# Topology and Polarization of Optical Vortex Fields from Atomic Phased Arrays
## 2 Radiation of an Atomic Phased Array with Many Elements
We notice that in the momentum space, an infinite number of states of light \(|\Lambda\rangle_{j}\) lying on a cone with an opening angle \(2\theta_{k}\) form a vortex state of light. Hence, with infinitely many atoms,  
\[A^{\mu}_{m_{z}}(\vec{r})=\sum_{j}e^{il\phi_{j}}\langle 0|\hat{A}^{\mu}|m_{z} \rangle_{j}. \tag{2}\]1  
the total field generated by the atomic phased ring array can be expressed by two vector vortices with a potential \(A_{\Lambda=\pm 1}^{\mu}\) with opposite helicities,  
\[A^{\mu}_{m_{z}}(\vec{r})=\sum_{j}e^{il\phi_{j}}\langle 0|\hat{A}^{\mu}|m_{z} \rangle_{j}. \tag{2}\]2  
where \(\alpha_{\pm}=\frac{1}{2}(1\pm m_{z}\cos\theta_{k})\). Note that \((\alpha_{-}^{2}+\alpha_{+}^{2})^{-1/2}=[(1+\cos(\theta_{k})^{2})/2]^{-1/2}\) has to be factored in for proper normalization of the field amplitude. A vector vortex field formed by a cone of light rays with helicity \(\Lambda\) can be expressed as [17; 18]  
\[ \begin{split} A^{\mu}_{\Lambda}=e^{-i(\omega t-k_{z}z)}\sqrt{\frac{ \kappa}{2\pi}}A\\ \int\frac{d\phi_{j}}{2\pi}(-i)^{m_{\gamma}}e^{im_{\gamma}\phi_{j} }e^{i\vec{\kappa}\cdot\vec{\rho}}\vec{\epsilon}_{\Lambda},\end{split}  \tag{9}\]  
\(m_{\gamma}=l+\Lambda\), where \(m_{\gamma}\) is the total angular momentum projection of the vortex photon, while \(l\) can be interpreted as the orbital angular momentum projection of the photon in a paraxial limit \(\theta_{k}\to 0\). The quantities \(\kappa\) and \(\rho\) are the transverse components of the wave vector and position vector in an observation plane, respectively; \(\vec{\kappa}_{j}\cdot\vec{\rho}=\vec{k}_{j}\cdot\vec{\rho}=\kappa\rho\cos( \phi_{j})\) and \(A\) is an overall normalization factor. Consider the term with \(\Lambda=+1\). The unit vectors are  
\[ \begin{split}\epsilon^{\mu}_{\Lambda=+1}=e^{-i\phi_{j}}\cos^{2} \frac{\theta_{k}}{2}\eta^{\mu}_{+1}+e^{i\phi_{j}}\sin^{2}\frac{\theta_{k}}{2} \eta^{\mu}_{-1}+\frac{1}{\sqrt{2}}\sin\theta_{k}\eta^{\mu}_{0},\end{split}  \tag{10}\]  
where  
\[ \begin{split}\eta^{\mu}_{\pm 1}=\frac{1}{\sqrt{2}}\big{(}0;& \mp 1,&-i,& 0\big{)}\\ \eta^{\mu}_{0}=\frac{1}{\sqrt{2}}\big{(}0;& 0,& 0,& 1 \big{)}\end{split}  \tag{11}\]  
are the spherical-basis vectors with a \(z\)-axis orthogonal to the array, as shown in Figure 1, and \(\mu\) denotes the four-dimensional Lorentz space-time indices, where \(+1\), \(-1\), and \(0\) stand for left-circular, right-circular, and \(z\) unit vectors, respectively. Hence, we have  
\[ \begin{split} A^{\mu}_{m_{z}}=e^{-i(\omega t-k_{z}z)}\sqrt{ \frac{\kappa}{2\pi}}A\int\frac{d\phi_{j}}{2\pi}e^{i\vec{\kappa}_{j}\cdot\vec{ \rho}}\\ [\frac{1}{2}(1+m_{z}\cos\theta_{k})e^{i(m_{z}-1)\phi_{j}}(-i)^{l +1}e^{i(l+1)\phi_{j}}\epsilon^{\mu}_{+1}\\ +\frac{1}{2}(1-m_{z}\cos\theta_{k})e^{i(m_{z}+1)\phi_{j}}(-i)^{l -1}e^{i(l-1)\phi_{j}}\epsilon^{\mu}_{-1}].\end{split}  \tag{12}\]  
Substituting expression (11) for basis vectors, we obtain  
\[ \begin{split} A^{\mu}_{m_{z}}=e^{-i(\omega t-k_{z}z)}\sqrt{ \frac{\kappa}{2\pi}}A\int\frac{d\phi_{j}}{2\pi}e^{i\vec{\kappa}\cdot\vec{\rho }}e^{i(l+m_{z})\phi_{j}}\\ [\frac{1}{2}(1+m_{z}\cos\theta_{k})(-i)^{l+1}\\ (e^{-i\phi_{j}}\cos^{2}\frac{\theta_{k}}{2}\eta^{\mu}_{1}+e^{i \phi_{j}})\sin^{2}\frac{\theta_{k}}{2}\eta^{\mu}_{-1}+\frac{1}{\sqrt{2}}\sin \theta_{k}\eta^{\mu}_{0})\\ +\frac{1}{2}(1-m_{z}\cos\theta_{k})(-i)^{l-1}\\ (e^{i\phi_{j}}\cos^{2}\frac{\theta_{k}}{2}\eta^{\mu}_{-1}+e^{-i \phi_{j}})\sin^{2}\frac{\theta_{k}}{2}\eta^{\mu}_{1}-\frac{1}{\sqrt{2}}\sin \theta_{k}\eta^{\mu}_{0})].\end{split}  \tag{13}\]  
The integral over \(\phi_{jAs a result, a vector potential for radiation from continuously distributed atoms (\(N\rightarrow\infty\)) around the ring array is  
\[