# Topology and Polarization of Optical Vortex Fields from Atomic Phased Arrays
## 2 Radiation of an Atomic Phased Array with Many Elements  
Consider a circular array of \(N\) excited atoms, each carrying an angular momentum \(S\) and its projection \(m_{z}\). The array geometry is shown in Figure 1, along with the observation plane located at a distance \(z\) parallel to the atomic array. Let us denote a state vector of light emitted by the individual \(j\)-th atom located at a polar angle \(\phi_{j}\) on the ring as \(\left|\phi_{j},S,m_{z}\right\rangle\). Each adjacent atom radiates with a phase difference \(\Delta\phi=\frac{2\pi j}{N}\), where \(l\) is an externally controlled phase parameter. The phase difference could be achieved, depending on the specific application, through laser assistance [13] or through incoming OAM light modes. The state vector of a photon emitted by the \(j\)-th atom is  
\[\left|\phi_{j},S,m_{z}\right\rangle\to e^{il\phi_{j}}|m_{z}\rangle_{j}, \tag{1}\]  
where the position dependence for the individual source is indicated by the source number \(j\). Using the photon field operator \(\hat{A}^{\mu}\), the total field of the emitted light of an atomic phased ring array is  
\[A^{\mu}_{m_{z}}(\vec{r})=\sum_{j}e^{il\phi_{j}}\langle 0|\hat{A}^{\mu}|m_{z} \rangle_{j}. \tag{2}\]  
Figure 1: Phased atomic ring array with atoms polarized along \(z\)-axis in spin states \(\left|S,m_{z}\right\rangle\). Location in the parallel observation plane at a distance \(z\) is defined by the vector \(\vec{\rho}\). A vector \(\vec{r}_{j}\) defines position of the \(j\)-th atom, \(|\vec{r}_{j}|=R\). Shown schematically are spherical wave fronts of radiation from individual atoms and a helical wave front of the field resulting from their interference. For each atom, the polarization state of the emitted light can be expanded in the basis formed by left- and right-circularly polarized light in its propagation direction along \(\vec{r}-\vec{r}_{j}\),  
\[|m_{z}\rangle_{j}=\alpha_{+}^{(j)}|\Lambda=+1\rangle+\alpha_{-}^{(j)}|\Lambda=-1\rangle, \tag{3}\]  
where \(\Lambda=\pm 1\) defines the photon's helicity and the expansion coefficients are  
\[\alpha_{\pm}^{(j)}=\left\langle m_{z}^{(j)}\right|\Lambda=\pm 1\rangle. \tag{4}\]  
It should be noted that the observation of the light emitted by the atom is near a point on the \(z\)-axis aligned with the center of the ring array. The light propagating in the \(|\Lambda=\pm 1\rangle\) state in the above expansion has an angle \(\theta_{j}\) with respect to the z-axis. The angle is the same for all the atoms \(\theta_{j}\equiv\theta_{k}\). In order to choose the quantization along the z-axis, we have to rotate the coordinate system attached to the atom by a rotation operator \(R(-\phi_{j},\theta_{k},\phi_{j})\). Therefore, the coefficients become  
\[\alpha_{+}^{(j)}=\langle m_{z}|R(-\phi_{j},\theta_{k},\phi_{j})| \Lambda=+1\rangle \tag{5}\] \[\alpha_{-}^{(j)}=\langle m_{z}|R(-\phi_{j},\theta_{k},\phi_{j})| \Lambda=-1\rangle.\]  
The above expressions give Wigner \(d\)-matrices for the considered case of electric dipole emission (while generalization to higher multipoles is straightforward):  
\[\alpha_{+}^{(j)}=d_{1m_{z}}^{1}(\theta_{k})e^{i(m_{z}-1)\phi_{j}} \tag{6}\] \[\alpha_{-}^{(j)}=d_{-1m_{z}}^{1}(\theta_{k})e^{i(m_{z}+1)\phi_{j}},\]  
where \(d_{\pm 1m_{z}}^{1}(\theta_{k})=(1\pm m_{z}\cos\theta_{k})/2\). The total field of the emitted light of an atomic phased ring array can then be expressed in the basis formed by left- and right-circularly polarized light by the above coefficients. The angles \(\phi_{j}\) and \(\theta_{k}\) define the orientation of the wave vector \(\vec{k}_{j}\) of light emitted by the j-th atom in the spherical coordinate system. \[A_{m_{z}}^{\mu}=\sum_{j}e^{i\phi_{j}} \tag{7}\] \[A^{\mu}_{m_{z}}(\vec{r})=\sum_{j}e^{il\phi_{j}}\langle 0|\hat{A}^{\mu}|m_{z} \rangle_{j}. \tag{2}\]0  
If the number of atoms in the ring goes to infinity, it gives us the continuous case.