## 2 Materials and Methods
### Fourier and Statistical Optics Theory
\begin{split}&\langle U^{\mathrm{out}}\left(\mathbf{\rho}_{1}\right)U^{ \mathrm{out}*}\left(\mathbf{\rho}_{2}\right)\rangle=W^{\mathrm{out}}\left(\mathbf{\rho }_{1},\mathbf{\rho}_{2}\right)\\ &=\frac{1}{\left(\lambda^{2}f_{1}f_{2}\right)^{2}}\iiint_{- \infty}^{\infty}T\left(\mathbf{\rho}_{1}^{\prime}\right)T^{*}\left(\mathbf{\rho}_{2}^ {\prime}\right)\tilde{W}^{\mathrm{src}}\left(\frac{\mathbf{\rho}_{1}^{\prime}}{ \lambda f_{1}},\frac{\mathbf{\rho}_{2}^{\prime}}{\lambda f_{1}}\right)\exp\left[ -\mathrm{j}\frac{k}{f_{2}}\left(\mathbf{\rho}_{1}\cdot\mathbf{\rho}_{1}^{\prime}-\mathbf{ \rho}_{2}\cdot\mathbf{\rho}_{2}^{\prime}\right)\right]\mathrm{d}^{2}\rho_{1}^{ \prime}\mathrm{d}^{2}\rho_{2}^{\prime},\end{split}  \tag{9}\]  
where \(W^{\mathrm{out}}\) is the output CSD function [1; 5] and \(\tilde{W}^{\mathrm{src}}\) is the Fourier transform of the source CSD function. It is quite easy to show that an equivalent and more physical expression for \(W^{\mathrm{out}}\) is  
\[W^{\mathrm{out}}\left(\mathbf{\rho}_{1},\mathbf{\rho}_{2}\right)=\left[\frac{M}{\left( \lambda f_{2}\right)^{2}}\right]^{2}\iiint_{-\infty}^{\infty}W^{\mathrm{src}} \left(\mathbf{\rho}_{1}^{\prime},\mathbf{\rho}_{2}^{\prime}\right)\tilde{T}\left( \frac{\mathbf{\rho}_{1}-M\mathbf{\rho}_{1}^{\prime}}{\lambda f_{2}}\right)\tilde{T}^{*} \left(\frac{\mathbf{\rho}_{2}-M\mathbf{\rho}_{2}^{\prime}}{\lambda f_{2}}\right) \mathrm{d}^{2}\rho_{1}^{\prime}\mathrm{d}^{2}\rho_{2}^{\prime}, \tag{10}\]  
where \(M=-f_{2}/f_{1}\) is the magnification. This expression states that \(W^{\mathrm{out}}\) is the scaled convolution of \(W^{\mathrm{src}}\) with the Fourier transform of the CSD function of \(T\). In arriving at Equations (9) and (10), no assumptions have been made about the spatial statistics of \(U^{\mathrm{src}}\); thus, they are accurate for a source of any state of coherence. To make further progress, an analytical form for \(W^{\mathrm{src}}\) is needed. Because of its use in FSOC research, the physical source modeled here is the output of a multimode fiber (MMF) excited by a \"broadband\" (1-2 THz) laser [25; 26; 27]. This source can be approximated by a quasi-homogeneous CSD function [1; 5; 6], which takes the form  
\[W^{\mathrm{src}}\left(\mathbf{\rho}_{1},\mathbf{\rho}_{2}\right)=S^{\mathrm{src}} \left(\frac{\mathbf{\rho}_{1}+\mathbf{\rho}_{2}}{2}\right)\upmu^{\mathrm{src}}\left( \mathbf{\rho}_{1}-\mathbf{\rho}_{2}\right), \tag{11}\]  
where \(S^{\mathrm{src}}\) is the source's SD, \(\upmu^{\mathrm{src}}\) is the source's SDoC, and \(S^{\mathrm{src}}\) is a \"slow function\" (varies slowly) compared to \(\upmu^{\mathrm{src}}\). Taking the Fourier transform of Equation (11) and substituting the resulting expression into Equation (9) produces  
\[\begin{split}& W^{\mathrm{out}}\left(\mathbf{\rho}_{1},\mathbf{\rho}_{2} \right)=\frac{1}{\left(\lambda^{2}f_{1}f_{2}\right)^{2}}\iiint_{-\infty}^{ \infty}T\left(\mathbf{\rho}_{1}^{\prime}\right)T^{*}\left(\mathbf{\rho}_{2}^{\prime} \right)\tilde{\upmu}^{\mathrm{src}}\left(\frac{\mathbf{\rho}_{1}^{\prime}+\mathbf{ \rho}_{2}^Making the variable substitutions \(\mathbf{s}=\mathbf{\rho}_{1}^{\prime}-\mathbf{\rho}_{2}^{\prime}\) and \(\mathbf{t}=\left(\mathbf{\rho}_{1}^{\prime}+\mathbf{\rho}_{2}^{\prime}\right)/2\) and simplifying, yields  
\[ \begin{split}& W^{\text{out}}\left(\mathbf{\rho}_{1},\mathbf{\rho}_{2} \right)=\frac{1}{\left(\lambda^{2}f_{1}f_{2}\right)^{2}}\iint_{-\infty}^{ \infty}\tilde{\mu}^{\text{src}}\left(\frac{\mathbf{t}}{\lambda f_{1}}\right)\exp \left[-\mathrm{j}\frac{k}{f_{2}}\left(\mathbf{\rho}_{1}-\mathbf{\rho}_{2}\right)\cdot \mathbf{t}\right]\\ &\times\iint_{-\infty}^{\infty}T\left(\mathbf{t}+\frac{1}{2}\mathbf{s} \right)T^{*}\left(\mathbf{t}-\frac{1}{2}\mathbf{s}\right)\tilde{S}^{\text{src}}\left( \frac{\mathbf{s}}{\lambda f_{1}}\right)\exp\left[-\mathrm{j}\frac{k}{f_{2}}\left( \frac{\mathbf{\rho}_{1}+\mathbf{\rho}_{2}}{2}\right)\cdot\mathbf{s}\right]\mathrm{d}^{2} \mathrm{s}\mathrm{d}^{2}t\end{split} .