## 2 Materials and Methods
### Fourier and Statistical Optics Theory
Starting with the above expression for \(\mu^{\text{out}}\) given in Equation (16) and assuming that \(\tilde{\mu}^{\text{src}}\) is a slow function compared to \(T\) (this is actually a consequence of \(W^{\text{src}}\) assuming a quasi-homogeneous form) produces  
\[\mu^{\text{out}}\left(\mathbf{\rho}_{1}-\mathbf{\rho}_{2}\right)\approx\frac{\iint_{- \infty}^{\infty}\left|T\left(\lambda f_{1}\mathbf{u}\right)\right|^{2}\exp\left[ \mathrm{j}2\pi\left(\frac{\mathbf{\rho}_{1}-\mathbf{\rho}_{2}}{M}\right)\cdot\mathbf{u} \right]\mathrm{d}^{2}u}{\iint_{-\infty}^{\infty}\left|T\left(\lambda f_{1}\mathbf{u }\right)\right|^{2}\mathrm{d}^{2}u}. \tag{17}\]  
Letting \(\mathbf{\Delta\rho}=\mathbf{\rho}_{1}-\mathbf{\rho}_{2}\) for convenience and substituting \(\mathbf{t}=\lambda f_{1}\mathbf{u}\) into both the numerator and denominator integrals simplifies Equation (17) to  
\[\mu^{\text{out}}\left(\mathbf{\Delta\rho}\right)=\frac{\iint_{-\infty}^{\infty} \left|T\left(\mathbf{t}\right)\right|^{2}\exp\left[-\mathrm{j}2\pi\left(\frac{\mathbf{ \Delta\rho}}{\lambda f_{2}}\right)\cdot\mathbf{t}\right]\mathrm{d}^{2}t}{\iint_{- \infty}^{\infty}\left|T\left(\mathbf{t}\right)\right|^{2}\mathrm{d}^{2}t}. \tag{18}\]  
Applying the Fourier transform defined in Equation (6) to both sides of Equation (18) produces the desired result:  
\[\frac{\left|T\left(-\lambda f_{2}\mathbf{f}\right)\right|^{2}}{\iint_{-\infty}^{ \infty}\left|T\left(\lambda f_{2}\mathbf{f}\right)\right|^{2}\mathrm{d}^{2}f}=\iint_{- \infty}^{\infty}\mu^{\text{out}}\left(\mathbf{\Delta\rho}\right)\exp\left(-\mathrm{ j}2\pi\mathbf{f}\cdot\mathbf{\Delta\rho}\right)\mathrm{d}^{2}\Delta\rho=\Phi^{\text{out}} \left(\mathbf{f}\right). \tag{19}\]This expression states that the spatial power spectrum of the output field \(\Phi^{\text{out}}\) (the Fourier transform of \(\mu^{\text{out}}\)) is proportional to the magnitude squared of \(T\) rotated \(180^{\circ}\). The proportionality constant is there to normalize the area under the power spectrum; it is inconsequential in practice. The above interpretation of Equation (19) leads to the following steps for finding \(T\):  
1. Choose the desired output SDoC \(\mu^{\text{out}}\). 2. Compute the spatial Fourier transform of \(\mu^{\text{out}}\) to find \(\Phi^{\text{out}}\). 3. Take the square root of \(\Phi^{\text{out}}\) to find the magnitude of \(T\). 4. Normalize \(T\) such that \(\max_{\mathbf{\rho}\in\mathbb{R}^{2}}\left|T\left(\mathbf{\rho}\right)\right|=1\). 5. Rotate \(T\)\(180^{\circ}\). It should be stated that Equations (16) and (19) are accurate if the following three conditions hold:  
1. The diameters of lenses \(\text{L}_{1}\) and \(\text{L}_{2}\) are large enough to collect all the light emitted from the source and the light that passes through the spatial filter \(T\), respectively. 2. The random source field \(U^{\text{src}}\) is well approximated by a CSD function with a quasi-homogeneous form. 3. \(\hat{S}^{\text{src}}\) is a fast function (narrow function) compared to \(T\). The first item is easily met in practice by using ray tracing and other standard optical design techniques. The second criterion is true for \"spatially incoherent\" sources such as LEDs or, applicable here, the light emitted from a broadband-laser-excited MMF. The third condition is the hardest to satisfy in practice. The effects of violating this condition are shown in Section 3.1. Because of the assumptions that underpin the analysis (enumerated above), Equations (16) and (19) are approximations of the output PCS moments. Recall that the goal here was to find simple, physical, \"engineering\" expressions that could be used to design a spatial filter that would produce a desired PCS. Equations (16) and (19) achieve that objective.