# Flexible Construction of a Partially Coherent Optical Array
## 2 Methods  
We began with the construction of the partially coherent beam sources. In the space-frequency domain, partially coherent beams can be characterized by the cross-spectral density (CSD) function of the electric field, which is defined as [1]:  
\[W_{o}(\mathbf{r}_{1},\mathbf{r}_{2})=\int\mathrm{I}(\mathbf{v})H^{*}(\mathbf{r} _{1},\mathbf{v})H(\mathbf{r}_{2},\mathbf{v})d^{2}\mathbf{v} \tag{1}\]  
where \(\mathbf{r}_{1}\) and \(\mathbf{r}_{2}\) are two arbitrary spatial points at the source plane, with \"\(\bullet\)\" denoting the complex conjugate. \(\mathrm{I}(\mathbf{v})\) is a non-negative function. \(H(\mathbf{r},\mathbf{v})\) is an arbitrary kernel. Equation (1) can be rewritten as an alternative form to facilitate experimental generation [35]:  
\[W_{o}(\mathbf{r}_{1},\mathbf{r}_{2})=\int\int W_{i}(\mathbf{v}_{1},\mathbf{v} _{2})H^{*}(\mathbf{r}_{1},\mathbf{v}_{1})H(\mathbf{r}_{2},\mathbf{v}_{2})d^{2 }\mathbf{v}_{1}d^{2}\mathbf{v}_{2}, \tag{2}\]  
with  
\[W_{i}(\mathbf{v}_{1},\mathbf{v}_{2})=\sqrt{\mathrm{I}(\mathbf{v}_{1})\mathrm{ I}(\mathbf{v}_{2})}\delta(\mathbf{v}_{1}-\mathbf{v}_{2}), \tag{3}\]  
where \(W_{i}(\mathbf{v}_{1},\mathbf{v}_{2})\) and \(W_{o}(\mathbf{r}_{1},\mathbf{r}_{2})\) denote the CSD functions in the input and output planes, respectively. The terms \(\mathrm{I}(\mathbf{v})\), \(\delta(\mathbf{v}_{1}-\mathbf{v}_{2})\), and \(H(\mathbf{r},\mathbf{v})\) correspond to the intensity, the delta function, and the response function of the optical system, respectively. Based on Equation (2), one can achieve any partially coherent beams desired by flexibly editing the \(\mathrm{I}(\mathbf{v})\) or \(H(\mathbf{r},\mathbf{v})\) functions. A coherent lattice with its incoherent intensity distribution is expressed as [35]:  
\[\mathrm{I}(\mathbf{v})=C_{0}\sum_{m=1}^{M}c_{m}\exp\left[-\frac{(\mathbf{v}- \mathbf{v}_{0m})^{2}}{\omega_{m}^{2}/2}\right], \tag{4}\]  
where \(C_{0}\) is a constant factor, \(M\) is a non-negative integer denoting the number of Gaussian beam spots, \(c_{m}\) is the weight coefficient, which is used to adjust amplitude distribution, \(\omega_{m}\) is the beam waist size of the Gaussian beam spot, and \(\mathbf{v}_{0m}=\left(v_{0 mx},v_{0 my}\right)\) is the off-axis displacement of the \(m\)th spot. Additionally, by manipulating the weight coefficient \(c_{m}\), the beam waist size \(\omega_{m}\), and the off-axis displacement \(\mathbf{v}_{0m}\), arrays can be flexibly implemented through programming. Spiral coherent lattices (SCLs) can be obtained from the incoherent intensity distribution, which is shaped as an Archimedes spiral distribution, with its off-axis displacement \(\mathbf{v}_{0m}\) expressed as:  
\[\mathrm{v}_{0 mx}=a_{m}\cos(t_{m}),\mathrm{v}_{0 my}=a_{m}\sin(t_{m}), \tag{5}\]  
where \(t_{m}=2\pi(m-1)/(M-1)\) and \(a_{m}=b(m-1)/(M-1)\) denote the rotation angle and the distance from the center, respectively, with \(b\) as a constant that is set as 0.48 mm in the following. The response function of an optical system \(H(\mathbf{r},\mathbf{v})\) is a traditional Fourier transform system, which is expressed as:  
\[H(\mathbf{r},\mathbf{v})=-\frac{i}{\lambda f}T(\mathbf{r})\exp\left[\frac{i \pi}{\lambda f}\left(\mathbf{v}^{2}-2\mathbf{r}\cdot\mathbf{v}\right)\right], \tag{6}\]  
where \(\lambda\) represents the wavelength, \(f\) is the focal length of the Fourier transform lens, and \(T(\mathbf{r})=\exp\left(-\mathbf{r}^{2}/2\sigma_{0}^{2}\right)\) is the amplitude function. By substituting Equations (3), (4), and (6) into Equation (2), we can find the CSD function in the source plane:  
\[W(\mathbf{r}_{1},\mathbf{r}_{2})=\exp\left(-\frac{\mathbf{r}_{1}^{2}+\mathbf{ r}_{2}^{2}}{2\sigma_{0}^{2}}\right)\mu(\mathbf{r}_{1},\mathbf{r}_{2}), \tag{7}\]  
where \(\sigma_{0}\) denotes the transverse beam waist size and is taken as \(\sigma_{0}=1.5\) mm in the following simulations.