# Flexible Construction of a Partially Coherent Optical Array
## 2 Methods
\(\mu(\mathbf{r}_{1},\mathbf{r}_{2})\) denotes the complex degree of coherence (CDOC), which is calculatedfrom the Fourier transform of the intensity distribution of the incoherent source I(\(\mathbf{v}\)), with the CDOC of the SCLs being  
\[ \begin{array}{ll}\mu(\mathbf{r}_{1},\mathbf{r}_{2})&=\frac{1}{\lambda^{2}f^{2} }\int I(\mathbf{v})\exp\Bigl{[}-\frac{i2\pi\mathbf{v}\cdot(\mathbf{r}_{2}- \mathbf{r}_{1})}{\lambda f}\Bigr{]}d^{2}\mathbf{v}\\ &=C_{1}\sum\limits_{m=1}^{M}c_{m}\exp\biggl{\{}-\frac{(\mathbf{r}_{2}-\mathbf{ r}_{1})^{2}}{2\delta_{m}^{2}}-\frac{ik\alpha_{m}}{f}\bigl{[}\cos(t_{m})\cdot(x_{2}-x_ {1})+\sin(t_{m})\cdot(y_{2}-y_{1})\bigr{]}\biggr{\}},\end{array}  \tag{8}\]  
where \(C_{1}\) is a constant factor, \(k=2\pi/\lambda\) is the wavenumber, and \(\delta_{m}=\lambda f/\pi\omega_{m}\) denotes the transverse coherence width of the \(m\)th spot of the SCLs. The second-order statistical characteristics of SCL can be described by the two-point cross-spectral density function in the source plane:  
\[ \begin{array}{ll}W(\mathbf{r}_{1},\mathbf{r}_{2})&=C_{1}\exp\biggl{(}-\frac{ \mathbf{r}_{1}^{2}+\mathbf{r}_{2}^{2}}{2\delta_{0}^{2}}\biggr{)}\\ &\times\sum\limits_{m=1}^{M}c_{m}\exp\biggl{\{}-\frac{(\mathbf{r}_{2}-\mathbf{ r}_{1})^{2}}{2\delta_{m}^{2}}-\frac{ik\alpha_{m}}{f}\bigl{[}\cos(t_{m})\cdot(x_{2}-x_ {1})+\sin(t_{m})\cdot(y_{2}-y_{1})\bigr{]}\biggr{\}},\end{array}  \tag{9}\]  
where based on Equation (9), a uniform spiral coherent lattice (USCL) with \(\omega_{m}\equiv\omega_{0}\) and a non-uniform spiral coherent lattice (NUSCL) with \(\omega_{m}=\sqrt{m}\omega_{0}\) can be achieved by manipulating the parameter \(\omega_{m}\), with its transverse coherence width being \(\delta_{0}\) and \(\delta_{m}=\delta_{0}/\sqrt{m}\). The paraxial propagation of a coherent lattice is characterized by the generalized Huygens-Fresnel integral, and the CSD function of the beam after propagating through a stigmatic ABCD optical system can be calculated from [41; 52]:  
\[ \begin{array}{ll}W(\mathbf{\rho}_{1},\mathbf{\rho}_{2})&=\frac{1}{(\lambda B )^{2}}\exp\Bigl{[}-\frac{ikD}{2B}\bigl{(}\mathbf{\rho}_{1}^{2}-\mathbf{\rho}_ {2}^{2}\bigr{)}\Bigr{]}\int\int W(\mathbf{r}_{1},\mathbf{r}_{2})\exp\Bigl{[}- \frac{ikA}{2B}\bigl{(}\mathbf{r}_{1}^{2}-\mathbf{r}_{2}^{2}\bigr{)}\Bigr{]}\\ &\times\exp\Bigl{[}\frac{ik}{B}(\mathbf{r}_{1}\cdot\mathbf{\rho}_{1}-\mathbf{ r}_{2}\cdot\mathbf{\rho}_{2})\Bigr{]}d^{2}\mathbf{r}_{1}d^{2}\mathbf{r}_{2}, \end{array}  \tag{10}\]  
where \(\mathbf{\rho}_{1}\) and \(\mathbf{\rho}_{2}\) are two arbitrary transverse position vectors at the output plane. \(A\), \(B\), \(C\), and \(D\) are the transfer matrix elements of the optical system. The intensity and CDOC of the SCLs in the output plane are obtained as [53]:  
\[I(\mathbf{\rho})=W(\mathbf{\rho},\mathbf{\rho}), \tag{11}\]  
\[\mu(\mathbf{\rho}_{1},\mathbf{\rho}_{2})=\frac{W(\mathbf{\rho}_{1},\mathbf{ \rho}_{2})}{\sqrt{I(\mathbf{\rho}_{1})I(\mathbf{\rho}_{2})}}, \tag{12}\]  
Assuming that the SCLs pass through a thin lens, the propagation properties can be studied with the help of the transfer matrix of the optical system between the source plane and the output plane, which is written as:  
\[ \begin{array}{ll}\begin{pmatrix}A&B\\ C&D\end{pmatrix}=\begin{pmatrix}1&z\\ 0&1\end{pmatrix}\begin{pmatrix}1&0\\ -1/f_{1}&1\end{pmatrix}=\begin{pmatrix}1-z/f_{1}&z\\ -1/f_{1}&1\end{pmatrix}.\end{array}  \tag{13}\]  
where \(z\) is the distance from the focal lens to the output plane. In the experiment, we can calculate the square modulus of the CDOC using the following formula.