# Propagation of Optical Coherence Vortex Lattices in Turbulent Atmosphere
## 2 Formulation of the Propagation of OCLVs in Turbulent Atmosphere  
The second-order properties of a partially coherent beam in frequency domain can be studied by the cross-spectral density (CSD) function. To construct a bona fide field, the CSD function at source can be expanded as follows [45]:  
\[W(\mathbf{r}_{1},\mathbf{r}_{2};0)=\int p(\mathbf{v})H^{*}(\mathbf{r}_{1}, \mathbf{v})H(\mathbf{r}_{2},\mathbf{v})d^{2}\mathbf{v}, \tag{1}\]  
where \(\mathbf{r}_{1}=(x_{1},y_{1})\) and \(\mathbf{r}_{2}=(x_{2},y_{2})\) are the two arbitrary transverse position vectors at source plane, \(p(\mathbf{v})\) is a scalar nonnegative function, \(H(\mathbf{r},\mathbf{v})\) denotes an arbitrary suitable kernel function. To generate OCVLs, we define \(p(\mathbf{v})\) and \(H(\mathbf{r},\mathbf{v})\) as follows:  
\[p(\mathbf{v})=\frac{1}{M}\underset{m=-(M-1)/2}{\overset{(M-1)/2}{\sum}}\exp \left[-\frac{\left|\mathbf{v}-\mathbf{v}_{0m}\right|^{2}}{2w_{0}^{2}}\right], \tag{2}\]  
\[H(\mathbf{r},\mathbf{v})=-\frac{i}{\lambda f}\tau(\mathbf{r})\exp\left[\frac {i\pi}{\lambda f}\left(\mathbf{v}^{2}-2\mathbf{r}\cdot\mathbf{v}\right) \right]. \tag{3}\]  
where \(p(\mathbf{v})\) is superposed by the \(M\) space-shifted Gaussian functions, \(w_{0}\) and \(\mathbf{v}_{0m}~{}=~{}(md,md)\) denote the circular aperture radius and the position vector of the lattices, respectively, and \(d\) denotes the distance between adjacent spots. \(H(\mathbf{r},\mathbf{v})\) represents in the experiment the response function of an optical Fourier transform system consisting of a free-space distance of \(f\), a thin lens with focal distance \(f\), and an optical filter with transmission function \(\tau(\mathbf{r})\)[44]. Here we set \(\tau(\mathbf{r})\) in the cylindrical coordinates as follows:  
\[\tau(r,\theta)=\left(\frac{\sqrt{2}r}{\omega_{0}}\right)^{l}\exp\left(-\frac{ r^{2}}{\omega_{0}^{2}}\right)\exp(-il\theta), \tag{4}\]  
where \(l\) and \(\omega_{0}\) are the topological charge and beam width, respectively, \((r,\,\theta)\) are cylindrical polar coordinates. By substituting Equations (2)-(4) into Equation (1), the CSD of the OCVLs at source plane is obtained as follows:  
\[ \begin{array}{ll}W_{0}(r_{1},\theta_{1},r_{2},\theta_{2})&=\frac{2w_{0}^{2}\pi} {M\lambda^{2}f^{2}}\sum\limits_{m=-(M-1)/2}^{(M-1)/2}\left(\frac{2r_{1}r_{2}}{ \omega_{0}^{2}}\right)^{l}\exp\left(-\frac{r_{1}^{2}+r_{2}^{2}}{\omega_{0}^{2} }\right)\exp[il(\theta_{2}-\theta_{1})]\\ &\times\exp\left[-\frac{r_{1}^{2}+r_{2}^{2}-2r_{1}r_{2}\cos(\theta_{2}-\theta _{1})}{2\theta_{0}^{2}}\right]\\ &\times\exp\{-b[(r_{2}\cos\theta_{2}-r_{1}\cos\theta_{1}+r_{2}\sin\theta_{2}-r _{1}\sin\theta_{1})]\},\end{array}  \tag{5}\]  
where \(\delta_{0}=f/(w_{0}k)\) is the beam initial coherence width. For the convenience of calculation, we set \(a=1/2\delta_{0}^{2}=2w_{0}^{2}\pi^{2}/(\lambda^{2}f^{2})\), \(b=2i\pi md/(\lambda f)\). The spectral density of a propagating partially coherent beam in turbulent atmosphere can be obtained by using the convolution approach as follows [46]:  
\[S(\rho,z)=S_{f}(-\rho/z)\otimes p_{1}(-\rho/z)\otimes p_{2}(-\rho/z), \tag{6}\]  
where \(\rho\) is the position vector in the receiver plane with the propagation distance being \(z\). \(S_{f}(-\rho/z)\) is the spectral density of the incoherent portion on propagation in free space, \(p_{1}(-\rho/z)\) is the Fourier transform of the degree of coherence of the beam at source plane, \(p_{2}(-\rho/z)\) is the Fourier transform of second-order random phase correlation function, and \(\otimes\) is the symbol of convolution.