z}\cos^{2}\theta_{j}^{\prime})e^{i(m_{z}+1) \phi_{j}^{\prime}}\eta_{-1}^{\mu}\\ &\qquad\qquad+\frac{3}{16\sqrt{2}\pi}\sin 2\theta_{j}^{\prime}e^{im_{z} \phi_{j}^{\prime}}\eta_{0}^{\mu}].\end{split}  \tag{18}\]  
We also assume that \(z\gg R\gg\rho\), \(\frac{1}{|\vec{r}-\vec{r}_{j}|}\rightarrow\frac{1}{R}\). Then,  
\[ \begin{split}&\cos\phi_{j}^{\prime}=\frac{1}{R}(\rho\cos\phi_{p}-R \cos\phi_{j})\\ &\sin\phi_{j}^{\prime}=\frac{1}{R}(\rho\sin\phi_{p}-R\sin\phi_{j}) \end{split}  \tag{19}\]and  
\[e^{i2\Phi_{j}^{\prime}}=\frac{1}{R^{2}2}(\rho e^{i\Phi_{\rho}}-Re^{i\Phi_{j}})^{2} \approx e^{i2\Phi_{j}}; \tag{20}\]  
therefore, \(\Phi_{j}^{\prime}\approx\Phi_{j}\). Note that \(\theta_{k}\) is defined as \(\theta_{k}=\arctan(\frac{R}{z})\approx\frac{R}{z}\). Further expanding  
\[ \begin{split}\cos^{2}\theta_{j}^{\prime}&=\frac{z^ {2}}{z^{2}+\rho^{2}+R^{2}-2R\rho\cos(\phi_{\rho}-\phi_{j})}\approx 1-\theta_{k}^{2} \\ \sin^{2}\theta_{j}^{\prime}&=\frac{\rho^{2}+R^{2}-2 R\rho\cos(\phi_{\rho}-\phi_{j})}{z^{2}+\rho^{2}+R^{2}-2R\rho\cos(\phi_{\rho}-\phi_{j})} \approx\theta_{k}^{2}\end{split}  \tag{21}\]  
leads to approximately equal \(\theta_{j}^{\prime}\approx\theta_{k}\). The total vector field of the atomic ring array with a finite number of sources is  
\[ \begin{split} A_{m_{z}}^{\mu}(\vec{r},t)&=\frac{A_ {N}}{16\pi}\frac{\omega_{0}^{2}e^{2}}{\epsilon_{0}c^{2}}\frac{1}{z}e^{-i \omega t}\\ \sum_{j=0}^{N-1}&[(1+m_{z}\cos^{2}\theta_{k})e^{i( m_{z}-1)\phi_{j}}\eta_{+1}^{\mu}\\ &+(1-m_{z}\cos^{2}\theta_{k})e^{i(m_{z}+1)\phi_{j}}\eta_{-1}^{\mu }\\ &+\frac{1}{\sqrt{2}}\sin 2\theta_{k}e^{im_{z}\phi_{j}}\eta_{0}^{ \mu}]e^{i\Phi_{j}}e^{i\tilde{\jmath}|\vec{r}-\vec{r}_{j}|}.\end{split}  \tag{22}\]  
The sum over sources \(j\) can be evaluated using the Jacobi-Anger expansion:  
\[